LMFDB - Embedded newform 4225.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4225,2,Mod(1,4225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [3,1,5,-1,0,4,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	\(q+0.445042 q^{2} +3.24698 q^{3} -1.80194 q^{4} +1.44504 q^{6} +3.24698 q^{7} -1.69202 q^{8} +7.54288 q^{9} +0.692021 q^{11} -5.85086 q^{12} +1.44504 q^{14} +2.85086 q^{16} -3.74094 q^{17} +3.35690 q^{18} +1.53319 q^{19} +10.5429 q^{21} +0.307979 q^{22} +1.22521 q^{23} -5.49396 q^{24} +14.7506 q^{27} -5.85086 q^{28} -6.07069 q^{29} +8.45473 q^{31} +4.65279 q^{32} +2.24698 q^{33} -1.66487 q^{34} -13.5918 q^{36} -1.89008 q^{37} +0.682333 q^{38} +0.457123 q^{41} +4.69202 q^{42} +6.19806 q^{43} -1.24698 q^{44} +0.545269 q^{46} +11.5429 q^{47} +9.25667 q^{48} +3.54288 q^{49} -12.1468 q^{51} -0.801938 q^{53} +6.56465 q^{54} -5.49396 q^{56} +4.97823 q^{57} -2.70171 q^{58} -6.60388 q^{59} -4.19806 q^{61} +3.76271 q^{62} +24.4916 q^{63} -3.63102 q^{64} +1.00000 q^{66} -13.8116 q^{67} +6.74094 q^{68} +3.97823 q^{69} +9.87263 q^{71} -12.7627 q^{72} +8.05429 q^{73} -0.841166 q^{74} -2.76271 q^{76} +2.24698 q^{77} -16.5157 q^{79} +25.2664 q^{81} +0.203439 q^{82} +6.17092 q^{83} -18.9976 q^{84} +2.75840 q^{86} -19.7114 q^{87} -1.17092 q^{88} +10.5254 q^{89} -2.20775 q^{92} +27.4523 q^{93} +5.13706 q^{94} +15.1075 q^{96} -3.45473 q^{97} +1.57673 q^{98} +5.21983 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{3} - q^{4} + 4 q^{6} + 5 q^{7} + 4 q^{9} - 3 q^{11} - 4 q^{12} + 4 q^{14} - 5 q^{16} + 3 q^{17} + 6 q^{18} + 8 q^{19} + 13 q^{21} + 6 q^{22} + 2 q^{23} - 7 q^{24} + 8 q^{27} - 4 q^{28}+ \cdots + 17 q^{99}+O(q^{100}) $ 3 * q + q^2 + 5 * q^3 - q^4 + 4 * q^6 + 5 * q^7 + 4 * q^9 - 3 * q^11 - 4 * q^12 + 4 * q^14 - 5 * q^16 + 3 * q^17 + 6 * q^18 + 8 * q^19 + 13 * q^21 + 6 * q^22 + 2 * q^23 - 7 * q^24 + 8 * q^27 - 4 * q^28 - 6 * q^29 + 3 * q^31 - 4 * q^32 + 2 * q^33 - 6 * q^34 - 13 * q^36 - 5 * q^37 + 19 * q^38 + 20 * q^41 + 9 * q^42 + 23 * q^43 + q^44 + 24 * q^46 + 16 * q^47 + q^48 - 8 * q^49 - 9 * q^51 + 2 * q^53 - 2 * q^54 - 7 * q^56 + 18 * q^57 + 19 * q^58 - 11 * q^59 - 17 * q^61 - 6 * q^62 + 23 * q^63 + 4 * q^64 + 3 * q^66 - 15 * q^67 + 6 * q^68 + 15 * q^69 + 13 * q^71 - 21 * q^72 + 12 * q^73 - 11 * q^74 + 9 * q^76 + 2 * q^77 - 37 * q^79 + 27 * q^81 + 2 * q^82 + 29 * q^83 - 16 * q^84 + 10 * q^86 - 10 * q^87 - 14 * q^88 - 3 * q^89 + 11 * q^92 + 19 * q^93 + 10 * q^94 + 5 * q^96 + 12 * q^97 + 2 * q^98 + 17 * 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 $

$2$	0.445042	0.314692	0.157346	−	0.987544i	$-0.449706\pi$
$2$	0.445042	0.314692	0.157346	+	0.987544i	$0.449706\pi$
$3$	3.24698	1.87464	0.937322	−	0.348464i	$-0.113297\pi$
$3$	3.24698	1.87464	0.937322	+	0.348464i	$0.113297\pi$
$4$	−1.80194	−0.900969
$5$	0	0
$5$	0	0
$6$	1.44504	0.589936
$7$	3.24698	1.22724	0.613621	−	0.789600i	$-0.289712\pi$
$7$	3.24698	1.22724	0.613621	+	0.789600i	$0.289712\pi$
$8$	−1.69202	−0.598220
$9$	7.54288	2.51429
$10$	0	0
$11$	0.692021	0.208652	0.104326	−	0.994543i	$-0.466731\pi$
$11$	0.692021	0.208652	0.104326	+	0.994543i	$0.466731\pi$
$12$	−5.85086	−1.68900
$13$	0	0
$13$	0	0
$14$	1.44504	0.386204
$15$	0	0
$16$	2.85086	0.712714
$17$	−3.74094	−0.907311	−0.453655	−	0.891177i	$-0.649880\pi$
$17$	−3.74094	−0.907311	−0.453655	+	0.891177i	$0.649880\pi$
$18$	3.35690	0.791228
$19$	1.53319	0.351737	0.175869	−	0.984414i	$-0.443727\pi$
$19$	1.53319	0.351737	0.175869	+	0.984414i	$0.443727\pi$
$20$	0	0
$21$	10.5429	2.30064
$22$	0.307979	0.0656612
$23$	1.22521	0.255474	0.127737	−	0.991808i	$-0.459229\pi$
$23$	1.22521	0.255474	0.127737	+	0.991808i	$0.459229\pi$
$24$	−5.49396	−1.12145
$25$	0	0
$26$	0	0
$27$	14.7506	2.83876
$28$	−5.85086	−1.10571
$29$	−6.07069	−1.12730	−0.563649	−	0.826014i	$-0.690603\pi$
$29$	−6.07069	−1.12730	−0.563649	+	0.826014i	$0.690603\pi$
$30$	0	0
$31$	8.45473	1.51851	0.759257	−	0.650791i	$-0.225562\pi$
$31$	8.45473	1.51851	0.759257	+	0.650791i	$0.225562\pi$
$32$	4.65279	0.822505
$33$	2.24698	0.391149
$34$	−1.66487	−0.285524
$35$	0	0
$36$	−13.5918	−2.26530
$37$	−1.89008	−0.310728	−0.155364	−	0.987857i	$-0.549655\pi$
$37$	−1.89008	−0.310728	−0.155364	+	0.987857i	$0.549655\pi$
$38$	0.682333	0.110689
$39$	0	0
$40$	0	0
$41$	0.457123	0.0713907	0.0356953	−	0.999363i	$-0.488635\pi$
$41$	0.457123	0.0713907	0.0356953	+	0.999363i	$0.488635\pi$
$42$	4.69202	0.723995
$43$	6.19806	0.945196	0.472598	−	0.881278i	$-0.343316\pi$
$43$	6.19806	0.945196	0.472598	+	0.881278i	$0.343316\pi$
$44$	−1.24698	−0.187989
$45$	0	0
$46$	0.545269	0.0803956
$47$	11.5429	1.68370	0.841851	−	0.539710i	$-0.181466\pi$
$47$	11.5429	1.68370	0.841851	+	0.539710i	$0.181466\pi$
$48$	9.25667	1.33608
$49$	3.54288	0.506125
$50$	0	0
$51$	−12.1468	−1.70089
$52$	0	0
$53$	−0.801938	−0.110155	−0.0550773	−	0.998482i	$-0.517541\pi$
$53$	−0.801938	−0.110155	−0.0550773	+	0.998482i	$0.517541\pi$
$54$	6.56465	0.893335
$55$	0	0
$56$	−5.49396	−0.734161
$57$	4.97823	0.659383
$58$	−2.70171	−0.354752
$59$	−6.60388	−0.859751	−0.429876	−	0.902888i	$-0.641443\pi$
$59$	−6.60388	−0.859751	−0.429876	+	0.902888i	$0.641443\pi$
$60$	0	0
$61$	−4.19806	−0.537507	−0.268753	−	0.963209i	$-0.586612\pi$
$61$	−4.19806	−0.537507	−0.268753	+	0.963209i	$0.586612\pi$
$62$	3.76271	0.477865
$63$	24.4916	3.08565
$64$	−3.63102	−0.453878
$65$	0	0
$66$	1.00000	0.123091
$67$	−13.8116	−1.68736	−0.843679	−	0.536847i	$-0.819615\pi$
$67$	−13.8116	−1.68736	−0.843679	+	0.536847i	$0.819615\pi$
$68$	6.74094	0.817459
$69$	3.97823	0.478923
$70$	0	0
$71$	9.87263	1.17167	0.585833	−	0.810432i	$-0.300768\pi$
$71$	9.87263	1.17167	0.585833	+	0.810432i	$0.300768\pi$
$72$	−12.7627	−1.50410
$73$	8.05429	0.942684	0.471342	−	0.881951i	$-0.343770\pi$
$73$	8.05429	0.942684	0.471342	+	0.881951i	$0.343770\pi$
$74$	−0.841166	−0.0977836
$75$	0	0
$76$	−2.76271	−0.316904
$77$	2.24698	0.256067
$78$	0	0
$79$	−16.5157	−1.85816	−0.929082	−	0.369873i	$-0.879401\pi$
$79$	−16.5157	−1.85816	−0.929082	+	0.369873i	$0.879401\pi$
$80$	0	0
$81$	25.2664	2.80737
$82$	0.203439	0.0224661
$83$	6.17092	0.677346	0.338673	−	0.940904i	$-0.390022\pi$
$83$	6.17092	0.677346	0.338673	+	0.940904i	$0.390022\pi$
$84$	−18.9976	−2.07281
$85$	0	0
$86$	2.75840	0.297446
$87$	−19.7114	−2.11328
$88$	−1.17092	−0.124820
$89$	10.5254	1.11569	0.557846	−	0.829944i	$-0.311628\pi$
$89$	10.5254	1.11569	0.557846	+	0.829944i	$0.311628\pi$
$90$	0	0
$91$	0	0
$92$	−2.20775	−0.230174
$93$	27.4523	2.84667
$94$	5.13706	0.529848
$95$	0	0
$96$	15.1075	1.54191
$97$	−3.45473	−0.350775	−0.175387	−	0.984500i	$-0.556118\pi$
$97$	−3.45473	−0.350775	−0.175387	+	0.984500i	$0.556118\pi$
$98$	1.57673	0.159274
$99$	5.21983	0.524613
$100$	0	0
$101$	4.32304	0.430159	0.215079	−	0.976597i	$-0.430999\pi$
$101$	4.32304	0.430159	0.215079	+	0.976597i	$0.430999\pi$
$102$	−5.40581	−0.535255
$103$	4.25906	0.419658	0.209829	−	0.977738i	$-0.432709\pi$
$103$	4.25906	0.419658	0.209829	+	0.977738i	$0.432709\pi$
$104$	0	0
$105$	0	0
$106$	−0.356896	−0.0346648
$107$	11.2174	1.08443	0.542215	−	0.840240i	$-0.317586\pi$
$107$	11.2174	1.08443	0.542215	+	0.840240i	$0.317586\pi$
$108$	−26.5797	−2.55763
$109$	−2.37196	−0.227193	−0.113596	−	0.993527i	$-0.536237\pi$
$109$	−2.37196	−0.227193	−0.113596	+	0.993527i	$0.536237\pi$
$110$	0	0
$111$	−6.13706	−0.582504
$112$	9.25667	0.874673
$113$	1.72886	0.162637	0.0813186	−	0.996688i	$-0.474087\pi$
$113$	1.72886	0.162637	0.0813186	+	0.996688i	$0.474087\pi$
$114$	2.21552	0.207503
$115$	0	0
$116$	10.9390	1.01566
$117$	0	0
$118$	−2.93900	−0.270557
$119$	−12.1468	−1.11349
$120$	0	0
$121$	−10.5211	−0.956464
$122$	−1.86831	−0.169149
$123$	1.48427	0.133832
$124$	−15.2349	−1.36813
$125$	0	0
$126$	10.8998	0.971029
$127$	−3.78986	−0.336295	−0.168148	−	0.985762i	$-0.553779\pi$
$127$	−3.78986	−0.336295	−0.168148	+	0.985762i	$0.553779\pi$
$128$	−10.9215	−0.965337
$129$	20.1250	1.77191
$130$	0	0
$131$	5.48858	0.479540	0.239770	−	0.970830i	$-0.422928\pi$
$131$	5.48858	0.479540	0.239770	+	0.970830i	$0.422928\pi$
$132$	−4.04892	−0.352413
$133$	4.97823	0.431667
$134$	−6.14675	−0.530998
$135$	0	0
$136$	6.32975	0.542771
$137$	2.58211	0.220604	0.110302	−	0.993898i	$-0.464818\pi$
$137$	2.58211	0.220604	0.110302	+	0.993898i	$0.464818\pi$
$138$	1.77048	0.150713
$139$	−14.4330	−1.22419	−0.612094	−	0.790785i	$-0.709673\pi$
$139$	−14.4330	−1.22419	−0.612094	+	0.790785i	$0.709673\pi$
$140$	0	0
$141$	37.4795	3.15634
$142$	4.39373	0.368714
$143$	0	0
$144$	21.5036	1.79197
$145$	0	0
$146$	3.58450	0.296655
$147$	11.5036	0.948805
$148$	3.40581	0.279956
$149$	0.572417	0.0468942	0.0234471	−	0.999725i	$-0.492536\pi$
$149$	0.572417	0.0468942	0.0234471	+	0.999725i	$0.492536\pi$
$150$	0	0
$151$	−1.23490	−0.100495	−0.0502473	−	0.998737i	$-0.516001\pi$
$151$	−1.23490	−0.100495	−0.0502473	+	0.998737i	$0.516001\pi$
$152$	−2.59419	−0.210416
$153$	−28.2174	−2.28124
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	−11.5375	−0.920793	−0.460396	−	0.887713i	$-0.652293\pi$
$157$	−11.5375	−0.920793	−0.460396	+	0.887713i	$0.652293\pi$
$158$	−7.35019	−0.584750
$159$	−2.60388	−0.206501
$160$	0	0
$161$	3.97823	0.313528
$162$	11.2446	0.883458
$163$	−7.28382	−0.570512	−0.285256	−	0.958451i	$-0.592079\pi$
$163$	−7.28382	−0.570512	−0.285256	+	0.958451i	$0.592079\pi$
$164$	−0.823708	−0.0643208
$165$	0	0
$166$	2.74632	0.213155
$167$	−14.4480	−1.11802	−0.559011	−	0.829160i	$-0.688819\pi$
$167$	−14.4480	−1.11802	−0.559011	+	0.829160i	$0.688819\pi$
$168$	−17.8388	−1.37629
$169$	0	0
$170$	0	0
$171$	11.5646	0.884371
$172$	−11.1685	−0.851592
$173$	−1.81940	−0.138326	−0.0691631	−	0.997605i	$-0.522033\pi$
$173$	−1.81940	−0.138326	−0.0691631	+	0.997605i	$0.522033\pi$
$174$	−8.77240	−0.665034
$175$	0	0
$176$	1.97285	0.148709
$177$	−21.4426	−1.61173
$178$	4.68425	0.351100
$179$	−18.2470	−1.36384	−0.681922	−	0.731425i	$-0.738856\pi$
$179$	−18.2470	−1.36384	−0.681922	+	0.731425i	$0.738856\pi$
$180$	0	0
$181$	−11.8605	−0.881587	−0.440794	−	0.897608i	$-0.645303\pi$
$181$	−11.8605	−0.881587	−0.440794	+	0.897608i	$0.645303\pi$
$182$	0	0
$183$	−13.6310	−1.00763
$184$	−2.07308	−0.152830
$185$	0	0
$186$	12.2174	0.895826
$187$	−2.58881	−0.189313
$188$	−20.7995	−1.51696
$189$	47.8950	3.48385
$190$	0	0
$191$	8.56465	0.619716	0.309858	−	0.950783i	$-0.399718\pi$
$191$	8.56465	0.619716	0.309858	+	0.950783i	$0.399718\pi$
$192$	−11.7899	−0.850860
$193$	21.6993	1.56195	0.780976	−	0.624562i	$-0.214722\pi$
$193$	21.6993	1.56195	0.780976	+	0.624562i	$0.214722\pi$
$194$	−1.53750	−0.110386
$195$	0	0
$196$	−6.38404	−0.456003
$197$	21.8431	1.55626	0.778128	−	0.628106i	$-0.216170\pi$
$197$	21.8431	1.55626	0.778128	+	0.628106i	$0.216170\pi$
$198$	2.32304	0.165092
$199$	−3.67025	−0.260177	−0.130089	−	0.991502i	$-0.541526\pi$
$199$	−3.67025	−0.260177	−0.130089	+	0.991502i	$0.541526\pi$
$200$	0	0
$201$	−44.8461	−3.16320
$202$	1.92394	0.135368
$203$	−19.7114	−1.38347
$204$	21.8877	1.53244
$205$	0	0
$206$	1.89546	0.132063
$207$	9.24160	0.642336
$208$	0	0
$209$	1.06100	0.0733908
$210$	0	0
$211$	−15.2862	−1.05235	−0.526173	−	0.850378i	$-0.676374\pi$
$211$	−15.2862	−1.05235	−0.526173	+	0.850378i	$0.676374\pi$
$212$	1.44504	0.0992459
$213$	32.0562	2.19646
$214$	4.99223	0.341262
$215$	0	0
$216$	−24.9584	−1.69820
$217$	27.4523	1.86359
$218$	−1.05562	−0.0714958
$219$	26.1521	1.76720
$220$	0	0
$221$	0	0
$222$	−2.73125	−0.183310
$223$	−3.66786	−0.245618	−0.122809	−	0.992430i	$-0.539190\pi$
$223$	−3.66786	−0.245618	−0.122809	+	0.992430i	$0.539190\pi$
$224$	15.1075	1.00941
$225$	0	0
$226$	0.769414	0.0511806
$227$	18.5676	1.23238	0.616188	−	0.787599i	$-0.288676\pi$
$227$	18.5676	1.23238	0.616188	+	0.787599i	$0.288676\pi$
$228$	−8.97046	−0.594083
$229$	−16.0978	−1.06377	−0.531887	−	0.846815i	$-0.678517\pi$
$229$	−16.0978	−1.06377	−0.531887	+	0.846815i	$0.678517\pi$
$230$	0	0
$231$	7.29590	0.480035
$232$	10.2717	0.674372
$233$	20.3056	1.33026	0.665132	−	0.746726i	$-0.268375\pi$
$233$	20.3056	1.33026	0.665132	+	0.746726i	$0.268375\pi$
$234$	0	0
$235$	0	0
$236$	11.8998	0.774609
$237$	−53.6262	−3.48340
$238$	−5.40581	−0.350407
$239$	20.7168	1.34006	0.670028	−	0.742335i	$-0.266282\pi$
$239$	20.7168	1.34006	0.670028	+	0.742335i	$0.266282\pi$
$240$	0	0
$241$	25.8582	1.66567	0.832835	−	0.553521i	$-0.186716\pi$
$241$	25.8582	1.66567	0.832835	+	0.553521i	$0.186716\pi$
$242$	−4.68233	−0.300992
$243$	37.7875	2.42407
$244$	7.56465	0.484277
$245$	0	0
$246$	0.660563	0.0421159
$247$	0	0
$248$	−14.3056	−0.908406
$249$	20.0368	1.26978
$250$	0	0
$251$	−26.2989	−1.65997	−0.829985	−	0.557785i	$-0.811651\pi$
$251$	−26.2989	−1.65997	−0.829985	+	0.557785i	$0.811651\pi$
$252$	−44.1323	−2.78007
$253$	0.847871	0.0533052
$254$	−1.68664	−0.105829
$255$	0	0
$256$	2.40150	0.150094
$257$	6.04354	0.376986	0.188493	−	0.982075i	$-0.439640\pi$
$257$	6.04354	0.376986	0.188493	+	0.982075i	$0.439640\pi$
$258$	8.95646	0.557605
$259$	−6.13706	−0.381339
$260$	0	0
$261$	−45.7904	−2.83436
$262$	2.44265	0.150907
$263$	16.5429	1.02008	0.510039	−	0.860151i	$-0.329631\pi$
$263$	16.5429	1.02008	0.510039	+	0.860151i	$0.329631\pi$
$264$	−3.80194	−0.233993
$265$	0	0
$266$	2.21552	0.135842
$267$	34.1758	2.09153
$268$	24.8877	1.52026
$269$	−12.0954	−0.737472	−0.368736	−	0.929534i	$-0.620209\pi$
$269$	−12.0954	−0.737472	−0.368736	+	0.929534i	$0.620209\pi$
$270$	0	0
$271$	−13.2687	−0.806019	−0.403010	−	0.915196i	$-0.632036\pi$
$271$	−13.2687	−0.806019	−0.403010	+	0.915196i	$0.632036\pi$
$272$	−10.6649	−0.646653
$273$	0	0
$274$	1.14914	0.0694224
$275$	0	0
$276$	−7.16852	−0.431494
$277$	8.48858	0.510029	0.255015	−	0.966937i	$-0.417920\pi$
$277$	8.48858	0.510029	0.255015	+	0.966937i	$0.417920\pi$
$278$	−6.42327	−0.385242
$279$	63.7730	3.81799
$280$	0	0
$281$	−26.7603	−1.59639	−0.798193	−	0.602401i	$-0.794211\pi$
$281$	−26.7603	−1.59639	−0.798193	+	0.602401i	$0.794211\pi$
$282$	16.6799	0.993276
$283$	11.6649	0.693405	0.346702	−	0.937975i	$-0.387301\pi$
$283$	11.6649	0.693405	0.346702	+	0.937975i	$0.387301\pi$
$284$	−17.7899	−1.05563
$285$	0	0
$286$	0	0
$287$	1.48427	0.0876137
$288$	35.0954	2.06802
$289$	−3.00538	−0.176787
$290$	0	0
$291$	−11.2174	−0.657578
$292$	−14.5133	−0.849329
$293$	−10.6407	−0.621637	−0.310818	−	0.950469i	$-0.600603\pi$
$293$	−10.6407	−0.621637	−0.310818	+	0.950469i	$0.600603\pi$
$294$	5.11960	0.298581
$295$	0	0
$296$	3.19806	0.185884
$297$	10.2078	0.592314
$298$	0.254749	0.0147572
$299$	0	0
$300$	0	0
$301$	20.1250	1.15998
$302$	−0.549581	−0.0316249
$303$	14.0368	0.806395
$304$	4.37090	0.250688
$305$	0	0
$306$	−12.5579	−0.717890
$307$	−0.291585	−0.0166416	−0.00832082	−	0.999965i	$-0.502649\pi$
$307$	−0.291585	−0.0166416	−0.00832082	+	0.999965i	$0.502649\pi$
$308$	−4.04892	−0.230708
$309$	13.8291	0.786709
$310$	0	0
$311$	30.2707	1.71649	0.858246	−	0.513238i	$-0.171554\pi$
$311$	30.2707	1.71649	0.858246	+	0.513238i	$0.171554\pi$
$312$	0	0
$313$	−8.50902	−0.480959	−0.240479	−	0.970654i	$-0.577305\pi$
$313$	−8.50902	−0.480959	−0.240479	+	0.970654i	$0.577305\pi$
$314$	−5.13467	−0.289766
$315$	0	0
$316$	29.7603	1.67415
$317$	11.5550	0.648991	0.324496	−	0.945887i	$-0.394805\pi$
$317$	11.5550	0.648991	0.324496	+	0.945887i	$0.394805\pi$
$318$	−1.15883	−0.0649842
$319$	−4.20105	−0.235213
$320$	0	0
$321$	36.4228	2.03292
$322$	1.77048	0.0986649
$323$	−5.73556	−0.319135
$324$	−45.5284	−2.52936
$325$	0	0
$326$	−3.24160	−0.179536
$327$	−7.70171	−0.425906
$328$	−0.773463	−0.0427073
$329$	37.4795	2.06631
$330$	0	0
$331$	−27.4403	−1.50825	−0.754126	−	0.656729i	$-0.771939\pi$
$331$	−27.4403	−1.50825	−0.754126	+	0.656729i	$0.771939\pi$
$332$	−11.1196	−0.610268
$333$	−14.2567	−0.781261
$334$	−6.42998	−0.351833
$335$	0	0
$336$	30.0562	1.63970
$337$	34.7265	1.89167	0.945836	−	0.324646i	$-0.105245\pi$
$337$	34.7265	1.89167	0.945836	+	0.324646i	$0.105245\pi$
$338$	0	0
$339$	5.61356	0.304887
$340$	0	0
$341$	5.85086	0.316842
$342$	5.14675	0.278304
$343$	−11.2252	−0.606104
$344$	−10.4873	−0.565435
$345$	0	0
$346$	−0.809707	−0.0435301
$347$	−29.6262	−1.59042	−0.795210	−	0.606334i	$-0.792639\pi$
$347$	−29.6262	−1.59042	−0.795210	+	0.606334i	$0.792639\pi$
$348$	35.5187	1.90400
$349$	−15.9463	−0.853586	−0.426793	−	0.904349i	$-0.640357\pi$
$349$	−15.9463	−0.853586	−0.426793	+	0.904349i	$0.640357\pi$
$350$	0	0
$351$	0	0
$352$	3.21983	0.171618
$353$	−22.2828	−1.18599	−0.592996	−	0.805206i	$-0.702055\pi$
$353$	−22.2828	−1.18599	−0.592996	+	0.805206i	$0.702055\pi$
$354$	−9.54288	−0.507198
$355$	0	0
$356$	−18.9661	−1.00520
$357$	−39.4403	−2.08740
$358$	−8.12067	−0.429191
$359$	34.6896	1.83085	0.915424	−	0.402490i	$-0.131855\pi$
$359$	34.6896	1.83085	0.915424	+	0.402490i	$0.131855\pi$
$360$	0	0
$361$	−16.6493	−0.876281
$362$	−5.27844	−0.277429
$363$	−34.1618	−1.79303
$364$	0	0
$365$	0	0
$366$	−6.06638	−0.317095
$367$	−9.71246	−0.506986	−0.253493	−	0.967337i	$-0.581580\pi$
$367$	−9.71246	−0.506986	−0.253493	+	0.967337i	$0.581580\pi$
$368$	3.49289	0.182080
$369$	3.44803	0.179497
$370$	0	0
$371$	−2.60388	−0.135186
$372$	−49.4674	−2.56477
$373$	−15.0411	−0.778801	−0.389401	−	0.921069i	$-0.627318\pi$
$373$	−15.0411	−0.778801	−0.389401	+	0.921069i	$0.627318\pi$
$374$	−1.15213	−0.0595752
$375$	0	0
$376$	−19.5308	−1.00722
$377$	0	0
$378$	21.3153	1.09634
$379$	−13.1347	−0.674683	−0.337341	−	0.941382i	$-0.609528\pi$
$379$	−13.1347	−0.674683	−0.337341	+	0.941382i	$0.609528\pi$
$380$	0	0
$381$	−12.3056	−0.630434
$382$	3.81163	0.195020
$383$	3.02177	0.154405	0.0772026	−	0.997015i	$-0.475401\pi$
$383$	3.02177	0.154405	0.0772026	+	0.997015i	$0.475401\pi$
$384$	−35.4620	−1.80966
$385$	0	0
$386$	9.65710	0.491534
$387$	46.7512	2.37650
$388$	6.22521	0.316037
$389$	31.4282	1.59347	0.796736	−	0.604328i	$-0.206558\pi$
$389$	31.4282	1.59347	0.796736	+	0.604328i	$0.206558\pi$
$390$	0	0
$391$	−4.58343	−0.231794
$392$	−5.99462	−0.302774
$393$	17.8213	0.898966
$394$	9.72109	0.489741
$395$	0	0
$396$	−9.40581	−0.472660
$397$	−23.0489	−1.15679	−0.578396	−	0.815756i	$-0.696321\pi$
$397$	−23.0489	−1.15679	−0.578396	+	0.815756i	$0.696321\pi$
$398$	−1.63342	−0.0818757
$399$	16.1642	0.809223
$400$	0	0
$401$	−19.8103	−0.989279	−0.494640	−	0.869098i	$-0.664700\pi$
$401$	−19.8103	−0.989279	−0.494640	+	0.869098i	$0.664700\pi$
$402$	−19.9584	−0.995433
$403$	0	0
$404$	−7.78986	−0.387560
$405$	0	0
$406$	−8.77240	−0.435367
$407$	−1.30798	−0.0648341
$408$	20.5526	1.01750
$409$	−32.9705	−1.63028	−0.815142	−	0.579261i	$-0.803341\pi$
$409$	−32.9705	−1.63028	−0.815142	+	0.579261i	$0.803341\pi$
$410$	0	0
$411$	8.38404	0.413554
$412$	−7.67456	−0.378099
$413$	−21.4426	−1.05512
$414$	4.11290	0.202138
$415$	0	0
$416$	0	0
$417$	−46.8635	−2.29492
$418$	0.472189	0.0230955
$419$	21.6136	1.05589	0.527946	−	0.849278i	$-0.322962\pi$
$419$	21.6136	1.05589	0.527946	+	0.849278i	$0.322962\pi$
$420$	0	0
$421$	−4.11769	−0.200684	−0.100342	−	0.994953i	$-0.531994\pi$
$421$	−4.11769	−0.200684	−0.100342	+	0.994953i	$0.531994\pi$
$422$	−6.80300	−0.331165
$423$	87.0665	4.23332
$424$	1.35690	0.0658967
$425$	0	0
$426$	14.2664	0.691207
$427$	−13.6310	−0.659651
$428$	−20.2131	−0.977038
$429$	0	0
$430$	0	0
$431$	−18.8616	−0.908532	−0.454266	−	0.890866i	$-0.650098\pi$
$431$	−18.8616	−0.908532	−0.454266	+	0.890866i	$0.650098\pi$
$432$	42.0519	2.02322
$433$	−3.31336	−0.159230	−0.0796148	−	0.996826i	$-0.525369\pi$
$433$	−3.31336	−0.159230	−0.0796148	+	0.996826i	$0.525369\pi$
$434$	12.2174	0.586456
$435$	0	0
$436$	4.27413	0.204694
$437$	1.87848	0.0898597
$438$	11.6388	0.556123
$439$	−14.3894	−0.686770	−0.343385	−	0.939195i	$-0.611573\pi$
$439$	−14.3894	−0.686770	−0.343385	+	0.939195i	$0.611573\pi$
$440$	0	0
$441$	26.7235	1.27255
$442$	0	0
$443$	−12.4577	−0.591884	−0.295942	−	0.955206i	$-0.595633\pi$
$443$	−12.4577	−0.591884	−0.295942	+	0.955206i	$0.595633\pi$
$444$	11.0586	0.524818
$445$	0	0
$446$	−1.63235	−0.0772940
$447$	1.85862	0.0879099
$448$	−11.7899	−0.557018
$449$	−30.5478	−1.44164	−0.720819	−	0.693123i	$-0.756234\pi$
$449$	−30.5478	−1.44164	−0.720819	+	0.693123i	$0.756234\pi$
$450$	0	0
$451$	0.316339	0.0148958
$452$	−3.11529	−0.146531
$453$	−4.00969	−0.188392
$454$	8.26337	0.387819
$455$	0	0
$456$	−8.42327	−0.394456
$457$	9.79225	0.458062	0.229031	−	0.973419i	$-0.426444\pi$
$457$	9.79225	0.458062	0.229031	+	0.973419i	$0.426444\pi$
$458$	−7.16421	−0.334762
$459$	−55.1812	−2.57564
$460$	0	0
$461$	13.2620	0.617675	0.308838	−	0.951115i	$-0.400060\pi$
$461$	13.2620	0.617675	0.308838	+	0.951115i	$0.400060\pi$
$462$	3.24698	0.151063
$463$	−17.2403	−0.801224	−0.400612	−	0.916248i	$-0.631202\pi$
$463$	−17.2403	−0.801224	−0.400612	+	0.916248i	$0.631202\pi$
$464$	−17.3067	−0.803441
$465$	0	0
$466$	9.03684	0.418623
$467$	4.98254	0.230565	0.115282	−	0.993333i	$-0.463223\pi$
$467$	4.98254	0.230565	0.115282	+	0.993333i	$0.463223\pi$
$468$	0	0
$469$	−44.8461	−2.07080
$470$	0	0
$471$	−37.4620	−1.72616
$472$	11.1739	0.514320
$473$	4.28919	0.197217
$474$	−23.8659	−1.09620
$475$	0	0
$476$	21.8877	1.00322
$477$	−6.04892	−0.276961
$478$	9.21983	0.421705
$479$	5.11529	0.233724	0.116862	−	0.993148i	$-0.462717\pi$
$479$	5.11529	0.233724	0.116862	+	0.993148i	$0.462717\pi$
$480$	0	0
$481$	0	0
$482$	11.5080	0.524173
$483$	12.9172	0.587754
$484$	18.9584	0.861744
$485$	0	0
$486$	16.8170	0.762835
$487$	−6.33704	−0.287159	−0.143579	−	0.989639i	$-0.545861\pi$
$487$	−6.33704	−0.287159	−0.143579	+	0.989639i	$0.545861\pi$
$488$	7.10321	0.321547
$489$	−23.6504	−1.06951
$490$	0	0
$491$	−31.3381	−1.41427	−0.707135	−	0.707079i	$-0.750012\pi$
$491$	−31.3381	−1.41427	−0.707135	+	0.707079i	$0.750012\pi$
$492$	−2.67456	−0.120579
$493$	22.7101	1.02281
$494$	0	0
$495$	0	0
$496$	24.1032	1.08227
$497$	32.0562	1.43792
$498$	8.91723	0.399591
$499$	23.7047	1.06117	0.530584	−	0.847632i	$-0.321973\pi$
$499$	23.7047	1.06117	0.530584	+	0.847632i	$0.321973\pi$
$500$	0	0
$501$	−46.9124	−2.09589
$502$	−11.7041	−0.522380
$503$	−15.7125	−0.700584	−0.350292	−	0.936641i	$-0.613918\pi$
$503$	−15.7125	−0.700584	−0.350292	+	0.936641i	$0.613918\pi$
$504$	−41.4403	−1.84590
$505$	0	0
$506$	0.377338	0.0167747
$507$	0	0
$508$	6.82908	0.302992
$509$	−36.3551	−1.61141	−0.805706	−	0.592316i	$-0.798214\pi$
$509$	−36.3551	−1.61141	−0.805706	+	0.592316i	$0.798214\pi$
$510$	0	0
$511$	26.1521	1.15690
$512$	22.9119	1.01257
$513$	22.6155	0.998498
$514$	2.68963	0.118634
$515$	0	0
$516$	−36.2640	−1.59643
$517$	7.98792	0.351308
$518$	−2.73125	−0.120004
$519$	−5.90754	−0.259312
$520$	0	0
$521$	9.55735	0.418715	0.209358	−	0.977839i	$-0.432863\pi$
$521$	9.55735	0.418715	0.209358	+	0.977839i	$0.432863\pi$
$522$	−20.3787	−0.891950
$523$	−3.57242	−0.156211	−0.0781054	−	0.996945i	$-0.524887\pi$
$523$	−3.57242	−0.156211	−0.0781054	+	0.996945i	$0.524887\pi$
$524$	−9.89008	−0.432050
$525$	0	0
$526$	7.36227	0.321010
$527$	−31.6286	−1.37776
$528$	6.40581	0.278777
$529$	−21.4989	−0.934733
$530$	0	0
$531$	−49.8122	−2.16167
$532$	−8.97046	−0.388919
$533$	0	0
$534$	15.2097	0.658187
$535$	0	0
$536$	23.3696	1.00941
$537$	−59.2476	−2.55672
$538$	−5.38298	−0.232077
$539$	2.45175	0.105604
$540$	0	0
$541$	9.99330	0.429645	0.214823	−	0.976653i	$-0.431083\pi$
$541$	9.99330	0.429645	0.214823	+	0.976653i	$0.431083\pi$
$542$	−5.90515	−0.253648
$543$	−38.5109	−1.65266
$544$	−17.4058	−0.746268
$545$	0	0
$546$	0	0
$547$	17.9836	0.768923	0.384462	−	0.923141i	$-0.374387\pi$
$547$	17.9836	0.768923	0.384462	+	0.923141i	$0.374387\pi$
$548$	−4.65279	−0.198757
$549$	−31.6655	−1.35145
$550$	0	0
$551$	−9.30750	−0.396513
$552$	−6.73125	−0.286501
$553$	−53.6262	−2.28042
$554$	3.77777	0.160502
$555$	0	0
$556$	26.0073	1.10296
$557$	16.5670	0.701968	0.350984	−	0.936381i	$-0.385847\pi$
$557$	16.5670	0.701968	0.350984	+	0.936381i	$0.385847\pi$
$558$	28.3817	1.20149
$559$	0	0
$560$	0	0
$561$	−8.40581	−0.354894
$562$	−11.9095	−0.502370
$563$	−31.7356	−1.33749	−0.668747	−	0.743490i	$-0.733169\pi$
$563$	−31.7356	−1.33749	−0.668747	+	0.743490i	$0.733169\pi$
$564$	−67.5357	−2.84377
$565$	0	0
$566$	5.19136	0.218209
$567$	82.0393	3.44533
$568$	−16.7047	−0.700913
$569$	13.9336	0.584128	0.292064	−	0.956399i	$-0.405658\pi$
$569$	13.9336	0.584128	0.292064	+	0.956399i	$0.405658\pi$
$570$	0	0
$571$	−26.2959	−1.10045	−0.550225	−	0.835017i	$-0.685458\pi$
$571$	−26.2959	−1.10045	−0.550225	+	0.835017i	$0.685458\pi$
$572$	0	0
$573$	27.8092	1.16175
$574$	0.660563	0.0275713
$575$	0	0
$576$	−27.3884	−1.14118
$577$	−20.0339	−0.834020	−0.417010	−	0.908902i	$-0.636922\pi$
$577$	−20.0339	−0.834020	−0.417010	+	0.908902i	$0.636922\pi$
$578$	−1.33752	−0.0556334
$579$	70.4572	2.92810
$580$	0	0
$581$	20.0368	0.831268
$582$	−4.99223	−0.206935
$583$	−0.554958	−0.0229840
$584$	−13.6280	−0.563932
$585$	0	0
$586$	−4.73556	−0.195624
$587$	−20.0103	−0.825913	−0.412956	−	0.910751i	$-0.635504\pi$
$587$	−20.0103	−0.825913	−0.412956	+	0.910751i	$0.635504\pi$
$588$	−20.7289	−0.854844
$589$	12.9627	0.534118
$590$	0	0
$591$	70.9241	2.91743
$592$	−5.38835	−0.221460
$593$	−36.7023	−1.50718	−0.753591	−	0.657343i	$-0.771680\pi$
$593$	−36.7023	−1.50718	−0.753591	+	0.657343i	$0.771680\pi$
$594$	4.54288	0.186396
$595$	0	0
$596$	−1.03146	−0.0422502
$597$	−11.9172	−0.487740
$598$	0	0
$599$	13.5386	0.553171	0.276585	−	0.960989i	$-0.410797\pi$
$599$	13.5386	0.553171	0.276585	+	0.960989i	$0.410797\pi$
$600$	0	0
$601$	42.4965	1.73347	0.866734	−	0.498771i	$-0.166215\pi$
$601$	42.4965	1.73347	0.866734	+	0.498771i	$0.166215\pi$
$602$	8.95646	0.365038
$603$	−104.179	−4.24251
$604$	2.22521	0.0905425
$605$	0	0
$606$	6.24698	0.253766
$607$	12.4969	0.507235	0.253618	−	0.967305i	$-0.418380\pi$
$607$	12.4969	0.507235	0.253618	+	0.967305i	$0.418380\pi$
$608$	7.13361	0.289306
$609$	−64.0025	−2.59351
$610$	0	0
$611$	0	0
$612$	50.8461	2.05533
$613$	−30.0694	−1.21449	−0.607245	−	0.794515i	$-0.707725\pi$
$613$	−30.0694	−1.21449	−0.607245	+	0.794515i	$0.707725\pi$
$614$	−0.129768	−0.00523699
$615$	0	0
$616$	−3.80194	−0.153184
$617$	28.1196	1.13205	0.566026	−	0.824387i	$-0.308480\pi$
$617$	28.1196	1.13205	0.566026	+	0.824387i	$0.308480\pi$
$618$	6.15452	0.247571
$619$	−20.3424	−0.817631	−0.408815	−	0.912617i	$-0.634058\pi$
$619$	−20.3424	−0.817631	−0.408815	+	0.912617i	$0.634058\pi$
$620$	0	0
$621$	18.0726	0.725229
$622$	13.4717	0.540167
$623$	34.1758	1.36923
$624$	0	0
$625$	0	0
$626$	−3.78687	−0.151354
$627$	3.44504	0.137582
$628$	20.7899	0.829606
$629$	7.07069	0.281927
$630$	0	0
$631$	3.66355	0.145843	0.0729217	−	0.997338i	$-0.476768\pi$
$631$	3.66355	0.145843	0.0729217	+	0.997338i	$0.476768\pi$
$632$	27.9450	1.11159
$633$	−49.6340	−1.97277
$634$	5.14244	0.204232
$635$	0	0
$636$	4.69202	0.186051
$637$	0	0
$638$	−1.86964	−0.0740198
$639$	74.4680	2.94591
$640$	0	0
$641$	5.73556	0.226541	0.113271	−	0.993564i	$-0.463867\pi$
$641$	5.73556	0.226541	0.113271	+	0.993564i	$0.463867\pi$
$642$	16.2097	0.639745
$643$	−7.35929	−0.290222	−0.145111	−	0.989415i	$-0.546354\pi$
$643$	−7.35929	−0.290222	−0.145111	+	0.989415i	$0.546354\pi$
$644$	−7.16852	−0.282479
$645$	0	0
$646$	−2.55257	−0.100429
$647$	−49.2664	−1.93686	−0.968430	−	0.249285i	$-0.919804\pi$
$647$	−49.2664	−1.93686	−0.968430	+	0.249285i	$0.919804\pi$
$648$	−42.7512	−1.67943
$649$	−4.57002	−0.179389
$650$	0	0
$651$	89.1372	3.49356
$652$	13.1250	0.514014
$653$	−8.50796	−0.332942	−0.166471	−	0.986046i	$-0.553237\pi$
$653$	−8.50796	−0.332942	−0.166471	+	0.986046i	$0.553237\pi$
$654$	−3.42758	−0.134029
$655$	0	0
$656$	1.30319	0.0508811
$657$	60.7525	2.37018
$658$	16.6799	0.650252
$659$	−5.42221	−0.211219	−0.105610	−	0.994408i	$-0.533679\pi$
$659$	−5.42221	−0.211219	−0.105610	+	0.994408i	$0.533679\pi$
$660$	0	0
$661$	16.2784	0.633158	0.316579	−	0.948566i	$-0.397466\pi$
$661$	16.2784	0.633158	0.316579	+	0.948566i	$0.397466\pi$
$662$	−12.2121	−0.474635
$663$	0	0
$664$	−10.4413	−0.405202
$665$	0	0
$666$	−6.34481	−0.245857
$667$	−7.43786	−0.287995
$668$	26.0344	1.00730
$669$	−11.9095	−0.460446
$670$	0	0
$671$	−2.90515	−0.112152
$672$	49.0538	1.89229
$673$	22.8866	0.882215	0.441107	−	0.897454i	$-0.354586\pi$
$673$	22.8866	0.882215	0.441107	+	0.897454i	$0.354586\pi$
$674$	15.4547	0.595294
$675$	0	0
$676$	0	0
$677$	36.9711	1.42091	0.710456	−	0.703741i	$-0.248489\pi$
$677$	36.9711	1.42091	0.710456	+	0.703741i	$0.248489\pi$
$678$	2.49827	0.0959455
$679$	−11.2174	−0.430486
$680$	0	0
$681$	60.2887	2.31027
$682$	2.60388	0.0997075
$683$	−21.3177	−0.815698	−0.407849	−	0.913049i	$-0.633721\pi$
$683$	−21.3177	−0.815698	−0.407849	+	0.913049i	$0.633721\pi$
$684$	−20.8388	−0.796790
$685$	0	0
$686$	−4.99569	−0.190736
$687$	−52.2693	−1.99420
$688$	17.6698	0.673654
$689$	0	0
$690$	0	0
$691$	−20.9554	−0.797181	−0.398590	−	0.917129i	$-0.630500\pi$
$691$	−20.9554	−0.797181	−0.398590	+	0.917129i	$0.630500\pi$
$692$	3.27844	0.124628
$693$	16.9487	0.643827
$694$	−13.1849	−0.500493
$695$	0	0
$696$	33.3521	1.26421
$697$	−1.71007	−0.0647736
$698$	−7.09677	−0.268617
$699$	65.9318	2.49377
$700$	0	0
$701$	−36.9439	−1.39535	−0.697676	−	0.716413i	$-0.745782\pi$
$701$	−36.9439	−1.39535	−0.697676	+	0.716413i	$0.745782\pi$
$702$	0	0
$703$	−2.89785	−0.109295
$704$	−2.51275	−0.0947027
$705$	0	0
$706$	−9.91676	−0.373222
$707$	14.0368	0.527910
$708$	38.6383	1.45212
$709$	−6.41358	−0.240867	−0.120434	−	0.992721i	$-0.538428\pi$
$709$	−6.41358	−0.240867	−0.120434	+	0.992721i	$0.538428\pi$
$710$	0	0
$711$	−124.576	−4.67197
$712$	−17.8092	−0.667429
$713$	10.3588	0.387941
$714$	−17.5526	−0.656888
$715$	0	0
$716$	32.8799	1.22878
$717$	67.2669	2.51213
$718$	15.4383	0.576154
$719$	−13.7265	−0.511911	−0.255955	−	0.966689i	$-0.582390\pi$
$719$	−13.7265	−0.511911	−0.255955	+	0.966689i	$0.582390\pi$
$720$	0	0
$721$	13.8291	0.515022
$722$	−7.40965	−0.275759
$723$	83.9609	3.12254
$724$	21.3720	0.794283
$725$	0	0
$726$	−15.2034	−0.564253
$727$	−33.1008	−1.22764	−0.613821	−	0.789445i	$-0.710368\pi$
$727$	−33.1008	−1.22764	−0.613821	+	0.789445i	$0.710368\pi$
$728$	0	0
$729$	46.8961	1.73689
$730$	0	0
$731$	−23.1866	−0.857586
$732$	24.5623	0.907847
$733$	50.6902	1.87229	0.936143	−	0.351620i	$-0.114369\pi$
$733$	50.6902	1.87229	0.936143	+	0.351620i	$0.114369\pi$
$734$	−4.32245	−0.159545
$735$	0	0
$736$	5.70065	0.210129
$737$	−9.55794	−0.352071
$738$	1.53452	0.0564863
$739$	24.5448	0.902895	0.451447	−	0.892298i	$-0.350908\pi$
$739$	24.5448	0.902895	0.451447	+	0.892298i	$0.350908\pi$
$740$	0	0
$741$	0	0
$742$	−1.15883	−0.0425421
$743$	−8.90217	−0.326589	−0.163294	−	0.986577i	$-0.552212\pi$
$743$	−8.90217	−0.326589	−0.163294	+	0.986577i	$0.552212\pi$
$744$	−46.4499	−1.70294
$745$	0	0
$746$	−6.69394	−0.245083
$747$	46.5465	1.70305
$748$	4.66487	0.170565
$749$	36.4228	1.33086
$750$	0	0
$751$	−3.00059	−0.109493	−0.0547466	−	0.998500i	$-0.517435\pi$
$751$	−3.00059	−0.109493	−0.0547466	+	0.998500i	$0.517435\pi$
$752$	32.9071	1.20000
$753$	−85.3919	−3.11185
$754$	0	0
$755$	0	0
$756$	−86.3038	−3.13884
$757$	23.1691	0.842096	0.421048	−	0.907038i	$-0.361662\pi$
$757$	23.1691	0.842096	0.421048	+	0.907038i	$0.361662\pi$
$758$	−5.84548	−0.212317
$759$	2.75302	0.0999283
$760$	0	0
$761$	13.6431	0.494562	0.247281	−	0.968944i	$-0.420463\pi$
$761$	13.6431	0.494562	0.247281	+	0.968944i	$0.420463\pi$
$762$	−5.47650	−0.198393
$763$	−7.70171	−0.278821
$764$	−15.4330	−0.558345
$765$	0	0
$766$	1.34481	0.0485901
$767$	0	0
$768$	7.79763	0.281373
$769$	31.8853	1.14981	0.574907	−	0.818219i	$-0.305038\pi$
$769$	31.8853	1.14981	0.574907	+	0.818219i	$0.305038\pi$
$770$	0	0
$771$	19.6233	0.706714
$772$	−39.1008	−1.40727
$773$	−18.4168	−0.662407	−0.331204	−	0.943559i	$-0.607455\pi$
$773$	−18.4168	−0.662407	−0.331204	+	0.943559i	$0.607455\pi$
$774$	20.8062	0.747865
$775$	0	0
$776$	5.84548	0.209840
$777$	−19.9269	−0.714874
$778$	13.9869	0.501453
$779$	0.700856	0.0251108
$780$	0	0
$781$	6.83207	0.244471
$782$	−2.03982	−0.0729438
$783$	−89.5465	−3.20013
$784$	10.1002	0.360722
$785$	0	0
$786$	7.93123	0.282898
$787$	31.5230	1.12367	0.561837	−	0.827248i	$-0.310095\pi$
$787$	31.5230	1.12367	0.561837	+	0.827248i	$0.310095\pi$
$788$	−39.3599	−1.40214
$789$	53.7144	1.91228
$790$	0	0
$791$	5.61356	0.199595
$792$	−8.83207	−0.313834
$793$	0	0
$794$	−10.2577	−0.364033
$795$	0	0
$796$	6.61356	0.234412
$797$	−41.3967	−1.46635	−0.733173	−	0.680042i	$-0.761962\pi$
$797$	−41.3967	−1.46635	−0.733173	+	0.680042i	$0.761962\pi$
$798$	7.19375	0.254656
$799$	−43.1812	−1.52764
$800$	0	0
$801$	79.3919	2.80518
$802$	−8.81641	−0.311318
$803$	5.57374	0.196693
$804$	80.8098	2.84994
$805$	0	0
$806$	0	0
$807$	−39.2737	−1.38250
$808$	−7.31468	−0.257330
$809$	33.9748	1.19449	0.597245	−	0.802059i	$-0.296262\pi$
$809$	33.9748	1.19449	0.597245	+	0.802059i	$0.296262\pi$
$810$	0	0
$811$	−42.4174	−1.48948	−0.744739	−	0.667356i	$-0.767426\pi$
$811$	−42.4174	−1.48948	−0.744739	+	0.667356i	$0.767426\pi$
$812$	35.5187	1.24646
$813$	−43.0834	−1.51100
$814$	−0.582105	−0.0204028
$815$	0	0
$816$	−34.6286	−1.21224
$817$	9.50279	0.332461
$818$	−14.6732	−0.513038
$819$	0	0
$820$	0	0
$821$	−20.8823	−0.728798	−0.364399	−	0.931243i	$-0.618726\pi$
$821$	−20.8823	−0.728798	−0.364399	+	0.931243i	$0.618726\pi$
$822$	3.73125	0.130142
$823$	31.4196	1.09522	0.547608	−	0.836735i	$-0.315538\pi$
$823$	31.4196	1.09522	0.547608	+	0.836735i	$0.315538\pi$
$824$	−7.20642	−0.251048
$825$	0	0
$826$	−9.54288	−0.332039
$827$	30.8471	1.07266	0.536330	−	0.844008i	$-0.319810\pi$
$827$	30.8471	1.07266	0.536330	+	0.844008i	$0.319810\pi$
$828$	−16.6528	−0.578725
$829$	30.9004	1.07321	0.536607	−	0.843832i	$-0.319706\pi$
$829$	30.9004	1.07321	0.536607	+	0.843832i	$0.319706\pi$
$830$	0	0
$831$	27.5623	0.956124
$832$	0	0
$833$	−13.2537	−0.459213
$834$	−20.8562	−0.722192
$835$	0	0
$836$	−1.91185	−0.0661229
$837$	124.713	4.31070
$838$	9.61894	0.332281
$839$	−17.9782	−0.620677	−0.310339	−	0.950626i	$-0.600442\pi$
$839$	−17.9782	−0.620677	−0.310339	+	0.950626i	$0.600442\pi$
$840$	0	0
$841$	7.85325	0.270802
$842$	−1.83254	−0.0631536
$843$	−86.8902	−2.99266
$844$	27.5448	0.948131
$845$	0	0
$846$	38.7482	1.33219
$847$	−34.1618	−1.17381
$848$	−2.28621	−0.0785087
$849$	37.8756	1.29989
$850$	0	0
$851$	−2.31575	−0.0793828
$852$	−57.7633	−1.97894
$853$	4.04413	0.138468	0.0692342	−	0.997600i	$-0.477944\pi$
$853$	4.04413	0.138468	0.0692342	+	0.997600i	$0.477944\pi$
$854$	−6.06638	−0.207587
$855$	0	0
$856$	−18.9801	−0.648728
$857$	−25.7888	−0.880928	−0.440464	−	0.897770i	$-0.645186\pi$
$857$	−25.7888	−0.880928	−0.440464	+	0.897770i	$0.645186\pi$
$858$	0	0
$859$	−5.63043	−0.192108	−0.0960539	−	0.995376i	$-0.530622\pi$
$859$	−5.63043	−0.192108	−0.0960539	+	0.995376i	$0.530622\pi$
$860$	0	0
$861$	4.81940	0.164245
$862$	−8.39421	−0.285908
$863$	−20.6112	−0.701612	−0.350806	−	0.936448i	$-0.614092\pi$
$863$	−20.6112	−0.701612	−0.350806	+	0.936448i	$0.614092\pi$
$864$	68.6316	2.33489
$865$	0	0
$866$	−1.47458	−0.0501083
$867$	−9.75840	−0.331413
$868$	−49.4674	−1.67903
$869$	−11.4292	−0.387710
$870$	0	0
$871$	0	0
$872$	4.01341	0.135911
$873$	−26.0586	−0.881950
$874$	0.836001	0.0282781
$875$	0	0
$876$	−47.1245	−1.59219
$877$	3.46383	0.116965	0.0584826	−	0.998288i	$-0.481374\pi$
$877$	3.46383	0.116965	0.0584826	+	0.998288i	$0.481374\pi$
$878$	−6.40389	−0.216121
$879$	−34.5502	−1.16535
$880$	0	0
$881$	−28.2040	−0.950218	−0.475109	−	0.879927i	$-0.657591\pi$
$881$	−28.2040	−0.950218	−0.475109	+	0.879927i	$0.657591\pi$
$882$	11.8931	0.400460
$883$	−36.4577	−1.22690	−0.613450	−	0.789734i	$-0.710219\pi$
$883$	−36.4577	−1.22690	−0.613450	+	0.789734i	$0.710219\pi$
$884$	0	0
$885$	0	0
$886$	−5.54420	−0.186261
$887$	−6.17331	−0.207279	−0.103640	−	0.994615i	$-0.533049\pi$
$887$	−6.17331	−0.207279	−0.103640	+	0.994615i	$0.533049\pi$
$888$	10.3840	0.348466
$889$	−12.3056	−0.412716
$890$	0	0
$891$	17.4849	0.585765
$892$	6.60925	0.221294
$893$	17.6974	0.592221
$894$	0.827166	0.0276646
$895$	0	0
$896$	−35.4620	−1.18470
$897$	0	0
$898$	−13.5950	−0.453672
$899$	−51.3260	−1.71182
$900$	0	0
$901$	3.00000	0.0999445
$902$	0.140784	0.00468760
$903$	65.3454	2.17456
$904$	−2.92526	−0.0972928
$905$	0	0
$906$	−1.78448	−0.0592854
$907$	15.1860	0.504242	0.252121	−	0.967696i	$-0.418872\pi$
$907$	15.1860	0.504242	0.252121	+	0.967696i	$0.418872\pi$
$908$	−33.4577	−1.11033
$909$	32.6082	1.08155
$910$	0	0
$911$	37.0810	1.22855	0.614274	−	0.789093i	$-0.289449\pi$
$911$	37.0810	1.22855	0.614274	+	0.789093i	$0.289449\pi$
$912$	14.1922	0.469951
$913$	4.27041	0.141330
$914$	4.35796	0.144149
$915$	0	0
$916$	29.0073	0.958428
$917$	17.8213	0.588512
$918$	−24.5579	−0.810533
$919$	−30.5590	−1.00805	−0.504024	−	0.863689i	$-0.668148\pi$
$919$	−30.5590	−1.00805	−0.504024	+	0.863689i	$0.668148\pi$
$920$	0	0
$921$	−0.946771	−0.0311972
$922$	5.90217	0.194377
$923$	0	0
$924$	−13.1468	−0.432496
$925$	0	0
$926$	−7.67264	−0.252139
$927$	32.1256	1.05514
$928$	−28.2457	−0.927209
$929$	27.2489	0.894007	0.447004	−	0.894532i	$-0.352491\pi$
$929$	27.2489	0.894007	0.447004	+	0.894532i	$0.352491\pi$
$930$	0	0
$931$	5.43190	0.178023
$932$	−36.5894	−1.19853
$933$	98.2882	3.21781
$934$	2.21744	0.0725568
$935$	0	0
$936$	0	0
$937$	−57.8286	−1.88918	−0.944589	−	0.328255i	$-0.893540\pi$
$937$	−57.8286	−1.88918	−0.944589	+	0.328255i	$0.893540\pi$
$938$	−19.9584	−0.651664
$939$	−27.6286	−0.901626
$940$	0	0
$941$	40.1463	1.30873	0.654366	−	0.756178i	$-0.272936\pi$
$941$	40.1463	1.30873	0.654366	+	0.756178i	$0.272936\pi$
$942$	−16.6722	−0.543209
$943$	0.560072	0.0182385
$944$	−18.8267	−0.612757
$945$	0	0
$946$	1.90887	0.0620627
$947$	−0.987918	−0.0321030	−0.0160515	−	0.999871i	$-0.505110\pi$
$947$	−0.987918	−0.0321030	−0.0160515	+	0.999871i	$0.505110\pi$
$948$	96.6311	3.13843
$949$	0	0
$950$	0	0
$951$	37.5187	1.21663
$952$	20.5526	0.666112
$953$	43.3351	1.40376	0.701881	−	0.712294i	$-0.252344\pi$
$953$	43.3351	1.40376	0.701881	+	0.712294i	$0.252344\pi$
$954$	−2.69202	−0.0871574
$955$	0	0
$956$	−37.3303	−1.20735
$957$	−13.6407	−0.440942
$958$	2.27652	0.0735510
$959$	8.38404	0.270735
$960$	0	0
$961$	40.4825	1.30589
$962$	0	0
$963$	84.6118	2.72658
$964$	−46.5948	−1.50072
$965$	0	0
$966$	5.74871	0.184962
$967$	4.89977	0.157566	0.0787830	−	0.996892i	$-0.474897\pi$
$967$	4.89977	0.157566	0.0787830	+	0.996892i	$0.474897\pi$
$968$	17.8019	0.572176
$969$	−18.6233	−0.598265
$970$	0	0
$971$	−37.5153	−1.20392	−0.601961	−	0.798526i	$-0.705614\pi$
$971$	−37.5153	−1.20392	−0.601961	+	0.798526i	$0.705614\pi$
$972$	−68.0907	−2.18401
$973$	−46.8635	−1.50238
$974$	−2.82025	−0.0903666
$975$	0	0
$976$	−11.9681	−0.383088
$977$	−59.7663	−1.91209	−0.956046	−	0.293215i	$-0.905275\pi$
$977$	−59.7663	−1.91209	−0.956046	+	0.293215i	$0.905275\pi$
$978$	−10.5254	−0.336566
$979$	7.28382	0.232792
$980$	0	0
$981$	−17.8914	−0.571229
$982$	−13.9468	−0.445059
$983$	39.2452	1.25173	0.625863	−	0.779933i	$-0.284747\pi$
$983$	39.2452	1.25173	0.625863	+	0.779933i	$0.284747\pi$
$984$	−2.51142	−0.0800611
$985$	0	0
$986$	10.1069	0.321870
$987$	121.695	3.87360
$988$	0	0
$989$	7.59392	0.241473
$990$	0	0
$991$	−12.2577	−0.389380	−0.194690	−	0.980865i	$-0.562370\pi$
$991$	−12.2577	−0.389380	−0.194690	+	0.980865i	$0.562370\pi$
$992$	39.3381	1.24899
$993$	−89.0980	−2.82744
$994$	14.2664	0.452501
$995$	0	0
$996$	−36.1051	−1.14403
$997$	24.5687	0.778098	0.389049	−	0.921217i	$-0.372804\pi$
$997$	24.5687	0.778098	0.389049	+	0.921217i	$0.372804\pi$
$998$	10.5496	0.333941
$999$	−27.8799	−0.882082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bf.1.2		3
5.4	even	2		845.2.a.h.1.2	✓	3
13.12	even	2		4225.2.a.bd.1.2		3
15.14	odd	2		7605.2.a.by.1.2		3
65.4	even	6		845.2.e.j.146.2		6
65.9	even	6		845.2.e.l.146.2		6
65.19	odd	12		845.2.m.i.361.4		12
65.24	odd	12		845.2.m.i.316.4		12
65.29	even	6		845.2.e.l.191.2		6
65.34	odd	4		845.2.c.f.506.3		6
65.44	odd	4		845.2.c.f.506.4		6
65.49	even	6		845.2.e.j.191.2		6
65.54	odd	12		845.2.m.i.316.3		12
65.59	odd	12		845.2.m.i.361.3		12
65.64	even	2		845.2.a.j.1.2	yes	3
195.194	odd	2		7605.2.a.br.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.h.1.2	✓	3	5.4	even	2
845.2.a.j.1.2	yes	3	65.64	even	2
845.2.c.f.506.3		6	65.34	odd	4
845.2.c.f.506.4		6	65.44	odd	4
845.2.e.j.146.2		6	65.4	even	6
845.2.e.j.191.2		6	65.49	even	6
845.2.e.l.146.2		6	65.9	even	6
845.2.e.l.191.2		6	65.29	even	6
845.2.m.i.316.3		12	65.54	odd	12
845.2.m.i.316.4		12	65.24	odd	12
845.2.m.i.361.3		12	65.59	odd	12
845.2.m.i.361.4		12	65.19	odd	12
4225.2.a.bd.1.2		3	13.12	even	2
4225.2.a.bf.1.2		3	1.1	even	1	trivial
7605.2.a.br.1.2		3	195.194	odd	2
7605.2.a.by.1.2		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.445042 q^{2} +3.24698 q^{3} -1.80194 q^{4} +1.44504 q^{6} +3.24698 q^{7} -1.69202 q^{8} +7.54288 q^{9} +0.692021 q^{11} -5.85086 q^{12} +1.44504 q^{14} +2.85086 q^{16} -3.74094 q^{17} +3.35690 q^{18} +1.53319 q^{19} +10.5429 q^{21} +0.307979 q^{22} +1.22521 q^{23} -5.49396 q^{24} +14.7506 q^{27} -5.85086 q^{28} -6.07069 q^{29} +8.45473 q^{31} +4.65279 q^{32} +2.24698 q^{33} -1.66487 q^{34} -13.5918 q^{36} -1.89008 q^{37} +0.682333 q^{38} +0.457123 q^{41} +4.69202 q^{42} +6.19806 q^{43} -1.24698 q^{44} +0.545269 q^{46} +11.5429 q^{47} +9.25667 q^{48} +3.54288 q^{49} -12.1468 q^{51} -0.801938 q^{53} +6.56465 q^{54} -5.49396 q^{56} +4.97823 q^{57} -2.70171 q^{58} -6.60388 q^{59} -4.19806 q^{61} +3.76271 q^{62} +24.4916 q^{63} -3.63102 q^{64} +1.00000 q^{66} -13.8116 q^{67} +6.74094 q^{68} +3.97823 q^{69} +9.87263 q^{71} -12.7627 q^{72} +8.05429 q^{73} -0.841166 q^{74} -2.76271 q^{76} +2.24698 q^{77} -16.5157 q^{79} +25.2664 q^{81} +0.203439 q^{82} +6.17092 q^{83} -18.9976 q^{84} +2.75840 q^{86} -19.7114 q^{87} -1.17092 q^{88} +10.5254 q^{89} -2.20775 q^{92} +27.4523 q^{93} +5.13706 q^{94} +15.1075 q^{96} -3.45473 q^{97} +1.57673 q^{98} +5.21983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{3} - q^{4} + 4 q^{6} + 5 q^{7} + 4 q^{9} - 3 q^{11} - 4 q^{12} + 4 q^{14} - 5 q^{16} + 3 q^{17} + 6 q^{18} + 8 q^{19} + 13 q^{21} + 6 q^{22} + 2 q^{23} - 7 q^{24} + 8 q^{27} - 4 q^{28}+ \cdots + 17 q^{99}+O(q^{100}) \) 3 * q + q^2 + 5 * q^3 - q^4 + 4 * q^6 + 5 * q^7 + 4 * q^9 - 3 * q^11 - 4 * q^12 + 4 * q^14 - 5 * q^16 + 3 * q^17 + 6 * q^18 + 8 * q^19 + 13 * q^21 + 6 * q^22 + 2 * q^23 - 7 * q^24 + 8 * q^27 - 4 * q^28 - 6 * q^29 + 3 * q^31 - 4 * q^32 + 2 * q^33 - 6 * q^34 - 13 * q^36 - 5 * q^37 + 19 * q^38 + 20 * q^41 + 9 * q^42 + 23 * q^43 + q^44 + 24 * q^46 + 16 * q^47 + q^48 - 8 * q^49 - 9 * q^51 + 2 * q^53 - 2 * q^54 - 7 * q^56 + 18 * q^57 + 19 * q^58 - 11 * q^59 - 17 * q^61 - 6 * q^62 + 23 * q^63 + 4 * q^64 + 3 * q^66 - 15 * q^67 + 6 * q^68 + 15 * q^69 + 13 * q^71 - 21 * q^72 + 12 * q^73 - 11 * q^74 + 9 * q^76 + 2 * q^77 - 37 * q^79 + 27 * q^81 + 2 * q^82 + 29 * q^83 - 16 * q^84 + 10 * q^86 - 10 * q^87 - 14 * q^88 - 3 * q^89 + 11 * q^92 + 19 * q^93 + 10 * q^94 + 5 * q^96 + 12 * q^97 + 2 * q^98 + 17 * q^99

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