LMFDB - Embedded newform 9025.2.a.bw.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.71593280.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 13x^{2} - 5 $ x^6 - 8x^4 + 13x^2 - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44118$ 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)

$f(q)$	$=$	$q-2.44118 q^{2} -1.94894 q^{3} +3.95934 q^{4} +4.75770 q^{6} +4.16098 q^{7} -4.78310 q^{8} +0.798360 q^{9} +O(q^{10})$	\(q-2.44118 q^{2} -1.94894 q^{3} +3.95934 q^{4} +4.75770 q^{6} +4.16098 q^{7} -4.78310 q^{8} +0.798360 q^{9} -4.75770 q^{11} -7.71651 q^{12} -2.83416 q^{13} -10.1577 q^{14} +3.75770 q^{16} +4.16098 q^{17} -1.94894 q^{18} -8.10950 q^{21} +11.6144 q^{22} +7.31376 q^{23} +9.32196 q^{24} +6.91868 q^{26} +4.29086 q^{27} +16.4747 q^{28} +10.1577 q^{29} -2.04819 q^{31} +0.392983 q^{32} +9.27247 q^{33} -10.1577 q^{34} +3.16098 q^{36} -6.25981 q^{37} +5.52360 q^{39} -7.69650 q^{41} +19.7967 q^{42} -3.32196 q^{43} -18.8374 q^{44} -17.8542 q^{46} +8.15278 q^{47} -7.32353 q^{48} +10.3138 q^{49} -8.10950 q^{51} -11.2214 q^{52} +8.20875 q^{53} -10.4747 q^{54} -19.9024 q^{56} -24.7967 q^{58} +4.09639 q^{59} +5.76590 q^{61} +5.00000 q^{62} +3.32196 q^{63} -8.47474 q^{64} -22.6357 q^{66} -0.392983 q^{67} +16.4747 q^{68} -14.2541 q^{69} -0.412997 q^{71} -3.81864 q^{72} -2.31376 q^{73} +15.2813 q^{74} -19.7967 q^{77} -13.4841 q^{78} -5.64831 q^{79} -10.7577 q^{81} +18.7885 q^{82} -2.31376 q^{83} -32.1083 q^{84} +8.10950 q^{86} -19.7967 q^{87} +22.7566 q^{88} -12.2059 q^{89} -11.7929 q^{91} +28.9577 q^{92} +3.99180 q^{93} -19.9024 q^{94} -0.765900 q^{96} +18.4657 q^{97} -25.1777 q^{98} -3.79836 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9	$ 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9} - 10 q^{11} + 4 q^{16} + 4 q^{17} + 8 q^{23} + 14 q^{24} + 2 q^{26} + 42 q^{28} - 2 q^{36} - 10 q^{39} + 20 q^{42} + 22 q^{43} - 34 q^{44} + 34 q^{47} + 26 q^{49} - 6 q^{54} - 50 q^{58} + 10 q^{61} + 30 q^{62} - 22 q^{63} + 6 q^{64} - 58 q^{66} + 42 q^{68} + 22 q^{73} + 30 q^{74} - 20 q^{77} - 46 q^{81} + 20 q^{82} + 22 q^{83} - 20 q^{87} + 54 q^{92} + 30 q^{93} + 20 q^{96} - 24 q^{99}+O(q^{100}) $ 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9 - 10 * q^11 + 4 * q^16 + 4 * q^17 + 8 * q^23 + 14 * q^24 + 2 * q^26 + 42 * q^28 - 2 * q^36 - 10 * q^39 + 20 * q^42 + 22 * q^43 - 34 * q^44 + 34 * q^47 + 26 * q^49 - 6 * q^54 - 50 * q^58 + 10 * q^61 + 30 * q^62 - 22 * q^63 + 6 * q^64 - 58 * q^66 + 42 * q^68 + 22 * q^73 + 30 * q^74 - 20 * q^77 - 46 * q^81 + 20 * q^82 + 22 * q^83 - 20 * q^87 + 54 * q^92 + 30 * q^93 + 20 * q^96 - 24 * 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 $

$2$	−2.44118	−1.72617	−0.863086	−	0.505057i	$-0.831472\pi$
$2$	−2.44118	−1.72617	−0.863086	+	0.505057i	$0.831472\pi$
$3$	−1.94894	−1.12522	−0.562610	−	0.826722i	$-0.690203\pi$
$3$	−1.94894	−1.12522	−0.562610	+	0.826722i	$0.690203\pi$
$4$	3.95934	1.97967
$5$	0	0
$5$	0	0
$6$	4.75770	1.94232
$7$	4.16098	1.57270	0.786352	−	0.617779i	$-0.211967\pi$
$7$	4.16098	1.57270	0.786352	+	0.617779i	$0.211967\pi$
$8$	−4.78310	−1.69108
$9$	0.798360	0.266120
$10$	0	0
$11$	−4.75770	−1.43450	−0.717251	−	0.696815i	$-0.754600\pi$
$11$	−4.75770	−1.43450	−0.717251	+	0.696815i	$0.754600\pi$
$12$	−7.71651	−2.22757
$13$	−2.83416	−0.786054	−0.393027	−	0.919527i	$-0.628572\pi$
$13$	−2.83416	−0.786054	−0.393027	+	0.919527i	$0.628572\pi$
$14$	−10.1577	−2.71476
$15$	0	0
$16$	3.75770	0.939425
$17$	4.16098	1.00919	0.504593	−	0.863357i	$-0.331643\pi$
$17$	4.16098	1.00919	0.504593	+	0.863357i	$0.331643\pi$
$18$	−1.94894	−0.459369
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−8.10950	−1.76964
$22$	11.6144	2.47620
$23$	7.31376	1.52503	0.762513	−	0.646973i	$-0.223966\pi$
$23$	7.31376	1.52503	0.762513	+	0.646973i	$0.223966\pi$
$24$	9.32196	1.90284
$25$	0	0
$26$	6.91868	1.35687
$27$	4.29086	0.825776
$28$	16.4747	3.11343
$29$	10.1577	1.88624	0.943118	−	0.332459i	$-0.107878\pi$
$29$	10.1577	1.88624	0.943118	+	0.332459i	$0.107878\pi$
$30$	0	0
$31$	−2.04819	−0.367866	−0.183933	−	0.982939i	$-0.558883\pi$
$31$	−2.04819	−0.367866	−0.183933	+	0.982939i	$0.558883\pi$
$32$	0.392983	0.0694703
$33$	9.27247	1.61413
$34$	−10.1577	−1.74203
$35$	0	0
$36$	3.16098	0.526830
$37$	−6.25981	−1.02911	−0.514553	−	0.857458i	$-0.672042\pi$
$37$	−6.25981	−1.02911	−0.514553	+	0.857458i	$0.672042\pi$
$38$	0	0
$39$	5.52360	0.884484
$40$	0	0
$41$	−7.69650	−1.20199	−0.600996	−	0.799252i	$-0.705229\pi$
$41$	−7.69650	−1.20199	−0.600996	+	0.799252i	$0.705229\pi$
$42$	19.7967	3.05470
$43$	−3.32196	−0.506594	−0.253297	−	0.967388i	$-0.581515\pi$
$43$	−3.32196	−0.506594	−0.253297	+	0.967388i	$0.581515\pi$
$44$	−18.8374	−2.83984
$45$	0	0
$46$	−17.8542	−2.63246
$47$	8.15278	1.18921	0.594603	−	0.804020i	$-0.297309\pi$
$47$	8.15278	1.18921	0.594603	+	0.804020i	$0.297309\pi$
$48$	−7.32353	−1.05706
$49$	10.3138	1.47339
$50$	0	0
$51$	−8.10950	−1.13556
$52$	−11.2214	−1.55613
$53$	8.20875	1.12756	0.563779	−	0.825926i	$-0.309347\pi$
$53$	8.20875	1.12756	0.563779	+	0.825926i	$0.309347\pi$
$54$	−10.4747	−1.42543
$55$	0	0
$56$	−19.9024	−2.65957
$57$	0	0
$58$	−24.7967	−3.25597
$59$	4.09639	0.533304	0.266652	−	0.963793i	$-0.414083\pi$
$59$	4.09639	0.533304	0.266652	+	0.963793i	$0.414083\pi$
$60$	0	0
$61$	5.76590	0.738248	0.369124	−	0.929380i	$-0.379658\pi$
$61$	5.76590	0.738248	0.369124	+	0.929380i	$0.379658\pi$
$62$	5.00000	0.635001
$63$	3.32196	0.418528
$64$	−8.47474	−1.05934
$65$	0	0
$66$	−22.6357	−2.78627
$67$	−0.392983	−0.0480105	−0.0240053	−	0.999712i	$-0.507642\pi$
$67$	−0.392983	−0.0480105	−0.0240053	+	0.999712i	$0.507642\pi$
$68$	16.4747	1.99786
$69$	−14.2541	−1.71599
$70$	0	0
$71$	−0.412997	−0.0490137	−0.0245069	−	0.999700i	$-0.507802\pi$
$71$	−0.412997	−0.0490137	−0.0245069	+	0.999700i	$0.507802\pi$
$72$	−3.81864	−0.450031
$73$	−2.31376	−0.270806	−0.135403	−	0.990791i	$-0.543233\pi$
$73$	−2.31376	−0.270806	−0.135403	+	0.990791i	$0.543233\pi$
$74$	15.2813	1.77642
$75$	0	0
$76$	0	0
$77$	−19.7967	−2.25604
$78$	−13.4841	−1.52677
$79$	−5.64831	−0.635484	−0.317742	−	0.948177i	$-0.602925\pi$
$79$	−5.64831	−0.635484	−0.317742	+	0.948177i	$0.602925\pi$
$80$	0	0
$81$	−10.7577	−1.19530
$82$	18.7885	2.07484
$83$	−2.31376	−0.253969	−0.126984	−	0.991905i	$-0.540530\pi$
$83$	−2.31376	−0.253969	−0.126984	+	0.991905i	$0.540530\pi$
$84$	−32.1083	−3.50330
$85$	0	0
$86$	8.10950	0.874469
$87$	−19.7967	−2.12243
$88$	22.7566	2.42586
$89$	−12.2059	−1.29382	−0.646910	−	0.762566i	$-0.723939\pi$
$89$	−12.2059	−1.29382	−0.646910	+	0.762566i	$0.723939\pi$
$90$	0	0
$91$	−11.7929	−1.23623
$92$	28.9577	3.01905
$93$	3.99180	0.413931
$94$	−19.9024	−2.05277
$95$	0	0
$96$	−0.765900	−0.0781694
$97$	18.4657	1.87491	0.937454	−	0.348110i	$-0.113177\pi$
$97$	18.4657	1.87491	0.937454	+	0.348110i	$0.113177\pi$
$98$	−25.1777	−2.54333
$99$	−3.79836	−0.381750
$100$	0	0
$101$	0.242298	0.0241096	0.0120548	−	0.999927i	$-0.496163\pi$
$101$	0.242298	0.0241096	0.0120548	+	0.999927i	$0.496163\pi$
$102$	19.7967	1.96017
$103$	15.4530	1.52263	0.761317	−	0.648380i	$-0.224553\pi$
$103$	15.4530	1.52263	0.761317	+	0.648380i	$0.224553\pi$
$104$	13.5561	1.32928
$105$	0	0
$106$	−20.0390	−1.94636
$107$	−1.65521	−0.160015	−0.0800076	−	0.996794i	$-0.525494\pi$
$107$	−1.65521	−0.160015	−0.0800076	+	0.996794i	$0.525494\pi$
$108$	16.9890	1.63477
$109$	3.60011	0.344828	0.172414	−	0.985025i	$-0.444843\pi$
$109$	3.60011	0.344828	0.172414	+	0.985025i	$0.444843\pi$
$110$	0	0
$111$	12.2000	1.15797
$112$	15.6357	1.47744
$113$	−2.42116	−0.227764	−0.113882	−	0.993494i	$-0.536329\pi$
$113$	−2.42116	−0.227764	−0.113882	+	0.993494i	$0.536329\pi$
$114$	0	0
$115$	0	0
$116$	40.2178	3.73412
$117$	−2.26268	−0.209185
$118$	−10.0000	−0.920575
$119$	17.3138	1.58715
$120$	0	0
$121$	11.6357	1.05779
$122$	−14.0756	−1.27434
$123$	15.0000	1.35250
$124$	−8.10950	−0.728254
$125$	0	0
$126$	−8.10950	−0.722451
$127$	−16.1038	−1.42898	−0.714489	−	0.699647i	$-0.753341\pi$
$127$	−16.1038	−1.42898	−0.714489	+	0.699647i	$0.753341\pi$
$128$	19.9024	1.75914
$129$	6.47430	0.570030
$130$	0	0
$131$	−10.3138	−0.901118	−0.450559	−	0.892747i	$-0.648775\pi$
$131$	−10.3138	−0.901118	−0.450559	+	0.892747i	$0.648775\pi$
$132$	36.7129	3.19544
$133$	0	0
$134$	0.959341	0.0828745
$135$	0	0
$136$	−19.9024	−1.70661
$137$	10.6357	0.908671	0.454336	−	0.890831i	$-0.349877\pi$
$137$	10.6357	0.908671	0.454336	+	0.890831i	$0.349877\pi$
$138$	34.7967	2.96209
$139$	−3.80656	−0.322868	−0.161434	−	0.986884i	$-0.551612\pi$
$139$	−3.80656	−0.322868	−0.161434	+	0.986884i	$0.551612\pi$
$140$	0	0
$141$	−15.8893	−1.33812
$142$	1.00820	0.0846061
$143$	13.4841	1.12760
$144$	3.00000	0.250000
$145$	0	0
$146$	5.64831	0.467457
$147$	−20.1009	−1.65789
$148$	−24.7847	−2.03729
$149$	12.7984	1.04848	0.524241	−	0.851570i	$-0.324349\pi$
$149$	12.7984	1.04848	0.524241	+	0.851570i	$0.324349\pi$
$150$	0	0
$151$	22.7766	1.85353	0.926765	−	0.375641i	$-0.122577\pi$
$151$	22.7766	1.85353	0.926765	+	0.375641i	$0.122577\pi$
$152$	0	0
$153$	3.32196	0.268565
$154$	48.3273	3.89432
$155$	0	0
$156$	21.8698	1.75099
$157$	−15.6357	−1.24787	−0.623933	−	0.781478i	$-0.714466\pi$
$157$	−15.6357	−1.24787	−0.623933	+	0.781478i	$0.714466\pi$
$158$	13.7885	1.09695
$159$	−15.9983	−1.26875
$160$	0	0
$161$	30.4324	2.39841
$162$	26.2614	2.06329
$163$	−6.47474	−0.507141	−0.253571	−	0.967317i	$-0.581605\pi$
$163$	−6.47474	−0.507141	−0.253571	+	0.967317i	$0.581605\pi$
$164$	−30.4731	−2.37955
$165$	0	0
$166$	5.64831	0.438394
$167$	3.79862	0.293946	0.146973	−	0.989140i	$-0.453047\pi$
$167$	3.79862	0.293946	0.146973	+	0.989140i	$0.453047\pi$
$168$	38.7885	2.99260
$169$	−4.96754	−0.382118
$170$	0	0
$171$	0	0
$172$	−13.1528	−1.00289
$173$	6.43427	0.489189	0.244594	−	0.969626i	$-0.421345\pi$
$173$	6.43427	0.489189	0.244594	+	0.969626i	$0.421345\pi$
$174$	48.3273	3.66368
$175$	0	0
$176$	−17.8780	−1.34761
$177$	−7.98360	−0.600084
$178$	29.7967	2.23336
$179$	−8.60577	−0.643225	−0.321613	−	0.946871i	$-0.604225\pi$
$179$	−8.60577	−0.643225	−0.321613	+	0.946871i	$0.604225\pi$
$180$	0	0
$181$	4.50938	0.335180	0.167590	−	0.985857i	$-0.446402\pi$
$181$	4.50938	0.335180	0.167590	+	0.985857i	$0.446402\pi$
$182$	28.7885	2.13395
$183$	−11.2374	−0.830691
$184$	−34.9824	−2.57894
$185$	0	0
$186$	−9.74469	−0.714515
$187$	−19.7967	−1.44768
$188$	32.2797	2.35424
$189$	17.8542	1.29870
$190$	0	0
$191$	2.85542	0.206611	0.103305	−	0.994650i	$-0.467058\pi$
$191$	2.85542	0.206611	0.103305	+	0.994650i	$0.467058\pi$
$192$	16.5168	1.19199
$193$	−4.19564	−0.302009	−0.151004	−	0.988533i	$-0.548251\pi$
$193$	−4.19564	−0.302009	−0.151004	+	0.988533i	$0.548251\pi$
$194$	−45.0780	−3.23641
$195$	0	0
$196$	40.8357	2.91684
$197$	10.8390	0.772248	0.386124	−	0.922447i	$-0.373814\pi$
$197$	10.8390	0.772248	0.386124	+	0.922447i	$0.373814\pi$
$198$	9.27247	0.658966
$199$	−13.7885	−0.977441	−0.488721	−	0.872440i	$-0.662536\pi$
$199$	−13.7885	−0.977441	−0.488721	+	0.872440i	$0.662536\pi$
$200$	0	0
$201$	0.765900	0.0540224
$202$	−0.591493	−0.0416173
$203$	42.2659	2.96649
$204$	−32.1083	−2.24803
$205$	0	0
$206$	−37.7236	−2.62833
$207$	5.83902	0.405840
$208$	−10.6499	−0.738440
$209$	0	0
$210$	0	0
$211$	15.8060	1.08813	0.544065	−	0.839043i	$-0.316885\pi$
$211$	15.8060	1.08813	0.544065	+	0.839043i	$0.316885\pi$
$212$	32.5012	2.23219
$213$	0.804906	0.0551512
$214$	4.04066	0.276214
$215$	0	0
$216$	−20.5236	−1.39645
$217$	−8.52249	−0.578544
$218$	−8.78851	−0.595233
$219$	4.50938	0.304716
$220$	0	0
$221$	−11.7929	−0.793275
$222$	−29.7823	−1.99886
$223$	−3.91385	−0.262091	−0.131046	−	0.991376i	$-0.541833\pi$
$223$	−3.91385	−0.262091	−0.131046	+	0.991376i	$0.541833\pi$
$224$	1.63520	0.109256
$225$	0	0
$226$	5.91048	0.393160
$227$	−2.85417	−0.189438	−0.0947191	−	0.995504i	$-0.530195\pi$
$227$	−2.85417	−0.189438	−0.0947191	+	0.995504i	$0.530195\pi$
$228$	0	0
$229$	−0.0570553	−0.00377032	−0.00188516	−	0.999998i	$-0.500600\pi$
$229$	−0.0570553	−0.00377032	−0.00188516	+	0.999998i	$0.500600\pi$
$230$	0	0
$231$	38.5826	2.53855
$232$	−48.5852	−3.18978
$233$	−12.1105	−0.793383	−0.396692	−	0.917952i	$-0.629842\pi$
$233$	−12.1105	−0.793383	−0.396692	+	0.917952i	$0.629842\pi$
$234$	5.52360	0.361089
$235$	0	0
$236$	16.2190	1.05577
$237$	11.0082	0.715059
$238$	−42.2659	−2.73969
$239$	12.4911	0.807985	0.403992	−	0.914762i	$-0.367622\pi$
$239$	12.4911	0.807985	0.403992	+	0.914762i	$0.367622\pi$
$240$	0	0
$241$	23.9155	1.54053	0.770266	−	0.637723i	$-0.220123\pi$
$241$	23.9155	1.54053	0.770266	+	0.637723i	$0.220123\pi$
$242$	−28.4049	−1.82593
$243$	8.09352	0.519199
$244$	22.8292	1.46149
$245$	0	0
$246$	−36.6176	−2.33466
$247$	0	0
$248$	9.79671	0.622092
$249$	4.50938	0.285771
$250$	0	0
$251$	13.1528	0.830196	0.415098	−	0.909777i	$-0.363747\pi$
$251$	13.1528	0.830196	0.415098	+	0.909777i	$0.363747\pi$
$252$	13.1528	0.828547
$253$	−34.7967	−2.18765
$254$	39.3121	2.46666
$255$	0	0
$256$	−31.6357	−1.97723
$257$	1.75043	0.109189	0.0545944	−	0.998509i	$-0.482613\pi$
$257$	1.75043	0.109189	0.0545944	+	0.998509i	$0.482613\pi$
$258$	−15.8049	−0.983970
$259$	−26.0470	−1.61848
$260$	0	0
$261$	8.10950	0.501965
$262$	25.1777	1.55548
$263$	26.4747	1.63250	0.816251	−	0.577697i	$-0.196048\pi$
$263$	26.4747	1.63250	0.816251	+	0.577697i	$0.196048\pi$
$264$	−44.3511	−2.72962
$265$	0	0
$266$	0	0
$267$	23.7885	1.45583
$268$	−1.55595	−0.0950451
$269$	2.46119	0.150061	0.0750307	−	0.997181i	$-0.476095\pi$
$269$	2.46119	0.150061	0.0750307	+	0.997181i	$0.476095\pi$
$270$	0	0
$271$	−15.2423	−0.925904	−0.462952	−	0.886383i	$-0.653210\pi$
$271$	−15.2423	−0.925904	−0.462952	+	0.886383i	$0.653210\pi$
$272$	15.6357	0.948055
$273$	22.9836	1.39103
$274$	−25.9637	−1.56852
$275$	0	0
$276$	−56.4367	−3.39709
$277$	−19.6275	−1.17930	−0.589652	−	0.807657i	$-0.700735\pi$
$277$	−19.6275	−1.17930	−0.589652	+	0.807657i	$0.700735\pi$
$278$	9.29248	0.557326
$279$	−1.63520	−0.0978966
$280$	0	0
$281$	4.09639	0.244370	0.122185	−	0.992507i	$-0.461010\pi$
$281$	4.09639	0.244370	0.122185	+	0.992507i	$0.461010\pi$
$282$	38.7885	2.30982
$283$	−25.2633	−1.50174	−0.750872	−	0.660447i	$-0.770367\pi$
$283$	−25.2633	−1.50174	−0.750872	+	0.660447i	$0.770367\pi$
$284$	−1.63520	−0.0970310
$285$	0	0
$286$	−32.9170	−1.94642
$287$	−32.0250	−1.89038
$288$	0.313742	0.0184874
$289$	0.313764	0.0184567
$290$	0	0
$291$	−35.9885	−2.10968
$292$	−9.16098	−0.536106
$293$	−15.9052	−0.929195	−0.464597	−	0.885522i	$-0.653801\pi$
$293$	−15.9052	−0.929195	−0.464597	+	0.885522i	$0.653801\pi$
$294$	49.0698	2.86181
$295$	0	0
$296$	29.9413	1.74030
$297$	−20.4146	−1.18458
$298$	−31.2431	−1.80986
$299$	−20.7284	−1.19875
$300$	0	0
$301$	−13.8226	−0.796723
$302$	−55.6016	−3.19951
$303$	−0.472224	−0.0271286
$304$	0	0
$305$	0	0
$306$	−8.10950	−0.463589
$307$	−13.7378	−0.784057	−0.392028	−	0.919953i	$-0.628227\pi$
$307$	−13.7378	−0.784057	−0.392028	+	0.919953i	$0.628227\pi$
$308$	−78.3819	−4.46622
$309$	−30.1170	−1.71330
$310$	0	0
$311$	−3.74950	−0.212615	−0.106307	−	0.994333i	$-0.533903\pi$
$311$	−3.74950	−0.212615	−0.106307	+	0.994333i	$0.533903\pi$
$312$	−26.4199	−1.49573
$313$	−12.1105	−0.684524	−0.342262	−	0.939605i	$-0.611193\pi$
$313$	−12.1105	−0.684524	−0.342262	+	0.939605i	$0.611193\pi$
$314$	38.1696	2.15403
$315$	0	0
$316$	−22.3636	−1.25805
$317$	−26.4399	−1.48502	−0.742508	−	0.669838i	$-0.766364\pi$
$317$	−26.4399	−1.48502	−0.742508	+	0.669838i	$0.766364\pi$
$318$	39.0548	2.19008
$319$	−48.3273	−2.70581
$320$	0	0
$321$	3.22590	0.180052
$322$	−74.2909	−4.14007
$323$	0	0
$324$	−42.5934	−2.36630
$325$	0	0
$326$	15.8060	0.875413
$327$	−7.01640	−0.388008
$328$	36.8131	2.03266
$329$	33.9236	1.87027
$330$	0	0
$331$	−18.6802	−1.02676	−0.513378	−	0.858163i	$-0.671606\pi$
$331$	−18.6802	−1.02676	−0.513378	+	0.858163i	$0.671606\pi$
$332$	−9.16098	−0.502774
$333$	−4.99759	−0.273866
$334$	−9.27311	−0.507402
$335$	0	0
$336$	−30.4731	−1.66244
$337$	19.0131	1.03571	0.517855	−	0.855468i	$-0.326731\pi$
$337$	19.0131	1.03571	0.517855	+	0.855468i	$0.326731\pi$
$338$	12.1266	0.659602
$339$	4.71870	0.256284
$340$	0	0
$341$	9.74469	0.527705
$342$	0	0
$343$	13.7885	0.744509
$344$	15.8893	0.856692
$345$	0	0
$346$	−15.7072	−0.844424
$347$	2.85542	0.153287	0.0766434	−	0.997059i	$-0.475580\pi$
$347$	2.85542	0.153287	0.0766434	+	0.997059i	$0.475580\pi$
$348$	−78.3819	−4.20171
$349$	29.7967	1.59498	0.797491	−	0.603331i	$-0.206160\pi$
$349$	29.7967	1.59498	0.797491	+	0.603331i	$0.206160\pi$
$350$	0	0
$351$	−12.1610	−0.649105
$352$	−1.86970	−0.0996552
$353$	13.4911	0.718061	0.359031	−	0.933326i	$-0.383107\pi$
$353$	13.4911	0.718061	0.359031	+	0.933326i	$0.383107\pi$
$354$	19.4894	1.03585
$355$	0	0
$356$	−48.3273	−2.56134
$357$	−33.7435	−1.78589
$358$	21.0082	1.11032
$359$	6.95934	0.367300	0.183650	−	0.982992i	$-0.441209\pi$
$359$	6.95934	0.367300	0.183650	+	0.982992i	$0.441209\pi$
$360$	0	0
$361$	0	0
$362$	−11.0082	−0.578578
$363$	−22.6773	−1.19025
$364$	−46.6921	−2.44733
$365$	0	0
$366$	27.4324	1.43392
$367$	−28.9577	−1.51158	−0.755790	−	0.654815i	$-0.772747\pi$
$367$	−28.9577	−1.51158	−0.755790	+	0.654815i	$0.772747\pi$
$368$	27.4829	1.43265
$369$	−6.14458	−0.319874
$370$	0	0
$371$	34.1565	1.77331
$372$	15.8049	0.819446
$373$	25.8644	1.33921	0.669605	−	0.742718i	$-0.266464\pi$
$373$	25.8644	1.33921	0.669605	+	0.742718i	$0.266464\pi$
$374$	48.3273	2.49894
$375$	0	0
$376$	−38.9956	−2.01104
$377$	−28.7885	−1.48268
$378$	−43.5852	−2.24178
$379$	−14.1708	−0.727905	−0.363952	−	0.931418i	$-0.618573\pi$
$379$	−14.1708	−0.727905	−0.363952	+	0.931418i	$0.618573\pi$
$380$	0	0
$381$	31.3852	1.60791
$382$	−6.97057	−0.356646
$383$	36.9706	1.88911	0.944555	−	0.328354i	$-0.106494\pi$
$383$	36.9706	1.88911	0.944555	+	0.328354i	$0.106494\pi$
$384$	−38.7885	−1.97942
$385$	0	0
$386$	10.2423	0.521319
$387$	−2.65212	−0.134815
$388$	73.1120	3.71170
$389$	−28.2551	−1.43259	−0.716294	−	0.697799i	$-0.754163\pi$
$389$	−28.2551	−1.43259	−0.716294	+	0.697799i	$0.754163\pi$
$390$	0	0
$391$	30.4324	1.53903
$392$	−49.3317	−2.49163
$393$	20.1009	1.01396
$394$	−26.4600	−1.33303
$395$	0	0
$396$	−15.0390	−0.755738
$397$	−24.4242	−1.22582	−0.612909	−	0.790154i	$-0.710001\pi$
$397$	−24.4242	−1.22582	−0.612909	+	0.790154i	$0.710001\pi$
$398$	33.6602	1.68723
$399$	0	0
$400$	0	0
$401$	−25.9637	−1.29656	−0.648282	−	0.761400i	$-0.724512\pi$
$401$	−25.9637	−1.29656	−0.648282	+	0.761400i	$0.724512\pi$
$402$	−1.86970	−0.0932520
$403$	5.80491	0.289163
$404$	0.959341	0.0477290
$405$	0	0
$406$	−103.179	−5.12067
$407$	29.7823	1.47625
$408$	38.7885	1.92032
$409$	−4.01311	−0.198435	−0.0992177	−	0.995066i	$-0.531634\pi$
$409$	−4.01311	−0.198435	−0.0992177	+	0.995066i	$0.531634\pi$
$410$	0	0
$411$	−20.7284	−1.02246
$412$	61.1839	3.01431
$413$	17.0450	0.838729
$414$	−14.2541	−0.700550
$415$	0	0
$416$	−1.11378	−0.0546074
$417$	7.41875	0.363298
$418$	0	0
$419$	14.1187	0.689742	0.344871	−	0.938650i	$-0.387923\pi$
$419$	14.1187	0.689742	0.344871	+	0.938650i	$0.387923\pi$
$420$	0	0
$421$	−18.2672	−0.890288	−0.445144	−	0.895459i	$-0.646848\pi$
$421$	−18.2672	−0.890288	−0.445144	+	0.895459i	$0.646848\pi$
$422$	−38.5852	−1.87830
$423$	6.50886	0.316472
$424$	−39.2633	−1.90679
$425$	0	0
$426$	−1.96492	−0.0952005
$427$	23.9918	1.16104
$428$	−6.55354	−0.316777
$429$	−26.2797	−1.26879
$430$	0	0
$431$	19.1765	0.923697	0.461849	−	0.886959i	$-0.347186\pi$
$431$	19.1765	0.923697	0.461849	+	0.886959i	$0.347186\pi$
$432$	16.1238	0.775755
$433$	18.6201	0.894827	0.447413	−	0.894327i	$-0.352345\pi$
$433$	18.6201	0.894827	0.447413	+	0.894327i	$0.352345\pi$
$434$	20.8049	0.998667
$435$	0	0
$436$	14.2541	0.682646
$437$	0	0
$438$	−11.0082	−0.525992
$439$	−14.2541	−0.680310	−0.340155	−	0.940369i	$-0.610480\pi$
$439$	−14.2541	−0.680310	−0.340155	+	0.940369i	$0.610480\pi$
$440$	0	0
$441$	8.23410	0.392100
$442$	28.7885	1.36933
$443$	27.2797	1.29610	0.648048	−	0.761600i	$-0.275586\pi$
$443$	27.2797	1.29610	0.648048	+	0.761600i	$0.275586\pi$
$444$	48.3039	2.29240
$445$	0	0
$446$	9.55441	0.452414
$447$	−24.9432	−1.17977
$448$	−35.2633	−1.66603
$449$	−1.22220	−0.0576791	−0.0288396	−	0.999584i	$-0.509181\pi$
$449$	−1.22220	−0.0576791	−0.0288396	+	0.999584i	$0.509181\pi$
$450$	0	0
$451$	36.6176	1.72426
$452$	−9.58621	−0.450897
$453$	−44.3901	−2.08563
$454$	6.96754	0.327003
$455$	0	0
$456$	0	0
$457$	10.1692	0.475694	0.237847	−	0.971303i	$-0.423558\pi$
$457$	10.1692	0.475694	0.237847	+	0.971303i	$0.423558\pi$
$458$	0.139282	0.00650822
$459$	17.8542	0.833362
$460$	0	0
$461$	−15.4911	−0.721494	−0.360747	−	0.932664i	$-0.617478\pi$
$461$	−15.4911	−0.721494	−0.360747	+	0.932664i	$0.617478\pi$
$462$	−94.1868	−4.38197
$463$	−1.47474	−0.0685372	−0.0342686	−	0.999413i	$-0.510910\pi$
$463$	−1.47474	−0.0685372	−0.0342686	+	0.999413i	$0.510910\pi$
$464$	38.1696	1.77198
$465$	0	0
$466$	29.5638	1.36952
$467$	30.6357	1.41765	0.708826	−	0.705383i	$-0.249225\pi$
$467$	30.6357	1.41765	0.708826	+	0.705383i	$0.249225\pi$
$468$	−8.95872	−0.414117
$469$	−1.63520	−0.0755063
$470$	0	0
$471$	30.4731	1.40412
$472$	−19.5934	−0.901860
$473$	15.8049	0.726710
$474$	−26.8730	−1.23432
$475$	0	0
$476$	68.5511	3.14203
$477$	6.55354	0.300066
$478$	−30.4931	−1.39472
$479$	21.7560	0.994059	0.497030	−	0.867734i	$-0.334424\pi$
$479$	21.7560	0.994059	0.497030	+	0.867734i	$0.334424\pi$
$480$	0	0
$481$	17.7413	0.808934
$482$	−58.3819	−2.65922
$483$	−59.3109	−2.69874
$484$	46.0698	2.09408
$485$	0	0
$486$	−19.7577	−0.896228
$487$	6.99059	0.316774	0.158387	−	0.987377i	$-0.449371\pi$
$487$	6.99059	0.316774	0.158387	+	0.987377i	$0.449371\pi$
$488$	−27.5789	−1.24844
$489$	12.6189	0.570645
$490$	0	0
$491$	−40.9741	−1.84913	−0.924567	−	0.381019i	$-0.875573\pi$
$491$	−40.9741	−1.84913	−0.924567	+	0.381019i	$0.875573\pi$
$492$	59.3901	2.67751
$493$	42.2659	1.90356
$494$	0	0
$495$	0	0
$496$	−7.69650	−0.345583
$497$	−1.71847	−0.0770840
$498$	−11.0082	−0.493289
$499$	22.9675	1.02817	0.514084	−	0.857740i	$-0.328132\pi$
$499$	22.9675	1.02817	0.514084	+	0.857740i	$0.328132\pi$
$500$	0	0
$501$	−7.40328	−0.330754
$502$	−32.1083	−1.43306
$503$	17.5171	0.781047	0.390523	−	0.920593i	$-0.372294\pi$
$503$	17.5171	0.781047	0.390523	+	0.920593i	$0.372294\pi$
$504$	−15.8893	−0.707764
$505$	0	0
$506$	84.9449	3.77626
$507$	9.68143	0.429967
$508$	−63.7603	−2.82890
$509$	11.7929	0.522710	0.261355	−	0.965243i	$-0.415831\pi$
$509$	11.7929	0.522710	0.261355	+	0.965243i	$0.415831\pi$
$510$	0	0
$511$	−9.62753	−0.425897
$512$	37.4236	1.65391
$513$	0	0
$514$	−4.27311	−0.188479
$515$	0	0
$516$	25.6340	1.12847
$517$	−38.7885	−1.70592
$518$	63.5852	2.79377
$519$	−12.5400	−0.550445
$520$	0	0
$521$	−4.92238	−0.215653	−0.107827	−	0.994170i	$-0.534389\pi$
$521$	−4.92238	−0.215653	−0.107827	+	0.994170i	$0.534389\pi$
$522$	−19.7967	−0.866478
$523$	29.6630	1.29707	0.648537	−	0.761183i	$-0.275381\pi$
$523$	29.6630	1.29707	0.648537	+	0.761183i	$0.275381\pi$
$524$	−40.8357	−1.78392
$525$	0	0
$526$	−64.6295	−2.81798
$527$	−8.52249	−0.371246
$528$	34.8432	1.51635
$529$	30.4911	1.32570
$530$	0	0
$531$	3.27039	0.141923
$532$	0	0
$533$	21.8131	0.944830
$534$	−58.0719	−2.51302
$535$	0	0
$536$	1.87968	0.0811897
$537$	16.7721	0.723770
$538$	−6.00820	−0.259032
$539$	−49.0698	−2.11359
$540$	0	0
$541$	3.22590	0.138692	0.0693462	−	0.997593i	$-0.477909\pi$
$541$	3.22590	0.138692	0.0693462	+	0.997593i	$0.477909\pi$
$542$	37.2091	1.59827
$543$	−8.78851	−0.377151
$544$	1.63520	0.0701085
$545$	0	0
$546$	−56.1070	−2.40116
$547$	−13.5593	−0.579753	−0.289877	−	0.957064i	$-0.593614\pi$
$547$	−13.5593	−0.579753	−0.289877	+	0.957064i	$0.593614\pi$
$548$	42.1105	1.79887
$549$	4.60327	0.196463
$550$	0	0
$551$	0	0
$552$	68.1786	2.90188
$553$	−23.5025	−0.999428
$554$	47.9143	2.03568
$555$	0	0
$556$	−15.0715	−0.639173
$557$	22.1446	0.938296	0.469148	−	0.883120i	$-0.344561\pi$
$557$	22.1446	0.938296	0.469148	+	0.883120i	$0.344561\pi$
$558$	3.99180	0.168986
$559$	9.41497	0.398211
$560$	0	0
$561$	38.5826	1.62896
$562$	−10.0000	−0.421825
$563$	37.4428	1.57803	0.789013	−	0.614376i	$-0.210592\pi$
$563$	37.4428	1.57803	0.789013	+	0.614376i	$0.210592\pi$
$564$	−62.9110	−2.64903
$565$	0	0
$566$	61.6721	2.59227
$567$	−44.7626	−1.87985
$568$	1.97540	0.0828861
$569$	39.8048	1.66870	0.834351	−	0.551233i	$-0.185843\pi$
$569$	39.8048	1.66870	0.834351	+	0.551233i	$0.185843\pi$
$570$	0	0
$571$	−4.55441	−0.190596	−0.0952980	−	0.995449i	$-0.530380\pi$
$571$	−4.55441	−0.190596	−0.0952980	+	0.995449i	$0.530380\pi$
$572$	53.3881	2.23227
$573$	−5.56503	−0.232482
$574$	78.1786	3.26311
$575$	0	0
$576$	−6.76590	−0.281913
$577$	16.1774	0.673473	0.336737	−	0.941599i	$-0.390677\pi$
$577$	16.1774	0.673473	0.336737	+	0.941599i	$0.390677\pi$
$578$	−0.765953	−0.0318594
$579$	8.17704	0.339826
$580$	0	0
$581$	−9.62753	−0.399417
$582$	87.8543	3.64168
$583$	−39.0548	−1.61748
$584$	11.0670	0.457954
$585$	0	0
$586$	38.8275	1.60395
$587$	1.17738	0.0485956	0.0242978	−	0.999705i	$-0.492265\pi$
$587$	1.17738	0.0485956	0.0242978	+	0.999705i	$0.492265\pi$
$588$	−79.5863	−3.28208
$589$	0	0
$590$	0	0
$591$	−21.1246	−0.868949
$592$	−23.5225	−0.966769
$593$	−45.1351	−1.85348	−0.926738	−	0.375709i	$-0.877399\pi$
$593$	−45.1351	−1.85348	−0.926738	+	0.375709i	$0.877399\pi$
$594$	49.8357	2.04478
$595$	0	0
$596$	50.6731	2.07565
$597$	26.8730	1.09984
$598$	50.6016	2.06925
$599$	5.23531	0.213909	0.106954	−	0.994264i	$-0.465890\pi$
$599$	5.23531	0.213909	0.106954	+	0.994264i	$0.465890\pi$
$600$	0	0
$601$	−40.3010	−1.64391	−0.821957	−	0.569550i	$-0.807117\pi$
$601$	−40.3010	−1.64391	−0.821957	+	0.569550i	$0.807117\pi$
$602$	33.7435	1.37528
$603$	−0.313742	−0.0127766
$604$	90.1802	3.66938
$605$	0	0
$606$	1.15278	0.0468286
$607$	14.9768	0.607889	0.303944	−	0.952690i	$-0.401696\pi$
$607$	14.9768	0.607889	0.303944	+	0.952690i	$0.401696\pi$
$608$	0	0
$609$	−82.3737	−3.33795
$610$	0	0
$611$	−23.1063	−0.934780
$612$	13.1528	0.531670
$613$	−11.4747	−0.463461	−0.231730	−	0.972780i	$-0.574439\pi$
$613$	−11.4747	−0.463461	−0.231730	+	0.972780i	$0.574439\pi$
$614$	33.5364	1.35342
$615$	0	0
$616$	94.6896	3.81515
$617$	−6.64392	−0.267474	−0.133737	−	0.991017i	$-0.542698\pi$
$617$	−6.64392	−0.267474	−0.133737	+	0.991017i	$0.542698\pi$
$618$	73.5209	2.95745
$619$	−3.84556	−0.154566	−0.0772831	−	0.997009i	$-0.524625\pi$
$619$	−3.84556	−0.154566	−0.0772831	+	0.997009i	$0.524625\pi$
$620$	0	0
$621$	31.3823	1.25933
$622$	9.15320	0.367010
$623$	−50.7884	−2.03480
$624$	20.7560	0.830907
$625$	0	0
$626$	29.5638	1.18161
$627$	0	0
$628$	−61.9072	−2.47037
$629$	−26.0470	−1.03856
$630$	0	0
$631$	−34.5544	−1.37559	−0.687795	−	0.725905i	$-0.741421\pi$
$631$	−34.5544	−1.37559	−0.687795	+	0.725905i	$0.741421\pi$
$632$	27.0164	1.07465
$633$	−30.8049	−1.22439
$634$	64.5446	2.56339
$635$	0	0
$636$	−63.3429	−2.51171
$637$	−29.2309	−1.15817
$638$	117.975	4.67069
$639$	−0.329720	−0.0130435
$640$	0	0
$641$	1.23899	0.0489372	0.0244686	−	0.999701i	$-0.492211\pi$
$641$	1.23899	0.0489372	0.0244686	+	0.999701i	$0.492211\pi$
$642$	−7.87499	−0.310801
$643$	−31.2715	−1.23323	−0.616613	−	0.787267i	$-0.711496\pi$
$643$	−31.2715	−1.23323	−0.616613	+	0.787267i	$0.711496\pi$
$644$	120.492	4.74807
$645$	0	0
$646$	0	0
$647$	43.9577	1.72816	0.864078	−	0.503359i	$-0.167903\pi$
$647$	43.9577	1.72816	0.864078	+	0.503359i	$0.167903\pi$
$648$	51.4551	2.02135
$649$	−19.4894	−0.765025
$650$	0	0
$651$	16.6098	0.650990
$652$	−25.6357	−1.00397
$653$	−10.2633	−0.401632	−0.200816	−	0.979629i	$-0.564359\pi$
$653$	−10.2633	−0.401632	−0.200816	+	0.979629i	$0.564359\pi$
$654$	17.1283	0.669768
$655$	0	0
$656$	−28.9211	−1.12918
$657$	−1.84722	−0.0720668
$658$	−82.8134	−3.22840
$659$	−29.1508	−1.13555	−0.567777	−	0.823182i	$-0.692196\pi$
$659$	−29.1508	−1.13555	−0.567777	+	0.823182i	$0.692196\pi$
$660$	0	0
$661$	8.83542	0.343658	0.171829	−	0.985127i	$-0.445032\pi$
$661$	8.83542	0.343658	0.171829	+	0.985127i	$0.445032\pi$
$662$	45.6016	1.77236
$663$	22.9836	0.892609
$664$	11.0670	0.429481
$665$	0	0
$666$	12.2000	0.472740
$667$	74.2909	2.87656
$668$	15.0400	0.581917
$669$	7.62786	0.294910
$670$	0	0
$671$	−27.4324	−1.05902
$672$	−3.18690	−0.122937
$673$	−35.8317	−1.38121	−0.690605	−	0.723232i	$-0.742656\pi$
$673$	−35.8317	−1.38121	−0.690605	+	0.723232i	$0.742656\pi$
$674$	−46.4144	−1.78782
$675$	0	0
$676$	−19.6682	−0.756469
$677$	23.0743	0.886819	0.443409	−	0.896319i	$-0.353769\pi$
$677$	23.0743	0.886819	0.443409	+	0.896319i	$0.353769\pi$
$678$	−11.5192	−0.442391
$679$	76.8354	2.94867
$680$	0	0
$681$	5.56261	0.213160
$682$	−23.7885	−0.910909
$683$	−10.1737	−0.389285	−0.194642	−	0.980874i	$-0.562355\pi$
$683$	−10.1737	−0.389285	−0.194642	+	0.980874i	$0.562355\pi$
$684$	0	0
$685$	0	0
$686$	−33.6602	−1.28515
$687$	0.111197	0.00424244
$688$	−12.4829	−0.475908
$689$	−23.2649	−0.886322
$690$	0	0
$691$	36.5298	1.38966	0.694830	−	0.719174i	$-0.255480\pi$
$691$	36.5298	1.38966	0.694830	+	0.719174i	$0.255480\pi$
$692$	25.4755	0.968432
$693$	−15.8049	−0.600379
$694$	−6.97057	−0.264599
$695$	0	0
$696$	94.6896	3.58920
$697$	−32.0250	−1.21303
$698$	−72.7390	−2.75321
$699$	23.6026	0.892731
$700$	0	0
$701$	2.78230	0.105086	0.0525430	−	0.998619i	$-0.483267\pi$
$701$	2.78230	0.105086	0.0525430	+	0.998619i	$0.483267\pi$
$702$	29.6871	1.12047
$703$	0	0
$704$	40.3203	1.51963
$705$	0	0
$706$	−32.9343	−1.23950
$707$	1.00820	0.0379172
$708$	−31.6098	−1.18797
$709$	−22.7803	−0.855533	−0.427766	−	0.903889i	$-0.640699\pi$
$709$	−22.7803	−0.855533	−0.427766	+	0.903889i	$0.640699\pi$
$710$	0	0
$711$	−4.50938	−0.169115
$712$	58.3819	2.18796
$713$	−14.9800	−0.561005
$714$	82.3737	3.08276
$715$	0	0
$716$	−34.0732	−1.27337
$717$	−24.3445	−0.909161
$718$	−16.9890	−0.634023
$719$	14.9429	0.557278	0.278639	−	0.960396i	$-0.410117\pi$
$719$	14.9429	0.557278	0.278639	+	0.960396i	$0.410117\pi$
$720$	0	0
$721$	64.2998	2.39465
$722$	0	0
$723$	−46.6098	−1.73344
$724$	17.8542	0.663546
$725$	0	0
$726$	55.3593	2.05458
$727$	17.4488	0.647141	0.323571	−	0.946204i	$-0.395117\pi$
$727$	17.4488	0.647141	0.323571	+	0.946204i	$0.395117\pi$
$728$	56.4065	2.09056
$729$	16.4993	0.611087
$730$	0	0
$731$	−13.8226	−0.511248
$732$	−44.4926	−1.64450
$733$	32.2387	1.19076	0.595381	−	0.803443i	$-0.297001\pi$
$733$	32.2387	1.19076	0.595381	+	0.803443i	$0.297001\pi$
$734$	70.6908	2.60925
$735$	0	0
$736$	2.87419	0.105944
$737$	1.86970	0.0688712
$738$	15.0000	0.552158
$739$	−29.7967	−1.09609	−0.548045	−	0.836449i	$-0.684628\pi$
$739$	−29.7967	−1.09609	−0.548045	+	0.836449i	$0.684628\pi$
$740$	0	0
$741$	0	0
$742$	−83.3819	−3.06105
$743$	4.41013	0.161792	0.0808960	−	0.996723i	$-0.474222\pi$
$743$	4.41013	0.161792	0.0808960	+	0.996723i	$0.474222\pi$
$744$	−19.0932	−0.699990
$745$	0	0
$746$	−63.1396	−2.31171
$747$	−1.84722	−0.0675861
$748$	−78.3819	−2.86593
$749$	−6.88730	−0.251656
$750$	0	0
$751$	31.6953	1.15658	0.578288	−	0.815832i	$-0.303721\pi$
$751$	31.6953	1.15658	0.578288	+	0.815832i	$0.303721\pi$
$752$	30.6357	1.11717
$753$	−25.6340	−0.934153
$754$	70.2778	2.55937
$755$	0	0
$756$	70.6908	2.57100
$757$	7.65212	0.278121	0.139061	−	0.990284i	$-0.455592\pi$
$757$	7.65212	0.278121	0.139061	+	0.990284i	$0.455592\pi$
$758$	34.5934	1.25649
$759$	67.8166	2.46159
$760$	0	0
$761$	−24.5544	−0.890097	−0.445048	−	0.895507i	$-0.646813\pi$
$761$	−24.5544	−0.890097	−0.445048	+	0.895507i	$0.646813\pi$
$762$	−76.6169	−2.77554
$763$	14.9800	0.542312
$764$	11.3056	0.409021
$765$	0	0
$766$	−90.2517	−3.26093
$767$	−11.6098	−0.419206
$768$	61.6561	2.22482
$769$	28.7705	1.03749	0.518745	−	0.854929i	$-0.326400\pi$
$769$	28.7705	1.03749	0.518745	+	0.854929i	$0.326400\pi$
$770$	0	0
$771$	−3.41148	−0.122861
$772$	−16.6120	−0.597878
$773$	32.4812	1.16827	0.584134	−	0.811657i	$-0.301434\pi$
$773$	32.4812	1.16827	0.584134	+	0.811657i	$0.301434\pi$
$774$	6.47430	0.232714
$775$	0	0
$776$	−88.3232	−3.17062
$777$	50.7639	1.82115
$778$	68.9756	2.47289
$779$	0	0
$780$	0	0
$781$	1.96492	0.0703102
$782$	−74.2909	−2.65664
$783$	43.5852	1.55761
$784$	38.7560	1.38414
$785$	0	0
$786$	−49.0698	−1.75026
$787$	−17.2427	−0.614635	−0.307318	−	0.951607i	$-0.599431\pi$
$787$	−17.2427	−0.614635	−0.307318	+	0.951607i	$0.599431\pi$
$788$	42.9154	1.52880
$789$	−51.5976	−1.83692
$790$	0	0
$791$	−10.0744	−0.358205
$792$	18.1679	0.645569
$793$	−16.3415	−0.580303
$794$	59.6239	2.11597
$795$	0	0
$796$	−54.5934	−1.93501
$797$	−43.6034	−1.54451	−0.772255	−	0.635312i	$-0.780871\pi$
$797$	−43.6034	−1.54451	−0.772255	+	0.635312i	$0.780871\pi$
$798$	0	0
$799$	33.9236	1.20013
$800$	0	0
$801$	−9.74469	−0.344312
$802$	63.3819	2.23809
$803$	11.0082	0.388471
$804$	3.03246	0.106947
$805$	0	0
$806$	−14.1708	−0.499145
$807$	−4.79671	−0.168852
$808$	−1.15894	−0.0407712
$809$	32.3737	1.13820	0.569100	−	0.822268i	$-0.307292\pi$
$809$	32.3737	1.13820	0.569100	+	0.822268i	$0.307292\pi$
$810$	0	0
$811$	−42.2659	−1.48416	−0.742079	−	0.670312i	$-0.766160\pi$
$811$	−42.2659	−1.48416	−0.742079	+	0.670312i	$0.766160\pi$
$812$	167.345	5.87267
$813$	29.7063	1.04185
$814$	−72.7039	−2.54827
$815$	0	0
$816$	−30.4731	−1.06677
$817$	0	0
$818$	9.79671	0.342534
$819$	−9.41497	−0.328986
$820$	0	0
$821$	−33.5688	−1.17156	−0.585780	−	0.810470i	$-0.699212\pi$
$821$	−33.5688	−1.17156	−0.585780	+	0.810470i	$0.699212\pi$
$822$	50.6016	1.76493
$823$	36.1023	1.25845	0.629223	−	0.777225i	$-0.283373\pi$
$823$	36.1023	1.25845	0.629223	+	0.777225i	$0.283373\pi$
$824$	−73.9134	−2.57489
$825$	0	0
$826$	−41.6098	−1.44779
$827$	−43.2905	−1.50536	−0.752678	−	0.658389i	$-0.771238\pi$
$827$	−43.2905	−1.50536	−0.752678	+	0.658389i	$0.771238\pi$
$828$	23.1187	0.803429
$829$	−10.4706	−0.363660	−0.181830	−	0.983330i	$-0.558202\pi$
$829$	−10.4706	−0.363660	−0.181830	+	0.983330i	$0.558202\pi$
$830$	0	0
$831$	38.2528	1.32698
$832$	24.0188	0.832701
$833$	42.9154	1.48693
$834$	−18.1105	−0.627114
$835$	0	0
$836$	0	0
$837$	−8.78851	−0.303775
$838$	−34.4662	−1.19061
$839$	−30.5563	−1.05492	−0.527461	−	0.849579i	$-0.676856\pi$
$839$	−30.5563	−1.05492	−0.527461	+	0.849579i	$0.676856\pi$
$840$	0	0
$841$	74.1786	2.55788
$842$	44.5934	1.53679
$843$	−7.98360	−0.274970
$844$	62.5813	2.15414
$845$	0	0
$846$	−15.8893	−0.546284
$847$	48.4160	1.66359
$848$	30.8460	1.05926
$849$	49.2365	1.68979
$850$	0	0
$851$	−45.7828	−1.56941
$852$	3.18690	0.109181
$853$	−24.1269	−0.826088	−0.413044	−	0.910711i	$-0.635534\pi$
$853$	−24.1269	−0.826088	−0.413044	+	0.910711i	$0.635534\pi$
$854$	−58.5682	−2.00416
$855$	0	0
$856$	7.91703	0.270599
$857$	−35.6932	−1.21926	−0.609628	−	0.792687i	$-0.708681\pi$
$857$	−35.6932	−1.21926	−0.609628	+	0.792687i	$0.708681\pi$
$858$	64.1533	2.19016
$859$	−39.9236	−1.36217	−0.681087	−	0.732202i	$-0.738493\pi$
$859$	−39.9236	−1.36217	−0.681087	+	0.732202i	$0.738493\pi$
$860$	0	0
$861$	62.4147	2.12709
$862$	−46.8131	−1.59446
$863$	−4.90640	−0.167016	−0.0835079	−	0.996507i	$-0.526612\pi$
$863$	−4.90640	−0.167016	−0.0835079	+	0.996507i	$0.526612\pi$
$864$	1.68624	0.0573669
$865$	0	0
$866$	−45.4550	−1.54463
$867$	−0.611506	−0.0207678
$868$	−33.7435	−1.14533
$869$	26.8730	0.911602
$870$	0	0
$871$	1.11378	0.0377389
$872$	−17.2197	−0.583132
$873$	14.7423	0.498950
$874$	0	0
$875$	0	0
$876$	17.8542	0.603237
$877$	24.6103	0.831030	0.415515	−	0.909586i	$-0.363601\pi$
$877$	24.6103	0.831030	0.415515	+	0.909586i	$0.363601\pi$
$878$	34.7967	1.17433
$879$	30.9983	1.04555
$880$	0	0
$881$	32.0554	1.07997	0.539987	−	0.841673i	$-0.318429\pi$
$881$	32.0554	1.07997	0.539987	+	0.841673i	$0.318429\pi$
$882$	−20.1009	−0.676832
$883$	47.1105	1.58539	0.792697	−	0.609616i	$-0.208676\pi$
$883$	47.1105	1.58539	0.792697	+	0.609616i	$0.208676\pi$
$884$	−46.6921	−1.57042
$885$	0	0
$886$	−66.5944	−2.23728
$887$	−12.1426	−0.407709	−0.203855	−	0.979001i	$-0.565347\pi$
$887$	−12.1426	−0.407709	−0.203855	+	0.979001i	$0.565347\pi$
$888$	−58.3537	−1.95822
$889$	−67.0074	−2.24736
$890$	0	0
$891$	51.1819	1.71466
$892$	−15.4963	−0.518854
$893$	0	0
$894$	60.8908	2.03649
$895$	0	0
$896$	82.8134	2.76660
$897$	40.3983	1.34886
$898$	2.98360	0.0995641
$899$	−20.8049	−0.693882
$900$	0	0
$901$	34.1565	1.13792
$902$	−89.3901	−2.97637
$903$	26.9394	0.896488
$904$	11.5807	0.385167
$905$	0	0
$906$	108.364	3.60016
$907$	18.7002	0.620930	0.310465	−	0.950585i	$-0.399515\pi$
$907$	18.7002	0.620930	0.310465	+	0.950585i	$0.399515\pi$
$908$	−11.3006	−0.375025
$909$	0.193441	0.00641604
$910$	0	0
$911$	−2.04819	−0.0678597	−0.0339298	−	0.999424i	$-0.510802\pi$
$911$	−2.04819	−0.0678597	−0.0339298	+	0.999424i	$0.510802\pi$
$912$	0	0
$913$	11.0082	0.364318
$914$	−24.8248	−0.821130
$915$	0	0
$916$	−0.225901	−0.00746399
$917$	−42.9154	−1.41719
$918$	−43.5852	−1.43853
$919$	−27.4911	−0.906849	−0.453425	−	0.891295i	$-0.649798\pi$
$919$	−27.4911	−0.906849	−0.453425	+	0.891295i	$0.649798\pi$
$920$	0	0
$921$	26.7741	0.882237
$922$	37.8166	1.24542
$923$	1.17050	0.0385275
$924$	152.762	5.02549
$925$	0	0
$926$	3.60011	0.118307
$927$	12.3371	0.405203
$928$	3.99180	0.131037
$929$	25.6016	0.839962	0.419981	−	0.907533i	$-0.362037\pi$
$929$	25.6016	0.839962	0.419981	+	0.907533i	$0.362037\pi$
$930$	0	0
$931$	0	0
$932$	−47.9495	−1.57064
$933$	7.30755	0.239238
$934$	−74.7872	−2.44711
$935$	0	0
$936$	10.8226	0.353748
$937$	43.7544	1.42939	0.714697	−	0.699434i	$-0.246565\pi$
$937$	43.7544	1.42939	0.714697	+	0.699434i	$0.246565\pi$
$938$	3.99180	0.130337
$939$	23.6026	0.770240
$940$	0	0
$941$	−55.5275	−1.81014	−0.905072	−	0.425258i	$-0.860183\pi$
$941$	−55.5275	−1.81014	−0.905072	+	0.425258i	$0.860183\pi$
$942$	−74.3901	−2.42376
$943$	−56.2904	−1.83307
$944$	15.3930	0.500999
$945$	0	0
$946$	−38.5826	−1.25443
$947$	31.5347	1.02474	0.512370	−	0.858765i	$-0.328768\pi$
$947$	31.5347	1.02474	0.512370	+	0.858765i	$0.328768\pi$
$948$	43.5852	1.41558
$949$	6.55758	0.212868
$950$	0	0
$951$	51.5298	1.67097
$952$	−82.8134	−2.68400
$953$	12.1899	0.394870	0.197435	−	0.980316i	$-0.436739\pi$
$953$	12.1899	0.394870	0.197435	+	0.980316i	$0.436739\pi$
$954$	−15.9983	−0.517966
$955$	0	0
$956$	49.4567	1.59954
$957$	94.1868	3.04463
$958$	−53.1103	−1.71592
$959$	44.2551	1.42907
$960$	0	0
$961$	−26.8049	−0.864674
$962$	−43.3097	−1.39636
$963$	−1.32145	−0.0425833
$964$	94.6896	3.04975
$965$	0	0
$966$	144.788	4.65849
$967$	26.8472	0.863348	0.431674	−	0.902030i	$-0.357923\pi$
$967$	26.8472	0.863348	0.431674	+	0.902030i	$0.357923\pi$
$968$	−55.6548	−1.78881
$969$	0	0
$970$	0	0
$971$	−13.1152	−0.420885	−0.210443	−	0.977606i	$-0.567491\pi$
$971$	−13.1152	−0.420885	−0.210443	+	0.977606i	$0.567491\pi$
$972$	32.0450	1.02784
$973$	−15.8390	−0.507776
$974$	−17.0653	−0.546806
$975$	0	0
$976$	21.6665	0.693529
$977$	37.0106	1.18407	0.592037	−	0.805910i	$-0.298324\pi$
$977$	37.0106	1.18407	0.592037	+	0.805910i	$0.298324\pi$
$978$	−30.8049	−0.985032
$979$	58.0719	1.85599
$980$	0	0
$981$	2.87419	0.0917657
$982$	100.025	3.19192
$983$	26.3166	0.839370	0.419685	−	0.907670i	$-0.362140\pi$
$983$	26.3166	0.839370	0.419685	+	0.907670i	$0.362140\pi$
$984$	−71.7465	−2.28719
$985$	0	0
$986$	−103.179	−3.28588
$987$	−66.1150	−2.10446
$988$	0	0
$989$	−24.2960	−0.772569
$990$	0	0
$991$	31.1157	0.988423	0.494212	−	0.869342i	$-0.335457\pi$
$991$	31.1157	0.988423	0.494212	+	0.869342i	$0.335457\pi$
$992$	−0.804906	−0.0255558
$993$	36.4065	1.15533
$994$	4.19509	0.133060
$995$	0	0
$996$	17.8542	0.565732
$997$	17.6521	0.559048	0.279524	−	0.960139i	$-0.409823\pi$
$997$	17.6521	0.559048	0.279524	+	0.960139i	$0.409823\pi$
$998$	−56.0678	−1.77480
$999$	−26.8600	−0.849812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bw.1.1	yes	6
5.4	even	2		9025.2.a.bv.1.6	yes	6
19.18	odd	2	inner	9025.2.a.bw.1.6	yes	6
95.94	odd	2		9025.2.a.bv.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.bv.1.1	✓	6	95.94	odd	2
9025.2.a.bv.1.6	yes	6	5.4	even	2
9025.2.a.bw.1.1	yes	6	1.1	even	1	trivial
9025.2.a.bw.1.6	yes	6	19.18	odd	2	inner

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

\(f(q)\)	\(=\)	\(q-2.44118 q^{2} -1.94894 q^{3} +3.95934 q^{4} +4.75770 q^{6} +4.16098 q^{7} -4.78310 q^{8} +0.798360 q^{9} +O(q^{10})\)	\(q-2.44118 q^{2} -1.94894 q^{3} +3.95934 q^{4} +4.75770 q^{6} +4.16098 q^{7} -4.78310 q^{8} +0.798360 q^{9} -4.75770 q^{11} -7.71651 q^{12} -2.83416 q^{13} -10.1577 q^{14} +3.75770 q^{16} +4.16098 q^{17} -1.94894 q^{18} -8.10950 q^{21} +11.6144 q^{22} +7.31376 q^{23} +9.32196 q^{24} +6.91868 q^{26} +4.29086 q^{27} +16.4747 q^{28} +10.1577 q^{29} -2.04819 q^{31} +0.392983 q^{32} +9.27247 q^{33} -10.1577 q^{34} +3.16098 q^{36} -6.25981 q^{37} +5.52360 q^{39} -7.69650 q^{41} +19.7967 q^{42} -3.32196 q^{43} -18.8374 q^{44} -17.8542 q^{46} +8.15278 q^{47} -7.32353 q^{48} +10.3138 q^{49} -8.10950 q^{51} -11.2214 q^{52} +8.20875 q^{53} -10.4747 q^{54} -19.9024 q^{56} -24.7967 q^{58} +4.09639 q^{59} +5.76590 q^{61} +5.00000 q^{62} +3.32196 q^{63} -8.47474 q^{64} -22.6357 q^{66} -0.392983 q^{67} +16.4747 q^{68} -14.2541 q^{69} -0.412997 q^{71} -3.81864 q^{72} -2.31376 q^{73} +15.2813 q^{74} -19.7967 q^{77} -13.4841 q^{78} -5.64831 q^{79} -10.7577 q^{81} +18.7885 q^{82} -2.31376 q^{83} -32.1083 q^{84} +8.10950 q^{86} -19.7967 q^{87} +22.7566 q^{88} -12.2059 q^{89} -11.7929 q^{91} +28.9577 q^{92} +3.99180 q^{93} -19.9024 q^{94} -0.765900 q^{96} +18.4657 q^{97} -25.1777 q^{98} -3.79836 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9	\( 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9} - 10 q^{11} + 4 q^{16} + 4 q^{17} + 8 q^{23} + 14 q^{24} + 2 q^{26} + 42 q^{28} - 2 q^{36} - 10 q^{39} + 20 q^{42} + 22 q^{43} - 34 q^{44} + 34 q^{47} + 26 q^{49} - 6 q^{54} - 50 q^{58} + 10 q^{61} + 30 q^{62} - 22 q^{63} + 6 q^{64} - 58 q^{66} + 42 q^{68} + 22 q^{73} + 30 q^{74} - 20 q^{77} - 46 q^{81} + 20 q^{82} + 22 q^{83} - 20 q^{87} + 54 q^{92} + 30 q^{93} + 20 q^{96} - 24 q^{99}+O(q^{100}) \) 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9 - 10 * q^11 + 4 * q^16 + 4 * q^17 + 8 * q^23 + 14 * q^24 + 2 * q^26 + 42 * q^28 - 2 * q^36 - 10 * q^39 + 20 * q^42 + 22 * q^43 - 34 * q^44 + 34 * q^47 + 26 * q^49 - 6 * q^54 - 50 * q^58 + 10 * q^61 + 30 * q^62 - 22 * q^63 + 6 * q^64 - 58 * q^66 + 42 * q^68 + 22 * q^73 + 30 * q^74 - 20 * q^77 - 46 * q^81 + 20 * q^82 + 22 * q^83 - 20 * q^87 + 54 * q^92 + 30 * q^93 + 20 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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