LMFDB - Embedded newform 4025.2.a.m.1.3

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			1.3
Root			$-0.491918$ 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)

$f(q)$	$=$	$q+0.491918 q^{2} -2.84864 q^{3} -1.75802 q^{4} -1.40130 q^{6} -1.00000 q^{7} -1.84864 q^{8} +5.11474 q^{9} +O(q^{10})$	\(q+0.491918 q^{2} -2.84864 q^{3} -1.75802 q^{4} -1.40130 q^{6} -1.00000 q^{7} -1.84864 q^{8} +5.11474 q^{9} +3.49192 q^{11} +5.00795 q^{12} +0.0906212 q^{13} -0.491918 q^{14} +2.60665 q^{16} -2.50808 q^{17} +2.51603 q^{18} -3.09857 q^{19} +2.84864 q^{21} +1.71774 q^{22} +1.00000 q^{23} +5.26610 q^{24} +0.0445782 q^{26} -6.02411 q^{27} +1.75802 q^{28} -9.21331 q^{29} +7.09857 q^{31} +4.97954 q^{32} -9.94721 q^{33} -1.23377 q^{34} -8.99179 q^{36} -5.99179 q^{37} -1.52424 q^{38} -0.258147 q^{39} +11.1432 q^{41} +1.40130 q^{42} -10.8969 q^{43} -6.13885 q^{44} +0.491918 q^{46} -6.91733 q^{47} -7.42541 q^{48} +1.00000 q^{49} +7.14462 q^{51} -0.159314 q^{52} -7.43913 q^{53} -2.96337 q^{54} +1.84864 q^{56} +8.82671 q^{57} -4.53220 q^{58} +2.92554 q^{59} -10.8062 q^{61} +3.49192 q^{62} -5.11474 q^{63} -2.76378 q^{64} -4.89322 q^{66} +1.90938 q^{67} +4.40925 q^{68} -2.84864 q^{69} -15.0576 q^{71} -9.45529 q^{72} +8.09281 q^{73} -2.94747 q^{74} +5.44734 q^{76} -3.49192 q^{77} -0.126987 q^{78} -6.21761 q^{79} +1.81631 q^{81} +5.48152 q^{82} +8.85229 q^{83} -5.00795 q^{84} -5.36037 q^{86} +26.2454 q^{87} -6.45529 q^{88} +9.49192 q^{89} -0.0906212 q^{91} -1.75802 q^{92} -20.2213 q^{93} -3.40276 q^{94} -14.1849 q^{96} +9.20291 q^{97} +0.491918 q^{98} +17.8602 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.491918	0.347839	0.173919	−	0.984760i	$-0.444357\pi$
$2$	0.491918	0.347839	0.173919	+	0.984760i	$0.444357\pi$
$3$	−2.84864	−1.64466	−0.822331	−	0.569010i	$-0.807327\pi$
$3$	−2.84864	−1.64466	−0.822331	+	0.569010i	$0.807327\pi$
$4$	−1.75802	−0.879008
$5$	0	0
$5$	0	0
$6$	−1.40130	−0.572077
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.84864	−0.653592
$9$	5.11474	1.70491
$10$	0	0
$11$	3.49192	1.05285	0.526427	−	0.850221i	$-0.323532\pi$
$11$	3.49192	1.05285	0.526427	+	0.850221i	$0.323532\pi$
$12$	5.00795	1.44567
$13$	0.0906212	0.0251338	0.0125669	−	0.999921i	$-0.496000\pi$
$13$	0.0906212	0.0251338	0.0125669	+	0.999921i	$0.496000\pi$
$14$	−0.491918	−0.131471
$15$	0	0
$16$	2.60665	0.651663
$17$	−2.50808	−0.608299	−0.304150	−	0.952624i	$-0.598372\pi$
$17$	−2.50808	−0.608299	−0.304150	+	0.952624i	$0.598372\pi$
$18$	2.51603	0.593035
$19$	−3.09857	−0.710861	−0.355431	−	0.934703i	$-0.615666\pi$
$19$	−3.09857	−0.710861	−0.355431	+	0.934703i	$0.615666\pi$
$20$	0	0
$21$	2.84864	0.621624
$22$	1.71774	0.366223
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	5.26610	1.07494
$25$	0	0
$26$	0.0445782	0.00874251
$27$	−6.02411	−1.15934
$28$	1.75802	0.332234
$29$	−9.21331	−1.71087	−0.855434	−	0.517912i	$-0.826710\pi$
$29$	−9.21331	−1.71087	−0.855434	+	0.517912i	$0.826710\pi$
$30$	0	0
$31$	7.09857	1.27494	0.637471	−	0.770475i	$-0.279981\pi$
$31$	7.09857	1.27494	0.637471	+	0.770475i	$0.279981\pi$
$32$	4.97954	0.880266
$33$	−9.94721	−1.73159
$34$	−1.23377	−0.211590
$35$	0	0
$36$	−8.99179	−1.49863
$37$	−5.99179	−0.985044	−0.492522	−	0.870300i	$-0.663925\pi$
$37$	−5.99179	−0.985044	−0.492522	+	0.870300i	$0.663925\pi$
$38$	−1.52424	−0.247265
$39$	−0.258147	−0.0413366
$40$	0	0
$41$	11.1432	1.74027	0.870134	−	0.492815i	$-0.164032\pi$
$41$	11.1432	1.74027	0.870134	+	0.492815i	$0.164032\pi$
$42$	1.40130	0.216225
$43$	−10.8969	−1.66176	−0.830878	−	0.556454i	$-0.812161\pi$
$43$	−10.8969	−1.66176	−0.830878	+	0.556454i	$0.812161\pi$
$44$	−6.13885	−0.925466
$45$	0	0
$46$	0.491918	0.0725294
$47$	−6.91733	−1.00900	−0.504498	−	0.863413i	$-0.668322\pi$
$47$	−6.91733	−1.00900	−0.504498	+	0.863413i	$0.668322\pi$
$48$	−7.42541	−1.07177
$49$	1.00000	0.142857
$50$	0	0
$51$	7.14462	1.00045
$52$	−0.159314	−0.0220928
$53$	−7.43913	−1.02184	−0.510921	−	0.859627i	$-0.670696\pi$
$53$	−7.43913	−1.02184	−0.510921	+	0.859627i	$0.670696\pi$
$54$	−2.96337	−0.403264
$55$	0	0
$56$	1.84864	0.247035
$57$	8.82671	1.16913
$58$	−4.53220	−0.595106
$59$	2.92554	0.380873	0.190437	−	0.981700i	$-0.439010\pi$
$59$	2.92554	0.380873	0.190437	+	0.981700i	$0.439010\pi$
$60$	0	0
$61$	−10.8062	−1.38360	−0.691799	−	0.722090i	$-0.743182\pi$
$61$	−10.8062	−1.38360	−0.691799	+	0.722090i	$0.743182\pi$
$62$	3.49192	0.443474
$63$	−5.11474	−0.644396
$64$	−2.76378	−0.345473
$65$	0	0
$66$	−4.89322	−0.602313
$67$	1.90938	0.233268	0.116634	−	0.993175i	$-0.462790\pi$
$67$	1.90938	0.233268	0.116634	+	0.993175i	$0.462790\pi$
$68$	4.40925	0.534700
$69$	−2.84864	−0.342936
$70$	0	0
$71$	−15.0576	−1.78701	−0.893507	−	0.449050i	$-0.851763\pi$
$71$	−15.0576	−1.78701	−0.893507	+	0.449050i	$0.851763\pi$
$72$	−9.45529	−1.11432
$73$	8.09281	0.947191	0.473596	−	0.880742i	$-0.342956\pi$
$73$	8.09281	0.947191	0.473596	+	0.880742i	$0.342956\pi$
$74$	−2.94747	−0.342637
$75$	0	0
$76$	5.44734	0.624853
$77$	−3.49192	−0.397941
$78$	−0.126987	−0.0143785
$79$	−6.21761	−0.699536	−0.349768	−	0.936836i	$-0.613740\pi$
$79$	−6.21761	−0.699536	−0.349768	+	0.936836i	$0.613740\pi$
$80$	0	0
$81$	1.81631	0.201812
$82$	5.48152	0.605333
$83$	8.85229	0.971665	0.485832	−	0.874052i	$-0.338517\pi$
$83$	8.85229	0.971665	0.485832	+	0.874052i	$0.338517\pi$
$84$	−5.00795	−0.546412
$85$	0	0
$86$	−5.36037	−0.578023
$87$	26.2454	2.81380
$88$	−6.45529	−0.688136
$89$	9.49192	1.00614	0.503071	−	0.864245i	$-0.332204\pi$
$89$	9.49192	1.00614	0.503071	+	0.864245i	$0.332204\pi$
$90$	0	0
$91$	−0.0906212	−0.00949968
$92$	−1.75802	−0.183286
$93$	−20.2213	−2.09685
$94$	−3.40276	−0.350968
$95$	0	0
$96$	−14.1849	−1.44774
$97$	9.20291	0.934414	0.467207	−	0.884148i	$-0.345260\pi$
$97$	9.20291	0.934414	0.467207	+	0.884148i	$0.345260\pi$
$98$	0.491918	0.0496913
$99$	17.8602	1.79502
$100$	0	0
$101$	−6.43336	−0.640143	−0.320072	−	0.947393i	$-0.603707\pi$
$101$	−6.43336	−0.640143	−0.320072	+	0.947393i	$0.603707\pi$
$102$	3.51457	0.347994
$103$	12.6915	1.25053	0.625266	−	0.780412i	$-0.284991\pi$
$103$	12.6915	1.25053	0.625266	+	0.780412i	$0.284991\pi$
$104$	−0.167526	−0.0164272
$105$	0	0
$106$	−3.65944	−0.355437
$107$	−6.52180	−0.630486	−0.315243	−	0.949011i	$-0.602086\pi$
$107$	−6.52180	−0.630486	−0.315243	+	0.949011i	$0.602086\pi$
$108$	10.5905	1.01907
$109$	−2.48762	−0.238271	−0.119135	−	0.992878i	$-0.538012\pi$
$109$	−2.48762	−0.238271	−0.119135	+	0.992878i	$0.538012\pi$
$110$	0	0
$111$	17.0684	1.62006
$112$	−2.60665	−0.246306
$113$	2.01225	0.189297	0.0946484	−	0.995511i	$-0.469827\pi$
$113$	2.01225	0.189297	0.0946484	+	0.995511i	$0.469827\pi$
$114$	4.34202	0.406667
$115$	0	0
$116$	16.1971	1.50387
$117$	0.463503	0.0428509
$118$	1.43913	0.132482
$119$	2.50808	0.229915
$120$	0	0
$121$	1.19349	0.108499
$122$	−5.31579	−0.481269
$123$	−31.7428	−2.86215
$124$	−12.4794	−1.12068
$125$	0	0
$126$	−2.51603	−0.224146
$127$	19.1468	1.69900	0.849502	−	0.527586i	$-0.176903\pi$
$127$	19.1468	1.69900	0.849502	+	0.527586i	$0.176903\pi$
$128$	−11.3186	−1.00043
$129$	31.0412	2.73303
$130$	0	0
$131$	−0.410713	−0.0358842	−0.0179421	−	0.999839i	$-0.505711\pi$
$131$	−0.410713	−0.0358842	−0.0179421	+	0.999839i	$0.505711\pi$
$132$	17.4874	1.52208
$133$	3.09857	0.268680
$134$	0.939259	0.0811396
$135$	0	0
$136$	4.63653	0.397579
$137$	10.0861	0.861710	0.430855	−	0.902421i	$-0.358212\pi$
$137$	10.0861	0.861710	0.430855	+	0.902421i	$0.358212\pi$
$138$	−1.40130	−0.119286
$139$	−15.0034	−1.27257	−0.636285	−	0.771454i	$-0.719530\pi$
$139$	−15.0034	−1.27257	−0.636285	+	0.771454i	$0.719530\pi$
$140$	0	0
$141$	19.7050	1.65946
$142$	−7.40713	−0.621593
$143$	0.316442	0.0264622
$144$	13.3323	1.11103
$145$	0	0
$146$	3.98100	0.329470
$147$	−2.84864	−0.234952
$148$	10.5337	0.865862
$149$	18.9670	1.55384	0.776920	−	0.629600i	$-0.216781\pi$
$149$	18.9670	1.55384	0.776920	+	0.629600i	$0.216781\pi$
$150$	0	0
$151$	−2.29233	−0.186547	−0.0932735	−	0.995641i	$-0.529733\pi$
$151$	−2.29233	−0.186547	−0.0932735	+	0.995641i	$0.529733\pi$
$152$	5.72814	0.464613
$153$	−12.8282	−1.03710
$154$	−1.71774	−0.138419
$155$	0	0
$156$	0.453826	0.0363352
$157$	−8.08752	−0.645455	−0.322727	−	0.946492i	$-0.604600\pi$
$157$	−8.08752	−0.645455	−0.322727	+	0.946492i	$0.604600\pi$
$158$	−3.05856	−0.243326
$159$	21.1914	1.68059
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0.893477	0.0701982
$163$	11.8053	0.924660	0.462330	−	0.886708i	$-0.347014\pi$
$163$	11.8053	0.924660	0.462330	+	0.886708i	$0.347014\pi$
$164$	−19.5898	−1.52971
$165$	0	0
$166$	4.35460	0.337983
$167$	16.1752	1.25168	0.625838	−	0.779953i	$-0.284757\pi$
$167$	16.1752	1.25168	0.625838	+	0.779953i	$0.284757\pi$
$168$	−5.26610	−0.406288
$169$	−12.9918	−0.999368
$170$	0	0
$171$	−15.8484	−1.21196
$172$	19.1569	1.46070
$173$	−13.0268	−0.990408	−0.495204	−	0.868777i	$-0.664907\pi$
$173$	−13.0268	−0.990408	−0.495204	+	0.868777i	$0.664907\pi$
$174$	12.9106	0.978749
$175$	0	0
$176$	9.10222	0.686106
$177$	−8.33381	−0.626407
$178$	4.66925	0.349975
$179$	−0.778480	−0.0581864	−0.0290932	−	0.999577i	$-0.509262\pi$
$179$	−0.778480	−0.0581864	−0.0290932	+	0.999577i	$0.509262\pi$
$180$	0	0
$181$	1.74000	0.129333	0.0646665	−	0.997907i	$-0.479402\pi$
$181$	1.74000	0.129333	0.0646665	+	0.997907i	$0.479402\pi$
$182$	−0.0445782	−0.00330436
$183$	30.7831	2.27555
$184$	−1.84864	−0.136283
$185$	0	0
$186$	−9.94721	−0.729365
$187$	−8.75802	−0.640450
$188$	12.1608	0.886916
$189$	6.02411	0.438190
$190$	0	0
$191$	1.54689	0.111929	0.0559647	−	0.998433i	$-0.482177\pi$
$191$	1.54689	0.111929	0.0559647	+	0.998433i	$0.482177\pi$
$192$	7.87301	0.568186
$193$	1.76046	0.126721	0.0633604	−	0.997991i	$-0.479818\pi$
$193$	1.76046	0.126721	0.0633604	+	0.997991i	$0.479818\pi$
$194$	4.52708	0.325025
$195$	0	0
$196$	−1.75802	−0.125573
$197$	22.2796	1.58735	0.793676	−	0.608340i	$-0.208164\pi$
$197$	22.2796	1.58735	0.793676	+	0.608340i	$0.208164\pi$
$198$	8.78578	0.624378
$199$	10.0055	0.709271	0.354636	−	0.935005i	$-0.384605\pi$
$199$	10.0055	0.709271	0.354636	+	0.935005i	$0.384605\pi$
$200$	0	0
$201$	−5.43913	−0.383647
$202$	−3.16469	−0.222667
$203$	9.21331	0.646647
$204$	−12.5603	−0.879400
$205$	0	0
$206$	6.24319	0.434983
$207$	5.11474	0.355499
$208$	0.236218	0.0163788
$209$	−10.8200	−0.748432
$210$	0	0
$211$	22.4575	1.54604	0.773018	−	0.634384i	$-0.218746\pi$
$211$	22.4575	1.54604	0.773018	+	0.634384i	$0.218746\pi$
$212$	13.0781	0.898208
$213$	42.8938	2.93903
$214$	−3.20819	−0.219308
$215$	0	0
$216$	11.1364	0.757736
$217$	−7.09857	−0.481882
$218$	−1.22370	−0.0828798
$219$	−23.0535	−1.55781
$220$	0	0
$221$	−0.227285	−0.0152889
$222$	8.39628	0.563521
$223$	20.4025	1.36625	0.683126	−	0.730300i	$-0.260620\pi$
$223$	20.4025	1.36625	0.683126	+	0.730300i	$0.260620\pi$
$224$	−4.97954	−0.332709
$225$	0	0
$226$	0.989864	0.0658448
$227$	−6.65733	−0.441862	−0.220931	−	0.975289i	$-0.570910\pi$
$227$	−6.65733	−0.441862	−0.220931	+	0.975289i	$0.570910\pi$
$228$	−15.5175	−1.02767
$229$	14.5185	0.959408	0.479704	−	0.877430i	$-0.340744\pi$
$229$	14.5185	0.959408	0.479704	+	0.877430i	$0.340744\pi$
$230$	0	0
$231$	9.94721	0.654478
$232$	17.0321	1.11821
$233$	−24.1328	−1.58099	−0.790497	−	0.612466i	$-0.790178\pi$
$233$	−24.1328	−1.58099	−0.790497	+	0.612466i	$0.790178\pi$
$234$	0.228006	0.0149052
$235$	0	0
$236$	−5.14315	−0.334791
$237$	17.7117	1.15050
$238$	1.23377	0.0799735
$239$	30.2145	1.95441	0.977207	−	0.212288i	$-0.0680915\pi$
$239$	30.2145	1.95441	0.977207	+	0.212288i	$0.0680915\pi$
$240$	0	0
$241$	12.4092	0.799350	0.399675	−	0.916657i	$-0.369123\pi$
$241$	12.4092	0.799350	0.399675	+	0.916657i	$0.369123\pi$
$242$	0.587102	0.0377403
$243$	12.8983	0.827428
$244$	18.9976	1.21619
$245$	0	0
$246$	−15.6149	−0.995567
$247$	−0.280796	−0.0178666
$248$	−13.1227	−0.833291
$249$	−25.2170	−1.59806
$250$	0	0
$251$	14.4348	0.911118	0.455559	−	0.890206i	$-0.349439\pi$
$251$	14.4348	0.911118	0.455559	+	0.890206i	$0.349439\pi$
$252$	8.99179	0.566429
$253$	3.49192	0.219535
$254$	9.41866	0.590980
$255$	0	0
$256$	−0.0402772	−0.00251733
$257$	−13.0595	−0.814629	−0.407315	−	0.913288i	$-0.633535\pi$
$257$	−13.0595	−0.814629	−0.407315	+	0.913288i	$0.633535\pi$
$258$	15.2697	0.950653
$259$	5.99179	0.372312
$260$	0	0
$261$	−47.1236	−2.91688
$262$	−0.202037	−0.0124819
$263$	25.7168	1.58577	0.792884	−	0.609373i	$-0.208579\pi$
$263$	25.7168	1.58577	0.792884	+	0.609373i	$0.208579\pi$
$264$	18.3888	1.13175
$265$	0	0
$266$	1.52424	0.0934574
$267$	−27.0390	−1.65476
$268$	−3.35672	−0.205044
$269$	12.6387	0.770596	0.385298	−	0.922792i	$-0.374099\pi$
$269$	12.6387	0.770596	0.385298	+	0.922792i	$0.374099\pi$
$270$	0	0
$271$	−19.2511	−1.16942	−0.584712	−	0.811241i	$-0.698792\pi$
$271$	−19.2511	−1.16942	−0.584712	+	0.811241i	$0.698792\pi$
$272$	−6.53770	−0.396406
$273$	0.258147	0.0156238
$274$	4.96152	0.299736
$275$	0	0
$276$	5.00795	0.301443
$277$	−2.96191	−0.177964	−0.0889819	−	0.996033i	$-0.528361\pi$
$277$	−2.96191	−0.177964	−0.0889819	+	0.996033i	$0.528361\pi$
$278$	−7.38044	−0.442650
$279$	36.3073	2.17366
$280$	0	0
$281$	−20.1285	−1.20076	−0.600381	−	0.799714i	$-0.704985\pi$
$281$	−20.1285	−1.20076	−0.600381	+	0.799714i	$0.704985\pi$
$282$	9.69324	0.577224
$283$	6.77995	0.403026	0.201513	−	0.979486i	$-0.435414\pi$
$283$	6.77995	0.403026	0.201513	+	0.979486i	$0.435414\pi$
$284$	26.4716	1.57080
$285$	0	0
$286$	0.155664	0.00920458
$287$	−11.1432	−0.657759
$288$	25.4690	1.50078
$289$	−10.7095	−0.629972
$290$	0	0
$291$	−26.2158	−1.53679
$292$	−14.2273	−0.832589
$293$	7.28750	0.425741	0.212870	−	0.977080i	$-0.431719\pi$
$293$	7.28750	0.425741	0.212870	+	0.977080i	$0.431719\pi$
$294$	−1.40130	−0.0817253
$295$	0	0
$296$	11.0766	0.643817
$297$	−21.0357	−1.22062
$298$	9.33023	0.540486
$299$	0.0906212	0.00524076
$300$	0	0
$301$	10.8969	0.628085
$302$	−1.12764	−0.0648883
$303$	18.3263	1.05282
$304$	−8.07690	−0.463242
$305$	0	0
$306$	−6.31041	−0.360742
$307$	−0.378165	−0.0215830	−0.0107915	−	0.999942i	$-0.503435\pi$
$307$	−0.378165	−0.0215830	−0.0107915	+	0.999942i	$0.503435\pi$
$308$	6.13885	0.349793
$309$	−36.1535	−2.05670
$310$	0	0
$311$	3.04849	0.172864	0.0864320	−	0.996258i	$-0.472453\pi$
$311$	3.04849	0.172864	0.0864320	+	0.996258i	$0.472453\pi$
$312$	0.477220	0.0270173
$313$	−1.95874	−0.110715	−0.0553573	−	0.998467i	$-0.517630\pi$
$313$	−1.95874	−0.110715	−0.0553573	+	0.998467i	$0.517630\pi$
$314$	−3.97840	−0.224514
$315$	0	0
$316$	10.9307	0.614897
$317$	11.9116	0.669020	0.334510	−	0.942392i	$-0.391429\pi$
$317$	11.9116	0.669020	0.334510	+	0.942392i	$0.391429\pi$
$318$	10.4244	0.584573
$319$	−32.1721	−1.80129
$320$	0	0
$321$	18.5782	1.03694
$322$	−0.491918	−0.0274135
$323$	7.77147	0.432416
$324$	−3.19310	−0.177395
$325$	0	0
$326$	5.80723	0.321633
$327$	7.08632	0.391874
$328$	−20.5996	−1.13743
$329$	6.91733	0.381365
$330$	0	0
$331$	−29.8151	−1.63879	−0.819394	−	0.573231i	$-0.805690\pi$
$331$	−29.8151	−1.63879	−0.819394	+	0.573231i	$0.805690\pi$
$332$	−15.5625	−0.854101
$333$	−30.6464	−1.67941
$334$	7.95689	0.435381
$335$	0	0
$336$	7.42541	0.405089
$337$	−24.5176	−1.33556	−0.667780	−	0.744359i	$-0.732755\pi$
$337$	−24.5176	−1.33556	−0.667780	+	0.744359i	$0.732755\pi$
$338$	−6.39090	−0.347619
$339$	−5.73218	−0.311329
$340$	0	0
$341$	24.7876	1.34233
$342$	−7.79611	−0.421565
$343$	−1.00000	−0.0539949
$344$	20.1444	1.08611
$345$	0	0
$346$	−6.40811	−0.344502
$347$	−3.10897	−0.166898	−0.0834491	−	0.996512i	$-0.526594\pi$
$347$	−3.10897	−0.166898	−0.0834491	+	0.996512i	$0.526594\pi$
$348$	−46.1398	−2.47335
$349$	−12.9069	−0.690892	−0.345446	−	0.938439i	$-0.612272\pi$
$349$	−12.9069	−0.690892	−0.345446	+	0.938439i	$0.612272\pi$
$350$	0	0
$351$	−0.545912	−0.0291386
$352$	17.3881	0.926791
$353$	−2.99941	−0.159642	−0.0798212	−	0.996809i	$-0.525435\pi$
$353$	−2.99941	−0.159642	−0.0798212	+	0.996809i	$0.525435\pi$
$354$	−4.09955	−0.217889
$355$	0	0
$356$	−16.6869	−0.884406
$357$	−7.14462	−0.378133
$358$	−0.382949	−0.0202395
$359$	23.9240	1.26266	0.631331	−	0.775513i	$-0.282509\pi$
$359$	23.9240	1.26266	0.631331	+	0.775513i	$0.282509\pi$
$360$	0	0
$361$	−9.39885	−0.494676
$362$	0.855937	0.0449871
$363$	−3.39983	−0.178445
$364$	0.159314	0.00835030
$365$	0	0
$366$	15.1428	0.791525
$367$	4.46715	0.233184	0.116592	−	0.993180i	$-0.462803\pi$
$367$	4.46715	0.233184	0.116592	+	0.993180i	$0.462803\pi$
$368$	2.60665	0.135881
$369$	56.9943	2.96700
$370$	0	0
$371$	7.43913	0.386220
$372$	35.5493	1.84315
$373$	6.97012	0.360899	0.180450	−	0.983584i	$-0.442245\pi$
$373$	6.97012	0.360899	0.180450	+	0.983584i	$0.442245\pi$
$374$	−4.30823	−0.222773
$375$	0	0
$376$	12.7876	0.659472
$377$	−0.834921	−0.0430006
$378$	2.96337	0.152419
$379$	2.81559	0.144627	0.0723136	−	0.997382i	$-0.476962\pi$
$379$	2.81559	0.144627	0.0723136	+	0.997382i	$0.476962\pi$
$380$	0	0
$381$	−54.5423	−2.79429
$382$	0.760946	0.0389334
$383$	−22.6204	−1.15585	−0.577925	−	0.816090i	$-0.696137\pi$
$383$	−22.6204	−1.15585	−0.577925	+	0.816090i	$0.696137\pi$
$384$	32.2427	1.64538
$385$	0	0
$386$	0.866004	0.0440784
$387$	−55.7346	−2.83315
$388$	−16.1789	−0.821357
$389$	1.81876	0.0922147	0.0461073	−	0.998936i	$-0.485318\pi$
$389$	1.81876	0.0922147	0.0461073	+	0.998936i	$0.485318\pi$
$390$	0	0
$391$	−2.50808	−0.126839
$392$	−1.84864	−0.0933703
$393$	1.16997	0.0590173
$394$	10.9597	0.552143
$395$	0	0
$396$	−31.3986	−1.57784
$397$	−0.112028	−0.00562250	−0.00281125	−	0.999996i	$-0.500895\pi$
$397$	−0.112028	−0.00562250	−0.00281125	+	0.999996i	$0.500895\pi$
$398$	4.92189	0.246712
$399$	−8.82671	−0.441888
$400$	0	0
$401$	−12.9500	−0.646694	−0.323347	−	0.946280i	$-0.604808\pi$
$401$	−12.9500	−0.646694	−0.323347	+	0.946280i	$0.604808\pi$
$402$	−2.67561	−0.133447
$403$	0.643281	0.0320441
$404$	11.3100	0.562691
$405$	0	0
$406$	4.53220	0.224929
$407$	−20.9228	−1.03711
$408$	−13.2078	−0.653884
$409$	−4.69607	−0.232206	−0.116103	−	0.993237i	$-0.537040\pi$
$409$	−4.69607	−0.232206	−0.116103	+	0.993237i	$0.537040\pi$
$410$	0	0
$411$	−28.7315	−1.41722
$412$	−22.3119	−1.09923
$413$	−2.92554	−0.143957
$414$	2.51603	0.123656
$415$	0	0
$416$	0.451251	0.0221244
$417$	42.7392	2.09295
$418$	−5.32254	−0.260334
$419$	31.0443	1.51661	0.758307	−	0.651898i	$-0.226027\pi$
$419$	31.0443	1.51661	0.758307	+	0.651898i	$0.226027\pi$
$420$	0	0
$421$	1.60851	0.0783939	0.0391969	−	0.999232i	$-0.487520\pi$
$421$	1.60851	0.0783939	0.0391969	+	0.999232i	$0.487520\pi$
$422$	11.0472	0.537772
$423$	−35.3803	−1.72025
$424$	13.7523	0.667868
$425$	0	0
$426$	21.1002	1.02231
$427$	10.8062	0.522951
$428$	11.4654	0.554202
$429$	−0.901428	−0.0435213
$430$	0	0
$431$	8.58717	0.413629	0.206815	−	0.978380i	$-0.433690\pi$
$431$	8.58717	0.413629	0.206815	+	0.978380i	$0.433690\pi$
$432$	−15.7028	−0.755500
$433$	−30.2836	−1.45534	−0.727668	−	0.685929i	$-0.759396\pi$
$433$	−30.2836	−1.45534	−0.727668	+	0.685929i	$0.759396\pi$
$434$	−3.49192	−0.167617
$435$	0	0
$436$	4.37327	0.209442
$437$	−3.09857	−0.148225
$438$	−11.3404	−0.541867
$439$	−3.55550	−0.169695	−0.0848473	−	0.996394i	$-0.527040\pi$
$439$	−3.55550	−0.169695	−0.0848473	+	0.996394i	$0.527040\pi$
$440$	0	0
$441$	5.11474	0.243559
$442$	−0.111806	−0.00531806
$443$	−20.6250	−0.979923	−0.489962	−	0.871744i	$-0.662989\pi$
$443$	−20.6250	−0.979923	−0.489962	+	0.871744i	$0.662989\pi$
$444$	−30.0066	−1.42405
$445$	0	0
$446$	10.0364	0.475236
$447$	−54.0302	−2.55554
$448$	2.76378	0.130576
$449$	−37.5906	−1.77401	−0.887004	−	0.461762i	$-0.847218\pi$
$449$	−37.5906	−1.77401	−0.887004	+	0.461762i	$0.847218\pi$
$450$	0	0
$451$	38.9110	1.83225
$452$	−3.53757	−0.166393
$453$	6.53001	0.306807
$454$	−3.27486	−0.153697
$455$	0	0
$456$	−16.3174	−0.764132
$457$	11.8674	0.555132	0.277566	−	0.960707i	$-0.410472\pi$
$457$	11.8674	0.555132	0.277566	+	0.960707i	$0.410472\pi$
$458$	7.14191	0.333719
$459$	15.1090	0.705226
$460$	0	0
$461$	0.663355	0.0308955	0.0154478	−	0.999881i	$-0.495083\pi$
$461$	0.663355	0.0308955	0.0154478	+	0.999881i	$0.495083\pi$
$462$	4.89322	0.227653
$463$	7.69998	0.357849	0.178924	−	0.983863i	$-0.442738\pi$
$463$	7.69998	0.357849	0.178924	+	0.983863i	$0.442738\pi$
$464$	−24.0159	−1.11491
$465$	0	0
$466$	−11.8714	−0.549931
$467$	−10.7308	−0.496562	−0.248281	−	0.968688i	$-0.579866\pi$
$467$	−10.7308	−0.496562	−0.248281	+	0.968688i	$0.579866\pi$
$468$	−0.814846	−0.0376663
$469$	−1.90938	−0.0881669
$470$	0	0
$471$	23.0384	1.06155
$472$	−5.40827	−0.248936
$473$	−38.0510	−1.74959
$474$	8.71272	0.400188
$475$	0	0
$476$	−4.40925	−0.202098
$477$	−38.0492	−1.74215
$478$	14.8631	0.679821
$479$	26.0282	1.18926	0.594628	−	0.804001i	$-0.297299\pi$
$479$	26.0282	1.18926	0.594628	+	0.804001i	$0.297299\pi$
$480$	0	0
$481$	−0.542983	−0.0247579
$482$	6.10434	0.278045
$483$	2.84864	0.129617
$484$	−2.09818	−0.0953719
$485$	0	0
$486$	6.34493	0.287812
$487$	40.5982	1.83968	0.919840	−	0.392295i	$-0.128319\pi$
$487$	40.5982	1.83968	0.919840	+	0.392295i	$0.128319\pi$
$488$	19.9768	0.904309
$489$	−33.6289	−1.52075
$490$	0	0
$491$	26.8455	1.21152	0.605761	−	0.795647i	$-0.292869\pi$
$491$	26.8455	1.21152	0.605761	+	0.795647i	$0.292869\pi$
$492$	55.8044	2.51585
$493$	23.1077	1.04072
$494$	−0.138129	−0.00621471
$495$	0	0
$496$	18.5035	0.830833
$497$	15.0576	0.675428
$498$	−12.4047	−0.555867
$499$	7.49518	0.335530	0.167765	−	0.985827i	$-0.446345\pi$
$499$	7.49518	0.335530	0.167765	+	0.985827i	$0.446345\pi$
$500$	0	0
$501$	−46.0773	−2.05858
$502$	7.10076	0.316922
$503$	34.2914	1.52898	0.764489	−	0.644636i	$-0.222991\pi$
$503$	34.2914	1.52898	0.764489	+	0.644636i	$0.222991\pi$
$504$	9.45529	0.421172
$505$	0	0
$506$	1.71774	0.0763628
$507$	37.0089	1.64362
$508$	−33.6604	−1.49344
$509$	15.6030	0.691591	0.345795	−	0.938310i	$-0.387609\pi$
$509$	15.6030	0.691591	0.345795	+	0.938310i	$0.387609\pi$
$510$	0	0
$511$	−8.09281	−0.358005
$512$	22.6174	0.999559
$513$	18.6662	0.824131
$514$	−6.42421	−0.283360
$515$	0	0
$516$	−54.5710	−2.40235
$517$	−24.1548	−1.06232
$518$	2.94747	0.129504
$519$	37.1086	1.62889
$520$	0	0
$521$	13.3195	0.583538	0.291769	−	0.956489i	$-0.405756\pi$
$521$	13.3195	0.583538	0.291769	+	0.956489i	$0.405756\pi$
$522$	−23.1810	−1.01460
$523$	7.42137	0.324514	0.162257	−	0.986749i	$-0.448123\pi$
$523$	7.42137	0.324514	0.162257	+	0.986749i	$0.448123\pi$
$524$	0.722040	0.0315425
$525$	0	0
$526$	12.6506	0.551592
$527$	−17.8038	−0.775546
$528$	−25.9289	−1.12841
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.9634	0.649355
$532$	−5.44734	−0.236172
$533$	1.00981	0.0437395
$534$	−13.3010	−0.575591
$535$	0	0
$536$	−3.52975	−0.152462
$537$	2.21761	0.0956969
$538$	6.21722	0.268043
$539$	3.49192	0.150408
$540$	0	0
$541$	24.1054	1.03637	0.518186	−	0.855268i	$-0.326608\pi$
$541$	24.1054	1.03637	0.518186	+	0.855268i	$0.326608\pi$
$542$	−9.46999	−0.406771
$543$	−4.95663	−0.212709
$544$	−12.4891	−0.535465
$545$	0	0
$546$	0.126987	0.00543455
$547$	6.85392	0.293052	0.146526	−	0.989207i	$-0.453191\pi$
$547$	6.85392	0.293052	0.146526	+	0.989207i	$0.453191\pi$
$548$	−17.7315	−0.757450
$549$	−55.2711	−2.35891
$550$	0	0
$551$	28.5481	1.21619
$552$	5.26610	0.224140
$553$	6.21761	0.264400
$554$	−1.45702	−0.0619027
$555$	0	0
$556$	26.3762	1.11860
$557$	44.1468	1.87056	0.935279	−	0.353911i	$-0.115148\pi$
$557$	44.1468	1.87056	0.935279	+	0.353911i	$0.115148\pi$
$558$	17.8602	0.756084
$559$	−0.987487	−0.0417662
$560$	0	0
$561$	24.9484	1.05332
$562$	−9.90156	−0.417672
$563$	38.1593	1.60822	0.804112	−	0.594477i	$-0.202641\pi$
$563$	38.1593	1.60822	0.804112	+	0.594477i	$0.202641\pi$
$564$	−34.6416	−1.45868
$565$	0	0
$566$	3.33518	0.140188
$567$	−1.81631	−0.0762779
$568$	27.8361	1.16798
$569$	−24.5410	−1.02881	−0.514407	−	0.857546i	$-0.671988\pi$
$569$	−24.5410	−1.02881	−0.514407	+	0.857546i	$0.671988\pi$
$570$	0	0
$571$	21.9702	0.919424	0.459712	−	0.888068i	$-0.347953\pi$
$571$	21.9702	0.919424	0.459712	+	0.888068i	$0.347953\pi$
$572$	−0.556310	−0.0232605
$573$	−4.40654	−0.184086
$574$	−5.48152	−0.228794
$575$	0	0
$576$	−14.1360	−0.589001
$577$	−28.1365	−1.17134	−0.585670	−	0.810550i	$-0.699168\pi$
$577$	−28.1365	−1.17134	−0.585670	+	0.810550i	$0.699168\pi$
$578$	−5.26821	−0.219129
$579$	−5.01492	−0.208413
$580$	0	0
$581$	−8.85229	−0.367255
$582$	−12.8960	−0.534557
$583$	−25.9768	−1.07585
$584$	−14.9607	−0.619077
$585$	0	0
$586$	3.58486	0.148089
$587$	−10.4616	−0.431798	−0.215899	−	0.976416i	$-0.569268\pi$
$587$	−10.4616	−0.431798	−0.215899	+	0.976416i	$0.569268\pi$
$588$	5.00795	0.206524
$589$	−21.9954	−0.906306
$590$	0	0
$591$	−63.4664	−2.61066
$592$	−15.6185	−0.641917
$593$	−0.521798	−0.0214277	−0.0107138	−	0.999943i	$-0.503410\pi$
$593$	−0.521798	−0.0214277	−0.0107138	+	0.999943i	$0.503410\pi$
$594$	−10.3479	−0.424578
$595$	0	0
$596$	−33.3443	−1.36584
$597$	−28.5021	−1.16651
$598$	0.0445782	0.00182294
$599$	−16.1096	−0.658221	−0.329111	−	0.944291i	$-0.606749\pi$
$599$	−16.1096	−0.658221	−0.329111	+	0.944291i	$0.606749\pi$
$600$	0	0
$601$	−25.5988	−1.04420	−0.522098	−	0.852886i	$-0.674850\pi$
$601$	−25.5988	−1.04420	−0.522098	+	0.852886i	$0.674850\pi$
$602$	5.36037	0.218472
$603$	9.76597	0.397701
$604$	4.02995	0.163976
$605$	0	0
$606$	9.01505	0.366211
$607$	41.6131	1.68902	0.844512	−	0.535537i	$-0.179891\pi$
$607$	41.6131	1.68902	0.844512	+	0.535537i	$0.179891\pi$
$608$	−15.4295	−0.625747
$609$	−26.2454	−1.06352
$610$	0	0
$611$	−0.626857	−0.0253599
$612$	22.5521	0.911616
$613$	19.5591	0.789986	0.394993	−	0.918684i	$-0.370747\pi$
$613$	19.5591	0.789986	0.394993	+	0.918684i	$0.370747\pi$
$614$	−0.186026	−0.00750741
$615$	0	0
$616$	6.45529	0.260091
$617$	10.6338	0.428102	0.214051	−	0.976823i	$-0.431334\pi$
$617$	10.6338	0.428102	0.214051	+	0.976823i	$0.431334\pi$
$618$	−17.7846	−0.715401
$619$	17.0785	0.686443	0.343221	−	0.939255i	$-0.388482\pi$
$619$	17.0785	0.686443	0.343221	+	0.939255i	$0.388482\pi$
$620$	0	0
$621$	−6.02411	−0.241739
$622$	1.49961	0.0601288
$623$	−9.49192	−0.380286
$624$	−0.672900	−0.0269375
$625$	0	0
$626$	−0.963541	−0.0385108
$627$	30.8221	1.23092
$628$	14.2180	0.567360
$629$	15.0279	0.599201
$630$	0	0
$631$	−4.73462	−0.188482	−0.0942412	−	0.995549i	$-0.530042\pi$
$631$	−4.73462	−0.188482	−0.0942412	+	0.995549i	$0.530042\pi$
$632$	11.4941	0.457211
$633$	−63.9732	−2.54271
$634$	5.85952	0.232711
$635$	0	0
$636$	−37.2548	−1.47725
$637$	0.0906212	0.00359054
$638$	−15.8261	−0.626560
$639$	−77.0159	−3.04670
$640$	0	0
$641$	12.4452	0.491557	0.245778	−	0.969326i	$-0.420956\pi$
$641$	12.4452	0.491557	0.245778	+	0.969326i	$0.420956\pi$
$642$	9.13898	0.360687
$643$	32.4126	1.27823	0.639113	−	0.769112i	$-0.279301\pi$
$643$	32.4126	1.27823	0.639113	+	0.769112i	$0.279301\pi$
$644$	1.75802	0.0692755
$645$	0	0
$646$	3.82293	0.150411
$647$	−2.31279	−0.0909252	−0.0454626	−	0.998966i	$-0.514476\pi$
$647$	−2.31279	−0.0909252	−0.0454626	+	0.998966i	$0.514476\pi$
$648$	−3.35770	−0.131903
$649$	10.2158	0.401003
$650$	0	0
$651$	20.2213	0.792534
$652$	−20.7538	−0.812783
$653$	11.5109	0.450457	0.225229	−	0.974306i	$-0.427687\pi$
$653$	11.5109	0.450457	0.225229	+	0.974306i	$0.427687\pi$
$654$	3.48589	0.136309
$655$	0	0
$656$	29.0463	1.13407
$657$	41.3926	1.61488
$658$	3.40276	0.132653
$659$	32.8624	1.28014	0.640068	−	0.768318i	$-0.278906\pi$
$659$	32.8624	1.28014	0.640068	+	0.768318i	$0.278906\pi$
$660$	0	0
$661$	34.3227	1.33500	0.667499	−	0.744611i	$-0.267365\pi$
$661$	34.3227	1.33500	0.667499	+	0.744611i	$0.267365\pi$
$662$	−14.6666	−0.570034
$663$	0.647453	0.0251450
$664$	−16.3647	−0.635072
$665$	0	0
$666$	−15.0755	−0.584165
$667$	−9.21331	−0.356741
$668$	−28.4363	−1.10023
$669$	−58.1193	−2.24702
$670$	0	0
$671$	−37.7345	−1.45673
$672$	14.1849	0.547194
$673$	30.1251	1.16124	0.580619	−	0.814175i	$-0.302810\pi$
$673$	30.1251	1.16124	0.580619	+	0.814175i	$0.302810\pi$
$674$	−12.0607	−0.464560
$675$	0	0
$676$	22.8398	0.878453
$677$	−8.85453	−0.340307	−0.170154	−	0.985418i	$-0.554426\pi$
$677$	−8.85453	−0.340307	−0.170154	+	0.985418i	$0.554426\pi$
$678$	−2.81976	−0.108292
$679$	−9.20291	−0.353175
$680$	0	0
$681$	18.9643	0.726714
$682$	12.1935	0.466913
$683$	−14.6948	−0.562282	−0.281141	−	0.959666i	$-0.590713\pi$
$683$	−14.6948	−0.562282	−0.281141	+	0.959666i	$0.590713\pi$
$684$	27.8617	1.06532
$685$	0	0
$686$	−0.491918	−0.0187815
$687$	−41.3579	−1.57790
$688$	−28.4044	−1.08291
$689$	−0.674143	−0.0256828
$690$	0	0
$691$	14.8600	0.565300	0.282650	−	0.959223i	$-0.408786\pi$
$691$	14.8600	0.565300	0.282650	+	0.959223i	$0.408786\pi$
$692$	22.9013	0.870577
$693$	−17.8602	−0.678454
$694$	−1.52936	−0.0580537
$695$	0	0
$696$	−48.5182	−1.83908
$697$	−27.9479	−1.05860
$698$	−6.34916	−0.240319
$699$	68.7457	2.60020
$700$	0	0
$701$	−23.9640	−0.905106	−0.452553	−	0.891737i	$-0.649487\pi$
$701$	−23.9640	−0.905106	−0.452553	+	0.891737i	$0.649487\pi$
$702$	−0.268544	−0.0101356
$703$	18.5660	0.700229
$704$	−9.65090	−0.363732
$705$	0	0
$706$	−1.47546	−0.0555298
$707$	6.43336	0.241951
$708$	14.6510	0.550617
$709$	−52.0393	−1.95438	−0.977189	−	0.212372i	$-0.931881\pi$
$709$	−52.0393	−1.95438	−0.977189	+	0.212372i	$0.931881\pi$
$710$	0	0
$711$	−31.8014	−1.19265
$712$	−17.5471	−0.657606
$713$	7.09857	0.265844
$714$	−3.51457	−0.131529
$715$	0	0
$716$	1.36858	0.0511463
$717$	−86.0702	−3.21435
$718$	11.7687	0.439203
$719$	12.5518	0.468103	0.234052	−	0.972224i	$-0.424801\pi$
$719$	12.5518	0.468103	0.234052	+	0.972224i	$0.424801\pi$
$720$	0	0
$721$	−12.6915	−0.472657
$722$	−4.62347	−0.172068
$723$	−35.3494	−1.31466
$724$	−3.05895	−0.113685
$725$	0	0
$726$	−1.67244	−0.0620701
$727$	4.64035	0.172101	0.0860506	−	0.996291i	$-0.472575\pi$
$727$	4.64035	0.172101	0.0860506	+	0.996291i	$0.472575\pi$
$728$	0.167526	0.00620892
$729$	−42.1916	−1.56265
$730$	0	0
$731$	27.3302	1.01084
$732$	−54.1171	−2.00023
$733$	45.2256	1.67044	0.835222	−	0.549913i	$-0.185339\pi$
$733$	45.2256	1.67044	0.835222	+	0.549913i	$0.185339\pi$
$734$	2.19748	0.0811103
$735$	0	0
$736$	4.97954	0.183548
$737$	6.66740	0.245597
$738$	28.0365	1.03204
$739$	−17.0974	−0.628939	−0.314470	−	0.949268i	$-0.601827\pi$
$739$	−17.0974	−0.628939	−0.314470	+	0.949268i	$0.601827\pi$
$740$	0	0
$741$	0.799887	0.0293846
$742$	3.65944	0.134342
$743$	−2.50584	−0.0919303	−0.0459651	−	0.998943i	$-0.514636\pi$
$743$	−2.50584	−0.0919303	−0.0459651	+	0.998943i	$0.514636\pi$
$744$	37.3818	1.37048
$745$	0	0
$746$	3.42873	0.125535
$747$	45.2771	1.65660
$748$	15.3967	0.562960
$749$	6.52180	0.238301
$750$	0	0
$751$	39.0400	1.42459	0.712294	−	0.701881i	$-0.247656\pi$
$751$	39.0400	1.42459	0.712294	+	0.701881i	$0.247656\pi$
$752$	−18.0311	−0.657526
$753$	−41.1196	−1.49848
$754$	−0.410713	−0.0149573
$755$	0	0
$756$	−10.5905	−0.385172
$757$	14.4867	0.526528	0.263264	−	0.964724i	$-0.415201\pi$
$757$	14.4867	0.526528	0.263264	+	0.964724i	$0.415201\pi$
$758$	1.38504	0.0503070
$759$	−9.94721	−0.361061
$760$	0	0
$761$	−25.9120	−0.939308	−0.469654	−	0.882851i	$-0.655621\pi$
$761$	−25.9120	−0.939308	−0.469654	+	0.882851i	$0.655621\pi$
$762$	−26.8304	−0.971961
$763$	2.48762	0.0900578
$764$	−2.71946	−0.0983868
$765$	0	0
$766$	−11.1274	−0.402050
$767$	0.265116	0.00957279
$768$	0.114735	0.00414015
$769$	15.2909	0.551405	0.275703	−	0.961243i	$-0.411090\pi$
$769$	15.2909	0.551405	0.275703	+	0.961243i	$0.411090\pi$
$770$	0	0
$771$	37.2018	1.33979
$772$	−3.09492	−0.111389
$773$	−49.1805	−1.76890	−0.884449	−	0.466637i	$-0.845466\pi$
$773$	−49.1805	−1.76890	−0.884449	+	0.466637i	$0.845466\pi$
$774$	−27.4169	−0.985479
$775$	0	0
$776$	−17.0128	−0.610726
$777$	−17.0684	−0.612327
$778$	0.894680	0.0320758
$779$	−34.5279	−1.23709
$780$	0	0
$781$	−52.5801	−1.88146
$782$	−1.23377	−0.0441196
$783$	55.5020	1.98348
$784$	2.60665	0.0930948
$785$	0	0
$786$	0.575531	0.0205285
$787$	−53.4258	−1.90443	−0.952213	−	0.305436i	$-0.901198\pi$
$787$	−53.4258	−1.90443	−0.952213	+	0.305436i	$0.901198\pi$
$788$	−39.1678	−1.39530
$789$	−73.2579	−2.60805
$790$	0	0
$791$	−2.01225	−0.0715474
$792$	−33.0171	−1.17321
$793$	−0.979275	−0.0347751
$794$	−0.0551084	−0.00195572
$795$	0	0
$796$	−17.5898	−0.623455
$797$	1.02678	0.0363705	0.0181852	−	0.999835i	$-0.494211\pi$
$797$	1.02678	0.0363705	0.0181852	+	0.999835i	$0.494211\pi$
$798$	−4.34202	−0.153706
$799$	17.3492	0.613772
$800$	0	0
$801$	48.5487	1.71538
$802$	−6.37037	−0.224945
$803$	28.2594	0.997253
$804$	9.56208	0.337228
$805$	0	0
$806$	0.316442	0.0111462
$807$	−36.0031	−1.26737
$808$	11.8930	0.418393
$809$	31.7679	1.11690	0.558449	−	0.829539i	$-0.311397\pi$
$809$	31.7679	1.11690	0.558449	+	0.829539i	$0.311397\pi$
$810$	0	0
$811$	46.9026	1.64697	0.823486	−	0.567336i	$-0.192026\pi$
$811$	46.9026	1.64697	0.823486	+	0.567336i	$0.192026\pi$
$812$	−16.1971	−0.568408
$813$	54.8395	1.92331
$814$	−10.2923	−0.360746
$815$	0	0
$816$	18.6235	0.651954
$817$	33.7647	1.18128
$818$	−2.31008	−0.0807702
$819$	−0.463503	−0.0161961
$820$	0	0
$821$	−44.8218	−1.56429	−0.782146	−	0.623095i	$-0.785875\pi$
$821$	−44.8218	−1.56429	−0.782146	+	0.623095i	$0.785875\pi$
$822$	−14.1336	−0.492965
$823$	−8.83553	−0.307987	−0.153994	−	0.988072i	$-0.549214\pi$
$823$	−8.83553	−0.307987	−0.153994	+	0.988072i	$0.549214\pi$
$824$	−23.4620	−0.817337
$825$	0	0
$826$	−1.43913	−0.0500737
$827$	−17.1472	−0.596266	−0.298133	−	0.954524i	$-0.596364\pi$
$827$	−17.1472	−0.596266	−0.298133	+	0.954524i	$0.596364\pi$
$828$	−8.99179	−0.312486
$829$	34.9985	1.21555	0.607774	−	0.794110i	$-0.292063\pi$
$829$	34.9985	1.21555	0.607774	+	0.794110i	$0.292063\pi$
$830$	0	0
$831$	8.43740	0.292690
$832$	−0.250457	−0.00868304
$833$	−2.50808	−0.0868999
$834$	21.0242	0.728009
$835$	0	0
$836$	19.0217	0.657878
$837$	−42.7626	−1.47809
$838$	15.2713	0.527537
$839$	19.5745	0.675788	0.337894	−	0.941184i	$-0.390285\pi$
$839$	19.5745	0.675788	0.337894	+	0.941184i	$0.390285\pi$
$840$	0	0
$841$	55.8850	1.92707
$842$	0.791255	0.0272684
$843$	57.3387	1.97485
$844$	−39.4806	−1.35898
$845$	0	0
$846$	−17.4042	−0.598370
$847$	−1.19349	−0.0410089
$848$	−19.3912	−0.665898
$849$	−19.3136	−0.662841
$850$	0	0
$851$	−5.99179	−0.205396
$852$	−75.4079	−2.58343
$853$	−40.1780	−1.37567	−0.687833	−	0.725869i	$-0.741438\pi$
$853$	−40.1780	−1.37567	−0.687833	+	0.725869i	$0.741438\pi$
$854$	5.31579	0.181903
$855$	0	0
$856$	12.0564	0.412081
$857$	21.3042	0.727736	0.363868	−	0.931451i	$-0.381456\pi$
$857$	21.3042	0.727736	0.363868	+	0.931451i	$0.381456\pi$
$858$	−0.443429	−0.0151384
$859$	6.80677	0.232244	0.116122	−	0.993235i	$-0.462954\pi$
$859$	6.80677	0.232244	0.116122	+	0.993235i	$0.462954\pi$
$860$	0	0
$861$	31.7428	1.08179
$862$	4.22419	0.143876
$863$	2.83639	0.0965517	0.0482758	−	0.998834i	$-0.484627\pi$
$863$	2.83639	0.0965517	0.0482758	+	0.998834i	$0.484627\pi$
$864$	−29.9973	−1.02053
$865$	0	0
$866$	−14.8971	−0.506223
$867$	30.5076	1.03609
$868$	12.4794	0.423579
$869$	−21.7114	−0.736508
$870$	0	0
$871$	0.173030	0.00586290
$872$	4.59870	0.155732
$873$	47.0704	1.59309
$874$	−1.52424	−0.0515583
$875$	0	0
$876$	40.5284	1.36933
$877$	−27.2270	−0.919391	−0.459696	−	0.888077i	$-0.652041\pi$
$877$	−27.2270	−0.919391	−0.459696	+	0.888077i	$0.652041\pi$
$878$	−1.74901	−0.0590264
$879$	−20.7595	−0.700199
$880$	0	0
$881$	11.1245	0.374795	0.187398	−	0.982284i	$-0.439995\pi$
$881$	11.1245	0.374795	0.187398	+	0.982284i	$0.439995\pi$
$882$	2.51603	0.0847192
$883$	−3.22927	−0.108674	−0.0543368	−	0.998523i	$-0.517304\pi$
$883$	−3.22927	−0.108674	−0.0543368	+	0.998523i	$0.517304\pi$
$884$	0.399571	0.0134390
$885$	0	0
$886$	−10.1458	−0.340855
$887$	−3.56185	−0.119595	−0.0597977	−	0.998211i	$-0.519046\pi$
$887$	−3.56185	−0.119595	−0.0597977	+	0.998211i	$0.519046\pi$
$888$	−31.5533	−1.05886
$889$	−19.1468	−0.642163
$890$	0	0
$891$	6.34241	0.212479
$892$	−35.8679	−1.20095
$893$	21.4338	0.717256
$894$	−26.5784	−0.888916
$895$	0	0
$896$	11.3186	0.378129
$897$	−0.258147	−0.00861927
$898$	−18.4915	−0.617069
$899$	−65.4013	−2.18126
$900$	0	0
$901$	18.6579	0.621586
$902$	19.1410	0.637326
$903$	−31.0412	−1.03299
$904$	−3.71992	−0.123723
$905$	0	0
$906$	3.21223	0.106719
$907$	−49.8441	−1.65505	−0.827524	−	0.561431i	$-0.810251\pi$
$907$	−49.8441	−1.65505	−0.827524	+	0.561431i	$0.810251\pi$
$908$	11.7037	0.388401
$909$	−32.9049	−1.09139
$910$	0	0
$911$	28.2841	0.937093	0.468546	−	0.883439i	$-0.344778\pi$
$911$	28.2841	0.937093	0.468546	+	0.883439i	$0.344778\pi$
$912$	23.0082	0.761877
$913$	30.9115	1.02302
$914$	5.83778	0.193097
$915$	0	0
$916$	−25.5237	−0.843328
$917$	0.410713	0.0135629
$918$	7.43238	0.245305
$919$	43.9191	1.44876	0.724378	−	0.689403i	$-0.242127\pi$
$919$	43.9191	1.44876	0.724378	+	0.689403i	$0.242127\pi$
$920$	0	0
$921$	1.07725	0.0354968
$922$	0.326317	0.0107467
$923$	−1.36454	−0.0449144
$924$	−17.4874	−0.575292
$925$	0	0
$926$	3.78776	0.124474
$927$	64.9137	2.13205
$928$	−45.8780	−1.50602
$929$	−29.1178	−0.955324	−0.477662	−	0.878544i	$-0.658516\pi$
$929$	−29.1178	−0.955324	−0.477662	+	0.878544i	$0.658516\pi$
$930$	0	0
$931$	−3.09857	−0.101552
$932$	42.4259	1.38971
$933$	−8.68404	−0.284303
$934$	−5.27868	−0.172724
$935$	0	0
$936$	−0.856850	−0.0280070
$937$	−16.6409	−0.543635	−0.271817	−	0.962349i	$-0.587625\pi$
$937$	−16.6409	−0.543635	−0.271817	+	0.962349i	$0.587625\pi$
$938$	−0.939259	−0.0306679
$939$	5.57974	0.182088
$940$	0	0
$941$	30.4193	0.991639	0.495820	−	0.868425i	$-0.334868\pi$
$941$	30.4193	0.991639	0.495820	+	0.868425i	$0.334868\pi$
$942$	11.3330	0.369250
$943$	11.1432	0.362871
$944$	7.62587	0.248201
$945$	0	0
$946$	−18.7180	−0.608574
$947$	−8.12666	−0.264081	−0.132040	−	0.991244i	$-0.542153\pi$
$947$	−8.12666	−0.264081	−0.132040	+	0.991244i	$0.542153\pi$
$948$	−31.1375	−1.01130
$949$	0.733380	0.0238065
$950$	0	0
$951$	−33.9317	−1.10031
$952$	−4.63653	−0.150271
$953$	21.7485	0.704503	0.352251	−	0.935905i	$-0.385416\pi$
$953$	21.7485	0.704503	0.352251	+	0.935905i	$0.385416\pi$
$954$	−18.7171	−0.605988
$955$	0	0
$956$	−53.1176	−1.71795
$957$	91.6467	2.96252
$958$	12.8037	0.413670
$959$	−10.0861	−0.325696
$960$	0	0
$961$	19.3897	0.625475
$962$	−0.267103	−0.00861176
$963$	−33.3573	−1.07492
$964$	−21.8157	−0.702635
$965$	0	0
$966$	1.40130	0.0450860
$967$	37.1593	1.19496	0.597481	−	0.801883i	$-0.296168\pi$
$967$	37.1593	1.19496	0.597481	+	0.801883i	$0.296168\pi$
$968$	−2.20634	−0.0709144
$969$	−22.1381	−0.711178
$970$	0	0
$971$	−57.7694	−1.85391	−0.926955	−	0.375174i	$-0.877583\pi$
$971$	−57.7694	−1.85391	−0.926955	+	0.375174i	$0.877583\pi$
$972$	−22.6755	−0.727316
$973$	15.0034	0.480987
$974$	19.9710	0.639912
$975$	0	0
$976$	−28.1681	−0.901640
$977$	35.4485	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$977$	35.4485	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$978$	−16.5427	−0.528977
$979$	33.1450	1.05932
$980$	0	0
$981$	−12.7235	−0.406230
$982$	13.2058	0.421415
$983$	−46.9993	−1.49905	−0.749523	−	0.661978i	$-0.769717\pi$
$983$	−46.9993	−1.49905	−0.749523	+	0.661978i	$0.769717\pi$
$984$	58.6809	1.87068
$985$	0	0
$986$	11.3671	0.362003
$987$	−19.7050	−0.627216
$988$	0.493644	0.0157049
$989$	−10.8969	−0.346500
$990$	0	0
$991$	48.9104	1.55369	0.776846	−	0.629691i	$-0.216818\pi$
$991$	48.9104	1.55369	0.776846	+	0.629691i	$0.216818\pi$
$992$	35.3476	1.12229
$993$	84.9325	2.69525
$994$	7.40713	0.234940
$995$	0	0
$996$	44.3318	1.40471
$997$	34.1831	1.08259	0.541295	−	0.840833i	$-0.317934\pi$
$997$	34.1831	1.08259	0.541295	+	0.840833i	$0.317934\pi$
$998$	3.68702	0.116710
$999$	36.0952	1.14200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.m.1.3		4
5.4	even	2		805.2.a.i.1.2	✓	4
15.14	odd	2		7245.2.a.bd.1.3		4
35.34	odd	2		5635.2.a.u.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.a.i.1.2	✓	4	5.4	even	2
4025.2.a.m.1.3		4	1.1	even	1	trivial
5635.2.a.u.1.2		4	35.34	odd	2
7245.2.a.bd.1.3		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)

\(f(q)\)	\(=\)	\(q+0.491918 q^{2} -2.84864 q^{3} -1.75802 q^{4} -1.40130 q^{6} -1.00000 q^{7} -1.84864 q^{8} +5.11474 q^{9} +O(q^{10})\)	\(q+0.491918 q^{2} -2.84864 q^{3} -1.75802 q^{4} -1.40130 q^{6} -1.00000 q^{7} -1.84864 q^{8} +5.11474 q^{9} +3.49192 q^{11} +5.00795 q^{12} +0.0906212 q^{13} -0.491918 q^{14} +2.60665 q^{16} -2.50808 q^{17} +2.51603 q^{18} -3.09857 q^{19} +2.84864 q^{21} +1.71774 q^{22} +1.00000 q^{23} +5.26610 q^{24} +0.0445782 q^{26} -6.02411 q^{27} +1.75802 q^{28} -9.21331 q^{29} +7.09857 q^{31} +4.97954 q^{32} -9.94721 q^{33} -1.23377 q^{34} -8.99179 q^{36} -5.99179 q^{37} -1.52424 q^{38} -0.258147 q^{39} +11.1432 q^{41} +1.40130 q^{42} -10.8969 q^{43} -6.13885 q^{44} +0.491918 q^{46} -6.91733 q^{47} -7.42541 q^{48} +1.00000 q^{49} +7.14462 q^{51} -0.159314 q^{52} -7.43913 q^{53} -2.96337 q^{54} +1.84864 q^{56} +8.82671 q^{57} -4.53220 q^{58} +2.92554 q^{59} -10.8062 q^{61} +3.49192 q^{62} -5.11474 q^{63} -2.76378 q^{64} -4.89322 q^{66} +1.90938 q^{67} +4.40925 q^{68} -2.84864 q^{69} -15.0576 q^{71} -9.45529 q^{72} +8.09281 q^{73} -2.94747 q^{74} +5.44734 q^{76} -3.49192 q^{77} -0.126987 q^{78} -6.21761 q^{79} +1.81631 q^{81} +5.48152 q^{82} +8.85229 q^{83} -5.00795 q^{84} -5.36037 q^{86} +26.2454 q^{87} -6.45529 q^{88} +9.49192 q^{89} -0.0906212 q^{91} -1.75802 q^{92} -20.2213 q^{93} -3.40276 q^{94} -14.1849 q^{96} +9.20291 q^{97} +0.491918 q^{98} +17.8602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) 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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