LMFDB - Embedded newform 4225.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} +2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} -1.60555 q^{11} -3.30278 q^{12} +2.30278 q^{14} +0.302776 q^{16} -7.60555 q^{17} +4.60555 q^{18} +5.60555 q^{19} +1.00000 q^{21} +3.69722 q^{22} +3.00000 q^{23} +3.00000 q^{24} +5.00000 q^{27} -3.30278 q^{28} -6.21110 q^{29} +4.00000 q^{31} +5.30278 q^{32} +1.60555 q^{33} +17.5139 q^{34} -6.60555 q^{36} +3.60555 q^{37} -12.9083 q^{38} -3.00000 q^{41} -2.30278 q^{42} +10.2111 q^{43} -5.30278 q^{44} -6.90833 q^{46} +9.21110 q^{47} -0.302776 q^{48} -6.00000 q^{49} +7.60555 q^{51} +3.21110 q^{53} -11.5139 q^{54} +3.00000 q^{56} -5.60555 q^{57} +14.3028 q^{58} +10.8167 q^{59} -1.00000 q^{61} -9.21110 q^{62} +2.00000 q^{63} -12.8167 q^{64} -3.69722 q^{66} -7.00000 q^{67} -25.1194 q^{68} -3.00000 q^{69} -4.81665 q^{71} +6.00000 q^{72} -0.788897 q^{73} -8.30278 q^{74} +18.5139 q^{76} +1.60555 q^{77} +5.21110 q^{79} +1.00000 q^{81} +6.90833 q^{82} -9.21110 q^{83} +3.30278 q^{84} -23.5139 q^{86} +6.21110 q^{87} +4.81665 q^{88} +6.21110 q^{89} +9.90833 q^{92} -4.00000 q^{93} -21.2111 q^{94} -5.30278 q^{96} -8.39445 q^{97} +13.8167 q^{98} +3.21110 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 4 q^{11} - 3 q^{12} + q^{14} - 3 q^{16} - 8 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{21} + 11 q^{22} + 6 q^{23} + 6 q^{24} + 10 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 + 3 * q^4 + q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 4 * q^11 - 3 * q^12 + q^14 - 3 * q^16 - 8 * q^17 + 2 * q^18 + 4 * q^19 + 2 * q^21 + 11 * q^22 + 6 * q^23 + 6 * q^24 + 10 * q^27 - 3 * q^28 + 2 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^33 + 17 * q^34 - 6 * q^36 - 15 * q^38 - 6 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 3 * q^46 + 4 * q^47 + 3 * q^48 - 12 * q^49 + 8 * q^51 - 8 * q^53 - 5 * q^54 + 6 * q^56 - 4 * q^57 + 25 * q^58 - 2 * q^61 - 4 * q^62 + 4 * q^63 - 4 * q^64 - 11 * q^66 - 14 * q^67 - 25 * q^68 - 6 * q^69 + 12 * q^71 + 12 * q^72 - 16 * q^73 - 13 * q^74 + 19 * q^76 - 4 * q^77 - 4 * q^79 + 2 * q^81 + 3 * q^82 - 4 * q^83 + 3 * q^84 - 29 * q^86 - 2 * q^87 - 12 * q^88 - 2 * q^89 + 9 * q^92 - 8 * q^93 - 28 * q^94 - 7 * q^96 - 24 * q^97 + 6 * q^98 - 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	2.30278	0.940104
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−1.60555	−0.484092	−0.242046	−	0.970265i	$-0.577818\pi$
$11$	−1.60555	−0.484092	−0.242046	+	0.970265i	$0.577818\pi$
$12$	−3.30278	−0.953429
$13$	0	0
$13$	0	0
$14$	2.30278	0.615443
$15$	0	0
$16$	0.302776	0.0756939
$17$	−7.60555	−1.84462	−0.922309	−	0.386454i	$-0.873700\pi$
$17$	−7.60555	−1.84462	−0.922309	+	0.386454i	$0.873700\pi$
$18$	4.60555	1.08554
$19$	5.60555	1.28600	0.643001	−	0.765865i	$-0.277689\pi$
$19$	5.60555	1.28600	0.643001	+	0.765865i	$0.277689\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	3.69722	0.788251
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	0	0
$27$	5.00000	0.962250
$28$	−3.30278	−0.624166
$29$	−6.21110	−1.15337	−0.576686	−	0.816966i	$-0.695655\pi$
$29$	−6.21110	−1.15337	−0.576686	+	0.816966i	$0.695655\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.30278	0.937407
$33$	1.60555	0.279491
$34$	17.5139	3.00361
$35$	0	0
$36$	−6.60555	−1.10093
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	−12.9083	−2.09401
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−2.30278	−0.355326
$43$	10.2111	1.55718	0.778589	−	0.627534i	$-0.215936\pi$
$43$	10.2111	1.55718	0.778589	+	0.627534i	$0.215936\pi$
$44$	−5.30278	−0.799424
$45$	0	0
$46$	−6.90833	−1.01858
$47$	9.21110	1.34358	0.671789	−	0.740743i	$-0.265526\pi$
$47$	9.21110	1.34358	0.671789	+	0.740743i	$0.265526\pi$
$48$	−0.302776	−0.0437019
$49$	−6.00000	−0.857143
$50$	0	0
$51$	7.60555	1.06499
$52$	0	0
$53$	3.21110	0.441079	0.220539	−	0.975378i	$-0.429218\pi$
$53$	3.21110	0.441079	0.220539	+	0.975378i	$0.429218\pi$
$54$	−11.5139	−1.56684
$55$	0	0
$56$	3.00000	0.400892
$57$	−5.60555	−0.742473
$58$	14.3028	1.87805
$59$	10.8167	1.40821	0.704104	−	0.710097i	$-0.251349\pi$
$59$	10.8167	1.40821	0.704104	+	0.710097i	$0.251349\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−9.21110	−1.16981
$63$	2.00000	0.251976
$64$	−12.8167	−1.60208
$65$	0	0
$66$	−3.69722	−0.455097
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−25.1194	−3.04618
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−4.81665	−0.571632	−0.285816	−	0.958285i	$-0.592265\pi$
$71$	−4.81665	−0.571632	−0.285816	+	0.958285i	$0.592265\pi$
$72$	6.00000	0.707107
$73$	−0.788897	−0.0923335	−0.0461667	−	0.998934i	$-0.514701\pi$
$73$	−0.788897	−0.0923335	−0.0461667	+	0.998934i	$0.514701\pi$
$74$	−8.30278	−0.965178
$75$	0	0
$76$	18.5139	2.12369
$77$	1.60555	0.182970
$78$	0	0
$79$	5.21110	0.586295	0.293147	−	0.956067i	$-0.405297\pi$
$79$	5.21110	0.586295	0.293147	+	0.956067i	$0.405297\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.90833	0.762897
$83$	−9.21110	−1.01105	−0.505525	−	0.862812i	$-0.668701\pi$
$83$	−9.21110	−1.01105	−0.505525	+	0.862812i	$0.668701\pi$
$84$	3.30278	0.360362
$85$	0	0
$86$	−23.5139	−2.53557
$87$	6.21110	0.665900
$88$	4.81665	0.513457
$89$	6.21110	0.658376	0.329188	−	0.944264i	$-0.393225\pi$
$89$	6.21110	0.658376	0.329188	+	0.944264i	$0.393225\pi$
$90$	0	0
$91$	0	0
$92$	9.90833	1.03301
$93$	−4.00000	−0.414781
$94$	−21.2111	−2.18776
$95$	0	0
$96$	−5.30278	−0.541212
$97$	−8.39445	−0.852327	−0.426164	−	0.904646i	$-0.640135\pi$
$97$	−8.39445	−0.852327	−0.426164	+	0.904646i	$0.640135\pi$
$98$	13.8167	1.39569
$99$	3.21110	0.322728
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	−17.5139	−1.73413
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	−7.39445	−0.718212
$107$	−6.21110	−0.600450	−0.300225	−	0.953868i	$-0.597062\pi$
$107$	−6.21110	−0.600450	−0.300225	+	0.953868i	$0.597062\pi$
$108$	16.5139	1.58905
$109$	19.2111	1.84009	0.920045	−	0.391813i	$-0.128152\pi$
$109$	19.2111	1.84009	0.920045	+	0.391813i	$0.128152\pi$
$110$	0	0
$111$	−3.60555	−0.342224
$112$	−0.302776	−0.0286096
$113$	1.60555	0.151038	0.0755188	−	0.997144i	$-0.475939\pi$
$113$	1.60555	0.151038	0.0755188	+	0.997144i	$0.475939\pi$
$114$	12.9083	1.20898
$115$	0	0
$116$	−20.5139	−1.90467
$117$	0	0
$118$	−24.9083	−2.29300
$119$	7.60555	0.697200
$120$	0	0
$121$	−8.42221	−0.765655
$122$	2.30278	0.208484
$123$	3.00000	0.270501
$124$	13.2111	1.18639
$125$	0	0
$126$	−4.60555	−0.410295
$127$	4.21110	0.373675	0.186837	−	0.982391i	$-0.440176\pi$
$127$	4.21110	0.373675	0.186837	+	0.982391i	$0.440176\pi$
$128$	18.9083	1.67128
$129$	−10.2111	−0.899037
$130$	0	0
$131$	−21.2111	−1.85322	−0.926611	−	0.376021i	$-0.877292\pi$
$131$	−21.2111	−1.85322	−0.926611	+	0.376021i	$0.877292\pi$
$132$	5.30278	0.461547
$133$	−5.60555	−0.486063
$134$	16.1194	1.39251
$135$	0	0
$136$	22.8167	1.95651
$137$	−1.60555	−0.137172	−0.0685858	−	0.997645i	$-0.521849\pi$
$137$	−1.60555	−0.137172	−0.0685858	+	0.997645i	$0.521849\pi$
$138$	6.90833	0.588076
$139$	6.39445	0.542370	0.271185	−	0.962527i	$-0.412584\pi$
$139$	6.39445	0.542370	0.271185	+	0.962527i	$0.412584\pi$
$140$	0	0
$141$	−9.21110	−0.775715
$142$	11.0917	0.930793
$143$	0	0
$144$	−0.605551	−0.0504626
$145$	0	0
$146$	1.81665	0.150347
$147$	6.00000	0.494872
$148$	11.9083	0.978858
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	1.21110	0.0985581	0.0492791	−	0.998785i	$-0.484308\pi$
$151$	1.21110	0.0985581	0.0492791	+	0.998785i	$0.484308\pi$
$152$	−16.8167	−1.36401
$153$	15.2111	1.22974
$154$	−3.69722	−0.297931
$155$	0	0
$156$	0	0
$157$	−11.2111	−0.894743	−0.447372	−	0.894348i	$-0.647640\pi$
$157$	−11.2111	−0.894743	−0.447372	+	0.894348i	$0.647640\pi$
$158$	−12.0000	−0.954669
$159$	−3.21110	−0.254657
$160$	0	0
$161$	−3.00000	−0.236433
$162$	−2.30278	−0.180923
$163$	−3.78890	−0.296769	−0.148385	−	0.988930i	$-0.547407\pi$
$163$	−3.78890	−0.296769	−0.148385	+	0.988930i	$0.547407\pi$
$164$	−9.90833	−0.773710
$165$	0	0
$166$	21.2111	1.64630
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	−3.00000	−0.231455
$169$	0	0
$170$	0	0
$171$	−11.2111	−0.857334
$172$	33.7250	2.57151
$173$	−4.81665	−0.366203	−0.183102	−	0.983094i	$-0.558614\pi$
$173$	−4.81665	−0.366203	−0.183102	+	0.983094i	$0.558614\pi$
$174$	−14.3028	−1.08429
$175$	0	0
$176$	−0.486122	−0.0366428
$177$	−10.8167	−0.813029
$178$	−14.3028	−1.07204
$179$	22.8167	1.70540	0.852698	−	0.522404i	$-0.174965\pi$
$179$	22.8167	1.70540	0.852698	+	0.522404i	$0.174965\pi$
$180$	0	0
$181$	17.6333	1.31067	0.655337	−	0.755337i	$-0.272527\pi$
$181$	17.6333	1.31067	0.655337	+	0.755337i	$0.272527\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	−9.00000	−0.663489
$185$	0	0
$186$	9.21110	0.675391
$187$	12.2111	0.892964
$188$	30.4222	2.21877
$189$	−5.00000	−0.363696
$190$	0	0
$191$	16.8167	1.21681	0.608405	−	0.793627i	$-0.291810\pi$
$191$	16.8167	1.21681	0.608405	+	0.793627i	$0.291810\pi$
$192$	12.8167	0.924962
$193$	15.6056	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$193$	15.6056	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$194$	19.3305	1.38785
$195$	0	0
$196$	−19.8167	−1.41548
$197$	1.18335	0.0843099	0.0421550	−	0.999111i	$-0.486578\pi$
$197$	1.18335	0.0843099	0.0421550	+	0.999111i	$0.486578\pi$
$198$	−7.39445	−0.525501
$199$	12.8167	0.908549	0.454274	−	0.890862i	$-0.349899\pi$
$199$	12.8167	0.908549	0.454274	+	0.890862i	$0.349899\pi$
$200$	0	0
$201$	7.00000	0.493742
$202$	20.7250	1.45820
$203$	6.21110	0.435934
$204$	25.1194	1.75871
$205$	0	0
$206$	−9.21110	−0.641768
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−9.00000	−0.622543
$210$	0	0
$211$	−23.6056	−1.62507	−0.812537	−	0.582910i	$-0.801914\pi$
$211$	−23.6056	−1.62507	−0.812537	+	0.582910i	$0.801914\pi$
$212$	10.6056	0.728392
$213$	4.81665	0.330032
$214$	14.3028	0.977718
$215$	0	0
$216$	−15.0000	−1.02062
$217$	−4.00000	−0.271538
$218$	−44.2389	−2.99623
$219$	0.788897	0.0533087
$220$	0	0
$221$	0	0
$222$	8.30278	0.557246
$223$	−4.21110	−0.281996	−0.140998	−	0.990010i	$-0.545031\pi$
$223$	−4.21110	−0.281996	−0.140998	+	0.990010i	$0.545031\pi$
$224$	−5.30278	−0.354307
$225$	0	0
$226$	−3.69722	−0.245936
$227$	−27.4222	−1.82008	−0.910038	−	0.414525i	$-0.863948\pi$
$227$	−27.4222	−1.82008	−0.910038	+	0.414525i	$0.863948\pi$
$228$	−18.5139	−1.22611
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−1.60555	−0.105638
$232$	18.6333	1.22334
$233$	−15.2111	−0.996512	−0.498256	−	0.867030i	$-0.666026\pi$
$233$	−15.2111	−0.996512	−0.498256	+	0.867030i	$0.666026\pi$
$234$	0	0
$235$	0	0
$236$	35.7250	2.32550
$237$	−5.21110	−0.338497
$238$	−17.5139	−1.13526
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−1.78890	−0.115233	−0.0576165	−	0.998339i	$-0.518350\pi$
$241$	−1.78890	−0.115233	−0.0576165	+	0.998339i	$0.518350\pi$
$242$	19.3944	1.24672
$243$	−16.0000	−1.02640
$244$	−3.30278	−0.211439
$245$	0	0
$246$	−6.90833	−0.440459
$247$	0	0
$248$	−12.0000	−0.762001
$249$	9.21110	0.583730
$250$	0	0
$251$	−7.18335	−0.453409	−0.226704	−	0.973964i	$-0.572795\pi$
$251$	−7.18335	−0.453409	−0.226704	+	0.973964i	$0.572795\pi$
$252$	6.60555	0.416111
$253$	−4.81665	−0.302820
$254$	−9.69722	−0.608458
$255$	0	0
$256$	−17.9083	−1.11927
$257$	16.3944	1.02266	0.511329	−	0.859385i	$-0.329153\pi$
$257$	16.3944	1.02266	0.511329	+	0.859385i	$0.329153\pi$
$258$	23.5139	1.46391
$259$	−3.60555	−0.224038
$260$	0	0
$261$	12.4222	0.768915
$262$	48.8444	3.01762
$263$	−11.7889	−0.726935	−0.363467	−	0.931607i	$-0.618407\pi$
$263$	−11.7889	−0.726935	−0.363467	+	0.931607i	$0.618407\pi$
$264$	−4.81665	−0.296445
$265$	0	0
$266$	12.9083	0.791460
$267$	−6.21110	−0.380113
$268$	−23.1194	−1.41224
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	20.8167	1.26452	0.632261	−	0.774756i	$-0.282127\pi$
$271$	20.8167	1.26452	0.632261	+	0.774756i	$0.282127\pi$
$272$	−2.30278	−0.139626
$273$	0	0
$274$	3.69722	0.223357
$275$	0	0
$276$	−9.90833	−0.596411
$277$	−27.6056	−1.65866	−0.829328	−	0.558761i	$-0.811277\pi$
$277$	−27.6056	−1.65866	−0.829328	+	0.558761i	$0.811277\pi$
$278$	−14.7250	−0.883146
$279$	−8.00000	−0.478947
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	21.2111	1.26310
$283$	−5.00000	−0.297219	−0.148610	−	0.988896i	$-0.547480\pi$
$283$	−5.00000	−0.297219	−0.148610	+	0.988896i	$0.547480\pi$
$284$	−15.9083	−0.943986
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	−10.6056	−0.624938
$289$	40.8444	2.40261
$290$	0	0
$291$	8.39445	0.492091
$292$	−2.60555	−0.152478
$293$	10.3944	0.607250	0.303625	−	0.952792i	$-0.401803\pi$
$293$	10.3944	0.607250	0.303625	+	0.952792i	$0.401803\pi$
$294$	−13.8167	−0.805804
$295$	0	0
$296$	−10.8167	−0.628705
$297$	−8.02776	−0.465818
$298$	6.90833	0.400189
$299$	0	0
$300$	0	0
$301$	−10.2111	−0.588558
$302$	−2.78890	−0.160483
$303$	9.00000	0.517036
$304$	1.69722	0.0973425
$305$	0	0
$306$	−35.0278	−2.00240
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	5.30278	0.302154
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−9.21110	−0.522314	−0.261157	−	0.965296i	$-0.584104\pi$
$311$	−9.21110	−0.522314	−0.261157	+	0.965296i	$0.584104\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	25.8167	1.45692
$315$	0	0
$316$	17.2111	0.968200
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	7.39445	0.414660
$319$	9.97224	0.558338
$320$	0	0
$321$	6.21110	0.346670
$322$	6.90833	0.384986
$323$	−42.6333	−2.37218
$324$	3.30278	0.183488
$325$	0	0
$326$	8.72498	0.483232
$327$	−19.2111	−1.06238
$328$	9.00000	0.496942
$329$	−9.21110	−0.507825
$330$	0	0
$331$	−10.0278	−0.551175	−0.275588	−	0.961276i	$-0.588872\pi$
$331$	−10.0278	−0.551175	−0.275588	+	0.961276i	$0.588872\pi$
$332$	−30.4222	−1.66964
$333$	−7.21110	−0.395166
$334$	−20.7250	−1.13402
$335$	0	0
$336$	0.302776	0.0165178
$337$	25.6333	1.39634	0.698168	−	0.715934i	$-0.253999\pi$
$337$	25.6333	1.39634	0.698168	+	0.715934i	$0.253999\pi$
$338$	0	0
$339$	−1.60555	−0.0872016
$340$	0	0
$341$	−6.42221	−0.347782
$342$	25.8167	1.39600
$343$	13.0000	0.701934
$344$	−30.6333	−1.65164
$345$	0	0
$346$	11.0917	0.596292
$347$	−5.78890	−0.310764	−0.155382	−	0.987854i	$-0.549661\pi$
$347$	−5.78890	−0.310764	−0.155382	+	0.987854i	$0.549661\pi$
$348$	20.5139	1.09966
$349$	3.78890	0.202815	0.101408	−	0.994845i	$-0.467665\pi$
$349$	3.78890	0.202815	0.101408	+	0.994845i	$0.467665\pi$
$350$	0	0
$351$	0	0
$352$	−8.51388	−0.453791
$353$	16.8167	0.895060	0.447530	−	0.894269i	$-0.352304\pi$
$353$	16.8167	0.895060	0.447530	+	0.894269i	$0.352304\pi$
$354$	24.9083	1.32386
$355$	0	0
$356$	20.5139	1.08723
$357$	−7.60555	−0.402528
$358$	−52.5416	−2.77691
$359$	−18.4222	−0.972287	−0.486143	−	0.873879i	$-0.661597\pi$
$359$	−18.4222	−0.972287	−0.486143	+	0.873879i	$0.661597\pi$
$360$	0	0
$361$	12.4222	0.653800
$362$	−40.6056	−2.13418
$363$	8.42221	0.442051
$364$	0	0
$365$	0	0
$366$	−2.30278	−0.120368
$367$	−11.4222	−0.596234	−0.298117	−	0.954529i	$-0.596359\pi$
$367$	−11.4222	−0.596234	−0.298117	+	0.954529i	$0.596359\pi$
$368$	0.908327	0.0473498
$369$	6.00000	0.312348
$370$	0	0
$371$	−3.21110	−0.166712
$372$	−13.2111	−0.684964
$373$	20.3944	1.05598	0.527992	−	0.849249i	$-0.322945\pi$
$373$	20.3944	1.05598	0.527992	+	0.849249i	$0.322945\pi$
$374$	−28.1194	−1.45402
$375$	0	0
$376$	−27.6333	−1.42508
$377$	0	0
$378$	11.5139	0.592210
$379$	−9.60555	−0.493404	−0.246702	−	0.969091i	$-0.579347\pi$
$379$	−9.60555	−0.493404	−0.246702	+	0.969091i	$0.579347\pi$
$380$	0	0
$381$	−4.21110	−0.215741
$382$	−38.7250	−1.98134
$383$	−24.6333	−1.25870	−0.629352	−	0.777121i	$-0.716679\pi$
$383$	−24.6333	−1.25870	−0.629352	+	0.777121i	$0.716679\pi$
$384$	−18.9083	−0.964912
$385$	0	0
$386$	−35.9361	−1.82910
$387$	−20.4222	−1.03812
$388$	−27.7250	−1.40752
$389$	−15.2111	−0.771234	−0.385617	−	0.922659i	$-0.626011\pi$
$389$	−15.2111	−0.771234	−0.385617	+	0.922659i	$0.626011\pi$
$390$	0	0
$391$	−22.8167	−1.15389
$392$	18.0000	0.909137
$393$	21.2111	1.06996
$394$	−2.72498	−0.137283
$395$	0	0
$396$	10.6056	0.532949
$397$	22.0278	1.10554	0.552771	−	0.833333i	$-0.313571\pi$
$397$	22.0278	1.10554	0.552771	+	0.833333i	$0.313571\pi$
$398$	−29.5139	−1.47940
$399$	5.60555	0.280629
$400$	0	0
$401$	−12.2111	−0.609793	−0.304897	−	0.952385i	$-0.598622\pi$
$401$	−12.2111	−0.609793	−0.304897	+	0.952385i	$0.598622\pi$
$402$	−16.1194	−0.803964
$403$	0	0
$404$	−29.7250	−1.47887
$405$	0	0
$406$	−14.3028	−0.709835
$407$	−5.78890	−0.286945
$408$	−22.8167	−1.12959
$409$	−8.21110	−0.406013	−0.203006	−	0.979177i	$-0.565071\pi$
$409$	−8.21110	−0.406013	−0.203006	+	0.979177i	$0.565071\pi$
$410$	0	0
$411$	1.60555	0.0791960
$412$	13.2111	0.650864
$413$	−10.8167	−0.532253
$414$	13.8167	0.679051
$415$	0	0
$416$	0	0
$417$	−6.39445	−0.313138
$418$	20.7250	1.01369
$419$	17.2389	0.842173	0.421087	−	0.907020i	$-0.361649\pi$
$419$	17.2389	0.842173	0.421087	+	0.907020i	$0.361649\pi$
$420$	0	0
$421$	−32.4222	−1.58016	−0.790081	−	0.613003i	$-0.789961\pi$
$421$	−32.4222	−1.58016	−0.790081	+	0.613003i	$0.789961\pi$
$422$	54.3583	2.64612
$423$	−18.4222	−0.895718
$424$	−9.63331	−0.467835
$425$	0	0
$426$	−11.0917	−0.537393
$427$	1.00000	0.0483934
$428$	−20.5139	−0.991576
$429$	0	0
$430$	0	0
$431$	−29.2389	−1.40839	−0.704193	−	0.710008i	$-0.748691\pi$
$431$	−29.2389	−1.40839	−0.704193	+	0.710008i	$0.748691\pi$
$432$	1.51388	0.0728365
$433$	−3.60555	−0.173272	−0.0866359	−	0.996240i	$-0.527612\pi$
$433$	−3.60555	−0.173272	−0.0866359	+	0.996240i	$0.527612\pi$
$434$	9.21110	0.442147
$435$	0	0
$436$	63.4500	3.03870
$437$	16.8167	0.804450
$438$	−1.81665	−0.0868031
$439$	−27.2389	−1.30004	−0.650020	−	0.759917i	$-0.725239\pi$
$439$	−27.2389	−1.30004	−0.650020	+	0.759917i	$0.725239\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−6.42221	−0.305128	−0.152564	−	0.988294i	$-0.548753\pi$
$443$	−6.42221	−0.305128	−0.152564	+	0.988294i	$0.548753\pi$
$444$	−11.9083	−0.565144
$445$	0	0
$446$	9.69722	0.459177
$447$	3.00000	0.141895
$448$	12.8167	0.605530
$449$	−30.6333	−1.44568	−0.722838	−	0.691018i	$-0.757163\pi$
$449$	−30.6333	−1.44568	−0.722838	+	0.691018i	$0.757163\pi$
$450$	0	0
$451$	4.81665	0.226807
$452$	5.30278	0.249422
$453$	−1.21110	−0.0569026
$454$	63.1472	2.96364
$455$	0	0
$456$	16.8167	0.787512
$457$	−26.8167	−1.25443	−0.627215	−	0.778846i	$-0.715805\pi$
$457$	−26.8167	−1.25443	−0.627215	+	0.778846i	$0.715805\pi$
$458$	32.2389	1.50642
$459$	−38.0278	−1.77498
$460$	0	0
$461$	−36.2111	−1.68652	−0.843260	−	0.537507i	$-0.819366\pi$
$461$	−36.2111	−1.68652	−0.843260	+	0.537507i	$0.819366\pi$
$462$	3.69722	0.172010
$463$	−34.4222	−1.59974	−0.799868	−	0.600176i	$-0.795097\pi$
$463$	−34.4222	−1.59974	−0.799868	+	0.600176i	$0.795097\pi$
$464$	−1.88057	−0.0873033
$465$	0	0
$466$	35.0278	1.62263
$467$	−2.78890	−0.129055	−0.0645274	−	0.997916i	$-0.520554\pi$
$467$	−2.78890	−0.129055	−0.0645274	+	0.997916i	$0.520554\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	11.2111	0.516580
$472$	−32.4500	−1.49363
$473$	−16.3944	−0.753818
$474$	12.0000	0.551178
$475$	0	0
$476$	25.1194	1.15135
$477$	−6.42221	−0.294053
$478$	0	0
$479$	28.8167	1.31667	0.658333	−	0.752727i	$-0.271262\pi$
$479$	28.8167	1.31667	0.658333	+	0.752727i	$0.271262\pi$
$480$	0	0
$481$	0	0
$482$	4.11943	0.187635
$483$	3.00000	0.136505
$484$	−27.8167	−1.26439
$485$	0	0
$486$	36.8444	1.67130
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	3.00000	0.135804
$489$	3.78890	0.171340
$490$	0	0
$491$	−16.8167	−0.758925	−0.379462	−	0.925207i	$-0.623891\pi$
$491$	−16.8167	−0.758925	−0.379462	+	0.925207i	$0.623891\pi$
$492$	9.90833	0.446702
$493$	47.2389	2.12753
$494$	0	0
$495$	0	0
$496$	1.21110	0.0543801
$497$	4.81665	0.216056
$498$	−21.2111	−0.950492
$499$	−2.42221	−0.108433	−0.0542164	−	0.998529i	$-0.517266\pi$
$499$	−2.42221	−0.108433	−0.0542164	+	0.998529i	$0.517266\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	16.5416	0.738289
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	11.0917	0.493085
$507$	0	0
$508$	13.9083	0.617082
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	0.788897	0.0348988
$512$	3.42221	0.151242
$513$	28.0278	1.23746
$514$	−37.7527	−1.66520
$515$	0	0
$516$	−33.7250	−1.48466
$517$	−14.7889	−0.650415
$518$	8.30278	0.364803
$519$	4.81665	0.211428
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−28.6056	−1.25203
$523$	1.42221	0.0621887	0.0310943	−	0.999516i	$-0.490101\pi$
$523$	1.42221	0.0621887	0.0310943	+	0.999516i	$0.490101\pi$
$524$	−70.0555	−3.06039
$525$	0	0
$526$	27.1472	1.18367
$527$	−30.4222	−1.32521
$528$	0.486122	0.0211557
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−21.6333	−0.938806
$532$	−18.5139	−0.802678
$533$	0	0
$534$	14.3028	0.618942
$535$	0	0
$536$	21.0000	0.907062
$537$	−22.8167	−0.984611
$538$	20.7250	0.893517
$539$	9.63331	0.414936
$540$	0	0
$541$	25.6333	1.10206	0.551031	−	0.834485i	$-0.314235\pi$
$541$	25.6333	1.10206	0.551031	+	0.834485i	$0.314235\pi$
$542$	−47.9361	−2.05903
$543$	−17.6333	−0.756718
$544$	−40.3305	−1.72916
$545$	0	0
$546$	0	0
$547$	−32.8444	−1.40433	−0.702163	−	0.712016i	$-0.747782\pi$
$547$	−32.8444	−1.40433	−0.702163	+	0.712016i	$0.747782\pi$
$548$	−5.30278	−0.226523
$549$	2.00000	0.0853579
$550$	0	0
$551$	−34.8167	−1.48324
$552$	9.00000	0.383065
$553$	−5.21110	−0.221599
$554$	63.5694	2.70080
$555$	0	0
$556$	21.1194	0.895663
$557$	−1.60555	−0.0680294	−0.0340147	−	0.999421i	$-0.510829\pi$
$557$	−1.60555	−0.0680294	−0.0340147	+	0.999421i	$0.510829\pi$
$558$	18.4222	0.779874
$559$	0	0
$560$	0	0
$561$	−12.2111	−0.515553
$562$	−13.8167	−0.582820
$563$	−9.42221	−0.397099	−0.198549	−	0.980091i	$-0.563623\pi$
$563$	−9.42221	−0.397099	−0.198549	+	0.980091i	$0.563623\pi$
$564$	−30.4222	−1.28101
$565$	0	0
$566$	11.5139	0.483964
$567$	−1.00000	−0.0419961
$568$	14.4500	0.606307
$569$	−27.4222	−1.14960	−0.574799	−	0.818294i	$-0.694920\pi$
$569$	−27.4222	−1.14960	−0.574799	+	0.818294i	$0.694920\pi$
$570$	0	0
$571$	20.8444	0.872311	0.436156	−	0.899871i	$-0.356340\pi$
$571$	20.8444	0.872311	0.436156	+	0.899871i	$0.356340\pi$
$572$	0	0
$573$	−16.8167	−0.702526
$574$	−6.90833	−0.288348
$575$	0	0
$576$	25.6333	1.06805
$577$	−13.6333	−0.567562	−0.283781	−	0.958889i	$-0.591589\pi$
$577$	−13.6333	−0.567562	−0.283781	+	0.958889i	$0.591589\pi$
$578$	−94.0555	−3.91219
$579$	−15.6056	−0.648545
$580$	0	0
$581$	9.21110	0.382141
$582$	−19.3305	−0.801276
$583$	−5.15559	−0.213523
$584$	2.36669	0.0979344
$585$	0	0
$586$	−23.9361	−0.988790
$587$	−33.4222	−1.37948	−0.689741	−	0.724056i	$-0.742276\pi$
$587$	−33.4222	−1.37948	−0.689741	+	0.724056i	$0.742276\pi$
$588$	19.8167	0.817225
$589$	22.4222	0.923891
$590$	0	0
$591$	−1.18335	−0.0486764
$592$	1.09167	0.0448675
$593$	20.7889	0.853698	0.426849	−	0.904323i	$-0.359624\pi$
$593$	20.7889	0.853698	0.426849	+	0.904323i	$0.359624\pi$
$594$	18.4861	0.758495
$595$	0	0
$596$	−9.90833	−0.405861
$597$	−12.8167	−0.524551
$598$	0	0
$599$	−21.2111	−0.866662	−0.433331	−	0.901235i	$-0.642662\pi$
$599$	−21.2111	−0.866662	−0.433331	+	0.901235i	$0.642662\pi$
$600$	0	0
$601$	13.7889	0.562461	0.281230	−	0.959640i	$-0.409257\pi$
$601$	13.7889	0.562461	0.281230	+	0.959640i	$0.409257\pi$
$602$	23.5139	0.958354
$603$	14.0000	0.570124
$604$	4.00000	0.162758
$605$	0	0
$606$	−20.7250	−0.841895
$607$	34.2111	1.38859	0.694293	−	0.719693i	$-0.255717\pi$
$607$	34.2111	1.38859	0.694293	+	0.719693i	$0.255717\pi$
$608$	29.7250	1.20551
$609$	−6.21110	−0.251687
$610$	0	0
$611$	0	0
$612$	50.2389	2.03079
$613$	−5.60555	−0.226406	−0.113203	−	0.993572i	$-0.536111\pi$
$613$	−5.60555	−0.226406	−0.113203	+	0.993572i	$0.536111\pi$
$614$	36.8444	1.48692
$615$	0	0
$616$	−4.81665	−0.194069
$617$	−38.4500	−1.54794	−0.773969	−	0.633224i	$-0.781731\pi$
$617$	−38.4500	−1.54794	−0.773969	+	0.633224i	$0.781731\pi$
$618$	9.21110	0.370525
$619$	−14.4222	−0.579677	−0.289839	−	0.957076i	$-0.593602\pi$
$619$	−14.4222	−0.579677	−0.289839	+	0.957076i	$0.593602\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	21.2111	0.850488
$623$	−6.21110	−0.248843
$624$	0	0
$625$	0	0
$626$	32.2389	1.28852
$627$	9.00000	0.359425
$628$	−37.0278	−1.47757
$629$	−27.4222	−1.09339
$630$	0	0
$631$	36.0278	1.43424	0.717121	−	0.696949i	$-0.245459\pi$
$631$	36.0278	1.43424	0.717121	+	0.696949i	$0.245459\pi$
$632$	−15.6333	−0.621860
$633$	23.6056	0.938236
$634$	−13.8167	−0.548729
$635$	0	0
$636$	−10.6056	−0.420537
$637$	0	0
$638$	−22.9638	−0.909147
$639$	9.63331	0.381088
$640$	0	0
$641$	9.42221	0.372155	0.186077	−	0.982535i	$-0.440422\pi$
$641$	9.42221	0.372155	0.186077	+	0.982535i	$0.440422\pi$
$642$	−14.3028	−0.564486
$643$	2.63331	0.103848	0.0519238	−	0.998651i	$-0.483465\pi$
$643$	2.63331	0.103848	0.0519238	+	0.998651i	$0.483465\pi$
$644$	−9.90833	−0.390443
$645$	0	0
$646$	98.1749	3.86264
$647$	−39.4222	−1.54985	−0.774923	−	0.632055i	$-0.782212\pi$
$647$	−39.4222	−1.54985	−0.774923	+	0.632055i	$0.782212\pi$
$648$	−3.00000	−0.117851
$649$	−17.3667	−0.681702
$650$	0	0
$651$	4.00000	0.156772
$652$	−12.5139	−0.490081
$653$	7.18335	0.281106	0.140553	−	0.990073i	$-0.455112\pi$
$653$	7.18335	0.281106	0.140553	+	0.990073i	$0.455112\pi$
$654$	44.2389	1.72988
$655$	0	0
$656$	−0.908327	−0.0354642
$657$	1.57779	0.0615556
$658$	21.2111	0.826895
$659$	−34.8167	−1.35626	−0.678132	−	0.734940i	$-0.737210\pi$
$659$	−34.8167	−1.35626	−0.678132	+	0.734940i	$0.737210\pi$
$660$	0	0
$661$	4.63331	0.180215	0.0901074	−	0.995932i	$-0.471279\pi$
$661$	4.63331	0.180215	0.0901074	+	0.995932i	$0.471279\pi$
$662$	23.0917	0.897483
$663$	0	0
$664$	27.6333	1.07238
$665$	0	0
$666$	16.6056	0.643452
$667$	−18.6333	−0.721485
$668$	29.7250	1.15009
$669$	4.21110	0.162811
$670$	0	0
$671$	1.60555	0.0619816
$672$	5.30278	0.204559
$673$	17.6056	0.678644	0.339322	−	0.940670i	$-0.389802\pi$
$673$	17.6056	0.678644	0.339322	+	0.940670i	$0.389802\pi$
$674$	−59.0278	−2.27366
$675$	0	0
$676$	0	0
$677$	9.63331	0.370238	0.185119	−	0.982716i	$-0.440733\pi$
$677$	9.63331	0.370238	0.185119	+	0.982716i	$0.440733\pi$
$678$	3.69722	0.141991
$679$	8.39445	0.322149
$680$	0	0
$681$	27.4222	1.05082
$682$	14.7889	0.566296
$683$	36.2111	1.38558	0.692790	−	0.721140i	$-0.256381\pi$
$683$	36.2111	1.38558	0.692790	+	0.721140i	$0.256381\pi$
$684$	−37.0278	−1.41579
$685$	0	0
$686$	−29.9361	−1.14296
$687$	14.0000	0.534133
$688$	3.09167	0.117869
$689$	0	0
$690$	0	0
$691$	30.0278	1.14231	0.571155	−	0.820842i	$-0.306496\pi$
$691$	30.0278	1.14231	0.571155	+	0.820842i	$0.306496\pi$
$692$	−15.9083	−0.604744
$693$	−3.21110	−0.121980
$694$	13.3305	0.506020
$695$	0	0
$696$	−18.6333	−0.706294
$697$	22.8167	0.864242
$698$	−8.72498	−0.330245
$699$	15.2111	0.575337
$700$	0	0
$701$	−36.4222	−1.37565	−0.687824	−	0.725878i	$-0.741434\pi$
$701$	−36.4222	−1.37565	−0.687824	+	0.725878i	$0.741434\pi$
$702$	0	0
$703$	20.2111	0.762276
$704$	20.5778	0.775555
$705$	0	0
$706$	−38.7250	−1.45743
$707$	9.00000	0.338480
$708$	−35.7250	−1.34263
$709$	13.8444	0.519938	0.259969	−	0.965617i	$-0.416288\pi$
$709$	13.8444	0.519938	0.259969	+	0.965617i	$0.416288\pi$
$710$	0	0
$711$	−10.4222	−0.390863
$712$	−18.6333	−0.698313
$713$	12.0000	0.449404
$714$	17.5139	0.655440
$715$	0	0
$716$	75.3583	2.81627
$717$	0	0
$718$	42.4222	1.58318
$719$	−25.6056	−0.954926	−0.477463	−	0.878652i	$-0.658444\pi$
$719$	−25.6056	−0.954926	−0.477463	+	0.878652i	$0.658444\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	−28.6056	−1.06459
$723$	1.78890	0.0665298
$724$	58.2389	2.16443
$725$	0	0
$726$	−19.3944	−0.719796
$727$	−13.5778	−0.503573	−0.251786	−	0.967783i	$-0.581018\pi$
$727$	−13.5778	−0.503573	−0.251786	+	0.967783i	$0.581018\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−77.6611	−2.87240
$732$	3.30278	0.122074
$733$	−46.8444	−1.73024	−0.865119	−	0.501567i	$-0.832757\pi$
$733$	−46.8444	−1.73024	−0.865119	+	0.501567i	$0.832757\pi$
$734$	26.3028	0.970853
$735$	0	0
$736$	15.9083	0.586389
$737$	11.2389	0.413989
$738$	−13.8167	−0.508598
$739$	35.6056	1.30977	0.654886	−	0.755728i	$-0.272717\pi$
$739$	35.6056	1.30977	0.654886	+	0.755728i	$0.272717\pi$
$740$	0	0
$741$	0	0
$742$	7.39445	0.271459
$743$	36.6333	1.34395	0.671973	−	0.740576i	$-0.265447\pi$
$743$	36.6333	1.34395	0.671973	+	0.740576i	$0.265447\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−46.9638	−1.71947
$747$	18.4222	0.674033
$748$	40.3305	1.47463
$749$	6.21110	0.226949
$750$	0	0
$751$	46.4500	1.69498	0.847492	−	0.530809i	$-0.178112\pi$
$751$	46.4500	1.69498	0.847492	+	0.530809i	$0.178112\pi$
$752$	2.78890	0.101701
$753$	7.18335	0.261776
$754$	0	0
$755$	0	0
$756$	−16.5139	−0.600604
$757$	−0.816654	−0.0296818	−0.0148409	−	0.999890i	$-0.504724\pi$
$757$	−0.816654	−0.0296818	−0.0148409	+	0.999890i	$0.504724\pi$
$758$	22.1194	0.803414
$759$	4.81665	0.174833
$760$	0	0
$761$	−18.6333	−0.675457	−0.337728	−	0.941244i	$-0.609659\pi$
$761$	−18.6333	−0.675457	−0.337728	+	0.941244i	$0.609659\pi$
$762$	9.69722	0.351293
$763$	−19.2111	−0.695489
$764$	55.5416	2.00943
$765$	0	0
$766$	56.7250	2.04956
$767$	0	0
$768$	17.9083	0.646211
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	−16.3944	−0.590432
$772$	51.5416	1.85502
$773$	22.3944	0.805472	0.402736	−	0.915316i	$-0.368059\pi$
$773$	22.3944	0.805472	0.402736	+	0.915316i	$0.368059\pi$
$774$	47.0278	1.69038
$775$	0	0
$776$	25.1833	0.904029
$777$	3.60555	0.129348
$778$	35.0278	1.25581
$779$	−16.8167	−0.602519
$780$	0	0
$781$	7.73338	0.276722
$782$	52.5416	1.87889
$783$	−31.0555	−1.10983
$784$	−1.81665	−0.0648805
$785$	0	0
$786$	−48.8444	−1.74222
$787$	14.6333	0.521621	0.260811	−	0.965390i	$-0.416010\pi$
$787$	14.6333	0.521621	0.260811	+	0.965390i	$0.416010\pi$
$788$	3.90833	0.139228
$789$	11.7889	0.419696
$790$	0	0
$791$	−1.60555	−0.0570868
$792$	−9.63331	−0.342305
$793$	0	0
$794$	−50.7250	−1.80016
$795$	0	0
$796$	42.3305	1.50037
$797$	14.4500	0.511844	0.255922	−	0.966697i	$-0.417621\pi$
$797$	14.4500	0.511844	0.255922	+	0.966697i	$0.417621\pi$
$798$	−12.9083	−0.456950
$799$	−70.0555	−2.47839
$800$	0	0
$801$	−12.4222	−0.438917
$802$	28.1194	0.992932
$803$	1.26662	0.0446979
$804$	23.1194	0.815359
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	27.0000	0.949857
$809$	−55.0555	−1.93565	−0.967824	−	0.251627i	$-0.919034\pi$
$809$	−55.0555	−1.93565	−0.967824	+	0.251627i	$0.919034\pi$
$810$	0	0
$811$	46.4222	1.63010	0.815052	−	0.579388i	$-0.196708\pi$
$811$	46.4222	1.63010	0.815052	+	0.579388i	$0.196708\pi$
$812$	20.5139	0.719896
$813$	−20.8167	−0.730072
$814$	13.3305	0.467235
$815$	0	0
$816$	2.30278	0.0806133
$817$	57.2389	2.00253
$818$	18.9083	0.661114
$819$	0	0
$820$	0	0
$821$	−21.4222	−0.747640	−0.373820	−	0.927501i	$-0.621952\pi$
$821$	−21.4222	−0.747640	−0.373820	+	0.927501i	$0.621952\pi$
$822$	−3.69722	−0.128956
$823$	16.6333	0.579801	0.289900	−	0.957057i	$-0.406378\pi$
$823$	16.6333	0.579801	0.289900	+	0.957057i	$0.406378\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	24.9083	0.866672
$827$	−42.4222	−1.47516	−0.737582	−	0.675257i	$-0.764033\pi$
$827$	−42.4222	−1.47516	−0.737582	+	0.675257i	$0.764033\pi$
$828$	−19.8167	−0.688676
$829$	29.4222	1.02188	0.510938	−	0.859618i	$-0.329298\pi$
$829$	29.4222	1.02188	0.510938	+	0.859618i	$0.329298\pi$
$830$	0	0
$831$	27.6056	0.957626
$832$	0	0
$833$	45.6333	1.58110
$834$	14.7250	0.509884
$835$	0	0
$836$	−29.7250	−1.02806
$837$	20.0000	0.691301
$838$	−39.6972	−1.37132
$839$	20.0278	0.691435	0.345717	−	0.938339i	$-0.387636\pi$
$839$	20.0278	0.691435	0.345717	+	0.938339i	$0.387636\pi$
$840$	0	0
$841$	9.57779	0.330269
$842$	74.6611	2.57299
$843$	−6.00000	−0.206651
$844$	−77.9638	−2.68363
$845$	0	0
$846$	42.4222	1.45851
$847$	8.42221	0.289390
$848$	0.972244	0.0333870
$849$	5.00000	0.171600
$850$	0	0
$851$	10.8167	0.370790
$852$	15.9083	0.545010
$853$	47.2111	1.61648	0.808239	−	0.588855i	$-0.200421\pi$
$853$	47.2111	1.61648	0.808239	+	0.588855i	$0.200421\pi$
$854$	−2.30278	−0.0787994
$855$	0	0
$856$	18.6333	0.636873
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	10.7889	0.368112	0.184056	−	0.982916i	$-0.441077\pi$
$859$	10.7889	0.368112	0.184056	+	0.982916i	$0.441077\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	67.3305	2.29329
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	26.5139	0.902020
$865$	0	0
$866$	8.30278	0.282140
$867$	−40.8444	−1.38715
$868$	−13.2111	−0.448414
$869$	−8.36669	−0.283821
$870$	0	0
$871$	0	0
$872$	−57.6333	−1.95171
$873$	16.7889	0.568218
$874$	−38.7250	−1.30989
$875$	0	0
$876$	2.60555	0.0880334
$877$	−1.97224	−0.0665979	−0.0332990	−	0.999445i	$-0.510601\pi$
$877$	−1.97224	−0.0665979	−0.0332990	+	0.999445i	$0.510601\pi$
$878$	62.7250	2.11687
$879$	−10.3944	−0.350596
$880$	0	0
$881$	−21.8444	−0.735957	−0.367978	−	0.929834i	$-0.619950\pi$
$881$	−21.8444	−0.735957	−0.367978	+	0.929834i	$0.619950\pi$
$882$	−27.6333	−0.930462
$883$	−11.6333	−0.391492	−0.195746	−	0.980655i	$-0.562713\pi$
$883$	−11.6333	−0.391492	−0.195746	+	0.980655i	$0.562713\pi$
$884$	0	0
$885$	0	0
$886$	14.7889	0.496843
$887$	37.0555	1.24420	0.622101	−	0.782937i	$-0.286279\pi$
$887$	37.0555	1.24420	0.622101	+	0.782937i	$0.286279\pi$
$888$	10.8167	0.362983
$889$	−4.21110	−0.141236
$890$	0	0
$891$	−1.60555	−0.0537880
$892$	−13.9083	−0.465685
$893$	51.6333	1.72784
$894$	−6.90833	−0.231049
$895$	0	0
$896$	−18.9083	−0.631683
$897$	0	0
$898$	70.5416	2.35400
$899$	−24.8444	−0.828607
$900$	0	0
$901$	−24.4222	−0.813622
$902$	−11.0917	−0.369312
$903$	10.2111	0.339804
$904$	−4.81665	−0.160200
$905$	0	0
$906$	2.78890	0.0926549
$907$	38.2666	1.27062	0.635311	−	0.772256i	$-0.280872\pi$
$907$	38.2666	1.27062	0.635311	+	0.772256i	$0.280872\pi$
$908$	−90.5694	−3.00565
$909$	18.0000	0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	−1.69722	−0.0562007
$913$	14.7889	0.489441
$914$	61.7527	2.04260
$915$	0	0
$916$	−46.2389	−1.52777
$917$	21.2111	0.700452
$918$	87.5694	2.89022
$919$	−38.8167	−1.28044	−0.640222	−	0.768190i	$-0.721157\pi$
$919$	−38.8167	−1.28044	−0.640222	+	0.768190i	$0.721157\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	83.3860	2.74617
$923$	0	0
$924$	−5.30278	−0.174449
$925$	0	0
$926$	79.2666	2.60486
$927$	−8.00000	−0.262754
$928$	−32.9361	−1.08118
$929$	15.4222	0.505986	0.252993	−	0.967468i	$-0.418585\pi$
$929$	15.4222	0.505986	0.252993	+	0.967468i	$0.418585\pi$
$930$	0	0
$931$	−33.6333	−1.10229
$932$	−50.2389	−1.64563
$933$	9.21110	0.301558
$934$	6.42221	0.210141
$935$	0	0
$936$	0	0
$937$	−54.4777	−1.77971	−0.889855	−	0.456244i	$-0.849194\pi$
$937$	−54.4777	−1.77971	−0.889855	+	0.456244i	$0.849194\pi$
$938$	−16.1194	−0.526318
$939$	14.0000	0.456873
$940$	0	0
$941$	9.63331	0.314037	0.157018	−	0.987596i	$-0.449812\pi$
$941$	9.63331	0.314037	0.157018	+	0.987596i	$0.449812\pi$
$942$	−25.8167	−0.841152
$943$	−9.00000	−0.293080
$944$	3.27502	0.106593
$945$	0	0
$946$	37.7527	1.22745
$947$	−18.6333	−0.605501	−0.302751	−	0.953070i	$-0.597905\pi$
$947$	−18.6333	−0.605501	−0.302751	+	0.953070i	$0.597905\pi$
$948$	−17.2111	−0.558991
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	−22.8167	−0.739492
$953$	14.4500	0.468080	0.234040	−	0.972227i	$-0.424805\pi$
$953$	14.4500	0.468080	0.234040	+	0.972227i	$0.424805\pi$
$954$	14.7889	0.478808
$955$	0	0
$956$	0	0
$957$	−9.97224	−0.322357
$958$	−66.3583	−2.14394
$959$	1.60555	0.0518460
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.4222	0.400300
$964$	−5.90833	−0.190294
$965$	0	0
$966$	−6.90833	−0.222272
$967$	−44.4777	−1.43031	−0.715153	−	0.698967i	$-0.753643\pi$
$967$	−44.4777	−1.43031	−0.715153	+	0.698967i	$0.753643\pi$
$968$	25.2666	0.812100
$969$	42.6333	1.36958
$970$	0	0
$971$	−44.0278	−1.41292	−0.706459	−	0.707754i	$-0.749709\pi$
$971$	−44.0278	−1.41292	−0.706459	+	0.707754i	$0.749709\pi$
$972$	−52.8444	−1.69499
$973$	−6.39445	−0.204997
$974$	2.30278	0.0737857
$975$	0	0
$976$	−0.302776	−0.00969161
$977$	28.8167	0.921926	0.460963	−	0.887419i	$-0.347504\pi$
$977$	28.8167	0.921926	0.460963	+	0.887419i	$0.347504\pi$
$978$	−8.72498	−0.278994
$979$	−9.97224	−0.318714
$980$	0	0
$981$	−38.4222	−1.22673
$982$	38.7250	1.23576
$983$	18.4222	0.587577	0.293789	−	0.955870i	$-0.405084\pi$
$983$	18.4222	0.587577	0.293789	+	0.955870i	$0.405084\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	−108.780	−3.46428
$987$	9.21110	0.293193
$988$	0	0
$989$	30.6333	0.974083
$990$	0	0
$991$	40.0278	1.27152	0.635762	−	0.771885i	$-0.280686\pi$
$991$	40.0278	1.27152	0.635762	+	0.771885i	$0.280686\pi$
$992$	21.2111	0.673453
$993$	10.0278	0.318221
$994$	−11.0917	−0.351807
$995$	0	0
$996$	30.4222	0.963964
$997$	18.4500	0.584316	0.292158	−	0.956370i	$-0.405627\pi$
$997$	18.4500	0.584316	0.292158	+	0.956370i	$0.405627\pi$
$998$	5.57779	0.176562
$999$	18.0278	0.570373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.t.1.1		2
5.4	even	2		845.2.a.f.1.2		2
13.4	even	6		325.2.e.a.276.1		4
13.10	even	6		325.2.e.a.126.1		4
13.12	even	2		4225.2.a.x.1.2		2
15.14	odd	2		7605.2.a.bb.1.1		2
65.4	even	6		65.2.e.b.16.2	✓	4
65.9	even	6		845.2.e.d.146.1		4
65.17	odd	12		325.2.o.b.224.4		8
65.19	odd	12		845.2.m.d.361.1		8
65.23	odd	12		325.2.o.b.74.4		8
65.24	odd	12		845.2.m.d.316.1		8
65.29	even	6		845.2.e.d.191.1		4
65.34	odd	4		845.2.c.d.506.4		4
65.43	odd	12		325.2.o.b.224.1		8
65.44	odd	4		845.2.c.d.506.1		4
65.49	even	6		65.2.e.b.61.2	yes	4
65.54	odd	12		845.2.m.d.316.4		8
65.59	odd	12		845.2.m.d.361.4		8
65.62	odd	12		325.2.o.b.74.1		8
65.64	even	2		845.2.a.c.1.1		2
195.134	odd	6		585.2.j.d.406.1		4
195.179	odd	6		585.2.j.d.451.1		4
195.194	odd	2		7605.2.a.bg.1.2		2
260.179	odd	6		1040.2.q.o.321.2		4
260.199	odd	6		1040.2.q.o.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.b.16.2	✓	4	65.4	even	6
65.2.e.b.61.2	yes	4	65.49	even	6
325.2.e.a.126.1		4	13.10	even	6
325.2.e.a.276.1		4	13.4	even	6
325.2.o.b.74.1		8	65.62	odd	12
325.2.o.b.74.4		8	65.23	odd	12
325.2.o.b.224.1		8	65.43	odd	12
325.2.o.b.224.4		8	65.17	odd	12
585.2.j.d.406.1		4	195.134	odd	6
585.2.j.d.451.1		4	195.179	odd	6
845.2.a.c.1.1		2	65.64	even	2
845.2.a.f.1.2		2	5.4	even	2
845.2.c.d.506.1		4	65.44	odd	4
845.2.c.d.506.4		4	65.34	odd	4
845.2.e.d.146.1		4	65.9	even	6
845.2.e.d.191.1		4	65.29	even	6
845.2.m.d.316.1		8	65.24	odd	12
845.2.m.d.316.4		8	65.54	odd	12
845.2.m.d.361.1		8	65.19	odd	12
845.2.m.d.361.4		8	65.59	odd	12
1040.2.q.o.81.2		4	260.199	odd	6
1040.2.q.o.321.2		4	260.179	odd	6
4225.2.a.t.1.1		2	1.1	even	1	trivial
4225.2.a.x.1.2		2	13.12	even	2
7605.2.a.bb.1.1		2	15.14	odd	2
7605.2.a.bg.1.2		2	195.194	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} +2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} -1.60555 q^{11} -3.30278 q^{12} +2.30278 q^{14} +0.302776 q^{16} -7.60555 q^{17} +4.60555 q^{18} +5.60555 q^{19} +1.00000 q^{21} +3.69722 q^{22} +3.00000 q^{23} +3.00000 q^{24} +5.00000 q^{27} -3.30278 q^{28} -6.21110 q^{29} +4.00000 q^{31} +5.30278 q^{32} +1.60555 q^{33} +17.5139 q^{34} -6.60555 q^{36} +3.60555 q^{37} -12.9083 q^{38} -3.00000 q^{41} -2.30278 q^{42} +10.2111 q^{43} -5.30278 q^{44} -6.90833 q^{46} +9.21110 q^{47} -0.302776 q^{48} -6.00000 q^{49} +7.60555 q^{51} +3.21110 q^{53} -11.5139 q^{54} +3.00000 q^{56} -5.60555 q^{57} +14.3028 q^{58} +10.8167 q^{59} -1.00000 q^{61} -9.21110 q^{62} +2.00000 q^{63} -12.8167 q^{64} -3.69722 q^{66} -7.00000 q^{67} -25.1194 q^{68} -3.00000 q^{69} -4.81665 q^{71} +6.00000 q^{72} -0.788897 q^{73} -8.30278 q^{74} +18.5139 q^{76} +1.60555 q^{77} +5.21110 q^{79} +1.00000 q^{81} +6.90833 q^{82} -9.21110 q^{83} +3.30278 q^{84} -23.5139 q^{86} +6.21110 q^{87} +4.81665 q^{88} +6.21110 q^{89} +9.90833 q^{92} -4.00000 q^{93} -21.2111 q^{94} -5.30278 q^{96} -8.39445 q^{97} +13.8167 q^{98} +3.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 4 q^{11} - 3 q^{12} + q^{14} - 3 q^{16} - 8 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{21} + 11 q^{22} + 6 q^{23} + 6 q^{24} + 10 q^{27}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 + 3 * q^4 + q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + 4 * q^11 - 3 * q^12 + q^14 - 3 * q^16 - 8 * q^17 + 2 * q^18 + 4 * q^19 + 2 * q^21 + 11 * q^22 + 6 * q^23 + 6 * q^24 + 10 * q^27 - 3 * q^28 + 2 * q^29 + 8 * q^31 + 7 * q^32 - 4 * q^33 + 17 * q^34 - 6 * q^36 - 15 * q^38 - 6 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 3 * q^46 + 4 * q^47 + 3 * q^48 - 12 * q^49 + 8 * q^51 - 8 * q^53 - 5 * q^54 + 6 * q^56 - 4 * q^57 + 25 * q^58 - 2 * q^61 - 4 * q^62 + 4 * q^63 - 4 * q^64 - 11 * q^66 - 14 * q^67 - 25 * q^68 - 6 * q^69 + 12 * q^71 + 12 * q^72 - 16 * q^73 - 13 * q^74 + 19 * q^76 - 4 * q^77 - 4 * q^79 + 2 * q^81 + 3 * q^82 - 4 * q^83 + 3 * q^84 - 29 * q^86 - 2 * q^87 - 12 * q^88 - 2 * q^89 + 9 * q^92 - 8 * q^93 - 28 * q^94 - 7 * q^96 - 24 * q^97 + 6 * q^98 - 8 * q^99

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