LMFDB - Embedded newform 9025.2.a.bl.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.90211 q^{2} +1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} -0.726543 q^{8} +0.618034 q^{9} -5.85410 q^{11} +3.07768 q^{12} -3.07768 q^{13} +8.05748 q^{14} -4.61803 q^{16} -5.23607 q^{17} +1.17557 q^{18} +8.05748 q^{21} -11.1352 q^{22} -4.09017 q^{23} -1.38197 q^{24} -5.85410 q^{26} -4.53077 q^{27} +6.85410 q^{28} -2.80017 q^{29} +1.90211 q^{31} -7.33094 q^{32} -11.1352 q^{33} -9.95959 q^{34} +1.00000 q^{36} +2.80017 q^{37} -5.85410 q^{39} -6.88191 q^{41} +15.3262 q^{42} -0.381966 q^{43} -9.47214 q^{44} -7.77997 q^{46} +1.47214 q^{47} -8.78402 q^{48} +10.9443 q^{49} -9.95959 q^{51} -4.97980 q^{52} -11.1352 q^{53} -8.61803 q^{54} -3.07768 q^{56} -5.32624 q^{58} +14.0413 q^{59} +3.94427 q^{61} +3.61803 q^{62} +2.61803 q^{63} -4.70820 q^{64} -21.1803 q^{66} -5.98385 q^{67} -8.47214 q^{68} -7.77997 q^{69} +0.171513 q^{71} -0.449028 q^{72} +1.00000 q^{73} +5.32624 q^{74} -24.7984 q^{77} -11.1352 q^{78} -5.25731 q^{79} -10.4721 q^{81} -13.0902 q^{82} +8.76393 q^{83} +13.0373 q^{84} -0.726543 q^{86} -5.32624 q^{87} +4.25325 q^{88} +7.77997 q^{89} -13.0373 q^{91} -6.61803 q^{92} +3.61803 q^{93} +2.80017 q^{94} -13.9443 q^{96} +2.62866 q^{97} +20.8172 q^{98} -3.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * q^99

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 $

$2$	1.90211	1.34500	0.672499	−	0.740098i	$-0.265221\pi$
$2$	1.90211	1.34500	0.672499	+	0.740098i	$0.265221\pi$
$3$	1.90211	1.09819	0.549093	−	0.835761i	$-0.314973\pi$
$3$	1.90211	1.09819	0.549093	+	0.835761i	$0.314973\pi$
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	3.61803	1.47706
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	−0.726543	−0.256872
$9$	0.618034	0.206011
$10$	0	0
$11$	−5.85410	−1.76508	−0.882539	−	0.470239i	$-0.844168\pi$
$11$	−5.85410	−1.76508	−0.882539	+	0.470239i	$0.844168\pi$
$12$	3.07768	0.888451
$13$	−3.07768	−0.853596	−0.426798	−	0.904347i	$-0.640358\pi$
$13$	−3.07768	−0.853596	−0.426798	+	0.904347i	$0.640358\pi$
$14$	8.05748	2.15345
$15$	0	0
$16$	−4.61803	−1.15451
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	1.17557	0.277085
$19$	0	0
$19$	0	0
$20$	0	0
$21$	8.05748	1.75829
$22$	−11.1352	−2.37402
$23$	−4.09017	−0.852859	−0.426430	−	0.904521i	$-0.640229\pi$
$23$	−4.09017	−0.852859	−0.426430	+	0.904521i	$0.640229\pi$
$24$	−1.38197	−0.282093
$25$	0	0
$26$	−5.85410	−1.14808
$27$	−4.53077	−0.871947
$28$	6.85410	1.29530
$29$	−2.80017	−0.519978	−0.259989	−	0.965612i	$-0.583719\pi$
$29$	−2.80017	−0.519978	−0.259989	+	0.965612i	$0.583719\pi$
$30$	0	0
$31$	1.90211	0.341630	0.170815	−	0.985303i	$-0.445360\pi$
$31$	1.90211	0.341630	0.170815	+	0.985303i	$0.445360\pi$
$32$	−7.33094	−1.29594
$33$	−11.1352	−1.93838
$34$	−9.95959	−1.70806
$35$	0	0
$36$	1.00000	0.166667
$37$	2.80017	0.460345	0.230172	−	0.973150i	$-0.426071\pi$
$37$	2.80017	0.460345	0.230172	+	0.973150i	$0.426071\pi$
$38$	0	0
$39$	−5.85410	−0.937407
$40$	0	0
$41$	−6.88191	−1.07477	−0.537387	−	0.843336i	$-0.680588\pi$
$41$	−6.88191	−1.07477	−0.537387	+	0.843336i	$0.680588\pi$
$42$	15.3262	2.36489
$43$	−0.381966	−0.0582493	−0.0291246	−	0.999576i	$-0.509272\pi$
$43$	−0.381966	−0.0582493	−0.0291246	+	0.999576i	$0.509272\pi$
$44$	−9.47214	−1.42798
$45$	0	0
$46$	−7.77997	−1.14709
$47$	1.47214	0.214733	0.107367	−	0.994220i	$-0.465758\pi$
$47$	1.47214	0.214733	0.107367	+	0.994220i	$0.465758\pi$
$48$	−8.78402	−1.26786
$49$	10.9443	1.56347
$50$	0	0
$51$	−9.95959	−1.39462
$52$	−4.97980	−0.690574
$53$	−11.1352	−1.52953	−0.764766	−	0.644308i	$-0.777146\pi$
$53$	−11.1352	−1.52953	−0.764766	+	0.644308i	$0.777146\pi$
$54$	−8.61803	−1.17277
$55$	0	0
$56$	−3.07768	−0.411273
$57$	0	0
$58$	−5.32624	−0.699369
$59$	14.0413	1.82803	0.914013	−	0.405685i	$-0.132967\pi$
$59$	14.0413	1.82803	0.914013	+	0.405685i	$0.132967\pi$
$60$	0	0
$61$	3.94427	0.505012	0.252506	−	0.967595i	$-0.418745\pi$
$61$	3.94427	0.505012	0.252506	+	0.967595i	$0.418745\pi$
$62$	3.61803	0.459491
$63$	2.61803	0.329841
$64$	−4.70820	−0.588525
$65$	0	0
$66$	−21.1803	−2.60712
$67$	−5.98385	−0.731044	−0.365522	−	0.930803i	$-0.619110\pi$
$67$	−5.98385	−0.731044	−0.365522	+	0.930803i	$0.619110\pi$
$68$	−8.47214	−1.02740
$69$	−7.77997	−0.936598
$70$	0	0
$71$	0.171513	0.0203549	0.0101774	−	0.999948i	$-0.496760\pi$
$71$	0.171513	0.0203549	0.0101774	+	0.999948i	$0.496760\pi$
$72$	−0.449028	−0.0529185
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	5.32624	0.619163
$75$	0	0
$76$	0	0
$77$	−24.7984	−2.82604
$78$	−11.1352	−1.26081
$79$	−5.25731	−0.591494	−0.295747	−	0.955266i	$-0.595568\pi$
$79$	−5.25731	−0.591494	−0.295747	+	0.955266i	$0.595568\pi$
$80$	0	0
$81$	−10.4721	−1.16357
$82$	−13.0902	−1.44557
$83$	8.76393	0.961967	0.480983	−	0.876730i	$-0.340280\pi$
$83$	8.76393	0.961967	0.480983	+	0.876730i	$0.340280\pi$
$84$	13.0373	1.42248
$85$	0	0
$86$	−0.726543	−0.0783451
$87$	−5.32624	−0.571033
$88$	4.25325	0.453398
$89$	7.77997	0.824675	0.412337	−	0.911031i	$-0.364713\pi$
$89$	7.77997	0.824675	0.412337	+	0.911031i	$0.364713\pi$
$90$	0	0
$91$	−13.0373	−1.36668
$92$	−6.61803	−0.689978
$93$	3.61803	0.375173
$94$	2.80017	0.288815
$95$	0	0
$96$	−13.9443	−1.42318
$97$	2.62866	0.266900	0.133450	−	0.991056i	$-0.457395\pi$
$97$	2.62866	0.266900	0.133450	+	0.991056i	$0.457395\pi$
$98$	20.8172	2.10286
$99$	−3.61803	−0.363626
$100$	0	0
$101$	3.29180	0.327546	0.163773	−	0.986498i	$-0.447634\pi$
$101$	3.29180	0.327546	0.163773	+	0.986498i	$0.447634\pi$
$102$	−18.9443	−1.87576
$103$	14.9394	1.47202	0.736011	−	0.676970i	$-0.236707\pi$
$103$	14.9394	1.47202	0.736011	+	0.676970i	$0.236707\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	−21.1803	−2.05722
$107$	8.33499	0.805774	0.402887	−	0.915250i	$-0.368007\pi$
$107$	8.33499	0.805774	0.402887	+	0.915250i	$0.368007\pi$
$108$	−7.33094	−0.705420
$109$	−2.17963	−0.208770	−0.104385	−	0.994537i	$-0.533287\pi$
$109$	−2.17963	−0.208770	−0.104385	+	0.994537i	$0.533287\pi$
$110$	0	0
$111$	5.32624	0.505544
$112$	−19.5623	−1.84846
$113$	−18.4661	−1.73714	−0.868572	−	0.495562i	$-0.834962\pi$
$113$	−18.4661	−1.73714	−0.868572	+	0.495562i	$0.834962\pi$
$114$	0	0
$115$	0	0
$116$	−4.53077	−0.420671
$117$	−1.90211	−0.175850
$118$	26.7082	2.45869
$119$	−22.1803	−2.03327
$120$	0	0
$121$	23.2705	2.11550
$122$	7.50245	0.679240
$123$	−13.0902	−1.18030
$124$	3.07768	0.276384
$125$	0	0
$126$	4.97980	0.443636
$127$	−9.23305	−0.819301	−0.409650	−	0.912243i	$-0.634349\pi$
$127$	−9.23305	−0.819301	−0.409650	+	0.912243i	$0.634349\pi$
$128$	5.70634	0.504374
$129$	−0.726543	−0.0639685
$130$	0	0
$131$	−15.6180	−1.36455	−0.682277	−	0.731094i	$-0.739010\pi$
$131$	−15.6180	−1.36455	−0.682277	+	0.731094i	$0.739010\pi$
$132$	−18.0171	−1.56818
$133$	0	0
$134$	−11.3820	−0.983252
$135$	0	0
$136$	3.80423	0.326210
$137$	−9.76393	−0.834189	−0.417095	−	0.908863i	$-0.636952\pi$
$137$	−9.76393	−0.834189	−0.417095	+	0.908863i	$0.636952\pi$
$138$	−14.7984	−1.25972
$139$	−22.0902	−1.87366	−0.936832	−	0.349780i	$-0.886256\pi$
$139$	−22.0902	−1.87366	−0.936832	+	0.349780i	$0.886256\pi$
$140$	0	0
$141$	2.80017	0.235817
$142$	0.326238	0.0273773
$143$	18.0171	1.50666
$144$	−2.85410	−0.237842
$145$	0	0
$146$	1.90211	0.157420
$147$	20.8172	1.71698
$148$	4.53077	0.372427
$149$	7.09017	0.580849	0.290425	−	0.956898i	$-0.406203\pi$
$149$	7.09017	0.580849	0.290425	+	0.956898i	$0.406203\pi$
$150$	0	0
$151$	0.620541	0.0504989	0.0252495	−	0.999681i	$-0.491962\pi$
$151$	0.620541	0.0504989	0.0252495	+	0.999681i	$0.491962\pi$
$152$	0	0
$153$	−3.23607	−0.261621
$154$	−47.1693	−3.80101
$155$	0	0
$156$	−9.47214	−0.758378
$157$	16.2705	1.29853	0.649264	−	0.760563i	$-0.275077\pi$
$157$	16.2705	1.29853	0.649264	+	0.760563i	$0.275077\pi$
$158$	−10.0000	−0.795557
$159$	−21.1803	−1.67971
$160$	0	0
$161$	−17.3262	−1.36550
$162$	−19.9192	−1.56500
$163$	25.1803	1.97228	0.986138	−	0.165926i	$-0.0530613\pi$
$163$	25.1803	1.97228	0.986138	+	0.165926i	$0.0530613\pi$
$164$	−11.1352	−0.869510
$165$	0	0
$166$	16.6700	1.29384
$167$	−0.171513	−0.0132721	−0.00663605	−	0.999978i	$-0.502112\pi$
$167$	−0.171513	−0.0132721	−0.00663605	+	0.999978i	$0.502112\pi$
$168$	−5.85410	−0.451654
$169$	−3.52786	−0.271374
$170$	0	0
$171$	0	0
$172$	−0.618034	−0.0471246
$173$	13.2088	1.00425	0.502123	−	0.864796i	$-0.332553\pi$
$173$	13.2088	1.00425	0.502123	+	0.864796i	$0.332553\pi$
$174$	−10.1311	−0.768037
$175$	0	0
$176$	27.0344	2.03780
$177$	26.7082	2.00751
$178$	14.7984	1.10919
$179$	10.1311	0.757234	0.378617	−	0.925553i	$-0.376400\pi$
$179$	10.1311	0.757234	0.378617	+	0.925553i	$0.376400\pi$
$180$	0	0
$181$	−6.71040	−0.498780	−0.249390	−	0.968403i	$-0.580230\pi$
$181$	−6.71040	−0.498780	−0.249390	+	0.968403i	$0.580230\pi$
$182$	−24.7984	−1.83818
$183$	7.50245	0.554597
$184$	2.97168	0.219075
$185$	0	0
$186$	6.88191	0.504606
$187$	30.6525	2.24153
$188$	2.38197	0.173723
$189$	−19.1926	−1.39606
$190$	0	0
$191$	13.0000	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$192$	−8.95554	−0.646310
$193$	−1.45309	−0.104595	−0.0522977	−	0.998632i	$-0.516654\pi$
$193$	−1.45309	−0.104595	−0.0522977	+	0.998632i	$0.516654\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	17.7082	1.26487
$197$	−14.8885	−1.06076	−0.530382	−	0.847759i	$-0.677952\pi$
$197$	−14.8885	−1.06076	−0.530382	+	0.847759i	$0.677952\pi$
$198$	−6.88191	−0.489076
$199$	13.4164	0.951064	0.475532	−	0.879698i	$-0.342256\pi$
$199$	13.4164	0.951064	0.475532	+	0.879698i	$0.342256\pi$
$200$	0	0
$201$	−11.3820	−0.802822
$202$	6.26137	0.440548
$203$	−11.8617	−0.832529
$204$	−16.1150	−1.12827
$205$	0	0
$206$	28.4164	1.97986
$207$	−2.52786	−0.175699
$208$	14.2128	0.985484
$209$	0	0
$210$	0	0
$211$	−1.73060	−0.119139	−0.0595697	−	0.998224i	$-0.518973\pi$
$211$	−1.73060	−0.119139	−0.0595697	+	0.998224i	$0.518973\pi$
$212$	−18.0171	−1.23742
$213$	0.326238	0.0223535
$214$	15.8541	1.08376
$215$	0	0
$216$	3.29180	0.223978
$217$	8.05748	0.546977
$218$	−4.14590	−0.280796
$219$	1.90211	0.128533
$220$	0	0
$221$	16.1150	1.08401
$222$	10.1311	0.679955
$223$	14.8334	0.993317	0.496659	−	0.867946i	$-0.334560\pi$
$223$	14.8334	0.993317	0.496659	+	0.867946i	$0.334560\pi$
$224$	−31.0543	−2.07491
$225$	0	0
$226$	−35.1246	−2.33645
$227$	−17.3965	−1.15465	−0.577324	−	0.816515i	$-0.695903\pi$
$227$	−17.3965	−1.15465	−0.577324	+	0.816515i	$0.695903\pi$
$228$	0	0
$229$	−18.1459	−1.19911	−0.599557	−	0.800332i	$-0.704657\pi$
$229$	−18.1459	−1.19911	−0.599557	+	0.800332i	$0.704657\pi$
$230$	0	0
$231$	−47.1693	−3.10351
$232$	2.03444	0.133568
$233$	−1.76393	−0.115559	−0.0577795	−	0.998329i	$-0.518402\pi$
$233$	−1.76393	−0.115559	−0.0577795	+	0.998329i	$0.518402\pi$
$234$	−3.61803	−0.236518
$235$	0	0
$236$	22.7194	1.47890
$237$	−10.0000	−0.649570
$238$	−42.1895	−2.73474
$239$	−23.2148	−1.50164	−0.750820	−	0.660507i	$-0.770341\pi$
$239$	−23.2148	−1.50164	−0.750820	+	0.660507i	$0.770341\pi$
$240$	0	0
$241$	−6.32688	−0.407550	−0.203775	−	0.979018i	$-0.565321\pi$
$241$	−6.32688	−0.407550	−0.203775	+	0.979018i	$0.565321\pi$
$242$	44.2631	2.84534
$243$	−6.32688	−0.405870
$244$	6.38197	0.408564
$245$	0	0
$246$	−24.8990	−1.58750
$247$	0	0
$248$	−1.38197	−0.0877549
$249$	16.6700	1.05642
$250$	0	0
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	4.23607	0.266847
$253$	23.9443	1.50536
$254$	−17.5623	−1.10196
$255$	0	0
$256$	20.2705	1.26691
$257$	3.24920	0.202679	0.101340	−	0.994852i	$-0.467687\pi$
$257$	3.24920	0.202679	0.101340	+	0.994852i	$0.467687\pi$
$258$	−1.38197	−0.0860374
$259$	11.8617	0.737051
$260$	0	0
$261$	−1.73060	−0.107121
$262$	−29.7073	−1.83532
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	8.09017	0.497916
$265$	0	0
$266$	0	0
$267$	14.7984	0.905646
$268$	−9.68208	−0.591427
$269$	19.6417	1.19757	0.598787	−	0.800908i	$-0.295650\pi$
$269$	19.6417	1.19757	0.598787	+	0.800908i	$0.295650\pi$
$270$	0	0
$271$	8.61803	0.523508	0.261754	−	0.965135i	$-0.415699\pi$
$271$	8.61803	0.523508	0.261754	+	0.965135i	$0.415699\pi$
$272$	24.1803	1.46615
$273$	−24.7984	−1.50087
$274$	−18.5721	−1.12198
$275$	0	0
$276$	−12.5882	−0.757724
$277$	−24.8328	−1.49206	−0.746030	−	0.665913i	$-0.768042\pi$
$277$	−24.8328	−1.49206	−0.746030	+	0.665913i	$0.768042\pi$
$278$	−42.0180	−2.52007
$279$	1.17557	0.0703796
$280$	0	0
$281$	−25.0705	−1.49558	−0.747790	−	0.663935i	$-0.768885\pi$
$281$	−25.0705	−1.49558	−0.747790	+	0.663935i	$0.768885\pi$
$282$	5.32624	0.317173
$283$	28.2705	1.68051	0.840254	−	0.542193i	$-0.182406\pi$
$283$	28.2705	1.68051	0.840254	+	0.542193i	$0.182406\pi$
$284$	0.277515	0.0164675
$285$	0	0
$286$	34.2705	2.02646
$287$	−29.1522	−1.72080
$288$	−4.53077	−0.266978
$289$	10.4164	0.612730
$290$	0	0
$291$	5.00000	0.293105
$292$	1.61803	0.0946883
$293$	−13.8293	−0.807918	−0.403959	−	0.914777i	$-0.632366\pi$
$293$	−13.8293	−0.807918	−0.403959	+	0.914777i	$0.632366\pi$
$294$	39.5967	2.30933
$295$	0	0
$296$	−2.03444	−0.118250
$297$	26.5236	1.53905
$298$	13.4863	0.781241
$299$	12.5882	0.727997
$300$	0	0
$301$	−1.61803	−0.0932619
$302$	1.18034	0.0679209
$303$	6.26137	0.359706
$304$	0	0
$305$	0	0
$306$	−6.15537	−0.351879
$307$	−18.3601	−1.04787	−0.523933	−	0.851759i	$-0.675536\pi$
$307$	−18.3601	−1.04787	−0.523933	+	0.851759i	$0.675536\pi$
$308$	−40.1246	−2.28631
$309$	28.4164	1.61655
$310$	0	0
$311$	−2.76393	−0.156728	−0.0783641	−	0.996925i	$-0.524970\pi$
$311$	−2.76393	−0.156728	−0.0783641	+	0.996925i	$0.524970\pi$
$312$	4.25325	0.240793
$313$	−3.27051	−0.184860	−0.0924301	−	0.995719i	$-0.529463\pi$
$313$	−3.27051	−0.184860	−0.0924301	+	0.995719i	$0.529463\pi$
$314$	30.9483	1.74652
$315$	0	0
$316$	−8.50651	−0.478528
$317$	16.4580	0.924373	0.462186	−	0.886783i	$-0.347065\pi$
$317$	16.4580	0.924373	0.462186	+	0.886783i	$0.347065\pi$
$318$	−40.2874	−2.25921
$319$	16.3925	0.917802
$320$	0	0
$321$	15.8541	0.884890
$322$	−32.9565	−1.83659
$323$	0	0
$324$	−16.9443	−0.941348
$325$	0	0
$326$	47.8959	2.65271
$327$	−4.14590	−0.229269
$328$	5.00000	0.276079
$329$	6.23607	0.343806
$330$	0	0
$331$	−19.9192	−1.09486	−0.547429	−	0.836852i	$-0.684393\pi$
$331$	−19.9192	−1.09486	−0.547429	+	0.836852i	$0.684393\pi$
$332$	14.1803	0.778247
$333$	1.73060	0.0948363
$334$	−0.326238	−0.0178509
$335$	0	0
$336$	−37.2097	−2.02996
$337$	−28.4257	−1.54845	−0.774223	−	0.632913i	$-0.781859\pi$
$337$	−28.4257	−1.54845	−0.774223	+	0.632913i	$0.781859\pi$
$338$	−6.71040	−0.364997
$339$	−35.1246	−1.90771
$340$	0	0
$341$	−11.1352	−0.603003
$342$	0	0
$343$	16.7082	0.902158
$344$	0.277515	0.0149626
$345$	0	0
$346$	25.1246	1.35071
$347$	−16.4721	−0.884271	−0.442135	−	0.896948i	$-0.645779\pi$
$347$	−16.4721	−0.884271	−0.442135	+	0.896948i	$0.645779\pi$
$348$	−8.61803	−0.461975
$349$	−2.56231	−0.137157	−0.0685785	−	0.997646i	$-0.521846\pi$
$349$	−2.56231	−0.137157	−0.0685785	+	0.997646i	$0.521846\pi$
$350$	0	0
$351$	13.9443	0.744290
$352$	42.9161	2.28743
$353$	−3.67376	−0.195535	−0.0977673	−	0.995209i	$-0.531170\pi$
$353$	−3.67376	−0.195535	−0.0977673	+	0.995209i	$0.531170\pi$
$354$	50.8020	2.70010
$355$	0	0
$356$	12.5882	0.667176
$357$	−42.1895	−2.23291
$358$	19.2705	1.01848
$359$	7.38197	0.389605	0.194803	−	0.980842i	$-0.437593\pi$
$359$	7.38197	0.389605	0.194803	+	0.980842i	$0.437593\pi$
$360$	0	0
$361$	0	0
$362$	−12.7639	−0.670857
$363$	44.2631	2.32321
$364$	−21.0948	−1.10567
$365$	0	0
$366$	14.2705	0.745931
$367$	−31.0689	−1.62178	−0.810891	−	0.585197i	$-0.801017\pi$
$367$	−31.0689	−1.62178	−0.810891	+	0.585197i	$0.801017\pi$
$368$	18.8885	0.984633
$369$	−4.25325	−0.221416
$370$	0	0
$371$	−47.1693	−2.44891
$372$	5.85410	0.303521
$373$	−5.42882	−0.281094	−0.140547	−	0.990074i	$-0.544886\pi$
$373$	−5.42882	−0.281094	−0.140547	+	0.990074i	$0.544886\pi$
$374$	58.3045	3.01485
$375$	0	0
$376$	−1.06957	−0.0551588
$377$	8.61803	0.443851
$378$	−36.5066	−1.87770
$379$	−12.8658	−0.660870	−0.330435	−	0.943829i	$-0.607195\pi$
$379$	−12.8658	−0.660870	−0.330435	+	0.943829i	$0.607195\pi$
$380$	0	0
$381$	−17.5623	−0.899744
$382$	24.7275	1.26517
$383$	4.25325	0.217331	0.108666	−	0.994078i	$-0.465342\pi$
$383$	4.25325	0.217331	0.108666	+	0.994078i	$0.465342\pi$
$384$	10.8541	0.553896
$385$	0	0
$386$	−2.76393	−0.140680
$387$	−0.236068	−0.0120000
$388$	4.25325	0.215926
$389$	−20.8541	−1.05734	−0.528672	−	0.848826i	$-0.677310\pi$
$389$	−20.8541	−1.05734	−0.528672	+	0.848826i	$0.677310\pi$
$390$	0	0
$391$	21.4164	1.08307
$392$	−7.95148	−0.401610
$393$	−29.7073	−1.49853
$394$	−28.3197	−1.42673
$395$	0	0
$396$	−5.85410	−0.294180
$397$	23.8328	1.19613	0.598067	−	0.801446i	$-0.295936\pi$
$397$	23.8328	1.19613	0.598067	+	0.801446i	$0.295936\pi$
$398$	25.5195	1.27918
$399$	0	0
$400$	0	0
$401$	9.06154	0.452512	0.226256	−	0.974068i	$-0.427351\pi$
$401$	9.06154	0.452512	0.226256	+	0.974068i	$0.427351\pi$
$402$	−21.6498	−1.07979
$403$	−5.85410	−0.291614
$404$	5.32624	0.264990
$405$	0	0
$406$	−22.5623	−1.11975
$407$	−16.3925	−0.812545
$408$	7.23607	0.358239
$409$	21.7153	1.07375	0.536876	−	0.843661i	$-0.319604\pi$
$409$	21.7153	1.07375	0.536876	+	0.843661i	$0.319604\pi$
$410$	0	0
$411$	−18.5721	−0.916094
$412$	24.1724	1.19089
$413$	59.4800	2.92682
$414$	−4.80828	−0.236314
$415$	0	0
$416$	22.5623	1.10621
$417$	−42.0180	−2.05763
$418$	0	0
$419$	26.1803	1.27899	0.639497	−	0.768794i	$-0.279143\pi$
$419$	26.1803	1.27899	0.639497	+	0.768794i	$0.279143\pi$
$420$	0	0
$421$	−11.5842	−0.564579	−0.282289	−	0.959329i	$-0.591094\pi$
$421$	−11.5842	−0.564579	−0.282289	+	0.959329i	$0.591094\pi$
$422$	−3.29180	−0.160242
$423$	0.909830	0.0442375
$424$	8.09017	0.392893
$425$	0	0
$426$	0.620541	0.0300653
$427$	16.7082	0.808567
$428$	13.4863	0.651885
$429$	34.2705	1.65460
$430$	0	0
$431$	6.88191	0.331490	0.165745	−	0.986169i	$-0.446997\pi$
$431$	6.88191	0.331490	0.165745	+	0.986169i	$0.446997\pi$
$432$	20.9232	1.00667
$433$	6.04937	0.290714	0.145357	−	0.989379i	$-0.453567\pi$
$433$	6.04937	0.290714	0.145357	+	0.989379i	$0.453567\pi$
$434$	15.3262	0.735683
$435$	0	0
$436$	−3.52671	−0.168899
$437$	0	0
$438$	3.61803	0.172876
$439$	5.64083	0.269222	0.134611	−	0.990899i	$-0.457022\pi$
$439$	5.64083	0.269222	0.134611	+	0.990899i	$0.457022\pi$
$440$	0	0
$441$	6.76393	0.322092
$442$	30.6525	1.45799
$443$	−34.7771	−1.65231	−0.826155	−	0.563443i	$-0.809476\pi$
$443$	−34.7771	−1.65231	−0.826155	+	0.563443i	$0.809476\pi$
$444$	8.61803	0.408994
$445$	0	0
$446$	28.2148	1.33601
$447$	13.4863	0.637880
$448$	−19.9443	−0.942278
$449$	5.64083	0.266207	0.133104	−	0.991102i	$-0.457506\pi$
$449$	5.64083	0.266207	0.133104	+	0.991102i	$0.457506\pi$
$450$	0	0
$451$	40.2874	1.89706
$452$	−29.8788	−1.40538
$453$	1.18034	0.0554572
$454$	−33.0902	−1.55300
$455$	0	0
$456$	0	0
$457$	−5.23607	−0.244933	−0.122466	−	0.992473i	$-0.539080\pi$
$457$	−5.23607	−0.244933	−0.122466	+	0.992473i	$0.539080\pi$
$458$	−34.5155	−1.61281
$459$	23.7234	1.10731
$460$	0	0
$461$	−8.00000	−0.372597	−0.186299	−	0.982493i	$-0.559649\pi$
$461$	−8.00000	−0.372597	−0.186299	+	0.982493i	$0.559649\pi$
$462$	−89.7214	−4.17422
$463$	−0.145898	−0.00678046	−0.00339023	−	0.999994i	$-0.501079\pi$
$463$	−0.145898	−0.00678046	−0.00339023	+	0.999994i	$0.501079\pi$
$464$	12.9313	0.600319
$465$	0	0
$466$	−3.35520	−0.155427
$467$	9.18034	0.424815	0.212408	−	0.977181i	$-0.431870\pi$
$467$	9.18034	0.424815	0.212408	+	0.977181i	$0.431870\pi$
$468$	−3.07768	−0.142266
$469$	−25.3480	−1.17046
$470$	0	0
$471$	30.9483	1.42602
$472$	−10.2016	−0.469568
$473$	2.23607	0.102815
$474$	−19.0211	−0.873669
$475$	0	0
$476$	−35.8885	−1.64495
$477$	−6.88191	−0.315101
$478$	−44.1571	−2.01970
$479$	−11.4377	−0.522602	−0.261301	−	0.965257i	$-0.584151\pi$
$479$	−11.4377	−0.522602	−0.261301	+	0.965257i	$0.584151\pi$
$480$	0	0
$481$	−8.61803	−0.392949
$482$	−12.0344	−0.548154
$483$	−32.9565	−1.49957
$484$	37.6525	1.71148
$485$	0	0
$486$	−12.0344	−0.545893
$487$	21.0292	0.952926	0.476463	−	0.879195i	$-0.341919\pi$
$487$	21.0292	0.952926	0.476463	+	0.879195i	$0.341919\pi$
$488$	−2.86568	−0.129723
$489$	47.8959	2.16593
$490$	0	0
$491$	37.2705	1.68199	0.840997	−	0.541039i	$-0.181969\pi$
$491$	37.2705	1.68199	0.840997	+	0.541039i	$0.181969\pi$
$492$	−21.1803	−0.954883
$493$	14.6619	0.660338
$494$	0	0
$495$	0	0
$496$	−8.78402	−0.394414
$497$	0.726543	0.0325899
$498$	31.7082	1.42088
$499$	−29.5279	−1.32185	−0.660924	−	0.750453i	$-0.729836\pi$
$499$	−29.5279	−1.32185	−0.660924	+	0.750453i	$0.729836\pi$
$500$	0	0
$501$	−0.326238	−0.0145752
$502$	−5.70634	−0.254686
$503$	−31.6525	−1.41131	−0.705657	−	0.708554i	$-0.749348\pi$
$503$	−31.6525	−1.41131	−0.705657	+	0.708554i	$0.749348\pi$
$504$	−1.90211	−0.0847268
$505$	0	0
$506$	45.5447	2.02471
$507$	−6.71040	−0.298019
$508$	−14.9394	−0.662828
$509$	30.1563	1.33665	0.668327	−	0.743868i	$-0.267011\pi$
$509$	30.1563	1.33665	0.668327	+	0.743868i	$0.267011\pi$
$510$	0	0
$511$	4.23607	0.187393
$512$	27.1441	1.19961
$513$	0	0
$514$	6.18034	0.272603
$515$	0	0
$516$	−1.17557	−0.0517516
$517$	−8.61803	−0.379021
$518$	22.5623	0.991331
$519$	25.1246	1.10285
$520$	0	0
$521$	−32.6789	−1.43169	−0.715845	−	0.698259i	$-0.753958\pi$
$521$	−32.6789	−1.43169	−0.715845	+	0.698259i	$0.753958\pi$
$522$	−3.29180	−0.144078
$523$	6.53888	0.285925	0.142963	−	0.989728i	$-0.454337\pi$
$523$	6.53888	0.285925	0.142963	+	0.989728i	$0.454337\pi$
$524$	−25.2705	−1.10395
$525$	0	0
$526$	11.4127	0.497616
$527$	−9.95959	−0.433847
$528$	51.4226	2.23788
$529$	−6.27051	−0.272631
$530$	0	0
$531$	8.67802	0.376594
$532$	0	0
$533$	21.1803	0.917422
$534$	28.1482	1.21809
$535$	0	0
$536$	4.34752	0.187784
$537$	19.2705	0.831584
$538$	37.3607	1.61073
$539$	−64.0689	−2.75964
$540$	0	0
$541$	−24.5967	−1.05750	−0.528748	−	0.848779i	$-0.677338\pi$
$541$	−24.5967	−1.05750	−0.528748	+	0.848779i	$0.677338\pi$
$542$	16.3925	0.704117
$543$	−12.7639	−0.547753
$544$	38.3853	1.64576
$545$	0	0
$546$	−47.1693	−2.01866
$547$	21.2663	0.909280	0.454640	−	0.890675i	$-0.349768\pi$
$547$	21.2663	0.909280	0.454640	+	0.890675i	$0.349768\pi$
$548$	−15.7984	−0.674873
$549$	2.43769	0.104038
$550$	0	0
$551$	0	0
$552$	5.65248	0.240585
$553$	−22.2703	−0.947031
$554$	−47.2348	−2.00682
$555$	0	0
$556$	−35.7426	−1.51583
$557$	31.0689	1.31643	0.658215	−	0.752830i	$-0.271312\pi$
$557$	31.0689	1.31643	0.658215	+	0.752830i	$0.271312\pi$
$558$	2.23607	0.0946603
$559$	1.17557	0.0497213
$560$	0	0
$561$	58.3045	2.46162
$562$	−47.6869	−2.01155
$563$	4.35926	0.183721	0.0918604	−	0.995772i	$-0.470719\pi$
$563$	4.35926	0.183721	0.0918604	+	0.995772i	$0.470719\pi$
$564$	4.53077	0.190780
$565$	0	0
$566$	53.7737	2.26028
$567$	−44.3607	−1.86297
$568$	−0.124612	−0.00522859
$569$	−25.7970	−1.08147	−0.540734	−	0.841194i	$-0.681853\pi$
$569$	−25.7970	−1.08147	−0.540734	+	0.841194i	$0.681853\pi$
$570$	0	0
$571$	−25.8541	−1.08196	−0.540980	−	0.841035i	$-0.681947\pi$
$571$	−25.8541	−1.08196	−0.540980	+	0.841035i	$0.681947\pi$
$572$	29.1522	1.21892
$573$	24.7275	1.03300
$574$	−55.4508	−2.31447
$575$	0	0
$576$	−2.90983	−0.121243
$577$	−8.52786	−0.355020	−0.177510	−	0.984119i	$-0.556804\pi$
$577$	−8.52786	−0.355020	−0.177510	+	0.984119i	$0.556804\pi$
$578$	19.8132	0.824120
$579$	−2.76393	−0.114865
$580$	0	0
$581$	37.1246	1.54019
$582$	9.51057	0.394226
$583$	65.1864	2.69974
$584$	−0.726543	−0.0300645
$585$	0	0
$586$	−26.3050	−1.08665
$587$	−36.4164	−1.50307	−0.751533	−	0.659695i	$-0.770685\pi$
$587$	−36.4164	−1.50307	−0.751533	+	0.659695i	$0.770685\pi$
$588$	33.6830	1.38906
$589$	0	0
$590$	0	0
$591$	−28.3197	−1.16492
$592$	−12.9313	−0.531472
$593$	−20.7082	−0.850384	−0.425192	−	0.905103i	$-0.639793\pi$
$593$	−20.7082	−0.850384	−0.425192	+	0.905103i	$0.639793\pi$
$594$	50.4508	2.07002
$595$	0	0
$596$	11.4721	0.469917
$597$	25.5195	1.04444
$598$	23.9443	0.979154
$599$	−30.0503	−1.22782	−0.613911	−	0.789375i	$-0.710405\pi$
$599$	−30.0503	−1.22782	−0.613911	+	0.789375i	$0.710405\pi$
$600$	0	0
$601$	10.5146	0.428900	0.214450	−	0.976735i	$-0.431204\pi$
$601$	10.5146	0.428900	0.214450	+	0.976735i	$0.431204\pi$
$602$	−3.07768	−0.125437
$603$	−3.69822	−0.150603
$604$	1.00406	0.0408545
$605$	0	0
$606$	11.9098	0.483804
$607$	16.2210	0.658389	0.329194	−	0.944262i	$-0.393223\pi$
$607$	16.2210	0.658389	0.329194	+	0.944262i	$0.393223\pi$
$608$	0	0
$609$	−22.5623	−0.914271
$610$	0	0
$611$	−4.53077	−0.183295
$612$	−5.23607	−0.211656
$613$	−26.5279	−1.07145	−0.535725	−	0.844392i	$-0.679962\pi$
$613$	−26.5279	−1.07145	−0.535725	+	0.844392i	$0.679962\pi$
$614$	−34.9230	−1.40938
$615$	0	0
$616$	18.0171	0.725929
$617$	−21.6180	−0.870309	−0.435155	−	0.900356i	$-0.643306\pi$
$617$	−21.6180	−0.870309	−0.435155	+	0.900356i	$0.643306\pi$
$618$	54.0512	2.17426
$619$	39.5410	1.58929	0.794644	−	0.607076i	$-0.207658\pi$
$619$	39.5410	1.58929	0.794644	+	0.607076i	$0.207658\pi$
$620$	0	0
$621$	18.5316	0.743648
$622$	−5.25731	−0.210799
$623$	32.9565	1.32037
$624$	27.0344	1.08224
$625$	0	0
$626$	−6.22088	−0.248636
$627$	0	0
$628$	26.3262	1.05053
$629$	−14.6619	−0.584607
$630$	0	0
$631$	−41.1803	−1.63936	−0.819682	−	0.572819i	$-0.805850\pi$
$631$	−41.1803	−1.63936	−0.819682	+	0.572819i	$0.805850\pi$
$632$	3.81966	0.151938
$633$	−3.29180	−0.130837
$634$	31.3050	1.24328
$635$	0	0
$636$	−34.2705	−1.35891
$637$	−33.6830	−1.33457
$638$	31.1803	1.23444
$639$	0.106001	0.00419334
$640$	0	0
$641$	−36.6952	−1.44937	−0.724686	−	0.689079i	$-0.758015\pi$
$641$	−36.6952	−1.44937	−0.724686	+	0.689079i	$0.758015\pi$
$642$	30.1563	1.19017
$643$	38.4721	1.51719	0.758596	−	0.651561i	$-0.225885\pi$
$643$	38.4721	1.51719	0.758596	+	0.651561i	$0.225885\pi$
$644$	−28.0344	−1.10471
$645$	0	0
$646$	0	0
$647$	−8.52786	−0.335265	−0.167632	−	0.985850i	$-0.553612\pi$
$647$	−8.52786	−0.335265	−0.167632	+	0.985850i	$0.553612\pi$
$648$	7.60845	0.298888
$649$	−82.1994	−3.22661
$650$	0	0
$651$	15.3262	0.600683
$652$	40.7426	1.59561
$653$	13.7984	0.539972	0.269986	−	0.962864i	$-0.412981\pi$
$653$	13.7984	0.539972	0.269986	+	0.962864i	$0.412981\pi$
$654$	−7.88597	−0.308366
$655$	0	0
$656$	31.7809	1.24084
$657$	0.618034	0.0241118
$658$	11.8617	0.462417
$659$	7.26543	0.283021	0.141510	−	0.989937i	$-0.454804\pi$
$659$	7.26543	0.283021	0.141510	+	0.989937i	$0.454804\pi$
$660$	0	0
$661$	−29.3238	−1.14056	−0.570281	−	0.821450i	$-0.693166\pi$
$661$	−29.3238	−1.14056	−0.570281	+	0.821450i	$0.693166\pi$
$662$	−37.8885	−1.47258
$663$	30.6525	1.19044
$664$	−6.36737	−0.247102
$665$	0	0
$666$	3.29180	0.127555
$667$	11.4532	0.443468
$668$	−0.277515	−0.0107374
$669$	28.2148	1.09085
$670$	0	0
$671$	−23.0902	−0.891386
$672$	−59.0689	−2.27863
$673$	−24.2380	−0.934304	−0.467152	−	0.884177i	$-0.654720\pi$
$673$	−24.2380	−0.934304	−0.467152	+	0.884177i	$0.654720\pi$
$674$	−54.0689	−2.08266
$675$	0	0
$676$	−5.70820	−0.219546
$677$	−33.5770	−1.29047	−0.645235	−	0.763985i	$-0.723240\pi$
$677$	−33.5770	−1.29047	−0.645235	+	0.763985i	$0.723240\pi$
$678$	−66.8110	−2.56586
$679$	11.1352	0.427328
$680$	0	0
$681$	−33.0902	−1.26802
$682$	−21.1803	−0.811037
$683$	−28.0422	−1.07300	−0.536502	−	0.843899i	$-0.680255\pi$
$683$	−28.0422	−1.07300	−0.536502	+	0.843899i	$0.680255\pi$
$684$	0	0
$685$	0	0
$686$	31.7809	1.21340
$687$	−34.5155	−1.31685
$688$	1.76393	0.0672493
$689$	34.2705	1.30560
$690$	0	0
$691$	−12.2361	−0.465482	−0.232741	−	0.972539i	$-0.574769\pi$
$691$	−12.2361	−0.465482	−0.232741	+	0.972539i	$0.574769\pi$
$692$	21.3723	0.812452
$693$	−15.3262	−0.582196
$694$	−31.3319	−1.18934
$695$	0	0
$696$	3.86974	0.146682
$697$	36.0341	1.36489
$698$	−4.87380	−0.184476
$699$	−3.35520	−0.126905
$700$	0	0
$701$	−31.9098	−1.20522	−0.602609	−	0.798037i	$-0.705872\pi$
$701$	−31.9098	−1.20522	−0.602609	+	0.798037i	$0.705872\pi$
$702$	26.5236	1.00107
$703$	0	0
$704$	27.5623	1.03879
$705$	0	0
$706$	−6.98791	−0.262993
$707$	13.9443	0.524428
$708$	43.2148	1.62411
$709$	36.1803	1.35878	0.679391	−	0.733777i	$-0.262244\pi$
$709$	36.1803	1.35878	0.679391	+	0.733777i	$0.262244\pi$
$710$	0	0
$711$	−3.24920	−0.121854
$712$	−5.65248	−0.211835
$713$	−7.77997	−0.291362
$714$	−80.2492	−3.00325
$715$	0	0
$716$	16.3925	0.612616
$717$	−44.1571	−1.64908
$718$	14.0413	0.524018
$719$	−30.9098	−1.15274	−0.576371	−	0.817188i	$-0.695532\pi$
$719$	−30.9098	−1.15274	−0.576371	+	0.817188i	$0.695532\pi$
$720$	0	0
$721$	63.2843	2.35683
$722$	0	0
$723$	−12.0344	−0.447566
$724$	−10.8576	−0.403521
$725$	0	0
$726$	84.1935	3.12471
$727$	−34.3607	−1.27437	−0.637184	−	0.770712i	$-0.719901\pi$
$727$	−34.3607	−1.27437	−0.637184	+	0.770712i	$0.719901\pi$
$728$	9.47214	0.351061
$729$	19.3820	0.717851
$730$	0	0
$731$	2.00000	0.0739727
$732$	12.1392	0.448679
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	−59.0965	−2.18129
$735$	0	0
$736$	29.9848	1.10525
$737$	35.0301	1.29035
$738$	−8.09017	−0.297803
$739$	−27.7639	−1.02131	−0.510656	−	0.859785i	$-0.670598\pi$
$739$	−27.7639	−1.02131	−0.510656	+	0.859785i	$0.670598\pi$
$740$	0	0
$741$	0	0
$742$	−89.7214	−3.29377
$743$	5.98385	0.219526	0.109763	−	0.993958i	$-0.464991\pi$
$743$	5.98385	0.219526	0.109763	+	0.993958i	$0.464991\pi$
$744$	−2.62866	−0.0963712
$745$	0	0
$746$	−10.3262	−0.378070
$747$	5.41641	0.198176
$748$	49.5967	1.81344
$749$	35.3076	1.29011
$750$	0	0
$751$	50.4590	1.84127	0.920637	−	0.390419i	$-0.127670\pi$
$751$	50.4590	1.84127	0.920637	+	0.390419i	$0.127670\pi$
$752$	−6.79837	−0.247911
$753$	−5.70634	−0.207951
$754$	16.3925	0.596979
$755$	0	0
$756$	−31.0543	−1.12944
$757$	−2.97871	−0.108263	−0.0541316	−	0.998534i	$-0.517239\pi$
$757$	−2.97871	−0.108263	−0.0541316	+	0.998534i	$0.517239\pi$
$758$	−24.4721	−0.888868
$759$	45.5447	1.65317
$760$	0	0
$761$	−18.8197	−0.682212	−0.341106	−	0.940025i	$-0.610802\pi$
$761$	−18.8197	−0.682212	−0.341106	+	0.940025i	$0.610802\pi$
$762$	−33.4055	−1.21015
$763$	−9.23305	−0.334259
$764$	21.0344	0.760999
$765$	0	0
$766$	8.09017	0.292310
$767$	−43.2148	−1.56040
$768$	38.5568	1.39130
$769$	−13.2705	−0.478547	−0.239273	−	0.970952i	$-0.576909\pi$
$769$	−13.2705	−0.478547	−0.239273	+	0.970952i	$0.576909\pi$
$770$	0	0
$771$	6.18034	0.222580
$772$	−2.35114	−0.0846194
$773$	32.5074	1.16921	0.584606	−	0.811318i	$-0.301249\pi$
$773$	32.5074	1.16921	0.584606	+	0.811318i	$0.301249\pi$
$774$	−0.449028	−0.0161400
$775$	0	0
$776$	−1.90983	−0.0685589
$777$	22.5623	0.809418
$778$	−39.6669	−1.42213
$779$	0	0
$780$	0	0
$781$	−1.00406	−0.0359280
$782$	40.7364	1.45673
$783$	12.6869	0.453393
$784$	−50.5410	−1.80504
$785$	0	0
$786$	−56.5066	−2.01552
$787$	−20.1312	−0.717599	−0.358800	−	0.933415i	$-0.616814\pi$
$787$	−20.1312	−0.717599	−0.358800	+	0.933415i	$0.616814\pi$
$788$	−24.0902	−0.858177
$789$	11.4127	0.406302
$790$	0	0
$791$	−78.2237	−2.78131
$792$	2.62866	0.0934052
$793$	−12.1392	−0.431076
$794$	45.3327	1.60880
$795$	0	0
$796$	21.7082	0.769427
$797$	23.7234	0.840326	0.420163	−	0.907449i	$-0.361973\pi$
$797$	23.7234	0.840326	0.420163	+	0.907449i	$0.361973\pi$
$798$	0	0
$799$	−7.70820	−0.272697
$800$	0	0
$801$	4.80828	0.169892
$802$	17.2361	0.608627
$803$	−5.85410	−0.206587
$804$	−18.4164	−0.649497
$805$	0	0
$806$	−11.1352	−0.392219
$807$	37.3607	1.31516
$808$	−2.39163	−0.0841372
$809$	−53.2148	−1.87093	−0.935466	−	0.353417i	$-0.885020\pi$
$809$	−53.2148	−1.87093	−0.935466	+	0.353417i	$0.885020\pi$
$810$	0	0
$811$	−13.1433	−0.461523	−0.230761	−	0.973010i	$-0.574122\pi$
$811$	−13.1433	−0.461523	−0.230761	+	0.973010i	$0.574122\pi$
$812$	−19.1926	−0.673530
$813$	16.3925	0.574909
$814$	−31.1803	−1.09287
$815$	0	0
$816$	45.9937	1.61010
$817$	0	0
$818$	41.3050	1.44419
$819$	−8.05748	−0.281551
$820$	0	0
$821$	−27.4508	−0.958041	−0.479021	−	0.877804i	$-0.659008\pi$
$821$	−27.4508	−0.958041	−0.479021	+	0.877804i	$0.659008\pi$
$822$	−35.3262	−1.23214
$823$	16.7639	0.584354	0.292177	−	0.956364i	$-0.405620\pi$
$823$	16.7639	0.584354	0.292177	+	0.956364i	$0.405620\pi$
$824$	−10.8541	−0.378121
$825$	0	0
$826$	113.138	3.93657
$827$	9.23305	0.321065	0.160532	−	0.987031i	$-0.448679\pi$
$827$	9.23305	0.321065	0.160532	+	0.987031i	$0.448679\pi$
$828$	−4.09017	−0.142143
$829$	25.9686	0.901925	0.450963	−	0.892543i	$-0.351081\pi$
$829$	25.9686	0.901925	0.450963	+	0.892543i	$0.351081\pi$
$830$	0	0
$831$	−47.2348	−1.63856
$832$	14.4904	0.502363
$833$	−57.3050	−1.98550
$834$	−79.9230	−2.76751
$835$	0	0
$836$	0	0
$837$	−8.61803	−0.297883
$838$	49.7980	1.72024
$839$	−4.08174	−0.140917	−0.0704587	−	0.997515i	$-0.522446\pi$
$839$	−4.08174	−0.140917	−0.0704587	+	0.997515i	$0.522446\pi$
$840$	0	0
$841$	−21.1591	−0.729623
$842$	−22.0344	−0.759357
$843$	−47.6869	−1.64242
$844$	−2.80017	−0.0963858
$845$	0	0
$846$	1.73060	0.0594992
$847$	98.5755	3.38709
$848$	51.4226	1.76586
$849$	53.7737	1.84551
$850$	0	0
$851$	−11.4532	−0.392610
$852$	0.527864	0.0180843
$853$	15.4721	0.529756	0.264878	−	0.964282i	$-0.414668\pi$
$853$	15.4721	0.529756	0.264878	+	0.964282i	$0.414668\pi$
$854$	31.7809	1.08752
$855$	0	0
$856$	−6.05573	−0.206981
$857$	−6.22088	−0.212501	−0.106251	−	0.994339i	$-0.533885\pi$
$857$	−6.22088	−0.212501	−0.106251	+	0.994339i	$0.533885\pi$
$858$	65.1864	2.22543
$859$	−22.7639	−0.776695	−0.388348	−	0.921513i	$-0.626954\pi$
$859$	−22.7639	−0.776695	−0.388348	+	0.921513i	$0.626954\pi$
$860$	0	0
$861$	−55.4508	−1.88976
$862$	13.0902	0.445853
$863$	−36.3772	−1.23829	−0.619147	−	0.785275i	$-0.712521\pi$
$863$	−36.3772	−1.23829	−0.619147	+	0.785275i	$0.712521\pi$
$864$	33.2148	1.12999
$865$	0	0
$866$	11.5066	0.391009
$867$	19.8132	0.672891
$868$	13.0373	0.442514
$869$	30.7768	1.04403
$870$	0	0
$871$	18.4164	0.624016
$872$	1.58359	0.0536272
$873$	1.62460	0.0549843
$874$	0	0
$875$	0	0
$876$	3.07768	0.103985
$877$	11.9677	0.404121	0.202060	−	0.979373i	$-0.435236\pi$
$877$	11.9677	0.404121	0.202060	+	0.979373i	$0.435236\pi$
$878$	10.7295	0.362103
$879$	−26.3050	−0.887244
$880$	0	0
$881$	10.3262	0.347900	0.173950	−	0.984755i	$-0.444347\pi$
$881$	10.3262	0.347900	0.173950	+	0.984755i	$0.444347\pi$
$882$	12.8658	0.433213
$883$	5.78522	0.194688	0.0973440	−	0.995251i	$-0.468965\pi$
$883$	5.78522	0.194688	0.0973440	+	0.995251i	$0.468965\pi$
$884$	26.0746	0.876982
$885$	0	0
$886$	−66.1500	−2.22235
$887$	14.3188	0.480780	0.240390	−	0.970676i	$-0.422725\pi$
$887$	14.3188	0.480780	0.240390	+	0.970676i	$0.422725\pi$
$888$	−3.86974	−0.129860
$889$	−39.1118	−1.31177
$890$	0	0
$891$	61.3050	2.05379
$892$	24.0009	0.803610
$893$	0	0
$894$	25.6525	0.857947
$895$	0	0
$896$	24.1724	0.807545
$897$	23.9443	0.799476
$898$	10.7295	0.358048
$899$	−5.32624	−0.177640
$900$	0	0
$901$	58.3045	1.94240
$902$	76.6312	2.55154
$903$	−3.07768	−0.102419
$904$	13.4164	0.446223
$905$	0	0
$906$	2.24514	0.0745898
$907$	51.9371	1.72454	0.862272	−	0.506446i	$-0.169041\pi$
$907$	51.9371	1.72454	0.862272	+	0.506446i	$0.169041\pi$
$908$	−28.1482	−0.934130
$909$	2.03444	0.0674782
$910$	0	0
$911$	34.3035	1.13653	0.568264	−	0.822847i	$-0.307615\pi$
$911$	34.3035	1.13653	0.568264	+	0.822847i	$0.307615\pi$
$912$	0	0
$913$	−51.3050	−1.69795
$914$	−9.95959	−0.329434
$915$	0	0
$916$	−29.3607	−0.970104
$917$	−66.1591	−2.18476
$918$	45.1246	1.48933
$919$	7.36068	0.242806	0.121403	−	0.992603i	$-0.461261\pi$
$919$	7.36068	0.242806	0.121403	+	0.992603i	$0.461261\pi$
$920$	0	0
$921$	−34.9230	−1.15075
$922$	−15.2169	−0.501142
$923$	−0.527864	−0.0173749
$924$	−76.3215	−2.51079
$925$	0	0
$926$	−0.277515	−0.00911969
$927$	9.23305	0.303253
$928$	20.5279	0.673860
$929$	37.5623	1.23238	0.616190	−	0.787598i	$-0.288675\pi$
$929$	37.5623	1.23238	0.616190	+	0.787598i	$0.288675\pi$
$930$	0	0
$931$	0	0
$932$	−2.85410	−0.0934892
$933$	−5.25731	−0.172117
$934$	17.4620	0.571376
$935$	0	0
$936$	1.38197	0.0451710
$937$	−23.2016	−0.757964	−0.378982	−	0.925404i	$-0.623726\pi$
$937$	−23.2016	−0.757964	−0.378982	+	0.925404i	$0.623726\pi$
$938$	−48.2148	−1.57427
$939$	−6.22088	−0.203011
$940$	0	0
$941$	−50.7615	−1.65478	−0.827389	−	0.561629i	$-0.810175\pi$
$941$	−50.7615	−1.65478	−0.827389	+	0.561629i	$0.810175\pi$
$942$	58.8673	1.91800
$943$	28.1482	0.916631
$944$	−64.8434	−2.11047
$945$	0	0
$946$	4.25325	0.138285
$947$	−1.41641	−0.0460271	−0.0230135	−	0.999735i	$-0.507326\pi$
$947$	−1.41641	−0.0460271	−0.0230135	+	0.999735i	$0.507326\pi$
$948$	−16.1803	−0.525513
$949$	−3.07768	−0.0999058
$950$	0	0
$951$	31.3050	1.01513
$952$	16.1150	0.522289
$953$	−42.2300	−1.36796	−0.683982	−	0.729499i	$-0.739753\pi$
$953$	−42.2300	−1.36796	−0.683982	+	0.729499i	$0.739753\pi$
$954$	−13.0902	−0.423810
$955$	0	0
$956$	−37.5623	−1.21485
$957$	31.1803	1.00792
$958$	−21.7558	−0.702898
$959$	−41.3607	−1.33561
$960$	0	0
$961$	−27.3820	−0.883289
$962$	−16.3925	−0.528515
$963$	5.15131	0.165999
$964$	−10.2371	−0.329715
$965$	0	0
$966$	−62.6869	−2.01692
$967$	29.0557	0.934369	0.467185	−	0.884160i	$-0.345268\pi$
$967$	29.0557	0.934369	0.467185	+	0.884160i	$0.345268\pi$
$968$	−16.9070	−0.543412
$969$	0	0
$970$	0	0
$971$	−27.4216	−0.880002	−0.440001	−	0.897997i	$-0.645022\pi$
$971$	−27.4216	−0.880002	−0.440001	+	0.897997i	$0.645022\pi$
$972$	−10.2371	−0.328355
$973$	−93.5755	−2.99989
$974$	40.0000	1.28168
$975$	0	0
$976$	−18.2148	−0.583041
$977$	36.7607	1.17608	0.588039	−	0.808832i	$-0.299900\pi$
$977$	36.7607	1.17608	0.588039	+	0.808832i	$0.299900\pi$
$978$	91.1033	2.91316
$979$	−45.5447	−1.45562
$980$	0	0
$981$	−1.34708	−0.0430091
$982$	70.8927	2.26228
$983$	19.6417	0.626472	0.313236	−	0.949675i	$-0.398587\pi$
$983$	19.6417	0.626472	0.313236	+	0.949675i	$0.398587\pi$
$984$	9.51057	0.303186
$985$	0	0
$986$	27.8885	0.888152
$987$	11.8617	0.377562
$988$	0	0
$989$	1.56231	0.0496784
$990$	0	0
$991$	57.6839	1.83239	0.916195	−	0.400732i	$-0.131244\pi$
$991$	57.6839	1.83239	0.916195	+	0.400732i	$0.131244\pi$
$992$	−13.9443	−0.442731
$993$	−37.8885	−1.20236
$994$	1.38197	0.0438333
$995$	0	0
$996$	26.9726	0.854660
$997$	36.9443	1.17004	0.585018	−	0.811020i	$-0.301087\pi$
$997$	36.9443	1.17004	0.585018	+	0.811020i	$0.301087\pi$
$998$	−56.1653	−1.77788
$999$	−12.6869	−0.401396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bl.1.4		4
5.4	even	2		1805.2.a.l.1.1	✓	4
19.18	odd	2	inner	9025.2.a.bl.1.1		4
95.94	odd	2		1805.2.a.l.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.l.1.1	✓	4	5.4	even	2
1805.2.a.l.1.4	yes	4	95.94	odd	2
9025.2.a.bl.1.1		4	19.18	odd	2	inner
9025.2.a.bl.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90211 q^{2} +1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} -0.726543 q^{8} +0.618034 q^{9} -5.85410 q^{11} +3.07768 q^{12} -3.07768 q^{13} +8.05748 q^{14} -4.61803 q^{16} -5.23607 q^{17} +1.17557 q^{18} +8.05748 q^{21} -11.1352 q^{22} -4.09017 q^{23} -1.38197 q^{24} -5.85410 q^{26} -4.53077 q^{27} +6.85410 q^{28} -2.80017 q^{29} +1.90211 q^{31} -7.33094 q^{32} -11.1352 q^{33} -9.95959 q^{34} +1.00000 q^{36} +2.80017 q^{37} -5.85410 q^{39} -6.88191 q^{41} +15.3262 q^{42} -0.381966 q^{43} -9.47214 q^{44} -7.77997 q^{46} +1.47214 q^{47} -8.78402 q^{48} +10.9443 q^{49} -9.95959 q^{51} -4.97980 q^{52} -11.1352 q^{53} -8.61803 q^{54} -3.07768 q^{56} -5.32624 q^{58} +14.0413 q^{59} +3.94427 q^{61} +3.61803 q^{62} +2.61803 q^{63} -4.70820 q^{64} -21.1803 q^{66} -5.98385 q^{67} -8.47214 q^{68} -7.77997 q^{69} +0.171513 q^{71} -0.449028 q^{72} +1.00000 q^{73} +5.32624 q^{74} -24.7984 q^{77} -11.1352 q^{78} -5.25731 q^{79} -10.4721 q^{81} -13.0902 q^{82} +8.76393 q^{83} +13.0373 q^{84} -0.726543 q^{86} -5.32624 q^{87} +4.25325 q^{88} +7.77997 q^{89} -13.0373 q^{91} -6.61803 q^{92} +3.61803 q^{93} +2.80017 q^{94} -13.9443 q^{96} +2.62866 q^{97} +20.8172 q^{98} -3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more