LMFDB - Embedded newform 4025.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7537.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 4x + 3 $ x^4 - x^3 - 5x^2 + 4x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.15976$ of defining polynomial
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15976 q^{2} -2.43525 q^{3} +2.66454 q^{4} +5.25956 q^{6} -1.00000 q^{7} -1.43525 q^{8} +2.93047 q^{9} +O(q^{10})$	\(q-2.15976 q^{2} -2.43525 q^{3} +2.66454 q^{4} +5.25956 q^{6} -1.00000 q^{7} -1.43525 q^{8} +2.93047 q^{9} +0.840244 q^{11} -6.48885 q^{12} +4.09980 q^{13} +2.15976 q^{14} -2.22929 q^{16} -5.15976 q^{17} -6.32909 q^{18} +4.38905 q^{19} +2.43525 q^{21} -1.81472 q^{22} +1.00000 q^{23} +3.49521 q^{24} -8.85457 q^{26} +0.169334 q^{27} -2.66454 q^{28} +0.458580 q^{29} -0.389046 q^{31} +7.68523 q^{32} -2.04621 q^{33} +11.1438 q^{34} +7.80836 q^{36} +10.8084 q^{37} -9.47927 q^{38} -9.98406 q^{39} -5.24361 q^{41} -5.25956 q^{42} +1.90341 q^{43} +2.23887 q^{44} -2.15976 q^{46} +8.58865 q^{47} +5.42889 q^{48} +1.00000 q^{49} +12.5653 q^{51} +10.9241 q^{52} +3.11355 q^{53} -0.365720 q^{54} +1.43525 q^{56} -10.6884 q^{57} -0.990422 q^{58} +4.21971 q^{59} +6.00321 q^{61} +0.840244 q^{62} -2.93047 q^{63} -12.1396 q^{64} +4.41931 q^{66} -2.09980 q^{67} -13.7484 q^{68} -2.43525 q^{69} -12.9814 q^{71} -4.20596 q^{72} -13.1932 q^{73} -23.3434 q^{74} +11.6948 q^{76} -0.840244 q^{77} +21.5631 q^{78} +11.4633 q^{79} -9.20377 q^{81} +11.3249 q^{82} +4.95115 q^{83} +6.48885 q^{84} -4.11091 q^{86} -1.11676 q^{87} -1.20596 q^{88} +6.84024 q^{89} -4.09980 q^{91} +2.66454 q^{92} +0.947426 q^{93} -18.5494 q^{94} -18.7155 q^{96} +8.02610 q^{97} -2.15976 q^{98} +2.46231 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - q^2 - 4 * q^3 + 3 * q^4 - 4 * q^7 + 6 * q^9	$ 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9} + 11 q^{11} - 3 q^{12} + 3 q^{13} + q^{14} - 7 q^{16} - 13 q^{17} - 10 q^{18} + 8 q^{19} + 4 q^{21} + 8 q^{22} + 4 q^{23} + 14 q^{24} - q^{26} - 7 q^{27} - 3 q^{28} - 2 q^{29} + 8 q^{31} + 4 q^{32} - 12 q^{33} + 14 q^{34} - 7 q^{36} + 5 q^{37} - 15 q^{38} - 17 q^{39} + 23 q^{41} - 2 q^{43} + 7 q^{44} - q^{46} - 2 q^{47} - 7 q^{48} + 4 q^{49} + 12 q^{51} + 15 q^{52} + q^{53} + 10 q^{54} + 7 q^{57} - 4 q^{58} + 15 q^{59} + q^{61} + 11 q^{62} - 6 q^{63} - 16 q^{64} - 11 q^{66} + 5 q^{67} - 11 q^{68} - 4 q^{69} - 8 q^{71} - 13 q^{72} - 3 q^{73} - 36 q^{74} + 20 q^{76} - 11 q^{77} + 27 q^{78} - 12 q^{81} + 28 q^{82} - 5 q^{83} + 3 q^{84} + 16 q^{86} + 30 q^{87} - q^{88} + 35 q^{89} - 3 q^{91} + 3 q^{92} - 23 q^{93} - 13 q^{94} - 29 q^{96} + 11 q^{97} - q^{98} + 8 q^{99}+O(q^{100}) $ 4 * q - q^2 - 4 * q^3 + 3 * q^4 - 4 * q^7 + 6 * q^9 + 11 * q^11 - 3 * q^12 + 3 * q^13 + q^14 - 7 * q^16 - 13 * q^17 - 10 * q^18 + 8 * q^19 + 4 * q^21 + 8 * q^22 + 4 * q^23 + 14 * q^24 - q^26 - 7 * q^27 - 3 * q^28 - 2 * q^29 + 8 * q^31 + 4 * q^32 - 12 * q^33 + 14 * q^34 - 7 * q^36 + 5 * q^37 - 15 * q^38 - 17 * q^39 + 23 * q^41 - 2 * q^43 + 7 * q^44 - q^46 - 2 * q^47 - 7 * q^48 + 4 * q^49 + 12 * q^51 + 15 * q^52 + q^53 + 10 * q^54 + 7 * q^57 - 4 * q^58 + 15 * q^59 + q^61 + 11 * q^62 - 6 * q^63 - 16 * q^64 - 11 * q^66 + 5 * q^67 - 11 * q^68 - 4 * q^69 - 8 * q^71 - 13 * q^72 - 3 * q^73 - 36 * q^74 + 20 * q^76 - 11 * q^77 + 27 * q^78 - 12 * q^81 + 28 * q^82 - 5 * q^83 + 3 * q^84 + 16 * q^86 + 30 * q^87 - q^88 + 35 * q^89 - 3 * q^91 + 3 * q^92 - 23 * q^93 - 13 * q^94 - 29 * q^96 + 11 * q^97 - q^98 + 8 * 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 $

$2$	−2.15976	−1.52718	−0.763589	−	0.645703i	$-0.776565\pi$
$2$	−2.15976	−1.52718	−0.763589	+	0.645703i	$0.776565\pi$
$3$	−2.43525	−1.40599	−0.702997	−	0.711192i	$-0.748156\pi$
$3$	−2.43525	−1.40599	−0.702997	+	0.711192i	$0.748156\pi$
$4$	2.66454	1.33227
$5$	0	0
$5$	0	0
$6$	5.25956	2.14720
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.43525	−0.507439
$9$	2.93047	0.976822
$10$	0	0
$11$	0.840244	0.253343	0.126672	−	0.991945i	$-0.459571\pi$
$11$	0.840244	0.253343	0.126672	+	0.991945i	$0.459571\pi$
$12$	−6.48885	−1.87317
$13$	4.09980	1.13708	0.568540	−	0.822656i	$-0.307508\pi$
$13$	4.09980	1.13708	0.568540	+	0.822656i	$0.307508\pi$
$14$	2.15976	0.577219
$15$	0	0
$16$	−2.22929	−0.557323
$17$	−5.15976	−1.25142	−0.625712	−	0.780054i	$-0.715192\pi$
$17$	−5.15976	−1.25142	−0.625712	+	0.780054i	$0.715192\pi$
$18$	−6.32909	−1.49178
$19$	4.38905	1.00692	0.503458	−	0.864020i	$-0.332061\pi$
$19$	4.38905	1.00692	0.503458	+	0.864020i	$0.332061\pi$
$20$	0	0
$21$	2.43525	0.531416
$22$	−1.81472	−0.386900
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	3.49521	0.713457
$25$	0	0
$26$	−8.85457	−1.73652
$27$	0.169334	0.0325884
$28$	−2.66454	−0.503552
$29$	0.458580	0.0851562	0.0425781	−	0.999093i	$-0.486443\pi$
$29$	0.458580	0.0851562	0.0425781	+	0.999093i	$0.486443\pi$
$30$	0	0
$31$	−0.389046	−0.0698747	−0.0349374	−	0.999390i	$-0.511123\pi$
$31$	−0.389046	−0.0698747	−0.0349374	+	0.999390i	$0.511123\pi$
$32$	7.68523	1.35857
$33$	−2.04621	−0.356199
$34$	11.1438	1.91115
$35$	0	0
$36$	7.80836	1.30139
$37$	10.8084	1.77688	0.888441	−	0.458990i	$-0.151789\pi$
$37$	10.8084	1.77688	0.888441	+	0.458990i	$0.151789\pi$
$38$	−9.47927	−1.53774
$39$	−9.98406	−1.59873
$40$	0	0
$41$	−5.24361	−0.818915	−0.409457	−	0.912329i	$-0.634282\pi$
$41$	−5.24361	−0.818915	−0.409457	+	0.912329i	$0.634282\pi$
$42$	−5.25956	−0.811567
$43$	1.90341	0.290268	0.145134	−	0.989412i	$-0.453639\pi$
$43$	1.90341	0.290268	0.145134	+	0.989412i	$0.453639\pi$
$44$	2.23887	0.337522
$45$	0	0
$46$	−2.15976	−0.318439
$47$	8.58865	1.25278	0.626391	−	0.779509i	$-0.284531\pi$
$47$	8.58865	1.25278	0.626391	+	0.779509i	$0.284531\pi$
$48$	5.42889	0.783593
$49$	1.00000	0.142857
$50$	0	0
$51$	12.5653	1.75950
$52$	10.9241	1.51490
$53$	3.11355	0.427679	0.213839	−	0.976869i	$-0.431403\pi$
$53$	3.11355	0.427679	0.213839	+	0.976869i	$0.431403\pi$
$54$	−0.365720	−0.0497682
$55$	0	0
$56$	1.43525	0.191794
$57$	−10.6884	−1.41572
$58$	−0.990422	−0.130049
$59$	4.21971	0.549360	0.274680	−	0.961536i	$-0.411428\pi$
$59$	4.21971	0.549360	0.274680	+	0.961536i	$0.411428\pi$
$60$	0	0
$61$	6.00321	0.768633	0.384316	−	0.923201i	$-0.374437\pi$
$61$	6.00321	0.768633	0.384316	+	0.923201i	$0.374437\pi$
$62$	0.840244	0.106711
$63$	−2.93047	−0.369204
$64$	−12.1396	−1.51746
$65$	0	0
$66$	4.41931	0.543980
$67$	−2.09980	−0.256531	−0.128266	−	0.991740i	$-0.540941\pi$
$67$	−2.09980	−0.256531	−0.128266	+	0.991740i	$0.540941\pi$
$68$	−13.7484	−1.66724
$69$	−2.43525	−0.293170
$70$	0	0
$71$	−12.9814	−1.54061	−0.770305	−	0.637675i	$-0.779896\pi$
$71$	−12.9814	−1.54061	−0.770305	+	0.637675i	$0.779896\pi$
$72$	−4.20596	−0.495678
$73$	−13.1932	−1.54415	−0.772076	−	0.635531i	$-0.780781\pi$
$73$	−13.1932	−1.54415	−0.772076	+	0.635531i	$0.780781\pi$
$74$	−23.3434	−2.71362
$75$	0	0
$76$	11.6948	1.34149
$77$	−0.840244	−0.0957547
$78$	21.5631	2.44154
$79$	11.4633	1.28972	0.644862	−	0.764299i	$-0.276915\pi$
$79$	11.4633	1.28972	0.644862	+	0.764299i	$0.276915\pi$
$80$	0	0
$81$	−9.20377	−1.02264
$82$	11.3249	1.25063
$83$	4.95115	0.543460	0.271730	−	0.962374i	$-0.412404\pi$
$83$	4.95115	0.543460	0.271730	+	0.962374i	$0.412404\pi$
$84$	6.48885	0.707991
$85$	0	0
$86$	−4.11091	−0.443291
$87$	−1.11676	−0.119729
$88$	−1.20596	−0.128556
$89$	6.84024	0.725064	0.362532	−	0.931971i	$-0.381912\pi$
$89$	6.84024	0.725064	0.362532	+	0.931971i	$0.381912\pi$
$90$	0	0
$91$	−4.09980	−0.429776
$92$	2.66454	0.277798
$93$	0.947426	0.0982435
$94$	−18.5494	−1.91322
$95$	0	0
$96$	−18.7155	−1.91014
$97$	8.02610	0.814927	0.407463	−	0.913222i	$-0.366413\pi$
$97$	8.02610	0.814927	0.407463	+	0.913222i	$0.366413\pi$
$98$	−2.15976	−0.218168
$99$	2.46231	0.247471
$100$	0	0
$101$	17.9177	1.78288	0.891441	−	0.453137i	$-0.149695\pi$
$101$	17.9177	1.78288	0.891441	+	0.453137i	$0.149695\pi$
$102$	−27.1380	−2.68706
$103$	−1.93368	−0.190531	−0.0952655	−	0.995452i	$-0.530370\pi$
$103$	−1.93368	−0.190531	−0.0952655	+	0.995452i	$0.530370\pi$
$104$	−5.88426	−0.576999
$105$	0	0
$106$	−6.72450	−0.653141
$107$	−11.4751	−1.10934	−0.554670	−	0.832071i	$-0.687155\pi$
$107$	−11.4751	−1.10934	−0.554670	+	0.832071i	$0.687155\pi$
$108$	0.451198	0.0434166
$109$	−7.84499	−0.751414	−0.375707	−	0.926739i	$-0.622600\pi$
$109$	−7.84499	−0.751414	−0.375707	+	0.926739i	$0.622600\pi$
$110$	0	0
$111$	−26.3211	−2.49829
$112$	2.22929	0.210648
$113$	−17.4936	−1.64566	−0.822829	−	0.568289i	$-0.807606\pi$
$113$	−17.4936	−1.64566	−0.822829	+	0.568289i	$0.807606\pi$
$114$	23.0844	2.16205
$115$	0	0
$116$	1.22191	0.113451
$117$	12.0143	1.11072
$118$	−9.11355	−0.838970
$119$	5.15976	0.472994
$120$	0	0
$121$	−10.2940	−0.935817
$122$	−12.9655	−1.17384
$123$	12.7695	1.15139
$124$	−1.03663	−0.0930922
$125$	0	0
$126$	6.32909	0.563840
$127$	−0.727714	−0.0645742	−0.0322871	−	0.999479i	$-0.510279\pi$
$127$	−0.727714	−0.0645742	−0.0322871	+	0.999479i	$0.510279\pi$
$128$	10.8482	0.958855
$129$	−4.63530	−0.408115
$130$	0	0
$131$	−4.06053	−0.354770	−0.177385	−	0.984142i	$-0.556764\pi$
$131$	−4.06053	−0.354770	−0.177385	+	0.984142i	$0.556764\pi$
$132$	−5.45222	−0.474554
$133$	−4.38905	−0.380579
$134$	4.53505	0.391769
$135$	0	0
$136$	7.40556	0.635022
$137$	−6.19266	−0.529075	−0.264537	−	0.964375i	$-0.585219\pi$
$137$	−6.19266	−0.529075	−0.264537	+	0.964375i	$0.585219\pi$
$138$	5.25956	0.447723
$139$	16.7813	1.42337	0.711686	−	0.702498i	$-0.247932\pi$
$139$	16.7813	1.42337	0.711686	+	0.702498i	$0.247932\pi$
$140$	0	0
$141$	−20.9155	−1.76141
$142$	28.0367	2.35279
$143$	3.44483	0.288071
$144$	−6.53286	−0.544405
$145$	0	0
$146$	28.4942	2.35819
$147$	−2.43525	−0.200856
$148$	28.7994	2.36729
$149$	12.8816	1.05530	0.527652	−	0.849461i	$-0.323073\pi$
$149$	12.8816	1.05530	0.527652	+	0.849461i	$0.323073\pi$
$150$	0	0
$151$	−11.6142	−0.945148	−0.472574	−	0.881291i	$-0.656675\pi$
$151$	−11.6142	−0.945148	−0.472574	+	0.881291i	$0.656675\pi$
$152$	−6.29940	−0.510949
$153$	−15.1205	−1.22242
$154$	1.81472	0.146234
$155$	0	0
$156$	−26.6030	−2.12994
$157$	−13.6163	−1.08670	−0.543348	−	0.839507i	$-0.682844\pi$
$157$	−13.6163	−1.08670	−0.543348	+	0.839507i	$0.682844\pi$
$158$	−24.7580	−1.96964
$159$	−7.58228	−0.601314
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	19.8779	1.56175
$163$	13.8015	1.08102	0.540510	−	0.841337i	$-0.318231\pi$
$163$	13.8015	1.08102	0.540510	+	0.841337i	$0.318231\pi$
$164$	−13.9718	−1.09102
$165$	0	0
$166$	−10.6933	−0.829959
$167$	−17.9018	−1.38528	−0.692641	−	0.721282i	$-0.743553\pi$
$167$	−17.9018	−1.38528	−0.692641	+	0.721282i	$0.743553\pi$
$168$	−3.49521	−0.269661
$169$	3.80836	0.292951
$170$	0	0
$171$	12.8619	0.983578
$172$	5.07173	0.386716
$173$	−22.1513	−1.68413	−0.842067	−	0.539373i	$-0.818661\pi$
$173$	−22.1513	−1.68413	−0.842067	+	0.539373i	$0.818661\pi$
$174$	2.41193	0.182848
$175$	0	0
$176$	−1.87315	−0.141194
$177$	−10.2761	−0.772397
$178$	−14.7733	−1.10730
$179$	6.34978	0.474605	0.237302	−	0.971436i	$-0.423737\pi$
$179$	6.34978	0.474605	0.237302	+	0.971436i	$0.423737\pi$
$180$	0	0
$181$	3.02486	0.224836	0.112418	−	0.993661i	$-0.464140\pi$
$181$	3.02486	0.224836	0.112418	+	0.993661i	$0.464140\pi$
$182$	8.85457	0.656344
$183$	−14.6194	−1.08069
$184$	−1.43525	−0.105808
$185$	0	0
$186$	−2.04621	−0.150035
$187$	−4.33546	−0.317040
$188$	22.8848	1.66905
$189$	−0.169334	−0.0123172
$190$	0	0
$191$	−18.4990	−1.33854	−0.669270	−	0.743019i	$-0.733393\pi$
$191$	−18.4990	−1.33854	−0.669270	+	0.743019i	$0.733393\pi$
$192$	29.5631	2.13353
$193$	0.339624	0.0244467	0.0122233	−	0.999925i	$-0.496109\pi$
$193$	0.339624	0.0244467	0.0122233	+	0.999925i	$0.496109\pi$
$194$	−17.3344	−1.24454
$195$	0	0
$196$	2.66454	0.190325
$197$	−5.48665	−0.390908	−0.195454	−	0.980713i	$-0.562618\pi$
$197$	−5.48665	−0.390908	−0.195454	+	0.980713i	$0.562618\pi$
$198$	−5.31798	−0.377932
$199$	−4.49301	−0.318501	−0.159251	−	0.987238i	$-0.550908\pi$
$199$	−4.49301	−0.318501	−0.159251	+	0.987238i	$0.550908\pi$
$200$	0	0
$201$	5.11355	0.360682
$202$	−38.6979	−2.72278
$203$	−0.458580	−0.0321860
$204$	33.4809	2.34413
$205$	0	0
$206$	4.17627	0.290975
$207$	2.93047	0.203681
$208$	−9.13964	−0.633720
$209$	3.68787	0.255095
$210$	0	0
$211$	−8.08707	−0.556737	−0.278368	−	0.960474i	$-0.589794\pi$
$211$	−8.08707	−0.556737	−0.278368	+	0.960474i	$0.589794\pi$
$212$	8.29619	0.569784
$213$	31.6131	2.16609
$214$	24.7834	1.69416
$215$	0	0
$216$	−0.243038	−0.0165366
$217$	0.389046	0.0264102
$218$	16.9433	1.14754
$219$	32.1289	2.17107
$220$	0	0
$221$	−21.1540	−1.42297
$222$	56.8472	3.81533
$223$	7.25217	0.485641	0.242821	−	0.970071i	$-0.421927\pi$
$223$	7.25217	0.485641	0.242821	+	0.970071i	$0.421927\pi$
$224$	−7.68523	−0.513491
$225$	0	0
$226$	37.7819	2.51321
$227$	7.56379	0.502026	0.251013	−	0.967984i	$-0.419236\pi$
$227$	7.56379	0.502026	0.251013	+	0.967984i	$0.419236\pi$
$228$	−28.4798	−1.88612
$229$	8.67508	0.573265	0.286633	−	0.958041i	$-0.407464\pi$
$229$	8.67508	0.573265	0.286633	+	0.958041i	$0.407464\pi$
$230$	0	0
$231$	2.04621	0.134631
$232$	−0.658180	−0.0432116
$233$	26.3403	1.72561	0.862804	−	0.505539i	$-0.168706\pi$
$233$	26.3403	1.72561	0.862804	+	0.505539i	$0.168706\pi$
$234$	−25.9480	−1.69627
$235$	0	0
$236$	11.2436	0.731897
$237$	−27.9161	−1.81335
$238$	−11.1438	−0.722346
$239$	14.0531	0.909023	0.454511	−	0.890741i	$-0.349814\pi$
$239$	14.0531	0.909023	0.454511	+	0.890741i	$0.349814\pi$
$240$	0	0
$241$	−5.74840	−0.370287	−0.185144	−	0.982711i	$-0.559275\pi$
$241$	−5.74840	−0.370287	−0.185144	+	0.982711i	$0.559275\pi$
$242$	22.2325	1.42916
$243$	21.9055	1.40524
$244$	15.9958	1.02403
$245$	0	0
$246$	−27.5791	−1.75838
$247$	17.9942	1.14494
$248$	0.558380	0.0354572
$249$	−12.0573	−0.764101
$250$	0	0
$251$	11.8912	0.750566	0.375283	−	0.926910i	$-0.377546\pi$
$251$	11.8912	0.750566	0.375283	+	0.926910i	$0.377546\pi$
$252$	−7.80836	−0.491880
$253$	0.840244	0.0528257
$254$	1.57168	0.0986163
$255$	0	0
$256$	0.849823	0.0531139
$257$	0.0274947	0.00171507	0.000857536	−	1.00000i	$-0.499727\pi$
$257$	0.0274947	0.00171507	0.000857536		1.00000i	$0.499727\pi$
$258$	10.0111	0.623264
$259$	−10.8084	−0.671599
$260$	0	0
$261$	1.34385	0.0831825
$262$	8.76975	0.541797
$263$	−1.59128	−0.0981228	−0.0490614	−	0.998796i	$-0.515623\pi$
$263$	−1.59128	−0.0981228	−0.0490614	+	0.998796i	$0.515623\pi$
$264$	2.93683	0.180749
$265$	0	0
$266$	9.47927	0.581211
$267$	−16.6577	−1.01944
$268$	−5.59501	−0.341770
$269$	−9.88747	−0.602850	−0.301425	−	0.953490i	$-0.597462\pi$
$269$	−9.88747	−0.602850	−0.301425	+	0.953490i	$0.597462\pi$
$270$	0	0
$271$	−5.68743	−0.345487	−0.172743	−	0.984967i	$-0.555263\pi$
$271$	−5.68743	−0.345487	−0.172743	+	0.984967i	$0.555263\pi$
$272$	11.5026	0.697447
$273$	9.98406	0.604263
$274$	13.3746	0.807991
$275$	0	0
$276$	−6.48885	−0.390583
$277$	21.4432	1.28840	0.644199	−	0.764858i	$-0.277191\pi$
$277$	21.4432	1.28840	0.644199	+	0.764858i	$0.277191\pi$
$278$	−36.2435	−2.17374
$279$	−1.14009	−0.0682552
$280$	0	0
$281$	−20.2458	−1.20776	−0.603882	−	0.797074i	$-0.706380\pi$
$281$	−20.2458	−1.20776	−0.603882	+	0.797074i	$0.706380\pi$
$282$	45.1725	2.68998
$283$	21.4592	1.27561	0.637807	−	0.770196i	$-0.279842\pi$
$283$	21.4592	1.27561	0.637807	+	0.770196i	$0.279842\pi$
$284$	−34.5896	−2.05251
$285$	0	0
$286$	−7.44000	−0.439936
$287$	5.24361	0.309521
$288$	22.5213	1.32708
$289$	9.62308	0.566063
$290$	0	0
$291$	−19.5456	−1.14578
$292$	−35.1540	−2.05723
$293$	−31.9755	−1.86803	−0.934014	−	0.357236i	$-0.883719\pi$
$293$	−31.9755	−1.86803	−0.934014	+	0.357236i	$0.883719\pi$
$294$	5.25956	0.306743
$295$	0	0
$296$	−15.5127	−0.901660
$297$	0.142282	0.00825604
$298$	−27.8211	−1.61164
$299$	4.09980	0.237098
$300$	0	0
$301$	−1.90341	−0.109711
$302$	25.0838	1.44341
$303$	−43.6342	−2.50672
$304$	−9.78446	−0.561177
$305$	0	0
$306$	32.6566	1.86685
$307$	18.8950	1.07839	0.539197	−	0.842180i	$-0.318728\pi$
$307$	18.8950	1.07839	0.539197	+	0.842180i	$0.318728\pi$
$308$	−2.23887	−0.127571
$309$	4.70900	0.267886
$310$	0	0
$311$	18.9585	1.07504	0.537520	−	0.843251i	$-0.319361\pi$
$311$	18.9585	1.07504	0.537520	+	0.843251i	$0.319361\pi$
$312$	14.3297	0.811257
$313$	−21.6546	−1.22399	−0.611995	−	0.790862i	$-0.709633\pi$
$313$	−21.6546	−1.22399	−0.611995	+	0.790862i	$0.709633\pi$
$314$	29.4078	1.65958
$315$	0	0
$316$	30.5445	1.71826
$317$	−17.3928	−0.976879	−0.488439	−	0.872598i	$-0.662434\pi$
$317$	−17.3928	−0.976879	−0.488439	+	0.872598i	$0.662434\pi$
$318$	16.3759	0.918313
$319$	0.385320	0.0215738
$320$	0	0
$321$	27.9448	1.55973
$322$	2.15976	0.120358
$323$	−22.6464	−1.26008
$324$	−24.5239	−1.36244
$325$	0	0
$326$	−29.8080	−1.65091
$327$	19.1045	1.05648
$328$	7.52592	0.415549
$329$	−8.58865	−0.473507
$330$	0	0
$331$	33.2775	1.82910	0.914549	−	0.404474i	$-0.132546\pi$
$331$	33.2775	1.82910	0.914549	+	0.404474i	$0.132546\pi$
$332$	13.1926	0.724036
$333$	31.6735	1.73570
$334$	38.6635	2.11557
$335$	0	0
$336$	−5.42889	−0.296170
$337$	23.6352	1.28749	0.643746	−	0.765239i	$-0.277379\pi$
$337$	23.6352	1.28749	0.643746	+	0.765239i	$0.277379\pi$
$338$	−8.22512	−0.447388
$339$	42.6013	2.31379
$340$	0	0
$341$	−0.326894	−0.0177023
$342$	−27.7787	−1.50210
$343$	−1.00000	−0.0539949
$344$	−2.73188	−0.147293
$345$	0	0
$346$	47.8415	2.57197
$347$	12.8737	0.691098	0.345549	−	0.938401i	$-0.387693\pi$
$347$	12.8737	0.691098	0.345549	+	0.938401i	$0.387693\pi$
$348$	−2.97566	−0.159512
$349$	−5.89603	−0.315607	−0.157804	−	0.987471i	$-0.550441\pi$
$349$	−5.89603	−0.315607	−0.157804	+	0.987471i	$0.550441\pi$
$350$	0	0
$351$	0.694236	0.0370556
$352$	6.45747	0.344184
$353$	11.0131	0.586167	0.293084	−	0.956087i	$-0.405319\pi$
$353$	11.0131	0.586167	0.293084	+	0.956087i	$0.405319\pi$
$354$	22.1938	1.17959
$355$	0	0
$356$	18.2261	0.965983
$357$	−12.5653	−0.665027
$358$	−13.7140	−0.724806
$359$	−12.5699	−0.663416	−0.331708	−	0.943382i	$-0.607625\pi$
$359$	−12.5699	−0.663416	−0.331708	+	0.943382i	$0.607625\pi$
$360$	0	0
$361$	0.263724	0.0138802
$362$	−6.53295	−0.343364
$363$	25.0685	1.31575
$364$	−10.9241	−0.572578
$365$	0	0
$366$	31.5742	1.65041
$367$	12.5302	0.654072	0.327036	−	0.945012i	$-0.393950\pi$
$367$	12.5302	0.654072	0.327036	+	0.945012i	$0.393950\pi$
$368$	−2.22929	−0.116210
$369$	−15.3662	−0.799934
$370$	0	0
$371$	−3.11355	−0.161647
$372$	2.52446	0.130887
$373$	−0.634854	−0.0328715	−0.0164357	−	0.999865i	$-0.505232\pi$
$373$	−0.634854	−0.0328715	−0.0164357	+	0.999865i	$0.505232\pi$
$374$	9.36352	0.484176
$375$	0	0
$376$	−12.3269	−0.635711
$377$	1.88009	0.0968294
$378$	0.365720	0.0188106
$379$	38.8982	1.99807	0.999033	−	0.0439648i	$-0.0139990\pi$
$379$	38.8982	1.99807	0.999033	+	0.0439648i	$0.0139990\pi$
$380$	0	0
$381$	1.77217	0.0907910
$382$	39.9533	2.04419
$383$	22.4953	1.14945	0.574727	−	0.818345i	$-0.305108\pi$
$383$	22.4953	1.14945	0.574727	+	0.818345i	$0.305108\pi$
$384$	−26.4181	−1.34814
$385$	0	0
$386$	−0.733505	−0.0373344
$387$	5.57789	0.283540
$388$	21.3859	1.08570
$389$	−6.19960	−0.314332	−0.157166	−	0.987572i	$-0.550236\pi$
$389$	−6.19960	−0.314332	−0.157166	+	0.987572i	$0.550236\pi$
$390$	0	0
$391$	−5.15976	−0.260940
$392$	−1.43525	−0.0724913
$393$	9.88843	0.498805
$394$	11.8498	0.596985
$395$	0	0
$396$	6.56093	0.329699
$397$	33.3709	1.67484	0.837419	−	0.546562i	$-0.184064\pi$
$397$	33.3709	1.67484	0.837419	+	0.546562i	$0.184064\pi$
$398$	9.70381	0.486408
$399$	10.6884	0.535092
$400$	0	0
$401$	24.7675	1.23683	0.618414	−	0.785852i	$-0.287775\pi$
$401$	24.7675	1.23683	0.618414	+	0.785852i	$0.287775\pi$
$402$	−11.0440	−0.550825
$403$	−1.59501	−0.0794531
$404$	47.7426	2.37528
$405$	0	0
$406$	0.990422	0.0491538
$407$	9.08166	0.450161
$408$	−18.0344	−0.892838
$409$	−10.3588	−0.512209	−0.256104	−	0.966649i	$-0.582439\pi$
$409$	−10.3588	−0.512209	−0.256104	+	0.966649i	$0.582439\pi$
$410$	0	0
$411$	15.0807	0.743876
$412$	−5.15237	−0.253839
$413$	−4.21971	−0.207638
$414$	−6.32909	−0.311058
$415$	0	0
$416$	31.5079	1.54480
$417$	−40.8668	−2.00125
$418$	−7.96490	−0.389576
$419$	−6.15177	−0.300534	−0.150267	−	0.988645i	$-0.548013\pi$
$419$	−6.15177	−0.300534	−0.150267	+	0.988645i	$0.548013\pi$
$420$	0	0
$421$	−14.2382	−0.693928	−0.346964	−	0.937879i	$-0.612787\pi$
$421$	−14.2382	−0.693928	−0.346964	+	0.937879i	$0.612787\pi$
$422$	17.4661	0.850236
$423$	25.1687	1.22375
$424$	−4.46873	−0.217021
$425$	0	0
$426$	−68.2765	−3.30801
$427$	−6.00321	−0.290516
$428$	−30.5759	−1.47794
$429$	−8.38905	−0.405027
$430$	0	0
$431$	−12.3488	−0.594822	−0.297411	−	0.954750i	$-0.596123\pi$
$431$	−12.3488	−0.594822	−0.297411	+	0.954750i	$0.596123\pi$
$432$	−0.377495	−0.0181622
$433$	33.7886	1.62378	0.811888	−	0.583813i	$-0.198440\pi$
$433$	33.7886	1.62378	0.811888	+	0.583813i	$0.198440\pi$
$434$	−0.840244	−0.0403330
$435$	0	0
$436$	−20.9033	−1.00109
$437$	4.38905	0.209957
$438$	−69.3905	−3.31561
$439$	32.5085	1.55155	0.775773	−	0.631012i	$-0.217360\pi$
$439$	32.5085	1.55155	0.775773	+	0.631012i	$0.217360\pi$
$440$	0	0
$441$	2.93047	0.139546
$442$	45.6874	2.17313
$443$	4.20281	0.199682	0.0998408	−	0.995003i	$-0.468167\pi$
$443$	4.20281	0.199682	0.0998408	+	0.995003i	$0.468167\pi$
$444$	−70.1338	−3.32840
$445$	0	0
$446$	−15.6629	−0.741661
$447$	−31.3700	−1.48375
$448$	12.1396	0.573544
$449$	15.1301	0.714035	0.357017	−	0.934098i	$-0.383794\pi$
$449$	15.1301	0.714035	0.357017	+	0.934098i	$0.383794\pi$
$450$	0	0
$451$	−4.40591	−0.207466
$452$	−46.6125	−2.19247
$453$	28.2835	1.32887
$454$	−16.3359	−0.766684
$455$	0	0
$456$	15.3406	0.718391
$457$	−41.3561	−1.93456	−0.967279	−	0.253716i	$-0.918347\pi$
$457$	−41.3561	−1.93456	−0.967279	+	0.253716i	$0.918347\pi$
$458$	−18.7361	−0.875478
$459$	−0.873723	−0.0407819
$460$	0	0
$461$	28.5376	1.32913	0.664564	−	0.747231i	$-0.268617\pi$
$461$	28.5376	1.32913	0.664564	+	0.747231i	$0.268617\pi$
$462$	−4.41931	−0.205605
$463$	38.1719	1.77400	0.886999	−	0.461772i	$-0.152786\pi$
$463$	38.1719	1.77400	0.886999	+	0.461772i	$0.152786\pi$
$464$	−1.02231	−0.0474595
$465$	0	0
$466$	−56.8885	−2.63531
$467$	−14.0213	−0.648827	−0.324413	−	0.945915i	$-0.605167\pi$
$467$	−14.0213	−0.648827	−0.324413	+	0.945915i	$0.605167\pi$
$468$	32.0127	1.47979
$469$	2.09980	0.0969597
$470$	0	0
$471$	33.1591	1.52789
$472$	−6.05636	−0.278767
$473$	1.59933	0.0735374
$474$	60.2920	2.76930
$475$	0	0
$476$	13.7484	0.630157
$477$	9.12414	0.417766
$478$	−30.3514	−1.38824
$479$	−16.4713	−0.752592	−0.376296	−	0.926499i	$-0.622802\pi$
$479$	−16.4713	−0.752592	−0.376296	+	0.926499i	$0.622802\pi$
$480$	0	0
$481$	44.3121	2.02046
$482$	12.4151	0.565494
$483$	2.43525	0.110808
$484$	−27.4288	−1.24676
$485$	0	0
$486$	−47.3106	−2.14605
$487$	−9.33485	−0.423002	−0.211501	−	0.977378i	$-0.567835\pi$
$487$	−9.33485	−0.423002	−0.211501	+	0.977378i	$0.567835\pi$
$488$	−8.61614	−0.390034
$489$	−33.6103	−1.51991
$490$	0	0
$491$	27.9517	1.26144	0.630722	−	0.776009i	$-0.282759\pi$
$491$	27.9517	1.26144	0.630722	+	0.776009i	$0.282759\pi$
$492$	34.0250	1.53396
$493$	−2.36616	−0.106567
$494$	−38.8631	−1.74853
$495$	0	0
$496$	0.867296	0.0389428
$497$	12.9814	0.582296
$498$	26.0409	1.16692
$499$	34.1740	1.52984	0.764919	−	0.644127i	$-0.222779\pi$
$499$	34.1740	1.52984	0.764919	+	0.644127i	$0.222779\pi$
$500$	0	0
$501$	43.5954	1.94770
$502$	−25.6821	−1.14625
$503$	19.8376	0.884515	0.442258	−	0.896888i	$-0.354178\pi$
$503$	19.8376	0.884515	0.442258	+	0.896888i	$0.354178\pi$
$504$	4.20596	0.187349
$505$	0	0
$506$	−1.81472	−0.0806742
$507$	−9.27432	−0.411887
$508$	−1.93903	−0.0860304
$509$	14.2548	0.631833	0.315917	−	0.948787i	$-0.397688\pi$
$509$	14.2548	0.631833	0.315917	+	0.948787i	$0.397688\pi$
$510$	0	0
$511$	13.1932	0.583634
$512$	−23.5318	−1.03997
$513$	0.743215	0.0328137
$514$	−0.0593818	−0.00261922
$515$	0	0
$516$	−12.3510	−0.543721
$517$	7.21656	0.317384
$518$	23.3434	1.02565
$519$	53.9441	2.36788
$520$	0	0
$521$	−1.05235	−0.0461043	−0.0230522	−	0.999734i	$-0.507338\pi$
$521$	−1.05235	−0.0461043	−0.0230522	+	0.999734i	$0.507338\pi$
$522$	−2.90240	−0.127034
$523$	30.8731	1.34998	0.674992	−	0.737825i	$-0.264147\pi$
$523$	30.8731	1.34998	0.674992	+	0.737825i	$0.264147\pi$
$524$	−10.8195	−0.472651
$525$	0	0
$526$	3.43678	0.149851
$527$	2.00738	0.0874429
$528$	4.56159	0.198518
$529$	1.00000	0.0434783
$530$	0	0
$531$	12.3657	0.536627
$532$	−11.6948	−0.507034
$533$	−21.4978	−0.931171
$534$	35.9766	1.55686
$535$	0	0
$536$	3.01375	0.130174
$537$	−15.4633	−0.667292
$538$	21.3545	0.920659
$539$	0.840244	0.0361919
$540$	0	0
$541$	−30.9709	−1.33154	−0.665772	−	0.746155i	$-0.731898\pi$
$541$	−30.9709	−1.33154	−0.665772	+	0.746155i	$0.731898\pi$
$542$	12.2835	0.527620
$543$	−7.36629	−0.316118
$544$	−39.6539	−1.70015
$545$	0	0
$546$	−21.5631	−0.922816
$547$	−20.3743	−0.871140	−0.435570	−	0.900155i	$-0.643453\pi$
$547$	−20.3743	−0.871140	−0.435570	+	0.900155i	$0.643453\pi$
$548$	−16.5006	−0.704872
$549$	17.5922	0.750817
$550$	0	0
$551$	2.01273	0.0857452
$552$	3.49521	0.148766
$553$	−11.4633	−0.487470
$554$	−46.3121	−1.96761
$555$	0	0
$556$	44.7145	1.89632
$557$	8.29157	0.351325	0.175663	−	0.984450i	$-0.443793\pi$
$557$	8.29157	0.351325	0.175663	+	0.984450i	$0.443793\pi$
$558$	2.46231	0.104238
$559$	7.80361	0.330058
$560$	0	0
$561$	10.5579	0.445756
$562$	43.7260	1.84447
$563$	−31.4861	−1.32698	−0.663491	−	0.748184i	$-0.730926\pi$
$563$	−31.4861	−1.32698	−0.663491	+	0.748184i	$0.730926\pi$
$564$	−55.7304	−2.34667
$565$	0	0
$566$	−46.3465	−1.94809
$567$	9.20377	0.386522
$568$	18.6316	0.781766
$569$	−17.2974	−0.725146	−0.362573	−	0.931955i	$-0.618102\pi$
$569$	−17.2974	−0.725146	−0.362573	+	0.931955i	$0.618102\pi$
$570$	0	0
$571$	−28.2162	−1.18081	−0.590405	−	0.807107i	$-0.701032\pi$
$571$	−28.2162	−1.18081	−0.590405	+	0.807107i	$0.701032\pi$
$572$	9.17891	0.383790
$573$	45.0498	1.88198
$574$	−11.3249	−0.472693
$575$	0	0
$576$	−35.5748	−1.48228
$577$	44.3581	1.84665	0.923326	−	0.384018i	$-0.125460\pi$
$577$	44.3581	1.84665	0.923326	+	0.384018i	$0.125460\pi$
$578$	−20.7835	−0.864480
$579$	−0.827071	−0.0343719
$580$	0	0
$581$	−4.95115	−0.205408
$582$	42.2137	1.74981
$583$	2.61614	0.108349
$584$	18.9357	0.783563
$585$	0	0
$586$	69.0593	2.85281
$587$	−33.0232	−1.36301	−0.681507	−	0.731811i	$-0.738675\pi$
$587$	−33.0232	−1.36301	−0.681507	+	0.731811i	$0.738675\pi$
$588$	−6.48885	−0.267595
$589$	−1.70754	−0.0703580
$590$	0	0
$591$	13.3614	0.549614
$592$	−24.0950	−0.990297
$593$	−5.47510	−0.224835	−0.112418	−	0.993661i	$-0.535859\pi$
$593$	−5.47510	−0.224835	−0.112418	+	0.993661i	$0.535859\pi$
$594$	−0.307294	−0.0126084
$595$	0	0
$596$	34.3236	1.40595
$597$	10.9416	0.447811
$598$	−8.85457	−0.362090
$599$	4.39436	0.179549	0.0897744	−	0.995962i	$-0.471385\pi$
$599$	4.39436	0.179549	0.0897744	+	0.995962i	$0.471385\pi$
$600$	0	0
$601$	10.3218	0.421034	0.210517	−	0.977590i	$-0.432485\pi$
$601$	10.3218	0.421034	0.210517	+	0.977590i	$0.432485\pi$
$602$	4.11091	0.167548
$603$	−6.15339	−0.250585
$604$	−30.9465	−1.25919
$605$	0	0
$606$	94.2393	3.82821
$607$	−12.5078	−0.507675	−0.253838	−	0.967247i	$-0.581693\pi$
$607$	−12.5078	−0.507675	−0.253838	+	0.967247i	$0.581693\pi$
$608$	33.7308	1.36797
$609$	1.11676	0.0452534
$610$	0	0
$611$	35.2117	1.42451
$612$	−40.2892	−1.62860
$613$	−19.9926	−0.807493	−0.403746	−	0.914871i	$-0.632292\pi$
$613$	−19.9926	−0.807493	−0.403746	+	0.914871i	$0.632292\pi$
$614$	−40.8085	−1.64690
$615$	0	0
$616$	1.20596	0.0485897
$617$	−17.8958	−0.720458	−0.360229	−	0.932864i	$-0.617301\pi$
$617$	−17.8958	−0.720458	−0.360229	+	0.932864i	$0.617301\pi$
$618$	−10.1703	−0.409109
$619$	−20.5217	−0.824835	−0.412418	−	0.910995i	$-0.635316\pi$
$619$	−20.5217	−0.824835	−0.412418	+	0.910995i	$0.635316\pi$
$620$	0	0
$621$	0.169334	0.00679514
$622$	−40.9458	−1.64178
$623$	−6.84024	−0.274049
$624$	22.2574	0.891007
$625$	0	0
$626$	46.7686	1.86925
$627$	−8.98090	−0.358663
$628$	−36.2812	−1.44778
$629$	−55.7685	−2.22364
$630$	0	0
$631$	40.5972	1.61615	0.808074	−	0.589081i	$-0.200510\pi$
$631$	40.5972	1.61615	0.808074	+	0.589081i	$0.200510\pi$
$632$	−16.4528	−0.654457
$633$	19.6941	0.782769
$634$	37.5643	1.49187
$635$	0	0
$636$	−20.2033	−0.801114
$637$	4.09980	0.162440
$638$	−0.832196	−0.0329470
$639$	−38.0416	−1.50490
$640$	0	0
$641$	1.40652	0.0555542	0.0277771	−	0.999614i	$-0.491157\pi$
$641$	1.40652	0.0555542	0.0277771	+	0.999614i	$0.491157\pi$
$642$	−60.3539	−2.38198
$643$	25.0516	0.987939	0.493970	−	0.869479i	$-0.335545\pi$
$643$	25.0516	0.987939	0.493970	+	0.869479i	$0.335545\pi$
$644$	−2.66454	−0.104998
$645$	0	0
$646$	48.9107	1.92437
$647$	−8.92894	−0.351033	−0.175516	−	0.984477i	$-0.556159\pi$
$647$	−8.92894	−0.351033	−0.175516	+	0.984477i	$0.556159\pi$
$648$	13.2098	0.518928
$649$	3.54559	0.139177
$650$	0	0
$651$	−0.947426	−0.0371326
$652$	36.7748	1.44021
$653$	−15.6539	−0.612585	−0.306293	−	0.951937i	$-0.599089\pi$
$653$	−15.6539	−0.612585	−0.306293	+	0.951937i	$0.599089\pi$
$654$	−41.2611	−1.61344
$655$	0	0
$656$	11.6895	0.456400
$657$	−38.6623	−1.50836
$658$	18.5494	0.723130
$659$	34.7506	1.35369	0.676846	−	0.736125i	$-0.263346\pi$
$659$	34.7506	1.35369	0.676846	+	0.736125i	$0.263346\pi$
$660$	0	0
$661$	−24.1501	−0.939332	−0.469666	−	0.882844i	$-0.655626\pi$
$661$	−24.1501	−0.939332	−0.469666	+	0.882844i	$0.655626\pi$
$662$	−71.8713	−2.79336
$663$	51.5153	2.00069
$664$	−7.10616	−0.275773
$665$	0	0
$666$	−68.4071	−2.65072
$667$	0.458580	0.0177563
$668$	−47.7001	−1.84557
$669$	−17.6609	−0.682809
$670$	0	0
$671$	5.04417	0.194728
$672$	18.7155	0.721966
$673$	19.4458	0.749580	0.374790	−	0.927110i	$-0.377715\pi$
$673$	19.4458	0.749580	0.374790	+	0.927110i	$0.377715\pi$
$674$	−51.0463	−1.96623
$675$	0	0
$676$	10.1475	0.390290
$677$	38.8756	1.49411	0.747055	−	0.664762i	$-0.231467\pi$
$677$	38.8756	1.49411	0.747055	+	0.664762i	$0.231467\pi$
$678$	−92.0085	−3.53357
$679$	−8.02610	−0.308013
$680$	0	0
$681$	−18.4198	−0.705846
$682$	0.706010	0.0270345
$683$	−10.8663	−0.415789	−0.207894	−	0.978151i	$-0.566661\pi$
$683$	−10.8663	−0.415789	−0.207894	+	0.978151i	$0.566661\pi$
$684$	34.2712	1.31039
$685$	0	0
$686$	2.15976	0.0824599
$687$	−21.1260	−0.806008
$688$	−4.24326	−0.161773
$689$	12.7649	0.486305
$690$	0	0
$691$	−28.8349	−1.09693	−0.548466	−	0.836173i	$-0.684788\pi$
$691$	−28.8349	−1.09693	−0.548466	+	0.836173i	$0.684788\pi$
$692$	−59.0232	−2.24373
$693$	−2.46231	−0.0935353
$694$	−27.8041	−1.05543
$695$	0	0
$696$	1.60284	0.0607553
$697$	27.0558	1.02481
$698$	12.7340	0.481989
$699$	−64.1452	−2.42620
$700$	0	0
$701$	−35.3788	−1.33624	−0.668119	−	0.744054i	$-0.732900\pi$
$701$	−35.3788	−1.33624	−0.668119	+	0.744054i	$0.732900\pi$
$702$	−1.49938	−0.0565904
$703$	47.4384	1.78917
$704$	−10.2003	−0.384437
$705$	0	0
$706$	−23.7856	−0.895182
$707$	−17.9177	−0.673866
$708$	−27.3811	−1.02904
$709$	−11.3929	−0.427869	−0.213935	−	0.976848i	$-0.568628\pi$
$709$	−11.3929	−0.427869	−0.213935	+	0.976848i	$0.568628\pi$
$710$	0	0
$711$	33.5929	1.25983
$712$	−9.81749	−0.367926
$713$	−0.389046	−0.0145699
$714$	27.1380	1.01561
$715$	0	0
$716$	16.9193	0.632303
$717$	−34.2230	−1.27808
$718$	27.1480	1.01315
$719$	−36.0051	−1.34276	−0.671382	−	0.741112i	$-0.734299\pi$
$719$	−36.0051	−1.34276	−0.671382	+	0.741112i	$0.734299\pi$
$720$	0	0
$721$	1.93368	0.0720140
$722$	−0.569580	−0.0211976
$723$	13.9988	0.520622
$724$	8.05986	0.299543
$725$	0	0
$726$	−54.1418	−2.00939
$727$	−41.2129	−1.52850	−0.764250	−	0.644920i	$-0.776891\pi$
$727$	−41.2129	−1.52850	−0.764250	+	0.644920i	$0.776891\pi$
$728$	5.88426	0.218085
$729$	−25.7342	−0.953119
$730$	0	0
$731$	−9.82115	−0.363248
$732$	−38.9539	−1.43978
$733$	16.0478	0.592740	0.296370	−	0.955073i	$-0.404224\pi$
$733$	16.0478	0.592740	0.296370	+	0.955073i	$0.404224\pi$
$734$	−27.0622	−0.998885
$735$	0	0
$736$	7.68523	0.283281
$737$	−1.76434	−0.0649905
$738$	33.1873	1.22164
$739$	26.4821	0.974160	0.487080	−	0.873357i	$-0.338062\pi$
$739$	26.4821	0.974160	0.487080	+	0.873357i	$0.338062\pi$
$740$	0	0
$741$	−43.8205	−1.60979
$742$	6.72450	0.246864
$743$	−48.9865	−1.79714	−0.898570	−	0.438829i	$-0.855393\pi$
$743$	−48.9865	−1.79714	−0.898570	+	0.438829i	$0.855393\pi$
$744$	−1.35980	−0.0498526
$745$	0	0
$746$	1.37113	0.0502006
$747$	14.5092	0.530863
$748$	−11.5520	−0.422383
$749$	11.4751	0.419291
$750$	0	0
$751$	−6.12774	−0.223604	−0.111802	−	0.993730i	$-0.535662\pi$
$751$	−6.12774	−0.223604	−0.111802	+	0.993730i	$0.535662\pi$
$752$	−19.1466	−0.698204
$753$	−28.9581	−1.05529
$754$	−4.06053	−0.147876
$755$	0	0
$756$	−0.451198	−0.0164099
$757$	−3.93157	−0.142895	−0.0714477	−	0.997444i	$-0.522762\pi$
$757$	−3.93157	−0.142895	−0.0714477	+	0.997444i	$0.522762\pi$
$758$	−84.0106	−3.05140
$759$	−2.04621	−0.0742727
$760$	0	0
$761$	36.2107	1.31264	0.656318	−	0.754484i	$-0.272113\pi$
$761$	36.2107	1.31264	0.656318	+	0.754484i	$0.272113\pi$
$762$	−3.82745	−0.138654
$763$	7.84499	0.284008
$764$	−49.2914	−1.78330
$765$	0	0
$766$	−48.5843	−1.75542
$767$	17.3000	0.624666
$768$	−2.06953	−0.0746779
$769$	−39.7761	−1.43436	−0.717181	−	0.696887i	$-0.754568\pi$
$769$	−39.7761	−1.43436	−0.717181	+	0.696887i	$0.754568\pi$
$770$	0	0
$771$	−0.0669566	−0.00241138
$772$	0.904944	0.0325696
$773$	27.6920	0.996012	0.498006	−	0.867173i	$-0.334066\pi$
$773$	27.6920	0.996012	0.498006	+	0.867173i	$0.334066\pi$
$774$	−12.0469	−0.433016
$775$	0	0
$776$	−11.5195	−0.413526
$777$	26.3211	0.944264
$778$	13.3896	0.480041
$779$	−23.0145	−0.824578
$780$	0	0
$781$	−10.9076	−0.390303
$782$	11.1438	0.398502
$783$	0.0776533	0.00277510
$784$	−2.22929	−0.0796175
$785$	0	0
$786$	−21.3566	−0.761764
$787$	50.8088	1.81114	0.905569	−	0.424200i	$-0.139445\pi$
$787$	50.8088	1.81114	0.905569	+	0.424200i	$0.139445\pi$
$788$	−14.6194	−0.520795
$789$	3.87518	0.137960
$790$	0	0
$791$	17.4936	0.622000
$792$	−3.53404	−0.125577
$793$	24.6120	0.873997
$794$	−72.0730	−2.55778
$795$	0	0
$796$	−11.9718	−0.424331
$797$	10.1513	0.359578	0.179789	−	0.983705i	$-0.442458\pi$
$797$	10.1513	0.359578	0.179789	+	0.983705i	$0.442458\pi$
$798$	−23.0844	−0.817180
$799$	−44.3153	−1.56776
$800$	0	0
$801$	20.0451	0.708259
$802$	−53.4917	−1.88886
$803$	−11.0855	−0.391200
$804$	13.6253	0.480526
$805$	0	0
$806$	3.44483	0.121339
$807$	24.0785	0.847604
$808$	−25.7165	−0.904704
$809$	−53.7243	−1.88885	−0.944424	−	0.328731i	$-0.893379\pi$
$809$	−53.7243	−1.88885	−0.944424	+	0.328731i	$0.893379\pi$
$810$	0	0
$811$	31.9201	1.12087	0.560433	−	0.828200i	$-0.310635\pi$
$811$	31.9201	1.12087	0.560433	+	0.828200i	$0.310635\pi$
$812$	−1.22191	−0.0428806
$813$	13.8503	0.485752
$814$	−19.6142	−0.687476
$815$	0	0
$816$	−28.0117	−0.980607
$817$	8.35417	0.292275
$818$	22.3724	0.782234
$819$	−12.0143	−0.419814
$820$	0	0
$821$	−19.3032	−0.673687	−0.336843	−	0.941561i	$-0.609359\pi$
$821$	−19.3032	−0.673687	−0.336843	+	0.941561i	$0.609359\pi$
$822$	−32.5706	−1.13603
$823$	14.3814	0.501305	0.250653	−	0.968077i	$-0.419355\pi$
$823$	14.3814	0.501305	0.250653	+	0.968077i	$0.419355\pi$
$824$	2.77532	0.0966829
$825$	0	0
$826$	9.11355	0.317101
$827$	35.5456	1.23604	0.618020	−	0.786162i	$-0.287935\pi$
$827$	35.5456	1.23604	0.618020	+	0.786162i	$0.287935\pi$
$828$	7.80836	0.271359
$829$	−2.78964	−0.0968883	−0.0484442	−	0.998826i	$-0.515426\pi$
$829$	−2.78964	−0.0968883	−0.0484442	+	0.998826i	$0.515426\pi$
$830$	0	0
$831$	−52.2197	−1.81148
$832$	−49.7701	−1.72547
$833$	−5.15976	−0.178775
$834$	88.2622	3.05627
$835$	0	0
$836$	9.82650	0.339857
$837$	−0.0658787	−0.00227710
$838$	13.2863	0.458968
$839$	16.4482	0.567854	0.283927	−	0.958846i	$-0.408363\pi$
$839$	16.4482	0.567854	0.283927	+	0.958846i	$0.408363\pi$
$840$	0	0
$841$	−28.7897	−0.992748
$842$	30.7510	1.05975
$843$	49.3037	1.69811
$844$	−21.5484	−0.741725
$845$	0	0
$846$	−54.3583	−1.86888
$847$	10.2940	0.353706
$848$	−6.94100	−0.238355
$849$	−52.2585	−1.79351
$850$	0	0
$851$	10.8084	0.370506
$852$	84.2344	2.88582
$853$	21.1625	0.724589	0.362294	−	0.932064i	$-0.381994\pi$
$853$	21.1625	0.724589	0.362294	+	0.932064i	$0.381994\pi$
$854$	12.9655	0.443669
$855$	0	0
$856$	16.4697	0.562922
$857$	27.9577	0.955017	0.477509	−	0.878627i	$-0.341540\pi$
$857$	27.9577	0.955017	0.477509	+	0.878627i	$0.341540\pi$
$858$	18.1183	0.618548
$859$	46.5912	1.58967	0.794835	−	0.606825i	$-0.207557\pi$
$859$	46.5912	1.58967	0.794835	+	0.606825i	$0.207557\pi$
$860$	0	0
$861$	−12.7695	−0.435184
$862$	26.6704	0.908399
$863$	21.9288	0.746467	0.373233	−	0.927738i	$-0.378249\pi$
$863$	21.9288	0.746467	0.373233	+	0.927738i	$0.378249\pi$
$864$	1.30137	0.0442736
$865$	0	0
$866$	−72.9751	−2.47980
$867$	−23.4346	−0.795882
$868$	1.03663	0.0351855
$869$	9.63199	0.326743
$870$	0	0
$871$	−8.60876	−0.291697
$872$	11.2596	0.381297
$873$	23.5202	0.796038
$874$	−9.47927	−0.320641
$875$	0	0
$876$	85.6089	2.89246
$877$	−19.8568	−0.670515	−0.335258	−	0.942126i	$-0.608823\pi$
$877$	−19.8568	−0.670515	−0.335258	+	0.942126i	$0.608823\pi$
$878$	−70.2104	−2.36949
$879$	77.8685	2.62644
$880$	0	0
$881$	−13.5673	−0.457094	−0.228547	−	0.973533i	$-0.573397\pi$
$881$	−13.5673	−0.457094	−0.228547	+	0.973533i	$0.573397\pi$
$882$	−6.32909	−0.213112
$883$	47.9700	1.61432	0.807159	−	0.590334i	$-0.201004\pi$
$883$	47.9700	1.61432	0.807159	+	0.590334i	$0.201004\pi$
$884$	−56.3657	−1.89578
$885$	0	0
$886$	−9.07705	−0.304949
$887$	4.69122	0.157516	0.0787578	−	0.996894i	$-0.474905\pi$
$887$	4.69122	0.157516	0.0787578	+	0.996894i	$0.474905\pi$
$888$	37.7775	1.26773
$889$	0.727714	0.0244067
$890$	0	0
$891$	−7.73341	−0.259079
$892$	19.3237	0.647007
$893$	37.6960	1.26145
$894$	67.7516	2.26595
$895$	0	0
$896$	−10.8482	−0.362413
$897$	−9.98406	−0.333358
$898$	−32.6774	−1.09046
$899$	−0.178409	−0.00595027
$900$	0	0
$901$	−16.0651	−0.535207
$902$	9.51570	0.316838
$903$	4.63530	0.154253
$904$	25.1078	0.835072
$905$	0	0
$906$	−61.0854	−2.02943
$907$	13.4385	0.446219	0.223110	−	0.974793i	$-0.428379\pi$
$907$	13.4385	0.446219	0.223110	+	0.974793i	$0.428379\pi$
$908$	20.1541	0.668836
$909$	52.5073	1.74156
$910$	0	0
$911$	39.3395	1.30338	0.651688	−	0.758487i	$-0.274061\pi$
$911$	39.3395	1.30338	0.651688	+	0.758487i	$0.274061\pi$
$912$	23.8276	0.789012
$913$	4.16018	0.137682
$914$	89.3191	2.95441
$915$	0	0
$916$	23.1151	0.763746
$917$	4.06053	0.134091
$918$	1.88703	0.0622812
$919$	19.9555	0.658270	0.329135	−	0.944283i	$-0.393243\pi$
$919$	19.9555	0.658270	0.329135	+	0.944283i	$0.393243\pi$
$920$	0	0
$921$	−46.0141	−1.51622
$922$	−61.6342	−2.02982
$923$	−53.2212	−1.75180
$924$	5.45222	0.179365
$925$	0	0
$926$	−82.4419	−2.70921
$927$	−5.66658	−0.186115
$928$	3.52430	0.115691
$929$	−9.43328	−0.309496	−0.154748	−	0.987954i	$-0.549457\pi$
$929$	−9.43328	−0.309496	−0.154748	+	0.987954i	$0.549457\pi$
$930$	0	0
$931$	4.38905	0.143845
$932$	70.1848	2.29898
$933$	−46.1689	−1.51150
$934$	30.2825	0.990874
$935$	0	0
$936$	−17.2436	−0.563625
$937$	31.1805	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$937$	31.1805	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$938$	−4.53505	−0.148075
$939$	52.7344	1.72092
$940$	0	0
$941$	36.5848	1.19263	0.596315	−	0.802751i	$-0.296631\pi$
$941$	36.5848	1.19263	0.596315	+	0.802751i	$0.296631\pi$
$942$	−71.6155	−2.33336
$943$	−5.24361	−0.170755
$944$	−9.40696	−0.306171
$945$	0	0
$946$	−3.45417	−0.112305
$947$	−0.720998	−0.0234293	−0.0117146	−	0.999931i	$-0.503729\pi$
$947$	−0.720998	−0.0234293	−0.0117146	+	0.999931i	$0.503729\pi$
$948$	−74.3837	−2.41587
$949$	−54.0896	−1.75582
$950$	0	0
$951$	42.3560	1.37349
$952$	−7.40556	−0.240016
$953$	33.6160	1.08893	0.544464	−	0.838784i	$-0.316733\pi$
$953$	33.6160	1.08893	0.544464	+	0.838784i	$0.316733\pi$
$954$	−19.7059	−0.638003
$955$	0	0
$956$	37.4452	1.21107
$957$	−0.938351	−0.0303326
$958$	35.5739	1.14934
$959$	6.19266	0.199971
$960$	0	0
$961$	−30.8486	−0.995118
$962$	−95.7033	−3.08560
$963$	−33.6274	−1.08363
$964$	−15.3169	−0.493323
$965$	0	0
$966$	−5.25956	−0.169223
$967$	10.0952	0.324639	0.162320	−	0.986738i	$-0.448102\pi$
$967$	10.0952	0.324639	0.162320	+	0.986738i	$0.448102\pi$
$968$	14.7745	0.474870
$969$	55.1498	1.77167
$970$	0	0
$971$	32.5160	1.04349	0.521744	−	0.853102i	$-0.325281\pi$
$971$	32.5160	1.04349	0.521744	+	0.853102i	$0.325281\pi$
$972$	58.3682	1.87216
$973$	−16.7813	−0.537984
$974$	20.1610	0.646000
$975$	0	0
$976$	−13.3829	−0.428376
$977$	−38.2250	−1.22293	−0.611463	−	0.791273i	$-0.709419\pi$
$977$	−38.2250	−1.22293	−0.611463	+	0.791273i	$0.709419\pi$
$978$	72.5900	2.32117
$979$	5.74748	0.183690
$980$	0	0
$981$	−22.9895	−0.733997
$982$	−60.3689	−1.92645
$983$	−51.5206	−1.64325	−0.821627	−	0.570026i	$-0.806933\pi$
$983$	−51.5206	−1.64325	−0.821627	+	0.570026i	$0.806933\pi$
$984$	−18.3275	−0.584260
$985$	0	0
$986$	5.11033	0.162746
$987$	20.9155	0.665749
$988$	47.9464	1.52538
$989$	1.90341	0.0605250
$990$	0	0
$991$	−51.0003	−1.62008	−0.810039	−	0.586376i	$-0.800554\pi$
$991$	−51.0003	−1.62008	−0.810039	+	0.586376i	$0.800554\pi$
$992$	−2.98991	−0.0949297
$993$	−81.0393	−2.57170
$994$	−28.0367	−0.889270
$995$	0	0
$996$	−32.1273	−1.01799
$997$	−27.3713	−0.866859	−0.433430	−	0.901187i	$-0.642697\pi$
$997$	−27.3713	−0.866859	−0.433430	+	0.901187i	$0.642697\pi$
$998$	−73.8075	−2.33633
$999$	1.83022	0.0579057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.m.1.1		4
5.4	even	2		805.2.a.i.1.4	✓	4
15.14	odd	2		7245.2.a.bd.1.1		4
35.34	odd	2		5635.2.a.u.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.a.i.1.4	✓	4	5.4	even	2
4025.2.a.m.1.1		4	1.1	even	1	trivial
5635.2.a.u.1.4		4	35.34	odd	2
7245.2.a.bd.1.1		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15976 q^{2} -2.43525 q^{3} +2.66454 q^{4} +5.25956 q^{6} -1.00000 q^{7} -1.43525 q^{8} +2.93047 q^{9} +O(q^{10})\)	\(q-2.15976 q^{2} -2.43525 q^{3} +2.66454 q^{4} +5.25956 q^{6} -1.00000 q^{7} -1.43525 q^{8} +2.93047 q^{9} +0.840244 q^{11} -6.48885 q^{12} +4.09980 q^{13} +2.15976 q^{14} -2.22929 q^{16} -5.15976 q^{17} -6.32909 q^{18} +4.38905 q^{19} +2.43525 q^{21} -1.81472 q^{22} +1.00000 q^{23} +3.49521 q^{24} -8.85457 q^{26} +0.169334 q^{27} -2.66454 q^{28} +0.458580 q^{29} -0.389046 q^{31} +7.68523 q^{32} -2.04621 q^{33} +11.1438 q^{34} +7.80836 q^{36} +10.8084 q^{37} -9.47927 q^{38} -9.98406 q^{39} -5.24361 q^{41} -5.25956 q^{42} +1.90341 q^{43} +2.23887 q^{44} -2.15976 q^{46} +8.58865 q^{47} +5.42889 q^{48} +1.00000 q^{49} +12.5653 q^{51} +10.9241 q^{52} +3.11355 q^{53} -0.365720 q^{54} +1.43525 q^{56} -10.6884 q^{57} -0.990422 q^{58} +4.21971 q^{59} +6.00321 q^{61} +0.840244 q^{62} -2.93047 q^{63} -12.1396 q^{64} +4.41931 q^{66} -2.09980 q^{67} -13.7484 q^{68} -2.43525 q^{69} -12.9814 q^{71} -4.20596 q^{72} -13.1932 q^{73} -23.3434 q^{74} +11.6948 q^{76} -0.840244 q^{77} +21.5631 q^{78} +11.4633 q^{79} -9.20377 q^{81} +11.3249 q^{82} +4.95115 q^{83} +6.48885 q^{84} -4.11091 q^{86} -1.11676 q^{87} -1.20596 q^{88} +6.84024 q^{89} -4.09980 q^{91} +2.66454 q^{92} +0.947426 q^{93} -18.5494 q^{94} -18.7155 q^{96} +8.02610 q^{97} -2.15976 q^{98} +2.46231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - q^2 - 4 * q^3 + 3 * q^4 - 4 * q^7 + 6 * q^9	\( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9} + 11 q^{11} - 3 q^{12} + 3 q^{13} + q^{14} - 7 q^{16} - 13 q^{17} - 10 q^{18} + 8 q^{19} + 4 q^{21} + 8 q^{22} + 4 q^{23} + 14 q^{24} - q^{26} - 7 q^{27} - 3 q^{28} - 2 q^{29} + 8 q^{31} + 4 q^{32} - 12 q^{33} + 14 q^{34} - 7 q^{36} + 5 q^{37} - 15 q^{38} - 17 q^{39} + 23 q^{41} - 2 q^{43} + 7 q^{44} - q^{46} - 2 q^{47} - 7 q^{48} + 4 q^{49} + 12 q^{51} + 15 q^{52} + q^{53} + 10 q^{54} + 7 q^{57} - 4 q^{58} + 15 q^{59} + q^{61} + 11 q^{62} - 6 q^{63} - 16 q^{64} - 11 q^{66} + 5 q^{67} - 11 q^{68} - 4 q^{69} - 8 q^{71} - 13 q^{72} - 3 q^{73} - 36 q^{74} + 20 q^{76} - 11 q^{77} + 27 q^{78} - 12 q^{81} + 28 q^{82} - 5 q^{83} + 3 q^{84} + 16 q^{86} + 30 q^{87} - q^{88} + 35 q^{89} - 3 q^{91} + 3 q^{92} - 23 q^{93} - 13 q^{94} - 29 q^{96} + 11 q^{97} - q^{98} + 8 q^{99}+O(q^{100}) \) 4 * q - q^2 - 4 * q^3 + 3 * q^4 - 4 * q^7 + 6 * q^9 + 11 * q^11 - 3 * q^12 + 3 * q^13 + q^14 - 7 * q^16 - 13 * q^17 - 10 * q^18 + 8 * q^19 + 4 * q^21 + 8 * q^22 + 4 * q^23 + 14 * q^24 - q^26 - 7 * q^27 - 3 * q^28 - 2 * q^29 + 8 * q^31 + 4 * q^32 - 12 * q^33 + 14 * q^34 - 7 * q^36 + 5 * q^37 - 15 * q^38 - 17 * q^39 + 23 * q^41 - 2 * q^43 + 7 * q^44 - q^46 - 2 * q^47 - 7 * q^48 + 4 * q^49 + 12 * q^51 + 15 * q^52 + q^53 + 10 * q^54 + 7 * q^57 - 4 * q^58 + 15 * q^59 + q^61 + 11 * q^62 - 6 * q^63 - 16 * q^64 - 11 * q^66 + 5 * q^67 - 11 * q^68 - 4 * q^69 - 8 * q^71 - 13 * q^72 - 3 * q^73 - 36 * q^74 + 20 * q^76 - 11 * q^77 + 27 * q^78 - 12 * q^81 + 28 * q^82 - 5 * q^83 + 3 * q^84 + 16 * q^86 + 30 * q^87 - q^88 + 35 * q^89 - 3 * q^91 + 3 * q^92 - 23 * q^93 - 13 * q^94 - 29 * q^96 + 11 * q^97 - q^98 + 8 * q^99

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