LMFDB - Embedded newform 4025.2.a.be.1.6

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:	$21$
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-1.53505 q^{2} -0.943091 q^{3} +0.356385 q^{4} +1.44769 q^{6} +1.00000 q^{7} +2.52303 q^{8} -2.11058 q^{9} +O(q^{10})$	\(q-1.53505 q^{2} -0.943091 q^{3} +0.356385 q^{4} +1.44769 q^{6} +1.00000 q^{7} +2.52303 q^{8} -2.11058 q^{9} +3.95214 q^{11} -0.336104 q^{12} -5.22032 q^{13} -1.53505 q^{14} -4.58576 q^{16} -3.97709 q^{17} +3.23985 q^{18} -4.56548 q^{19} -0.943091 q^{21} -6.06674 q^{22} +1.00000 q^{23} -2.37945 q^{24} +8.01347 q^{26} +4.81974 q^{27} +0.356385 q^{28} +5.14238 q^{29} -0.386311 q^{31} +1.99331 q^{32} -3.72723 q^{33} +6.10504 q^{34} -0.752179 q^{36} -9.97114 q^{37} +7.00825 q^{38} +4.92324 q^{39} +7.11955 q^{41} +1.44769 q^{42} -5.94427 q^{43} +1.40848 q^{44} -1.53505 q^{46} -2.97825 q^{47} +4.32479 q^{48} +1.00000 q^{49} +3.75076 q^{51} -1.86045 q^{52} +12.1283 q^{53} -7.39855 q^{54} +2.52303 q^{56} +4.30566 q^{57} -7.89382 q^{58} -1.77541 q^{59} +1.66138 q^{61} +0.593008 q^{62} -2.11058 q^{63} +6.11168 q^{64} +5.72149 q^{66} -4.98306 q^{67} -1.41738 q^{68} -0.943091 q^{69} +4.87787 q^{71} -5.32506 q^{72} -4.39185 q^{73} +15.3062 q^{74} -1.62707 q^{76} +3.95214 q^{77} -7.55743 q^{78} +10.3896 q^{79} +1.78628 q^{81} -10.9289 q^{82} -10.9799 q^{83} -0.336104 q^{84} +9.12477 q^{86} -4.84973 q^{87} +9.97138 q^{88} +9.53321 q^{89} -5.22032 q^{91} +0.356385 q^{92} +0.364327 q^{93} +4.57176 q^{94} -1.87987 q^{96} +15.5382 q^{97} -1.53505 q^{98} -8.34130 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9}+O(q^{10}) $ 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9	$ 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9} + 7 q^{11} - 22 q^{12} - 3 q^{13} + 2 q^{14} + 56 q^{16} + 7 q^{17} + 24 q^{19} - q^{21} + 4 q^{22} + 21 q^{23} + 24 q^{24} - 2 q^{26} - 19 q^{27} + 30 q^{28} + 11 q^{29} + 46 q^{31} - 6 q^{32} - 3 q^{33} + 28 q^{34} + 58 q^{36} + 24 q^{37} - 4 q^{38} + 31 q^{39} + 14 q^{41} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 2 q^{46} - 25 q^{47} - 36 q^{48} + 21 q^{49} + 17 q^{51} - 8 q^{52} + 22 q^{53} - 6 q^{54} + 6 q^{56} + 40 q^{57} + 6 q^{58} + 10 q^{59} + 38 q^{61} - 54 q^{62} + 30 q^{63} + 100 q^{64} + 38 q^{66} + 12 q^{67} + 18 q^{68} - q^{69} + 56 q^{71} + 42 q^{72} - 40 q^{73} - 20 q^{74} + 60 q^{76} + 7 q^{77} + 38 q^{78} + 49 q^{79} + 57 q^{81} + 16 q^{82} - 2 q^{83} - 22 q^{84} + 16 q^{86} - 23 q^{87} + 12 q^{88} + 28 q^{89} - 3 q^{91} + 30 q^{92} - 30 q^{93} + 66 q^{94} + 46 q^{96} - q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) $ 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9 + 7 * q^11 - 22 * q^12 - 3 * q^13 + 2 * q^14 + 56 * q^16 + 7 * q^17 + 24 * q^19 - q^21 + 4 * q^22 + 21 * q^23 + 24 * q^24 - 2 * q^26 - 19 * q^27 + 30 * q^28 + 11 * q^29 + 46 * q^31 - 6 * q^32 - 3 * q^33 + 28 * q^34 + 58 * q^36 + 24 * q^37 - 4 * q^38 + 31 * q^39 + 14 * q^41 + 6 * q^42 + 18 * q^43 + 12 * q^44 + 2 * q^46 - 25 * q^47 - 36 * q^48 + 21 * q^49 + 17 * q^51 - 8 * q^52 + 22 * q^53 - 6 * q^54 + 6 * q^56 + 40 * q^57 + 6 * q^58 + 10 * q^59 + 38 * q^61 - 54 * q^62 + 30 * q^63 + 100 * q^64 + 38 * q^66 + 12 * q^67 + 18 * q^68 - q^69 + 56 * q^71 + 42 * q^72 - 40 * q^73 - 20 * q^74 + 60 * q^76 + 7 * q^77 + 38 * q^78 + 49 * q^79 + 57 * q^81 + 16 * q^82 - 2 * q^83 - 22 * q^84 + 16 * q^86 - 23 * q^87 + 12 * q^88 + 28 * q^89 - 3 * q^91 + 30 * q^92 - 30 * q^93 + 66 * q^94 + 46 * q^96 - q^97 + 2 * q^98 - 2 * 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 $

$2$	−1.53505	−1.08545	−0.542723	−	0.839912i	$-0.682607\pi$
$2$	−1.53505	−1.08545	−0.542723	+	0.839912i	$0.682607\pi$
$3$	−0.943091	−0.544494	−0.272247	−	0.962227i	$-0.587767\pi$
$3$	−0.943091	−0.544494	−0.272247	+	0.962227i	$0.587767\pi$
$4$	0.356385	0.178193
$5$	0	0
$5$	0	0
$6$	1.44769	0.591019
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.52303	0.892027
$9$	−2.11058	−0.703526
$10$	0	0
$11$	3.95214	1.19161	0.595807	−	0.803127i	$-0.296832\pi$
$11$	3.95214	1.19161	0.595807	+	0.803127i	$0.296832\pi$
$12$	−0.336104	−0.0970248
$13$	−5.22032	−1.44786	−0.723928	−	0.689875i	$-0.757665\pi$
$13$	−5.22032	−1.44786	−0.723928	+	0.689875i	$0.757665\pi$
$14$	−1.53505	−0.410260
$15$	0	0
$16$	−4.58576	−1.14644
$17$	−3.97709	−0.964586	−0.482293	−	0.876010i	$-0.660196\pi$
$17$	−3.97709	−0.964586	−0.482293	+	0.876010i	$0.660196\pi$
$18$	3.23985	0.763640
$19$	−4.56548	−1.04739	−0.523697	−	0.851905i	$-0.675447\pi$
$19$	−4.56548	−1.04739	−0.523697	+	0.851905i	$0.675447\pi$
$20$	0	0
$21$	−0.943091	−0.205799
$22$	−6.06674	−1.29343
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.37945	−0.485703
$25$	0	0
$26$	8.01347	1.57157
$27$	4.81974	0.927560
$28$	0.356385	0.0673505
$29$	5.14238	0.954916	0.477458	−	0.878655i	$-0.341558\pi$
$29$	5.14238	0.954916	0.477458	+	0.878655i	$0.341558\pi$
$30$	0	0
$31$	−0.386311	−0.0693835	−0.0346918	−	0.999398i	$-0.511045\pi$
$31$	−0.386311	−0.0693835	−0.0346918	+	0.999398i	$0.511045\pi$
$32$	1.99331	0.352371
$33$	−3.72723	−0.648827
$34$	6.10504	1.04701
$35$	0	0
$36$	−0.752179	−0.125363
$37$	−9.97114	−1.63924	−0.819622	−	0.572904i	$-0.805817\pi$
$37$	−9.97114	−1.63924	−0.819622	+	0.572904i	$0.805817\pi$
$38$	7.00825	1.13689
$39$	4.92324	0.788349
$40$	0	0
$41$	7.11955	1.11189	0.555944	−	0.831220i	$-0.312357\pi$
$41$	7.11955	1.11189	0.555944	+	0.831220i	$0.312357\pi$
$42$	1.44769	0.223384
$43$	−5.94427	−0.906493	−0.453246	−	0.891385i	$-0.649734\pi$
$43$	−5.94427	−0.906493	−0.453246	+	0.891385i	$0.649734\pi$
$44$	1.40848	0.212337
$45$	0	0
$46$	−1.53505	−0.226331
$47$	−2.97825	−0.434422	−0.217211	−	0.976125i	$-0.569696\pi$
$47$	−2.97825	−0.434422	−0.217211	+	0.976125i	$0.569696\pi$
$48$	4.32479	0.624230
$49$	1.00000	0.142857
$50$	0	0
$51$	3.75076	0.525211
$52$	−1.86045	−0.257997
$53$	12.1283	1.66595	0.832976	−	0.553309i	$-0.186635\pi$
$53$	12.1283	1.66595	0.832976	+	0.553309i	$0.186635\pi$
$54$	−7.39855	−1.00682
$55$	0	0
$56$	2.52303	0.337155
$57$	4.30566	0.570299
$58$	−7.89382	−1.03651
$59$	−1.77541	−0.231139	−0.115569	−	0.993299i	$-0.536869\pi$
$59$	−1.77541	−0.231139	−0.115569	+	0.993299i	$0.536869\pi$
$60$	0	0
$61$	1.66138	0.212718	0.106359	−	0.994328i	$-0.466081\pi$
$61$	1.66138	0.212718	0.106359	+	0.994328i	$0.466081\pi$
$62$	0.593008	0.0753121
$63$	−2.11058	−0.265908
$64$	6.11168	0.763960
$65$	0	0
$66$	5.72149	0.704266
$67$	−4.98306	−0.608777	−0.304389	−	0.952548i	$-0.598452\pi$
$67$	−4.98306	−0.608777	−0.304389	+	0.952548i	$0.598452\pi$
$68$	−1.41738	−0.171882
$69$	−0.943091	−0.113535
$70$	0	0
$71$	4.87787	0.578896	0.289448	−	0.957194i	$-0.406528\pi$
$71$	4.87787	0.578896	0.289448	+	0.957194i	$0.406528\pi$
$72$	−5.32506	−0.627565
$73$	−4.39185	−0.514027	−0.257013	−	0.966408i	$-0.582738\pi$
$73$	−4.39185	−0.514027	−0.257013	+	0.966408i	$0.582738\pi$
$74$	15.3062	1.77931
$75$	0	0
$76$	−1.62707	−0.186638
$77$	3.95214	0.450388
$78$	−7.55743	−0.855710
$79$	10.3896	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$79$	10.3896	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$80$	0	0
$81$	1.78628	0.198476
$82$	−10.9289	−1.20689
$83$	−10.9799	−1.20520	−0.602602	−	0.798042i	$-0.705869\pi$
$83$	−10.9799	−1.20520	−0.602602	+	0.798042i	$0.705869\pi$
$84$	−0.336104	−0.0366719
$85$	0	0
$86$	9.12477	0.983949
$87$	−4.84973	−0.519946
$88$	9.97138	1.06295
$89$	9.53321	1.01052	0.505259	−	0.862968i	$-0.331397\pi$
$89$	9.53321	1.01052	0.505259	+	0.862968i	$0.331397\pi$
$90$	0	0
$91$	−5.22032	−0.547238
$92$	0.356385	0.0371557
$93$	0.364327	0.0377789
$94$	4.57176	0.471541
$95$	0	0
$96$	−1.87987	−0.191864
$97$	15.5382	1.57766	0.788831	−	0.614610i	$-0.210687\pi$
$97$	15.5382	1.57766	0.788831	+	0.614610i	$0.210687\pi$
$98$	−1.53505	−0.155064
$99$	−8.34130	−0.838332
$100$	0	0
$101$	−18.4035	−1.83121	−0.915607	−	0.402074i	$-0.868290\pi$
$101$	−18.4035	−1.83121	−0.915607	+	0.402074i	$0.868290\pi$
$102$	−5.75761	−0.570088
$103$	−11.4267	−1.12591	−0.562954	−	0.826488i	$-0.690335\pi$
$103$	−11.4267	−1.12591	−0.562954	+	0.826488i	$0.690335\pi$
$104$	−13.1711	−1.29153
$105$	0	0
$106$	−18.6176	−1.80830
$107$	−5.82585	−0.563206	−0.281603	−	0.959531i	$-0.590866\pi$
$107$	−5.82585	−0.563206	−0.281603	+	0.959531i	$0.590866\pi$
$108$	1.71768	0.165284
$109$	−15.5062	−1.48522	−0.742611	−	0.669723i	$-0.766413\pi$
$109$	−15.5062	−1.48522	−0.742611	+	0.669723i	$0.766413\pi$
$110$	0	0
$111$	9.40369	0.892559
$112$	−4.58576	−0.433314
$113$	−13.9712	−1.31430	−0.657151	−	0.753759i	$-0.728239\pi$
$113$	−13.9712	−1.31430	−0.657151	+	0.753759i	$0.728239\pi$
$114$	−6.60942	−0.619029
$115$	0	0
$116$	1.83267	0.170159
$117$	11.0179	1.01861
$118$	2.72535	0.250889
$119$	−3.97709	−0.364579
$120$	0	0
$121$	4.61940	0.419945
$122$	−2.55030	−0.230894
$123$	−6.71439	−0.605416
$124$	−0.137676	−0.0123636
$125$	0	0
$126$	3.23985	0.288629
$127$	0.459101	0.0407386	0.0203693	−	0.999793i	$-0.493516\pi$
$127$	0.459101	0.0407386	0.0203693	+	0.999793i	$0.493516\pi$
$128$	−13.3684	−1.18161
$129$	5.60599	0.493580
$130$	0	0
$131$	−7.57134	−0.661511	−0.330756	−	0.943716i	$-0.607304\pi$
$131$	−7.57134	−0.661511	−0.330756	+	0.943716i	$0.607304\pi$
$132$	−1.32833	−0.115616
$133$	−4.56548	−0.395877
$134$	7.64925	0.660795
$135$	0	0
$136$	−10.0343	−0.860437
$137$	13.9999	1.19609	0.598045	−	0.801462i	$-0.295944\pi$
$137$	13.9999	1.19609	0.598045	+	0.801462i	$0.295944\pi$
$138$	1.44769	0.123236
$139$	2.35719	0.199934	0.0999672	−	0.994991i	$-0.468126\pi$
$139$	2.35719	0.199934	0.0999672	+	0.994991i	$0.468126\pi$
$140$	0	0
$141$	2.80876	0.236540
$142$	−7.48778	−0.628360
$143$	−20.6314	−1.72529
$144$	9.67861	0.806551
$145$	0	0
$146$	6.74171	0.557948
$147$	−0.943091	−0.0777848
$148$	−3.55356	−0.292101
$149$	14.5128	1.18893	0.594466	−	0.804121i	$-0.297363\pi$
$149$	14.5128	1.18893	0.594466	+	0.804121i	$0.297363\pi$
$150$	0	0
$151$	16.8601	1.37205	0.686026	−	0.727577i	$-0.259354\pi$
$151$	16.8601	1.37205	0.686026	+	0.727577i	$0.259354\pi$
$152$	−11.5189	−0.934303
$153$	8.39396	0.678612
$154$	−6.06674	−0.488872
$155$	0	0
$156$	1.75457	0.140478
$157$	13.9252	1.11135	0.555676	−	0.831399i	$-0.312460\pi$
$157$	13.9252	1.11135	0.555676	+	0.831399i	$0.312460\pi$
$158$	−15.9486	−1.26880
$159$	−11.4381	−0.907101
$160$	0	0
$161$	1.00000	0.0788110
$162$	−2.74204	−0.215435
$163$	9.63742	0.754861	0.377431	−	0.926038i	$-0.376808\pi$
$163$	9.63742	0.754861	0.377431	+	0.926038i	$0.376808\pi$
$164$	2.53730	0.198130
$165$	0	0
$166$	16.8548	1.30818
$167$	−21.6043	−1.67179	−0.835894	−	0.548890i	$-0.815051\pi$
$167$	−21.6043	−1.67179	−0.835894	+	0.548890i	$0.815051\pi$
$168$	−2.37945	−0.183579
$169$	14.2518	1.09629
$170$	0	0
$171$	9.63581	0.736869
$172$	−2.11845	−0.161530
$173$	−7.29689	−0.554772	−0.277386	−	0.960758i	$-0.589468\pi$
$173$	−7.29689	−0.554772	−0.277386	+	0.960758i	$0.589468\pi$
$174$	7.44459	0.564373
$175$	0	0
$176$	−18.1236	−1.36611
$177$	1.67437	0.125854
$178$	−14.6340	−1.09686
$179$	7.24292	0.541362	0.270681	−	0.962669i	$-0.412751\pi$
$179$	7.24292	0.541362	0.270681	+	0.962669i	$0.412751\pi$
$180$	0	0
$181$	−5.48395	−0.407619	−0.203809	−	0.979011i	$-0.565332\pi$
$181$	−5.48395	−0.407619	−0.203809	+	0.979011i	$0.565332\pi$
$182$	8.01347	0.593998
$183$	−1.56683	−0.115824
$184$	2.52303	0.186001
$185$	0	0
$186$	−0.559260	−0.0410070
$187$	−15.7180	−1.14941
$188$	−1.06140	−0.0774108
$189$	4.81974	0.350585
$190$	0	0
$191$	16.4704	1.19175	0.595877	−	0.803075i	$-0.296804\pi$
$191$	16.4704	1.19175	0.595877	+	0.803075i	$0.296804\pi$
$192$	−5.76387	−0.415972
$193$	15.5952	1.12257	0.561285	−	0.827623i	$-0.310307\pi$
$193$	15.5952	1.12257	0.561285	+	0.827623i	$0.310307\pi$
$194$	−23.8519	−1.71247
$195$	0	0
$196$	0.356385	0.0254561
$197$	20.2326	1.44152	0.720758	−	0.693186i	$-0.243794\pi$
$197$	20.2326	1.44152	0.720758	+	0.693186i	$0.243794\pi$
$198$	12.8043	0.909964
$199$	−3.02265	−0.214270	−0.107135	−	0.994244i	$-0.534168\pi$
$199$	−3.02265	−0.214270	−0.107135	+	0.994244i	$0.534168\pi$
$200$	0	0
$201$	4.69948	0.331475
$202$	28.2503	1.98768
$203$	5.14238	0.360924
$204$	1.33671	0.0935887
$205$	0	0
$206$	17.5406	1.22211
$207$	−2.11058	−0.146695
$208$	23.9391	1.65988
$209$	−18.0434	−1.24809
$210$	0	0
$211$	−10.0153	−0.689482	−0.344741	−	0.938698i	$-0.612033\pi$
$211$	−10.0153	−0.689482	−0.344741	+	0.938698i	$0.612033\pi$
$212$	4.32235	0.296860
$213$	−4.60027	−0.315205
$214$	8.94299	0.611330
$215$	0	0
$216$	12.1604	0.827409
$217$	−0.386311	−0.0262245
$218$	23.8028	1.61213
$219$	4.14191	0.279884
$220$	0	0
$221$	20.7617	1.39658
$222$	−14.4352	−0.968824
$223$	1.25173	0.0838220	0.0419110	−	0.999121i	$-0.486655\pi$
$223$	1.25173	0.0838220	0.0419110	+	0.999121i	$0.486655\pi$
$224$	1.99331	0.133184
$225$	0	0
$226$	21.4466	1.42660
$227$	29.8582	1.98176	0.990878	−	0.134758i	$-0.0430258\pi$
$227$	29.8582	1.98176	0.990878	+	0.134758i	$0.0430258\pi$
$228$	1.53447	0.101623
$229$	−6.51352	−0.430426	−0.215213	−	0.976567i	$-0.569044\pi$
$229$	−6.51352	−0.430426	−0.215213	+	0.976567i	$0.569044\pi$
$230$	0	0
$231$	−3.72723	−0.245234
$232$	12.9744	0.851811
$233$	−18.8200	−1.23294	−0.616468	−	0.787380i	$-0.711437\pi$
$233$	−18.8200	−1.23294	−0.616468	+	0.787380i	$0.711437\pi$
$234$	−16.9131	−1.10564
$235$	0	0
$236$	−0.632730	−0.0411872
$237$	−9.79833	−0.636470
$238$	6.10504	0.395731
$239$	0.545414	0.0352799	0.0176400	−	0.999844i	$-0.494385\pi$
$239$	0.545414	0.0352799	0.0176400	+	0.999844i	$0.494385\pi$
$240$	0	0
$241$	17.3989	1.12076	0.560380	−	0.828235i	$-0.310655\pi$
$241$	17.3989	1.12076	0.560380	+	0.828235i	$0.310655\pi$
$242$	−7.09102	−0.455828
$243$	−16.1439	−1.03563
$244$	0.592091	0.0379047
$245$	0	0
$246$	10.3069	0.657146
$247$	23.8333	1.51648
$248$	−0.974676	−0.0618920
$249$	10.3551	0.656226
$250$	0	0
$251$	26.8568	1.69519	0.847594	−	0.530645i	$-0.178050\pi$
$251$	26.8568	1.69519	0.847594	+	0.530645i	$0.178050\pi$
$252$	−0.752179	−0.0473828
$253$	3.95214	0.248469
$254$	−0.704743	−0.0442195
$255$	0	0
$256$	8.29779	0.518612
$257$	5.20491	0.324674	0.162337	−	0.986735i	$-0.448097\pi$
$257$	5.20491	0.324674	0.162337	+	0.986735i	$0.448097\pi$
$258$	−8.60549	−0.535754
$259$	−9.97114	−0.619576
$260$	0	0
$261$	−10.8534	−0.671808
$262$	11.6224	0.718035
$263$	28.8301	1.77774	0.888871	−	0.458157i	$-0.151490\pi$
$263$	28.8301	1.77774	0.888871	+	0.458157i	$0.151490\pi$
$264$	−9.40392	−0.578771
$265$	0	0
$266$	7.00825	0.429703
$267$	−8.99069	−0.550221
$268$	−1.77589	−0.108480
$269$	1.45127	0.0884857	0.0442428	−	0.999021i	$-0.485912\pi$
$269$	1.45127	0.0884857	0.0442428	+	0.999021i	$0.485912\pi$
$270$	0	0
$271$	−1.42456	−0.0865359	−0.0432680	−	0.999064i	$-0.513777\pi$
$271$	−1.42456	−0.0865359	−0.0432680	+	0.999064i	$0.513777\pi$
$272$	18.2380	1.10584
$273$	4.92324	0.297968
$274$	−21.4905	−1.29829
$275$	0	0
$276$	−0.336104	−0.0202311
$277$	19.4044	1.16590	0.582950	−	0.812508i	$-0.301898\pi$
$277$	19.4044	1.16590	0.582950	+	0.812508i	$0.301898\pi$
$278$	−3.61841	−0.217018
$279$	0.815340	0.0488131
$280$	0	0
$281$	−14.9059	−0.889210	−0.444605	−	0.895727i	$-0.646656\pi$
$281$	−14.9059	−0.889210	−0.444605	+	0.895727i	$0.646656\pi$
$282$	−4.31159	−0.256751
$283$	2.34950	0.139663	0.0698315	−	0.997559i	$-0.477754\pi$
$283$	2.34950	0.139663	0.0698315	+	0.997559i	$0.477754\pi$
$284$	1.73840	0.103155
$285$	0	0
$286$	31.6703	1.87271
$287$	7.11955	0.420254
$288$	−4.20704	−0.247902
$289$	−1.18276	−0.0695742
$290$	0	0
$291$	−14.6539	−0.859028
$292$	−1.56519	−0.0915958
$293$	−2.30320	−0.134555	−0.0672773	−	0.997734i	$-0.521431\pi$
$293$	−2.30320	−0.134555	−0.0672773	+	0.997734i	$0.521431\pi$
$294$	1.44769	0.0844312
$295$	0	0
$296$	−25.1575	−1.46225
$297$	19.0483	1.10529
$298$	−22.2779	−1.29052
$299$	−5.22032	−0.301899
$300$	0	0
$301$	−5.94427	−0.342622
$302$	−25.8811	−1.48929
$303$	17.3562	0.997085
$304$	20.9362	1.20077
$305$	0	0
$306$	−12.8852	−0.736596
$307$	−23.6213	−1.34814	−0.674068	−	0.738669i	$-0.735455\pi$
$307$	−23.6213	−1.34814	−0.674068	+	0.738669i	$0.735455\pi$
$308$	1.40848	0.0802558
$309$	10.7764	0.613050
$310$	0	0
$311$	−15.4294	−0.874919	−0.437460	−	0.899238i	$-0.644122\pi$
$311$	−15.4294	−0.874919	−0.437460	+	0.899238i	$0.644122\pi$
$312$	12.4215	0.703229
$313$	2.23117	0.126113	0.0630566	−	0.998010i	$-0.479915\pi$
$313$	2.23117	0.126113	0.0630566	+	0.998010i	$0.479915\pi$
$314$	−21.3759	−1.20631
$315$	0	0
$316$	3.70269	0.208293
$317$	−8.96613	−0.503588	−0.251794	−	0.967781i	$-0.581021\pi$
$317$	−8.96613	−0.503588	−0.251794	+	0.967781i	$0.581021\pi$
$318$	17.5581	0.984609
$319$	20.3234	1.13789
$320$	0	0
$321$	5.49431	0.306662
$322$	−1.53505	−0.0855451
$323$	18.1573	1.01030
$324$	0.636604	0.0353669
$325$	0	0
$326$	−14.7939	−0.819361
$327$	14.6237	0.808694
$328$	17.9629	0.991834
$329$	−2.97825	−0.164196
$330$	0	0
$331$	−23.6082	−1.29762	−0.648811	−	0.760950i	$-0.724733\pi$
$331$	−23.6082	−1.29762	−0.648811	+	0.760950i	$0.724733\pi$
$332$	−3.91308	−0.214758
$333$	21.0449	1.15325
$334$	33.1637	1.81464
$335$	0	0
$336$	4.32479	0.235937
$337$	20.5537	1.11963	0.559816	−	0.828617i	$-0.310872\pi$
$337$	20.5537	1.11963	0.559816	+	0.828617i	$0.310872\pi$
$338$	−21.8772	−1.18996
$339$	13.1761	0.715630
$340$	0	0
$341$	−1.52675	−0.0826784
$342$	−14.7915	−0.799831
$343$	1.00000	0.0539949
$344$	−14.9976	−0.808616
$345$	0	0
$346$	11.2011	0.602175
$347$	19.5522	1.04962	0.524808	−	0.851221i	$-0.324137\pi$
$347$	19.5522	1.04962	0.524808	+	0.851221i	$0.324137\pi$
$348$	−1.72837	−0.0926505
$349$	23.4093	1.25307	0.626536	−	0.779392i	$-0.284472\pi$
$349$	23.4093	1.25307	0.626536	+	0.779392i	$0.284472\pi$
$350$	0	0
$351$	−25.1606	−1.34297
$352$	7.87784	0.419890
$353$	27.3762	1.45709	0.728544	−	0.684999i	$-0.240197\pi$
$353$	27.3762	1.45709	0.728544	+	0.684999i	$0.240197\pi$
$354$	−2.57025	−0.136607
$355$	0	0
$356$	3.39749	0.180067
$357$	3.75076	0.198511
$358$	−11.1183	−0.587619
$359$	−27.0736	−1.42889	−0.714444	−	0.699692i	$-0.753320\pi$
$359$	−27.0736	−1.42889	−0.714444	+	0.699692i	$0.753320\pi$
$360$	0	0
$361$	1.84361	0.0970323
$362$	8.41814	0.442448
$363$	−4.35651	−0.228658
$364$	−1.86045	−0.0975138
$365$	0	0
$366$	2.40517	0.125720
$367$	−4.22104	−0.220336	−0.110168	−	0.993913i	$-0.535139\pi$
$367$	−4.22104	−0.220336	−0.110168	+	0.993913i	$0.535139\pi$
$368$	−4.58576	−0.239049
$369$	−15.0264	−0.782242
$370$	0	0
$371$	12.1283	0.629671
$372$	0.129841	0.00673192
$373$	1.61136	0.0834330	0.0417165	−	0.999129i	$-0.486717\pi$
$373$	1.61136	0.0834330	0.0417165	+	0.999129i	$0.486717\pi$
$374$	24.1280	1.24763
$375$	0	0
$376$	−7.51422	−0.387516
$377$	−26.8449	−1.38258
$378$	−7.39855	−0.380541
$379$	6.26192	0.321653	0.160827	−	0.986983i	$-0.448584\pi$
$379$	6.26192	0.321653	0.160827	+	0.986983i	$0.448584\pi$
$380$	0	0
$381$	−0.432974	−0.0221819
$382$	−25.2829	−1.29359
$383$	25.0713	1.28108	0.640542	−	0.767923i	$-0.278710\pi$
$383$	25.0713	1.28108	0.640542	+	0.767923i	$0.278710\pi$
$384$	12.6076	0.643379
$385$	0	0
$386$	−23.9395	−1.21849
$387$	12.5459	0.637742
$388$	5.53757	0.281128
$389$	9.56706	0.485069	0.242535	−	0.970143i	$-0.422021\pi$
$389$	9.56706	0.485069	0.242535	+	0.970143i	$0.422021\pi$
$390$	0	0
$391$	−3.97709	−0.201130
$392$	2.52303	0.127432
$393$	7.14047	0.360189
$394$	−31.0582	−1.56469
$395$	0	0
$396$	−2.97272	−0.149385
$397$	14.1377	0.709550	0.354775	−	0.934952i	$-0.384558\pi$
$397$	14.1377	0.709550	0.354775	+	0.934952i	$0.384558\pi$
$398$	4.63993	0.232579
$399$	4.30566	0.215553
$400$	0	0
$401$	−6.72338	−0.335750	−0.167875	−	0.985808i	$-0.553690\pi$
$401$	−6.72338	−0.335750	−0.167875	+	0.985808i	$0.553690\pi$
$402$	−7.21394	−0.359799
$403$	2.01667	0.100457
$404$	−6.55873	−0.326309
$405$	0	0
$406$	−7.89382	−0.391764
$407$	−39.4073	−1.95335
$408$	9.46329	0.468503
$409$	16.0860	0.795401	0.397700	−	0.917515i	$-0.369808\pi$
$409$	16.0860	0.795401	0.397700	+	0.917515i	$0.369808\pi$
$410$	0	0
$411$	−13.2032	−0.651264
$412$	−4.07231	−0.200628
$413$	−1.77541	−0.0873623
$414$	3.23985	0.159230
$415$	0	0
$416$	−10.4057	−0.510183
$417$	−2.22305	−0.108863
$418$	27.6976	1.35473
$419$	18.9705	0.926769	0.463384	−	0.886157i	$-0.346635\pi$
$419$	18.9705	0.926769	0.463384	+	0.886157i	$0.346635\pi$
$420$	0	0
$421$	−13.6098	−0.663301	−0.331650	−	0.943402i	$-0.607605\pi$
$421$	−13.6098	−0.663301	−0.331650	+	0.943402i	$0.607605\pi$
$422$	15.3740	0.748396
$423$	6.28583	0.305627
$424$	30.6002	1.48608
$425$	0	0
$426$	7.06166	0.342138
$427$	1.66138	0.0803998
$428$	−2.07625	−0.100359
$429$	19.4573	0.939408
$430$	0	0
$431$	−6.92519	−0.333575	−0.166787	−	0.985993i	$-0.553339\pi$
$431$	−6.92519	−0.333575	−0.166787	+	0.985993i	$0.553339\pi$
$432$	−22.1022	−1.06339
$433$	33.1282	1.59204	0.796020	−	0.605270i	$-0.206935\pi$
$433$	33.1282	1.59204	0.796020	+	0.605270i	$0.206935\pi$
$434$	0.593008	0.0284653
$435$	0	0
$436$	−5.52617	−0.264655
$437$	−4.56548	−0.218397
$438$	−6.35805	−0.303799
$439$	26.9149	1.28458	0.642290	−	0.766462i	$-0.277985\pi$
$439$	26.9149	1.28458	0.642290	+	0.766462i	$0.277985\pi$
$440$	0	0
$441$	−2.11058	−0.100504
$442$	−31.8703	−1.51591
$443$	29.7660	1.41422	0.707112	−	0.707101i	$-0.249998\pi$
$443$	29.7660	1.41422	0.707112	+	0.707101i	$0.249998\pi$
$444$	3.35134	0.159047
$445$	0	0
$446$	−1.92147	−0.0909842
$447$	−13.6869	−0.647367
$448$	6.11168	0.288750
$449$	33.8429	1.59715	0.798573	−	0.601898i	$-0.205588\pi$
$449$	33.8429	1.59715	0.798573	+	0.601898i	$0.205588\pi$
$450$	0	0
$451$	28.1375	1.32494
$452$	−4.97914	−0.234199
$453$	−15.9006	−0.747074
$454$	−45.8339	−2.15109
$455$	0	0
$456$	10.8633	0.508722
$457$	14.5218	0.679301	0.339651	−	0.940552i	$-0.389691\pi$
$457$	14.5218	0.679301	0.339651	+	0.940552i	$0.389691\pi$
$458$	9.99859	0.467204
$459$	−19.1685	−0.894711
$460$	0	0
$461$	20.9108	0.973914	0.486957	−	0.873426i	$-0.338107\pi$
$461$	20.9108	0.973914	0.486957	+	0.873426i	$0.338107\pi$
$462$	5.72149	0.266188
$463$	26.1755	1.21648	0.608238	−	0.793755i	$-0.291877\pi$
$463$	26.1755	1.21648	0.608238	+	0.793755i	$0.291877\pi$
$464$	−23.5817	−1.09475
$465$	0	0
$466$	28.8896	1.33829
$467$	20.2614	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$467$	20.2614	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$468$	3.92662	0.181508
$469$	−4.98306	−0.230096
$470$	0	0
$471$	−13.1327	−0.605124
$472$	−4.47943	−0.206182
$473$	−23.4926	−1.08019
$474$	15.0409	0.690853
$475$	0	0
$476$	−1.41738	−0.0649653
$477$	−25.5978	−1.17204
$478$	−0.837239	−0.0382944
$479$	4.20344	0.192060	0.0960300	−	0.995378i	$-0.469386\pi$
$479$	4.20344	0.192060	0.0960300	+	0.995378i	$0.469386\pi$
$480$	0	0
$481$	52.0526	2.37339
$482$	−26.7082	−1.21653
$483$	−0.943091	−0.0429121
$484$	1.64629	0.0748311
$485$	0	0
$486$	24.7817	1.12412
$487$	19.8775	0.900734	0.450367	−	0.892844i	$-0.351293\pi$
$487$	19.8775	0.900734	0.450367	+	0.892844i	$0.351293\pi$
$488$	4.19172	0.189750
$489$	−9.08897	−0.411017
$490$	0	0
$491$	41.1248	1.85594	0.927969	−	0.372657i	$-0.121553\pi$
$491$	41.1248	1.85594	0.927969	+	0.372657i	$0.121553\pi$
$492$	−2.39291	−0.107881
$493$	−20.4517	−0.921098
$494$	−36.5853	−1.64605
$495$	0	0
$496$	1.77153	0.0795440
$497$	4.87787	0.218802
$498$	−15.8956	−0.712298
$499$	−7.44404	−0.333241	−0.166620	−	0.986021i	$-0.553285\pi$
$499$	−7.44404	−0.333241	−0.166620	+	0.986021i	$0.553285\pi$
$500$	0	0
$501$	20.3748	0.910279
$502$	−41.2266	−1.84004
$503$	−17.9002	−0.798129	−0.399065	−	0.916923i	$-0.630665\pi$
$503$	−17.9002	−0.798129	−0.399065	+	0.916923i	$0.630665\pi$
$504$	−5.32506	−0.237197
$505$	0	0
$506$	−6.06674	−0.269699
$507$	−13.4407	−0.596923
$508$	0.163617	0.00725931
$509$	2.09056	0.0926626	0.0463313	−	0.998926i	$-0.485247\pi$
$509$	2.09056	0.0926626	0.0463313	+	0.998926i	$0.485247\pi$
$510$	0	0
$511$	−4.39185	−0.194284
$512$	13.9992	0.618684
$513$	−22.0044	−0.971520
$514$	−7.98982	−0.352416
$515$	0	0
$516$	1.99789	0.0879523
$517$	−11.7704	−0.517664
$518$	15.3062	0.672517
$519$	6.88163	0.302070
$520$	0	0
$521$	−33.4451	−1.46526	−0.732628	−	0.680630i	$-0.761706\pi$
$521$	−33.4451	−1.46526	−0.732628	+	0.680630i	$0.761706\pi$
$522$	16.6605	0.729212
$523$	3.46041	0.151313	0.0756566	−	0.997134i	$-0.475895\pi$
$523$	3.46041	0.151313	0.0756566	+	0.997134i	$0.475895\pi$
$524$	−2.69831	−0.117876
$525$	0	0
$526$	−44.2558	−1.92964
$527$	1.53639	0.0669264
$528$	17.0922	0.743841
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.74715	0.162612
$532$	−1.62707	−0.0705424
$533$	−37.1664	−1.60985
$534$	13.8012	0.597235
$535$	0	0
$536$	−12.5724	−0.543046
$537$	−6.83074	−0.294768
$538$	−2.22778	−0.0960464
$539$	3.95214	0.170231
$540$	0	0
$541$	16.4764	0.708374	0.354187	−	0.935175i	$-0.384758\pi$
$541$	16.4764	0.708374	0.354187	+	0.935175i	$0.384758\pi$
$542$	2.18678	0.0939301
$543$	5.17186	0.221946
$544$	−7.92758	−0.339892
$545$	0	0
$546$	−7.55743	−0.323428
$547$	−9.04704	−0.386824	−0.193412	−	0.981118i	$-0.561955\pi$
$547$	−9.04704	−0.386824	−0.193412	+	0.981118i	$0.561955\pi$
$548$	4.98935	0.213134
$549$	−3.50647	−0.149653
$550$	0	0
$551$	−23.4774	−1.00017
$552$	−2.37945	−0.101276
$553$	10.3896	0.441810
$554$	−29.7868	−1.26552
$555$	0	0
$556$	0.840068	0.0356268
$557$	−4.94450	−0.209505	−0.104753	−	0.994498i	$-0.533405\pi$
$557$	−4.94450	−0.209505	−0.104753	+	0.994498i	$0.533405\pi$
$558$	−1.25159	−0.0529840
$559$	31.0310	1.31247
$560$	0	0
$561$	14.8235	0.625849
$562$	22.8813	0.965189
$563$	−15.2161	−0.641282	−0.320641	−	0.947201i	$-0.603898\pi$
$563$	−15.2161	−0.641282	−0.320641	+	0.947201i	$0.603898\pi$
$564$	1.00100	0.0421497
$565$	0	0
$566$	−3.60660	−0.151597
$567$	1.78628	0.0750168
$568$	12.3070	0.516391
$569$	−44.5700	−1.86847	−0.934235	−	0.356658i	$-0.883916\pi$
$569$	−44.5700	−1.86847	−0.934235	+	0.356658i	$0.883916\pi$
$570$	0	0
$571$	24.1323	1.00991	0.504953	−	0.863147i	$-0.331510\pi$
$571$	24.1323	1.00991	0.504953	+	0.863147i	$0.331510\pi$
$572$	−7.35274	−0.307433
$573$	−15.5331	−0.648903
$574$	−10.9289	−0.456163
$575$	0	0
$576$	−12.8992	−0.537466
$577$	−3.97602	−0.165524	−0.0827619	−	0.996569i	$-0.526374\pi$
$577$	−3.97602	−0.165524	−0.0827619	+	0.996569i	$0.526374\pi$
$578$	1.81560	0.0755191
$579$	−14.7077	−0.611232
$580$	0	0
$581$	−10.9799	−0.455524
$582$	22.4945	0.932428
$583$	47.9328	1.98517
$584$	−11.0808	−0.458526
$585$	0	0
$586$	3.53554	0.146052
$587$	15.2089	0.627738	0.313869	−	0.949466i	$-0.398375\pi$
$587$	15.2089	0.627738	0.313869	+	0.949466i	$0.398375\pi$
$588$	−0.336104	−0.0138607
$589$	1.76370	0.0726718
$590$	0	0
$591$	−19.0812	−0.784897
$592$	45.7252	1.87930
$593$	−33.3570	−1.36981	−0.684904	−	0.728634i	$-0.740156\pi$
$593$	−33.3570	−1.36981	−0.684904	+	0.728634i	$0.740156\pi$
$594$	−29.2401	−1.19974
$595$	0	0
$596$	5.17214	0.211859
$597$	2.85064	0.116669
$598$	8.01347	0.327695
$599$	13.1463	0.537143	0.268571	−	0.963260i	$-0.413448\pi$
$599$	13.1463	0.537143	0.268571	+	0.963260i	$0.413448\pi$
$600$	0	0
$601$	5.48882	0.223894	0.111947	−	0.993714i	$-0.464291\pi$
$601$	5.48882	0.223894	0.111947	+	0.993714i	$0.464291\pi$
$602$	9.12477	0.371898
$603$	10.5171	0.428291
$604$	6.00867	0.244489
$605$	0	0
$606$	−26.6426	−1.08228
$607$	−44.7445	−1.81612	−0.908062	−	0.418836i	$-0.862438\pi$
$607$	−44.7445	−1.81612	−0.908062	+	0.418836i	$0.862438\pi$
$608$	−9.10042	−0.369071
$609$	−4.84973	−0.196521
$610$	0	0
$611$	15.5474	0.628981
$612$	2.99148	0.120924
$613$	−33.5368	−1.35454	−0.677270	−	0.735735i	$-0.736837\pi$
$613$	−33.5368	−1.35454	−0.677270	+	0.735735i	$0.736837\pi$
$614$	36.2599	1.46333
$615$	0	0
$616$	9.97138	0.401758
$617$	−37.1716	−1.49647	−0.748237	−	0.663431i	$-0.769099\pi$
$617$	−37.1716	−1.49647	−0.748237	+	0.663431i	$0.769099\pi$
$618$	−16.5424	−0.665432
$619$	31.7772	1.27723	0.638617	−	0.769525i	$-0.279507\pi$
$619$	31.7772	1.27723	0.638617	+	0.769525i	$0.279507\pi$
$620$	0	0
$621$	4.81974	0.193410
$622$	23.6849	0.949677
$623$	9.53321	0.381940
$624$	−22.5768	−0.903795
$625$	0	0
$626$	−3.42496	−0.136889
$627$	17.0166	0.679577
$628$	4.96273	0.198034
$629$	39.6561	1.58119
$630$	0	0
$631$	−28.8102	−1.14692	−0.573459	−	0.819234i	$-0.694399\pi$
$631$	−28.8102	−1.14692	−0.573459	+	0.819234i	$0.694399\pi$
$632$	26.2133	1.04271
$633$	9.44535	0.375419
$634$	13.7635	0.546617
$635$	0	0
$636$	−4.07637	−0.161639
$637$	−5.22032	−0.206837
$638$	−31.1975	−1.23512
$639$	−10.2951	−0.407269
$640$	0	0
$641$	18.8199	0.743341	0.371671	−	0.928365i	$-0.378785\pi$
$641$	18.8199	0.743341	0.371671	+	0.928365i	$0.378785\pi$
$642$	−8.43405	−0.332865
$643$	−4.20086	−0.165666	−0.0828329	−	0.996563i	$-0.526397\pi$
$643$	−4.20086	−0.165666	−0.0828329	+	0.996563i	$0.526397\pi$
$644$	0.356385	0.0140435
$645$	0	0
$646$	−27.8724	−1.09663
$647$	12.2761	0.482624	0.241312	−	0.970448i	$-0.422422\pi$
$647$	12.2761	0.482624	0.241312	+	0.970448i	$0.422422\pi$
$648$	4.50685	0.177046
$649$	−7.01667	−0.275429
$650$	0	0
$651$	0.364327	0.0142791
$652$	3.43463	0.134511
$653$	33.9257	1.32762	0.663809	−	0.747903i	$-0.268939\pi$
$653$	33.9257	1.32762	0.663809	+	0.747903i	$0.268939\pi$
$654$	−22.4482	−0.877794
$655$	0	0
$656$	−32.6486	−1.27471
$657$	9.26934	0.361631
$658$	4.57176	0.178226
$659$	−22.9926	−0.895665	−0.447833	−	0.894117i	$-0.647804\pi$
$659$	−22.9926	−0.895665	−0.447833	+	0.894117i	$0.647804\pi$
$660$	0	0
$661$	−0.220429	−0.00857368	−0.00428684	−	0.999991i	$-0.501365\pi$
$661$	−0.220429	−0.00857368	−0.00428684	+	0.999991i	$0.501365\pi$
$662$	36.2397	1.40850
$663$	−19.5802	−0.760430
$664$	−27.7027	−1.07507
$665$	0	0
$666$	−32.3050	−1.25179
$667$	5.14238	0.199114
$668$	−7.69944	−0.297900
$669$	−1.18049	−0.0456406
$670$	0	0
$671$	6.56600	0.253478
$672$	−1.87987	−0.0725177
$673$	17.5898	0.678035	0.339018	−	0.940780i	$-0.389905\pi$
$673$	17.5898	0.678035	0.339018	+	0.940780i	$0.389905\pi$
$674$	−31.5510	−1.21530
$675$	0	0
$676$	5.07912	0.195351
$677$	−35.3647	−1.35918	−0.679588	−	0.733594i	$-0.737841\pi$
$677$	−35.3647	−1.35918	−0.679588	+	0.733594i	$0.737841\pi$
$678$	−20.2261	−0.776777
$679$	15.5382	0.596300
$680$	0	0
$681$	−28.1590	−1.07905
$682$	2.34365	0.0897429
$683$	−33.5832	−1.28503	−0.642513	−	0.766274i	$-0.722108\pi$
$683$	−33.5832	−1.28503	−0.642513	+	0.766274i	$0.722108\pi$
$684$	3.43406	0.131305
$685$	0	0
$686$	−1.53505	−0.0586086
$687$	6.14284	0.234364
$688$	27.2590	1.03924
$689$	−63.3137	−2.41206
$690$	0	0
$691$	33.1530	1.26120	0.630599	−	0.776109i	$-0.282809\pi$
$691$	33.1530	1.26120	0.630599	+	0.776109i	$0.282809\pi$
$692$	−2.60050	−0.0988563
$693$	−8.34130	−0.316860
$694$	−30.0136	−1.13930
$695$	0	0
$696$	−12.2360	−0.463806
$697$	−28.3151	−1.07251
$698$	−35.9345	−1.36014
$699$	17.7489	0.671326
$700$	0	0
$701$	5.66567	0.213990	0.106995	−	0.994260i	$-0.465877\pi$
$701$	5.66567	0.213990	0.106995	+	0.994260i	$0.465877\pi$
$702$	38.6228	1.45773
$703$	45.5230	1.71693
$704$	24.1542	0.910346
$705$	0	0
$706$	−42.0239	−1.58159
$707$	−18.4035	−0.692134
$708$	0.596722	0.0224262
$709$	16.5570	0.621810	0.310905	−	0.950441i	$-0.399368\pi$
$709$	16.5570	0.621810	0.310905	+	0.950441i	$0.399368\pi$
$710$	0	0
$711$	−21.9280	−0.822366
$712$	24.0526	0.901410
$713$	−0.386311	−0.0144675
$714$	−5.75761	−0.215473
$715$	0	0
$716$	2.58127	0.0964666
$717$	−0.514375	−0.0192097
$718$	41.5594	1.55098
$719$	18.8863	0.704340	0.352170	−	0.935936i	$-0.385444\pi$
$719$	18.8863	0.704340	0.352170	+	0.935936i	$0.385444\pi$
$720$	0	0
$721$	−11.4267	−0.425553
$722$	−2.83004	−0.105323
$723$	−16.4087	−0.610247
$724$	−1.95440	−0.0726346
$725$	0	0
$726$	6.68748	0.248196
$727$	−5.57932	−0.206926	−0.103463	−	0.994633i	$-0.532992\pi$
$727$	−5.57932	−0.206926	−0.103463	+	0.994633i	$0.532992\pi$
$728$	−13.1711	−0.488152
$729$	9.86628	0.365418
$730$	0	0
$731$	23.6409	0.874390
$732$	−0.558396	−0.0206389
$733$	13.7692	0.508577	0.254289	−	0.967128i	$-0.418159\pi$
$733$	13.7692	0.508577	0.254289	+	0.967128i	$0.418159\pi$
$734$	6.47951	0.239163
$735$	0	0
$736$	1.99331	0.0734744
$737$	−19.6937	−0.725428
$738$	23.0663	0.849081
$739$	−27.6925	−1.01868	−0.509342	−	0.860564i	$-0.670111\pi$
$739$	−27.6925	−1.01868	−0.509342	+	0.860564i	$0.670111\pi$
$740$	0	0
$741$	−22.4770	−0.825712
$742$	−18.6176	−0.683474
$743$	30.2412	1.10944	0.554721	−	0.832037i	$-0.312825\pi$
$743$	30.2412	1.10944	0.554721	+	0.832037i	$0.312825\pi$
$744$	0.919208	0.0336998
$745$	0	0
$746$	−2.47352	−0.0905620
$747$	23.1740	0.847893
$748$	−5.60166	−0.204817
$749$	−5.82585	−0.212872
$750$	0	0
$751$	−10.6759	−0.389567	−0.194784	−	0.980846i	$-0.562401\pi$
$751$	−10.6759	−0.389567	−0.194784	+	0.980846i	$0.562401\pi$
$752$	13.6575	0.498039
$753$	−25.3284	−0.923020
$754$	41.2083	1.50072
$755$	0	0
$756$	1.71768	0.0624716
$757$	28.7122	1.04356	0.521781	−	0.853080i	$-0.325268\pi$
$757$	28.7122	1.04356	0.521781	+	0.853080i	$0.325268\pi$
$758$	−9.61237	−0.349137
$759$	−3.72723	−0.135290
$760$	0	0
$761$	21.3408	0.773602	0.386801	−	0.922163i	$-0.373580\pi$
$761$	21.3408	0.773602	0.386801	+	0.922163i	$0.373580\pi$
$762$	0.664637	0.0240773
$763$	−15.5062	−0.561361
$764$	5.86980	0.212362
$765$	0	0
$766$	−38.4858	−1.39055
$767$	9.26822	0.334656
$768$	−7.82557	−0.282381
$769$	−14.9967	−0.540794	−0.270397	−	0.962749i	$-0.587155\pi$
$769$	−14.9967	−0.540794	−0.270397	+	0.962749i	$0.587155\pi$
$770$	0	0
$771$	−4.90871	−0.176783
$772$	5.55791	0.200034
$773$	37.3605	1.34376	0.671882	−	0.740658i	$-0.265486\pi$
$773$	37.3605	1.34376	0.671882	+	0.740658i	$0.265486\pi$
$774$	−19.2585	−0.692234
$775$	0	0
$776$	39.2033	1.40732
$777$	9.40369	0.337356
$778$	−14.6859	−0.526516
$779$	−32.5042	−1.16458
$780$	0	0
$781$	19.2780	0.689821
$782$	6.10504	0.218316
$783$	24.7849	0.885741
$784$	−4.58576	−0.163777
$785$	0	0
$786$	−10.9610	−0.390965
$787$	19.4172	0.692147	0.346074	−	0.938207i	$-0.387515\pi$
$787$	19.4172	0.692147	0.346074	+	0.938207i	$0.387515\pi$
$788$	7.21061	0.256868
$789$	−27.1894	−0.967970
$790$	0	0
$791$	−13.9712	−0.496760
$792$	−21.0454	−0.747815
$793$	−8.67294	−0.307985
$794$	−21.7021	−0.770178
$795$	0	0
$796$	−1.07723	−0.0381813
$797$	42.2449	1.49639	0.748195	−	0.663479i	$-0.230921\pi$
$797$	42.2449	1.49639	0.748195	+	0.663479i	$0.230921\pi$
$798$	−6.60942	−0.233971
$799$	11.8448	0.419037
$800$	0	0
$801$	−20.1206	−0.710926
$802$	10.3207	0.364438
$803$	−17.3572	−0.612522
$804$	1.67482	0.0590665
$805$	0	0
$806$	−3.09569	−0.109041
$807$	−1.36868	−0.0481799
$808$	−46.4326	−1.63349
$809$	51.1689	1.79900	0.899502	−	0.436917i	$-0.143930\pi$
$809$	51.1689	1.79900	0.899502	+	0.436917i	$0.143930\pi$
$810$	0	0
$811$	27.2951	0.958462	0.479231	−	0.877689i	$-0.340916\pi$
$811$	27.2951	0.958462	0.479231	+	0.877689i	$0.340916\pi$
$812$	1.83267	0.0643140
$813$	1.34349	0.0471183
$814$	60.4923	2.12025
$815$	0	0
$816$	−17.2001	−0.602123
$817$	27.1385	0.949454
$818$	−24.6928	−0.863364
$819$	11.0179	0.384997
$820$	0	0
$821$	−49.5791	−1.73032	−0.865161	−	0.501495i	$-0.832784\pi$
$821$	−49.5791	−1.73032	−0.865161	+	0.501495i	$0.832784\pi$
$822$	20.2675	0.706912
$823$	−23.8598	−0.831699	−0.415850	−	0.909433i	$-0.636516\pi$
$823$	−23.8598	−0.831699	−0.415850	+	0.909433i	$0.636516\pi$
$824$	−28.8300	−1.00434
$825$	0	0
$826$	2.72535	0.0948270
$827$	−28.3613	−0.986217	−0.493109	−	0.869968i	$-0.664140\pi$
$827$	−28.3613	−0.986217	−0.493109	+	0.869968i	$0.664140\pi$
$828$	−0.752179	−0.0261400
$829$	−36.9907	−1.28474	−0.642370	−	0.766395i	$-0.722049\pi$
$829$	−36.9907	−1.28474	−0.642370	+	0.766395i	$0.722049\pi$
$830$	0	0
$831$	−18.3002	−0.634825
$832$	−31.9050	−1.10611
$833$	−3.97709	−0.137798
$834$	3.41249	0.118165
$835$	0	0
$836$	−6.43040	−0.222400
$837$	−1.86192	−0.0643574
$838$	−29.1207	−1.00596
$839$	−24.0825	−0.831419	−0.415710	−	0.909497i	$-0.636467\pi$
$839$	−24.0825	−0.831419	−0.415710	+	0.909497i	$0.636467\pi$
$840$	0	0
$841$	−2.55594	−0.0881359
$842$	20.8917	0.719977
$843$	14.0576	0.484169
$844$	−3.56931	−0.122861
$845$	0	0
$846$	−9.64907	−0.331742
$847$	4.61940	0.158724
$848$	−55.6176	−1.90991
$849$	−2.21579	−0.0760457
$850$	0	0
$851$	−9.97114	−0.341806
$852$	−1.63947	−0.0561673
$853$	−10.5707	−0.361933	−0.180966	−	0.983489i	$-0.557923\pi$
$853$	−10.5707	−0.361933	−0.180966	+	0.983489i	$0.557923\pi$
$854$	−2.55030	−0.0872696
$855$	0	0
$856$	−14.6988	−0.502395
$857$	−29.9631	−1.02352	−0.511759	−	0.859129i	$-0.671006\pi$
$857$	−29.9631	−1.02352	−0.511759	+	0.859129i	$0.671006\pi$
$858$	−29.8680	−1.01968
$859$	3.04269	0.103815	0.0519076	−	0.998652i	$-0.483470\pi$
$859$	3.04269	0.103815	0.0519076	+	0.998652i	$0.483470\pi$
$860$	0	0
$861$	−6.71439	−0.228826
$862$	10.6305	0.362077
$863$	8.55787	0.291313	0.145657	−	0.989335i	$-0.453471\pi$
$863$	8.55787	0.291313	0.145657	+	0.989335i	$0.453471\pi$
$864$	9.60725	0.326845
$865$	0	0
$866$	−50.8535	−1.72807
$867$	1.11545	0.0378828
$868$	−0.137676	−0.00467301
$869$	41.0611	1.39290
$870$	0	0
$871$	26.0132	0.881422
$872$	−39.1226	−1.32486
$873$	−32.7945	−1.10993
$874$	7.00825	0.237058
$875$	0	0
$876$	1.47612	0.0498733
$877$	23.6746	0.799434	0.399717	−	0.916639i	$-0.369108\pi$
$877$	23.6746	0.799434	0.399717	+	0.916639i	$0.369108\pi$
$878$	−41.3158	−1.39434
$879$	2.17213	0.0732641
$880$	0	0
$881$	−49.1808	−1.65694	−0.828472	−	0.560030i	$-0.810789\pi$
$881$	−49.1808	−1.65694	−0.828472	+	0.560030i	$0.810789\pi$
$882$	3.23985	0.109091
$883$	0.611783	0.0205881	0.0102941	−	0.999947i	$-0.496723\pi$
$883$	0.611783	0.0205881	0.0102941	+	0.999947i	$0.496723\pi$
$884$	7.39916	0.248861
$885$	0	0
$886$	−45.6923	−1.53506
$887$	−47.2610	−1.58687	−0.793434	−	0.608656i	$-0.791709\pi$
$887$	−47.2610	−1.58687	−0.793434	+	0.608656i	$0.791709\pi$
$888$	23.7258	0.796187
$889$	0.459101	0.0153977
$890$	0	0
$891$	7.05963	0.236507
$892$	0.446098	0.0149365
$893$	13.5971	0.455011
$894$	21.0101	0.702681
$895$	0	0
$896$	−13.3684	−0.446606
$897$	4.92324	0.164382
$898$	−51.9507	−1.73362
$899$	−1.98656	−0.0662554
$900$	0	0
$901$	−48.2354	−1.60695
$902$	−43.1925	−1.43815
$903$	5.60599	0.186556
$904$	−35.2499	−1.17239
$905$	0	0
$906$	24.4082	0.810908
$907$	29.8332	0.990595	0.495297	−	0.868724i	$-0.335059\pi$
$907$	29.8332	0.990595	0.495297	+	0.868724i	$0.335059\pi$
$908$	10.6410	0.353134
$909$	38.8420	1.28831
$910$	0	0
$911$	16.6250	0.550810	0.275405	−	0.961328i	$-0.411188\pi$
$911$	16.6250	0.550810	0.275405	+	0.961328i	$0.411188\pi$
$912$	−19.7447	−0.653814
$913$	−43.3942	−1.43614
$914$	−22.2917	−0.737344
$915$	0	0
$916$	−2.32132	−0.0766986
$917$	−7.57134	−0.250028
$918$	29.4247	0.971160
$919$	34.7111	1.14501	0.572507	−	0.819900i	$-0.305971\pi$
$919$	34.7111	1.14501	0.572507	+	0.819900i	$0.305971\pi$
$920$	0	0
$921$	22.2770	0.734052
$922$	−32.0992	−1.05713
$923$	−25.4640	−0.838159
$924$	−1.32833	−0.0436988
$925$	0	0
$926$	−40.1807	−1.32042
$927$	24.1170	0.792106
$928$	10.2504	0.336485
$929$	34.1318	1.11983	0.559914	−	0.828551i	$-0.310834\pi$
$929$	34.1318	1.11983	0.559914	+	0.828551i	$0.310834\pi$
$930$	0	0
$931$	−4.56548	−0.149628
$932$	−6.70715	−0.219700
$933$	14.5513	0.476388
$934$	−31.1023	−1.01770
$935$	0	0
$936$	27.7986	0.908624
$937$	−28.7762	−0.940077	−0.470039	−	0.882646i	$-0.655760\pi$
$937$	−28.7762	−0.940077	−0.470039	+	0.882646i	$0.655760\pi$
$938$	7.64925	0.249757
$939$	−2.10420	−0.0686678
$940$	0	0
$941$	−23.4919	−0.765815	−0.382908	−	0.923787i	$-0.625077\pi$
$941$	−23.4919	−0.765815	−0.382908	+	0.923787i	$0.625077\pi$
$942$	20.1594	0.656829
$943$	7.11955	0.231845
$944$	8.14161	0.264987
$945$	0	0
$946$	36.0623	1.17249
$947$	3.07257	0.0998453	0.0499226	−	0.998753i	$-0.484103\pi$
$947$	3.07257	0.0998453	0.0499226	+	0.998753i	$0.484103\pi$
$948$	−3.49198	−0.113414
$949$	22.9269	0.744237
$950$	0	0
$951$	8.45588	0.274201
$952$	−10.0343	−0.325215
$953$	47.6974	1.54507	0.772535	−	0.634973i	$-0.218989\pi$
$953$	47.6974	1.54507	0.772535	+	0.634973i	$0.218989\pi$
$954$	39.2939	1.27219
$955$	0	0
$956$	0.194378	0.00628662
$957$	−19.1668	−0.619575
$958$	−6.45250	−0.208471
$959$	13.9999	0.452080
$960$	0	0
$961$	−30.8508	−0.995186
$962$	−79.9034	−2.57619
$963$	12.2959	0.396230
$964$	6.20070	0.199711
$965$	0	0
$966$	1.44769	0.0465788
$967$	22.5354	0.724689	0.362344	−	0.932044i	$-0.381976\pi$
$967$	22.5354	0.724689	0.362344	+	0.932044i	$0.381976\pi$
$968$	11.6549	0.374603
$969$	−17.1240	−0.550102
$970$	0	0
$971$	−29.3868	−0.943068	−0.471534	−	0.881848i	$-0.656300\pi$
$971$	−29.3868	−0.943068	−0.471534	+	0.881848i	$0.656300\pi$
$972$	−5.75343	−0.184541
$973$	2.35719	0.0755681
$974$	−30.5129	−0.977697
$975$	0	0
$976$	−7.61869	−0.243868
$977$	16.7844	0.536979	0.268490	−	0.963283i	$-0.413475\pi$
$977$	16.7844	0.536979	0.268490	+	0.963283i	$0.413475\pi$
$978$	13.9520	0.446137
$979$	37.6766	1.20415
$980$	0	0
$981$	32.7270	1.04489
$982$	−63.1288	−2.01452
$983$	25.7012	0.819740	0.409870	−	0.912144i	$-0.365574\pi$
$983$	25.7012	0.819740	0.409870	+	0.912144i	$0.365574\pi$
$984$	−16.9406	−0.540047
$985$	0	0
$986$	31.3944	0.999802
$987$	2.80876	0.0894038
$988$	8.49383	0.270225
$989$	−5.94427	−0.189017
$990$	0	0
$991$	45.2305	1.43679	0.718397	−	0.695633i	$-0.244876\pi$
$991$	45.2305	1.43679	0.718397	+	0.695633i	$0.244876\pi$
$992$	−0.770038	−0.0244487
$993$	22.2646	0.706547
$994$	−7.48778	−0.237498
$995$	0	0
$996$	3.69039	0.116935
$997$	11.8939	0.376682	0.188341	−	0.982104i	$-0.439689\pi$
$997$	11.8939	0.376682	0.188341	+	0.982104i	$0.439689\pi$
$998$	11.4270	0.361715
$999$	−48.0583	−1.52050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.be.1.6		21
5.2	odd	4		805.2.c.c.484.12	✓	42
5.3	odd	4		805.2.c.c.484.31	yes	42
5.4	even	2		4025.2.a.bd.1.16		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.c.c.484.12	✓	42	5.2	odd	4
805.2.c.c.484.31	yes	42	5.3	odd	4
4025.2.a.bd.1.16		21	5.4	even	2
4025.2.a.be.1.6		21	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.53505 q^{2} -0.943091 q^{3} +0.356385 q^{4} +1.44769 q^{6} +1.00000 q^{7} +2.52303 q^{8} -2.11058 q^{9} +O(q^{10})\)	\(q-1.53505 q^{2} -0.943091 q^{3} +0.356385 q^{4} +1.44769 q^{6} +1.00000 q^{7} +2.52303 q^{8} -2.11058 q^{9} +3.95214 q^{11} -0.336104 q^{12} -5.22032 q^{13} -1.53505 q^{14} -4.58576 q^{16} -3.97709 q^{17} +3.23985 q^{18} -4.56548 q^{19} -0.943091 q^{21} -6.06674 q^{22} +1.00000 q^{23} -2.37945 q^{24} +8.01347 q^{26} +4.81974 q^{27} +0.356385 q^{28} +5.14238 q^{29} -0.386311 q^{31} +1.99331 q^{32} -3.72723 q^{33} +6.10504 q^{34} -0.752179 q^{36} -9.97114 q^{37} +7.00825 q^{38} +4.92324 q^{39} +7.11955 q^{41} +1.44769 q^{42} -5.94427 q^{43} +1.40848 q^{44} -1.53505 q^{46} -2.97825 q^{47} +4.32479 q^{48} +1.00000 q^{49} +3.75076 q^{51} -1.86045 q^{52} +12.1283 q^{53} -7.39855 q^{54} +2.52303 q^{56} +4.30566 q^{57} -7.89382 q^{58} -1.77541 q^{59} +1.66138 q^{61} +0.593008 q^{62} -2.11058 q^{63} +6.11168 q^{64} +5.72149 q^{66} -4.98306 q^{67} -1.41738 q^{68} -0.943091 q^{69} +4.87787 q^{71} -5.32506 q^{72} -4.39185 q^{73} +15.3062 q^{74} -1.62707 q^{76} +3.95214 q^{77} -7.55743 q^{78} +10.3896 q^{79} +1.78628 q^{81} -10.9289 q^{82} -10.9799 q^{83} -0.336104 q^{84} +9.12477 q^{86} -4.84973 q^{87} +9.97138 q^{88} +9.53321 q^{89} -5.22032 q^{91} +0.356385 q^{92} +0.364327 q^{93} +4.57176 q^{94} -1.87987 q^{96} +15.5382 q^{97} -1.53505 q^{98} -8.34130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9}+O(q^{10}) \) 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9	\( 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9} + 7 q^{11} - 22 q^{12} - 3 q^{13} + 2 q^{14} + 56 q^{16} + 7 q^{17} + 24 q^{19} - q^{21} + 4 q^{22} + 21 q^{23} + 24 q^{24} - 2 q^{26} - 19 q^{27} + 30 q^{28} + 11 q^{29} + 46 q^{31} - 6 q^{32} - 3 q^{33} + 28 q^{34} + 58 q^{36} + 24 q^{37} - 4 q^{38} + 31 q^{39} + 14 q^{41} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 2 q^{46} - 25 q^{47} - 36 q^{48} + 21 q^{49} + 17 q^{51} - 8 q^{52} + 22 q^{53} - 6 q^{54} + 6 q^{56} + 40 q^{57} + 6 q^{58} + 10 q^{59} + 38 q^{61} - 54 q^{62} + 30 q^{63} + 100 q^{64} + 38 q^{66} + 12 q^{67} + 18 q^{68} - q^{69} + 56 q^{71} + 42 q^{72} - 40 q^{73} - 20 q^{74} + 60 q^{76} + 7 q^{77} + 38 q^{78} + 49 q^{79} + 57 q^{81} + 16 q^{82} - 2 q^{83} - 22 q^{84} + 16 q^{86} - 23 q^{87} + 12 q^{88} + 28 q^{89} - 3 q^{91} + 30 q^{92} - 30 q^{93} + 66 q^{94} + 46 q^{96} - q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9 + 7 * q^11 - 22 * q^12 - 3 * q^13 + 2 * q^14 + 56 * q^16 + 7 * q^17 + 24 * q^19 - q^21 + 4 * q^22 + 21 * q^23 + 24 * q^24 - 2 * q^26 - 19 * q^27 + 30 * q^28 + 11 * q^29 + 46 * q^31 - 6 * q^32 - 3 * q^33 + 28 * q^34 + 58 * q^36 + 24 * q^37 - 4 * q^38 + 31 * q^39 + 14 * q^41 + 6 * q^42 + 18 * q^43 + 12 * q^44 + 2 * q^46 - 25 * q^47 - 36 * q^48 + 21 * q^49 + 17 * q^51 - 8 * q^52 + 22 * q^53 - 6 * q^54 + 6 * q^56 + 40 * q^57 + 6 * q^58 + 10 * q^59 + 38 * q^61 - 54 * q^62 + 30 * q^63 + 100 * q^64 + 38 * q^66 + 12 * q^67 + 18 * q^68 - q^69 + 56 * q^71 + 42 * q^72 - 40 * q^73 - 20 * q^74 + 60 * q^76 + 7 * q^77 + 38 * q^78 + 49 * q^79 + 57 * q^81 + 16 * q^82 - 2 * q^83 - 22 * q^84 + 16 * q^86 - 23 * q^87 + 12 * q^88 + 28 * q^89 - 3 * q^91 + 30 * q^92 - 30 * q^93 + 66 * q^94 + 46 * q^96 - q^97 + 2 * q^98 - 2 * q^99

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