LMFDB - Embedded newform 8649.2.a.bu.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 2883)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.959557 q^{2} -1.07925 q^{4} +4.21749 q^{5} -1.12111 q^{7} +2.95472 q^{8} +O(q^{10})$	$q-0.959557 q^{2} -1.07925 q^{4} +4.21749 q^{5} -1.12111 q^{7} +2.95472 q^{8} -4.04692 q^{10} +3.26504 q^{11} -5.17039 q^{13} +1.07577 q^{14} -0.676715 q^{16} +3.37303 q^{17} +2.36547 q^{19} -4.55173 q^{20} -3.13299 q^{22} -6.68428 q^{23} +12.7872 q^{25} +4.96128 q^{26} +1.20996 q^{28} -1.71816 q^{29} -5.26009 q^{32} -3.23661 q^{34} -4.72828 q^{35} +4.47243 q^{37} -2.26981 q^{38} +12.4615 q^{40} -0.225396 q^{41} -4.92722 q^{43} -3.52380 q^{44} +6.41394 q^{46} +10.5773 q^{47} -5.74311 q^{49} -12.2701 q^{50} +5.58015 q^{52} -4.54963 q^{53} +13.7703 q^{55} -3.31256 q^{56} +1.64867 q^{58} -3.23106 q^{59} +11.8692 q^{61} +6.40078 q^{64} -21.8061 q^{65} +12.6641 q^{67} -3.64034 q^{68} +4.53705 q^{70} +2.57266 q^{71} -5.02651 q^{73} -4.29155 q^{74} -2.55294 q^{76} -3.66047 q^{77} -3.11460 q^{79} -2.85404 q^{80} +0.216280 q^{82} +2.10327 q^{83} +14.2257 q^{85} +4.72795 q^{86} +9.64727 q^{88} +13.4532 q^{89} +5.79658 q^{91} +7.21402 q^{92} -10.1495 q^{94} +9.97637 q^{95} -18.2326 q^{97} +5.51084 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	$ 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29} + 48 q^{32} + 48 q^{35} + 64 q^{37} + 24 q^{38} + 32 q^{43} - 48 q^{44} + 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} + 64 q^{52} - 80 q^{53} + 48 q^{56} + 32 q^{58} + 32 q^{61} + 56 q^{64} - 16 q^{65} - 16 q^{67} - 80 q^{68} + 8 q^{70} + 32 q^{73} + 56 q^{76} - 96 q^{77} + 32 q^{79} + 72 q^{80} + 8 q^{82} - 48 q^{83} + 96 q^{85} + 32 q^{86} + 96 q^{88} + 16 q^{89} - 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) $ 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

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 $

$2$	−0.959557	−0.678509	−0.339255	−	0.940695i	$-0.610175\pi$
$2$	−0.959557	−0.678509	−0.339255	+	0.940695i	$0.610175\pi$
$3$	0	0
$3$	0	0
$4$	−1.07925	−0.539626
$5$	4.21749	1.88612	0.943060	−	0.332623i	$-0.107934\pi$
$5$	4.21749	1.88612	0.943060	+	0.332623i	$0.107934\pi$
$6$	0	0
$7$	−1.12111	−0.423740	−0.211870	−	0.977298i	$-0.567955\pi$
$7$	−1.12111	−0.423740	−0.211870	+	0.977298i	$0.567955\pi$
$8$	2.95472	1.04465
$9$	0	0
$10$	−4.04692	−1.27975
$11$	3.26504	0.984447	0.492223	−	0.870469i	$-0.336184\pi$
$11$	3.26504	0.984447	0.492223	+	0.870469i	$0.336184\pi$
$12$	0	0
$13$	−5.17039	−1.43401	−0.717004	−	0.697069i	$-0.754487\pi$
$13$	−5.17039	−1.43401	−0.717004	+	0.697069i	$0.754487\pi$
$14$	1.07577	0.287511
$15$	0	0
$16$	−0.676715	−0.169179
$17$	3.37303	0.818079	0.409039	−	0.912517i	$-0.365864\pi$
$17$	3.37303	0.818079	0.409039	+	0.912517i	$0.365864\pi$
$18$	0	0
$19$	2.36547	0.542677	0.271339	−	0.962484i	$-0.412534\pi$
$19$	2.36547	0.542677	0.271339	+	0.962484i	$0.412534\pi$
$20$	−4.55173	−1.01780
$21$	0	0
$22$	−3.13299	−0.667956
$23$	−6.68428	−1.39377	−0.696884	−	0.717184i	$-0.745431\pi$
$23$	−6.68428	−1.39377	−0.696884	+	0.717184i	$0.745431\pi$
$24$	0	0
$25$	12.7872	2.55745
$26$	4.96128	0.972987
$27$	0	0
$28$	1.20996	0.228661
$29$	−1.71816	−0.319054	−0.159527	−	0.987194i	$-0.550997\pi$
$29$	−1.71816	−0.319054	−0.159527	+	0.987194i	$0.550997\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−5.26009	−0.929860
$33$	0	0
$34$	−3.23661	−0.555074
$35$	−4.72828	−0.799224
$36$	0	0
$37$	4.47243	0.735262	0.367631	−	0.929972i	$-0.380169\pi$
$37$	4.47243	0.735262	0.367631	+	0.929972i	$0.380169\pi$
$38$	−2.26981	−0.368211
$39$	0	0
$40$	12.4615	1.97033
$41$	−0.225396	−0.0352009	−0.0176004	−	0.999845i	$-0.505603\pi$
$41$	−0.225396	−0.0352009	−0.0176004	+	0.999845i	$0.505603\pi$
$42$	0	0
$43$	−4.92722	−0.751394	−0.375697	−	0.926743i	$-0.622597\pi$
$43$	−4.92722	−0.751394	−0.375697	+	0.926743i	$0.622597\pi$
$44$	−3.52380	−0.531233
$45$	0	0
$46$	6.41394	0.945685
$47$	10.5773	1.54286	0.771428	−	0.636316i	$-0.219543\pi$
$47$	10.5773	1.54286	0.771428	+	0.636316i	$0.219543\pi$
$48$	0	0
$49$	−5.74311	−0.820444
$50$	−12.2701	−1.73525
$51$	0	0
$52$	5.58015	0.773827
$53$	−4.54963	−0.624940	−0.312470	−	0.949928i	$-0.601156\pi$
$53$	−4.54963	−0.624940	−0.312470	+	0.949928i	$0.601156\pi$
$54$	0	0
$55$	13.7703	1.85678
$56$	−3.31256	−0.442660
$57$	0	0
$58$	1.64867	0.216481
$59$	−3.23106	−0.420648	−0.210324	−	0.977632i	$-0.567452\pi$
$59$	−3.23106	−0.420648	−0.210324	+	0.977632i	$0.567452\pi$
$60$	0	0
$61$	11.8692	1.51969	0.759846	−	0.650103i	$-0.225274\pi$
$61$	11.8692	1.51969	0.759846	+	0.650103i	$0.225274\pi$
$62$	0	0
$63$	0	0
$64$	6.40078	0.800097
$65$	−21.8061	−2.70471
$66$	0	0
$67$	12.6641	1.54717	0.773584	−	0.633693i	$-0.218462\pi$
$67$	12.6641	1.54717	0.773584	+	0.633693i	$0.218462\pi$
$68$	−3.64034	−0.441456
$69$	0	0
$70$	4.53705	0.542281
$71$	2.57266	0.305318	0.152659	−	0.988279i	$-0.451216\pi$
$71$	2.57266	0.305318	0.152659	+	0.988279i	$0.451216\pi$
$72$	0	0
$73$	−5.02651	−0.588308	−0.294154	−	0.955758i	$-0.595038\pi$
$73$	−5.02651	−0.588308	−0.294154	+	0.955758i	$0.595038\pi$
$74$	−4.29155	−0.498882
$75$	0	0
$76$	−2.55294	−0.292842
$77$	−3.66047	−0.417149
$78$	0	0
$79$	−3.11460	−0.350419	−0.175210	−	0.984531i	$-0.556060\pi$
$79$	−3.11460	−0.350419	−0.175210	+	0.984531i	$0.556060\pi$
$80$	−2.85404	−0.319091
$81$	0	0
$82$	0.216280	0.0238841
$83$	2.10327	0.230864	0.115432	−	0.993315i	$-0.463175\pi$
$83$	2.10327	0.230864	0.115432	+	0.993315i	$0.463175\pi$
$84$	0	0
$85$	14.2257	1.54299
$86$	4.72795	0.509828
$87$	0	0
$88$	9.64727	1.02840
$89$	13.4532	1.42604	0.713018	−	0.701145i	$-0.247328\pi$
$89$	13.4532	1.42604	0.713018	+	0.701145i	$0.247328\pi$
$90$	0	0
$91$	5.79658	0.607646
$92$	7.21402	0.752113
$93$	0	0
$94$	−10.1495	−1.04684
$95$	9.97637	1.02355
$96$	0	0
$97$	−18.2326	−1.85124	−0.925619	−	0.378457i	$-0.876455\pi$
$97$	−18.2326	−1.85124	−0.925619	+	0.378457i	$0.876455\pi$
$98$	5.51084	0.556679
$99$	0	0
$100$	−13.8006	−1.38006
$101$	0.162666	0.0161859	0.00809294	−	0.999967i	$-0.497424\pi$
$101$	0.162666	0.0161859	0.00809294	+	0.999967i	$0.497424\pi$
$102$	0	0
$103$	15.1665	1.49440	0.747199	−	0.664601i	$-0.231398\pi$
$103$	15.1665	1.49440	0.747199	+	0.664601i	$0.231398\pi$
$104$	−15.2770	−1.49804
$105$	0	0
$106$	4.36563	0.424028
$107$	8.69867	0.840932	0.420466	−	0.907308i	$-0.361867\pi$
$107$	8.69867	0.840932	0.420466	+	0.907308i	$0.361867\pi$
$108$	0	0
$109$	3.21102	0.307560	0.153780	−	0.988105i	$-0.450855\pi$
$109$	3.21102	0.307560	0.153780	+	0.988105i	$0.450855\pi$
$110$	−13.2134	−1.25984
$111$	0	0
$112$	0.758672	0.0716878
$113$	−2.75576	−0.259240	−0.129620	−	0.991564i	$-0.541376\pi$
$113$	−2.75576	−0.259240	−0.129620	+	0.991564i	$0.541376\pi$
$114$	0	0
$115$	−28.1909	−2.62881
$116$	1.85432	0.172169
$117$	0	0
$118$	3.10038	0.285414
$119$	−3.78153	−0.346653
$120$	0	0
$121$	−0.339513	−0.0308648
$122$	−11.3892	−1.03113
$123$	0	0
$124$	0	0
$125$	32.8426	2.93753
$126$	0	0
$127$	15.4071	1.36716	0.683579	−	0.729877i	$-0.260423\pi$
$127$	15.4071	1.36716	0.683579	+	0.729877i	$0.260423\pi$
$128$	4.37826	0.386987
$129$	0	0
$130$	20.9242	1.83517
$131$	4.68764	0.409561	0.204781	−	0.978808i	$-0.434352\pi$
$131$	4.68764	0.409561	0.204781	+	0.978808i	$0.434352\pi$
$132$	0	0
$133$	−2.65196	−0.229954
$134$	−12.1519	−1.04977
$135$	0	0
$136$	9.96633	0.854606
$137$	−4.95674	−0.423483	−0.211742	−	0.977326i	$-0.567914\pi$
$137$	−4.95674	−0.423483	−0.211742	+	0.977326i	$0.567914\pi$
$138$	0	0
$139$	12.0823	1.02480	0.512402	−	0.858746i	$-0.328756\pi$
$139$	12.0823	1.02480	0.512402	+	0.858746i	$0.328756\pi$
$140$	5.10300	0.431282
$141$	0	0
$142$	−2.46861	−0.207161
$143$	−16.8815	−1.41170
$144$	0	0
$145$	−7.24631	−0.601773
$146$	4.82322	0.399172
$147$	0	0
$148$	−4.82687	−0.396766
$149$	−0.854945	−0.0700398	−0.0350199	−	0.999387i	$-0.511149\pi$
$149$	−0.854945	−0.0700398	−0.0350199	+	0.999387i	$0.511149\pi$
$150$	0	0
$151$	−8.22405	−0.669264	−0.334632	−	0.942349i	$-0.608612\pi$
$151$	−8.22405	−0.669264	−0.334632	+	0.942349i	$0.608612\pi$
$152$	6.98930	0.566907
$153$	0	0
$154$	3.51243	0.283040
$155$	0	0
$156$	0	0
$157$	15.3118	1.22201	0.611006	−	0.791626i	$-0.290765\pi$
$157$	15.3118	1.22201	0.611006	+	0.791626i	$0.290765\pi$
$158$	2.98863	0.237763
$159$	0	0
$160$	−22.1844	−1.75383
$161$	7.49382	0.590595
$162$	0	0
$163$	−2.24364	−0.175736	−0.0878678	−	0.996132i	$-0.528005\pi$
$163$	−2.24364	−0.175736	−0.0878678	+	0.996132i	$0.528005\pi$
$164$	0.243258	0.0189953
$165$	0	0
$166$	−2.01821	−0.156643
$167$	−2.72510	−0.210875	−0.105437	−	0.994426i	$-0.533624\pi$
$167$	−2.72510	−0.210875	−0.105437	+	0.994426i	$0.533624\pi$
$168$	0	0
$169$	13.7329	1.05638
$170$	−13.6504	−1.04694
$171$	0	0
$172$	5.31771	0.405471
$173$	25.0837	1.90708	0.953538	−	0.301271i	$-0.0974110\pi$
$173$	25.0837	1.90708	0.953538	+	0.301271i	$0.0974110\pi$
$174$	0	0
$175$	−14.3359	−1.08369
$176$	−2.20950	−0.166547
$177$	0	0
$178$	−12.9091	−0.967579
$179$	−9.85007	−0.736229	−0.368115	−	0.929780i	$-0.619997\pi$
$179$	−9.85007	−0.736229	−0.368115	+	0.929780i	$0.619997\pi$
$180$	0	0
$181$	8.33201	0.619313	0.309657	−	0.950848i	$-0.399786\pi$
$181$	8.33201	0.619313	0.309657	+	0.950848i	$0.399786\pi$
$182$	−5.56214	−0.412294
$183$	0	0
$184$	−19.7501	−1.45600
$185$	18.8624	1.38679
$186$	0	0
$187$	11.0131	0.805355
$188$	−11.4156	−0.832565
$189$	0	0
$190$	−9.57289	−0.694491
$191$	−2.88541	−0.208781	−0.104390	−	0.994536i	$-0.533289\pi$
$191$	−2.88541	−0.208781	−0.104390	+	0.994536i	$0.533289\pi$
$192$	0	0
$193$	6.08279	0.437849	0.218925	−	0.975742i	$-0.429745\pi$
$193$	6.08279	0.437849	0.218925	+	0.975742i	$0.429745\pi$
$194$	17.4952	1.25608
$195$	0	0
$196$	6.19826	0.442733
$197$	−14.6092	−1.04086	−0.520430	−	0.853905i	$-0.674228\pi$
$197$	−14.6092	−1.04086	−0.520430	+	0.853905i	$0.674228\pi$
$198$	0	0
$199$	6.11463	0.433454	0.216727	−	0.976232i	$-0.430462\pi$
$199$	6.11463	0.433454	0.216727	+	0.976232i	$0.430462\pi$
$200$	37.7827	2.67164
$201$	0	0
$202$	−0.156087	−0.0109823
$203$	1.92624	0.135196
$204$	0	0
$205$	−0.950604	−0.0663931
$206$	−14.5531	−1.01396
$207$	0	0
$208$	3.49888	0.242604
$209$	7.72337	0.534237
$210$	0	0
$211$	9.35472	0.644005	0.322003	−	0.946739i	$-0.395644\pi$
$211$	9.35472	0.644005	0.322003	+	0.946739i	$0.395644\pi$
$212$	4.91020	0.337234
$213$	0	0
$214$	−8.34686	−0.570580
$215$	−20.7805	−1.41722
$216$	0	0
$217$	0	0
$218$	−3.08116	−0.208682
$219$	0	0
$220$	−14.8616	−1.00197
$221$	−17.4399	−1.17313
$222$	0	0
$223$	−4.90156	−0.328232	−0.164116	−	0.986441i	$-0.552477\pi$
$223$	−4.90156	−0.328232	−0.164116	+	0.986441i	$0.552477\pi$
$224$	5.89714	0.394019
$225$	0	0
$226$	2.64431	0.175897
$227$	13.3316	0.884847	0.442423	−	0.896806i	$-0.354119\pi$
$227$	13.3316	0.884847	0.442423	+	0.896806i	$0.354119\pi$
$228$	0	0
$229$	−0.728245	−0.0481238	−0.0240619	−	0.999710i	$-0.507660\pi$
$229$	−0.728245	−0.0481238	−0.0240619	+	0.999710i	$0.507660\pi$
$230$	27.0508	1.78367
$231$	0	0
$232$	−5.07666	−0.333299
$233$	−1.83966	−0.120520	−0.0602601	−	0.998183i	$-0.519193\pi$
$233$	−1.83966	−0.120520	−0.0602601	+	0.998183i	$0.519193\pi$
$234$	0	0
$235$	44.6096	2.91001
$236$	3.48712	0.226992
$237$	0	0
$238$	3.62860	0.235207
$239$	−13.3304	−0.862272	−0.431136	−	0.902287i	$-0.641887\pi$
$239$	−13.3304	−0.862272	−0.431136	+	0.902287i	$0.641887\pi$
$240$	0	0
$241$	2.63348	0.169637	0.0848187	−	0.996396i	$-0.472969\pi$
$241$	2.63348	0.169637	0.0848187	+	0.996396i	$0.472969\pi$
$242$	0.325782	0.0209421
$243$	0	0
$244$	−12.8098	−0.820065
$245$	−24.2215	−1.54746
$246$	0	0
$247$	−12.2304	−0.778203
$248$	0	0
$249$	0	0
$250$	−31.5144	−1.99314
$251$	15.0740	0.951465	0.475733	−	0.879590i	$-0.342183\pi$
$251$	15.0740	0.951465	0.475733	+	0.879590i	$0.342183\pi$
$252$	0	0
$253$	−21.8244	−1.37209
$254$	−14.7840	−0.927628
$255$	0	0
$256$	−17.0027	−1.06267
$257$	−12.9199	−0.805923	−0.402962	−	0.915217i	$-0.632019\pi$
$257$	−12.9199	−0.805923	−0.402962	+	0.915217i	$0.632019\pi$
$258$	0	0
$259$	−5.01408	−0.311560
$260$	23.5342	1.45953
$261$	0	0
$262$	−4.49806	−0.277891
$263$	−23.6830	−1.46036	−0.730178	−	0.683257i	$-0.760563\pi$
$263$	−23.6830	−1.46036	−0.730178	+	0.683257i	$0.760563\pi$
$264$	0	0
$265$	−19.1880	−1.17871
$266$	2.54470	0.156026
$267$	0	0
$268$	−13.6678	−0.834892
$269$	2.91961	0.178012	0.0890059	−	0.996031i	$-0.471631\pi$
$269$	2.91961	0.178012	0.0890059	+	0.996031i	$0.471631\pi$
$270$	0	0
$271$	27.8075	1.68919	0.844593	−	0.535409i	$-0.179843\pi$
$271$	27.8075	1.68919	0.844593	+	0.535409i	$0.179843\pi$
$272$	−2.28258	−0.138402
$273$	0	0
$274$	4.75628	0.287337
$275$	41.7509	2.51767
$276$	0	0
$277$	18.6154	1.11849	0.559245	−	0.829003i	$-0.311091\pi$
$277$	18.6154	1.11849	0.559245	+	0.829003i	$0.311091\pi$
$278$	−11.5936	−0.695339
$279$	0	0
$280$	−13.9707	−0.834910
$281$	18.2964	1.09147	0.545735	−	0.837958i	$-0.316250\pi$
$281$	18.2964	1.09147	0.545735	+	0.837958i	$0.316250\pi$
$282$	0	0
$283$	9.00205	0.535116	0.267558	−	0.963542i	$-0.413783\pi$
$283$	9.00205	0.535116	0.267558	+	0.963542i	$0.413783\pi$
$284$	−2.77654	−0.164758
$285$	0	0
$286$	16.1988	0.957854
$287$	0.252693	0.0149160
$288$	0	0
$289$	−5.62270	−0.330747
$290$	6.95325	0.408309
$291$	0	0
$292$	5.42486	0.317466
$293$	−24.4153	−1.42636	−0.713179	−	0.700982i	$-0.752746\pi$
$293$	−24.4153	−1.42636	−0.713179	+	0.700982i	$0.752746\pi$
$294$	0	0
$295$	−13.6270	−0.793393
$296$	13.2147	0.768092
$297$	0	0
$298$	0.820368	0.0475226
$299$	34.5603	1.99868
$300$	0	0
$301$	5.52396	0.318396
$302$	7.89144	0.454101
$303$	0	0
$304$	−1.60075	−0.0918094
$305$	50.0582	2.86632
$306$	0	0
$307$	−28.8213	−1.64492	−0.822460	−	0.568824i	$-0.807399\pi$
$307$	−28.8213	−1.64492	−0.822460	+	0.568824i	$0.807399\pi$
$308$	3.95057	0.225104
$309$	0	0
$310$	0	0
$311$	−30.7091	−1.74135	−0.870676	−	0.491856i	$-0.836319\pi$
$311$	−30.7091	−1.74135	−0.870676	+	0.491856i	$0.836319\pi$
$312$	0	0
$313$	−2.92508	−0.165335	−0.0826677	−	0.996577i	$-0.526344\pi$
$313$	−2.92508	−0.165335	−0.0826677	+	0.996577i	$0.526344\pi$
$314$	−14.6925	−0.829146
$315$	0	0
$316$	3.36143	0.189095
$317$	4.50032	0.252763	0.126382	−	0.991982i	$-0.459664\pi$
$317$	4.50032	0.252763	0.126382	+	0.991982i	$0.459664\pi$
$318$	0	0
$319$	−5.60985	−0.314091
$320$	26.9952	1.50908
$321$	0	0
$322$	−7.19074	−0.400724
$323$	7.97881	0.443953
$324$	0	0
$325$	−66.1150	−3.66740
$326$	2.15290	0.119238
$327$	0	0
$328$	−0.665980	−0.0367726
$329$	−11.8583	−0.653770
$330$	0	0
$331$	23.2329	1.27699	0.638497	−	0.769624i	$-0.279556\pi$
$331$	23.2329	1.27699	0.638497	+	0.769624i	$0.279556\pi$
$332$	−2.26996	−0.124580
$333$	0	0
$334$	2.61489	0.143080
$335$	53.4108	2.91815
$336$	0	0
$337$	35.0362	1.90854	0.954272	−	0.298940i	$-0.0966330\pi$
$337$	35.0362	1.90854	0.954272	+	0.298940i	$0.0966330\pi$
$338$	−13.1775	−0.716762
$339$	0	0
$340$	−15.3531	−0.832639
$341$	0	0
$342$	0	0
$343$	14.2864	0.771395
$344$	−14.5585	−0.784944
$345$	0	0
$346$	−24.0692	−1.29397
$347$	−19.1134	−1.02606	−0.513031	−	0.858370i	$-0.671477\pi$
$347$	−19.1134	−1.02606	−0.513031	+	0.858370i	$0.671477\pi$
$348$	0	0
$349$	−6.51273	−0.348618	−0.174309	−	0.984691i	$-0.555769\pi$
$349$	−6.51273	−0.348618	−0.174309	+	0.984691i	$0.555769\pi$
$350$	13.7561	0.735295
$351$	0	0
$352$	−17.1744	−0.915398
$353$	−34.5222	−1.83743	−0.918715	−	0.394921i	$-0.870772\pi$
$353$	−34.5222	−1.83743	−0.918715	+	0.394921i	$0.870772\pi$
$354$	0	0
$355$	10.8502	0.575867
$356$	−14.5194	−0.769526
$357$	0	0
$358$	9.45170	0.499538
$359$	11.4931	0.606582	0.303291	−	0.952898i	$-0.401915\pi$
$359$	11.4931	0.606582	0.303291	+	0.952898i	$0.401915\pi$
$360$	0	0
$361$	−13.4045	−0.705502
$362$	−7.99503	−0.420210
$363$	0	0
$364$	−6.25596	−0.327902
$365$	−21.1993	−1.10962
$366$	0	0
$367$	−17.6487	−0.921257	−0.460628	−	0.887593i	$-0.652376\pi$
$367$	−17.6487	−0.921257	−0.460628	+	0.887593i	$0.652376\pi$
$368$	4.52335	0.235796
$369$	0	0
$370$	−18.0996	−0.940952
$371$	5.10064	0.264812
$372$	0	0
$373$	6.23272	0.322718	0.161359	−	0.986896i	$-0.448412\pi$
$373$	6.23272	0.322718	0.161359	+	0.986896i	$0.448412\pi$
$374$	−10.5677	−0.546441
$375$	0	0
$376$	31.2529	1.61174
$377$	8.88354	0.457525
$378$	0	0
$379$	19.6849	1.01114	0.505572	−	0.862784i	$-0.331281\pi$
$379$	19.6849	1.01114	0.505572	+	0.862784i	$0.331281\pi$
$380$	−10.7670	−0.552336
$381$	0	0
$382$	2.76871	0.141660
$383$	−11.6071	−0.593097	−0.296548	−	0.955018i	$-0.595836\pi$
$383$	−11.6071	−0.593097	−0.296548	+	0.955018i	$0.595836\pi$
$384$	0	0
$385$	−15.4380	−0.786794
$386$	−5.83679	−0.297085
$387$	0	0
$388$	19.6775	0.998975
$389$	4.68307	0.237441	0.118721	−	0.992928i	$-0.462121\pi$
$389$	4.68307	0.237441	0.118721	+	0.992928i	$0.462121\pi$
$390$	0	0
$391$	−22.5462	−1.14021
$392$	−16.9693	−0.857077
$393$	0	0
$394$	14.0183	0.706232
$395$	−13.1358	−0.660933
$396$	0	0
$397$	−14.0219	−0.703740	−0.351870	−	0.936049i	$-0.614454\pi$
$397$	−14.0219	−0.703740	−0.351870	+	0.936049i	$0.614454\pi$
$398$	−5.86733	−0.294103
$399$	0	0
$400$	−8.65332	−0.432666
$401$	16.7259	0.835254	0.417627	−	0.908619i	$-0.362862\pi$
$401$	16.7259	0.835254	0.417627	+	0.908619i	$0.362862\pi$
$402$	0	0
$403$	0	0
$404$	−0.175557	−0.00873431
$405$	0	0
$406$	−1.84834	−0.0917315
$407$	14.6027	0.723827
$408$	0	0
$409$	−33.6920	−1.66596	−0.832982	−	0.553300i	$-0.813368\pi$
$409$	−33.6920	−1.66596	−0.832982	+	0.553300i	$0.813368\pi$
$410$	0.912159	0.0450483
$411$	0	0
$412$	−16.3684	−0.806415
$413$	3.62237	0.178245
$414$	0	0
$415$	8.87054	0.435438
$416$	27.1967	1.33343
$417$	0	0
$418$	−7.41101	−0.362484
$419$	−18.0980	−0.884146	−0.442073	−	0.896979i	$-0.645757\pi$
$419$	−18.0980	−0.884146	−0.442073	+	0.896979i	$0.645757\pi$
$420$	0	0
$421$	−23.8424	−1.16201	−0.581003	−	0.813901i	$-0.697340\pi$
$421$	−23.8424	−1.16201	−0.581003	+	0.813901i	$0.697340\pi$
$422$	−8.97638	−0.436963
$423$	0	0
$424$	−13.4429	−0.652844
$425$	43.1317	2.09219
$426$	0	0
$427$	−13.3067	−0.643955
$428$	−9.38804	−0.453788
$429$	0	0
$430$	19.9401	0.961596
$431$	24.5384	1.18197	0.590987	−	0.806681i	$-0.298739\pi$
$431$	24.5384	1.18197	0.590987	+	0.806681i	$0.298739\pi$
$432$	0	0
$433$	−26.1896	−1.25859	−0.629296	−	0.777165i	$-0.716657\pi$
$433$	−26.1896	−1.25859	−0.629296	+	0.777165i	$0.716657\pi$
$434$	0	0
$435$	0	0
$436$	−3.46550	−0.165967
$437$	−15.8115	−0.756366
$438$	0	0
$439$	31.4438	1.50073	0.750366	−	0.661023i	$-0.229877\pi$
$439$	31.4438	1.50073	0.750366	+	0.661023i	$0.229877\pi$
$440$	40.6873	1.93969
$441$	0	0
$442$	16.7345	0.795980
$443$	24.3720	1.15795	0.578975	−	0.815345i	$-0.303453\pi$
$443$	24.3720	1.15795	0.578975	+	0.815345i	$0.303453\pi$
$444$	0	0
$445$	56.7388	2.68968
$446$	4.70332	0.222709
$447$	0	0
$448$	−7.17598	−0.339033
$449$	−4.88854	−0.230704	−0.115352	−	0.993325i	$-0.536800\pi$
$449$	−4.88854	−0.230704	−0.115352	+	0.993325i	$0.536800\pi$
$450$	0	0
$451$	−0.735926	−0.0346534
$452$	2.97416	0.139893
$453$	0	0
$454$	−12.7924	−0.600377
$455$	24.4470	1.14609
$456$	0	0
$457$	2.56232	0.119860	0.0599302	−	0.998203i	$-0.480912\pi$
$457$	2.56232	0.119860	0.0599302	+	0.998203i	$0.480912\pi$
$458$	0.698792	0.0326524
$459$	0	0
$460$	30.4251	1.41858
$461$	−12.1455	−0.565672	−0.282836	−	0.959168i	$-0.591275\pi$
$461$	−12.1455	−0.565672	−0.282836	+	0.959168i	$0.591275\pi$
$462$	0	0
$463$	1.94465	0.0903758	0.0451879	−	0.998979i	$-0.485611\pi$
$463$	1.94465	0.0903758	0.0451879	+	0.998979i	$0.485611\pi$
$464$	1.16270	0.0539771
$465$	0	0
$466$	1.76526	0.0817740
$467$	−15.9289	−0.737104	−0.368552	−	0.929607i	$-0.620146\pi$
$467$	−15.9289	−0.737104	−0.368552	+	0.929607i	$0.620146\pi$
$468$	0	0
$469$	−14.1979	−0.655597
$470$	−42.8055	−1.97447
$471$	0	0
$472$	−9.54686	−0.439430
$473$	−16.0876	−0.739708
$474$	0	0
$475$	30.2479	1.38787
$476$	4.08122	0.187063
$477$	0	0
$478$	12.7913	0.585059
$479$	31.8590	1.45568	0.727838	−	0.685749i	$-0.240525\pi$
$479$	31.8590	1.45568	0.727838	+	0.685749i	$0.240525\pi$
$480$	0	0
$481$	−23.1242	−1.05437
$482$	−2.52697	−0.115101
$483$	0	0
$484$	0.366420	0.0166554
$485$	−76.8958	−3.49166
$486$	0	0
$487$	24.7322	1.12072	0.560362	−	0.828248i	$-0.310662\pi$
$487$	24.7322	1.12072	0.560362	+	0.828248i	$0.310662\pi$
$488$	35.0701	1.58755
$489$	0	0
$490$	23.2419	1.04996
$491$	27.7737	1.25341	0.626705	−	0.779256i	$-0.284403\pi$
$491$	27.7737	1.25341	0.626705	+	0.779256i	$0.284403\pi$
$492$	0	0
$493$	−5.79538	−0.261011
$494$	11.7358	0.528018
$495$	0	0
$496$	0	0
$497$	−2.88423	−0.129376
$498$	0	0
$499$	5.73749	0.256845	0.128423	−	0.991720i	$-0.459009\pi$
$499$	5.73749	0.256845	0.128423	+	0.991720i	$0.459009\pi$
$500$	−35.4454	−1.58517
$501$	0	0
$502$	−14.4644	−0.645578
$503$	11.3924	0.507964	0.253982	−	0.967209i	$-0.418260\pi$
$503$	11.3924	0.507964	0.253982	+	0.967209i	$0.418260\pi$
$504$	0	0
$505$	0.686043	0.0305285
$506$	20.9418	0.930976
$507$	0	0
$508$	−16.6281	−0.737753
$509$	−5.82971	−0.258397	−0.129199	−	0.991619i	$-0.541240\pi$
$509$	−5.82971	−0.258397	−0.129199	+	0.991619i	$0.541240\pi$
$510$	0	0
$511$	5.63527	0.249290
$512$	7.55858	0.334045
$513$	0	0
$514$	12.3974	0.546826
$515$	63.9645	2.81861
$516$	0	0
$517$	34.5353	1.51886
$518$	4.81130	0.211396
$519$	0	0
$520$	−64.4308	−2.82548
$521$	33.1783	1.45357	0.726784	−	0.686866i	$-0.241014\pi$
$521$	33.1783	1.45357	0.726784	+	0.686866i	$0.241014\pi$
$522$	0	0
$523$	21.3557	0.933818	0.466909	−	0.884305i	$-0.345368\pi$
$523$	21.3557	0.933818	0.466909	+	0.884305i	$0.345368\pi$
$524$	−5.05914	−0.221010
$525$	0	0
$526$	22.7252	0.990865
$527$	0	0
$528$	0	0
$529$	21.6796	0.942591
$530$	18.4120	0.799767
$531$	0	0
$532$	2.86213	0.124089
$533$	1.16538	0.0504783
$534$	0	0
$535$	36.6866	1.58610
$536$	37.4189	1.61625
$537$	0	0
$538$	−2.80153	−0.120783
$539$	−18.7515	−0.807684
$540$	0	0
$541$	20.7036	0.890115	0.445058	−	0.895502i	$-0.353183\pi$
$541$	20.7036	0.890115	0.445058	+	0.895502i	$0.353183\pi$
$542$	−26.6829	−1.14613
$543$	0	0
$544$	−17.7424	−0.760699
$545$	13.5425	0.580095
$546$	0	0
$547$	15.1871	0.649354	0.324677	−	0.945825i	$-0.394744\pi$
$547$	15.1871	0.649354	0.324677	+	0.945825i	$0.394744\pi$
$548$	5.34957	0.228522
$549$	0	0
$550$	−40.0623	−1.70826
$551$	−4.06425	−0.173143
$552$	0	0
$553$	3.49181	0.148487
$554$	−17.8625	−0.758905
$555$	0	0
$556$	−13.0398	−0.553010
$557$	−28.0751	−1.18958	−0.594789	−	0.803882i	$-0.702764\pi$
$557$	−28.0751	−1.18958	−0.594789	+	0.803882i	$0.702764\pi$
$558$	0	0
$559$	25.4757	1.07751
$560$	3.19970	0.135212
$561$	0	0
$562$	−17.5564	−0.740573
$563$	−10.4612	−0.440888	−0.220444	−	0.975400i	$-0.570751\pi$
$563$	−10.4612	−0.440888	−0.220444	+	0.975400i	$0.570751\pi$
$564$	0	0
$565$	−11.6224	−0.488958
$566$	−8.63798	−0.363081
$567$	0	0
$568$	7.60147	0.318951
$569$	37.9146	1.58946	0.794732	−	0.606961i	$-0.207611\pi$
$569$	37.9146	1.58946	0.794732	+	0.606961i	$0.207611\pi$
$570$	0	0
$571$	32.0047	1.33936	0.669678	−	0.742651i	$-0.266432\pi$
$571$	32.0047	1.33936	0.669678	+	0.742651i	$0.266432\pi$
$572$	18.2194	0.761792
$573$	0	0
$574$	−0.242474	−0.0101207
$575$	−85.4735	−3.56449
$576$	0	0
$577$	1.04946	0.0436896	0.0218448	−	0.999761i	$-0.493046\pi$
$577$	1.04946	0.0436896	0.0218448	+	0.999761i	$0.493046\pi$
$578$	5.39530	0.224415
$579$	0	0
$580$	7.82059	0.324732
$581$	−2.35800	−0.0978264
$582$	0	0
$583$	−14.8547	−0.615220
$584$	−14.8519	−0.614576
$585$	0	0
$586$	23.4279	0.967797
$587$	−33.6958	−1.39077	−0.695387	−	0.718636i	$-0.744767\pi$
$587$	−33.6958	−1.39077	−0.695387	+	0.718636i	$0.744767\pi$
$588$	0	0
$589$	0	0
$590$	13.0758	0.538324
$591$	0	0
$592$	−3.02656	−0.124391
$593$	12.0587	0.495193	0.247597	−	0.968863i	$-0.420359\pi$
$593$	12.0587	0.495193	0.247597	+	0.968863i	$0.420359\pi$
$594$	0	0
$595$	−15.9486	−0.653829
$596$	0.922700	0.0377953
$597$	0	0
$598$	−33.1626	−1.35612
$599$	−11.4313	−0.467068	−0.233534	−	0.972349i	$-0.575029\pi$
$599$	−11.4313	−0.467068	−0.233534	+	0.972349i	$0.575029\pi$
$600$	0	0
$601$	−20.7591	−0.846779	−0.423390	−	0.905948i	$-0.639160\pi$
$601$	−20.7591	−0.846779	−0.423390	+	0.905948i	$0.639160\pi$
$602$	−5.30055	−0.216034
$603$	0	0
$604$	8.87581	0.361152
$605$	−1.43189	−0.0582148
$606$	0	0
$607$	19.7507	0.801658	0.400829	−	0.916153i	$-0.368722\pi$
$607$	19.7507	0.801658	0.400829	+	0.916153i	$0.368722\pi$
$608$	−12.4426	−0.504614
$609$	0	0
$610$	−48.0337	−1.94483
$611$	−54.6887	−2.21247
$612$	0	0
$613$	−0.175159	−0.00707460	−0.00353730	−	0.999994i	$-0.501126\pi$
$613$	−0.175159	−0.00707460	−0.00353730	+	0.999994i	$0.501126\pi$
$614$	27.6557	1.11609
$615$	0	0
$616$	−10.8157	−0.435775
$617$	−14.6688	−0.590545	−0.295272	−	0.955413i	$-0.595410\pi$
$617$	−14.6688	−0.590545	−0.295272	+	0.955413i	$0.595410\pi$
$618$	0	0
$619$	29.7394	1.19533	0.597664	−	0.801747i	$-0.296096\pi$
$619$	29.7394	1.19533	0.597664	+	0.801747i	$0.296096\pi$
$620$	0	0
$621$	0	0
$622$	29.4671	1.18152
$623$	−15.0825	−0.604269
$624$	0	0
$625$	74.5773	2.98309
$626$	2.80678	0.112182
$627$	0	0
$628$	−16.5252	−0.659429
$629$	15.0856	0.601503
$630$	0	0
$631$	−30.8180	−1.22685	−0.613423	−	0.789755i	$-0.710208\pi$
$631$	−30.8180	−1.22685	−0.613423	+	0.789755i	$0.710208\pi$
$632$	−9.20274	−0.366065
$633$	0	0
$634$	−4.31831	−0.171502
$635$	64.9792	2.57862
$636$	0	0
$637$	29.6941	1.17652
$638$	5.38297	0.213114
$639$	0	0
$640$	18.4653	0.729904
$641$	12.7553	0.503804	0.251902	−	0.967753i	$-0.418944\pi$
$641$	12.7553	0.503804	0.251902	+	0.967753i	$0.418944\pi$
$642$	0	0
$643$	−34.7654	−1.37101	−0.685507	−	0.728066i	$-0.740419\pi$
$643$	−34.7654	−1.37101	−0.685507	+	0.728066i	$0.740419\pi$
$644$	−8.08771	−0.318700
$645$	0	0
$646$	−7.65612	−0.301226
$647$	33.5817	1.32023	0.660116	−	0.751163i	$-0.270507\pi$
$647$	33.5817	1.32023	0.660116	+	0.751163i	$0.270507\pi$
$648$	0	0
$649$	−10.5495	−0.414106
$650$	63.4411	2.48836
$651$	0	0
$652$	2.42145	0.0948315
$653$	36.2582	1.41889	0.709447	−	0.704759i	$-0.248945\pi$
$653$	36.2582	1.41889	0.709447	+	0.704759i	$0.248945\pi$
$654$	0	0
$655$	19.7701	0.772482
$656$	0.152529	0.00595524
$657$	0	0
$658$	11.3787	0.443589
$659$	15.3566	0.598208	0.299104	−	0.954221i	$-0.403312\pi$
$659$	15.3566	0.598208	0.299104	+	0.954221i	$0.403312\pi$
$660$	0	0
$661$	−40.4321	−1.57263	−0.786313	−	0.617828i	$-0.788013\pi$
$661$	−40.4321	−1.57263	−0.786313	+	0.617828i	$0.788013\pi$
$662$	−22.2933	−0.866452
$663$	0	0
$664$	6.21457	0.241172
$665$	−11.1846	−0.433721
$666$	0	0
$667$	11.4846	0.444687
$668$	2.94107	0.113793
$669$	0	0
$670$	−51.2507	−1.97999
$671$	38.7534	1.49606
$672$	0	0
$673$	−3.36630	−0.129761	−0.0648806	−	0.997893i	$-0.520667\pi$
$673$	−3.36630	−0.129761	−0.0648806	+	0.997893i	$0.520667\pi$
$674$	−33.6192	−1.29496
$675$	0	0
$676$	−14.8213	−0.570049
$677$	−20.1190	−0.773237	−0.386618	−	0.922240i	$-0.626357\pi$
$677$	−20.1190	−0.773237	−0.386618	+	0.922240i	$0.626357\pi$
$678$	0	0
$679$	20.4407	0.784443
$680$	42.0329	1.61189
$681$	0	0
$682$	0	0
$683$	21.2388	0.812679	0.406340	−	0.913722i	$-0.366805\pi$
$683$	21.2388	0.812679	0.406340	+	0.913722i	$0.366805\pi$
$684$	0	0
$685$	−20.9050	−0.798740
$686$	−13.7086	−0.523398
$687$	0	0
$688$	3.33433	0.127120
$689$	23.5234	0.896169
$690$	0	0
$691$	43.8098	1.66660	0.833301	−	0.552820i	$-0.186448\pi$
$691$	43.8098	1.66660	0.833301	+	0.552820i	$0.186448\pi$
$692$	−27.0716	−1.02911
$693$	0	0
$694$	18.3404	0.696192
$695$	50.9568	1.93290
$696$	0	0
$697$	−0.760265	−0.0287971
$698$	6.24933	0.236541
$699$	0	0
$700$	15.4720	0.584788
$701$	−10.1560	−0.383587	−0.191794	−	0.981435i	$-0.561430\pi$
$701$	−10.1560	−0.383587	−0.191794	+	0.981435i	$0.561430\pi$
$702$	0	0
$703$	10.5794	0.399010
$704$	20.8988	0.787653
$705$	0	0
$706$	33.1260	1.24671
$707$	−0.182367	−0.00685860
$708$	0	0
$709$	25.9285	0.973764	0.486882	−	0.873468i	$-0.338134\pi$
$709$	25.9285	0.973764	0.486882	+	0.873468i	$0.338134\pi$
$710$	−10.4113	−0.390731
$711$	0	0
$712$	39.7504	1.48971
$713$	0	0
$714$	0	0
$715$	−71.1977	−2.66264
$716$	10.6307	0.397288
$717$	0	0
$718$	−11.0283	−0.411572
$719$	10.3370	0.385506	0.192753	−	0.981247i	$-0.438258\pi$
$719$	10.3370	0.385506	0.192753	+	0.981247i	$0.438258\pi$
$720$	0	0
$721$	−17.0033	−0.633236
$722$	12.8624	0.478689
$723$	0	0
$724$	−8.99233	−0.334197
$725$	−21.9705	−0.815963
$726$	0	0
$727$	27.6617	1.02592	0.512958	−	0.858414i	$-0.328550\pi$
$727$	27.6617	1.02592	0.512958	+	0.858414i	$0.328550\pi$
$728$	17.1272	0.634778
$729$	0	0
$730$	20.3419	0.752887
$731$	−16.6196	−0.614700
$732$	0	0
$733$	49.6297	1.83312	0.916558	−	0.399902i	$-0.130956\pi$
$733$	49.6297	1.83312	0.916558	+	0.399902i	$0.130956\pi$
$734$	16.9350	0.625081
$735$	0	0
$736$	35.1599	1.29601
$737$	41.3489	1.52311
$738$	0	0
$739$	−24.8244	−0.913182	−0.456591	−	0.889677i	$-0.650930\pi$
$739$	−24.8244	−0.913182	−0.456591	+	0.889677i	$0.650930\pi$
$740$	−20.3573	−0.748349
$741$	0	0
$742$	−4.89436	−0.179677
$743$	−19.8531	−0.728341	−0.364170	−	0.931332i	$-0.618647\pi$
$743$	−19.8531	−0.728341	−0.364170	+	0.931332i	$0.618647\pi$
$744$	0	0
$745$	−3.60572	−0.132104
$746$	−5.98065	−0.218967
$747$	0	0
$748$	−11.8859	−0.434590
$749$	−9.75216	−0.356336
$750$	0	0
$751$	13.8041	0.503719	0.251860	−	0.967764i	$-0.418958\pi$
$751$	13.8041	0.503719	0.251860	+	0.967764i	$0.418958\pi$
$752$	−7.15781	−0.261019
$753$	0	0
$754$	−8.52426	−0.310435
$755$	−34.6849	−1.26231
$756$	0	0
$757$	−4.61382	−0.167692	−0.0838461	−	0.996479i	$-0.526720\pi$
$757$	−4.61382	−0.167692	−0.0838461	+	0.996479i	$0.526720\pi$
$758$	−18.8888	−0.686071
$759$	0	0
$760$	29.4773	1.06926
$761$	−24.6310	−0.892873	−0.446436	−	0.894815i	$-0.647307\pi$
$761$	−24.6310	−0.892873	−0.446436	+	0.894815i	$0.647307\pi$
$762$	0	0
$763$	−3.59991	−0.130325
$764$	3.11408	0.112664
$765$	0	0
$766$	11.1377	0.402421
$767$	16.7058	0.603213
$768$	0	0
$769$	14.5696	0.525393	0.262697	−	0.964878i	$-0.415388\pi$
$769$	14.5696	0.525393	0.262697	+	0.964878i	$0.415388\pi$
$770$	14.8136	0.533847
$771$	0	0
$772$	−6.56486	−0.236275
$773$	−5.96258	−0.214459	−0.107229	−	0.994234i	$-0.534198\pi$
$773$	−5.96258	−0.214459	−0.107229	+	0.994234i	$0.534198\pi$
$774$	0	0
$775$	0	0
$776$	−53.8721	−1.93390
$777$	0	0
$778$	−4.49368	−0.161106
$779$	−0.533168	−0.0191027
$780$	0	0
$781$	8.39983	0.300570
$782$	21.6344	0.773645
$783$	0	0
$784$	3.88645	0.138802
$785$	64.5773	2.30486
$786$	0	0
$787$	5.76315	0.205434	0.102717	−	0.994711i	$-0.467246\pi$
$787$	5.76315	0.205434	0.102717	+	0.994711i	$0.467246\pi$
$788$	15.7669	0.561674
$789$	0	0
$790$	12.6045	0.448449
$791$	3.08951	0.109850
$792$	0	0
$793$	−61.3683	−2.17925
$794$	13.4548	0.477494
$795$	0	0
$796$	−6.59922	−0.233903
$797$	−35.3270	−1.25135	−0.625673	−	0.780086i	$-0.715176\pi$
$797$	−35.3270	−1.25135	−0.625673	+	0.780086i	$0.715176\pi$
$798$	0	0
$799$	35.6775	1.26218
$800$	−67.2620	−2.37807
$801$	0	0
$802$	−16.0495	−0.566727
$803$	−16.4117	−0.579158
$804$	0	0
$805$	31.6051	1.11393
$806$	0	0
$807$	0	0
$808$	0.480632	0.0169086
$809$	−9.54009	−0.335412	−0.167706	−	0.985837i	$-0.553636\pi$
$809$	−9.54009	−0.335412	−0.167706	+	0.985837i	$0.553636\pi$
$810$	0	0
$811$	−9.88307	−0.347041	−0.173521	−	0.984830i	$-0.555514\pi$
$811$	−9.88307	−0.347041	−0.173521	+	0.984830i	$0.555514\pi$
$812$	−2.07890	−0.0729551
$813$	0	0
$814$	−14.0121	−0.491123
$815$	−9.46255	−0.331459
$816$	0	0
$817$	−11.6552	−0.407764
$818$	32.3294	1.13037
$819$	0	0
$820$	1.02594	0.0358274
$821$	38.7863	1.35365	0.676825	−	0.736144i	$-0.263355\pi$
$821$	38.7863	1.35365	0.676825	+	0.736144i	$0.263355\pi$
$822$	0	0
$823$	−54.8713	−1.91269	−0.956347	−	0.292232i	$-0.905602\pi$
$823$	−54.8713	−1.91269	−0.956347	+	0.292232i	$0.905602\pi$
$824$	44.8126	1.56112
$825$	0	0
$826$	−3.47587	−0.120941
$827$	−38.7335	−1.34690	−0.673448	−	0.739235i	$-0.735187\pi$
$827$	−38.7335	−1.34690	−0.673448	+	0.739235i	$0.735187\pi$
$828$	0	0
$829$	7.17950	0.249354	0.124677	−	0.992197i	$-0.460211\pi$
$829$	7.17950	0.249354	0.124677	+	0.992197i	$0.460211\pi$
$830$	−8.51178	−0.295448
$831$	0	0
$832$	−33.0945	−1.14735
$833$	−19.3717	−0.671188
$834$	0	0
$835$	−11.4931	−0.397735
$836$	−8.33545	−0.288288
$837$	0	0
$838$	17.3661	0.599901
$839$	−11.7368	−0.405198	−0.202599	−	0.979262i	$-0.564939\pi$
$839$	−11.7368	−0.405198	−0.202599	+	0.979262i	$0.564939\pi$
$840$	0	0
$841$	−26.0479	−0.898205
$842$	22.8781	0.788432
$843$	0	0
$844$	−10.0961	−0.347522
$845$	57.9185	1.99246
$846$	0	0
$847$	0.380632	0.0130787
$848$	3.07881	0.105727
$849$	0	0
$850$	−41.3873	−1.41957
$851$	−29.8949	−1.02479
$852$	0	0
$853$	31.6293	1.08297	0.541484	−	0.840711i	$-0.317863\pi$
$853$	31.6293	1.08297	0.541484	+	0.840711i	$0.317863\pi$
$854$	12.7685	0.436929
$855$	0	0
$856$	25.7021	0.878479
$857$	43.2062	1.47590	0.737948	−	0.674858i	$-0.235795\pi$
$857$	43.2062	1.47590	0.737948	+	0.674858i	$0.235795\pi$
$858$	0	0
$859$	−6.92780	−0.236374	−0.118187	−	0.992991i	$-0.537708\pi$
$859$	−6.92780	−0.236374	−0.118187	+	0.992991i	$0.537708\pi$
$860$	22.4274	0.764768
$861$	0	0
$862$	−23.5460	−0.801980
$863$	41.5436	1.41416	0.707081	−	0.707133i	$-0.250012\pi$
$863$	41.5436	1.41416	0.707081	+	0.707133i	$0.250012\pi$
$864$	0	0
$865$	105.790	3.59698
$866$	25.1304	0.853966
$867$	0	0
$868$	0	0
$869$	−10.1693	−0.344969
$870$	0	0
$871$	−65.4784	−2.21865
$872$	9.48765	0.321292
$873$	0	0
$874$	15.1720	0.513201
$875$	−36.8202	−1.24475
$876$	0	0
$877$	16.0855	0.543170	0.271585	−	0.962414i	$-0.412452\pi$
$877$	16.0855	0.543170	0.271585	+	0.962414i	$0.412452\pi$
$878$	−30.1721	−1.01826
$879$	0	0
$880$	−9.31856	−0.314129
$881$	−5.06226	−0.170552	−0.0852760	−	0.996357i	$-0.527177\pi$
$881$	−5.06226	−0.170552	−0.0852760	+	0.996357i	$0.527177\pi$
$882$	0	0
$883$	23.9347	0.805467	0.402734	−	0.915317i	$-0.368060\pi$
$883$	23.9347	0.805467	0.402734	+	0.915317i	$0.368060\pi$
$884$	18.8220	0.633052
$885$	0	0
$886$	−23.3863	−0.785680
$887$	−22.9409	−0.770279	−0.385139	−	0.922858i	$-0.625847\pi$
$887$	−22.9409	−0.770279	−0.385139	+	0.922858i	$0.625847\pi$
$888$	0	0
$889$	−17.2730	−0.579319
$890$	−54.4441	−1.82497
$891$	0	0
$892$	5.29001	0.177123
$893$	25.0203	0.837273
$894$	0	0
$895$	−41.5426	−1.38862
$896$	−4.90851	−0.163982
$897$	0	0
$898$	4.69083	0.156535
$899$	0	0
$900$	0	0
$901$	−15.3460	−0.511250
$902$	0.706163	0.0235126
$903$	0	0
$904$	−8.14249	−0.270815
$905$	35.1402	1.16810
$906$	0	0
$907$	27.2867	0.906038	0.453019	−	0.891501i	$-0.350347\pi$
$907$	27.2867	0.906038	0.453019	+	0.891501i	$0.350347\pi$
$908$	−14.3881	−0.477486
$909$	0	0
$910$	−23.4583	−0.777635
$911$	38.6961	1.28206	0.641029	−	0.767517i	$-0.278508\pi$
$911$	38.6961	1.28206	0.641029	+	0.767517i	$0.278508\pi$
$912$	0	0
$913$	6.86727	0.227273
$914$	−2.45870	−0.0813264
$915$	0	0
$916$	0.785959	0.0259688
$917$	−5.25537	−0.173547
$918$	0	0
$919$	34.4336	1.13586	0.567929	−	0.823077i	$-0.307745\pi$
$919$	34.4336	1.13586	0.567929	+	0.823077i	$0.307745\pi$
$920$	−83.2961	−2.74619
$921$	0	0
$922$	11.6543	0.383814
$923$	−13.3016	−0.437829
$924$	0	0
$925$	57.1900	1.88040
$926$	−1.86601	−0.0613208
$927$	0	0
$928$	9.03765	0.296675
$929$	3.72381	0.122174	0.0610871	−	0.998132i	$-0.480543\pi$
$929$	3.72381	0.122174	0.0610871	+	0.998132i	$0.480543\pi$
$930$	0	0
$931$	−13.5852	−0.445236
$932$	1.98546	0.0650358
$933$	0	0
$934$	15.2847	0.500132
$935$	46.4475	1.51900
$936$	0	0
$937$	21.7976	0.712097	0.356048	−	0.934468i	$-0.384124\pi$
$937$	21.7976	0.712097	0.356048	+	0.934468i	$0.384124\pi$
$938$	13.6237	0.444829
$939$	0	0
$940$	−48.1450	−1.57032
$941$	29.3671	0.957339	0.478669	−	0.877995i	$-0.341119\pi$
$941$	29.3671	0.957339	0.478669	+	0.877995i	$0.341119\pi$
$942$	0	0
$943$	1.50661	0.0490619
$944$	2.18651	0.0711647
$945$	0	0
$946$	15.4369	0.501898
$947$	11.9073	0.386935	0.193468	−	0.981107i	$-0.438027\pi$
$947$	11.9073	0.386935	0.193468	+	0.981107i	$0.438027\pi$
$948$	0	0
$949$	25.9890	0.843638
$950$	−29.0246	−0.941681
$951$	0	0
$952$	−11.1734	−0.362131
$953$	−40.2144	−1.30267	−0.651337	−	0.758789i	$-0.725791\pi$
$953$	−40.2144	−1.30267	−0.651337	+	0.758789i	$0.725791\pi$
$954$	0	0
$955$	−12.1692	−0.393786
$956$	14.3869	0.465304
$957$	0	0
$958$	−30.5705	−0.987689
$959$	5.55706	0.179447
$960$	0	0
$961$	0	0
$962$	22.1890	0.715401
$963$	0	0
$964$	−2.84219	−0.0915407
$965$	25.6541	0.825836
$966$	0	0
$967$	15.3870	0.494813	0.247406	−	0.968912i	$-0.420422\pi$
$967$	15.3870	0.494813	0.247406	+	0.968912i	$0.420422\pi$
$968$	−1.00316	−0.0322429
$969$	0	0
$970$	73.7858	2.36912
$971$	15.9535	0.511973	0.255986	−	0.966680i	$-0.417600\pi$
$971$	15.9535	0.511973	0.255986	+	0.966680i	$0.417600\pi$
$972$	0	0
$973$	−13.5456	−0.434250
$974$	−23.7319	−0.760421
$975$	0	0
$976$	−8.03205	−0.257100
$977$	19.6437	0.628458	0.314229	−	0.949347i	$-0.398254\pi$
$977$	19.6437	0.628458	0.314229	+	0.949347i	$0.398254\pi$
$978$	0	0
$979$	43.9253	1.40386
$980$	26.1411	0.835047
$981$	0	0
$982$	−26.6505	−0.850450
$983$	42.8679	1.36727	0.683636	−	0.729823i	$-0.260397\pi$
$983$	42.8679	1.36727	0.683636	+	0.729823i	$0.260397\pi$
$984$	0	0
$985$	−61.6140	−1.96319
$986$	5.56100	0.177098
$987$	0	0
$988$	13.1997	0.419938
$989$	32.9349	1.04727
$990$	0	0
$991$	−34.6394	−1.10036	−0.550179	−	0.835047i	$-0.685441\pi$
$991$	−34.6394	−1.10036	−0.550179	+	0.835047i	$0.685441\pi$
$992$	0	0
$993$	0	0
$994$	2.76759	0.0877825
$995$	25.7884	0.817547
$996$	0	0
$997$	46.4170	1.47004	0.735021	−	0.678045i	$-0.237172\pi$
$997$	46.4170	1.47004	0.735021	+	0.678045i	$0.237172\pi$
$998$	−5.50545	−0.174272
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bu.1.10		24
3.2	odd	2		2883.2.a.v.1.15	yes	24
31.30	odd	2		8649.2.a.bv.1.10		24
93.92	even	2		2883.2.a.u.1.15	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.2.a.u.1.15	✓	24	93.92	even	2
2883.2.a.v.1.15	yes	24	3.2	odd	2
8649.2.a.bu.1.10		24	1.1	even	1	trivial
8649.2.a.bv.1.10		24	31.30	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-0.959557 q^{2} -1.07925 q^{4} +4.21749 q^{5} -1.12111 q^{7} +2.95472 q^{8} +O(q^{10})\)	\(q-0.959557 q^{2} -1.07925 q^{4} +4.21749 q^{5} -1.12111 q^{7} +2.95472 q^{8} -4.04692 q^{10} +3.26504 q^{11} -5.17039 q^{13} +1.07577 q^{14} -0.676715 q^{16} +3.37303 q^{17} +2.36547 q^{19} -4.55173 q^{20} -3.13299 q^{22} -6.68428 q^{23} +12.7872 q^{25} +4.96128 q^{26} +1.20996 q^{28} -1.71816 q^{29} -5.26009 q^{32} -3.23661 q^{34} -4.72828 q^{35} +4.47243 q^{37} -2.26981 q^{38} +12.4615 q^{40} -0.225396 q^{41} -4.92722 q^{43} -3.52380 q^{44} +6.41394 q^{46} +10.5773 q^{47} -5.74311 q^{49} -12.2701 q^{50} +5.58015 q^{52} -4.54963 q^{53} +13.7703 q^{55} -3.31256 q^{56} +1.64867 q^{58} -3.23106 q^{59} +11.8692 q^{61} +6.40078 q^{64} -21.8061 q^{65} +12.6641 q^{67} -3.64034 q^{68} +4.53705 q^{70} +2.57266 q^{71} -5.02651 q^{73} -4.29155 q^{74} -2.55294 q^{76} -3.66047 q^{77} -3.11460 q^{79} -2.85404 q^{80} +0.216280 q^{82} +2.10327 q^{83} +14.2257 q^{85} +4.72795 q^{86} +9.64727 q^{88} +13.4532 q^{89} +5.79658 q^{91} +7.21402 q^{92} -10.1495 q^{94} +9.97637 q^{95} -18.2326 q^{97} +5.51084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8}+O(q^{10}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8	\( 24 q + 32 q^{4} + 16 q^{7} + 24 q^{8} - 8 q^{10} - 16 q^{11} + 32 q^{13} + 24 q^{14} + 48 q^{16} - 32 q^{17} + 32 q^{19} + 24 q^{20} + 32 q^{22} - 32 q^{23} + 40 q^{25} - 16 q^{26} + 8 q^{28} - 48 q^{29} + 48 q^{32} + 48 q^{35} + 64 q^{37} + 24 q^{38} + 32 q^{43} - 48 q^{44} + 32 q^{46} + 48 q^{47} + 56 q^{49} + 24 q^{50} + 64 q^{52} - 80 q^{53} + 48 q^{56} + 32 q^{58} + 32 q^{61} + 56 q^{64} - 16 q^{65} - 16 q^{67} - 80 q^{68} + 8 q^{70} + 32 q^{73} + 56 q^{76} - 96 q^{77} + 32 q^{79} + 72 q^{80} + 8 q^{82} - 48 q^{83} + 96 q^{85} + 32 q^{86} + 96 q^{88} + 16 q^{89} - 32 q^{92} + 48 q^{94} + 48 q^{95} + 16 q^{97} + 24 q^{98}+O(q^{100}) \) 24 * q + 32 * q^4 + 16 * q^7 + 24 * q^8 - 8 * q^10 - 16 * q^11 + 32 * q^13 + 24 * q^14 + 48 * q^16 - 32 * q^17 + 32 * q^19 + 24 * q^20 + 32 * q^22 - 32 * q^23 + 40 * q^25 - 16 * q^26 + 8 * q^28 - 48 * q^29 + 48 * q^32 + 48 * q^35 + 64 * q^37 + 24 * q^38 + 32 * q^43 - 48 * q^44 + 32 * q^46 + 48 * q^47 + 56 * q^49 + 24 * q^50 + 64 * q^52 - 80 * q^53 + 48 * q^56 + 32 * q^58 + 32 * q^61 + 56 * q^64 - 16 * q^65 - 16 * q^67 - 80 * q^68 + 8 * q^70 + 32 * q^73 + 56 * q^76 - 96 * q^77 + 32 * q^79 + 72 * q^80 + 8 * q^82 - 48 * q^83 + 96 * q^85 + 32 * q^86 + 96 * q^88 + 16 * q^89 - 32 * q^92 + 48 * q^94 + 48 * q^95 + 16 * q^97 + 24 * q^98

Introduction

L-functions

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Database

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