LMFDB - Embedded newform 6025.2.a.l.1.20

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:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-0.599456 q^{2} +2.83593 q^{3} -1.64065 q^{4} -1.70002 q^{6} +0.360434 q^{7} +2.18241 q^{8} +5.04251 q^{9} +O(q^{10})$	\(q-0.599456 q^{2} +2.83593 q^{3} -1.64065 q^{4} -1.70002 q^{6} +0.360434 q^{7} +2.18241 q^{8} +5.04251 q^{9} -0.625458 q^{11} -4.65278 q^{12} +4.68460 q^{13} -0.216064 q^{14} +1.97304 q^{16} -8.17080 q^{17} -3.02276 q^{18} -4.27121 q^{19} +1.02216 q^{21} +0.374934 q^{22} -2.92878 q^{23} +6.18917 q^{24} -2.80821 q^{26} +5.79241 q^{27} -0.591346 q^{28} -9.18926 q^{29} -2.11268 q^{31} -5.54758 q^{32} -1.77376 q^{33} +4.89804 q^{34} -8.27300 q^{36} +2.51815 q^{37} +2.56040 q^{38} +13.2852 q^{39} -2.70037 q^{41} -0.612743 q^{42} +1.59843 q^{43} +1.02616 q^{44} +1.75568 q^{46} -5.73910 q^{47} +5.59542 q^{48} -6.87009 q^{49} -23.1718 q^{51} -7.68580 q^{52} +9.95812 q^{53} -3.47230 q^{54} +0.786614 q^{56} -12.1128 q^{57} +5.50856 q^{58} -6.89519 q^{59} +4.62027 q^{61} +1.26646 q^{62} +1.81749 q^{63} -0.620559 q^{64} +1.06329 q^{66} +6.13471 q^{67} +13.4054 q^{68} -8.30582 q^{69} -12.7519 q^{71} +11.0048 q^{72} -0.559355 q^{73} -1.50952 q^{74} +7.00756 q^{76} -0.225436 q^{77} -7.96390 q^{78} +4.10565 q^{79} +1.29937 q^{81} +1.61875 q^{82} -15.1584 q^{83} -1.67702 q^{84} -0.958189 q^{86} -26.0601 q^{87} -1.36501 q^{88} -13.6969 q^{89} +1.68849 q^{91} +4.80511 q^{92} -5.99142 q^{93} +3.44034 q^{94} -15.7325 q^{96} +13.5129 q^{97} +4.11832 q^{98} -3.15388 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.599456	−0.423880	−0.211940	−	0.977283i	$-0.567978\pi$
$2$	−0.599456	−0.423880	−0.211940	+	0.977283i	$0.567978\pi$
$3$	2.83593	1.63733	0.818663	−	0.574274i	$-0.194716\pi$
$3$	2.83593	1.63733	0.818663	+	0.574274i	$0.194716\pi$
$4$	−1.64065	−0.820326
$5$	0	0
$5$	0	0
$6$	−1.70002	−0.694029
$7$	0.360434	0.136231	0.0681155	−	0.997677i	$-0.478301\pi$
$7$	0.360434	0.136231	0.0681155	+	0.997677i	$0.478301\pi$
$8$	2.18241	0.771599
$9$	5.04251	1.68084
$10$	0	0
$11$	−0.625458	−0.188583	−0.0942913	−	0.995545i	$-0.530059\pi$
$11$	−0.625458	−0.188583	−0.0942913	+	0.995545i	$0.530059\pi$
$12$	−4.65278	−1.34314
$13$	4.68460	1.29927	0.649637	−	0.760245i	$-0.274921\pi$
$13$	4.68460	1.29927	0.649637	+	0.760245i	$0.274921\pi$
$14$	−0.216064	−0.0577456
$15$	0	0
$16$	1.97304	0.493261
$17$	−8.17080	−1.98171	−0.990855	−	0.134929i	$-0.956919\pi$
$17$	−8.17080	−1.98171	−0.990855	+	0.134929i	$0.956919\pi$
$18$	−3.02276	−0.712472
$19$	−4.27121	−0.979882	−0.489941	−	0.871756i	$-0.662982\pi$
$19$	−4.27121	−0.979882	−0.489941	+	0.871756i	$0.662982\pi$
$20$	0	0
$21$	1.02216	0.223055
$22$	0.374934	0.0799363
$23$	−2.92878	−0.610693	−0.305347	−	0.952241i	$-0.598772\pi$
$23$	−2.92878	−0.610693	−0.305347	+	0.952241i	$0.598772\pi$
$24$	6.18917	1.26336
$25$	0	0
$26$	−2.80821	−0.550736
$27$	5.79241	1.11475
$28$	−0.591346	−0.111754
$29$	−9.18926	−1.70640	−0.853201	−	0.521582i	$-0.825342\pi$
$29$	−9.18926	−1.70640	−0.853201	+	0.521582i	$0.825342\pi$
$30$	0	0
$31$	−2.11268	−0.379449	−0.189724	−	0.981837i	$-0.560759\pi$
$31$	−2.11268	−0.379449	−0.189724	+	0.981837i	$0.560759\pi$
$32$	−5.54758	−0.980682
$33$	−1.77376	−0.308771
$34$	4.89804	0.840007
$35$	0	0
$36$	−8.27300	−1.37883
$37$	2.51815	0.413981	0.206990	−	0.978343i	$-0.433633\pi$
$37$	2.51815	0.413981	0.206990	+	0.978343i	$0.433633\pi$
$38$	2.56040	0.415352
$39$	13.2852	2.12733
$40$	0	0
$41$	−2.70037	−0.421727	−0.210864	−	0.977516i	$-0.567628\pi$
$41$	−2.70037	−0.421727	−0.210864	+	0.977516i	$0.567628\pi$
$42$	−0.612743	−0.0945483
$43$	1.59843	0.243758	0.121879	−	0.992545i	$-0.461108\pi$
$43$	1.59843	0.243758	0.121879	+	0.992545i	$0.461108\pi$
$44$	1.02616	0.154699
$45$	0	0
$46$	1.75568	0.258860
$47$	−5.73910	−0.837134	−0.418567	−	0.908186i	$-0.637468\pi$
$47$	−5.73910	−0.837134	−0.418567	+	0.908186i	$0.637468\pi$
$48$	5.59542	0.807629
$49$	−6.87009	−0.981441
$50$	0	0
$51$	−23.1718	−3.24471
$52$	−7.68580	−1.06583
$53$	9.95812	1.36785	0.683927	−	0.729551i	$-0.260271\pi$
$53$	9.95812	1.36785	0.683927	+	0.729551i	$0.260271\pi$
$54$	−3.47230	−0.472520
$55$	0	0
$56$	0.786614	0.105116
$57$	−12.1128	−1.60439
$58$	5.50856	0.723309
$59$	−6.89519	−0.897678	−0.448839	−	0.893613i	$-0.648162\pi$
$59$	−6.89519	−0.897678	−0.448839	+	0.893613i	$0.648162\pi$
$60$	0	0
$61$	4.62027	0.591564	0.295782	−	0.955255i	$-0.404420\pi$
$61$	4.62027	0.591564	0.295782	+	0.955255i	$0.404420\pi$
$62$	1.26646	0.160841
$63$	1.81749	0.228982
$64$	−0.620559	−0.0775699
$65$	0	0
$66$	1.06329	0.130882
$67$	6.13471	0.749474	0.374737	−	0.927131i	$-0.377733\pi$
$67$	6.13471	0.749474	0.374737	+	0.927131i	$0.377733\pi$
$68$	13.4054	1.62565
$69$	−8.30582	−0.999904
$70$	0	0
$71$	−12.7519	−1.51337	−0.756687	−	0.653778i	$-0.773183\pi$
$71$	−12.7519	−1.51337	−0.756687	+	0.653778i	$0.773183\pi$
$72$	11.0048	1.29693
$73$	−0.559355	−0.0654675	−0.0327338	−	0.999464i	$-0.510421\pi$
$73$	−0.559355	−0.0654675	−0.0327338	+	0.999464i	$0.510421\pi$
$74$	−1.50952	−0.175478
$75$	0	0
$76$	7.00756	0.803823
$77$	−0.225436	−0.0256908
$78$	−7.96390	−0.901734
$79$	4.10565	0.461922	0.230961	−	0.972963i	$-0.425813\pi$
$79$	4.10565	0.461922	0.230961	+	0.972963i	$0.425813\pi$
$80$	0	0
$81$	1.29937	0.144374
$82$	1.61875	0.178761
$83$	−15.1584	−1.66385	−0.831923	−	0.554891i	$-0.812760\pi$
$83$	−15.1584	−1.66385	−0.831923	+	0.554891i	$0.812760\pi$
$84$	−1.67702	−0.182978
$85$	0	0
$86$	−0.958189	−0.103324
$87$	−26.0601	−2.79394
$88$	−1.36501	−0.145510
$89$	−13.6969	−1.45187	−0.725934	−	0.687764i	$-0.758592\pi$
$89$	−13.6969	−1.45187	−0.725934	+	0.687764i	$0.758592\pi$
$90$	0	0
$91$	1.68849	0.177002
$92$	4.80511	0.500968
$93$	−5.99142	−0.621281
$94$	3.44034	0.354844
$95$	0	0
$96$	−15.7325	−1.60570
$97$	13.5129	1.37203	0.686016	−	0.727586i	$-0.259358\pi$
$97$	13.5129	1.37203	0.686016	+	0.727586i	$0.259358\pi$
$98$	4.11832	0.416013
$99$	−3.15388	−0.316976
$100$	0	0
$101$	5.92525	0.589585	0.294792	−	0.955561i	$-0.404750\pi$
$101$	5.92525	0.589585	0.294792	+	0.955561i	$0.404750\pi$
$102$	13.8905	1.37536
$103$	−6.48880	−0.639360	−0.319680	−	0.947526i	$-0.603575\pi$
$103$	−6.48880	−0.639360	−0.319680	+	0.947526i	$0.603575\pi$
$104$	10.2237	1.00252
$105$	0	0
$106$	−5.96946	−0.579805
$107$	−8.70727	−0.841764	−0.420882	−	0.907115i	$-0.638279\pi$
$107$	−8.70727	−0.841764	−0.420882	+	0.907115i	$0.638279\pi$
$108$	−9.50334	−0.914459
$109$	6.50110	0.622693	0.311346	−	0.950297i	$-0.399220\pi$
$109$	6.50110	0.622693	0.311346	+	0.950297i	$0.399220\pi$
$110$	0	0
$111$	7.14129	0.677822
$112$	0.711151	0.0671975
$113$	−6.93590	−0.652474	−0.326237	−	0.945288i	$-0.605781\pi$
$113$	−6.93590	−0.652474	−0.326237	+	0.945288i	$0.605781\pi$
$114$	7.26112	0.680066
$115$	0	0
$116$	15.0764	1.39981
$117$	23.6221	2.18387
$118$	4.13337	0.380507
$119$	−2.94503	−0.269971
$120$	0	0
$121$	−10.6088	−0.964437
$122$	−2.76965	−0.250752
$123$	−7.65807	−0.690505
$124$	3.46617	0.311272
$125$	0	0
$126$	−1.08951	−0.0970608
$127$	5.45809	0.484327	0.242164	−	0.970235i	$-0.422143\pi$
$127$	5.45809	0.484327	0.242164	+	0.970235i	$0.422143\pi$
$128$	11.4672	1.01356
$129$	4.53304	0.399112
$130$	0	0
$131$	−11.5419	−1.00842	−0.504210	−	0.863581i	$-0.668216\pi$
$131$	−11.5419	−1.00842	−0.504210	+	0.863581i	$0.668216\pi$
$132$	2.91012	0.253293
$133$	−1.53949	−0.133490
$134$	−3.67749	−0.317687
$135$	0	0
$136$	−17.8321	−1.52909
$137$	14.6748	1.25375	0.626875	−	0.779120i	$-0.284334\pi$
$137$	14.6748	1.25375	0.626875	+	0.779120i	$0.284334\pi$
$138$	4.97898	0.423839
$139$	12.9136	1.09532	0.547659	−	0.836701i	$-0.315519\pi$
$139$	12.9136	1.09532	0.547659	+	0.836701i	$0.315519\pi$
$140$	0	0
$141$	−16.2757	−1.37066
$142$	7.64421	0.641488
$143$	−2.93002	−0.245020
$144$	9.94909	0.829091
$145$	0	0
$146$	0.335309	0.0277503
$147$	−19.4831	−1.60694
$148$	−4.13140	−0.339599
$149$	−7.82834	−0.641323	−0.320661	−	0.947194i	$-0.603905\pi$
$149$	−7.82834	−0.641323	−0.320661	+	0.947194i	$0.603905\pi$
$150$	0	0
$151$	6.54793	0.532863	0.266431	−	0.963854i	$-0.414155\pi$
$151$	6.54793	0.532863	0.266431	+	0.963854i	$0.414155\pi$
$152$	−9.32153	−0.756076
$153$	−41.2013	−3.33093
$154$	0.135139	0.0108898
$155$	0	0
$156$	−21.7964	−1.74511
$157$	−10.2687	−0.819532	−0.409766	−	0.912191i	$-0.634390\pi$
$157$	−10.2687	−0.819532	−0.409766	+	0.912191i	$0.634390\pi$
$158$	−2.46116	−0.195799
$159$	28.2406	2.23962
$160$	0	0
$161$	−1.05563	−0.0831954
$162$	−0.778914	−0.0611972
$163$	14.8492	1.16308	0.581540	−	0.813518i	$-0.302450\pi$
$163$	14.8492	1.16308	0.581540	+	0.813518i	$0.302450\pi$
$164$	4.43037	0.345954
$165$	0	0
$166$	9.08678	0.705271
$167$	12.9361	1.00102	0.500511	−	0.865730i	$-0.333145\pi$
$167$	12.9361	1.00102	0.500511	+	0.865730i	$0.333145\pi$
$168$	2.23078	0.172109
$169$	8.94547	0.688113
$170$	0	0
$171$	−21.5376	−1.64702
$172$	−2.62247	−0.199961
$173$	23.4676	1.78421	0.892103	−	0.451833i	$-0.149230\pi$
$173$	23.4676	1.78421	0.892103	+	0.451833i	$0.149230\pi$
$174$	15.6219	1.18429
$175$	0	0
$176$	−1.23406	−0.0930204
$177$	−19.5543	−1.46979
$178$	8.21069	0.615418
$179$	13.1703	0.984392	0.492196	−	0.870484i	$-0.336194\pi$
$179$	13.1703	0.984392	0.492196	+	0.870484i	$0.336194\pi$
$180$	0	0
$181$	3.77626	0.280687	0.140344	−	0.990103i	$-0.455179\pi$
$181$	3.77626	0.280687	0.140344	+	0.990103i	$0.455179\pi$
$182$	−1.01217	−0.0750273
$183$	13.1028	0.968584
$184$	−6.39181	−0.471210
$185$	0	0
$186$	3.59159	0.263348
$187$	5.11049	0.373716
$188$	9.41587	0.686723
$189$	2.08778	0.151864
$190$	0	0
$191$	−0.804301	−0.0581972	−0.0290986	−	0.999577i	$-0.509264\pi$
$191$	−0.804301	−0.0581972	−0.0290986	+	0.999577i	$0.509264\pi$
$192$	−1.75986	−0.127007
$193$	−22.5417	−1.62259	−0.811293	−	0.584640i	$-0.801236\pi$
$193$	−22.5417	−1.62259	−0.811293	+	0.584640i	$0.801236\pi$
$194$	−8.10042	−0.581576
$195$	0	0
$196$	11.2714	0.805102
$197$	−5.11900	−0.364714	−0.182357	−	0.983232i	$-0.558373\pi$
$197$	−5.11900	−0.364714	−0.182357	+	0.983232i	$0.558373\pi$
$198$	1.89061	0.134360
$199$	7.35358	0.521281	0.260641	−	0.965436i	$-0.416066\pi$
$199$	7.35358	0.521281	0.260641	+	0.965436i	$0.416066\pi$
$200$	0	0
$201$	17.3976	1.22713
$202$	−3.55193	−0.249913
$203$	−3.31212	−0.232465
$204$	38.0169	2.66172
$205$	0	0
$206$	3.88975	0.271012
$207$	−14.7684	−1.02648
$208$	9.24292	0.640881
$209$	2.67146	0.184789
$210$	0	0
$211$	16.9013	1.16353	0.581767	−	0.813355i	$-0.302362\pi$
$211$	16.9013	1.16353	0.581767	+	0.813355i	$0.302362\pi$
$212$	−16.3378	−1.12209
$213$	−36.1635	−2.47789
$214$	5.21963	0.356807
$215$	0	0
$216$	12.6414	0.860141
$217$	−0.761481	−0.0516927
$218$	−3.89713	−0.263947
$219$	−1.58629	−0.107192
$220$	0	0
$221$	−38.2769	−2.57479
$222$	−4.28089	−0.287315
$223$	−21.6084	−1.44701	−0.723503	−	0.690321i	$-0.757469\pi$
$223$	−21.6084	−1.44701	−0.723503	+	0.690321i	$0.757469\pi$
$224$	−1.99953	−0.133599
$225$	0	0
$226$	4.15777	0.276570
$227$	−6.07474	−0.403195	−0.201597	−	0.979468i	$-0.564613\pi$
$227$	−6.07474	−0.403195	−0.201597	+	0.979468i	$0.564613\pi$
$228$	19.8730	1.31612
$229$	14.2052	0.938709	0.469354	−	0.883010i	$-0.344487\pi$
$229$	14.2052	0.938709	0.469354	+	0.883010i	$0.344487\pi$
$230$	0	0
$231$	−0.639321	−0.0420642
$232$	−20.0547	−1.31666
$233$	10.4653	0.685604	0.342802	−	0.939408i	$-0.388624\pi$
$233$	10.4653	0.685604	0.342802	+	0.939408i	$0.388624\pi$
$234$	−14.1604	−0.925696
$235$	0	0
$236$	11.3126	0.736389
$237$	11.6434	0.756317
$238$	1.76542	0.114435
$239$	19.5952	1.26751	0.633755	−	0.773534i	$-0.281513\pi$
$239$	19.5952	1.26751	0.633755	+	0.773534i	$0.281513\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	6.35951	0.408805
$243$	−13.6923	−0.878363
$244$	−7.58025	−0.485276
$245$	0	0
$246$	4.59068	0.292691
$247$	−20.0089	−1.27314
$248$	−4.61074	−0.292782
$249$	−42.9881	−2.72426
$250$	0	0
$251$	−9.43169	−0.595323	−0.297661	−	0.954672i	$-0.596207\pi$
$251$	−9.43169	−0.595323	−0.297661	+	0.954672i	$0.596207\pi$
$252$	−2.98187	−0.187840
$253$	1.83183	0.115166
$254$	−3.27189	−0.205296
$255$	0	0
$256$	−5.63294	−0.352059
$257$	18.5042	1.15426	0.577131	−	0.816652i	$-0.304172\pi$
$257$	18.5042	1.15426	0.577131	+	0.816652i	$0.304172\pi$
$258$	−2.71736	−0.169175
$259$	0.907625	0.0563971
$260$	0	0
$261$	−46.3369	−2.86818
$262$	6.91886	0.427449
$263$	−23.9444	−1.47647	−0.738237	−	0.674541i	$-0.764341\pi$
$263$	−23.9444	−1.47647	−0.738237	+	0.674541i	$0.764341\pi$
$264$	−3.87106	−0.238247
$265$	0	0
$266$	0.922855	0.0565838
$267$	−38.8435	−2.37718
$268$	−10.0649	−0.614813
$269$	−22.0724	−1.34578	−0.672890	−	0.739743i	$-0.734947\pi$
$269$	−22.0724	−1.34578	−0.672890	+	0.739743i	$0.734947\pi$
$270$	0	0
$271$	−8.04970	−0.488984	−0.244492	−	0.969651i	$-0.578621\pi$
$271$	−8.04970	−0.488984	−0.244492	+	0.969651i	$0.578621\pi$
$272$	−16.1214	−0.977501
$273$	4.78843	0.289809
$274$	−8.79688	−0.531439
$275$	0	0
$276$	13.6270	0.820247
$277$	−16.6952	−1.00312	−0.501558	−	0.865124i	$-0.667240\pi$
$277$	−16.6952	−1.00312	−0.501558	+	0.865124i	$0.667240\pi$
$278$	−7.74115	−0.464283
$279$	−10.6532	−0.637791
$280$	0	0
$281$	5.81800	0.347073	0.173536	−	0.984827i	$-0.444481\pi$
$281$	5.81800	0.347073	0.173536	+	0.984827i	$0.444481\pi$
$282$	9.75657	0.580995
$283$	−12.2054	−0.725536	−0.362768	−	0.931880i	$-0.618168\pi$
$283$	−12.2054	−0.725536	−0.362768	+	0.931880i	$0.618168\pi$
$284$	20.9214	1.24146
$285$	0	0
$286$	1.75642	0.103859
$287$	−0.973304	−0.0574523
$288$	−27.9737	−1.64837
$289$	49.7620	2.92718
$290$	0	0
$291$	38.3218	2.24646
$292$	0.917706	0.0537047
$293$	−0.0653701	−0.00381896	−0.00190948	−	0.999998i	$-0.500608\pi$
$293$	−0.0653701	−0.00381896	−0.00190948	+	0.999998i	$0.500608\pi$
$294$	11.6793	0.681149
$295$	0	0
$296$	5.49563	0.319427
$297$	−3.62291	−0.210223
$298$	4.69275	0.271844
$299$	−13.7202	−0.793458
$300$	0	0
$301$	0.576128	0.0332075
$302$	−3.92520	−0.225870
$303$	16.8036	0.965342
$304$	−8.42728	−0.483338
$305$	0	0
$306$	24.6984	1.41191
$307$	−8.55277	−0.488133	−0.244066	−	0.969759i	$-0.578481\pi$
$307$	−8.55277	−0.488133	−0.244066	+	0.969759i	$0.578481\pi$
$308$	0.369862	0.0210748
$309$	−18.4018	−1.04684
$310$	0	0
$311$	8.65202	0.490611	0.245306	−	0.969446i	$-0.421112\pi$
$311$	8.65202	0.490611	0.245306	+	0.969446i	$0.421112\pi$
$312$	28.9938	1.64145
$313$	−1.44232	−0.0815247	−0.0407623	−	0.999169i	$-0.512979\pi$
$313$	−1.44232	−0.0815247	−0.0407623	+	0.999169i	$0.512979\pi$
$314$	6.15564	0.347383
$315$	0	0
$316$	−6.73595	−0.378927
$317$	−9.49802	−0.533462	−0.266731	−	0.963771i	$-0.585943\pi$
$317$	−9.49802	−0.533462	−0.266731	+	0.963771i	$0.585943\pi$
$318$	−16.9290	−0.949330
$319$	5.74749	0.321798
$320$	0	0
$321$	−24.6932	−1.37824
$322$	0.632805	0.0352648
$323$	34.8992	1.94184
$324$	−2.13181	−0.118434
$325$	0	0
$326$	−8.90145	−0.493006
$327$	18.4367	1.01955
$328$	−5.89332	−0.325404
$329$	−2.06857	−0.114044
$330$	0	0
$331$	−16.0047	−0.879696	−0.439848	−	0.898072i	$-0.644968\pi$
$331$	−16.0047	−0.879696	−0.439848	+	0.898072i	$0.644968\pi$
$332$	24.8696	1.36490
$333$	12.6978	0.695834
$334$	−7.75460	−0.424313
$335$	0	0
$336$	2.01678	0.110024
$337$	−3.38373	−0.184323	−0.0921616	−	0.995744i	$-0.529378\pi$
$337$	−3.38373	−0.184323	−0.0921616	+	0.995744i	$0.529378\pi$
$338$	−5.36242	−0.291677
$339$	−19.6697	−1.06831
$340$	0	0
$341$	1.32139	0.0715574
$342$	12.9108	0.698139
$343$	−4.99925	−0.269934
$344$	3.48843	0.188084
$345$	0	0
$346$	−14.0678	−0.756288
$347$	−34.0410	−1.82742	−0.913709	−	0.406369i	$-0.866795\pi$
$347$	−34.0410	−1.82742	−0.913709	+	0.406369i	$0.866795\pi$
$348$	42.7556	2.29194
$349$	5.88943	0.315254	0.157627	−	0.987499i	$-0.449616\pi$
$349$	5.88943	0.315254	0.157627	+	0.987499i	$0.449616\pi$
$350$	0	0
$351$	27.1351	1.44837
$352$	3.46977	0.184940
$353$	−33.8088	−1.79946	−0.899731	−	0.436444i	$-0.856238\pi$
$353$	−33.8088	−1.79946	−0.899731	+	0.436444i	$0.856238\pi$
$354$	11.7219	0.623015
$355$	0	0
$356$	22.4719	1.19101
$357$	−8.35191	−0.442030
$358$	−7.89500	−0.417264
$359$	12.9670	0.684371	0.342186	−	0.939632i	$-0.388833\pi$
$359$	12.9670	0.684371	0.342186	+	0.939632i	$0.388833\pi$
$360$	0	0
$361$	−0.756797	−0.0398314
$362$	−2.26370	−0.118978
$363$	−30.0858	−1.57910
$364$	−2.77022	−0.145199
$365$	0	0
$366$	−7.85453	−0.410563
$367$	9.73813	0.508326	0.254163	−	0.967161i	$-0.418200\pi$
$367$	9.73813	0.508326	0.254163	+	0.967161i	$0.418200\pi$
$368$	−5.77862	−0.301231
$369$	−13.6166	−0.708854
$370$	0	0
$371$	3.58924	0.186344
$372$	9.82983	0.509653
$373$	10.8997	0.564367	0.282184	−	0.959360i	$-0.408941\pi$
$373$	10.8997	0.564367	0.282184	+	0.959360i	$0.408941\pi$
$374$	−3.06352	−0.158411
$375$	0	0
$376$	−12.5251	−0.645932
$377$	−43.0480	−2.21708
$378$	−1.25153	−0.0643719
$379$	−32.8409	−1.68692	−0.843462	−	0.537189i	$-0.819486\pi$
$379$	−32.8409	−1.68692	−0.843462	+	0.537189i	$0.819486\pi$
$380$	0	0
$381$	15.4788	0.793002
$382$	0.482143	0.0246686
$383$	−15.9330	−0.814138	−0.407069	−	0.913397i	$-0.633449\pi$
$383$	−15.9330	−0.814138	−0.407069	+	0.913397i	$0.633449\pi$
$384$	32.5201	1.65953
$385$	0	0
$386$	13.5127	0.687781
$387$	8.06010	0.409718
$388$	−22.1700	−1.12551
$389$	−35.3117	−1.79037	−0.895186	−	0.445692i	$-0.852958\pi$
$389$	−35.3117	−1.79037	−0.895186	+	0.445692i	$0.852958\pi$
$390$	0	0
$391$	23.9305	1.21022
$392$	−14.9934	−0.757279
$393$	−32.7320	−1.65111
$394$	3.06862	0.154595
$395$	0	0
$396$	5.17441	0.260024
$397$	22.1785	1.11310	0.556552	−	0.830812i	$-0.312124\pi$
$397$	22.1785	1.11310	0.556552	+	0.830812i	$0.312124\pi$
$398$	−4.40815	−0.220961
$399$	−4.36588	−0.218567
$400$	0	0
$401$	28.6544	1.43093	0.715466	−	0.698647i	$-0.246214\pi$
$401$	28.6544	1.43093	0.715466	+	0.698647i	$0.246214\pi$
$402$	−10.4291	−0.520157
$403$	−9.89706	−0.493008
$404$	−9.72128	−0.483652
$405$	0	0
$406$	1.98547	0.0985372
$407$	−1.57499	−0.0780696
$408$	−50.5705	−2.50361
$409$	19.8415	0.981098	0.490549	−	0.871414i	$-0.336796\pi$
$409$	19.8415	0.981098	0.490549	+	0.871414i	$0.336796\pi$
$410$	0	0
$411$	41.6167	2.05280
$412$	10.6459	0.524484
$413$	−2.48526	−0.122292
$414$	8.85301	0.435102
$415$	0	0
$416$	−25.9882	−1.27418
$417$	36.6221	1.79339
$418$	−1.60142	−0.0783281
$419$	−34.1544	−1.66855	−0.834277	−	0.551346i	$-0.814115\pi$
$419$	−34.1544	−1.66855	−0.834277	+	0.551346i	$0.814115\pi$
$420$	0	0
$421$	−21.6854	−1.05688	−0.528440	−	0.848971i	$-0.677223\pi$
$421$	−21.6854	−1.05688	−0.528440	+	0.848971i	$0.677223\pi$
$422$	−10.1316	−0.493198
$423$	−28.9395	−1.40709
$424$	21.7327	1.05543
$425$	0	0
$426$	21.6785	1.05032
$427$	1.66530	0.0805895
$428$	14.2856	0.690521
$429$	−8.30933	−0.401178
$430$	0	0
$431$	34.0237	1.63886	0.819431	−	0.573177i	$-0.194289\pi$
$431$	34.0237	1.63886	0.819431	+	0.573177i	$0.194289\pi$
$432$	11.4287	0.549863
$433$	−18.2413	−0.876620	−0.438310	−	0.898824i	$-0.644423\pi$
$433$	−18.2413	−0.876620	−0.438310	+	0.898824i	$0.644423\pi$
$434$	0.456475	0.0219115
$435$	0	0
$436$	−10.6660	−0.510811
$437$	12.5094	0.598407
$438$	0.950912	0.0454363
$439$	−33.1350	−1.58145	−0.790723	−	0.612174i	$-0.790295\pi$
$439$	−33.1350	−1.58145	−0.790723	+	0.612174i	$0.790295\pi$
$440$	0	0
$441$	−34.6425	−1.64964
$442$	22.9453	1.09140
$443$	17.0218	0.808731	0.404366	−	0.914597i	$-0.367492\pi$
$443$	17.0218	0.808731	0.404366	+	0.914597i	$0.367492\pi$
$444$	−11.7164	−0.556035
$445$	0	0
$446$	12.9533	0.613357
$447$	−22.2006	−1.05005
$448$	−0.223670	−0.0105674
$449$	11.7660	0.555273	0.277636	−	0.960686i	$-0.410449\pi$
$449$	11.7660	0.555273	0.277636	+	0.960686i	$0.410449\pi$
$450$	0	0
$451$	1.68897	0.0795304
$452$	11.3794	0.535242
$453$	18.5695	0.872470
$454$	3.64154	0.170906
$455$	0	0
$456$	−26.4352	−1.23794
$457$	−38.1291	−1.78360	−0.891801	−	0.452428i	$-0.850558\pi$
$457$	−38.1291	−1.78360	−0.891801	+	0.452428i	$0.850558\pi$
$458$	−8.51542	−0.397899
$459$	−47.3287	−2.20911
$460$	0	0
$461$	4.15125	0.193343	0.0966715	−	0.995316i	$-0.469180\pi$
$461$	4.15125	0.193343	0.0966715	+	0.995316i	$0.469180\pi$
$462$	0.383245	0.0178302
$463$	−32.8714	−1.52766	−0.763831	−	0.645416i	$-0.776684\pi$
$463$	−32.8714	−1.52766	−0.763831	+	0.645416i	$0.776684\pi$
$464$	−18.1308	−0.841702
$465$	0	0
$466$	−6.27348	−0.290614
$467$	−27.9382	−1.29283	−0.646413	−	0.762988i	$-0.723732\pi$
$467$	−27.9382	−1.29283	−0.646413	+	0.762988i	$0.723732\pi$
$468$	−38.7557	−1.79148
$469$	2.21116	0.102102
$470$	0	0
$471$	−29.1214	−1.34184
$472$	−15.0482	−0.692647
$473$	−0.999750	−0.0459686
$474$	−6.97968	−0.320587
$475$	0	0
$476$	4.83177	0.221464
$477$	50.2139	2.29914
$478$	−11.7465	−0.537272
$479$	12.5295	0.572487	0.286244	−	0.958157i	$-0.407593\pi$
$479$	12.5295	0.572487	0.286244	+	0.958157i	$0.407593\pi$
$480$	0	0
$481$	11.7965	0.537875
$482$	0.599456	0.0273045
$483$	−2.99370	−0.136218
$484$	17.4054	0.791153
$485$	0	0
$486$	8.20795	0.372320
$487$	−20.2102	−0.915812	−0.457906	−	0.889001i	$-0.651400\pi$
$487$	−20.2102	−0.915812	−0.457906	+	0.889001i	$0.651400\pi$
$488$	10.0833	0.456451
$489$	42.1113	1.90434
$490$	0	0
$491$	39.0790	1.76361	0.881805	−	0.471614i	$-0.156329\pi$
$491$	39.0790	1.76361	0.881805	+	0.471614i	$0.156329\pi$
$492$	12.5642	0.566439
$493$	75.0836	3.38160
$494$	11.9945	0.539656
$495$	0	0
$496$	−4.16841	−0.187167
$497$	−4.59622	−0.206168
$498$	25.7695	1.15476
$499$	−23.2259	−1.03973	−0.519867	−	0.854247i	$-0.674018\pi$
$499$	−23.2259	−1.03973	−0.519867	+	0.854247i	$0.674018\pi$
$500$	0	0
$501$	36.6858	1.63900
$502$	5.65388	0.252345
$503$	20.7067	0.923267	0.461633	−	0.887071i	$-0.347264\pi$
$503$	20.7067	0.923267	0.461633	+	0.887071i	$0.347264\pi$
$504$	3.96651	0.176682
$505$	0	0
$506$	−1.09810	−0.0488165
$507$	25.3687	1.12667
$508$	−8.95483	−0.397306
$509$	13.4785	0.597425	0.298712	−	0.954343i	$-0.403443\pi$
$509$	13.4785	0.597425	0.298712	+	0.954343i	$0.403443\pi$
$510$	0	0
$511$	−0.201610	−0.00891871
$512$	−19.5576	−0.864332
$513$	−24.7406	−1.09232
$514$	−11.0925	−0.489268
$515$	0	0
$516$	−7.43714	−0.327402
$517$	3.58957	0.157869
$518$	−0.544081	−0.0239056
$519$	66.5524	2.92133
$520$	0	0
$521$	−2.39561	−0.104953	−0.0524767	−	0.998622i	$-0.516712\pi$
$521$	−2.39561	−0.104953	−0.0524767	+	0.998622i	$0.516712\pi$
$522$	27.7770	1.21576
$523$	−17.2203	−0.752991	−0.376496	−	0.926418i	$-0.622871\pi$
$523$	−17.2203	−0.752991	−0.376496	+	0.926418i	$0.622871\pi$
$524$	18.9362	0.827233
$525$	0	0
$526$	14.3536	0.625847
$527$	17.2623	0.751957
$528$	−3.49970	−0.152305
$529$	−14.4222	−0.627054
$530$	0	0
$531$	−34.7691	−1.50885
$532$	2.52576	0.109506
$533$	−12.6502	−0.547939
$534$	23.2850	1.00764
$535$	0	0
$536$	13.3885	0.578294
$537$	37.3500	1.61177
$538$	13.2315	0.570448
$539$	4.29695	0.185083
$540$	0	0
$541$	9.63512	0.414246	0.207123	−	0.978315i	$-0.433590\pi$
$541$	9.63512	0.414246	0.207123	+	0.978315i	$0.433590\pi$
$542$	4.82544	0.207270
$543$	10.7092	0.459577
$544$	45.3282	1.94343
$545$	0	0
$546$	−2.87046	−0.122844
$547$	−10.4736	−0.447818	−0.223909	−	0.974610i	$-0.571882\pi$
$547$	−10.4736	−0.447818	−0.223909	+	0.974610i	$0.571882\pi$
$548$	−24.0762	−1.02848
$549$	23.2977	0.994323
$550$	0	0
$551$	39.2492	1.67207
$552$	−18.1267	−0.771525
$553$	1.47982	0.0629281
$554$	10.0080	0.425200
$555$	0	0
$556$	−21.1867	−0.898518
$557$	−12.5042	−0.529821	−0.264910	−	0.964273i	$-0.585342\pi$
$557$	−12.5042	−0.529821	−0.264910	+	0.964273i	$0.585342\pi$
$558$	6.38613	0.270347
$559$	7.48800	0.316709
$560$	0	0
$561$	14.4930	0.611895
$562$	−3.48764	−0.147117
$563$	24.8151	1.04583	0.522915	−	0.852385i	$-0.324844\pi$
$563$	24.8151	1.04583	0.522915	+	0.852385i	$0.324844\pi$
$564$	26.7028	1.12439
$565$	0	0
$566$	7.31660	0.307540
$567$	0.468336	0.0196682
$568$	−27.8299	−1.16772
$569$	46.6025	1.95368	0.976838	−	0.213979i	$-0.0686423\pi$
$569$	46.6025	1.95368	0.976838	+	0.213979i	$0.0686423\pi$
$570$	0	0
$571$	−16.5906	−0.694295	−0.347147	−	0.937811i	$-0.612850\pi$
$571$	−16.5906	−0.694295	−0.347147	+	0.937811i	$0.612850\pi$
$572$	4.80714	0.200997
$573$	−2.28094	−0.0952878
$574$	0.583453	0.0243529
$575$	0	0
$576$	−3.12917	−0.130382
$577$	38.4798	1.60194	0.800968	−	0.598707i	$-0.204319\pi$
$577$	38.4798	1.60194	0.800968	+	0.598707i	$0.204319\pi$
$578$	−29.8301	−1.24077
$579$	−63.9266	−2.65670
$580$	0	0
$581$	−5.46358	−0.226668
$582$	−22.9722	−0.952230
$583$	−6.22838	−0.257953
$584$	−1.22074	−0.0505147
$585$	0	0
$586$	0.0391865	0.00161878
$587$	35.9523	1.48391	0.741956	−	0.670449i	$-0.233898\pi$
$587$	35.9523	1.48391	0.741956	+	0.670449i	$0.233898\pi$
$588$	31.9650	1.31821
$589$	9.02370	0.371815
$590$	0	0
$591$	−14.5171	−0.597155
$592$	4.96842	0.204201
$593$	42.5918	1.74903	0.874517	−	0.484994i	$-0.161178\pi$
$593$	42.5918	1.74903	0.874517	+	0.484994i	$0.161178\pi$
$594$	2.17178	0.0891090
$595$	0	0
$596$	12.8436	0.526094
$597$	20.8542	0.853508
$598$	8.22464	0.336330
$599$	−21.1344	−0.863529	−0.431765	−	0.901986i	$-0.642109\pi$
$599$	−21.1344	−0.863529	−0.431765	+	0.901986i	$0.642109\pi$
$600$	0	0
$601$	10.3494	0.422161	0.211081	−	0.977469i	$-0.432302\pi$
$601$	10.3494	0.422161	0.211081	+	0.977469i	$0.432302\pi$
$602$	−0.345363	−0.0140760
$603$	30.9343	1.25974
$604$	−10.7429	−0.437121
$605$	0	0
$606$	−10.0730	−0.409189
$607$	8.32484	0.337895	0.168948	−	0.985625i	$-0.445963\pi$
$607$	8.32484	0.337895	0.168948	+	0.985625i	$0.445963\pi$
$608$	23.6948	0.960953
$609$	−9.39294	−0.380621
$610$	0	0
$611$	−26.8854	−1.08767
$612$	67.5971	2.73245
$613$	19.5004	0.787613	0.393806	−	0.919193i	$-0.371158\pi$
$613$	19.5004	0.787613	0.393806	+	0.919193i	$0.371158\pi$
$614$	5.12701	0.206909
$615$	0	0
$616$	−0.491994	−0.0198230
$617$	17.9240	0.721591	0.360796	−	0.932645i	$-0.382505\pi$
$617$	17.9240	0.721591	0.360796	+	0.932645i	$0.382505\pi$
$618$	11.0311	0.443734
$619$	−34.3316	−1.37991	−0.689953	−	0.723855i	$-0.742369\pi$
$619$	−34.3316	−1.37991	−0.689953	+	0.723855i	$0.742369\pi$
$620$	0	0
$621$	−16.9647	−0.680771
$622$	−5.18651	−0.207960
$623$	−4.93682	−0.197790
$624$	26.2123	1.04933
$625$	0	0
$626$	0.864607	0.0345566
$627$	7.57607	0.302559
$628$	16.8474	0.672284
$629$	−20.5753	−0.820390
$630$	0	0
$631$	−1.90161	−0.0757020	−0.0378510	−	0.999283i	$-0.512051\pi$
$631$	−1.90161	−0.0757020	−0.0378510	+	0.999283i	$0.512051\pi$
$632$	8.96023	0.356419
$633$	47.9310	1.90508
$634$	5.69364	0.226123
$635$	0	0
$636$	−46.3329	−1.83722
$637$	−32.1836	−1.27516
$638$	−3.44537	−0.136403
$639$	−64.3016	−2.54373
$640$	0	0
$641$	−14.3867	−0.568240	−0.284120	−	0.958789i	$-0.591701\pi$
$641$	−14.3867	−0.568240	−0.284120	+	0.958789i	$0.591701\pi$
$642$	14.8025	0.584209
$643$	6.97932	0.275238	0.137619	−	0.990485i	$-0.456055\pi$
$643$	6.97932	0.275238	0.137619	+	0.990485i	$0.456055\pi$
$644$	1.73192	0.0682473
$645$	0	0
$646$	−20.9205	−0.823107
$647$	14.3296	0.563356	0.281678	−	0.959509i	$-0.409109\pi$
$647$	14.3296	0.563356	0.281678	+	0.959509i	$0.409109\pi$
$648$	2.83575	0.111399
$649$	4.31265	0.169286
$650$	0	0
$651$	−2.15951	−0.0846378
$652$	−24.3624	−0.954105
$653$	2.72501	0.106638	0.0533188	−	0.998578i	$-0.483020\pi$
$653$	2.72501	0.106638	0.0533188	+	0.998578i	$0.483020\pi$
$654$	−11.0520	−0.432167
$655$	0	0
$656$	−5.32795	−0.208022
$657$	−2.82055	−0.110040
$658$	1.24001	0.0483408
$659$	35.4962	1.38274	0.691368	−	0.722502i	$-0.257008\pi$
$659$	35.4962	1.38274	0.691368	+	0.722502i	$0.257008\pi$
$660$	0	0
$661$	19.0204	0.739810	0.369905	−	0.929070i	$-0.379390\pi$
$661$	19.0204	0.739810	0.369905	+	0.929070i	$0.379390\pi$
$662$	9.59410	0.372885
$663$	−108.551	−4.21576
$664$	−33.0818	−1.28382
$665$	0	0
$666$	−7.61176	−0.294950
$667$	26.9133	1.04209
$668$	−21.2236	−0.821165
$669$	−61.2800	−2.36922
$670$	0	0
$671$	−2.88978	−0.111559
$672$	−5.67054	−0.218746
$673$	−6.00845	−0.231609	−0.115804	−	0.993272i	$-0.536945\pi$
$673$	−6.00845	−0.231609	−0.115804	+	0.993272i	$0.536945\pi$
$674$	2.02840	0.0781309
$675$	0	0
$676$	−14.6764	−0.564477
$677$	−3.66159	−0.140726	−0.0703632	−	0.997521i	$-0.522416\pi$
$677$	−3.66159	−0.140726	−0.0703632	+	0.997521i	$0.522416\pi$
$678$	11.7911	0.452836
$679$	4.87052	0.186913
$680$	0	0
$681$	−17.2276	−0.660161
$682$	−0.792117	−0.0303317
$683$	0.903303	0.0345639	0.0172820	−	0.999851i	$-0.494499\pi$
$683$	0.903303	0.0345639	0.0172820	+	0.999851i	$0.494499\pi$
$684$	35.3357	1.35109
$685$	0	0
$686$	2.99683	0.114419
$687$	40.2851	1.53697
$688$	3.15377	0.120236
$689$	46.6498	1.77722
$690$	0	0
$691$	7.44227	0.283117	0.141559	−	0.989930i	$-0.454789\pi$
$691$	7.44227	0.283117	0.141559	+	0.989930i	$0.454789\pi$
$692$	−38.5021	−1.46363
$693$	−1.13676	−0.0431820
$694$	20.4061	0.774605
$695$	0	0
$696$	−56.8739	−2.15580
$697$	22.0642	0.835741
$698$	−3.53046	−0.133630
$699$	29.6789	1.12256
$700$	0	0
$701$	16.7088	0.631082	0.315541	−	0.948912i	$-0.397814\pi$
$701$	16.7088	0.631082	0.315541	+	0.948912i	$0.397814\pi$
$702$	−16.2663	−0.613933
$703$	−10.7555	−0.405652
$704$	0.388133	0.0146283
$705$	0	0
$706$	20.2669	0.762755
$707$	2.13566	0.0803197
$708$	32.0818	1.20571
$709$	−6.73868	−0.253076	−0.126538	−	0.991962i	$-0.540387\pi$
$709$	−6.73868	−0.253076	−0.126538	+	0.991962i	$0.540387\pi$
$710$	0	0
$711$	20.7028	0.776415
$712$	−29.8923	−1.12026
$713$	6.18758	0.231727
$714$	5.00660	0.187367
$715$	0	0
$716$	−21.6078	−0.807523
$717$	55.5708	2.07533
$718$	−7.77314	−0.290091
$719$	−20.5983	−0.768188	−0.384094	−	0.923294i	$-0.625486\pi$
$719$	−20.5983	−0.768188	−0.384094	+	0.923294i	$0.625486\pi$
$720$	0	0
$721$	−2.33878	−0.0871007
$722$	0.453667	0.0168837
$723$	−2.83593	−0.105469
$724$	−6.19554	−0.230255
$725$	0	0
$726$	18.0351	0.669347
$727$	−6.45223	−0.239300	−0.119650	−	0.992816i	$-0.538177\pi$
$727$	−6.45223	−0.239300	−0.119650	+	0.992816i	$0.538177\pi$
$728$	3.68497	0.136574
$729$	−42.7286	−1.58254
$730$	0	0
$731$	−13.0605	−0.483058
$732$	−21.4971	−0.794555
$733$	−41.0167	−1.51499	−0.757494	−	0.652842i	$-0.773576\pi$
$733$	−41.0167	−1.51499	−0.757494	+	0.652842i	$0.773576\pi$
$734$	−5.83758	−0.215469
$735$	0	0
$736$	16.2476	0.598896
$737$	−3.83700	−0.141338
$738$	8.16258	0.300469
$739$	52.6069	1.93518	0.967589	−	0.252531i	$-0.0812631\pi$
$739$	52.6069	1.93518	0.967589	+	0.252531i	$0.0812631\pi$
$740$	0	0
$741$	−56.7438	−2.08454
$742$	−2.15159	−0.0789875
$743$	36.7709	1.34899	0.674497	−	0.738278i	$-0.264361\pi$
$743$	36.7709	1.34899	0.674497	+	0.738278i	$0.264361\pi$
$744$	−13.0757	−0.479380
$745$	0	0
$746$	−6.53392	−0.239224
$747$	−76.4362	−2.79665
$748$	−8.38454	−0.306569
$749$	−3.13839	−0.114674
$750$	0	0
$751$	42.6384	1.55590	0.777949	−	0.628327i	$-0.216260\pi$
$751$	42.6384	1.55590	0.777949	+	0.628327i	$0.216260\pi$
$752$	−11.3235	−0.412926
$753$	−26.7476	−0.974737
$754$	25.8054	0.939777
$755$	0	0
$756$	−3.42532	−0.124578
$757$	38.5991	1.40291	0.701453	−	0.712715i	$-0.252535\pi$
$757$	38.5991	1.40291	0.701453	+	0.712715i	$0.252535\pi$
$758$	19.6867	0.715052
$759$	5.19494	0.188564
$760$	0	0
$761$	13.8171	0.500868	0.250434	−	0.968134i	$-0.419427\pi$
$761$	13.8171	0.500868	0.250434	+	0.968134i	$0.419427\pi$
$762$	−9.27885	−0.336137
$763$	2.34322	0.0848301
$764$	1.31958	0.0477407
$765$	0	0
$766$	9.55113	0.345096
$767$	−32.3012	−1.16633
$768$	−15.9746	−0.576435
$769$	−30.8304	−1.11177	−0.555886	−	0.831259i	$-0.687621\pi$
$769$	−30.8304	−1.11177	−0.555886	+	0.831259i	$0.687621\pi$
$770$	0	0
$771$	52.4767	1.88990
$772$	36.9830	1.33105
$773$	−2.27083	−0.0816762	−0.0408381	−	0.999166i	$-0.513003\pi$
$773$	−2.27083	−0.0816762	−0.0408381	+	0.999166i	$0.513003\pi$
$774$	−4.83168	−0.173671
$775$	0	0
$776$	29.4908	1.05866
$777$	2.57396	0.0923404
$778$	21.1678	0.758902
$779$	11.5338	0.413243
$780$	0	0
$781$	7.97578	0.285396
$782$	−14.3453	−0.512986
$783$	−53.2280	−1.90221
$784$	−13.5550	−0.484107
$785$	0	0
$786$	19.6214	0.699873
$787$	−5.81025	−0.207113	−0.103557	−	0.994624i	$-0.533022\pi$
$787$	−5.81025	−0.207113	−0.103557	+	0.994624i	$0.533022\pi$
$788$	8.39850	0.299184
$789$	−67.9046	−2.41747
$790$	0	0
$791$	−2.49993	−0.0888873
$792$	−6.88305	−0.244579
$793$	21.6441	0.768604
$794$	−13.2950	−0.471822
$795$	0	0
$796$	−12.0647	−0.427621
$797$	17.8987	0.634003	0.317002	−	0.948425i	$-0.397324\pi$
$797$	17.8987	0.634003	0.317002	+	0.948425i	$0.397324\pi$
$798$	2.61715	0.0926462
$799$	46.8931	1.65896
$800$	0	0
$801$	−69.0668	−2.44035
$802$	−17.1771	−0.606543
$803$	0.349853	0.0123460
$804$	−28.5435	−1.00665
$805$	0	0
$806$	5.93286	0.208976
$807$	−62.5959	−2.20348
$808$	12.9313	0.454923
$809$	−30.9388	−1.08775	−0.543876	−	0.839166i	$-0.683044\pi$
$809$	−30.9388	−1.08775	−0.543876	+	0.839166i	$0.683044\pi$
$810$	0	0
$811$	14.9632	0.525429	0.262714	−	0.964874i	$-0.415382\pi$
$811$	14.9632	0.525429	0.262714	+	0.964874i	$0.415382\pi$
$812$	5.43403	0.190697
$813$	−22.8284	−0.800627
$814$	0.944140	0.0330921
$815$	0	0
$816$	−45.7191	−1.60049
$817$	−6.82722	−0.238854
$818$	−11.8941	−0.415867
$819$	8.51421	0.297511
$820$	0	0
$821$	30.0601	1.04911	0.524553	−	0.851378i	$-0.324232\pi$
$821$	30.0601	1.04911	0.524553	+	0.851378i	$0.324232\pi$
$822$	−24.9474	−0.870139
$823$	−29.1394	−1.01573	−0.507867	−	0.861435i	$-0.669566\pi$
$823$	−29.1394	−1.01573	−0.507867	+	0.861435i	$0.669566\pi$
$824$	−14.1612	−0.493330
$825$	0	0
$826$	1.48980	0.0518369
$827$	−29.5419	−1.02727	−0.513637	−	0.858008i	$-0.671702\pi$
$827$	−29.5419	−1.02727	−0.513637	+	0.858008i	$0.671702\pi$
$828$	24.2298	0.842044
$829$	−51.4599	−1.78728	−0.893638	−	0.448789i	$-0.851855\pi$
$829$	−51.4599	−1.78728	−0.893638	+	0.448789i	$0.851855\pi$
$830$	0	0
$831$	−47.3464	−1.64243
$832$	−2.90707	−0.100785
$833$	56.1341	1.94493
$834$	−21.9534	−0.760183
$835$	0	0
$836$	−4.38293	−0.151587
$837$	−12.2375	−0.422991
$838$	20.4741	0.707266
$839$	10.3376	0.356893	0.178446	−	0.983950i	$-0.442893\pi$
$839$	10.3376	0.356893	0.178446	+	0.983950i	$0.442893\pi$
$840$	0	0
$841$	55.4425	1.91181
$842$	12.9994	0.447990
$843$	16.4994	0.568271
$844$	−27.7292	−0.954478
$845$	0	0
$846$	17.3479	0.596435
$847$	−3.82377	−0.131386
$848$	19.6478	0.674709
$849$	−34.6137	−1.18794
$850$	0	0
$851$	−7.37510	−0.252815
$852$	59.3318	2.03267
$853$	19.2879	0.660404	0.330202	−	0.943910i	$-0.392883\pi$
$853$	19.2879	0.660404	0.330202	+	0.943910i	$0.392883\pi$
$854$	−0.998274	−0.0341602
$855$	0	0
$856$	−19.0029	−0.649504
$857$	−2.52022	−0.0860889	−0.0430445	−	0.999073i	$-0.513706\pi$
$857$	−2.52022	−0.0860889	−0.0430445	+	0.999073i	$0.513706\pi$
$858$	4.98108	0.170051
$859$	41.4410	1.41395	0.706974	−	0.707240i	$-0.250060\pi$
$859$	41.4410	1.41395	0.706974	+	0.707240i	$0.250060\pi$
$860$	0	0
$861$	−2.76022	−0.0940682
$862$	−20.3957	−0.694680
$863$	−53.8341	−1.83253	−0.916266	−	0.400570i	$-0.868812\pi$
$863$	−53.8341	−1.83253	−0.916266	+	0.400570i	$0.868812\pi$
$864$	−32.1339	−1.09322
$865$	0	0
$866$	10.9349	0.371581
$867$	141.122	4.79274
$868$	1.24933	0.0424049
$869$	−2.56791	−0.0871105
$870$	0	0
$871$	28.7387	0.973773
$872$	14.1881	0.480469
$873$	68.1392	2.30616
$874$	−7.49885	−0.253653
$875$	0	0
$876$	2.60255	0.0879321
$877$	−36.8178	−1.24325	−0.621624	−	0.783316i	$-0.713527\pi$
$877$	−36.8178	−1.24325	−0.621624	+	0.783316i	$0.713527\pi$
$878$	19.8630	0.670343
$879$	−0.185385	−0.00625288
$880$	0	0
$881$	40.7279	1.37216	0.686080	−	0.727526i	$-0.259330\pi$
$881$	40.7279	1.37216	0.686080	+	0.727526i	$0.259330\pi$
$882$	20.7666	0.699249
$883$	28.6309	0.963507	0.481753	−	0.876307i	$-0.340000\pi$
$883$	28.6309	0.963507	0.481753	+	0.876307i	$0.340000\pi$
$884$	62.7991	2.11216
$885$	0	0
$886$	−10.2038	−0.342805
$887$	−4.30342	−0.144495	−0.0722473	−	0.997387i	$-0.523017\pi$
$887$	−4.30342	−0.144495	−0.0722473	+	0.997387i	$0.523017\pi$
$888$	15.5852	0.523006
$889$	1.96728	0.0659804
$890$	0	0
$891$	−0.812699	−0.0272264
$892$	35.4519	1.18702
$893$	24.5129	0.820293
$894$	13.3083	0.445097
$895$	0	0
$896$	4.13315	0.138079
$897$	−38.9095	−1.29915
$898$	−7.05321	−0.235369
$899$	19.4140	0.647492
$900$	0	0
$901$	−81.3659	−2.71069
$902$	−1.01246	−0.0337113
$903$	1.63386	0.0543714
$904$	−15.1370	−0.503448
$905$	0	0
$906$	−11.1316	−0.369822
$907$	45.3984	1.50743	0.753714	−	0.657202i	$-0.228260\pi$
$907$	45.3984	1.50743	0.753714	+	0.657202i	$0.228260\pi$
$908$	9.96654	0.330751
$909$	29.8781	0.990995
$910$	0	0
$911$	−14.5189	−0.481032	−0.240516	−	0.970645i	$-0.577317\pi$
$911$	−14.5189	−0.481032	−0.240516	+	0.970645i	$0.577317\pi$
$912$	−23.8992	−0.791381
$913$	9.48091	0.313772
$914$	22.8567	0.756032
$915$	0	0
$916$	−23.3059	−0.770047
$917$	−4.16009	−0.137378
$918$	28.3715	0.936398
$919$	12.2387	0.403718	0.201859	−	0.979415i	$-0.435302\pi$
$919$	12.2387	0.403718	0.201859	+	0.979415i	$0.435302\pi$
$920$	0	0
$921$	−24.2551	−0.799232
$922$	−2.48849	−0.0819542
$923$	−59.7376	−1.96629
$924$	1.04890	0.0345064
$925$	0	0
$926$	19.7050	0.647545
$927$	−32.7198	−1.07466
$928$	50.9781	1.67344
$929$	−47.9762	−1.57405	−0.787024	−	0.616922i	$-0.788379\pi$
$929$	−47.9762	−1.57405	−0.787024	+	0.616922i	$0.788379\pi$
$930$	0	0
$931$	29.3436	0.961696
$932$	−17.1699	−0.562419
$933$	24.5366	0.803291
$934$	16.7477	0.548002
$935$	0	0
$936$	51.5532	1.68507
$937$	−20.7632	−0.678306	−0.339153	−	0.940731i	$-0.610140\pi$
$937$	−20.7632	−0.678306	−0.339153	+	0.940731i	$0.610140\pi$
$938$	−1.32549	−0.0432788
$939$	−4.09032	−0.133482
$940$	0	0
$941$	−38.2342	−1.24640	−0.623199	−	0.782063i	$-0.714167\pi$
$941$	−38.2342	−1.24640	−0.623199	+	0.782063i	$0.714167\pi$
$942$	17.4570	0.568779
$943$	7.90879	0.257546
$944$	−13.6045	−0.442790
$945$	0	0
$946$	0.599306	0.0194851
$947$	−17.4619	−0.567437	−0.283718	−	0.958908i	$-0.591568\pi$
$947$	−17.4619	−0.567437	−0.283718	+	0.958908i	$0.591568\pi$
$948$	−19.1027	−0.620427
$949$	−2.62035	−0.0850602
$950$	0	0
$951$	−26.9357	−0.873451
$952$	−6.42727	−0.208309
$953$	23.5351	0.762377	0.381188	−	0.924497i	$-0.375515\pi$
$953$	23.5351	0.762377	0.381188	+	0.924497i	$0.375515\pi$
$954$	−30.1011	−0.974558
$955$	0	0
$956$	−32.1490	−1.03977
$957$	16.2995	0.526888
$958$	−7.51089	−0.242666
$959$	5.28928	0.170800
$960$	0	0
$961$	−26.5366	−0.856019
$962$	−7.07149	−0.227994
$963$	−43.9065	−1.41487
$964$	1.64065	0.0528419
$965$	0	0
$966$	1.79459	0.0577400
$967$	56.8777	1.82906	0.914531	−	0.404515i	$-0.132560\pi$
$967$	56.8777	1.82906	0.914531	+	0.404515i	$0.132560\pi$
$968$	−23.1528	−0.744158
$969$	98.9717	3.17943
$970$	0	0
$971$	−31.8532	−1.02222	−0.511108	−	0.859516i	$-0.670765\pi$
$971$	−31.8532	−1.02222	−0.511108	+	0.859516i	$0.670765\pi$
$972$	22.4643	0.720544
$973$	4.65450	0.149216
$974$	12.1151	0.388194
$975$	0	0
$976$	9.11599	0.291796
$977$	−52.0439	−1.66503	−0.832516	−	0.554002i	$-0.813100\pi$
$977$	−52.0439	−1.66503	−0.832516	+	0.554002i	$0.813100\pi$
$978$	−25.2439	−0.807211
$979$	8.56683	0.273797
$980$	0	0
$981$	32.7819	1.04664
$982$	−23.4261	−0.747558
$983$	9.89877	0.315722	0.157861	−	0.987461i	$-0.449540\pi$
$983$	9.89877	0.315722	0.157861	+	0.987461i	$0.449540\pi$
$984$	−16.7131	−0.532793
$985$	0	0
$986$	−45.0093	−1.43339
$987$	−5.86631	−0.186727
$988$	32.8276	1.04439
$989$	−4.68145	−0.148862
$990$	0	0
$991$	24.2687	0.770921	0.385461	−	0.922724i	$-0.374043\pi$
$991$	24.2687	0.770921	0.385461	+	0.922724i	$0.374043\pi$
$992$	11.7203	0.372119
$993$	−45.3882	−1.44035
$994$	2.75523	0.0873906
$995$	0	0
$996$	70.5285	2.23478
$997$	21.7614	0.689191	0.344596	−	0.938751i	$-0.388016\pi$
$997$	21.7614	0.689191	0.344596	+	0.938751i	$0.388016\pi$
$998$	13.9229	0.440722
$999$	14.5862	0.461485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.l.1.20	✓	40
5.4	even	2		6025.2.a.o.1.21	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.20	✓	40	1.1	even	1	trivial
6025.2.a.o.1.21	yes	40	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.599456 q^{2} +2.83593 q^{3} -1.64065 q^{4} -1.70002 q^{6} +0.360434 q^{7} +2.18241 q^{8} +5.04251 q^{9} +O(q^{10})\)	\(q-0.599456 q^{2} +2.83593 q^{3} -1.64065 q^{4} -1.70002 q^{6} +0.360434 q^{7} +2.18241 q^{8} +5.04251 q^{9} -0.625458 q^{11} -4.65278 q^{12} +4.68460 q^{13} -0.216064 q^{14} +1.97304 q^{16} -8.17080 q^{17} -3.02276 q^{18} -4.27121 q^{19} +1.02216 q^{21} +0.374934 q^{22} -2.92878 q^{23} +6.18917 q^{24} -2.80821 q^{26} +5.79241 q^{27} -0.591346 q^{28} -9.18926 q^{29} -2.11268 q^{31} -5.54758 q^{32} -1.77376 q^{33} +4.89804 q^{34} -8.27300 q^{36} +2.51815 q^{37} +2.56040 q^{38} +13.2852 q^{39} -2.70037 q^{41} -0.612743 q^{42} +1.59843 q^{43} +1.02616 q^{44} +1.75568 q^{46} -5.73910 q^{47} +5.59542 q^{48} -6.87009 q^{49} -23.1718 q^{51} -7.68580 q^{52} +9.95812 q^{53} -3.47230 q^{54} +0.786614 q^{56} -12.1128 q^{57} +5.50856 q^{58} -6.89519 q^{59} +4.62027 q^{61} +1.26646 q^{62} +1.81749 q^{63} -0.620559 q^{64} +1.06329 q^{66} +6.13471 q^{67} +13.4054 q^{68} -8.30582 q^{69} -12.7519 q^{71} +11.0048 q^{72} -0.559355 q^{73} -1.50952 q^{74} +7.00756 q^{76} -0.225436 q^{77} -7.96390 q^{78} +4.10565 q^{79} +1.29937 q^{81} +1.61875 q^{82} -15.1584 q^{83} -1.67702 q^{84} -0.958189 q^{86} -26.0601 q^{87} -1.36501 q^{88} -13.6969 q^{89} +1.68849 q^{91} +4.80511 q^{92} -5.99142 q^{93} +3.44034 q^{94} -15.7325 q^{96} +13.5129 q^{97} +4.11832 q^{98} -3.15388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) \) 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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