LMFDB - Embedded newform 4225.2.a.z.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 325)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	\(q+2.41421 q^{2} +2.82843 q^{3} +3.82843 q^{4} +6.82843 q^{6} +2.41421 q^{7} +4.41421 q^{8} +5.00000 q^{9} -6.41421 q^{11} +10.8284 q^{12} +5.82843 q^{14} +3.00000 q^{16} +3.82843 q^{17} +12.0711 q^{18} +3.65685 q^{19} +6.82843 q^{21} -15.4853 q^{22} +3.17157 q^{23} +12.4853 q^{24} +5.65685 q^{27} +9.24264 q^{28} -0.171573 q^{29} -1.24264 q^{31} -1.58579 q^{32} -18.1421 q^{33} +9.24264 q^{34} +19.1421 q^{36} -6.00000 q^{37} +8.82843 q^{38} +5.65685 q^{41} +16.4853 q^{42} -0.828427 q^{43} -24.5563 q^{44} +7.65685 q^{46} +3.58579 q^{47} +8.48528 q^{48} -1.17157 q^{49} +10.8284 q^{51} -3.00000 q^{53} +13.6569 q^{54} +10.6569 q^{56} +10.3431 q^{57} -0.414214 q^{58} -13.2426 q^{59} +1.00000 q^{61} -3.00000 q^{62} +12.0711 q^{63} -9.82843 q^{64} -43.7990 q^{66} -2.75736 q^{67} +14.6569 q^{68} +8.97056 q^{69} +9.31371 q^{71} +22.0711 q^{72} -6.00000 q^{73} -14.4853 q^{74} +14.0000 q^{76} -15.4853 q^{77} -6.00000 q^{79} +1.00000 q^{81} +13.6569 q^{82} +6.89949 q^{83} +26.1421 q^{84} -2.00000 q^{86} -0.485281 q^{87} -28.3137 q^{88} +8.48528 q^{89} +12.1421 q^{92} -3.51472 q^{93} +8.65685 q^{94} -4.48528 q^{96} -0.828427 q^{97} -2.82843 q^{98} -32.0711 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 8 q^{6} + 2 q^{7} + 6 q^{8} + 10 q^{9} - 10 q^{11} + 16 q^{12} + 6 q^{14} + 6 q^{16} + 2 q^{17} + 10 q^{18} - 4 q^{19} + 8 q^{21} - 14 q^{22} + 12 q^{23} + 8 q^{24} + 10 q^{28}+ \cdots - 50 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 8 * q^6 + 2 * q^7 + 6 * q^8 + 10 * q^9 - 10 * q^11 + 16 * q^12 + 6 * q^14 + 6 * q^16 + 2 * q^17 + 10 * q^18 - 4 * q^19 + 8 * q^21 - 14 * q^22 + 12 * q^23 + 8 * q^24 + 10 * q^28 - 6 * q^29 + 6 * q^31 - 6 * q^32 - 8 * q^33 + 10 * q^34 + 10 * q^36 - 12 * q^37 + 12 * q^38 + 16 * q^42 + 4 * q^43 - 18 * q^44 + 4 * q^46 + 10 * q^47 - 8 * q^49 + 16 * q^51 - 6 * q^53 + 16 * q^54 + 10 * q^56 + 32 * q^57 + 2 * q^58 - 18 * q^59 + 2 * q^61 - 6 * q^62 + 10 * q^63 - 14 * q^64 - 48 * q^66 - 14 * q^67 + 18 * q^68 - 16 * q^69 - 4 * q^71 + 30 * q^72 - 12 * q^73 - 12 * q^74 + 28 * q^76 - 14 * q^77 - 12 * q^79 + 2 * q^81 + 16 * q^82 - 6 * q^83 + 24 * q^84 - 4 * q^86 + 16 * q^87 - 34 * q^88 - 4 * q^92 - 24 * q^93 + 6 * q^94 + 8 * q^96 + 4 * q^97 - 50 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	6.82843	2.78769
$7$	2.41421	0.912487	0.456243	−	0.889855i	$-0.349195\pi$
$7$	2.41421	0.912487	0.456243	+	0.889855i	$0.349195\pi$
$8$	4.41421	1.56066
$9$	5.00000	1.66667
$10$	0	0
$11$	−6.41421	−1.93396	−0.966979	−	0.254856i	$-0.917972\pi$
$11$	−6.41421	−1.93396	−0.966979	+	0.254856i	$0.917972\pi$
$12$	10.8284	3.12590
$13$	0	0
$13$	0	0
$14$	5.82843	1.55771
$15$	0	0
$16$	3.00000	0.750000
$17$	3.82843	0.928530	0.464265	−	0.885696i	$-0.346319\pi$
$17$	3.82843	0.928530	0.464265	+	0.885696i	$0.346319\pi$
$18$	12.0711	2.84518
$19$	3.65685	0.838940	0.419470	−	0.907769i	$-0.362216\pi$
$19$	3.65685	0.838940	0.419470	+	0.907769i	$0.362216\pi$
$20$	0	0
$21$	6.82843	1.49008
$22$	−15.4853	−3.30147
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	12.4853	2.54855
$25$	0	0
$26$	0	0
$27$	5.65685	1.08866
$28$	9.24264	1.74669
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	−1.24264	−0.223185	−0.111592	−	0.993754i	$-0.535595\pi$
$31$	−1.24264	−0.223185	−0.111592	+	0.993754i	$0.535595\pi$
$32$	−1.58579	−0.280330
$33$	−18.1421	−3.15814
$34$	9.24264	1.58510
$35$	0	0
$36$	19.1421	3.19036
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	8.82843	1.43216
$39$	0	0
$40$	0	0
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	16.4853	2.54373
$43$	−0.828427	−0.126334	−0.0631670	−	0.998003i	$-0.520120\pi$
$43$	−0.828427	−0.126334	−0.0631670	+	0.998003i	$0.520120\pi$
$44$	−24.5563	−3.70201
$45$	0	0
$46$	7.65685	1.12894
$47$	3.58579	0.523041	0.261520	−	0.965198i	$-0.415776\pi$
$47$	3.58579	0.523041	0.261520	+	0.965198i	$0.415776\pi$
$48$	8.48528	1.22474
$49$	−1.17157	−0.167368
$50$	0	0
$51$	10.8284	1.51628
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	13.6569	1.85846
$55$	0	0
$56$	10.6569	1.42408
$57$	10.3431	1.36998
$58$	−0.414214	−0.0543889
$59$	−13.2426	−1.72404	−0.862022	−	0.506870i	$-0.830802\pi$
$59$	−13.2426	−1.72404	−0.862022	+	0.506870i	$0.830802\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−3.00000	−0.381000
$63$	12.0711	1.52081
$64$	−9.82843	−1.22855
$65$	0	0
$66$	−43.7990	−5.39128
$67$	−2.75736	−0.336865	−0.168433	−	0.985713i	$-0.553871\pi$
$67$	−2.75736	−0.336865	−0.168433	+	0.985713i	$0.553871\pi$
$68$	14.6569	1.77740
$69$	8.97056	1.07993
$70$	0	0
$71$	9.31371	1.10533	0.552667	−	0.833402i	$-0.313610\pi$
$71$	9.31371	1.10533	0.552667	+	0.833402i	$0.313610\pi$
$72$	22.0711	2.60110
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−14.4853	−1.68388
$75$	0	0
$76$	14.0000	1.60591
$77$	−15.4853	−1.76471
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	13.6569	1.50815
$83$	6.89949	0.757318	0.378659	−	0.925536i	$-0.376385\pi$
$83$	6.89949	0.757318	0.378659	+	0.925536i	$0.376385\pi$
$84$	26.1421	2.85234
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−0.485281	−0.0520276
$88$	−28.3137	−3.01825
$89$	8.48528	0.899438	0.449719	−	0.893170i	$-0.351524\pi$
$89$	8.48528	0.899438	0.449719	+	0.893170i	$0.351524\pi$
$90$	0	0
$91$	0	0
$92$	12.1421	1.26591
$93$	−3.51472	−0.364459
$94$	8.65685	0.892886
$95$	0	0
$96$	−4.48528	−0.457777
$97$	−0.828427	−0.0841140	−0.0420570	−	0.999115i	$-0.513391\pi$
$97$	−0.828427	−0.0841140	−0.0420570	+	0.999115i	$0.513391\pi$
$98$	−2.82843	−0.285714
$99$	−32.0711	−3.22326
$100$	0	0
$101$	−6.65685	−0.662382	−0.331191	−	0.943564i	$-0.607450\pi$
$101$	−6.65685	−0.662382	−0.331191	+	0.943564i	$0.607450\pi$
$102$	26.1421	2.58846
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	0	0
$106$	−7.24264	−0.703467
$107$	−16.1421	−1.56052	−0.780260	−	0.625456i	$-0.784913\pi$
$107$	−16.1421	−1.56052	−0.780260	+	0.625456i	$0.784913\pi$
$108$	21.6569	2.08393
$109$	16.4853	1.57900	0.789502	−	0.613748i	$-0.210339\pi$
$109$	16.4853	1.57900	0.789502	+	0.613748i	$0.210339\pi$
$110$	0	0
$111$	−16.9706	−1.61077
$112$	7.24264	0.684365
$113$	11.6569	1.09658	0.548292	−	0.836287i	$-0.315278\pi$
$113$	11.6569	1.09658	0.548292	+	0.836287i	$0.315278\pi$
$114$	24.9706	2.33871
$115$	0	0
$116$	−0.656854	−0.0609874
$117$	0	0
$118$	−31.9706	−2.94313
$119$	9.24264	0.847271
$120$	0	0
$121$	30.1421	2.74019
$122$	2.41421	0.218573
$123$	16.0000	1.44267
$124$	−4.75736	−0.427223
$125$	0	0
$126$	29.1421	2.59619
$127$	3.65685	0.324493	0.162247	−	0.986750i	$-0.448126\pi$
$127$	3.65685	0.324493	0.162247	+	0.986750i	$0.448126\pi$
$128$	−20.5563	−1.81694
$129$	−2.34315	−0.206302
$130$	0	0
$131$	8.48528	0.741362	0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528	0.741362	0.370681	+	0.928760i	$0.379124\pi$
$132$	−69.4558	−6.04536
$133$	8.82843	0.765522
$134$	−6.65685	−0.575065
$135$	0	0
$136$	16.8995	1.44912
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	21.6569	1.84355
$139$	−18.4853	−1.56790	−0.783951	−	0.620823i	$-0.786798\pi$
$139$	−18.4853	−1.56790	−0.783951	+	0.620823i	$0.786798\pi$
$140$	0	0
$141$	10.1421	0.854122
$142$	22.4853	1.88692
$143$	0	0
$144$	15.0000	1.25000
$145$	0	0
$146$	−14.4853	−1.19881
$147$	−3.31371	−0.273310
$148$	−22.9706	−1.88817
$149$	−8.82843	−0.723253	−0.361626	−	0.932323i	$-0.617778\pi$
$149$	−8.82843	−0.723253	−0.361626	+	0.932323i	$0.617778\pi$
$150$	0	0
$151$	8.75736	0.712664	0.356332	−	0.934359i	$-0.384027\pi$
$151$	8.75736	0.712664	0.356332	+	0.934359i	$0.384027\pi$
$152$	16.1421	1.30930
$153$	19.1421	1.54755
$154$	−37.3848	−3.01255
$155$	0	0
$156$	0	0
$157$	−23.4853	−1.87433	−0.937165	−	0.348887i	$-0.886560\pi$
$157$	−23.4853	−1.87433	−0.937165	+	0.348887i	$0.886560\pi$
$158$	−14.4853	−1.15239
$159$	−8.48528	−0.672927
$160$	0	0
$161$	7.65685	0.603445
$162$	2.41421	0.189679
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	21.6569	1.69112
$165$	0	0
$166$	16.6569	1.29282
$167$	−5.31371	−0.411187	−0.205594	−	0.978637i	$-0.565912\pi$
$167$	−5.31371	−0.411187	−0.205594	+	0.978637i	$0.565912\pi$
$168$	30.1421	2.32552
$169$	0	0
$170$	0	0
$171$	18.2843	1.39823
$172$	−3.17157	−0.241830
$173$	−18.6569	−1.41845	−0.709227	−	0.704980i	$-0.750956\pi$
$173$	−18.6569	−1.41845	−0.709227	+	0.704980i	$0.750956\pi$
$174$	−1.17157	−0.0888167
$175$	0	0
$176$	−19.2426	−1.45047
$177$	−37.4558	−2.81535
$178$	20.4853	1.53544
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	−17.4853	−1.29967	−0.649835	−	0.760075i	$-0.725162\pi$
$181$	−17.4853	−1.29967	−0.649835	+	0.760075i	$0.725162\pi$
$182$	0	0
$183$	2.82843	0.209083
$184$	14.0000	1.03209
$185$	0	0
$186$	−8.48528	−0.622171
$187$	−24.5563	−1.79574
$188$	13.7279	1.00121
$189$	13.6569	0.993390
$190$	0	0
$191$	25.3137	1.83164	0.915818	−	0.401594i	$-0.131544\pi$
$191$	25.3137	1.83164	0.915818	+	0.401594i	$0.131544\pi$
$192$	−27.7990	−2.00622
$193$	1.65685	0.119263	0.0596315	−	0.998220i	$-0.481007\pi$
$193$	1.65685	0.119263	0.0596315	+	0.998220i	$0.481007\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−4.48528	−0.320377
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−77.4264	−5.50246
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−7.79899	−0.550098
$202$	−16.0711	−1.13076
$203$	−0.414214	−0.0290721
$204$	41.4558	2.90249
$205$	0	0
$206$	−32.9706	−2.29717
$207$	15.8579	1.10220
$208$	0	0
$209$	−23.4558	−1.62247
$210$	0	0
$211$	17.7990	1.22533	0.612666	−	0.790342i	$-0.290097\pi$
$211$	17.7990	1.22533	0.612666	+	0.790342i	$0.290097\pi$
$212$	−11.4853	−0.788812
$213$	26.3431	1.80500
$214$	−38.9706	−2.66397
$215$	0	0
$216$	24.9706	1.69903
$217$	−3.00000	−0.203653
$218$	39.7990	2.69553
$219$	−16.9706	−1.14676
$220$	0	0
$221$	0	0
$222$	−40.9706	−2.74976
$223$	−10.9706	−0.734643	−0.367322	−	0.930094i	$-0.619725\pi$
$223$	−10.9706	−0.734643	−0.367322	+	0.930094i	$0.619725\pi$
$224$	−3.82843	−0.255798
$225$	0	0
$226$	28.1421	1.87199
$227$	−8.89949	−0.590680	−0.295340	−	0.955392i	$-0.595433\pi$
$227$	−8.89949	−0.590680	−0.295340	+	0.955392i	$0.595433\pi$
$228$	39.5980	2.62244
$229$	4.82843	0.319071	0.159536	−	0.987192i	$-0.449000\pi$
$229$	4.82843	0.319071	0.159536	+	0.987192i	$0.449000\pi$
$230$	0	0
$231$	−43.7990	−2.88176
$232$	−0.757359	−0.0497231
$233$	−20.6274	−1.35135	−0.675674	−	0.737201i	$-0.736147\pi$
$233$	−20.6274	−1.35135	−0.675674	+	0.737201i	$0.736147\pi$
$234$	0	0
$235$	0	0
$236$	−50.6985	−3.30019
$237$	−16.9706	−1.10236
$238$	22.3137	1.44638
$239$	−23.5858	−1.52564	−0.762819	−	0.646612i	$-0.776185\pi$
$239$	−23.5858	−1.52564	−0.762819	+	0.646612i	$0.776185\pi$
$240$	0	0
$241$	−4.97056	−0.320182	−0.160091	−	0.987102i	$-0.551179\pi$
$241$	−4.97056	−0.320182	−0.160091	+	0.987102i	$0.551179\pi$
$242$	72.7696	4.67780
$243$	−14.1421	−0.907218
$244$	3.82843	0.245090
$245$	0	0
$246$	38.6274	2.46279
$247$	0	0
$248$	−5.48528	−0.348316
$249$	19.5147	1.23670
$250$	0	0
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	46.2132	2.91116
$253$	−20.3431	−1.27896
$254$	8.82843	0.553945
$255$	0	0
$256$	−29.9706	−1.87316
$257$	6.65685	0.415243	0.207622	−	0.978209i	$-0.433428\pi$
$257$	6.65685	0.415243	0.207622	+	0.978209i	$0.433428\pi$
$258$	−5.65685	−0.352180
$259$	−14.4853	−0.900072
$260$	0	0
$261$	−0.857864	−0.0531005
$262$	20.4853	1.26558
$263$	26.1421	1.61199	0.805997	−	0.591920i	$-0.201630\pi$
$263$	26.1421	1.61199	0.805997	+	0.591920i	$0.201630\pi$
$264$	−80.0833	−4.92878
$265$	0	0
$266$	21.3137	1.30683
$267$	24.0000	1.46878
$268$	−10.5563	−0.644832
$269$	15.1421	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$269$	15.1421	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$270$	0	0
$271$	25.7279	1.56286	0.781430	−	0.623993i	$-0.214491\pi$
$271$	25.7279	1.56286	0.781430	+	0.623993i	$0.214491\pi$
$272$	11.4853	0.696397
$273$	0	0
$274$	6.82843	0.412520
$275$	0	0
$276$	34.3431	2.06721
$277$	−9.31371	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$277$	−9.31371	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$278$	−44.6274	−2.67657
$279$	−6.21320	−0.371975
$280$	0	0
$281$	−4.82843	−0.288040	−0.144020	−	0.989575i	$-0.546003\pi$
$281$	−4.82843	−0.288040	−0.144020	+	0.989575i	$0.546003\pi$
$282$	24.4853	1.45808
$283$	22.4853	1.33661	0.668306	−	0.743887i	$-0.267020\pi$
$283$	22.4853	1.33661	0.668306	+	0.743887i	$0.267020\pi$
$284$	35.6569	2.11585
$285$	0	0
$286$	0	0
$287$	13.6569	0.806139
$288$	−7.92893	−0.467217
$289$	−2.34315	−0.137832
$290$	0	0
$291$	−2.34315	−0.137358
$292$	−22.9706	−1.34425
$293$	8.82843	0.515762	0.257881	−	0.966177i	$-0.416976\pi$
$293$	8.82843	0.515762	0.257881	+	0.966177i	$0.416976\pi$
$294$	−8.00000	−0.466569
$295$	0	0
$296$	−26.4853	−1.53943
$297$	−36.2843	−2.10543
$298$	−21.3137	−1.23467
$299$	0	0
$300$	0	0
$301$	−2.00000	−0.115278
$302$	21.1421	1.21659
$303$	−18.8284	−1.08166
$304$	10.9706	0.629205
$305$	0	0
$306$	46.2132	2.64183
$307$	−9.31371	−0.531561	−0.265781	−	0.964034i	$-0.585630\pi$
$307$	−9.31371	−0.531561	−0.265781	+	0.964034i	$0.585630\pi$
$308$	−59.2843	−3.37803
$309$	−38.6274	−2.19744
$310$	0	0
$311$	9.51472	0.539530	0.269765	−	0.962926i	$-0.413054\pi$
$311$	9.51472	0.539530	0.269765	+	0.962926i	$0.413054\pi$
$312$	0	0
$313$	−2.85786	−0.161536	−0.0807680	−	0.996733i	$-0.525737\pi$
$313$	−2.85786	−0.161536	−0.0807680	+	0.996733i	$0.525737\pi$
$314$	−56.6985	−3.19968
$315$	0	0
$316$	−22.9706	−1.29220
$317$	20.1421	1.13130	0.565648	−	0.824647i	$-0.308626\pi$
$317$	20.1421	1.13130	0.565648	+	0.824647i	$0.308626\pi$
$318$	−20.4853	−1.14876
$319$	1.10051	0.0616165
$320$	0	0
$321$	−45.6569	−2.54832
$322$	18.4853	1.03014
$323$	14.0000	0.778981
$324$	3.82843	0.212690
$325$	0	0
$326$	24.1421	1.33711
$327$	46.6274	2.57850
$328$	24.9706	1.37877
$329$	8.65685	0.477268
$330$	0	0
$331$	−27.6569	−1.52016	−0.760079	−	0.649831i	$-0.774840\pi$
$331$	−27.6569	−1.52016	−0.760079	+	0.649831i	$0.774840\pi$
$332$	26.4142	1.44967
$333$	−30.0000	−1.64399
$334$	−12.8284	−0.701940
$335$	0	0
$336$	20.4853	1.11756
$337$	20.7990	1.13299	0.566497	−	0.824064i	$-0.308298\pi$
$337$	20.7990	1.13299	0.566497	+	0.824064i	$0.308298\pi$
$338$	0	0
$339$	32.9706	1.79072
$340$	0	0
$341$	7.97056	0.431630
$342$	44.1421	2.38693
$343$	−19.7279	−1.06521
$344$	−3.65685	−0.197164
$345$	0	0
$346$	−45.0416	−2.42145
$347$	19.4558	1.04444	0.522222	−	0.852809i	$-0.325103\pi$
$347$	19.4558	1.04444	0.522222	+	0.852809i	$0.325103\pi$
$348$	−1.85786	−0.0995920
$349$	−35.4558	−1.89791	−0.948954	−	0.315415i	$-0.897856\pi$
$349$	−35.4558	−1.89791	−0.948954	+	0.315415i	$0.897856\pi$
$350$	0	0
$351$	0	0
$352$	10.1716	0.542147
$353$	26.1421	1.39141	0.695703	−	0.718330i	$-0.255093\pi$
$353$	26.1421	1.39141	0.695703	+	0.718330i	$0.255093\pi$
$354$	−90.4264	−4.80611
$355$	0	0
$356$	32.4853	1.72172
$357$	26.1421	1.38359
$358$	1.65685	0.0875675
$359$	−21.3848	−1.12865	−0.564323	−	0.825554i	$-0.690863\pi$
$359$	−21.3848	−1.12865	−0.564323	+	0.825554i	$0.690863\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	−42.2132	−2.21868
$363$	85.2548	4.47472
$364$	0	0
$365$	0	0
$366$	6.82843	0.356928
$367$	34.8284	1.81803	0.909015	−	0.416764i	$-0.136836\pi$
$367$	34.8284	1.81803	0.909015	+	0.416764i	$0.136836\pi$
$368$	9.51472	0.495989
$369$	28.2843	1.47242
$370$	0	0
$371$	−7.24264	−0.376019
$372$	−13.4558	−0.697653
$373$	−1.14214	−0.0591375	−0.0295688	−	0.999563i	$-0.509413\pi$
$373$	−1.14214	−0.0591375	−0.0295688	+	0.999563i	$0.509413\pi$
$374$	−59.2843	−3.06552
$375$	0	0
$376$	15.8284	0.816289
$377$	0	0
$378$	32.9706	1.69582
$379$	31.8701	1.63705	0.818527	−	0.574467i	$-0.194791\pi$
$379$	31.8701	1.63705	0.818527	+	0.574467i	$0.194791\pi$
$380$	0	0
$381$	10.3431	0.529895
$382$	61.1127	3.12680
$383$	27.6569	1.41320	0.706600	−	0.707614i	$-0.250228\pi$
$383$	27.6569	1.41320	0.706600	+	0.707614i	$0.250228\pi$
$384$	−58.1421	−2.96705
$385$	0	0
$386$	4.00000	0.203595
$387$	−4.14214	−0.210557
$388$	−3.17157	−0.161012
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	12.1421	0.614054
$392$	−5.17157	−0.261204
$393$	24.0000	1.21064
$394$	−28.9706	−1.45952
$395$	0	0
$396$	−122.782	−6.17001
$397$	7.65685	0.384286	0.192143	−	0.981367i	$-0.438456\pi$
$397$	7.65685	0.384286	0.192143	+	0.981367i	$0.438456\pi$
$398$	24.1421	1.21014
$399$	24.9706	1.25009
$400$	0	0
$401$	3.85786	0.192653	0.0963263	−	0.995350i	$-0.469291\pi$
$401$	3.85786	0.192653	0.0963263	+	0.995350i	$0.469291\pi$
$402$	−18.8284	−0.939077
$403$	0	0
$404$	−25.4853	−1.26794
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	38.4853	1.90764
$408$	47.7990	2.36640
$409$	10.8284	0.535431	0.267716	−	0.963498i	$-0.413731\pi$
$409$	10.8284	0.535431	0.267716	+	0.963498i	$0.413731\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	−52.2843	−2.57586
$413$	−31.9706	−1.57317
$414$	38.2843	1.88157
$415$	0	0
$416$	0	0
$417$	−52.2843	−2.56037
$418$	−56.6274	−2.76974
$419$	22.8284	1.11524	0.557621	−	0.830096i	$-0.311714\pi$
$419$	22.8284	1.11524	0.557621	+	0.830096i	$0.311714\pi$
$420$	0	0
$421$	16.9706	0.827095	0.413547	−	0.910483i	$-0.364290\pi$
$421$	16.9706	0.827095	0.413547	+	0.910483i	$0.364290\pi$
$422$	42.9706	2.09177
$423$	17.9289	0.871735
$424$	−13.2426	−0.643119
$425$	0	0
$426$	63.5980	3.08133
$427$	2.41421	0.116832
$428$	−61.7990	−2.98717
$429$	0	0
$430$	0	0
$431$	8.34315	0.401875	0.200938	−	0.979604i	$-0.435601\pi$
$431$	8.34315	0.401875	0.200938	+	0.979604i	$0.435601\pi$
$432$	16.9706	0.816497
$433$	−16.3431	−0.785401	−0.392701	−	0.919666i	$-0.628459\pi$
$433$	−16.3431	−0.785401	−0.392701	+	0.919666i	$0.628459\pi$
$434$	−7.24264	−0.347658
$435$	0	0
$436$	63.1127	3.02255
$437$	11.5980	0.554807
$438$	−40.9706	−1.95765
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−5.85786	−0.278946
$442$	0	0
$443$	29.7990	1.41579	0.707896	−	0.706316i	$-0.249644\pi$
$443$	29.7990	1.41579	0.707896	+	0.706316i	$0.249644\pi$
$444$	−64.9706	−3.08337
$445$	0	0
$446$	−26.4853	−1.25411
$447$	−24.9706	−1.18107
$448$	−23.7279	−1.12104
$449$	5.17157	0.244062	0.122031	−	0.992526i	$-0.461059\pi$
$449$	5.17157	0.244062	0.122031	+	0.992526i	$0.461059\pi$
$450$	0	0
$451$	−36.2843	−1.70856
$452$	44.6274	2.09910
$453$	24.7696	1.16378
$454$	−21.4853	−1.00835
$455$	0	0
$456$	45.6569	2.13808
$457$	8.48528	0.396925	0.198462	−	0.980109i	$-0.436405\pi$
$457$	8.48528	0.396925	0.198462	+	0.980109i	$0.436405\pi$
$458$	11.6569	0.544689
$459$	21.6569	1.01086
$460$	0	0
$461$	19.4558	0.906149	0.453074	−	0.891473i	$-0.350327\pi$
$461$	19.4558	0.906149	0.453074	+	0.891473i	$0.350327\pi$
$462$	−105.740	−4.91948
$463$	28.5563	1.32713	0.663563	−	0.748120i	$-0.269043\pi$
$463$	28.5563	1.32713	0.663563	+	0.748120i	$0.269043\pi$
$464$	−0.514719	−0.0238952
$465$	0	0
$466$	−49.7990	−2.30689
$467$	−31.1127	−1.43972	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−31.1127	−1.43972	−0.719862	+	0.694117i	$0.755795\pi$
$468$	0	0
$469$	−6.65685	−0.307385
$470$	0	0
$471$	−66.4264	−3.06077
$472$	−58.4558	−2.69065
$473$	5.31371	0.244325
$474$	−40.9706	−1.88184
$475$	0	0
$476$	35.3848	1.62186
$477$	−15.0000	−0.686803
$478$	−56.9411	−2.60443
$479$	−14.2721	−0.652108	−0.326054	−	0.945351i	$-0.605719\pi$
$479$	−14.2721	−0.652108	−0.326054	+	0.945351i	$0.605719\pi$
$480$	0	0
$481$	0	0
$482$	−12.0000	−0.546585
$483$	21.6569	0.985421
$484$	115.397	5.24532
$485$	0	0
$486$	−34.1421	−1.54872
$487$	−42.2132	−1.91286	−0.956431	−	0.291957i	$-0.905694\pi$
$487$	−42.2132	−1.91286	−0.956431	+	0.291957i	$0.905694\pi$
$488$	4.41421	0.199822
$489$	28.2843	1.27906
$490$	0	0
$491$	11.1716	0.504166	0.252083	−	0.967706i	$-0.418884\pi$
$491$	11.1716	0.504166	0.252083	+	0.967706i	$0.418884\pi$
$492$	61.2548	2.76158
$493$	−0.656854	−0.0295832
$494$	0	0
$495$	0	0
$496$	−3.72792	−0.167389
$497$	22.4853	1.00860
$498$	47.1127	2.11117
$499$	−2.55635	−0.114438	−0.0572190	−	0.998362i	$-0.518223\pi$
$499$	−2.55635	−0.114438	−0.0572190	+	0.998362i	$0.518223\pi$
$500$	0	0
$501$	−15.0294	−0.671466
$502$	−13.6569	−0.609535
$503$	14.6274	0.652204	0.326102	−	0.945335i	$-0.394265\pi$
$503$	14.6274	0.652204	0.326102	+	0.945335i	$0.394265\pi$
$504$	53.2843	2.37347
$505$	0	0
$506$	−49.1127	−2.18333
$507$	0	0
$508$	14.0000	0.621150
$509$	−14.6274	−0.648349	−0.324174	−	0.945997i	$-0.605086\pi$
$509$	−14.6274	−0.648349	−0.324174	+	0.945997i	$0.605086\pi$
$510$	0	0
$511$	−14.4853	−0.640791
$512$	−31.2426	−1.38074
$513$	20.6863	0.913322
$514$	16.0711	0.708864
$515$	0	0
$516$	−8.97056	−0.394907
$517$	−23.0000	−1.01154
$518$	−34.9706	−1.53652
$519$	−52.7696	−2.31633
$520$	0	0
$521$	0.343146	0.0150335	0.00751674	−	0.999972i	$-0.497607\pi$
$521$	0.343146	0.0150335	0.00751674	+	0.999972i	$0.497607\pi$
$522$	−2.07107	−0.0906482
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	32.4853	1.41913
$525$	0	0
$526$	63.1127	2.75184
$527$	−4.75736	−0.207234
$528$	−54.4264	−2.36861
$529$	−12.9411	−0.562658
$530$	0	0
$531$	−66.2132	−2.87341
$532$	33.7990	1.46537
$533$	0	0
$534$	57.9411	2.50736
$535$	0	0
$536$	−12.1716	−0.525732
$537$	1.94113	0.0837657
$538$	36.5563	1.57606
$539$	7.51472	0.323682
$540$	0	0
$541$	−33.7990	−1.45313	−0.726566	−	0.687097i	$-0.758885\pi$
$541$	−33.7990	−1.45313	−0.726566	+	0.687097i	$0.758885\pi$
$542$	62.1127	2.66797
$543$	−49.4558	−2.12235
$544$	−6.07107	−0.260295
$545$	0	0
$546$	0	0
$547$	−28.4853	−1.21794	−0.608971	−	0.793192i	$-0.708418\pi$
$547$	−28.4853	−1.21794	−0.608971	+	0.793192i	$0.708418\pi$
$548$	10.8284	0.462567
$549$	5.00000	0.213395
$550$	0	0
$551$	−0.627417	−0.0267289
$552$	39.5980	1.68540
$553$	−14.4853	−0.615977
$554$	−22.4853	−0.955308
$555$	0	0
$556$	−70.7696	−3.00130
$557$	27.3137	1.15732	0.578659	−	0.815569i	$-0.303576\pi$
$557$	27.3137	1.15732	0.578659	+	0.815569i	$0.303576\pi$
$558$	−15.0000	−0.635001
$559$	0	0
$560$	0	0
$561$	−69.4558	−2.93243
$562$	−11.6569	−0.491715
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	38.8284	1.63497
$565$	0	0
$566$	54.2843	2.28174
$567$	2.41421	0.101387
$568$	41.1127	1.72505
$569$	−35.2843	−1.47919	−0.739597	−	0.673050i	$-0.764984\pi$
$569$	−35.2843	−1.47919	−0.739597	+	0.673050i	$0.764984\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	71.5980	2.99105
$574$	32.9706	1.37616
$575$	0	0
$576$	−49.1421	−2.04759
$577$	−7.17157	−0.298556	−0.149278	−	0.988795i	$-0.547695\pi$
$577$	−7.17157	−0.298556	−0.149278	+	0.988795i	$0.547695\pi$
$578$	−5.65685	−0.235294
$579$	4.68629	0.194756
$580$	0	0
$581$	16.6569	0.691043
$582$	−5.65685	−0.234484
$583$	19.2426	0.796949
$584$	−26.4853	−1.09597
$585$	0	0
$586$	21.3137	0.880461
$587$	14.4142	0.594938	0.297469	−	0.954731i	$-0.403857\pi$
$587$	14.4142	0.594938	0.297469	+	0.954731i	$0.403857\pi$
$588$	−12.6863	−0.523174
$589$	−4.54416	−0.187239
$590$	0	0
$591$	−33.9411	−1.39615
$592$	−18.0000	−0.739795
$593$	19.6569	0.807210	0.403605	−	0.914933i	$-0.367757\pi$
$593$	19.6569	0.807210	0.403605	+	0.914933i	$0.367757\pi$
$594$	−87.5980	−3.59419
$595$	0	0
$596$	−33.7990	−1.38446
$597$	28.2843	1.15760
$598$	0	0
$599$	−3.51472	−0.143608	−0.0718038	−	0.997419i	$-0.522876\pi$
$599$	−3.51472	−0.143608	−0.0718038	+	0.997419i	$0.522876\pi$
$600$	0	0
$601$	13.8284	0.564073	0.282037	−	0.959404i	$-0.408990\pi$
$601$	13.8284	0.564073	0.282037	+	0.959404i	$0.408990\pi$
$602$	−4.82843	−0.196792
$603$	−13.7868	−0.561442
$604$	33.5269	1.36419
$605$	0	0
$606$	−45.4558	−1.84652
$607$	21.5147	0.873255	0.436628	−	0.899642i	$-0.356173\pi$
$607$	21.5147	0.873255	0.436628	+	0.899642i	$0.356173\pi$
$608$	−5.79899	−0.235180
$609$	−1.17157	−0.0474745
$610$	0	0
$611$	0	0
$612$	73.2843	2.96234
$613$	−12.8284	−0.518135	−0.259068	−	0.965859i	$-0.583415\pi$
$613$	−12.8284	−0.518135	−0.259068	+	0.965859i	$0.583415\pi$
$614$	−22.4853	−0.907432
$615$	0	0
$616$	−68.3553	−2.75412
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−93.2548	−3.75126
$619$	−22.9706	−0.923265	−0.461632	−	0.887071i	$-0.652736\pi$
$619$	−22.9706	−0.923265	−0.461632	+	0.887071i	$0.652736\pi$
$620$	0	0
$621$	17.9411	0.719953
$622$	22.9706	0.921036
$623$	20.4853	0.820725
$624$	0	0
$625$	0	0
$626$	−6.89949	−0.275759
$627$	−66.3431	−2.64949
$628$	−89.9117	−3.58787
$629$	−22.9706	−0.915896
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	−26.4853	−1.05353
$633$	50.3431	2.00096
$634$	48.6274	1.93124
$635$	0	0
$636$	−32.4853	−1.28813
$637$	0	0
$638$	2.65685	0.105186
$639$	46.5685	1.84222
$640$	0	0
$641$	28.1127	1.11038	0.555192	−	0.831722i	$-0.312645\pi$
$641$	28.1127	1.11038	0.555192	+	0.831722i	$0.312645\pi$
$642$	−110.225	−4.35025
$643$	−1.02944	−0.0405970	−0.0202985	−	0.999794i	$-0.506462\pi$
$643$	−1.02944	−0.0405970	−0.0202985	+	0.999794i	$0.506462\pi$
$644$	29.3137	1.15512
$645$	0	0
$646$	33.7990	1.32980
$647$	−18.8284	−0.740222	−0.370111	−	0.928988i	$-0.620680\pi$
$647$	−18.8284	−0.740222	−0.370111	+	0.928988i	$0.620680\pi$
$648$	4.41421	0.173407
$649$	84.9411	3.33423
$650$	0	0
$651$	−8.48528	−0.332564
$652$	38.2843	1.49933
$653$	22.4558	0.878765	0.439383	−	0.898300i	$-0.355197\pi$
$653$	22.4558	0.878765	0.439383	+	0.898300i	$0.355197\pi$
$654$	112.569	4.40178
$655$	0	0
$656$	16.9706	0.662589
$657$	−30.0000	−1.17041
$658$	20.8995	0.814747
$659$	−8.62742	−0.336076	−0.168038	−	0.985780i	$-0.553743\pi$
$659$	−8.62742	−0.336076	−0.168038	+	0.985780i	$0.553743\pi$
$660$	0	0
$661$	6.82843	0.265595	0.132798	−	0.991143i	$-0.457604\pi$
$661$	6.82843	0.265595	0.132798	+	0.991143i	$0.457604\pi$
$662$	−66.7696	−2.59507
$663$	0	0
$664$	30.4558	1.18192
$665$	0	0
$666$	−72.4264	−2.80647
$667$	−0.544156	−0.0210698
$668$	−20.3431	−0.787100
$669$	−31.0294	−1.19967
$670$	0	0
$671$	−6.41421	−0.247618
$672$	−10.8284	−0.417716
$673$	−24.4558	−0.942704	−0.471352	−	0.881945i	$-0.656234\pi$
$673$	−24.4558	−0.942704	−0.471352	+	0.881945i	$0.656234\pi$
$674$	50.2132	1.93414
$675$	0	0
$676$	0	0
$677$	41.3137	1.58781	0.793907	−	0.608039i	$-0.208043\pi$
$677$	41.3137	1.58781	0.793907	+	0.608039i	$0.208043\pi$
$678$	79.5980	3.05694
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	−25.1716	−0.964577
$682$	19.2426	0.736839
$683$	21.5858	0.825957	0.412979	−	0.910741i	$-0.364488\pi$
$683$	21.5858	0.825957	0.412979	+	0.910741i	$0.364488\pi$
$684$	70.0000	2.67652
$685$	0	0
$686$	−47.6274	−1.81842
$687$	13.6569	0.521041
$688$	−2.48528	−0.0947505
$689$	0	0
$690$	0	0
$691$	−8.89949	−0.338553	−0.169276	−	0.985569i	$-0.554143\pi$
$691$	−8.89949	−0.338553	−0.169276	+	0.985569i	$0.554143\pi$
$692$	−71.4264	−2.71522
$693$	−77.4264	−2.94119
$694$	46.9706	1.78298
$695$	0	0
$696$	−2.14214	−0.0811974
$697$	21.6569	0.820312
$698$	−85.5980	−3.23993
$699$	−58.3431	−2.20674
$700$	0	0
$701$	−15.8284	−0.597831	−0.298916	−	0.954280i	$-0.596625\pi$
$701$	−15.8284	−0.597831	−0.298916	+	0.954280i	$0.596625\pi$
$702$	0	0
$703$	−21.9411	−0.827525
$704$	63.0416	2.37597
$705$	0	0
$706$	63.1127	2.37528
$707$	−16.0711	−0.604415
$708$	−143.397	−5.38919
$709$	49.2548	1.84980	0.924902	−	0.380205i	$-0.124147\pi$
$709$	49.2548	1.84980	0.924902	+	0.380205i	$0.124147\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	37.4558	1.40372
$713$	−3.94113	−0.147596
$714$	63.1127	2.36193
$715$	0	0
$716$	2.62742	0.0981912
$717$	−66.7107	−2.49136
$718$	−51.6274	−1.92672
$719$	−49.4558	−1.84439	−0.922196	−	0.386723i	$-0.873607\pi$
$719$	−49.4558	−1.84439	−0.922196	+	0.386723i	$0.873607\pi$
$720$	0	0
$721$	−32.9706	−1.22789
$722$	−13.5858	−0.505611
$723$	−14.0589	−0.522855
$724$	−66.9411	−2.48785
$725$	0	0
$726$	205.823	7.63882
$727$	−16.6863	−0.618860	−0.309430	−	0.950922i	$-0.600138\pi$
$727$	−16.6863	−0.618860	−0.309430	+	0.950922i	$0.600138\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	−3.17157	−0.117305
$732$	10.8284	0.400230
$733$	−35.7990	−1.32227	−0.661133	−	0.750269i	$-0.729924\pi$
$733$	−35.7990	−1.32227	−0.661133	+	0.750269i	$0.729924\pi$
$734$	84.0833	3.10357
$735$	0	0
$736$	−5.02944	−0.185388
$737$	17.6863	0.651483
$738$	68.2843	2.51358
$739$	27.7279	1.01999	0.509994	−	0.860178i	$-0.329648\pi$
$739$	27.7279	1.01999	0.509994	+	0.860178i	$0.329648\pi$
$740$	0	0
$741$	0	0
$742$	−17.4853	−0.641905
$743$	−14.6985	−0.539235	−0.269618	−	0.962967i	$-0.586897\pi$
$743$	−14.6985	−0.539235	−0.269618	+	0.962967i	$0.586897\pi$
$744$	−15.5147	−0.568797
$745$	0	0
$746$	−2.75736	−0.100954
$747$	34.4975	1.26220
$748$	−94.0122	−3.43743
$749$	−38.9706	−1.42395
$750$	0	0
$751$	31.4558	1.14784	0.573920	−	0.818911i	$-0.305422\pi$
$751$	31.4558	1.14784	0.573920	+	0.818911i	$0.305422\pi$
$752$	10.7574	0.392281
$753$	−16.0000	−0.583072
$754$	0	0
$755$	0	0
$756$	52.2843	1.90156
$757$	16.3137	0.592932	0.296466	−	0.955043i	$-0.404192\pi$
$757$	16.3137	0.592932	0.296466	+	0.955043i	$0.404192\pi$
$758$	76.9411	2.79463
$759$	−57.5391	−2.08854
$760$	0	0
$761$	21.7990	0.790213	0.395106	−	0.918635i	$-0.370708\pi$
$761$	21.7990	0.790213	0.395106	+	0.918635i	$0.370708\pi$
$762$	24.9706	0.904588
$763$	39.7990	1.44082
$764$	96.9117	3.50614
$765$	0	0
$766$	66.7696	2.41248
$767$	0	0
$768$	−84.7696	−3.05886
$769$	−4.97056	−0.179243	−0.0896215	−	0.995976i	$-0.528566\pi$
$769$	−4.97056	−0.179243	−0.0896215	+	0.995976i	$0.528566\pi$
$770$	0	0
$771$	18.8284	0.678089
$772$	6.34315	0.228295
$773$	−5.79899	−0.208575	−0.104288	−	0.994547i	$-0.533256\pi$
$773$	−5.79899	−0.208575	−0.104288	+	0.994547i	$0.533256\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−3.65685	−0.131273
$777$	−40.9706	−1.46981
$778$	69.1127	2.47781
$779$	20.6863	0.741163
$780$	0	0
$781$	−59.7401	−2.13767
$782$	29.3137	1.04826
$783$	−0.970563	−0.0346851
$784$	−3.51472	−0.125526
$785$	0	0
$786$	57.9411	2.06669
$787$	38.7574	1.38155	0.690775	−	0.723069i	$-0.257269\pi$
$787$	38.7574	1.38155	0.690775	+	0.723069i	$0.257269\pi$
$788$	−45.9411	−1.63658
$789$	73.9411	2.63237
$790$	0	0
$791$	28.1421	1.00062
$792$	−141.569	−5.03042
$793$	0	0
$794$	18.4853	0.656018
$795$	0	0
$796$	38.2843	1.35695
$797$	11.4853	0.406830	0.203415	−	0.979093i	$-0.434796\pi$
$797$	11.4853	0.406830	0.203415	+	0.979093i	$0.434796\pi$
$798$	60.2843	2.13404
$799$	13.7279	0.485659
$800$	0	0
$801$	42.4264	1.49906
$802$	9.31371	0.328878
$803$	38.4853	1.35812
$804$	−29.8579	−1.05301
$805$	0	0
$806$	0	0
$807$	42.8284	1.50763
$808$	−29.3848	−1.03375
$809$	17.3137	0.608718	0.304359	−	0.952557i	$-0.401558\pi$
$809$	17.3137	0.608718	0.304359	+	0.952557i	$0.401558\pi$
$810$	0	0
$811$	−9.58579	−0.336602	−0.168301	−	0.985736i	$-0.553828\pi$
$811$	−9.58579	−0.336602	−0.168301	+	0.985736i	$0.553828\pi$
$812$	−1.58579	−0.0556502
$813$	72.7696	2.55214
$814$	92.9117	3.25655
$815$	0	0
$816$	32.4853	1.13721
$817$	−3.02944	−0.105987
$818$	26.1421	0.914038
$819$	0	0
$820$	0	0
$821$	20.8284	0.726917	0.363459	−	0.931610i	$-0.381596\pi$
$821$	20.8284	0.726917	0.363459	+	0.931610i	$0.381596\pi$
$822$	19.3137	0.673643
$823$	20.2843	0.707065	0.353533	−	0.935422i	$-0.384980\pi$
$823$	20.2843	0.707065	0.353533	+	0.935422i	$0.384980\pi$
$824$	−60.2843	−2.10010
$825$	0	0
$826$	−77.1838	−2.68557
$827$	−32.3553	−1.12511	−0.562553	−	0.826761i	$-0.690181\pi$
$827$	−32.3553	−1.12511	−0.562553	+	0.826761i	$0.690181\pi$
$828$	60.7107	2.10984
$829$	7.97056	0.276829	0.138415	−	0.990374i	$-0.455799\pi$
$829$	7.97056	0.276829	0.138415	+	0.990374i	$0.455799\pi$
$830$	0	0
$831$	−26.3431	−0.913834
$832$	0	0
$833$	−4.48528	−0.155406
$834$	−126.225	−4.37083
$835$	0	0
$836$	−89.7990	−3.10576
$837$	−7.02944	−0.242973
$838$	55.1127	1.90384
$839$	1.02944	0.0355401	0.0177701	−	0.999842i	$-0.494343\pi$
$839$	1.02944	0.0355401	0.0177701	+	0.999842i	$0.494343\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	40.9706	1.41194
$843$	−13.6569	−0.470367
$844$	68.1421	2.34555
$845$	0	0
$846$	43.2843	1.48814
$847$	72.7696	2.50039
$848$	−9.00000	−0.309061
$849$	63.5980	2.18268
$850$	0	0
$851$	−19.0294	−0.652321
$852$	100.853	3.45516
$853$	−42.4264	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$854$	5.82843	0.199445
$855$	0	0
$856$	−71.2548	−2.43544
$857$	4.62742	0.158070	0.0790348	−	0.996872i	$-0.474816\pi$
$857$	4.62742	0.158070	0.0790348	+	0.996872i	$0.474816\pi$
$858$	0	0
$859$	24.1421	0.823719	0.411860	−	0.911247i	$-0.364879\pi$
$859$	24.1421	0.823719	0.411860	+	0.911247i	$0.364879\pi$
$860$	0	0
$861$	38.6274	1.31642
$862$	20.1421	0.686044
$863$	29.1838	0.993427	0.496713	−	0.867915i	$-0.334540\pi$
$863$	29.1838	0.993427	0.496713	+	0.867915i	$0.334540\pi$
$864$	−8.97056	−0.305185
$865$	0	0
$866$	−39.4558	−1.34076
$867$	−6.62742	−0.225079
$868$	−11.4853	−0.389836
$869$	38.4853	1.30552
$870$	0	0
$871$	0	0
$872$	72.7696	2.46429
$873$	−4.14214	−0.140190
$874$	28.0000	0.947114
$875$	0	0
$876$	−64.9706	−2.19515
$877$	−15.7990	−0.533494	−0.266747	−	0.963767i	$-0.585949\pi$
$877$	−15.7990	−0.533494	−0.266747	+	0.963767i	$0.585949\pi$
$878$	33.7990	1.14066
$879$	24.9706	0.842236
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	−14.1421	−0.476190
$883$	−32.9706	−1.10955	−0.554774	−	0.832001i	$-0.687195\pi$
$883$	−32.9706	−1.10955	−0.554774	+	0.832001i	$0.687195\pi$
$884$	0	0
$885$	0	0
$886$	71.9411	2.41691
$887$	−8.62742	−0.289680	−0.144840	−	0.989455i	$-0.546267\pi$
$887$	−8.62742	−0.289680	−0.144840	+	0.989455i	$0.546267\pi$
$888$	−74.9117	−2.51387
$889$	8.82843	0.296096
$890$	0	0
$891$	−6.41421	−0.214884
$892$	−42.0000	−1.40626
$893$	13.1127	0.438800
$894$	−60.2843	−2.01621
$895$	0	0
$896$	−49.6274	−1.65794
$897$	0	0
$898$	12.4853	0.416639
$899$	0.213203	0.00711073
$900$	0	0
$901$	−11.4853	−0.382630
$902$	−87.5980	−2.91669
$903$	−5.65685	−0.188248
$904$	51.4558	1.71140
$905$	0	0
$906$	59.7990	1.98669
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−34.0711	−1.13069
$909$	−33.2843	−1.10397
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	31.0294	1.02749
$913$	−44.2548	−1.46462
$914$	20.4853	0.677593
$915$	0	0
$916$	18.4853	0.610771
$917$	20.4853	0.676484
$918$	52.2843	1.72564
$919$	−18.9706	−0.625781	−0.312891	−	0.949789i	$-0.601297\pi$
$919$	−18.9706	−0.625781	−0.312891	+	0.949789i	$0.601297\pi$
$920$	0	0
$921$	−26.3431	−0.868036
$922$	46.9706	1.54689
$923$	0	0
$924$	−167.681	−5.51631
$925$	0	0
$926$	68.9411	2.26555
$927$	−68.2843	−2.24275
$928$	0.272078	0.00893140
$929$	−30.2843	−0.993595	−0.496797	−	0.867867i	$-0.665491\pi$
$929$	−30.2843	−0.993595	−0.496797	+	0.867867i	$0.665491\pi$
$930$	0	0
$931$	−4.28427	−0.140411
$932$	−78.9706	−2.58677
$933$	26.9117	0.881049
$934$	−75.1127	−2.45776
$935$	0	0
$936$	0	0
$937$	−1.97056	−0.0643755	−0.0321877	−	0.999482i	$-0.510247\pi$
$937$	−1.97056	−0.0643755	−0.0321877	+	0.999482i	$0.510247\pi$
$938$	−16.0711	−0.524739
$939$	−8.08326	−0.263787
$940$	0	0
$941$	35.6569	1.16238	0.581190	−	0.813768i	$-0.302587\pi$
$941$	35.6569	1.16238	0.581190	+	0.813768i	$0.302587\pi$
$942$	−160.368	−5.22506
$943$	17.9411	0.584243
$944$	−39.7279	−1.29303
$945$	0	0
$946$	12.8284	0.417088
$947$	−26.4142	−0.858347	−0.429173	−	0.903222i	$-0.641195\pi$
$947$	−26.4142	−0.858347	−0.429173	+	0.903222i	$0.641195\pi$
$948$	−64.9706	−2.11015
$949$	0	0
$950$	0	0
$951$	56.9706	1.84740
$952$	40.7990	1.32230
$953$	47.3431	1.53359	0.766797	−	0.641889i	$-0.221849\pi$
$953$	47.3431	1.53359	0.766797	+	0.641889i	$0.221849\pi$
$954$	−36.2132	−1.17245
$955$	0	0
$956$	−90.2965	−2.92040
$957$	3.11270	0.100619
$958$	−34.4558	−1.11322
$959$	6.82843	0.220501
$960$	0	0
$961$	−29.4558	−0.950189
$962$	0	0
$963$	−80.7107	−2.60087
$964$	−19.0294	−0.612897
$965$	0	0
$966$	52.2843	1.68222
$967$	43.7279	1.40620	0.703098	−	0.711093i	$-0.251800\pi$
$967$	43.7279	1.40620	0.703098	+	0.711093i	$0.251800\pi$
$968$	133.054	4.27651
$969$	39.5980	1.27207
$970$	0	0
$971$	36.8284	1.18188	0.590940	−	0.806715i	$-0.298757\pi$
$971$	36.8284	1.18188	0.590940	+	0.806715i	$0.298757\pi$
$972$	−54.1421	−1.73661
$973$	−44.6274	−1.43069
$974$	−101.912	−3.26546
$975$	0	0
$976$	3.00000	0.0960277
$977$	−8.48528	−0.271468	−0.135734	−	0.990745i	$-0.543339\pi$
$977$	−8.48528	−0.271468	−0.135734	+	0.990745i	$0.543339\pi$
$978$	68.2843	2.18349
$979$	−54.4264	−1.73948
$980$	0	0
$981$	82.4264	2.63167
$982$	26.9706	0.860665
$983$	26.6985	0.851549	0.425775	−	0.904829i	$-0.360002\pi$
$983$	26.6985	0.851549	0.425775	+	0.904829i	$0.360002\pi$
$984$	70.6274	2.25152
$985$	0	0
$986$	−1.58579	−0.0505017
$987$	24.4853	0.779375
$988$	0	0
$989$	−2.62742	−0.0835470
$990$	0	0
$991$	−13.5147	−0.429309	−0.214655	−	0.976690i	$-0.568863\pi$
$991$	−13.5147	−0.429309	−0.214655	+	0.976690i	$0.568863\pi$
$992$	1.97056	0.0625654
$993$	−78.2254	−2.48241
$994$	54.2843	1.72179
$995$	0	0
$996$	74.7107	2.36730
$997$	−12.6569	−0.400847	−0.200423	−	0.979709i	$-0.564232\pi$
$997$	−12.6569	−0.400847	−0.200423	+	0.979709i	$0.564232\pi$
$998$	−6.17157	−0.195358
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.z.1.2		2
5.4	even	2		4225.2.a.s.1.1		2
13.12	even	2		325.2.a.f.1.1	✓	2
39.38	odd	2		2925.2.a.bd.1.2		2
52.51	odd	2		5200.2.a.bt.1.1		2
65.12	odd	4		325.2.b.d.274.1		4
65.38	odd	4		325.2.b.d.274.4		4
65.64	even	2		325.2.a.h.1.2	yes	2
195.38	even	4		2925.2.c.q.2224.1		4
195.77	even	4		2925.2.c.q.2224.4		4
195.194	odd	2		2925.2.a.w.1.1		2
260.259	odd	2		5200.2.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.f.1.1	✓	2	13.12	even	2
325.2.a.h.1.2	yes	2	65.64	even	2
325.2.b.d.274.1		4	65.12	odd	4
325.2.b.d.274.4		4	65.38	odd	4
2925.2.a.w.1.1		2	195.194	odd	2
2925.2.a.bd.1.2		2	39.38	odd	2
2925.2.c.q.2224.1		4	195.38	even	4
2925.2.c.q.2224.4		4	195.77	even	4
4225.2.a.s.1.1		2	5.4	even	2
4225.2.a.z.1.2		2	1.1	even	1	trivial
5200.2.a.br.1.2		2	260.259	odd	2
5200.2.a.bt.1.1		2	52.51	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +2.82843 q^{3} +3.82843 q^{4} +6.82843 q^{6} +2.41421 q^{7} +4.41421 q^{8} +5.00000 q^{9} -6.41421 q^{11} +10.8284 q^{12} +5.82843 q^{14} +3.00000 q^{16} +3.82843 q^{17} +12.0711 q^{18} +3.65685 q^{19} +6.82843 q^{21} -15.4853 q^{22} +3.17157 q^{23} +12.4853 q^{24} +5.65685 q^{27} +9.24264 q^{28} -0.171573 q^{29} -1.24264 q^{31} -1.58579 q^{32} -18.1421 q^{33} +9.24264 q^{34} +19.1421 q^{36} -6.00000 q^{37} +8.82843 q^{38} +5.65685 q^{41} +16.4853 q^{42} -0.828427 q^{43} -24.5563 q^{44} +7.65685 q^{46} +3.58579 q^{47} +8.48528 q^{48} -1.17157 q^{49} +10.8284 q^{51} -3.00000 q^{53} +13.6569 q^{54} +10.6569 q^{56} +10.3431 q^{57} -0.414214 q^{58} -13.2426 q^{59} +1.00000 q^{61} -3.00000 q^{62} +12.0711 q^{63} -9.82843 q^{64} -43.7990 q^{66} -2.75736 q^{67} +14.6569 q^{68} +8.97056 q^{69} +9.31371 q^{71} +22.0711 q^{72} -6.00000 q^{73} -14.4853 q^{74} +14.0000 q^{76} -15.4853 q^{77} -6.00000 q^{79} +1.00000 q^{81} +13.6569 q^{82} +6.89949 q^{83} +26.1421 q^{84} -2.00000 q^{86} -0.485281 q^{87} -28.3137 q^{88} +8.48528 q^{89} +12.1421 q^{92} -3.51472 q^{93} +8.65685 q^{94} -4.48528 q^{96} -0.828427 q^{97} -2.82843 q^{98} -32.0711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 8 q^{6} + 2 q^{7} + 6 q^{8} + 10 q^{9} - 10 q^{11} + 16 q^{12} + 6 q^{14} + 6 q^{16} + 2 q^{17} + 10 q^{18} - 4 q^{19} + 8 q^{21} - 14 q^{22} + 12 q^{23} + 8 q^{24} + 10 q^{28}+ \cdots - 50 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 8 * q^6 + 2 * q^7 + 6 * q^8 + 10 * q^9 - 10 * q^11 + 16 * q^12 + 6 * q^14 + 6 * q^16 + 2 * q^17 + 10 * q^18 - 4 * q^19 + 8 * q^21 - 14 * q^22 + 12 * q^23 + 8 * q^24 + 10 * q^28 - 6 * q^29 + 6 * q^31 - 6 * q^32 - 8 * q^33 + 10 * q^34 + 10 * q^36 - 12 * q^37 + 12 * q^38 + 16 * q^42 + 4 * q^43 - 18 * q^44 + 4 * q^46 + 10 * q^47 - 8 * q^49 + 16 * q^51 - 6 * q^53 + 16 * q^54 + 10 * q^56 + 32 * q^57 + 2 * q^58 - 18 * q^59 + 2 * q^61 - 6 * q^62 + 10 * q^63 - 14 * q^64 - 48 * q^66 - 14 * q^67 + 18 * q^68 - 16 * q^69 - 4 * q^71 + 30 * q^72 - 12 * q^73 - 12 * q^74 + 28 * q^76 - 14 * q^77 - 12 * q^79 + 2 * q^81 + 16 * q^82 - 6 * q^83 + 24 * q^84 - 4 * q^86 + 16 * q^87 - 34 * q^88 - 4 * q^92 - 24 * q^93 + 6 * q^94 + 8 * q^96 + 4 * q^97 - 50 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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