LMFDB - Embedded newform 5225.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.71457$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.65432 q^{2} -0.121872 q^{3} +5.04540 q^{4} +0.323487 q^{6} +3.36889 q^{7} -8.08346 q^{8} -2.98515 q^{9} +1.00000 q^{11} -0.614893 q^{12} +2.15993 q^{13} -8.94209 q^{14} +11.3653 q^{16} -3.67288 q^{17} +7.92353 q^{18} -1.00000 q^{19} -0.410573 q^{21} -2.65432 q^{22} -3.15468 q^{23} +0.985147 q^{24} -5.73313 q^{26} +0.729422 q^{27} +16.9974 q^{28} +7.17849 q^{29} +4.65295 q^{31} -14.0001 q^{32} -0.121872 q^{33} +9.74900 q^{34} -15.0613 q^{36} -2.27446 q^{37} +2.65432 q^{38} -0.263235 q^{39} -11.3852 q^{41} +1.08979 q^{42} -9.38838 q^{43} +5.04540 q^{44} +8.37352 q^{46} +5.77094 q^{47} -1.38511 q^{48} +4.34940 q^{49} +0.447622 q^{51} +10.8977 q^{52} +5.65820 q^{53} -1.93612 q^{54} -27.2322 q^{56} +0.121872 q^{57} -19.0540 q^{58} -13.7944 q^{59} +6.98152 q^{61} -12.3504 q^{62} -10.0566 q^{63} +14.4301 q^{64} +0.323487 q^{66} +4.81332 q^{67} -18.5312 q^{68} +0.384467 q^{69} +15.2629 q^{71} +24.1303 q^{72} +8.08806 q^{73} +6.03713 q^{74} -5.04540 q^{76} +3.36889 q^{77} +0.698709 q^{78} +13.4291 q^{79} +8.86655 q^{81} +30.2199 q^{82} -9.96666 q^{83} -2.07150 q^{84} +24.9197 q^{86} -0.874858 q^{87} -8.08346 q^{88} -4.61626 q^{89} +7.27655 q^{91} -15.9166 q^{92} -0.567064 q^{93} -15.3179 q^{94} +1.70622 q^{96} +4.09907 q^{97} -11.5447 q^{98} -2.98515 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * 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 $

$2$	−2.65432	−1.87689	−0.938443	−	0.345435i	$-0.887732\pi$
$2$	−2.65432	−1.87689	−0.938443	+	0.345435i	$0.887732\pi$
$3$	−0.121872	−0.0703629	−0.0351814	−	0.999381i	$-0.511201\pi$
$3$	−0.121872	−0.0703629	−0.0351814	+	0.999381i	$0.511201\pi$
$4$	5.04540	2.52270
$5$	0	0
$5$	0	0
$6$	0.323487	0.132063
$7$	3.36889	1.27332	0.636660	−	0.771145i	$-0.280316\pi$
$7$	3.36889	1.27332	0.636660	+	0.771145i	$0.280316\pi$
$8$	−8.08346	−2.85793
$9$	−2.98515	−0.995049
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.614893	−0.177504
$13$	2.15993	0.599056	0.299528	−	0.954087i	$-0.403171\pi$
$13$	2.15993	0.599056	0.299528	+	0.954087i	$0.403171\pi$
$14$	−8.94209	−2.38987
$15$	0	0
$16$	11.3653	2.84131
$17$	−3.67288	−0.890805	−0.445402	−	0.895330i	$-0.646939\pi$
$17$	−3.67288	−0.890805	−0.445402	+	0.895330i	$0.646939\pi$
$18$	7.92353	1.86759
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.410573	−0.0895944
$22$	−2.65432	−0.565902
$23$	−3.15468	−0.657797	−0.328898	−	0.944365i	$-0.606677\pi$
$23$	−3.15468	−0.657797	−0.328898	+	0.944365i	$0.606677\pi$
$24$	0.985147	0.201092
$25$	0	0
$26$	−5.73313	−1.12436
$27$	0.729422	0.140377
$28$	16.9974	3.21220
$29$	7.17849	1.33301	0.666506	−	0.745499i	$-0.267789\pi$
$29$	7.17849	1.33301	0.666506	+	0.745499i	$0.267789\pi$
$30$	0	0
$31$	4.65295	0.835694	0.417847	−	0.908517i	$-0.362785\pi$
$31$	4.65295	0.835694	0.417847	+	0.908517i	$0.362785\pi$
$32$	−14.0001	−2.47489
$33$	−0.121872	−0.0212152
$34$	9.74900	1.67194
$35$	0	0
$36$	−15.0613	−2.51021
$37$	−2.27446	−0.373918	−0.186959	−	0.982368i	$-0.559863\pi$
$37$	−2.27446	−0.373918	−0.186959	+	0.982368i	$0.559863\pi$
$38$	2.65432	0.430587
$39$	−0.263235	−0.0421513
$40$	0	0
$41$	−11.3852	−1.77807	−0.889034	−	0.457841i	$-0.848623\pi$
$41$	−11.3852	−1.77807	−0.889034	+	0.457841i	$0.848623\pi$
$42$	1.08979	0.168158
$43$	−9.38838	−1.43171	−0.715857	−	0.698247i	$-0.753964\pi$
$43$	−9.38838	−1.43171	−0.715857	+	0.698247i	$0.753964\pi$
$44$	5.04540	0.760623
$45$	0	0
$46$	8.37352	1.23461
$47$	5.77094	0.841778	0.420889	−	0.907112i	$-0.361718\pi$
$47$	5.77094	0.841778	0.420889	+	0.907112i	$0.361718\pi$
$48$	−1.38511	−0.199923
$49$	4.34940	0.621342
$50$	0	0
$51$	0.447622	0.0626796
$52$	10.8977	1.51124
$53$	5.65820	0.777213	0.388607	−	0.921404i	$-0.372957\pi$
$53$	5.65820	0.777213	0.388607	+	0.921404i	$0.372957\pi$
$54$	−1.93612	−0.263472
$55$	0	0
$56$	−27.2322	−3.63906
$57$	0.121872	0.0161423
$58$	−19.0540	−2.50191
$59$	−13.7944	−1.79588	−0.897939	−	0.440121i	$-0.854936\pi$
$59$	−13.7944	−1.79588	−0.897939	+	0.440121i	$0.854936\pi$
$60$	0	0
$61$	6.98152	0.893892	0.446946	−	0.894561i	$-0.352512\pi$
$61$	6.98152	0.893892	0.446946	+	0.894561i	$0.352512\pi$
$62$	−12.3504	−1.56850
$63$	−10.0566	−1.26702
$64$	14.4301	1.80377
$65$	0	0
$66$	0.323487	0.0398185
$67$	4.81332	0.588041	0.294020	−	0.955799i	$-0.405007\pi$
$67$	4.81332	0.588041	0.294020	+	0.955799i	$0.405007\pi$
$68$	−18.5312	−2.24723
$69$	0.384467	0.0462844
$70$	0	0
$71$	15.2629	1.81137	0.905685	−	0.423951i	$-0.139357\pi$
$71$	15.2629	1.81137	0.905685	+	0.423951i	$0.139357\pi$
$72$	24.1303	2.84378
$73$	8.08806	0.946636	0.473318	−	0.880892i	$-0.343056\pi$
$73$	8.08806	0.946636	0.473318	+	0.880892i	$0.343056\pi$
$74$	6.03713	0.701802
$75$	0	0
$76$	−5.04540	−0.578747
$77$	3.36889	0.383920
$78$	0.698709	0.0791132
$79$	13.4291	1.51090	0.755448	−	0.655209i	$-0.227419\pi$
$79$	13.4291	1.51090	0.755448	+	0.655209i	$0.227419\pi$
$80$	0	0
$81$	8.86655	0.985172
$82$	30.2199	3.33723
$83$	−9.96666	−1.09398	−0.546992	−	0.837138i	$-0.684227\pi$
$83$	−9.96666	−1.09398	−0.546992	+	0.837138i	$0.684227\pi$
$84$	−2.07150	−0.226020
$85$	0	0
$86$	24.9197	2.68716
$87$	−0.874858	−0.0937946
$88$	−8.08346	−0.861699
$89$	−4.61626	−0.489323	−0.244661	−	0.969609i	$-0.578677\pi$
$89$	−4.61626	−0.489323	−0.244661	+	0.969609i	$0.578677\pi$
$90$	0	0
$91$	7.27655	0.762790
$92$	−15.9166	−1.65942
$93$	−0.567064	−0.0588018
$94$	−15.3179	−1.57992
$95$	0	0
$96$	1.70622	0.174140
$97$	4.09907	0.416197	0.208099	−	0.978108i	$-0.433273\pi$
$97$	4.09907	0.416197	0.208099	+	0.978108i	$0.433273\pi$
$98$	−11.5447	−1.16619
$99$	−2.98515	−0.300019
$100$	0	0
$101$	−10.8730	−1.08190	−0.540950	−	0.841055i	$-0.681935\pi$
$101$	−10.8730	−1.08190	−0.540950	+	0.841055i	$0.681935\pi$
$102$	−1.18813	−0.117642
$103$	−8.85864	−0.872867	−0.436434	−	0.899736i	$-0.643759\pi$
$103$	−8.85864	−0.872867	−0.436434	+	0.899736i	$0.643759\pi$
$104$	−17.4597	−1.71206
$105$	0	0
$106$	−15.0186	−1.45874
$107$	18.1669	1.75626	0.878131	−	0.478421i	$-0.158791\pi$
$107$	18.1669	1.75626	0.878131	+	0.478421i	$0.158791\pi$
$108$	3.68023	0.354130
$109$	−0.211303	−0.0202392	−0.0101196	−	0.999949i	$-0.503221\pi$
$109$	−0.211303	−0.0202392	−0.0101196	+	0.999949i	$0.503221\pi$
$110$	0	0
$111$	0.277193	0.0263100
$112$	38.2883	3.61790
$113$	0.198345	0.0186587	0.00932935	−	0.999956i	$-0.497030\pi$
$113$	0.198345	0.0186587	0.00932935	+	0.999956i	$0.497030\pi$
$114$	−0.323487	−0.0302973
$115$	0	0
$116$	36.2184	3.36279
$117$	−6.44770	−0.596090
$118$	36.6147	3.37066
$119$	−12.3735	−1.13428
$120$	0	0
$121$	1.00000	0.0909091
$122$	−18.5312	−1.67773
$123$	1.38754	0.125110
$124$	23.4760	2.10821
$125$	0	0
$126$	26.6935	2.37804
$127$	3.60331	0.319742	0.159871	−	0.987138i	$-0.448892\pi$
$127$	3.60331	0.319742	0.159871	+	0.987138i	$0.448892\pi$
$128$	−10.3020	−0.910578
$129$	1.14418	0.100739
$130$	0	0
$131$	16.8741	1.47430	0.737150	−	0.675729i	$-0.236171\pi$
$131$	16.8741	1.47430	0.737150	+	0.675729i	$0.236171\pi$
$132$	−0.614893	−0.0535196
$133$	−3.36889	−0.292119
$134$	−12.7761	−1.10369
$135$	0	0
$136$	29.6896	2.54586
$137$	6.10839	0.521875	0.260937	−	0.965356i	$-0.415968\pi$
$137$	6.10839	0.521875	0.260937	+	0.965356i	$0.415968\pi$
$138$	−1.02050	−0.0868706
$139$	11.3347	0.961398	0.480699	−	0.876886i	$-0.340383\pi$
$139$	11.3347	0.961398	0.480699	+	0.876886i	$0.340383\pi$
$140$	0	0
$141$	−0.703316	−0.0592299
$142$	−40.5125	−3.39973
$143$	2.15993	0.180622
$144$	−33.9270	−2.82725
$145$	0	0
$146$	−21.4683	−1.77673
$147$	−0.530070	−0.0437194
$148$	−11.4755	−0.943284
$149$	6.75698	0.553554	0.276777	−	0.960934i	$-0.410734\pi$
$149$	6.75698	0.553554	0.276777	+	0.960934i	$0.410734\pi$
$150$	0	0
$151$	6.46231	0.525895	0.262948	−	0.964810i	$-0.415305\pi$
$151$	6.46231	0.525895	0.262948	+	0.964810i	$0.415305\pi$
$152$	8.08346	0.655655
$153$	10.9641	0.886395
$154$	−8.94209	−0.720574
$155$	0	0
$156$	−1.32813	−0.106335
$157$	−0.248382	−0.0198230	−0.00991152	−	0.999951i	$-0.503155\pi$
$157$	−0.248382	−0.0198230	−0.00991152	+	0.999951i	$0.503155\pi$
$158$	−35.6452	−2.83578
$159$	−0.689576	−0.0546869
$160$	0	0
$161$	−10.6278	−0.837585
$162$	−23.5346	−1.84905
$163$	16.9710	1.32927	0.664637	−	0.747167i	$-0.268586\pi$
$163$	16.9710	1.32927	0.664637	+	0.747167i	$0.268586\pi$
$164$	−57.4428	−4.48553
$165$	0	0
$166$	26.4547	2.05328
$167$	0.865538	0.0669773	0.0334887	−	0.999439i	$-0.489338\pi$
$167$	0.865538	0.0669773	0.0334887	+	0.999439i	$0.489338\pi$
$168$	3.31885	0.256055
$169$	−8.33471	−0.641132
$170$	0	0
$171$	2.98515	0.228280
$172$	−47.3681	−3.61178
$173$	4.14483	0.315125	0.157563	−	0.987509i	$-0.449636\pi$
$173$	4.14483	0.315125	0.157563	+	0.987509i	$0.449636\pi$
$174$	2.32215	0.176042
$175$	0	0
$176$	11.3653	0.856688
$177$	1.68115	0.126363
$178$	12.2530	0.918403
$179$	2.04584	0.152913	0.0764567	−	0.997073i	$-0.475639\pi$
$179$	2.04584	0.152913	0.0764567	+	0.997073i	$0.475639\pi$
$180$	0	0
$181$	5.27519	0.392101	0.196051	−	0.980594i	$-0.437188\pi$
$181$	5.27519	0.392101	0.196051	+	0.980594i	$0.437188\pi$
$182$	−19.3143	−1.43167
$183$	−0.850852	−0.0628968
$184$	25.5007	1.87994
$185$	0	0
$186$	1.50517	0.110364
$187$	−3.67288	−0.268588
$188$	29.1167	2.12355
$189$	2.45734	0.178745
$190$	0	0
$191$	−16.1899	−1.17146	−0.585729	−	0.810507i	$-0.699192\pi$
$191$	−16.1899	−1.17146	−0.585729	+	0.810507i	$0.699192\pi$
$192$	−1.75863	−0.126918
$193$	20.0620	1.44409	0.722045	−	0.691846i	$-0.243202\pi$
$193$	20.0620	1.44409	0.722045	+	0.691846i	$0.243202\pi$
$194$	−10.8802	−0.781155
$195$	0	0
$196$	21.9444	1.56746
$197$	23.8398	1.69851	0.849257	−	0.527979i	$-0.177050\pi$
$197$	23.8398	1.69851	0.849257	+	0.527979i	$0.177050\pi$
$198$	7.92353	0.563101
$199$	−1.90194	−0.134825	−0.0674125	−	0.997725i	$-0.521474\pi$
$199$	−1.90194	−0.134825	−0.0674125	+	0.997725i	$0.521474\pi$
$200$	0	0
$201$	−0.586609	−0.0413762
$202$	28.8603	2.03060
$203$	24.1835	1.69735
$204$	2.25843	0.158122
$205$	0	0
$206$	23.5136	1.63827
$207$	9.41719	0.654540
$208$	24.5481	1.70211
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	12.3048	0.847100	0.423550	−	0.905873i	$-0.360784\pi$
$211$	12.3048	0.847100	0.423550	+	0.905873i	$0.360784\pi$
$212$	28.5479	1.96068
$213$	−1.86012	−0.127453
$214$	−48.2207	−3.29630
$215$	0	0
$216$	−5.89625	−0.401189
$217$	15.6753	1.06411
$218$	0.560865	0.0379866
$219$	−0.985709	−0.0666080
$220$	0	0
$221$	−7.93316	−0.533642
$222$	−0.735757	−0.0493808
$223$	−22.8473	−1.52997	−0.764985	−	0.644048i	$-0.777254\pi$
$223$	−22.8473	−1.52997	−0.764985	+	0.644048i	$0.777254\pi$
$224$	−47.1647	−3.15132
$225$	0	0
$226$	−0.526470	−0.0350202
$227$	−4.00654	−0.265923	−0.132962	−	0.991121i	$-0.542449\pi$
$227$	−4.00654	−0.265923	−0.132962	+	0.991121i	$0.542449\pi$
$228$	0.614893	0.0407223
$229$	−24.6024	−1.62577	−0.812887	−	0.582422i	$-0.802105\pi$
$229$	−24.6024	−1.62577	−0.812887	+	0.582422i	$0.802105\pi$
$230$	0	0
$231$	−0.410573	−0.0270137
$232$	−58.0270	−3.80966
$233$	−16.2994	−1.06781	−0.533903	−	0.845545i	$-0.679275\pi$
$233$	−16.2994	−1.06781	−0.533903	+	0.845545i	$0.679275\pi$
$234$	17.1143	1.11879
$235$	0	0
$236$	−69.5982	−4.53046
$237$	−1.63664	−0.106311
$238$	32.8433	2.12891
$239$	−15.4182	−0.997324	−0.498662	−	0.866797i	$-0.666175\pi$
$239$	−15.4182	−0.997324	−0.498662	+	0.866797i	$0.666175\pi$
$240$	0	0
$241$	24.1441	1.55526	0.777629	−	0.628723i	$-0.216422\pi$
$241$	24.1441	1.55526	0.777629	+	0.628723i	$0.216422\pi$
$242$	−2.65432	−0.170626
$243$	−3.26885	−0.209697
$244$	35.2245	2.25502
$245$	0	0
$246$	−3.68296	−0.234817
$247$	−2.15993	−0.137433
$248$	−37.6119	−2.38836
$249$	1.21466	0.0769758
$250$	0	0
$251$	−19.9099	−1.25670	−0.628350	−	0.777931i	$-0.716269\pi$
$251$	−19.9099	−1.25670	−0.628350	+	0.777931i	$0.716269\pi$
$252$	−50.7397	−3.19630
$253$	−3.15468	−0.198333
$254$	−9.56433	−0.600119
$255$	0	0
$256$	−1.51547	−0.0947166
$257$	22.3126	1.39182	0.695911	−	0.718128i	$-0.255001\pi$
$257$	22.3126	1.39182	0.695911	+	0.718128i	$0.255001\pi$
$258$	−3.03702	−0.189076
$259$	−7.66239	−0.476118
$260$	0	0
$261$	−21.4289	−1.32641
$262$	−44.7893	−2.76709
$263$	−13.7963	−0.850716	−0.425358	−	0.905025i	$-0.639852\pi$
$263$	−13.7963	−0.850716	−0.425358	+	0.905025i	$0.639852\pi$
$264$	0.985147	0.0606316
$265$	0	0
$266$	8.94209	0.548275
$267$	0.562593	0.0344301
$268$	24.2851	1.48345
$269$	−11.2028	−0.683048	−0.341524	−	0.939873i	$-0.610943\pi$
$269$	−11.2028	−0.683048	−0.341524	+	0.939873i	$0.610943\pi$
$270$	0	0
$271$	17.1234	1.04017	0.520085	−	0.854115i	$-0.325900\pi$
$271$	17.1234	1.04017	0.520085	+	0.854115i	$0.325900\pi$
$272$	−41.7433	−2.53106
$273$	−0.886808	−0.0536721
$274$	−16.2136	−0.979499
$275$	0	0
$276$	1.93979	0.116762
$277$	13.7127	0.823919	0.411960	−	0.911202i	$-0.364844\pi$
$277$	13.7127	0.823919	0.411960	+	0.911202i	$0.364844\pi$
$278$	−30.0859	−1.80443
$279$	−13.8897	−0.831557
$280$	0	0
$281$	25.8599	1.54267	0.771337	−	0.636427i	$-0.219588\pi$
$281$	25.8599	1.54267	0.771337	+	0.636427i	$0.219588\pi$
$282$	1.86682	0.111168
$283$	−28.8711	−1.71621	−0.858103	−	0.513477i	$-0.828357\pi$
$283$	−28.8711	−1.71621	−0.858103	+	0.513477i	$0.828357\pi$
$284$	77.0073	4.56954
$285$	0	0
$286$	−5.73313	−0.339007
$287$	−38.3554	−2.26405
$288$	41.7923	2.46264
$289$	−3.50993	−0.206467
$290$	0	0
$291$	−0.499562	−0.0292848
$292$	40.8075	2.38808
$293$	−6.37971	−0.372707	−0.186353	−	0.982483i	$-0.559667\pi$
$293$	−6.37971	−0.372707	−0.186353	+	0.982483i	$0.559667\pi$
$294$	1.40697	0.0820563
$295$	0	0
$296$	18.3855	1.06863
$297$	0.729422	0.0423254
$298$	−17.9352	−1.03896
$299$	−6.81389	−0.394057
$300$	0	0
$301$	−31.6284	−1.82303
$302$	−17.1530	−0.987045
$303$	1.32511	0.0761256
$304$	−11.3653	−0.651842
$305$	0	0
$306$	−29.1022	−1.66366
$307$	−10.4299	−0.595264	−0.297632	−	0.954681i	$-0.596197\pi$
$307$	−10.4299	−0.595264	−0.297632	+	0.954681i	$0.596197\pi$
$308$	16.9974	0.968515
$309$	1.07962	0.0614174
$310$	0	0
$311$	−8.26224	−0.468508	−0.234254	−	0.972175i	$-0.575265\pi$
$311$	−8.26224	−0.468508	−0.234254	+	0.972175i	$0.575265\pi$
$312$	2.12785	0.120466
$313$	24.1167	1.36315	0.681577	−	0.731746i	$-0.261294\pi$
$313$	24.1167	1.36315	0.681577	+	0.731746i	$0.261294\pi$
$314$	0.659285	0.0372056
$315$	0	0
$316$	67.7554	3.81154
$317$	11.7330	0.658989	0.329495	−	0.944157i	$-0.393122\pi$
$317$	11.7330	0.658989	0.329495	+	0.944157i	$0.393122\pi$
$318$	1.83035	0.102641
$319$	7.17849	0.401918
$320$	0	0
$321$	−2.21404	−0.123576
$322$	28.2095	1.57205
$323$	3.67288	0.204365
$324$	44.7353	2.48529
$325$	0	0
$326$	−45.0465	−2.49489
$327$	0.0257519	0.00142409
$328$	92.0317	5.08160
$329$	19.4416	1.07185
$330$	0	0
$331$	−6.95223	−0.382129	−0.191065	−	0.981577i	$-0.561194\pi$
$331$	−6.95223	−0.382129	−0.191065	+	0.981577i	$0.561194\pi$
$332$	−50.2858	−2.75979
$333$	6.78959	0.372067
$334$	−2.29741	−0.125709
$335$	0	0
$336$	−4.66627	−0.254566
$337$	−29.4416	−1.60379	−0.801893	−	0.597468i	$-0.796174\pi$
$337$	−29.4416	−1.60379	−0.801893	+	0.597468i	$0.796174\pi$
$338$	22.1230	1.20333
$339$	−0.0241727	−0.00131288
$340$	0	0
$341$	4.65295	0.251971
$342$	−7.92353	−0.428455
$343$	−8.92959	−0.482152
$344$	75.8905	4.09174
$345$	0	0
$346$	−11.0017	−0.591454
$347$	17.2180	0.924311	0.462155	−	0.886799i	$-0.347076\pi$
$347$	17.2180	0.924311	0.462155	+	0.886799i	$0.347076\pi$
$348$	−4.41401	−0.236616
$349$	4.32405	0.231461	0.115731	−	0.993281i	$-0.463079\pi$
$349$	4.32405	0.231461	0.115731	+	0.993281i	$0.463079\pi$
$350$	0	0
$351$	1.57550	0.0840939
$352$	−14.0001	−0.746207
$353$	−23.4421	−1.24770	−0.623848	−	0.781545i	$-0.714432\pi$
$353$	−23.4421	−1.24770	−0.623848	+	0.781545i	$0.714432\pi$
$354$	−4.46231	−0.237169
$355$	0	0
$356$	−23.2909	−1.23441
$357$	1.50799	0.0798111
$358$	−5.43031	−0.287001
$359$	8.20544	0.433066	0.216533	−	0.976275i	$-0.430525\pi$
$359$	8.20544	0.433066	0.216533	+	0.976275i	$0.430525\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−14.0020	−0.735930
$363$	−0.121872	−0.00639662
$364$	36.7131	1.92429
$365$	0	0
$366$	2.25843	0.118050
$367$	−33.1494	−1.73039	−0.865193	−	0.501439i	$-0.832804\pi$
$367$	−33.1494	−1.73039	−0.865193	+	0.501439i	$0.832804\pi$
$368$	−35.8538	−1.86901
$369$	33.9865	1.76926
$370$	0	0
$371$	19.0618	0.989640
$372$	−2.86107	−0.148339
$373$	16.5840	0.858685	0.429343	−	0.903142i	$-0.358745\pi$
$373$	16.5840	0.858685	0.429343	+	0.903142i	$0.358745\pi$
$374$	9.74900	0.504109
$375$	0	0
$376$	−46.6492	−2.40575
$377$	15.5050	0.798550
$378$	−6.52256	−0.335484
$379$	5.79467	0.297652	0.148826	−	0.988863i	$-0.452451\pi$
$379$	5.79467	0.297652	0.148826	+	0.988863i	$0.452451\pi$
$380$	0	0
$381$	−0.439143	−0.0224980
$382$	42.9730	2.19869
$383$	−5.89016	−0.300973	−0.150487	−	0.988612i	$-0.548084\pi$
$383$	−5.89016	−0.300973	−0.150487	+	0.988612i	$0.548084\pi$
$384$	1.25553	0.0640709
$385$	0	0
$386$	−53.2508	−2.71039
$387$	28.0257	1.42463
$388$	20.6814	1.04994
$389$	13.8055	0.699964	0.349982	−	0.936756i	$-0.386188\pi$
$389$	13.8055	0.699964	0.349982	+	0.936756i	$0.386188\pi$
$390$	0	0
$391$	11.5868	0.585968
$392$	−35.1581	−1.77575
$393$	−2.05649	−0.103736
$394$	−63.2784	−3.18792
$395$	0	0
$396$	−15.0613	−0.756857
$397$	−6.48471	−0.325458	−0.162729	−	0.986671i	$-0.552030\pi$
$397$	−6.48471	−0.325458	−0.162729	+	0.986671i	$0.552030\pi$
$398$	5.04835	0.253051
$399$	0.410573	0.0205544
$400$	0	0
$401$	3.70806	0.185172	0.0925858	−	0.995705i	$-0.470487\pi$
$401$	3.70806	0.185172	0.0925858	+	0.995705i	$0.470487\pi$
$402$	1.55705	0.0776584
$403$	10.0500	0.500628
$404$	−54.8584	−2.72931
$405$	0	0
$406$	−64.1908	−3.18573
$407$	−2.27446	−0.112741
$408$	−3.61833	−0.179134
$409$	31.0569	1.53567	0.767833	−	0.640650i	$-0.221335\pi$
$409$	31.0569	1.53567	0.767833	+	0.640650i	$0.221335\pi$
$410$	0	0
$411$	−0.744442	−0.0367206
$412$	−44.6954	−2.20198
$413$	−46.4717	−2.28673
$414$	−24.9962	−1.22850
$415$	0	0
$416$	−30.2392	−1.48260
$417$	−1.38138	−0.0676467
$418$	2.65432	0.129827
$419$	−11.2871	−0.551410	−0.275705	−	0.961242i	$-0.588911\pi$
$419$	−11.2871	−0.551410	−0.275705	+	0.961242i	$0.588911\pi$
$420$	0	0
$421$	34.5194	1.68237	0.841186	−	0.540746i	$-0.181858\pi$
$421$	34.5194	1.68237	0.841186	+	0.540746i	$0.181858\pi$
$422$	−32.6609	−1.58991
$423$	−17.2271	−0.837611
$424$	−45.7378	−2.22122
$425$	0	0
$426$	4.93734	0.239215
$427$	23.5199	1.13821
$428$	91.6593	4.43052
$429$	−0.263235	−0.0127091
$430$	0	0
$431$	18.1654	0.874996	0.437498	−	0.899219i	$-0.355865\pi$
$431$	18.1654	0.874996	0.437498	+	0.899219i	$0.355865\pi$
$432$	8.29007	0.398856
$433$	−10.7804	−0.518075	−0.259037	−	0.965867i	$-0.583405\pi$
$433$	−10.7804	−0.518075	−0.259037	+	0.965867i	$0.583405\pi$
$434$	−41.6071	−1.99720
$435$	0	0
$436$	−1.06611	−0.0510573
$437$	3.15468	0.150909
$438$	2.61638	0.125016
$439$	−1.38742	−0.0662180	−0.0331090	−	0.999452i	$-0.510541\pi$
$439$	−1.38742	−0.0662180	−0.0331090	+	0.999452i	$0.510541\pi$
$440$	0	0
$441$	−12.9836	−0.618266
$442$	21.0571	1.00159
$443$	7.26792	0.345310	0.172655	−	0.984982i	$-0.444766\pi$
$443$	7.26792	0.345310	0.172655	+	0.984982i	$0.444766\pi$
$444$	1.39855	0.0663722
$445$	0	0
$446$	60.6441	2.87158
$447$	−0.823487	−0.0389496
$448$	48.6135	2.29677
$449$	21.4298	1.01134	0.505668	−	0.862728i	$-0.331246\pi$
$449$	21.4298	1.01134	0.505668	+	0.862728i	$0.331246\pi$
$450$	0	0
$451$	−11.3852	−0.536108
$452$	1.00073	0.0470703
$453$	−0.787575	−0.0370035
$454$	10.6346	0.499108
$455$	0	0
$456$	−0.985147	−0.0461337
$457$	17.6312	0.824754	0.412377	−	0.911013i	$-0.364699\pi$
$457$	17.6312	0.824754	0.412377	+	0.911013i	$0.364699\pi$
$458$	65.3026	3.05139
$459$	−2.67908	−0.125049
$460$	0	0
$461$	34.2073	1.59320	0.796598	−	0.604510i	$-0.206631\pi$
$461$	34.2073	1.59320	0.796598	+	0.604510i	$0.206631\pi$
$462$	1.08979	0.0507017
$463$	11.0009	0.511253	0.255626	−	0.966776i	$-0.417718\pi$
$463$	11.0009	0.511253	0.255626	+	0.966776i	$0.417718\pi$
$464$	81.5854	3.78751
$465$	0	0
$466$	43.2637	2.00415
$467$	−4.38453	−0.202892	−0.101446	−	0.994841i	$-0.532347\pi$
$467$	−4.38453	−0.202892	−0.101446	+	0.994841i	$0.532347\pi$
$468$	−32.5312	−1.50376
$469$	16.2155	0.748764
$470$	0	0
$471$	0.0302708	0.00139481
$472$	111.506	5.13250
$473$	−9.38838	−0.431678
$474$	4.34415	0.199534
$475$	0	0
$476$	−62.4294	−2.86145
$477$	−16.8905	−0.773365
$478$	40.9249	1.87186
$479$	−14.3683	−0.656505	−0.328253	−	0.944590i	$-0.606460\pi$
$479$	−14.3683	−0.656505	−0.328253	+	0.944590i	$0.606460\pi$
$480$	0	0
$481$	−4.91266	−0.223998
$482$	−64.0861	−2.91904
$483$	1.29523	0.0589349
$484$	5.04540	0.229336
$485$	0	0
$486$	8.67657	0.393577
$487$	−1.01712	−0.0460899	−0.0230449	−	0.999734i	$-0.507336\pi$
$487$	−1.01712	−0.0460899	−0.0230449	+	0.999734i	$0.507336\pi$
$488$	−56.4348	−2.55468
$489$	−2.06829	−0.0935314
$490$	0	0
$491$	−2.41045	−0.108782	−0.0543910	−	0.998520i	$-0.517322\pi$
$491$	−2.41045	−0.108782	−0.0543910	+	0.998520i	$0.517322\pi$
$492$	7.00067	0.315615
$493$	−26.3658	−1.18745
$494$	5.73313	0.257946
$495$	0	0
$496$	52.8820	2.37447
$497$	51.4189	2.30645
$498$	−3.22409	−0.144475
$499$	4.07426	0.182389	0.0911945	−	0.995833i	$-0.470931\pi$
$499$	4.07426	0.182389	0.0911945	+	0.995833i	$0.470931\pi$
$500$	0	0
$501$	−0.105485	−0.00471272
$502$	52.8471	2.35868
$503$	19.9553	0.889762	0.444881	−	0.895590i	$-0.353246\pi$
$503$	19.9553	0.889762	0.444881	+	0.895590i	$0.353246\pi$
$504$	81.2923	3.62104
$505$	0	0
$506$	8.37352	0.372249
$507$	1.01577	0.0451118
$508$	18.1801	0.806613
$509$	−8.79239	−0.389716	−0.194858	−	0.980831i	$-0.562425\pi$
$509$	−8.79239	−0.389716	−0.194858	+	0.980831i	$0.562425\pi$
$510$	0	0
$511$	27.2478	1.20537
$512$	24.6266	1.08835
$513$	−0.729422	−0.0322048
$514$	−59.2247	−2.61229
$515$	0	0
$516$	5.77285	0.254135
$517$	5.77094	0.253806
$518$	20.3384	0.893618
$519$	−0.505138	−0.0221731
$520$	0	0
$521$	27.3483	1.19815	0.599075	−	0.800693i	$-0.295535\pi$
$521$	27.3483	1.19815	0.599075	+	0.800693i	$0.295535\pi$
$522$	56.8790	2.48953
$523$	41.7043	1.82360	0.911800	−	0.410634i	$-0.134693\pi$
$523$	41.7043	1.82360	0.911800	+	0.410634i	$0.134693\pi$
$524$	85.1368	3.71922
$525$	0	0
$526$	36.6197	1.59670
$527$	−17.0897	−0.744441
$528$	−1.38511	−0.0602790
$529$	−13.0480	−0.567304
$530$	0	0
$531$	41.1783	1.78699
$532$	−16.9974	−0.736930
$533$	−24.5912	−1.06516
$534$	−1.49330	−0.0646214
$535$	0	0
$536$	−38.9083	−1.68058
$537$	−0.249331	−0.0107594
$538$	29.7358	1.28200
$539$	4.34940	0.187342
$540$	0	0
$541$	−20.7584	−0.892472	−0.446236	−	0.894915i	$-0.647236\pi$
$541$	−20.7584	−0.892472	−0.446236	+	0.894915i	$0.647236\pi$
$542$	−45.4508	−1.95228
$543$	−0.642898	−0.0275894
$544$	51.4207	2.20464
$545$	0	0
$546$	2.35387	0.100736
$547$	−5.39396	−0.230629	−0.115315	−	0.993329i	$-0.536788\pi$
$547$	−5.39396	−0.230629	−0.115315	+	0.993329i	$0.536788\pi$
$548$	30.8193	1.31653
$549$	−20.8409	−0.889466
$550$	0	0
$551$	−7.17849	−0.305814
$552$	−3.10783	−0.132278
$553$	45.2412	1.92385
$554$	−36.3980	−1.54640
$555$	0	0
$556$	57.1881	2.42532
$557$	38.0076	1.61043	0.805217	−	0.592981i	$-0.202049\pi$
$557$	38.0076	1.61043	0.805217	+	0.592981i	$0.202049\pi$
$558$	36.8678	1.56074
$559$	−20.2782	−0.857677
$560$	0	0
$561$	0.447622	0.0188986
$562$	−68.6404	−2.89542
$563$	26.4313	1.11395	0.556973	−	0.830530i	$-0.311963\pi$
$563$	26.4313	1.11395	0.556973	+	0.830530i	$0.311963\pi$
$564$	−3.54851	−0.149419
$565$	0	0
$566$	76.6330	3.22112
$567$	29.8704	1.25444
$568$	−123.377	−5.17677
$569$	40.6105	1.70248	0.851239	−	0.524778i	$-0.175852\pi$
$569$	40.6105	1.70248	0.851239	+	0.524778i	$0.175852\pi$
$570$	0	0
$571$	8.44299	0.353328	0.176664	−	0.984271i	$-0.443469\pi$
$571$	8.44299	0.353328	0.176664	+	0.984271i	$0.443469\pi$
$572$	10.8977	0.455656
$573$	1.97309	0.0824271
$574$	101.807	4.24936
$575$	0	0
$576$	−43.0761	−1.79484
$577$	−20.9771	−0.873288	−0.436644	−	0.899634i	$-0.643833\pi$
$577$	−20.9771	−0.873288	−0.436644	+	0.899634i	$0.643833\pi$
$578$	9.31648	0.387514
$579$	−2.44499	−0.101610
$580$	0	0
$581$	−33.5766	−1.39299
$582$	1.32600	0.0549643
$583$	5.65820	0.234339
$584$	−65.3795	−2.70542
$585$	0	0
$586$	16.9338	0.699528
$587$	21.1452	0.872756	0.436378	−	0.899764i	$-0.356261\pi$
$587$	21.1452	0.872756	0.436378	+	0.899764i	$0.356261\pi$
$588$	−2.67441	−0.110291
$589$	−4.65295	−0.191721
$590$	0	0
$591$	−2.90540	−0.119512
$592$	−25.8498	−1.06242
$593$	−27.7731	−1.14051	−0.570253	−	0.821469i	$-0.693155\pi$
$593$	−27.7731	−1.14051	−0.570253	+	0.821469i	$0.693155\pi$
$594$	−1.93612	−0.0794399
$595$	0	0
$596$	34.0917	1.39645
$597$	0.231793	0.00948667
$598$	18.0862	0.739600
$599$	23.0129	0.940281	0.470140	−	0.882592i	$-0.344203\pi$
$599$	23.0129	0.940281	0.470140	+	0.882592i	$0.344203\pi$
$600$	0	0
$601$	23.3322	0.951738	0.475869	−	0.879516i	$-0.342134\pi$
$601$	23.3322	0.951738	0.475869	+	0.879516i	$0.342134\pi$
$602$	83.9517	3.42162
$603$	−14.3685	−0.585129
$604$	32.6049	1.32668
$605$	0	0
$606$	−3.51726	−0.142879
$607$	3.22857	0.131044	0.0655218	−	0.997851i	$-0.479129\pi$
$607$	3.22857	0.131044	0.0655218	+	0.997851i	$0.479129\pi$
$608$	14.0001	0.567778
$609$	−2.94730	−0.119430
$610$	0	0
$611$	12.4648	0.504273
$612$	55.3182	2.23611
$613$	17.2202	0.695519	0.347759	−	0.937584i	$-0.386943\pi$
$613$	17.2202	0.695519	0.347759	+	0.937584i	$0.386943\pi$
$614$	27.6842	1.11724
$615$	0	0
$616$	−27.2322	−1.09722
$617$	27.4776	1.10621	0.553103	−	0.833113i	$-0.313443\pi$
$617$	27.4776	1.10621	0.553103	+	0.833113i	$0.313443\pi$
$618$	−2.86565	−0.115274
$619$	−9.53870	−0.383393	−0.191696	−	0.981454i	$-0.561399\pi$
$619$	−9.53870	−0.383393	−0.191696	+	0.981454i	$0.561399\pi$
$620$	0	0
$621$	−2.30109	−0.0923397
$622$	21.9306	0.879337
$623$	−15.5517	−0.623064
$624$	−2.99173	−0.119765
$625$	0	0
$626$	−64.0133	−2.55848
$627$	0.121872	0.00486710
$628$	−1.25319	−0.0500076
$629$	8.35381	0.333088
$630$	0	0
$631$	1.16647	0.0464363	0.0232182	−	0.999730i	$-0.492609\pi$
$631$	1.16647	0.0464363	0.0232182	+	0.999730i	$0.492609\pi$
$632$	−108.554	−4.31804
$633$	−1.49962	−0.0596044
$634$	−31.1430	−1.23685
$635$	0	0
$636$	−3.47919	−0.137959
$637$	9.39438	0.372219
$638$	−19.0540	−0.754355
$639$	−45.5619	−1.80240
$640$	0	0
$641$	10.0249	0.395960	0.197980	−	0.980206i	$-0.436562\pi$
$641$	10.0249	0.395960	0.197980	+	0.980206i	$0.436562\pi$
$642$	5.87676	0.231937
$643$	40.7197	1.60583	0.802915	−	0.596094i	$-0.203281\pi$
$643$	40.7197	1.60583	0.802915	+	0.596094i	$0.203281\pi$
$644$	−53.6213	−2.11298
$645$	0	0
$646$	−9.74900	−0.383569
$647$	16.6460	0.654423	0.327211	−	0.944951i	$-0.393891\pi$
$647$	16.6460	0.654423	0.327211	+	0.944951i	$0.393891\pi$
$648$	−71.6723	−2.81555
$649$	−13.7944	−0.541477
$650$	0	0
$651$	−1.91038	−0.0748735
$652$	85.6256	3.35336
$653$	34.7416	1.35954	0.679771	−	0.733424i	$-0.262079\pi$
$653$	34.7416	1.35954	0.679771	+	0.733424i	$0.262079\pi$
$654$	−0.0683538	−0.00267285
$655$	0	0
$656$	−129.396	−5.05205
$657$	−24.1441	−0.941949
$658$	−51.6043	−2.01175
$659$	−22.9537	−0.894148	−0.447074	−	0.894497i	$-0.647534\pi$
$659$	−22.9537	−0.894148	−0.447074	+	0.894497i	$0.647534\pi$
$660$	0	0
$661$	−22.7141	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$661$	−22.7141	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$662$	18.4534	0.717213
$663$	0.966831	0.0375486
$664$	80.5651	3.12653
$665$	0	0
$666$	−18.0217	−0.698328
$667$	−22.6459	−0.876851
$668$	4.36698	0.168964
$669$	2.78445	0.107653
$670$	0	0
$671$	6.98152	0.269518
$672$	5.74806	0.221736
$673$	−20.9448	−0.807364	−0.403682	−	0.914899i	$-0.632270\pi$
$673$	−20.9448	−0.807364	−0.403682	+	0.914899i	$0.632270\pi$
$674$	78.1473	3.01012
$675$	0	0
$676$	−42.0519	−1.61738
$677$	−11.4153	−0.438726	−0.219363	−	0.975643i	$-0.570398\pi$
$677$	−11.4153	−0.438726	−0.219363	+	0.975643i	$0.570398\pi$
$678$	0.0641619	0.00246412
$679$	13.8093	0.529952
$680$	0	0
$681$	0.488285	0.0187111
$682$	−12.3504	−0.472921
$683$	34.9550	1.33752	0.668759	−	0.743479i	$-0.266826\pi$
$683$	34.9550	1.33752	0.668759	+	0.743479i	$0.266826\pi$
$684$	15.0613	0.575882
$685$	0	0
$686$	23.7020	0.904945
$687$	2.99835	0.114394
$688$	−106.701	−4.06795
$689$	12.2213	0.465594
$690$	0	0
$691$	−15.8717	−0.603789	−0.301894	−	0.953341i	$-0.597619\pi$
$691$	−15.8717	−0.603789	−0.301894	+	0.953341i	$0.597619\pi$
$692$	20.9123	0.794967
$693$	−10.0566	−0.382019
$694$	−45.7020	−1.73483
$695$	0	0
$696$	7.07187	0.268059
$697$	41.8165	1.58391
$698$	−11.4774	−0.434426
$699$	1.98644	0.0751339
$700$	0	0
$701$	−18.5830	−0.701869	−0.350935	−	0.936400i	$-0.614136\pi$
$701$	−18.5830	−0.701869	−0.350935	+	0.936400i	$0.614136\pi$
$702$	−4.18188	−0.157835
$703$	2.27446	0.0857828
$704$	14.4301	0.543857
$705$	0	0
$706$	62.2228	2.34178
$707$	−36.6298	−1.37760
$708$	8.48208	0.318776
$709$	23.8063	0.894064	0.447032	−	0.894518i	$-0.352481\pi$
$709$	23.8063	0.894064	0.447032	+	0.894518i	$0.352481\pi$
$710$	0	0
$711$	−40.0880	−1.50342
$712$	37.3153	1.39845
$713$	−14.6786	−0.549717
$714$	−4.00267	−0.149796
$715$	0	0
$716$	10.3221	0.385755
$717$	1.87905	0.0701745
$718$	−21.7798	−0.812816
$719$	37.0144	1.38040	0.690202	−	0.723616i	$-0.257522\pi$
$719$	37.0144	1.38040	0.690202	+	0.723616i	$0.257522\pi$
$720$	0	0
$721$	−29.8437	−1.11144
$722$	−2.65432	−0.0987835
$723$	−2.94249	−0.109432
$724$	26.6154	0.989154
$725$	0	0
$726$	0.323487	0.0120057
$727$	22.0744	0.818694	0.409347	−	0.912379i	$-0.365757\pi$
$727$	22.0744	0.818694	0.409347	+	0.912379i	$0.365757\pi$
$728$	−58.8197	−2.18000
$729$	−26.2013	−0.970417
$730$	0	0
$731$	34.4824	1.27538
$732$	−4.29289	−0.158670
$733$	18.7090	0.691031	0.345515	−	0.938413i	$-0.387704\pi$
$733$	18.7090	0.691031	0.345515	+	0.938413i	$0.387704\pi$
$734$	87.9891	3.24774
$735$	0	0
$736$	44.1658	1.62797
$737$	4.81332	0.177301
$738$	−90.2109	−3.32071
$739$	−5.91630	−0.217635	−0.108817	−	0.994062i	$-0.534706\pi$
$739$	−5.91630	−0.217635	−0.108817	+	0.994062i	$0.534706\pi$
$740$	0	0
$741$	0.263235	0.00967017
$742$	−50.5961	−1.85744
$743$	−14.0422	−0.515159	−0.257579	−	0.966257i	$-0.582925\pi$
$743$	−14.0422	−0.515159	−0.257579	+	0.966257i	$0.582925\pi$
$744$	4.58384	0.168052
$745$	0	0
$746$	−44.0191	−1.61165
$747$	29.7520	1.08857
$748$	−18.5312	−0.677566
$749$	61.2023	2.23628
$750$	0	0
$751$	4.91166	0.179229	0.0896145	−	0.995977i	$-0.471437\pi$
$751$	4.91166	0.179229	0.0896145	+	0.995977i	$0.471437\pi$
$752$	65.5882	2.39176
$753$	2.42646	0.0884250
$754$	−41.1553	−1.49879
$755$	0	0
$756$	12.3983	0.450920
$757$	−49.2829	−1.79122	−0.895609	−	0.444843i	$-0.853259\pi$
$757$	−49.2829	−1.79122	−0.895609	+	0.444843i	$0.853259\pi$
$758$	−15.3809	−0.558659
$759$	0.384467	0.0139553
$760$	0	0
$761$	−7.25922	−0.263146	−0.131573	−	0.991306i	$-0.542003\pi$
$761$	−7.25922	−0.263146	−0.131573	+	0.991306i	$0.542003\pi$
$762$	1.16562	0.0422261
$763$	−0.711856	−0.0257709
$764$	−81.6843	−2.95524
$765$	0	0
$766$	15.6344	0.564892
$767$	−29.7949	−1.07583
$768$	0.184693	0.00666453
$769$	−21.5123	−0.775755	−0.387877	−	0.921711i	$-0.626792\pi$
$769$	−21.5123	−0.775755	−0.387877	+	0.921711i	$0.626792\pi$
$770$	0	0
$771$	−2.71928	−0.0979325
$772$	101.221	3.64301
$773$	−4.54038	−0.163306	−0.0816530	−	0.996661i	$-0.526020\pi$
$773$	−4.54038	−0.163306	−0.0816530	+	0.996661i	$0.526020\pi$
$774$	−74.3891	−2.67386
$775$	0	0
$776$	−33.1346	−1.18946
$777$	0.933831	0.0335010
$778$	−36.6440	−1.31375
$779$	11.3852	0.407917
$780$	0	0
$781$	15.2629	0.546149
$782$	−30.7550	−1.09980
$783$	5.23615	0.187125
$784$	49.4320	1.76543
$785$	0	0
$786$	5.45856	0.194701
$787$	7.25028	0.258444	0.129222	−	0.991616i	$-0.458752\pi$
$787$	7.25028	0.258444	0.129222	+	0.991616i	$0.458752\pi$
$788$	120.281	4.28484
$789$	1.68138	0.0598588
$790$	0	0
$791$	0.668201	0.0237585
$792$	24.1303	0.857433
$793$	15.0796	0.535491
$794$	17.2125	0.610848
$795$	0	0
$796$	−9.59605	−0.340123
$797$	−18.8674	−0.668319	−0.334159	−	0.942517i	$-0.608452\pi$
$797$	−18.8674	−0.668319	−0.334159	+	0.942517i	$0.608452\pi$
$798$	−1.08979	−0.0385782
$799$	−21.1960	−0.749860
$800$	0	0
$801$	13.7802	0.486900
$802$	−9.84236	−0.347546
$803$	8.08806	0.285422
$804$	−2.95968	−0.104380
$805$	0	0
$806$	−26.6760	−0.939622
$807$	1.36531	0.0480612
$808$	87.8911	3.09200
$809$	−19.4948	−0.685399	−0.342700	−	0.939445i	$-0.611341\pi$
$809$	−19.4948	−0.685399	−0.342700	+	0.939445i	$0.611341\pi$
$810$	0	0
$811$	4.28635	0.150514	0.0752570	−	0.997164i	$-0.476022\pi$
$811$	4.28635	0.150514	0.0752570	+	0.997164i	$0.476022\pi$
$812$	122.016	4.28191
$813$	−2.08686	−0.0731893
$814$	6.03713	0.211601
$815$	0	0
$816$	5.08734	0.178092
$817$	9.38838	0.328458
$818$	−82.4350	−2.88227
$819$	−21.7216	−0.759014
$820$	0	0
$821$	5.37419	0.187561	0.0937803	−	0.995593i	$-0.470105\pi$
$821$	5.37419	0.187561	0.0937803	+	0.995593i	$0.470105\pi$
$822$	1.97598	0.0689204
$823$	−26.3196	−0.917445	−0.458723	−	0.888580i	$-0.651693\pi$
$823$	−26.3196	−0.917445	−0.458723	+	0.888580i	$0.651693\pi$
$824$	71.6084	2.49460
$825$	0	0
$826$	123.351	4.29192
$827$	8.11186	0.282077	0.141039	−	0.990004i	$-0.454956\pi$
$827$	8.11186	0.282077	0.141039	+	0.990004i	$0.454956\pi$
$828$	47.5135	1.65121
$829$	−21.5948	−0.750017	−0.375009	−	0.927021i	$-0.622360\pi$
$829$	−21.5948	−0.750017	−0.375009	+	0.927021i	$0.622360\pi$
$830$	0	0
$831$	−1.67120	−0.0579733
$832$	31.1681	1.08056
$833$	−15.9748	−0.553495
$834$	3.66663	0.126965
$835$	0	0
$836$	−5.04540	−0.174499
$837$	3.39396	0.117313
$838$	29.9595	1.03493
$839$	−4.35109	−0.150216	−0.0751082	−	0.997175i	$-0.523930\pi$
$839$	−4.35109	−0.150216	−0.0751082	+	0.997175i	$0.523930\pi$
$840$	0	0
$841$	22.5308	0.776923
$842$	−91.6254	−3.15762
$843$	−3.15160	−0.108547
$844$	62.0828	2.13698
$845$	0	0
$846$	45.7262	1.57210
$847$	3.36889	0.115756
$848$	64.3069	2.20831
$849$	3.51858	0.120757
$850$	0	0
$851$	7.17519	0.245962
$852$	−9.38504	−0.321526
$853$	−19.5022	−0.667744	−0.333872	−	0.942618i	$-0.608355\pi$
$853$	−19.5022	−0.667744	−0.333872	+	0.942618i	$0.608355\pi$
$854$	−62.4294	−2.13629
$855$	0	0
$856$	−146.851	−5.01928
$857$	−13.3462	−0.455898	−0.227949	−	0.973673i	$-0.573202\pi$
$857$	−13.3462	−0.455898	−0.227949	+	0.973673i	$0.573202\pi$
$858$	0.698709	0.0238535
$859$	21.6016	0.737036	0.368518	−	0.929621i	$-0.379865\pi$
$859$	21.6016	0.737036	0.368518	+	0.929621i	$0.379865\pi$
$860$	0	0
$861$	4.67445	0.159305
$862$	−48.2167	−1.64227
$863$	2.72425	0.0927345	0.0463673	−	0.998924i	$-0.485236\pi$
$863$	2.72425	0.0927345	0.0463673	+	0.998924i	$0.485236\pi$
$864$	−10.2120	−0.347418
$865$	0	0
$866$	28.6147	0.972367
$867$	0.427763	0.0145276
$868$	79.0879	2.68442
$869$	13.4291	0.455552
$870$	0	0
$871$	10.3964	0.352269
$872$	1.70806	0.0578422
$873$	−12.2363	−0.414137
$874$	−8.37352	−0.283239
$875$	0	0
$876$	−4.97329	−0.168032
$877$	−1.29130	−0.0436041	−0.0218021	−	0.999762i	$-0.506940\pi$
$877$	−1.29130	−0.0436041	−0.0218021	+	0.999762i	$0.506940\pi$
$878$	3.68266	0.124284
$879$	0.777508	0.0262247
$880$	0	0
$881$	24.9873	0.841844	0.420922	−	0.907097i	$-0.361707\pi$
$881$	24.9873	0.841844	0.420922	+	0.907097i	$0.361707\pi$
$882$	34.4626	1.16041
$883$	−15.5046	−0.521772	−0.260886	−	0.965370i	$-0.584015\pi$
$883$	−15.5046	−0.521772	−0.260886	+	0.965370i	$0.584015\pi$
$884$	−40.0260	−1.34622
$885$	0	0
$886$	−19.2914	−0.648107
$887$	9.98029	0.335105	0.167553	−	0.985863i	$-0.446414\pi$
$887$	9.98029	0.335105	0.167553	+	0.985863i	$0.446414\pi$
$888$	−2.24068	−0.0751921
$889$	12.1391	0.407134
$890$	0	0
$891$	8.86655	0.297040
$892$	−115.274	−3.85966
$893$	−5.77094	−0.193117
$894$	2.18580	0.0731040
$895$	0	0
$896$	−34.7063	−1.15946
$897$	0.830422	0.0277270
$898$	−56.8815	−1.89816
$899$	33.4012	1.11399
$900$	0	0
$901$	−20.7819	−0.692345
$902$	30.2199	1.00621
$903$	3.85461	0.128274
$904$	−1.60331	−0.0533253
$905$	0	0
$906$	2.09047	0.0694513
$907$	−11.8887	−0.394758	−0.197379	−	0.980327i	$-0.563243\pi$
$907$	−11.8887	−0.394758	−0.197379	+	0.980327i	$0.563243\pi$
$908$	−20.2146	−0.670845
$909$	32.4574	1.07654
$910$	0	0
$911$	−23.7166	−0.785767	−0.392883	−	0.919588i	$-0.628522\pi$
$911$	−23.7166	−0.785767	−0.392883	+	0.919588i	$0.628522\pi$
$912$	1.38511	0.0458655
$913$	−9.96666	−0.329848
$914$	−46.7989	−1.54797
$915$	0	0
$916$	−124.129	−4.10134
$917$	56.8471	1.87726
$918$	7.11113	0.234702
$919$	18.1426	0.598470	0.299235	−	0.954179i	$-0.403269\pi$
$919$	18.1426	0.598470	0.299235	+	0.954179i	$0.403269\pi$
$920$	0	0
$921$	1.27111	0.0418845
$922$	−90.7972	−2.99024
$923$	32.9667	1.08511
$924$	−2.07150	−0.0681475
$925$	0	0
$926$	−29.1997	−0.959563
$927$	26.4443	0.868546
$928$	−100.500	−3.29906
$929$	−50.5019	−1.65691	−0.828457	−	0.560053i	$-0.810781\pi$
$929$	−50.5019	−1.65691	−0.828457	+	0.560053i	$0.810781\pi$
$930$	0	0
$931$	−4.34940	−0.142546
$932$	−82.2368	−2.69376
$933$	1.00694	0.0329656
$934$	11.6379	0.380805
$935$	0	0
$936$	52.1197	1.70359
$937$	39.6587	1.29559	0.647797	−	0.761813i	$-0.275691\pi$
$937$	39.6587	1.29559	0.647797	+	0.761813i	$0.275691\pi$
$938$	−43.0412	−1.40534
$939$	−2.93915	−0.0959154
$940$	0	0
$941$	−34.8809	−1.13709	−0.568543	−	0.822653i	$-0.692493\pi$
$941$	−34.8809	−1.13709	−0.568543	+	0.822653i	$0.692493\pi$
$942$	−0.0803484	−0.00261789
$943$	35.9166	1.16961
$944$	−156.777	−5.10265
$945$	0	0
$946$	24.9197	0.810210
$947$	30.3067	0.984835	0.492417	−	0.870359i	$-0.336113\pi$
$947$	30.3067	0.984835	0.492417	+	0.870359i	$0.336113\pi$
$948$	−8.25748	−0.268191
$949$	17.4696	0.567088
$950$	0	0
$951$	−1.42992	−0.0463684
$952$	100.021	3.24169
$953$	6.65804	0.215675	0.107838	−	0.994169i	$-0.465607\pi$
$953$	6.65804	0.215675	0.107838	+	0.994169i	$0.465607\pi$
$954$	44.8329	1.45152
$955$	0	0
$956$	−77.7912	−2.51595
$957$	−0.874858	−0.0282801
$958$	38.1381	1.23219
$959$	20.5785	0.664513
$960$	0	0
$961$	−9.35006	−0.301615
$962$	13.0398	0.420419
$963$	−54.2309	−1.74757
$964$	121.817	3.92345
$965$	0	0
$966$	−3.43794	−0.110614
$967$	20.0020	0.643221	0.321611	−	0.946872i	$-0.395776\pi$
$967$	20.0020	0.643221	0.321611	+	0.946872i	$0.395776\pi$
$968$	−8.08346	−0.259812
$969$	−0.447622	−0.0143797
$970$	0	0
$971$	−0.491941	−0.0157871	−0.00789357	−	0.999969i	$-0.502513\pi$
$971$	−0.491941	−0.0157871	−0.00789357	+	0.999969i	$0.502513\pi$
$972$	−16.4927	−0.529002
$973$	38.1853	1.22417
$974$	2.69975	0.0865055
$975$	0	0
$976$	79.3467	2.53983
$977$	−23.6515	−0.756679	−0.378339	−	0.925667i	$-0.623505\pi$
$977$	−23.6515	−0.756679	−0.378339	+	0.925667i	$0.623505\pi$
$978$	5.48991	0.175548
$979$	−4.61626	−0.147536
$980$	0	0
$981$	0.630771	0.0201390
$982$	6.39809	0.204171
$983$	−39.5609	−1.26180	−0.630899	−	0.775865i	$-0.717314\pi$
$983$	−39.5609	−1.26180	−0.630899	+	0.775865i	$0.717314\pi$
$984$	−11.2161	−0.357556
$985$	0	0
$986$	69.9831	2.22872
$987$	−2.36939	−0.0754186
$988$	−10.8977	−0.346702
$989$	29.6173	0.941777
$990$	0	0
$991$	−13.6096	−0.432322	−0.216161	−	0.976358i	$-0.569354\pi$
$991$	−13.6096	−0.432322	−0.216161	+	0.976358i	$0.569354\pi$
$992$	−65.1417	−2.06825
$993$	0.847283	0.0268877
$994$	−136.482	−4.32895
$995$	0	0
$996$	6.12843	0.194187
$997$	−41.1821	−1.30425	−0.652125	−	0.758111i	$-0.726122\pi$
$997$	−41.1821	−1.30425	−0.652125	+	0.758111i	$0.726122\pi$
$998$	−10.8144	−0.342323
$999$	−1.65904	−0.0524897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.h.1.1		5
5.4	even	2		209.2.a.c.1.5	✓	5
15.14	odd	2		1881.2.a.k.1.1		5
20.19	odd	2		3344.2.a.t.1.3		5
55.54	odd	2		2299.2.a.n.1.1		5
95.94	odd	2		3971.2.a.h.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.5	✓	5	5.4	even	2
1881.2.a.k.1.1		5	15.14	odd	2
2299.2.a.n.1.1		5	55.54	odd	2
3344.2.a.t.1.3		5	20.19	odd	2
3971.2.a.h.1.1		5	95.94	odd	2
5225.2.a.h.1.1		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65432 q^{2} -0.121872 q^{3} +5.04540 q^{4} +0.323487 q^{6} +3.36889 q^{7} -8.08346 q^{8} -2.98515 q^{9} +1.00000 q^{11} -0.614893 q^{12} +2.15993 q^{13} -8.94209 q^{14} +11.3653 q^{16} -3.67288 q^{17} +7.92353 q^{18} -1.00000 q^{19} -0.410573 q^{21} -2.65432 q^{22} -3.15468 q^{23} +0.985147 q^{24} -5.73313 q^{26} +0.729422 q^{27} +16.9974 q^{28} +7.17849 q^{29} +4.65295 q^{31} -14.0001 q^{32} -0.121872 q^{33} +9.74900 q^{34} -15.0613 q^{36} -2.27446 q^{37} +2.65432 q^{38} -0.263235 q^{39} -11.3852 q^{41} +1.08979 q^{42} -9.38838 q^{43} +5.04540 q^{44} +8.37352 q^{46} +5.77094 q^{47} -1.38511 q^{48} +4.34940 q^{49} +0.447622 q^{51} +10.8977 q^{52} +5.65820 q^{53} -1.93612 q^{54} -27.2322 q^{56} +0.121872 q^{57} -19.0540 q^{58} -13.7944 q^{59} +6.98152 q^{61} -12.3504 q^{62} -10.0566 q^{63} +14.4301 q^{64} +0.323487 q^{66} +4.81332 q^{67} -18.5312 q^{68} +0.384467 q^{69} +15.2629 q^{71} +24.1303 q^{72} +8.08806 q^{73} +6.03713 q^{74} -5.04540 q^{76} +3.36889 q^{77} +0.698709 q^{78} +13.4291 q^{79} +8.86655 q^{81} +30.2199 q^{82} -9.96666 q^{83} -2.07150 q^{84} +24.9197 q^{86} -0.874858 q^{87} -8.08346 q^{88} -4.61626 q^{89} +7.27655 q^{91} -15.9166 q^{92} -0.567064 q^{93} -15.3179 q^{94} +1.70622 q^{96} +4.09907 q^{97} -11.5447 q^{98} -2.98515 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - q^3 + 6 * q^4 - 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 + 5 * q^11 - 6 * q^12 - 4 * q^13 - 14 * q^14 + 8 * q^16 + 4 * q^17 + 20 * q^18 - 5 * q^19 + 10 * q^21 - 2 * q^22 - 3 * q^23 - 14 * q^24 - 6 * q^26 + 11 * q^27 + 10 * q^28 + 10 * q^29 + 11 * q^31 - 14 * q^32 - q^33 - 4 * q^34 - 26 * q^36 - q^37 + 2 * q^38 + 2 * q^39 + 2 * q^41 + 16 * q^42 - 20 * q^43 + 6 * q^44 - 4 * q^46 + 20 * q^47 - 4 * q^48 + 3 * q^49 + 24 * q^51 - 6 * q^52 + 14 * q^53 + 16 * q^54 - 38 * q^56 + q^57 + 6 * q^58 + 3 * q^59 - 10 * q^61 + 6 * q^62 - 24 * q^63 - 2 * q^66 - 9 * q^67 - 24 * q^68 - 5 * q^69 + 23 * q^71 + 12 * q^72 + 8 * q^74 - 6 * q^76 - 6 * q^77 + 22 * q^78 + 44 * q^79 + q^81 + 30 * q^82 + 14 * q^83 + 14 * q^84 + 52 * q^86 - 28 * q^87 - 6 * q^88 - 27 * q^89 + 24 * q^91 - 58 * q^92 + 27 * q^93 - 8 * q^94 + 50 * q^96 - 15 * q^97 + 10 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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