LMFDB - Embedded newform 6025.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.06849 q^{2} +2.17142 q^{3} +2.27866 q^{4} -4.49155 q^{6} +2.39120 q^{7} -0.576402 q^{8} +1.71504 q^{9} +O(q^{10})$	\(q-2.06849 q^{2} +2.17142 q^{3} +2.27866 q^{4} -4.49155 q^{6} +2.39120 q^{7} -0.576402 q^{8} +1.71504 q^{9} +4.70134 q^{11} +4.94791 q^{12} -0.253706 q^{13} -4.94619 q^{14} -3.36503 q^{16} +7.11822 q^{17} -3.54755 q^{18} +7.42666 q^{19} +5.19230 q^{21} -9.72469 q^{22} -0.555066 q^{23} -1.25161 q^{24} +0.524789 q^{26} -2.79017 q^{27} +5.44874 q^{28} -0.396086 q^{29} +9.01229 q^{31} +8.11335 q^{32} +10.2086 q^{33} -14.7240 q^{34} +3.90800 q^{36} +2.41780 q^{37} -15.3620 q^{38} -0.550901 q^{39} -1.30753 q^{41} -10.7402 q^{42} -1.26218 q^{43} +10.7128 q^{44} +1.14815 q^{46} +3.17874 q^{47} -7.30688 q^{48} -1.28214 q^{49} +15.4566 q^{51} -0.578109 q^{52} +6.62578 q^{53} +5.77145 q^{54} -1.37830 q^{56} +16.1264 q^{57} +0.819300 q^{58} -11.3512 q^{59} +5.87246 q^{61} -18.6419 q^{62} +4.10102 q^{63} -10.0523 q^{64} -21.1163 q^{66} -4.45188 q^{67} +16.2200 q^{68} -1.20528 q^{69} -4.24560 q^{71} -0.988554 q^{72} +2.46811 q^{73} -5.00120 q^{74} +16.9228 q^{76} +11.2419 q^{77} +1.13953 q^{78} -2.95686 q^{79} -11.2038 q^{81} +2.70462 q^{82} +7.18991 q^{83} +11.8315 q^{84} +2.61081 q^{86} -0.860067 q^{87} -2.70986 q^{88} -14.7384 q^{89} -0.606663 q^{91} -1.26481 q^{92} +19.5694 q^{93} -6.57520 q^{94} +17.6174 q^{96} -12.0608 q^{97} +2.65209 q^{98} +8.06301 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * 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 $

$2$	−2.06849	−1.46264	−0.731322	−	0.682032i	$-0.761096\pi$
$2$	−2.06849	−1.46264	−0.731322	+	0.682032i	$0.761096\pi$
$3$	2.17142	1.25367	0.626834	−	0.779153i	$-0.284351\pi$
$3$	2.17142	1.25367	0.626834	+	0.779153i	$0.284351\pi$
$4$	2.27866	1.13933
$5$	0	0
$5$	0	0
$6$	−4.49155	−1.83367
$7$	2.39120	0.903790	0.451895	−	0.892071i	$-0.350748\pi$
$7$	2.39120	0.903790	0.451895	+	0.892071i	$0.350748\pi$
$8$	−0.576402	−0.203789
$9$	1.71504	0.571681
$10$	0	0
$11$	4.70134	1.41751	0.708754	−	0.705455i	$-0.249257\pi$
$11$	4.70134	1.41751	0.708754	+	0.705455i	$0.249257\pi$
$12$	4.94791	1.42834
$13$	−0.253706	−0.0703654	−0.0351827	−	0.999381i	$-0.511201\pi$
$13$	−0.253706	−0.0703654	−0.0351827	+	0.999381i	$0.511201\pi$
$14$	−4.94619	−1.32192
$15$	0	0
$16$	−3.36503	−0.841258
$17$	7.11822	1.72642	0.863211	−	0.504843i	$-0.168450\pi$
$17$	7.11822	1.72642	0.863211	+	0.504843i	$0.168450\pi$
$18$	−3.54755	−0.836166
$19$	7.42666	1.70379	0.851896	−	0.523710i	$-0.175453\pi$
$19$	7.42666	1.70379	0.851896	+	0.523710i	$0.175453\pi$
$20$	0	0
$21$	5.19230	1.13305
$22$	−9.72469	−2.07331
$23$	−0.555066	−0.115739	−0.0578696	−	0.998324i	$-0.518431\pi$
$23$	−0.555066	−0.115739	−0.0578696	+	0.998324i	$0.518431\pi$
$24$	−1.25161	−0.255483
$25$	0	0
$26$	0.524789	0.102920
$27$	−2.79017	−0.536969
$28$	5.44874	1.02971
$29$	−0.396086	−0.0735513	−0.0367756	−	0.999324i	$-0.511709\pi$
$29$	−0.396086	−0.0735513	−0.0367756	+	0.999324i	$0.511709\pi$
$30$	0	0
$31$	9.01229	1.61866	0.809328	−	0.587357i	$-0.199832\pi$
$31$	9.01229	1.61866	0.809328	+	0.587357i	$0.199832\pi$
$32$	8.11335	1.43425
$33$	10.2086	1.77708
$34$	−14.7240	−2.52514
$35$	0	0
$36$	3.90800	0.651333
$37$	2.41780	0.397484	0.198742	−	0.980052i	$-0.436314\pi$
$37$	2.41780	0.397484	0.198742	+	0.980052i	$0.436314\pi$
$38$	−15.3620	−2.49204
$39$	−0.550901	−0.0882148
$40$	0	0
$41$	−1.30753	−0.204202	−0.102101	−	0.994774i	$-0.532556\pi$
$41$	−1.30753	−0.204202	−0.102101	+	0.994774i	$0.532556\pi$
$42$	−10.7402	−1.65725
$43$	−1.26218	−0.192480	−0.0962402	−	0.995358i	$-0.530682\pi$
$43$	−1.26218	−0.192480	−0.0962402	+	0.995358i	$0.530682\pi$
$44$	10.7128	1.61501
$45$	0	0
$46$	1.14815	0.169285
$47$	3.17874	0.463667	0.231834	−	0.972755i	$-0.425528\pi$
$47$	3.17874	0.463667	0.231834	+	0.972755i	$0.425528\pi$
$48$	−7.30688	−1.05466
$49$	−1.28214	−0.183163
$50$	0	0
$51$	15.4566	2.16436
$52$	−0.578109	−0.0801693
$53$	6.62578	0.910121	0.455060	−	0.890461i	$-0.349618\pi$
$53$	6.62578	0.910121	0.455060	+	0.890461i	$0.349618\pi$
$54$	5.77145	0.785395
$55$	0	0
$56$	−1.37830	−0.184182
$57$	16.1264	2.13599
$58$	0.819300	0.107579
$59$	−11.3512	−1.47780	−0.738902	−	0.673813i	$-0.764655\pi$
$59$	−11.3512	−1.47780	−0.738902	+	0.673813i	$0.764655\pi$
$60$	0	0
$61$	5.87246	0.751892	0.375946	−	0.926642i	$-0.377318\pi$
$61$	5.87246	0.751892	0.375946	+	0.926642i	$0.377318\pi$
$62$	−18.6419	−2.36752
$63$	4.10102	0.516680
$64$	−10.0523	−1.25654
$65$	0	0
$66$	−21.1163	−2.59924
$67$	−4.45188	−0.543884	−0.271942	−	0.962314i	$-0.587666\pi$
$67$	−4.45188	−0.543884	−0.271942	+	0.962314i	$0.587666\pi$
$68$	16.2200	1.96696
$69$	−1.20528	−0.145098
$70$	0	0
$71$	−4.24560	−0.503860	−0.251930	−	0.967745i	$-0.581065\pi$
$71$	−4.24560	−0.503860	−0.251930	+	0.967745i	$0.581065\pi$
$72$	−0.988554	−0.116502
$73$	2.46811	0.288870	0.144435	−	0.989514i	$-0.453864\pi$
$73$	2.46811	0.288870	0.144435	+	0.989514i	$0.453864\pi$
$74$	−5.00120	−0.581378
$75$	0	0
$76$	16.9228	1.94118
$77$	11.2419	1.28113
$78$	1.13953	0.129027
$79$	−2.95686	−0.332673	−0.166336	−	0.986069i	$-0.553194\pi$
$79$	−2.95686	−0.332673	−0.166336	+	0.986069i	$0.553194\pi$
$80$	0	0
$81$	−11.2038	−1.24486
$82$	2.70462	0.298675
$83$	7.18991	0.789195	0.394597	−	0.918854i	$-0.370884\pi$
$83$	7.18991	0.789195	0.394597	+	0.918854i	$0.370884\pi$
$84$	11.8315	1.29092
$85$	0	0
$86$	2.61081	0.281530
$87$	−0.860067	−0.0922088
$88$	−2.70986	−0.288873
$89$	−14.7384	−1.56227	−0.781135	−	0.624362i	$-0.785359\pi$
$89$	−14.7384	−1.56227	−0.781135	+	0.624362i	$0.785359\pi$
$90$	0	0
$91$	−0.606663	−0.0635956
$92$	−1.26481	−0.131865
$93$	19.5694	2.02925
$94$	−6.57520	−0.678180
$95$	0	0
$96$	17.6174	1.79807
$97$	−12.0608	−1.22459	−0.612293	−	0.790631i	$-0.709753\pi$
$97$	−12.0608	−1.22459	−0.612293	+	0.790631i	$0.709753\pi$
$98$	2.65209	0.267902
$99$	8.06301	0.810363
$100$	0	0
$101$	−10.6884	−1.06354	−0.531769	−	0.846889i	$-0.678473\pi$
$101$	−10.6884	−1.06354	−0.531769	+	0.846889i	$0.678473\pi$
$102$	−31.9719	−3.16569
$103$	5.14635	0.507085	0.253543	−	0.967324i	$-0.418404\pi$
$103$	5.14635	0.507085	0.253543	+	0.967324i	$0.418404\pi$
$104$	0.146237	0.0143397
$105$	0	0
$106$	−13.7054	−1.33118
$107$	11.9639	1.15659	0.578296	−	0.815827i	$-0.303718\pi$
$107$	11.9639	1.15659	0.578296	+	0.815827i	$0.303718\pi$
$108$	−6.35785	−0.611785
$109$	−9.37117	−0.897595	−0.448798	−	0.893633i	$-0.648148\pi$
$109$	−9.37117	−0.897595	−0.448798	+	0.893633i	$0.648148\pi$
$110$	0	0
$111$	5.25005	0.498313
$112$	−8.04648	−0.760321
$113$	10.9473	1.02984	0.514920	−	0.857238i	$-0.327822\pi$
$113$	10.9473	1.02984	0.514920	+	0.857238i	$0.327822\pi$
$114$	−33.3572	−3.12419
$115$	0	0
$116$	−0.902544	−0.0837991
$117$	−0.435117	−0.0402266
$118$	23.4799	2.16150
$119$	17.0211	1.56032
$120$	0	0
$121$	11.1026	1.00933
$122$	−12.1471	−1.09975
$123$	−2.83919	−0.256001
$124$	20.5359	1.84418
$125$	0	0
$126$	−8.48293	−0.755719
$127$	−11.3544	−1.00754	−0.503772	−	0.863837i	$-0.668055\pi$
$127$	−11.3544	−1.00754	−0.503772	+	0.863837i	$0.668055\pi$
$128$	4.56646	0.403622
$129$	−2.74071	−0.241306
$130$	0	0
$131$	12.3951	1.08296	0.541482	−	0.840712i	$-0.317863\pi$
$131$	12.3951	1.08296	0.541482	+	0.840712i	$0.317863\pi$
$132$	23.2618	2.02468
$133$	17.7587	1.53987
$134$	9.20869	0.795509
$135$	0	0
$136$	−4.10296	−0.351826
$137$	−18.0409	−1.54134	−0.770671	−	0.637233i	$-0.780079\pi$
$137$	−18.0409	−1.54134	−0.770671	+	0.637233i	$0.780079\pi$
$138$	2.49311	0.212227
$139$	−9.26721	−0.786035	−0.393017	−	0.919531i	$-0.628569\pi$
$139$	−9.26721	−0.786035	−0.393017	+	0.919531i	$0.628569\pi$
$140$	0	0
$141$	6.90237	0.581284
$142$	8.78199	0.736968
$143$	−1.19276	−0.0997435
$144$	−5.77118	−0.480932
$145$	0	0
$146$	−5.10526	−0.422514
$147$	−2.78406	−0.229625
$148$	5.50934	0.452865
$149$	−14.5537	−1.19229	−0.596144	−	0.802878i	$-0.703301\pi$
$149$	−14.5537	−1.19229	−0.596144	+	0.802878i	$0.703301\pi$
$150$	0	0
$151$	−2.42020	−0.196953	−0.0984764	−	0.995139i	$-0.531397\pi$
$151$	−2.42020	−0.196953	−0.0984764	+	0.995139i	$0.531397\pi$
$152$	−4.28074	−0.347214
$153$	12.2081	0.986963
$154$	−23.2537	−1.87384
$155$	0	0
$156$	−1.25532	−0.100506
$157$	−15.3397	−1.22424	−0.612120	−	0.790765i	$-0.709683\pi$
$157$	−15.3397	−1.22424	−0.612120	+	0.790765i	$0.709683\pi$
$158$	6.11624	0.486582
$159$	14.3873	1.14099
$160$	0	0
$161$	−1.32728	−0.104604
$162$	23.1749	1.82079
$163$	−2.99703	−0.234745	−0.117373	−	0.993088i	$-0.537447\pi$
$163$	−2.99703	−0.234745	−0.117373	+	0.993088i	$0.537447\pi$
$164$	−2.97942	−0.232653
$165$	0	0
$166$	−14.8723	−1.15431
$167$	6.93355	0.536535	0.268267	−	0.963345i	$-0.413549\pi$
$167$	6.93355	0.536535	0.268267	+	0.963345i	$0.413549\pi$
$168$	−2.99285	−0.230903
$169$	−12.9356	−0.995049
$170$	0	0
$171$	12.7370	0.974026
$172$	−2.87607	−0.219299
$173$	8.57752	0.652137	0.326068	−	0.945346i	$-0.394276\pi$
$173$	8.57752	0.652137	0.326068	+	0.945346i	$0.394276\pi$
$174$	1.77904	0.134869
$175$	0	0
$176$	−15.8202	−1.19249
$177$	−24.6482	−1.85267
$178$	30.4863	2.28505
$179$	−8.81455	−0.658831	−0.329415	−	0.944185i	$-0.606852\pi$
$179$	−8.81455	−0.658831	−0.329415	+	0.944185i	$0.606852\pi$
$180$	0	0
$181$	10.5046	0.780801	0.390401	−	0.920645i	$-0.372337\pi$
$181$	10.5046	0.780801	0.390401	+	0.920645i	$0.372337\pi$
$182$	1.25488	0.0930177
$183$	12.7515	0.942622
$184$	0.319941	0.0235864
$185$	0	0
$186$	−40.4792	−2.96808
$187$	33.4652	2.44722
$188$	7.24327	0.528270
$189$	−6.67188	−0.485308
$190$	0	0
$191$	−16.8933	−1.22236	−0.611178	−	0.791493i	$-0.709304\pi$
$191$	−16.8933	−1.22236	−0.611178	+	0.791493i	$0.709304\pi$
$192$	−21.8278	−1.57528
$193$	−19.3285	−1.39130	−0.695648	−	0.718383i	$-0.744883\pi$
$193$	−19.3285	−1.39130	−0.695648	+	0.718383i	$0.744883\pi$
$194$	24.9476	1.79114
$195$	0	0
$196$	−2.92156	−0.208683
$197$	15.8861	1.13183	0.565917	−	0.824462i	$-0.308522\pi$
$197$	15.8861	1.13183	0.565917	+	0.824462i	$0.308522\pi$
$198$	−16.6783	−1.18527
$199$	−22.5111	−1.59577	−0.797884	−	0.602811i	$-0.794047\pi$
$199$	−22.5111	−1.59577	−0.797884	+	0.602811i	$0.794047\pi$
$200$	0	0
$201$	−9.66689	−0.681850
$202$	22.1089	1.55558
$203$	−0.947122	−0.0664750
$204$	35.2203	2.46592
$205$	0	0
$206$	−10.6452	−0.741685
$207$	−0.951962	−0.0661659
$208$	0.853729	0.0591955
$209$	34.9153	2.41514
$210$	0	0
$211$	11.7192	0.806783	0.403392	−	0.915027i	$-0.367831\pi$
$211$	11.7192	0.806783	0.403392	+	0.915027i	$0.367831\pi$
$212$	15.0979	1.03693
$213$	−9.21896	−0.631673
$214$	−24.7472	−1.69168
$215$	0	0
$216$	1.60826	0.109428
$217$	21.5502	1.46293
$218$	19.3842	1.31286
$219$	5.35928	0.362147
$220$	0	0
$221$	−1.80594	−0.121480
$222$	−10.8597	−0.728854
$223$	2.48105	0.166143	0.0830717	−	0.996544i	$-0.473527\pi$
$223$	2.48105	0.166143	0.0830717	+	0.996544i	$0.473527\pi$
$224$	19.4007	1.29626
$225$	0	0
$226$	−22.6445	−1.50629
$227$	22.7571	1.51044	0.755221	−	0.655470i	$-0.227529\pi$
$227$	22.7571	1.51044	0.755221	+	0.655470i	$0.227529\pi$
$228$	36.7465	2.43359
$229$	−2.93838	−0.194173	−0.0970867	−	0.995276i	$-0.530952\pi$
$229$	−2.93838	−0.194173	−0.0970867	+	0.995276i	$0.530952\pi$
$230$	0	0
$231$	24.4108	1.60611
$232$	0.228305	0.0149889
$233$	4.41077	0.288959	0.144480	−	0.989508i	$-0.453849\pi$
$233$	4.41077	0.288959	0.144480	+	0.989508i	$0.453849\pi$
$234$	0.900035	0.0588372
$235$	0	0
$236$	−25.8656	−1.68371
$237$	−6.42057	−0.417061
$238$	−35.2081	−2.28220
$239$	−18.8141	−1.21698	−0.608490	−	0.793562i	$-0.708224\pi$
$239$	−18.8141	−1.21698	−0.608490	+	0.793562i	$0.708224\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−22.9657	−1.47629
$243$	−15.9575	−1.02367
$244$	13.3813	0.856652
$245$	0	0
$246$	5.87285	0.374439
$247$	−1.88419	−0.119888
$248$	−5.19470	−0.329864
$249$	15.6123	0.989388
$250$	0	0
$251$	−23.8009	−1.50230	−0.751149	−	0.660133i	$-0.770500\pi$
$251$	−23.8009	−1.50230	−0.751149	+	0.660133i	$0.770500\pi$
$252$	9.34482	0.588668
$253$	−2.60956	−0.164061
$254$	23.4866	1.47368
$255$	0	0
$256$	10.6590	0.666186
$257$	−2.94320	−0.183592	−0.0917959	−	0.995778i	$-0.529261\pi$
$257$	−2.94320	−0.183592	−0.0917959	+	0.995778i	$0.529261\pi$
$258$	5.66914	0.352945
$259$	5.78146	0.359242
$260$	0	0
$261$	−0.679304	−0.0420479
$262$	−25.6392	−1.58399
$263$	15.6487	0.964939	0.482470	−	0.875913i	$-0.339740\pi$
$263$	15.6487	0.964939	0.482470	+	0.875913i	$0.339740\pi$
$264$	−5.88424	−0.362150
$265$	0	0
$266$	−36.7337	−2.25228
$267$	−32.0033	−1.95857
$268$	−10.1443	−0.619663
$269$	20.6793	1.26084	0.630418	−	0.776255i	$-0.282883\pi$
$269$	20.6793	1.26084	0.630418	+	0.776255i	$0.282883\pi$
$270$	0	0
$271$	6.61504	0.401835	0.200917	−	0.979608i	$-0.435608\pi$
$271$	6.61504	0.401835	0.200917	+	0.979608i	$0.435608\pi$
$272$	−23.9530	−1.45237
$273$	−1.31732	−0.0797277
$274$	37.3175	2.25444
$275$	0	0
$276$	−2.74642	−0.165315
$277$	0.120199	0.00722208	0.00361104	−	0.999993i	$-0.498851\pi$
$277$	0.120199	0.00722208	0.00361104	+	0.999993i	$0.498851\pi$
$278$	19.1692	1.14969
$279$	15.4565	0.925355
$280$	0	0
$281$	10.4169	0.621421	0.310711	−	0.950505i	$-0.399433\pi$
$281$	10.4169	0.621421	0.310711	+	0.950505i	$0.399433\pi$
$282$	−14.2775	−0.850212
$283$	−10.1377	−0.602624	−0.301312	−	0.953526i	$-0.597425\pi$
$283$	−10.1377	−0.602624	−0.301312	+	0.953526i	$0.597425\pi$
$284$	−9.67427	−0.574062
$285$	0	0
$286$	2.46721	0.145889
$287$	−3.12657	−0.184556
$288$	13.9147	0.819934
$289$	33.6691	1.98053
$290$	0	0
$291$	−26.1890	−1.53522
$292$	5.62397	0.329118
$293$	−2.95196	−0.172455	−0.0862276	−	0.996275i	$-0.527481\pi$
$293$	−2.95196	−0.172455	−0.0862276	+	0.996275i	$0.527481\pi$
$294$	5.75880	0.335860
$295$	0	0
$296$	−1.39363	−0.0810028
$297$	−13.1176	−0.761159
$298$	30.1043	1.74389
$299$	0.140823	0.00814403
$300$	0	0
$301$	−3.01813	−0.173962
$302$	5.00616	0.288072
$303$	−23.2090	−1.33332
$304$	−24.9910	−1.43333
$305$	0	0
$306$	−25.2523	−1.44358
$307$	−3.12263	−0.178218	−0.0891089	−	0.996022i	$-0.528402\pi$
$307$	−3.12263	−0.178218	−0.0891089	+	0.996022i	$0.528402\pi$
$308$	25.6164	1.45963
$309$	11.1749	0.635716
$310$	0	0
$311$	3.28464	0.186255	0.0931275	−	0.995654i	$-0.470314\pi$
$311$	3.28464	0.186255	0.0931275	+	0.995654i	$0.470314\pi$
$312$	0.317540	0.0179772
$313$	−8.20499	−0.463773	−0.231887	−	0.972743i	$-0.574490\pi$
$313$	−8.20499	−0.463773	−0.231887	+	0.972743i	$0.574490\pi$
$314$	31.7300	1.79063
$315$	0	0
$316$	−6.73768	−0.379024
$317$	27.0918	1.52163	0.760813	−	0.648971i	$-0.224801\pi$
$317$	27.0918	1.52163	0.760813	+	0.648971i	$0.224801\pi$
$318$	−29.7600	−1.66886
$319$	−1.86214	−0.104260
$320$	0	0
$321$	25.9785	1.44998
$322$	2.74546	0.152998
$323$	52.8646	2.94147
$324$	−25.5295	−1.41831
$325$	0	0
$326$	6.19933	0.343349
$327$	−20.3487	−1.12529
$328$	0.753663	0.0416141
$329$	7.60102	0.419058
$330$	0	0
$331$	−32.4955	−1.78612	−0.893058	−	0.449942i	$-0.851444\pi$
$331$	−32.4955	−1.78612	−0.893058	+	0.449942i	$0.851444\pi$
$332$	16.3833	0.899153
$333$	4.14663	0.227234
$334$	−14.3420	−0.784759
$335$	0	0
$336$	−17.4723	−0.953190
$337$	31.6787	1.72565	0.862825	−	0.505502i	$-0.168693\pi$
$337$	31.6787	1.72565	0.862825	+	0.505502i	$0.168693\pi$
$338$	26.7573	1.45540
$339$	23.7712	1.29108
$340$	0	0
$341$	42.3699	2.29446
$342$	−26.3465	−1.42465
$343$	−19.8043	−1.06933
$344$	0.727522	0.0392254
$345$	0	0
$346$	−17.7425	−0.953844
$347$	0.679422	0.0364733	0.0182366	−	0.999834i	$-0.494195\pi$
$347$	0.679422	0.0364733	0.0182366	+	0.999834i	$0.494195\pi$
$348$	−1.95980	−0.105056
$349$	−1.62414	−0.0869384	−0.0434692	−	0.999055i	$-0.513841\pi$
$349$	−1.62414	−0.0869384	−0.0434692	+	0.999055i	$0.513841\pi$
$350$	0	0
$351$	0.707884	0.0377840
$352$	38.1436	2.03306
$353$	15.4623	0.822976	0.411488	−	0.911415i	$-0.365009\pi$
$353$	15.4623	0.822976	0.411488	+	0.911415i	$0.365009\pi$
$354$	50.9847	2.70980
$355$	0	0
$356$	−33.5838	−1.77994
$357$	36.9599	1.95613
$358$	18.2328	0.963635
$359$	−6.84097	−0.361052	−0.180526	−	0.983570i	$-0.557780\pi$
$359$	−6.84097	−0.361052	−0.180526	+	0.983570i	$0.557780\pi$
$360$	0	0
$361$	36.1553	1.90291
$362$	−21.7287	−1.14203
$363$	24.1084	1.26536
$364$	−1.38238	−0.0724563
$365$	0	0
$366$	−26.3765	−1.37872
$367$	−25.6144	−1.33706	−0.668531	−	0.743684i	$-0.733077\pi$
$367$	−25.6144	−1.33706	−0.668531	+	0.743684i	$0.733077\pi$
$368$	1.86781	0.0973666
$369$	−2.24247	−0.116738
$370$	0	0
$371$	15.8436	0.822558
$372$	44.5920	2.31199
$373$	−17.4408	−0.903053	−0.451526	−	0.892258i	$-0.649120\pi$
$373$	−17.4408	−0.903053	−0.451526	+	0.892258i	$0.649120\pi$
$374$	−69.2225	−3.57941
$375$	0	0
$376$	−1.83223	−0.0944902
$377$	0.100489	0.00517546
$378$	13.8007	0.709833
$379$	16.6567	0.855596	0.427798	−	0.903874i	$-0.359289\pi$
$379$	16.6567	0.855596	0.427798	+	0.903874i	$0.359289\pi$
$380$	0	0
$381$	−24.6552	−1.26312
$382$	34.9437	1.78787
$383$	−32.7982	−1.67591	−0.837954	−	0.545741i	$-0.816248\pi$
$383$	−32.7982	−1.67591	−0.837954	+	0.545741i	$0.816248\pi$
$384$	9.91568	0.506007
$385$	0	0
$386$	39.9808	2.03497
$387$	−2.16469	−0.110037
$388$	−27.4824	−1.39521
$389$	−27.5650	−1.39760	−0.698800	−	0.715317i	$-0.746282\pi$
$389$	−27.5650	−1.39760	−0.698800	+	0.715317i	$0.746282\pi$
$390$	0	0
$391$	−3.95108	−0.199815
$392$	0.739028	0.0373265
$393$	26.9149	1.35768
$394$	−32.8602	−1.65547
$395$	0	0
$396$	18.3728	0.923270
$397$	7.75624	0.389274	0.194637	−	0.980875i	$-0.437647\pi$
$397$	7.75624	0.389274	0.194637	+	0.980875i	$0.437647\pi$
$398$	46.5640	2.33404
$399$	38.5614	1.93049
$400$	0	0
$401$	−30.5064	−1.52341	−0.761707	−	0.647921i	$-0.775639\pi$
$401$	−30.5064	−1.52341	−0.761707	+	0.647921i	$0.775639\pi$
$402$	19.9959	0.997304
$403$	−2.28647	−0.113897
$404$	−24.3553	−1.21172
$405$	0	0
$406$	1.95911	0.0972292
$407$	11.3669	0.563437
$408$	−8.90922	−0.441072
$409$	−30.2831	−1.49740	−0.748702	−	0.662906i	$-0.769323\pi$
$409$	−30.2831	−1.49740	−0.748702	+	0.662906i	$0.769323\pi$
$410$	0	0
$411$	−39.1744	−1.93233
$412$	11.7268	0.577737
$413$	−27.1431	−1.33563
$414$	1.96913	0.0967772
$415$	0	0
$416$	−2.05840	−0.100922
$417$	−20.1230	−0.985426
$418$	−72.2220	−3.53249
$419$	−32.6700	−1.59603	−0.798016	−	0.602636i	$-0.794117\pi$
$419$	−32.6700	−1.59603	−0.798016	+	0.602636i	$0.794117\pi$
$420$	0	0
$421$	−29.5463	−1.44000	−0.719998	−	0.693976i	$-0.755857\pi$
$421$	−29.5463	−1.44000	−0.719998	+	0.693976i	$0.755857\pi$
$422$	−24.2411	−1.18004
$423$	5.45168	0.265070
$424$	−3.81911	−0.185472
$425$	0	0
$426$	19.0693	0.923912
$427$	14.0423	0.679552
$428$	27.2616	1.31774
$429$	−2.58998	−0.125045
$430$	0	0
$431$	13.0444	0.628327	0.314163	−	0.949369i	$-0.398276\pi$
$431$	13.0444	0.628327	0.314163	+	0.949369i	$0.398276\pi$
$432$	9.38903	0.451730
$433$	−25.3737	−1.21938	−0.609691	−	0.792639i	$-0.708707\pi$
$433$	−25.3737	−1.21938	−0.609691	+	0.792639i	$0.708707\pi$
$434$	−44.5765	−2.13974
$435$	0	0
$436$	−21.3537	−1.02266
$437$	−4.12228	−0.197196
$438$	−11.0856	−0.529692
$439$	−8.31550	−0.396877	−0.198439	−	0.980113i	$-0.563587\pi$
$439$	−8.31550	−0.396877	−0.198439	+	0.980113i	$0.563587\pi$
$440$	0	0
$441$	−2.19892	−0.104711
$442$	3.73556	0.177683
$443$	−36.3117	−1.72522	−0.862610	−	0.505869i	$-0.831172\pi$
$443$	−36.3117	−1.72522	−0.862610	+	0.505869i	$0.831172\pi$
$444$	11.9631	0.567742
$445$	0	0
$446$	−5.13203	−0.243009
$447$	−31.6022	−1.49473
$448$	−24.0372	−1.13565
$449$	−1.60114	−0.0755625	−0.0377812	−	0.999286i	$-0.512029\pi$
$449$	−1.60114	−0.0755625	−0.0377812	+	0.999286i	$0.512029\pi$
$450$	0	0
$451$	−6.14715	−0.289458
$452$	24.9452	1.17333
$453$	−5.25525	−0.246913
$454$	−47.0729	−2.20924
$455$	0	0
$456$	−9.29527	−0.435291
$457$	−15.9244	−0.744914	−0.372457	−	0.928050i	$-0.621485\pi$
$457$	−15.9244	−0.744914	−0.372457	+	0.928050i	$0.621485\pi$
$458$	6.07801	0.284007
$459$	−19.8611	−0.927036
$460$	0	0
$461$	36.2943	1.69039	0.845196	−	0.534456i	$-0.179483\pi$
$461$	36.2943	1.69039	0.845196	+	0.534456i	$0.179483\pi$
$462$	−50.4935	−2.34917
$463$	−21.1492	−0.982888	−0.491444	−	0.870909i	$-0.663531\pi$
$463$	−21.1492	−0.982888	−0.491444	+	0.870909i	$0.663531\pi$
$464$	1.33284	0.0618756
$465$	0	0
$466$	−9.12364	−0.422644
$467$	19.0450	0.881299	0.440649	−	0.897679i	$-0.354748\pi$
$467$	19.0450	0.881299	0.440649	+	0.897679i	$0.354748\pi$
$468$	−0.991482	−0.0458313
$469$	−10.6454	−0.491557
$470$	0	0
$471$	−33.3088	−1.53479
$472$	6.54287	0.301160
$473$	−5.93394	−0.272843
$474$	13.2809	0.610012
$475$	0	0
$476$	38.7853	1.77772
$477$	11.3635	0.520299
$478$	38.9167	1.78001
$479$	40.6363	1.85672	0.928360	−	0.371683i	$-0.121219\pi$
$479$	40.6363	1.85672	0.928360	+	0.371683i	$0.121219\pi$
$480$	0	0
$481$	−0.613410	−0.0279691
$482$	2.06849	0.0942172
$483$	−2.88207	−0.131139
$484$	25.2991	1.14996
$485$	0	0
$486$	33.0079	1.49727
$487$	35.0271	1.58723	0.793615	−	0.608420i	$-0.208197\pi$
$487$	35.0271	1.58723	0.793615	+	0.608420i	$0.208197\pi$
$488$	−3.38490	−0.153227
$489$	−6.50779	−0.294293
$490$	0	0
$491$	29.1537	1.31569	0.657844	−	0.753154i	$-0.271468\pi$
$491$	29.1537	1.31569	0.657844	+	0.753154i	$0.271468\pi$
$492$	−6.46955	−0.291670
$493$	−2.81943	−0.126981
$494$	3.89743	0.175354
$495$	0	0
$496$	−30.3267	−1.36171
$497$	−10.1521	−0.455384
$498$	−32.2939	−1.44712
$499$	−18.3419	−0.821094	−0.410547	−	0.911839i	$-0.634662\pi$
$499$	−18.3419	−0.821094	−0.410547	+	0.911839i	$0.634662\pi$
$500$	0	0
$501$	15.0556	0.672636
$502$	49.2319	2.19733
$503$	17.8382	0.795365	0.397682	−	0.917523i	$-0.369815\pi$
$503$	17.8382	0.795365	0.397682	+	0.917523i	$0.369815\pi$
$504$	−2.36384	−0.105294
$505$	0	0
$506$	5.39784	0.239963
$507$	−28.0886	−1.24746
$508$	−25.8729	−1.14792
$509$	38.3247	1.69871	0.849356	−	0.527820i	$-0.176991\pi$
$509$	38.3247	1.69871	0.849356	+	0.527820i	$0.176991\pi$
$510$	0	0
$511$	5.90175	0.261078
$512$	−31.1809	−1.37801
$513$	−20.7217	−0.914884
$514$	6.08798	0.268529
$515$	0	0
$516$	−6.24515	−0.274927
$517$	14.9444	0.657252
$518$	−11.9589	−0.525444
$519$	18.6254	0.817563
$520$	0	0
$521$	27.5109	1.20527	0.602637	−	0.798015i	$-0.294117\pi$
$521$	27.5109	1.20527	0.602637	+	0.798015i	$0.294117\pi$
$522$	1.40514	0.0615011
$523$	−18.3097	−0.800627	−0.400313	−	0.916378i	$-0.631099\pi$
$523$	−18.3097	−0.800627	−0.400313	+	0.916378i	$0.631099\pi$
$524$	28.2442	1.23385
$525$	0	0
$526$	−32.3692	−1.41136
$527$	64.1515	2.79448
$528$	−34.3522	−1.49499
$529$	−22.6919	−0.986604
$530$	0	0
$531$	−19.4679	−0.844833
$532$	40.4659	1.75442
$533$	0.331728	0.0143688
$534$	66.1985	2.86469
$535$	0	0
$536$	2.56608	0.110838
$537$	−19.1401	−0.825954
$538$	−42.7749	−1.84416
$539$	−6.02778	−0.259635
$540$	0	0
$541$	−13.1780	−0.566567	−0.283283	−	0.959036i	$-0.591424\pi$
$541$	−13.1780	−0.566567	−0.283283	+	0.959036i	$0.591424\pi$
$542$	−13.6831	−0.587741
$543$	22.8099	0.978865
$544$	57.7526	2.47612
$545$	0	0
$546$	2.72486	0.116613
$547$	17.1910	0.735035	0.367517	−	0.930017i	$-0.380208\pi$
$547$	17.1910	0.735035	0.367517	+	0.930017i	$0.380208\pi$
$548$	−41.1091	−1.75610
$549$	10.0715	0.429842
$550$	0	0
$551$	−2.94159	−0.125316
$552$	0.694725	0.0295694
$553$	−7.07046	−0.300667
$554$	−0.248631	−0.0105633
$555$	0	0
$556$	−21.1168	−0.895552
$557$	15.4599	0.655055	0.327528	−	0.944842i	$-0.393785\pi$
$557$	15.4599	0.655055	0.327528	+	0.944842i	$0.393785\pi$
$558$	−31.9716	−1.35346
$559$	0.320222	0.0135440
$560$	0	0
$561$	72.6668	3.06800
$562$	−21.5473	−0.908918
$563$	−1.45982	−0.0615243	−0.0307621	−	0.999527i	$-0.509793\pi$
$563$	−1.45982	−0.0615243	−0.0307621	+	0.999527i	$0.509793\pi$
$564$	15.7281	0.662274
$565$	0	0
$566$	20.9698	0.881425
$567$	−26.7905	−1.12509
$568$	2.44717	0.102681
$569$	41.2926	1.73108	0.865538	−	0.500843i	$-0.166977\pi$
$569$	41.2926	1.73108	0.865538	+	0.500843i	$0.166977\pi$
$570$	0	0
$571$	12.1353	0.507848	0.253924	−	0.967224i	$-0.418279\pi$
$571$	12.1353	0.507848	0.253924	+	0.967224i	$0.418279\pi$
$572$	−2.71789	−0.113641
$573$	−36.6824	−1.53243
$574$	6.46729	0.269940
$575$	0	0
$576$	−17.2402	−0.718341
$577$	9.01743	0.375400	0.187700	−	0.982226i	$-0.439897\pi$
$577$	9.01743	0.375400	0.187700	+	0.982226i	$0.439897\pi$
$578$	−69.6442	−2.89682
$579$	−41.9702	−1.74422
$580$	0	0
$581$	17.1925	0.713267
$582$	54.1717	2.24549
$583$	31.1501	1.29010
$584$	−1.42262	−0.0588685
$585$	0	0
$586$	6.10610	0.252241
$587$	−16.5160	−0.681687	−0.340843	−	0.940120i	$-0.610713\pi$
$587$	−16.5160	−0.681687	−0.340843	+	0.940120i	$0.610713\pi$
$588$	−6.34391	−0.261619
$589$	66.9312	2.75785
$590$	0	0
$591$	34.4952	1.41894
$592$	−8.13598	−0.334387
$593$	35.7781	1.46923	0.734615	−	0.678484i	$-0.237363\pi$
$593$	35.7781	1.46923	0.734615	+	0.678484i	$0.237363\pi$
$594$	27.1336	1.11330
$595$	0	0
$596$	−33.1630	−1.35841
$597$	−48.8809	−2.00056
$598$	−0.291292	−0.0119118
$599$	6.94282	0.283676	0.141838	−	0.989890i	$-0.454699\pi$
$599$	6.94282	0.283676	0.141838	+	0.989890i	$0.454699\pi$
$600$	0	0
$601$	−4.39599	−0.179316	−0.0896581	−	0.995973i	$-0.528577\pi$
$601$	−4.39599	−0.179316	−0.0896581	+	0.995973i	$0.528577\pi$
$602$	6.24297	0.254445
$603$	−7.63518	−0.310928
$604$	−5.51480	−0.224394
$605$	0	0
$606$	48.0077	1.95018
$607$	−14.8186	−0.601469	−0.300734	−	0.953708i	$-0.597232\pi$
$607$	−14.8186	−0.601469	−0.300734	+	0.953708i	$0.597232\pi$
$608$	60.2551	2.44367
$609$	−2.05660	−0.0833375
$610$	0	0
$611$	−0.806466	−0.0326261
$612$	27.8180	1.12448
$613$	22.8415	0.922558	0.461279	−	0.887255i	$-0.347391\pi$
$613$	22.8415	0.922558	0.461279	+	0.887255i	$0.347391\pi$
$614$	6.45913	0.260669
$615$	0	0
$616$	−6.47984	−0.261080
$617$	−35.8540	−1.44343	−0.721714	−	0.692191i	$-0.756645\pi$
$617$	−35.8540	−1.44343	−0.721714	+	0.692191i	$0.756645\pi$
$618$	−23.1151	−0.929827
$619$	14.5058	0.583039	0.291520	−	0.956565i	$-0.405839\pi$
$619$	14.5058	0.583039	0.291520	+	0.956565i	$0.405839\pi$
$620$	0	0
$621$	1.54873	0.0621484
$622$	−6.79426	−0.272425
$623$	−35.2426	−1.41197
$624$	1.85380	0.0742114
$625$	0	0
$626$	16.9719	0.678335
$627$	75.8156	3.02778
$628$	−34.9539	−1.39481
$629$	17.2104	0.686225
$630$	0	0
$631$	25.1454	1.00102	0.500512	−	0.865729i	$-0.333145\pi$
$631$	25.1454	1.00102	0.500512	+	0.865729i	$0.333145\pi$
$632$	1.70434	0.0677950
$633$	25.4473	1.01144
$634$	−56.0391	−2.22560
$635$	0	0
$636$	32.7838	1.29996
$637$	0.325286	0.0128883
$638$	3.85181	0.152495
$639$	−7.28139	−0.288047
$640$	0	0
$641$	29.0163	1.14608	0.573038	−	0.819529i	$-0.305765\pi$
$641$	29.0163	1.14608	0.573038	+	0.819529i	$0.305765\pi$
$642$	−53.7364	−2.12081
$643$	−9.88413	−0.389792	−0.194896	−	0.980824i	$-0.562437\pi$
$643$	−9.88413	−0.389792	−0.194896	+	0.980824i	$0.562437\pi$
$644$	−3.02441	−0.119178
$645$	0	0
$646$	−109.350	−4.30232
$647$	8.00749	0.314807	0.157404	−	0.987534i	$-0.449688\pi$
$647$	8.00749	0.314807	0.157404	+	0.987534i	$0.449688\pi$
$648$	6.45787	0.253689
$649$	−53.3660	−2.09480
$650$	0	0
$651$	46.7945	1.83402
$652$	−6.82920	−0.267452
$653$	34.5838	1.35337	0.676684	−	0.736274i	$-0.263416\pi$
$653$	34.5838	1.35337	0.676684	+	0.736274i	$0.263416\pi$
$654$	42.0911	1.64589
$655$	0	0
$656$	4.39988	0.171787
$657$	4.23291	0.165142
$658$	−15.7227	−0.612933
$659$	−17.2363	−0.671430	−0.335715	−	0.941964i	$-0.608978\pi$
$659$	−17.2363	−0.671430	−0.335715	+	0.941964i	$0.608978\pi$
$660$	0	0
$661$	2.76112	0.107395	0.0536975	−	0.998557i	$-0.482899\pi$
$661$	2.76112	0.107395	0.0536975	+	0.998557i	$0.482899\pi$
$662$	67.2167	2.61245
$663$	−3.92143	−0.152296
$664$	−4.14428	−0.160829
$665$	0	0
$666$	−8.57728	−0.332363
$667$	0.219854	0.00851277
$668$	15.7992	0.611289
$669$	5.38739	0.208288
$670$	0	0
$671$	27.6085	1.06581
$672$	42.1269	1.62508
$673$	20.9147	0.806202	0.403101	−	0.915155i	$-0.367932\pi$
$673$	20.9147	0.806202	0.403101	+	0.915155i	$0.367932\pi$
$674$	−65.5272	−2.52401
$675$	0	0
$676$	−29.4759	−1.13369
$677$	−6.28203	−0.241438	−0.120719	−	0.992687i	$-0.538520\pi$
$677$	−6.28203	−0.241438	−0.120719	+	0.992687i	$0.538520\pi$
$678$	−49.1706	−1.88838
$679$	−28.8398	−1.10677
$680$	0	0
$681$	49.4151	1.89359
$682$	−87.6418	−3.35598
$683$	−12.7629	−0.488358	−0.244179	−	0.969730i	$-0.578518\pi$
$683$	−12.7629	−0.488358	−0.244179	+	0.969730i	$0.578518\pi$
$684$	29.0234	1.10974
$685$	0	0
$686$	40.9650	1.56405
$687$	−6.38044	−0.243429
$688$	4.24727	0.161926
$689$	−1.68100	−0.0640410
$690$	0	0
$691$	−25.7879	−0.981016	−0.490508	−	0.871437i	$-0.663189\pi$
$691$	−25.7879	−0.981016	−0.490508	+	0.871437i	$0.663189\pi$
$692$	19.5452	0.742998
$693$	19.2803	0.732398
$694$	−1.40538	−0.0533474
$695$	0	0
$696$	0.495744	0.0187911
$697$	−9.30729	−0.352539
$698$	3.35953	0.127160
$699$	9.57761	0.362259
$700$	0	0
$701$	−1.87714	−0.0708986	−0.0354493	−	0.999371i	$-0.511286\pi$
$701$	−1.87714	−0.0708986	−0.0354493	+	0.999371i	$0.511286\pi$
$702$	−1.46425	−0.0552646
$703$	17.9562	0.677230
$704$	−47.2594	−1.78116
$705$	0	0
$706$	−31.9837	−1.20372
$707$	−25.5582	−0.961216
$708$	−56.1649	−2.11081
$709$	29.9700	1.12555	0.562773	−	0.826611i	$-0.309734\pi$
$709$	29.9700	1.12555	0.562773	+	0.826611i	$0.309734\pi$
$710$	0	0
$711$	−5.07115	−0.190183
$712$	8.49526	0.318373
$713$	−5.00241	−0.187342
$714$	−76.4513	−2.86112
$715$	0	0
$716$	−20.0854	−0.750625
$717$	−40.8531	−1.52569
$718$	14.1505	0.528091
$719$	31.6231	1.17934	0.589670	−	0.807644i	$-0.299258\pi$
$719$	31.6231	1.17934	0.589670	+	0.807644i	$0.299258\pi$
$720$	0	0
$721$	12.3060	0.458299
$722$	−74.7869	−2.78328
$723$	−2.17142	−0.0807558
$724$	23.9364	0.889589
$725$	0	0
$726$	−49.8681	−1.85078
$727$	5.41532	0.200843	0.100422	−	0.994945i	$-0.467981\pi$
$727$	5.41532	0.200843	0.100422	+	0.994945i	$0.467981\pi$
$728$	0.349682	0.0129601
$729$	−1.03905	−0.0384833
$730$	0	0
$731$	−8.98446	−0.332302
$732$	29.0564	1.07396
$733$	6.69162	0.247160	0.123580	−	0.992335i	$-0.460562\pi$
$733$	6.69162	0.247160	0.123580	+	0.992335i	$0.460562\pi$
$734$	52.9832	1.95565
$735$	0	0
$736$	−4.50344	−0.165999
$737$	−20.9298	−0.770961
$738$	4.63853	0.170747
$739$	−39.3132	−1.44616	−0.723080	−	0.690765i	$-0.757274\pi$
$739$	−39.3132	−1.44616	−0.723080	+	0.690765i	$0.757274\pi$
$740$	0	0
$741$	−4.09135	−0.150300
$742$	−32.7723	−1.20311
$743$	34.1723	1.25366	0.626829	−	0.779157i	$-0.284352\pi$
$743$	34.1723	1.25366	0.626829	+	0.779157i	$0.284352\pi$
$744$	−11.2799	−0.413540
$745$	0	0
$746$	36.0762	1.32084
$747$	12.3310	0.451168
$748$	76.2558	2.78819
$749$	28.6081	1.04532
$750$	0	0
$751$	−23.6027	−0.861276	−0.430638	−	0.902525i	$-0.641711\pi$
$751$	−23.6027	−0.861276	−0.430638	+	0.902525i	$0.641711\pi$
$752$	−10.6966	−0.390064
$753$	−51.6816	−1.88338
$754$	−0.207861	−0.00756986
$755$	0	0
$756$	−15.2029	−0.552925
$757$	43.9304	1.59668	0.798339	−	0.602209i	$-0.205712\pi$
$757$	43.9304	1.59668	0.798339	+	0.602209i	$0.205712\pi$
$758$	−34.4542	−1.25143
$759$	−5.66643	−0.205678
$760$	0	0
$761$	47.6870	1.72865	0.864325	−	0.502933i	$-0.167746\pi$
$761$	47.6870	1.72865	0.864325	+	0.502933i	$0.167746\pi$
$762$	50.9991	1.84750
$763$	−22.4084	−0.811238
$764$	−38.4941	−1.39267
$765$	0	0
$766$	67.8427	2.45126
$767$	2.87987	0.103986
$768$	23.1451	0.835175
$769$	49.5554	1.78701	0.893507	−	0.449049i	$-0.148237\pi$
$769$	49.5554	1.78701	0.893507	+	0.449049i	$0.148237\pi$
$770$	0	0
$771$	−6.39091	−0.230163
$772$	−44.0430	−1.58514
$773$	−42.2142	−1.51834	−0.759170	−	0.650892i	$-0.774395\pi$
$773$	−42.2142	−1.51834	−0.759170	+	0.650892i	$0.774395\pi$
$774$	4.47765	0.160946
$775$	0	0
$776$	6.95186	0.249557
$777$	12.5539	0.450370
$778$	57.0179	2.04419
$779$	−9.71059	−0.347918
$780$	0	0
$781$	−19.9600	−0.714226
$782$	8.17278	0.292258
$783$	1.10515	0.0394948
$784$	4.31444	0.154087
$785$	0	0
$786$	−55.6733	−1.98580
$787$	−14.5248	−0.517752	−0.258876	−	0.965911i	$-0.583352\pi$
$787$	−14.5248	−0.517752	−0.258876	+	0.965911i	$0.583352\pi$
$788$	36.1989	1.28953
$789$	33.9798	1.20971
$790$	0	0
$791$	26.1773	0.930759
$792$	−4.64753	−0.165143
$793$	−1.48988	−0.0529071
$794$	−16.0437	−0.569370
$795$	0	0
$796$	−51.2950	−1.81810
$797$	37.5458	1.32994	0.664970	−	0.746870i	$-0.268444\pi$
$797$	37.5458	1.32994	0.664970	+	0.746870i	$0.268444\pi$
$798$	−79.7640	−2.82362
$799$	22.6270	0.800485
$800$	0	0
$801$	−25.2770	−0.893121
$802$	63.1022	2.22821
$803$	11.6034	0.409476
$804$	−22.0275	−0.776851
$805$	0	0
$806$	4.72955	0.166591
$807$	44.9033	1.58067
$808$	6.16083	0.216737
$809$	3.94799	0.138804	0.0694019	−	0.997589i	$-0.477891\pi$
$809$	3.94799	0.138804	0.0694019	+	0.997589i	$0.477891\pi$
$810$	0	0
$811$	−34.9378	−1.22683	−0.613416	−	0.789760i	$-0.710205\pi$
$811$	−34.9378	−1.22683	−0.613416	+	0.789760i	$0.710205\pi$
$812$	−2.15817	−0.0757368
$813$	14.3640	0.503767
$814$	−23.5124	−0.824108
$815$	0	0
$816$	−52.0120	−1.82078
$817$	−9.37377	−0.327947
$818$	62.6404	2.19017
$819$	−1.04045	−0.0363564
$820$	0	0
$821$	−36.2650	−1.26566	−0.632829	−	0.774291i	$-0.718107\pi$
$821$	−36.2650	−1.26566	−0.632829	+	0.774291i	$0.718107\pi$
$822$	81.0319	2.82631
$823$	15.9422	0.555711	0.277856	−	0.960623i	$-0.410376\pi$
$823$	15.9422	0.555711	0.277856	+	0.960623i	$0.410376\pi$
$824$	−2.96637	−0.103338
$825$	0	0
$826$	56.1453	1.95355
$827$	12.0584	0.419312	0.209656	−	0.977775i	$-0.432766\pi$
$827$	12.0584	0.419312	0.209656	+	0.977775i	$0.432766\pi$
$828$	−2.16920	−0.0753848
$829$	22.2859	0.774023	0.387011	−	0.922075i	$-0.373507\pi$
$829$	22.2859	0.774023	0.387011	+	0.922075i	$0.373507\pi$
$830$	0	0
$831$	0.261003	0.00905409
$832$	2.55034	0.0884170
$833$	−9.12655	−0.316216
$834$	41.6242	1.44133
$835$	0	0
$836$	79.5600	2.75164
$837$	−25.1459	−0.869168
$838$	67.5776	2.33443
$839$	23.6447	0.816304	0.408152	−	0.912914i	$-0.366173\pi$
$839$	23.6447	0.816304	0.408152	+	0.912914i	$0.366173\pi$
$840$	0	0
$841$	−28.8431	−0.994590
$842$	61.1162	2.10620
$843$	22.6194	0.779055
$844$	26.7041	0.919192
$845$	0	0
$846$	−11.2768	−0.387703
$847$	26.5487	0.912223
$848$	−22.2960	−0.765647
$849$	−22.0132	−0.755490
$850$	0	0
$851$	−1.34204	−0.0460045
$852$	−21.0069	−0.719683
$853$	41.7485	1.42944	0.714721	−	0.699409i	$-0.246554\pi$
$853$	41.7485	1.42944	0.714721	+	0.699409i	$0.246554\pi$
$854$	−29.0463	−0.993944
$855$	0	0
$856$	−6.89600	−0.235701
$857$	−48.4136	−1.65378	−0.826888	−	0.562366i	$-0.809891\pi$
$857$	−48.4136	−1.65378	−0.826888	+	0.562366i	$0.809891\pi$
$858$	5.35734	0.182897
$859$	42.7773	1.45954	0.729771	−	0.683691i	$-0.239626\pi$
$859$	42.7773	1.45954	0.729771	+	0.683691i	$0.239626\pi$
$860$	0	0
$861$	−6.78909	−0.231372
$862$	−26.9822	−0.919019
$863$	−28.5676	−0.972451	−0.486226	−	0.873833i	$-0.661627\pi$
$863$	−28.5676	−0.972451	−0.486226	+	0.873833i	$0.661627\pi$
$864$	−22.6377	−0.770149
$865$	0	0
$866$	52.4853	1.78352
$867$	73.1095	2.48293
$868$	49.1056	1.66675
$869$	−13.9012	−0.471567
$870$	0	0
$871$	1.12947	0.0382706
$872$	5.40156	0.182920
$873$	−20.6848	−0.700073
$874$	8.52691	0.288427
$875$	0	0
$876$	12.2120	0.412604
$877$	−42.3416	−1.42978	−0.714888	−	0.699239i	$-0.753522\pi$
$877$	−42.3416	−1.42978	−0.714888	+	0.699239i	$0.753522\pi$
$878$	17.2006	0.580490
$879$	−6.40992	−0.216201
$880$	0	0
$881$	34.2562	1.15412	0.577061	−	0.816701i	$-0.304199\pi$
$881$	34.2562	1.15412	0.577061	+	0.816701i	$0.304199\pi$
$882$	4.54846	0.153155
$883$	46.5867	1.56777	0.783883	−	0.620908i	$-0.213236\pi$
$883$	46.5867	1.56777	0.783883	+	0.620908i	$0.213236\pi$
$884$	−4.11511	−0.138406
$885$	0	0
$886$	75.1105	2.52339
$887$	−33.8795	−1.13756	−0.568781	−	0.822489i	$-0.692585\pi$
$887$	−33.8795	−1.13756	−0.568781	+	0.822489i	$0.692585\pi$
$888$	−3.02614	−0.101551
$889$	−27.1508	−0.910608
$890$	0	0
$891$	−52.6727	−1.76460
$892$	5.65346	0.189292
$893$	23.6074	0.789993
$894$	65.3689	2.18626
$895$	0	0
$896$	10.9193	0.364789
$897$	0.305786	0.0102099
$898$	3.31194	0.110521
$899$	−3.56964	−0.119054
$900$	0	0
$901$	47.1638	1.57125
$902$	12.7153	0.423374
$903$	−6.55361	−0.218090
$904$	−6.31007	−0.209870
$905$	0	0
$906$	10.8704	0.361146
$907$	19.7428	0.655550	0.327775	−	0.944756i	$-0.393701\pi$
$907$	19.7428	0.655550	0.327775	+	0.944756i	$0.393701\pi$
$908$	51.8557	1.72089
$909$	−18.3311	−0.608005
$910$	0	0
$911$	−59.5956	−1.97449	−0.987245	−	0.159210i	$-0.949105\pi$
$911$	−59.5956	−1.97449	−0.987245	+	0.159210i	$0.949105\pi$
$912$	−54.2657	−1.79692
$913$	33.8022	1.11869
$914$	32.9396	1.08954
$915$	0	0
$916$	−6.69556	−0.221227
$917$	29.6392	0.978773
$918$	41.0825	1.35592
$919$	53.2098	1.75523	0.877615	−	0.479367i	$-0.159134\pi$
$919$	53.2098	1.75523	0.877615	+	0.479367i	$0.159134\pi$
$920$	0	0
$921$	−6.78052	−0.223426
$922$	−75.0744	−2.47244
$923$	1.07713	0.0354543
$924$	55.6238	1.82989
$925$	0	0
$926$	43.7470	1.43762
$927$	8.82622	0.289891
$928$	−3.21358	−0.105491
$929$	14.3392	0.470454	0.235227	−	0.971940i	$-0.424417\pi$
$929$	14.3392	0.470454	0.235227	+	0.971940i	$0.424417\pi$
$930$	0	0
$931$	−9.52201	−0.312071
$932$	10.0506	0.329219
$933$	7.13232	0.233502
$934$	−39.3945	−1.28903
$935$	0	0
$936$	0.250802	0.00819773
$937$	−21.6386	−0.706903	−0.353452	−	0.935453i	$-0.614992\pi$
$937$	−21.6386	−0.706903	−0.353452	+	0.935453i	$0.614992\pi$
$938$	22.0199	0.718974
$939$	−17.8164	−0.581417
$940$	0	0
$941$	24.6771	0.804450	0.402225	−	0.915541i	$-0.368237\pi$
$941$	24.6771	0.804450	0.402225	+	0.915541i	$0.368237\pi$
$942$	68.8990	2.24485
$943$	0.725766	0.0236342
$944$	38.1973	1.24322
$945$	0	0
$946$	12.2743	0.399072
$947$	13.1735	0.428081	0.214041	−	0.976825i	$-0.431338\pi$
$947$	13.1735	0.428081	0.214041	+	0.976825i	$0.431338\pi$
$948$	−14.6303	−0.475170
$949$	−0.626174	−0.0203265
$950$	0	0
$951$	58.8275	1.90761
$952$	−9.81101	−0.317977
$953$	−0.964370	−0.0312390	−0.0156195	−	0.999878i	$-0.504972\pi$
$953$	−0.964370	−0.0312390	−0.0156195	+	0.999878i	$0.504972\pi$
$954$	−23.5053	−0.761012
$955$	0	0
$956$	−42.8708	−1.38654
$957$	−4.04347	−0.130707
$958$	−84.0558	−2.71572
$959$	−43.1396	−1.39305
$960$	0	0
$961$	50.2214	1.62004
$962$	1.26883	0.0409089
$963$	20.5186	0.661202
$964$	−2.27866	−0.0733906
$965$	0	0
$966$	5.96153	0.191809
$967$	26.4652	0.851063	0.425532	−	0.904944i	$-0.360087\pi$
$967$	26.4652	0.851063	0.425532	+	0.904944i	$0.360087\pi$
$968$	−6.39958	−0.205690
$969$	114.791	3.68762
$970$	0	0
$971$	−17.6868	−0.567596	−0.283798	−	0.958884i	$-0.591595\pi$
$971$	−17.6868	−0.567596	−0.283798	+	0.958884i	$0.591595\pi$
$972$	−36.3616	−1.16630
$973$	−22.1598	−0.710411
$974$	−72.4533	−2.32155
$975$	0	0
$976$	−19.7610	−0.632535
$977$	26.2020	0.838276	0.419138	−	0.907922i	$-0.362332\pi$
$977$	26.2020	0.838276	0.419138	+	0.907922i	$0.362332\pi$
$978$	13.4613	0.430445
$979$	−69.2904	−2.21453
$980$	0	0
$981$	−16.0720	−0.513138
$982$	−60.3042	−1.92438
$983$	50.8568	1.62208	0.811041	−	0.584990i	$-0.198901\pi$
$983$	50.8568	1.62208	0.811041	+	0.584990i	$0.198901\pi$
$984$	1.63652	0.0521702
$985$	0	0
$986$	5.83196	0.185727
$987$	16.5050	0.525359
$988$	−4.29342	−0.136592
$989$	0.700592	0.0222775
$990$	0	0
$991$	−11.7515	−0.373298	−0.186649	−	0.982427i	$-0.559763\pi$
$991$	−11.7515	−0.373298	−0.186649	+	0.982427i	$0.559763\pi$
$992$	73.1199	2.32156
$993$	−70.5613	−2.23919
$994$	20.9995	0.666065
$995$	0	0
$996$	35.5750	1.12724
$997$	7.92973	0.251137	0.125568	−	0.992085i	$-0.459925\pi$
$997$	7.92973	0.251137	0.125568	+	0.992085i	$0.459925\pi$
$998$	37.9400	1.20097
$999$	−6.74608	−0.213437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.6	yes	40
5.4	even	2		6025.2.a.l.1.35	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.35	✓	40	5.4	even	2
6025.2.a.o.1.6	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.06849 q^{2} +2.17142 q^{3} +2.27866 q^{4} -4.49155 q^{6} +2.39120 q^{7} -0.576402 q^{8} +1.71504 q^{9} +O(q^{10})\)	\(q-2.06849 q^{2} +2.17142 q^{3} +2.27866 q^{4} -4.49155 q^{6} +2.39120 q^{7} -0.576402 q^{8} +1.71504 q^{9} +4.70134 q^{11} +4.94791 q^{12} -0.253706 q^{13} -4.94619 q^{14} -3.36503 q^{16} +7.11822 q^{17} -3.54755 q^{18} +7.42666 q^{19} +5.19230 q^{21} -9.72469 q^{22} -0.555066 q^{23} -1.25161 q^{24} +0.524789 q^{26} -2.79017 q^{27} +5.44874 q^{28} -0.396086 q^{29} +9.01229 q^{31} +8.11335 q^{32} +10.2086 q^{33} -14.7240 q^{34} +3.90800 q^{36} +2.41780 q^{37} -15.3620 q^{38} -0.550901 q^{39} -1.30753 q^{41} -10.7402 q^{42} -1.26218 q^{43} +10.7128 q^{44} +1.14815 q^{46} +3.17874 q^{47} -7.30688 q^{48} -1.28214 q^{49} +15.4566 q^{51} -0.578109 q^{52} +6.62578 q^{53} +5.77145 q^{54} -1.37830 q^{56} +16.1264 q^{57} +0.819300 q^{58} -11.3512 q^{59} +5.87246 q^{61} -18.6419 q^{62} +4.10102 q^{63} -10.0523 q^{64} -21.1163 q^{66} -4.45188 q^{67} +16.2200 q^{68} -1.20528 q^{69} -4.24560 q^{71} -0.988554 q^{72} +2.46811 q^{73} -5.00120 q^{74} +16.9228 q^{76} +11.2419 q^{77} +1.13953 q^{78} -2.95686 q^{79} -11.2038 q^{81} +2.70462 q^{82} +7.18991 q^{83} +11.8315 q^{84} +2.61081 q^{86} -0.860067 q^{87} -2.70986 q^{88} -14.7384 q^{89} -0.606663 q^{91} -1.26481 q^{92} +19.5694 q^{93} -6.57520 q^{94} +17.6174 q^{96} -12.0608 q^{97} +2.65209 q^{98} +8.06301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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