LMFDB - Embedded newform 5225.2.a.x.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [15,5,4,17,0,-1,15] 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:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 $ x^15 - 5x^14 - 11x^13 + 85x^12 + 6x^11 - 537x^10 + 327x^9 + 1556x^8 - 1451x^7 - 2033x^6 + 2316x^5 + 927x^4 - 1295x^3 + 17x^2 + 103x - 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.59045$ 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)

$f(q)$	$=$	\(q+2.59045 q^{2} -2.99450 q^{3} +4.71045 q^{4} -7.75712 q^{6} +1.08607 q^{7} +7.02129 q^{8} +5.96706 q^{9} +1.00000 q^{11} -14.1055 q^{12} +3.56681 q^{13} +2.81342 q^{14} +8.76742 q^{16} -2.11143 q^{17} +15.4574 q^{18} -1.00000 q^{19} -3.25225 q^{21} +2.59045 q^{22} +4.81445 q^{23} -21.0253 q^{24} +9.23965 q^{26} -8.88487 q^{27} +5.11589 q^{28} -3.35232 q^{29} +4.72904 q^{31} +8.66902 q^{32} -2.99450 q^{33} -5.46957 q^{34} +28.1075 q^{36} +5.10670 q^{37} -2.59045 q^{38} -10.6808 q^{39} -8.37205 q^{41} -8.42480 q^{42} +7.95355 q^{43} +4.71045 q^{44} +12.4716 q^{46} +1.41079 q^{47} -26.2541 q^{48} -5.82045 q^{49} +6.32269 q^{51} +16.8013 q^{52} +9.64120 q^{53} -23.0158 q^{54} +7.62562 q^{56} +2.99450 q^{57} -8.68403 q^{58} -0.737428 q^{59} -5.70238 q^{61} +12.2504 q^{62} +6.48066 q^{63} +4.92184 q^{64} -7.75712 q^{66} -2.12759 q^{67} -9.94579 q^{68} -14.4169 q^{69} +10.6171 q^{71} +41.8964 q^{72} +11.0070 q^{73} +13.2287 q^{74} -4.71045 q^{76} +1.08607 q^{77} -27.6682 q^{78} -9.96615 q^{79} +8.70462 q^{81} -21.6874 q^{82} -14.0187 q^{83} -15.3195 q^{84} +20.6033 q^{86} +10.0385 q^{87} +7.02129 q^{88} +9.82782 q^{89} +3.87381 q^{91} +22.6782 q^{92} -14.1611 q^{93} +3.65458 q^{94} -25.9594 q^{96} +8.89758 q^{97} -15.0776 q^{98} +5.96706 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 15 * q^7 + 15 * q^8 + 19 * q^9 + 15 * q^11 + 9 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 11 * q^17 + 16 * q^18 - 15 * q^19 + 5 * q^22 + 10 * q^23 + 17 * q^24 - 17 * q^26 + 13 * q^27 + 30 * q^28 + 5 * q^29 + 6 * q^31 + 40 * q^32 + 4 * q^33 + 17 * q^34 + 28 * q^36 + 13 * q^37 - 5 * q^38 - 22 * q^39 + 30 * q^42 + 36 * q^43 + 17 * q^44 + 13 * q^46 - 6 * q^47 + 14 * q^48 + 16 * q^49 + 4 * q^51 + 50 * q^52 + 9 * q^53 + 9 * q^54 - 18 * q^56 - 4 * q^57 + 2 * q^58 - 7 * q^59 - 2 * q^61 + 11 * q^62 + 39 * q^63 + 17 * q^64 - q^66 + 35 * q^67 - 18 * q^68 - 9 * q^69 + 13 * q^71 + 68 * q^72 + 2 * q^73 + 13 * q^74 - 17 * q^76 + 15 * q^77 - 10 * q^78 + 6 * q^79 + 11 * q^81 + 14 * q^82 + 30 * q^83 - 6 * q^84 - 25 * q^86 + 19 * q^87 + 15 * q^88 + 55 * q^89 + 26 * q^91 - 18 * q^92 + 14 * q^93 - 22 * q^94 - 17 * q^96 + 28 * q^97 - 22 * q^98 + 19 * 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 $

$2$	2.59045	1.83173	0.915863	−	0.401490i	$-0.131507\pi$
$2$	2.59045	1.83173	0.915863	+	0.401490i	$0.131507\pi$
$3$	−2.99450	−1.72888	−0.864439	−	0.502738i	$-0.832326\pi$
$3$	−2.99450	−1.72888	−0.864439	+	0.502738i	$0.832326\pi$
$4$	4.71045	2.35522
$5$	0	0
$5$	0	0
$6$	−7.75712	−3.16683
$7$	1.08607	0.410497	0.205248	−	0.978710i	$-0.434200\pi$
$7$	1.08607	0.410497	0.205248	+	0.978710i	$0.434200\pi$
$8$	7.02129	2.48240
$9$	5.96706	1.98902
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−14.1055	−4.07189
$13$	3.56681	0.989254	0.494627	−	0.869105i	$-0.335305\pi$
$13$	3.56681	0.989254	0.494627	+	0.869105i	$0.335305\pi$
$14$	2.81342	0.751918
$15$	0	0
$16$	8.76742	2.19185
$17$	−2.11143	−0.512097	−0.256049	−	0.966664i	$-0.582421\pi$
$17$	−2.11143	−0.512097	−0.256049	+	0.966664i	$0.582421\pi$
$18$	15.4574	3.64334
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.25225	−0.709699
$22$	2.59045	0.552286
$23$	4.81445	1.00388	0.501941	−	0.864902i	$-0.332619\pi$
$23$	4.81445	1.00388	0.501941	+	0.864902i	$0.332619\pi$
$24$	−21.0253	−4.29177
$25$	0	0
$26$	9.23965	1.81204
$27$	−8.88487	−1.70989
$28$	5.11589	0.966811
$29$	−3.35232	−0.622510	−0.311255	−	0.950326i	$-0.600749\pi$
$29$	−3.35232	−0.622510	−0.311255	+	0.950326i	$0.600749\pi$
$30$	0	0
$31$	4.72904	0.849361	0.424680	−	0.905343i	$-0.360386\pi$
$31$	4.72904	0.849361	0.424680	+	0.905343i	$0.360386\pi$
$32$	8.66902	1.53248
$33$	−2.99450	−0.521276
$34$	−5.46957	−0.938023
$35$	0	0
$36$	28.1075	4.68459
$37$	5.10670	0.839536	0.419768	−	0.907632i	$-0.362112\pi$
$37$	5.10670	0.839536	0.419768	+	0.907632i	$0.362112\pi$
$38$	−2.59045	−0.420227
$39$	−10.6808	−1.71030
$40$	0	0
$41$	−8.37205	−1.30749	−0.653747	−	0.756713i	$-0.726804\pi$
$41$	−8.37205	−1.30749	−0.653747	+	0.756713i	$0.726804\pi$
$42$	−8.42480	−1.29997
$43$	7.95355	1.21290	0.606452	−	0.795120i	$-0.292592\pi$
$43$	7.95355	1.21290	0.606452	+	0.795120i	$0.292592\pi$
$44$	4.71045	0.710127
$45$	0	0
$46$	12.4716	1.83884
$47$	1.41079	0.205785	0.102892	−	0.994693i	$-0.467190\pi$
$47$	1.41079	0.205785	0.102892	+	0.994693i	$0.467190\pi$
$48$	−26.2541	−3.78945
$49$	−5.82045	−0.831493
$50$	0	0
$51$	6.32269	0.885354
$52$	16.8013	2.32992
$53$	9.64120	1.32432	0.662160	−	0.749362i	$-0.269640\pi$
$53$	9.64120	1.32432	0.662160	+	0.749362i	$0.269640\pi$
$54$	−23.0158	−3.13206
$55$	0	0
$56$	7.62562	1.01902
$57$	2.99450	0.396632
$58$	−8.68403	−1.14027
$59$	−0.737428	−0.0960050	−0.0480025	−	0.998847i	$-0.515286\pi$
$59$	−0.737428	−0.0960050	−0.0480025	+	0.998847i	$0.515286\pi$
$60$	0	0
$61$	−5.70238	−0.730114	−0.365057	−	0.930985i	$-0.618951\pi$
$61$	−5.70238	−0.730114	−0.365057	+	0.930985i	$0.618951\pi$
$62$	12.2504	1.55580
$63$	6.48066	0.816486
$64$	4.92184	0.615230
$65$	0	0
$66$	−7.75712	−0.954836
$67$	−2.12759	−0.259927	−0.129964	−	0.991519i	$-0.541486\pi$
$67$	−2.12759	−0.259927	−0.129964	+	0.991519i	$0.541486\pi$
$68$	−9.94579	−1.20610
$69$	−14.4169	−1.73559
$70$	0	0
$71$	10.6171	1.26002	0.630008	−	0.776588i	$-0.283051\pi$
$71$	10.6171	1.26002	0.630008	+	0.776588i	$0.283051\pi$
$72$	41.8964	4.93754
$73$	11.0070	1.28828	0.644138	−	0.764910i	$-0.277216\pi$
$73$	11.0070	1.28828	0.644138	+	0.764910i	$0.277216\pi$
$74$	13.2287	1.53780
$75$	0	0
$76$	−4.71045	−0.540325
$77$	1.08607	0.123769
$78$	−27.6682	−3.13280
$79$	−9.96615	−1.12128	−0.560640	−	0.828060i	$-0.689445\pi$
$79$	−9.96615	−1.12128	−0.560640	+	0.828060i	$0.689445\pi$
$80$	0	0
$81$	8.70462	0.967179
$82$	−21.6874	−2.39497
$83$	−14.0187	−1.53875	−0.769377	−	0.638795i	$-0.779433\pi$
$83$	−14.0187	−1.53875	−0.769377	+	0.638795i	$0.779433\pi$
$84$	−15.3195	−1.67150
$85$	0	0
$86$	20.6033	2.22171
$87$	10.0385	1.07624
$88$	7.02129	0.748472
$89$	9.82782	1.04175	0.520873	−	0.853634i	$-0.325606\pi$
$89$	9.82782	1.04175	0.520873	+	0.853634i	$0.325606\pi$
$90$	0	0
$91$	3.87381	0.406086
$92$	22.6782	2.36437
$93$	−14.1611	−1.46844
$94$	3.65458	0.376941
$95$	0	0
$96$	−25.9594	−2.64947
$97$	8.89758	0.903412	0.451706	−	0.892167i	$-0.350816\pi$
$97$	8.89758	0.903412	0.451706	+	0.892167i	$0.350816\pi$
$98$	−15.0776	−1.52307
$99$	5.96706	0.599712
$100$	0	0
$101$	−7.16344	−0.712788	−0.356394	−	0.934336i	$-0.615994\pi$
$101$	−7.16344	−0.712788	−0.356394	+	0.934336i	$0.615994\pi$
$102$	16.3786	1.62173
$103$	−6.40667	−0.631268	−0.315634	−	0.948881i	$-0.602217\pi$
$103$	−6.40667	−0.631268	−0.315634	+	0.948881i	$0.602217\pi$
$104$	25.0436	2.45572
$105$	0	0
$106$	24.9751	2.42579
$107$	−8.12343	−0.785321	−0.392661	−	0.919683i	$-0.628445\pi$
$107$	−8.12343	−0.785321	−0.392661	+	0.919683i	$0.628445\pi$
$108$	−41.8517	−4.02718
$109$	5.30036	0.507683	0.253841	−	0.967246i	$-0.418306\pi$
$109$	5.30036	0.507683	0.253841	+	0.967246i	$0.418306\pi$
$110$	0	0
$111$	−15.2920	−1.45145
$112$	9.52205	0.899749
$113$	18.0156	1.69477	0.847384	−	0.530980i	$-0.178176\pi$
$113$	18.0156	1.69477	0.847384	+	0.530980i	$0.178176\pi$
$114$	7.75712	0.726521
$115$	0	0
$116$	−15.7909	−1.46615
$117$	21.2833	1.96765
$118$	−1.91027	−0.175855
$119$	−2.29317	−0.210214
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.7717	−1.33737
$123$	25.0701	2.26050
$124$	22.2759	2.00043
$125$	0	0
$126$	16.7878	1.49558
$127$	3.99191	0.354225	0.177112	−	0.984191i	$-0.443324\pi$
$127$	3.99191	0.354225	0.177112	+	0.984191i	$0.443324\pi$
$128$	−4.58824	−0.405547
$129$	−23.8169	−2.09696
$130$	0	0
$131$	11.9276	1.04212	0.521059	−	0.853521i	$-0.325537\pi$
$131$	11.9276	1.04212	0.521059	+	0.853521i	$0.325537\pi$
$132$	−14.1055	−1.22772
$133$	−1.08607	−0.0941744
$134$	−5.51143	−0.476115
$135$	0	0
$136$	−14.8250	−1.27123
$137$	−21.8741	−1.86883	−0.934415	−	0.356187i	$-0.884077\pi$
$137$	−21.8741	−1.86883	−0.934415	+	0.356187i	$0.884077\pi$
$138$	−37.3463	−3.17913
$139$	11.6552	0.988579	0.494290	−	0.869297i	$-0.335428\pi$
$139$	11.6552	0.988579	0.494290	+	0.869297i	$0.335428\pi$
$140$	0	0
$141$	−4.22461	−0.355777
$142$	27.5031	2.30801
$143$	3.56681	0.298271
$144$	52.3157	4.35964
$145$	0	0
$146$	28.5132	2.35977
$147$	17.4294	1.43755
$148$	24.0548	1.97729
$149$	9.67476	0.792587	0.396294	−	0.918124i	$-0.370296\pi$
$149$	9.67476	0.792587	0.396294	+	0.918124i	$0.370296\pi$
$150$	0	0
$151$	14.6123	1.18913	0.594565	−	0.804047i	$-0.297324\pi$
$151$	14.6123	1.18913	0.594565	+	0.804047i	$0.297324\pi$
$152$	−7.02129	−0.569502
$153$	−12.5990	−1.01857
$154$	2.81342	0.226712
$155$	0	0
$156$	−50.3114	−4.02814
$157$	7.58342	0.605223	0.302612	−	0.953114i	$-0.402142\pi$
$157$	7.58342	0.605223	0.302612	+	0.953114i	$0.402142\pi$
$158$	−25.8168	−2.05388
$159$	−28.8706	−2.28959
$160$	0	0
$161$	5.22884	0.412090
$162$	22.5489	1.77161
$163$	17.0674	1.33682	0.668411	−	0.743792i	$-0.266975\pi$
$163$	17.0674	1.33682	0.668411	+	0.743792i	$0.266975\pi$
$164$	−39.4361	−3.07944
$165$	0	0
$166$	−36.3148	−2.81858
$167$	6.96828	0.539222	0.269611	−	0.962969i	$-0.413105\pi$
$167$	6.96828	0.539222	0.269611	+	0.962969i	$0.413105\pi$
$168$	−22.8350	−1.76176
$169$	−0.277886	−0.0213758
$170$	0	0
$171$	−5.96706	−0.456312
$172$	37.4648	2.85666
$173$	25.6458	1.94982	0.974908	−	0.222607i	$-0.0714567\pi$
$173$	25.6458	1.94982	0.974908	+	0.222607i	$0.0714567\pi$
$174$	26.0044	1.97139
$175$	0	0
$176$	8.76742	0.660869
$177$	2.20823	0.165981
$178$	25.4585	1.90820
$179$	−15.6845	−1.17232	−0.586159	−	0.810196i	$-0.699361\pi$
$179$	−15.6845	−1.17232	−0.586159	+	0.810196i	$0.699361\pi$
$180$	0	0
$181$	−10.2376	−0.760953	−0.380477	−	0.924791i	$-0.624240\pi$
$181$	−10.2376	−0.760953	−0.380477	+	0.924791i	$0.624240\pi$
$182$	10.0349	0.743838
$183$	17.0758	1.26228
$184$	33.8036	2.49204
$185$	0	0
$186$	−36.6838	−2.68978
$187$	−2.11143	−0.154403
$188$	6.64544	0.484669
$189$	−9.64961	−0.701906
$190$	0	0
$191$	11.0398	0.798812	0.399406	−	0.916774i	$-0.369216\pi$
$191$	11.0398	0.798812	0.399406	+	0.916774i	$0.369216\pi$
$192$	−14.7385	−1.06366
$193$	15.4104	1.10926	0.554631	−	0.832097i	$-0.312860\pi$
$193$	15.4104	1.10926	0.554631	+	0.832097i	$0.312860\pi$
$194$	23.0488	1.65480
$195$	0	0
$196$	−27.4169	−1.95835
$197$	15.7536	1.12239	0.561197	−	0.827682i	$-0.310341\pi$
$197$	15.7536	1.12239	0.561197	+	0.827682i	$0.310341\pi$
$198$	15.4574	1.09851
$199$	8.77418	0.621985	0.310992	−	0.950412i	$-0.399339\pi$
$199$	8.77418	0.621985	0.310992	+	0.950412i	$0.399339\pi$
$200$	0	0
$201$	6.37109	0.449382
$202$	−18.5565	−1.30563
$203$	−3.64086	−0.255538
$204$	29.7827	2.08521
$205$	0	0
$206$	−16.5962	−1.15631
$207$	28.7281	1.99674
$208$	31.2717	2.16830
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	9.19454	0.632978	0.316489	−	0.948596i	$-0.397496\pi$
$211$	9.19454	0.632978	0.316489	+	0.948596i	$0.397496\pi$
$212$	45.4144	3.11907
$213$	−31.7929	−2.17842
$214$	−21.0434	−1.43849
$215$	0	0
$216$	−62.3832	−4.24464
$217$	5.13608	0.348660
$218$	13.7303	0.929936
$219$	−32.9606	−2.22727
$220$	0	0
$221$	−7.53107	−0.506595
$222$	−39.6133	−2.65867
$223$	16.3430	1.09441	0.547203	−	0.837000i	$-0.315692\pi$
$223$	16.3430	1.09441	0.547203	+	0.837000i	$0.315692\pi$
$224$	9.41517	0.629078
$225$	0	0
$226$	46.6687	3.10435
$227$	6.96435	0.462240	0.231120	−	0.972925i	$-0.425761\pi$
$227$	6.96435	0.462240	0.231120	+	0.972925i	$0.425761\pi$
$228$	14.1055	0.934157
$229$	6.67980	0.441413	0.220707	−	0.975340i	$-0.429164\pi$
$229$	6.67980	0.441413	0.220707	+	0.975340i	$0.429164\pi$
$230$	0	0
$231$	−3.25225	−0.213982
$232$	−23.5376	−1.54532
$233$	−4.46110	−0.292256	−0.146128	−	0.989266i	$-0.546681\pi$
$233$	−4.46110	−0.292256	−0.146128	+	0.989266i	$0.546681\pi$
$234$	55.1335	3.60419
$235$	0	0
$236$	−3.47362	−0.226113
$237$	29.8437	1.93856
$238$	−5.94034	−0.385055
$239$	3.20600	0.207379	0.103690	−	0.994610i	$-0.466935\pi$
$239$	3.20600	0.207379	0.103690	+	0.994610i	$0.466935\pi$
$240$	0	0
$241$	−18.5606	−1.19559	−0.597797	−	0.801647i	$-0.703957\pi$
$241$	−18.5606	−1.19559	−0.597797	+	0.801647i	$0.703957\pi$
$242$	2.59045	0.166521
$243$	0.588604	0.0377589
$244$	−26.8607	−1.71958
$245$	0	0
$246$	64.9430	4.14062
$247$	−3.56681	−0.226951
$248$	33.2039	2.10845
$249$	41.9791	2.66032
$250$	0	0
$251$	2.10322	0.132754	0.0663771	−	0.997795i	$-0.478856\pi$
$251$	2.10322	0.132754	0.0663771	+	0.997795i	$0.478856\pi$
$252$	30.5268	1.92301
$253$	4.81445	0.302682
$254$	10.3409	0.648843
$255$	0	0
$256$	−21.7293	−1.35808
$257$	−19.8087	−1.23563	−0.617817	−	0.786322i	$-0.711983\pi$
$257$	−19.8087	−1.23563	−0.617817	+	0.786322i	$0.711983\pi$
$258$	−61.6967	−3.84107
$259$	5.54624	0.344627
$260$	0	0
$261$	−20.0035	−1.23818
$262$	30.8978	1.90888
$263$	−13.1192	−0.808967	−0.404483	−	0.914545i	$-0.632549\pi$
$263$	−13.1192	−0.808967	−0.404483	+	0.914545i	$0.632549\pi$
$264$	−21.0253	−1.29402
$265$	0	0
$266$	−2.81342	−0.172502
$267$	−29.4295	−1.80105
$268$	−10.0219	−0.612186
$269$	−23.8316	−1.45304	−0.726520	−	0.687145i	$-0.758864\pi$
$269$	−23.8316	−1.45304	−0.726520	+	0.687145i	$0.758864\pi$
$270$	0	0
$271$	11.1137	0.675108	0.337554	−	0.941306i	$-0.390401\pi$
$271$	11.1137	0.675108	0.337554	+	0.941306i	$0.390401\pi$
$272$	−18.5118	−1.12244
$273$	−11.6001	−0.702072
$274$	−56.6638	−3.42319
$275$	0	0
$276$	−67.9100	−4.08770
$277$	−16.3055	−0.979700	−0.489850	−	0.871807i	$-0.662948\pi$
$277$	−16.3055	−0.979700	−0.489850	+	0.871807i	$0.662948\pi$
$278$	30.1922	1.81081
$279$	28.2185	1.68940
$280$	0	0
$281$	5.40191	0.322251	0.161125	−	0.986934i	$-0.448488\pi$
$281$	5.40191	0.322251	0.161125	+	0.986934i	$0.448488\pi$
$282$	−10.9437	−0.651685
$283$	3.52795	0.209715	0.104857	−	0.994487i	$-0.466561\pi$
$283$	3.52795	0.209715	0.104857	+	0.994487i	$0.466561\pi$
$284$	50.0112	2.96762
$285$	0	0
$286$	9.23965	0.546352
$287$	−9.09265	−0.536722
$288$	51.7285	3.04813
$289$	−12.5419	−0.737756
$290$	0	0
$291$	−26.6438	−1.56189
$292$	51.8480	3.03418
$293$	−30.9882	−1.81035	−0.905174	−	0.425040i	$-0.860260\pi$
$293$	−30.9882	−1.81035	−0.905174	+	0.425040i	$0.860260\pi$
$294$	45.1499	2.63320
$295$	0	0
$296$	35.8556	2.08406
$297$	−8.88487	−0.515553
$298$	25.0620	1.45180
$299$	17.1722	0.993095
$300$	0	0
$301$	8.63813	0.497893
$302$	37.8524	2.17816
$303$	21.4509	1.23232
$304$	−8.76742	−0.502846
$305$	0	0
$306$	−32.6372	−1.86575
$307$	−31.7550	−1.81236	−0.906178	−	0.422897i	$-0.861013\pi$
$307$	−31.7550	−1.81236	−0.906178	+	0.422897i	$0.861013\pi$
$308$	5.11589	0.291505
$309$	19.1848	1.09139
$310$	0	0
$311$	−19.8017	−1.12285	−0.561426	−	0.827527i	$-0.689747\pi$
$311$	−19.8017	−1.12285	−0.561426	+	0.827527i	$0.689747\pi$
$312$	−74.9931	−4.24565
$313$	−20.7094	−1.17057	−0.585283	−	0.810829i	$-0.699017\pi$
$313$	−20.7094	−1.17057	−0.585283	+	0.810829i	$0.699017\pi$
$314$	19.6445	1.10860
$315$	0	0
$316$	−46.9450	−2.64086
$317$	−32.8790	−1.84667	−0.923335	−	0.383996i	$-0.874548\pi$
$317$	−32.8790	−1.84667	−0.923335	+	0.383996i	$0.874548\pi$
$318$	−74.7880	−4.19390
$319$	−3.35232	−0.187694
$320$	0	0
$321$	24.3256	1.35772
$322$	13.5451	0.754837
$323$	2.11143	0.117483
$324$	41.0026	2.27792
$325$	0	0
$326$	44.2123	2.44869
$327$	−15.8720	−0.877721
$328$	−58.7826	−3.24572
$329$	1.53222	0.0844739
$330$	0	0
$331$	−19.8918	−1.09335	−0.546677	−	0.837343i	$-0.684108\pi$
$331$	−19.8918	−1.09335	−0.546677	+	0.837343i	$0.684108\pi$
$332$	−66.0344	−3.62411
$333$	30.4720	1.66985
$334$	18.0510	0.987707
$335$	0	0
$336$	−28.5138	−1.55556
$337$	19.9888	1.08886	0.544429	−	0.838807i	$-0.316746\pi$
$337$	19.9888	1.08886	0.544429	+	0.838807i	$0.316746\pi$
$338$	−0.719850	−0.0391547
$339$	−53.9479	−2.93005
$340$	0	0
$341$	4.72904	0.256092
$342$	−15.4574	−0.835840
$343$	−13.9239	−0.751822
$344$	55.8442	3.01092
$345$	0	0
$346$	66.4343	3.57153
$347$	11.6347	0.624582	0.312291	−	0.949987i	$-0.398904\pi$
$347$	11.6347	0.624582	0.312291	+	0.949987i	$0.398904\pi$
$348$	47.2860	2.53480
$349$	−33.9709	−1.81842	−0.909211	−	0.416335i	$-0.863314\pi$
$349$	−33.9709	−1.81842	−0.909211	+	0.416335i	$0.863314\pi$
$350$	0	0
$351$	−31.6906	−1.69152
$352$	8.66902	0.462060
$353$	−31.2710	−1.66439	−0.832194	−	0.554484i	$-0.812916\pi$
$353$	−31.2710	−1.66439	−0.832194	+	0.554484i	$0.812916\pi$
$354$	5.72032	0.304032
$355$	0	0
$356$	46.2934	2.45355
$357$	6.86690	0.363435
$358$	−40.6301	−2.14737
$359$	−9.02684	−0.476418	−0.238209	−	0.971214i	$-0.576560\pi$
$359$	−9.02684	−0.476418	−0.238209	+	0.971214i	$0.576560\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−26.5200	−1.39386
$363$	−2.99450	−0.157171
$364$	18.2474	0.956422
$365$	0	0
$366$	44.2340	2.31215
$367$	10.8245	0.565035	0.282517	−	0.959262i	$-0.408831\pi$
$367$	10.8245	0.565035	0.282517	+	0.959262i	$0.408831\pi$
$368$	42.2103	2.20036
$369$	−49.9565	−2.60063
$370$	0	0
$371$	10.4710	0.543629
$372$	−66.7053	−3.45851
$373$	21.7074	1.12397	0.561984	−	0.827148i	$-0.310038\pi$
$373$	21.7074	1.12397	0.561984	+	0.827148i	$0.310038\pi$
$374$	−5.46957	−0.282824
$375$	0	0
$376$	9.90555	0.510840
$377$	−11.9571	−0.615821
$378$	−24.9969	−1.28570
$379$	20.2102	1.03813	0.519063	−	0.854736i	$-0.326281\pi$
$379$	20.2102	1.03813	0.519063	+	0.854736i	$0.326281\pi$
$380$	0	0
$381$	−11.9538	−0.612412
$382$	28.5981	1.46321
$383$	34.9348	1.78509	0.892543	−	0.450962i	$-0.148919\pi$
$383$	34.9348	1.78509	0.892543	+	0.450962i	$0.148919\pi$
$384$	13.7395	0.701141
$385$	0	0
$386$	39.9198	2.03186
$387$	47.4593	2.41249
$388$	41.9116	2.12774
$389$	−32.2359	−1.63443	−0.817213	−	0.576336i	$-0.804482\pi$
$389$	−32.2359	−1.63443	−0.817213	+	0.576336i	$0.804482\pi$
$390$	0	0
$391$	−10.1654	−0.514086
$392$	−40.8670	−2.06410
$393$	−35.7172	−1.80169
$394$	40.8089	2.05592
$395$	0	0
$396$	28.1075	1.41246
$397$	−21.6075	−1.08445	−0.542224	−	0.840234i	$-0.682418\pi$
$397$	−21.6075	−1.08445	−0.542224	+	0.840234i	$0.682418\pi$
$398$	22.7291	1.13931
$399$	3.25225	0.162816
$400$	0	0
$401$	32.9816	1.64702	0.823512	−	0.567299i	$-0.192012\pi$
$401$	32.9816	1.64702	0.823512	+	0.567299i	$0.192012\pi$
$402$	16.5040	0.823145
$403$	16.8676	0.840234
$404$	−33.7430	−1.67878
$405$	0	0
$406$	−9.43148	−0.468076
$407$	5.10670	0.253130
$408$	44.3934	2.19780
$409$	8.12644	0.401826	0.200913	−	0.979609i	$-0.435609\pi$
$409$	8.12644	0.401826	0.200913	+	0.979609i	$0.435609\pi$
$410$	0	0
$411$	65.5021	3.23098
$412$	−30.1783	−1.48678
$413$	−0.800900	−0.0394097
$414$	74.4188	3.65749
$415$	0	0
$416$	30.9207	1.51601
$417$	−34.9015	−1.70913
$418$	−2.59045	−0.126703
$419$	29.7694	1.45433	0.727166	−	0.686462i	$-0.240837\pi$
$419$	29.7694	1.45433	0.727166	+	0.686462i	$0.240837\pi$
$420$	0	0
$421$	−2.95550	−0.144042	−0.0720211	−	0.997403i	$-0.522945\pi$
$421$	−2.95550	−0.144042	−0.0720211	+	0.997403i	$0.522945\pi$
$422$	23.8180	1.15944
$423$	8.41826	0.409310
$424$	67.6936	3.28749
$425$	0	0
$426$	−82.3581	−3.99026
$427$	−6.19319	−0.299709
$428$	−38.2650	−1.84961
$429$	−10.6808	−0.515675
$430$	0	0
$431$	−15.7212	−0.757262	−0.378631	−	0.925548i	$-0.623605\pi$
$431$	−15.7212	−0.757262	−0.378631	+	0.925548i	$0.623605\pi$
$432$	−77.8974	−3.74784
$433$	−17.1336	−0.823386	−0.411693	−	0.911323i	$-0.635062\pi$
$433$	−17.1336	−0.823386	−0.411693	+	0.911323i	$0.635062\pi$
$434$	13.3048	0.638649
$435$	0	0
$436$	24.9671	1.19571
$437$	−4.81445	−0.230306
$438$	−85.3829	−4.07975
$439$	6.13113	0.292623	0.146311	−	0.989239i	$-0.453260\pi$
$439$	6.13113	0.292623	0.146311	+	0.989239i	$0.453260\pi$
$440$	0	0
$441$	−34.7310	−1.65385
$442$	−19.5089	−0.927943
$443$	−16.9204	−0.803914	−0.401957	−	0.915659i	$-0.631670\pi$
$443$	−16.9204	−0.803914	−0.401957	+	0.915659i	$0.631670\pi$
$444$	−72.0323	−3.41850
$445$	0	0
$446$	42.3357	2.00465
$447$	−28.9711	−1.37029
$448$	5.34547	0.252550
$449$	−10.5785	−0.499230	−0.249615	−	0.968345i	$-0.580304\pi$
$449$	−10.5785	−0.499230	−0.249615	+	0.968345i	$0.580304\pi$
$450$	0	0
$451$	−8.37205	−0.394225
$452$	84.8617	3.99156
$453$	−43.7565	−2.05586
$454$	18.0408	0.846698
$455$	0	0
$456$	21.0253	0.984599
$457$	−0.207034	−0.00968465	−0.00484232	−	0.999988i	$-0.501541\pi$
$457$	−0.207034	−0.00968465	−0.00484232	+	0.999988i	$0.501541\pi$
$458$	17.3037	0.808549
$459$	18.7598	0.875633
$460$	0	0
$461$	−24.6854	−1.14971	−0.574856	−	0.818254i	$-0.694942\pi$
$461$	−24.6854	−1.14971	−0.574856	+	0.818254i	$0.694942\pi$
$462$	−8.42480	−0.391957
$463$	32.3068	1.50142	0.750712	−	0.660630i	$-0.229711\pi$
$463$	32.3068	1.50142	0.750712	+	0.660630i	$0.229711\pi$
$464$	−29.3912	−1.36445
$465$	0	0
$466$	−11.5563	−0.535334
$467$	−20.3410	−0.941269	−0.470634	−	0.882328i	$-0.655975\pi$
$467$	−20.3410	−0.941269	−0.470634	+	0.882328i	$0.655975\pi$
$468$	100.254	4.63425
$469$	−2.31072	−0.106699
$470$	0	0
$471$	−22.7086	−1.04636
$472$	−5.17769	−0.238323
$473$	7.95355	0.365705
$474$	77.3086	3.55090
$475$	0	0
$476$	−10.8018	−0.495102
$477$	57.5296	2.63410
$478$	8.30500	0.379862
$479$	6.75628	0.308702	0.154351	−	0.988016i	$-0.450671\pi$
$479$	6.75628	0.308702	0.154351	+	0.988016i	$0.450671\pi$
$480$	0	0
$481$	18.2146	0.830514
$482$	−48.0804	−2.19000
$483$	−15.6578	−0.712454
$484$	4.71045	0.214111
$485$	0	0
$486$	1.52475	0.0691641
$487$	22.2673	1.00903	0.504514	−	0.863403i	$-0.331672\pi$
$487$	22.2673	1.00903	0.504514	+	0.863403i	$0.331672\pi$
$488$	−40.0380	−1.81244
$489$	−51.1084	−2.31120
$490$	0	0
$491$	37.5046	1.69256	0.846278	−	0.532741i	$-0.178838\pi$
$491$	37.5046	1.69256	0.846278	+	0.532741i	$0.178838\pi$
$492$	118.092	5.32398
$493$	7.07819	0.318786
$494$	−9.23965	−0.415711
$495$	0	0
$496$	41.4615	1.86168
$497$	11.5309	0.517233
$498$	108.745	4.87298
$499$	−18.1313	−0.811670	−0.405835	−	0.913946i	$-0.633019\pi$
$499$	−18.1313	−0.811670	−0.405835	+	0.913946i	$0.633019\pi$
$500$	0	0
$501$	−20.8665	−0.932248
$502$	5.44830	0.243169
$503$	6.96999	0.310776	0.155388	−	0.987853i	$-0.450337\pi$
$503$	6.96999	0.310776	0.155388	+	0.987853i	$0.450337\pi$
$504$	45.5025	2.02684
$505$	0	0
$506$	12.4716	0.554431
$507$	0.832131	0.0369562
$508$	18.8037	0.834279
$509$	−27.9741	−1.23993	−0.619966	−	0.784629i	$-0.712854\pi$
$509$	−27.9741	−1.23993	−0.619966	+	0.784629i	$0.712854\pi$
$510$	0	0
$511$	11.9544	0.528833
$512$	−47.1122	−2.08209
$513$	8.88487	0.392277
$514$	−51.3135	−2.26334
$515$	0	0
$516$	−112.188	−4.93882
$517$	1.41079	0.0620464
$518$	14.3673	0.631262
$519$	−76.7966	−3.37100
$520$	0	0
$521$	−1.35523	−0.0593736	−0.0296868	−	0.999559i	$-0.509451\pi$
$521$	−1.35523	−0.0593736	−0.0296868	+	0.999559i	$0.509451\pi$
$522$	−51.8181	−2.26802
$523$	18.1213	0.792391	0.396196	−	0.918166i	$-0.370330\pi$
$523$	18.1213	0.792391	0.396196	+	0.918166i	$0.370330\pi$
$524$	56.1842	2.45442
$525$	0	0
$526$	−33.9848	−1.48181
$527$	−9.98505	−0.434955
$528$	−26.2541	−1.14256
$529$	0.178941	0.00778006
$530$	0	0
$531$	−4.40028	−0.190956
$532$	−5.11589	−0.221802
$533$	−29.8615	−1.29344
$534$	−76.2356	−3.29904
$535$	0	0
$536$	−14.9384	−0.645243
$537$	46.9674	2.02679
$538$	−61.7347	−2.66157
$539$	−5.82045	−0.250704
$540$	0	0
$541$	−8.41360	−0.361729	−0.180864	−	0.983508i	$-0.557890\pi$
$541$	−8.41360	−0.361729	−0.180864	+	0.983508i	$0.557890\pi$
$542$	28.7895	1.23661
$543$	30.6565	1.31560
$544$	−18.3040	−0.784779
$545$	0	0
$546$	−30.0496	−1.28601
$547$	20.0414	0.856907	0.428453	−	0.903564i	$-0.359059\pi$
$547$	20.0414	0.856907	0.428453	+	0.903564i	$0.359059\pi$
$548$	−103.037	−4.40151
$549$	−34.0264	−1.45221
$550$	0	0
$551$	3.35232	0.142814
$552$	−101.225	−4.30843
$553$	−10.8240	−0.460281
$554$	−42.2385	−1.79454
$555$	0	0
$556$	54.9011	2.32833
$557$	−16.5714	−0.702154	−0.351077	−	0.936347i	$-0.614184\pi$
$557$	−16.5714	−0.702154	−0.351077	+	0.936347i	$0.614184\pi$
$558$	73.0986	3.09451
$559$	28.3688	1.19987
$560$	0	0
$561$	6.32269	0.266944
$562$	13.9934	0.590275
$563$	−22.8289	−0.962123	−0.481061	−	0.876687i	$-0.659749\pi$
$563$	−22.8289	−0.962123	−0.481061	+	0.876687i	$0.659749\pi$
$564$	−19.8998	−0.837933
$565$	0	0
$566$	9.13898	0.384140
$567$	9.45384	0.397024
$568$	74.5456	3.12786
$569$	−12.4875	−0.523502	−0.261751	−	0.965135i	$-0.584300\pi$
$569$	−12.4875	−0.523502	−0.261751	+	0.965135i	$0.584300\pi$
$570$	0	0
$571$	−30.6254	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$571$	−30.6254	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$572$	16.8013	0.702496
$573$	−33.0587	−1.38105
$574$	−23.5541	−0.983129
$575$	0	0
$576$	29.3689	1.22370
$577$	14.1736	0.590053	0.295027	−	0.955489i	$-0.404671\pi$
$577$	14.1736	0.590053	0.295027	+	0.955489i	$0.404671\pi$
$578$	−32.4891	−1.35137
$579$	−46.1464	−1.91778
$580$	0	0
$581$	−15.2253	−0.631654
$582$	−69.0196	−2.86096
$583$	9.64120	0.399298
$584$	77.2835	3.19801
$585$	0	0
$586$	−80.2734	−3.31606
$587$	−39.0045	−1.60989	−0.804943	−	0.593352i	$-0.797804\pi$
$587$	−39.0045	−1.60989	−0.804943	+	0.593352i	$0.797804\pi$
$588$	82.1001	3.38575
$589$	−4.72904	−0.194857
$590$	0	0
$591$	−47.1741	−1.94048
$592$	44.7725	1.84014
$593$	43.5330	1.78769	0.893843	−	0.448381i	$-0.147999\pi$
$593$	43.5330	1.78769	0.893843	+	0.448381i	$0.147999\pi$
$594$	−23.0158	−0.944352
$595$	0	0
$596$	45.5725	1.86672
$597$	−26.2743	−1.07534
$598$	44.4838	1.81908
$599$	19.2721	0.787437	0.393719	−	0.919231i	$-0.371188\pi$
$599$	19.2721	0.787437	0.393719	+	0.919231i	$0.371188\pi$
$600$	0	0
$601$	−21.9919	−0.897067	−0.448533	−	0.893766i	$-0.648053\pi$
$601$	−21.9919	−0.897067	−0.448533	+	0.893766i	$0.648053\pi$
$602$	22.3767	0.912005
$603$	−12.6955	−0.517000
$604$	68.8304	2.80067
$605$	0	0
$606$	55.5677	2.25728
$607$	−17.9582	−0.728902	−0.364451	−	0.931223i	$-0.618743\pi$
$607$	−17.9582	−0.728902	−0.364451	+	0.931223i	$0.618743\pi$
$608$	−8.66902	−0.351575
$609$	10.9026	0.441795
$610$	0	0
$611$	5.03201	0.203573
$612$	−59.3471	−2.39896
$613$	−38.3600	−1.54935	−0.774673	−	0.632362i	$-0.782086\pi$
$613$	−38.3600	−1.54935	−0.774673	+	0.632362i	$0.782086\pi$
$614$	−82.2599	−3.31974
$615$	0	0
$616$	7.62562	0.307245
$617$	−37.0412	−1.49122	−0.745610	−	0.666382i	$-0.767842\pi$
$617$	−37.0412	−1.49122	−0.745610	+	0.666382i	$0.767842\pi$
$618$	49.6973	1.99912
$619$	−42.9798	−1.72750	−0.863752	−	0.503917i	$-0.831892\pi$
$619$	−42.9798	−1.72750	−0.863752	+	0.503917i	$0.831892\pi$
$620$	0	0
$621$	−42.7758	−1.71653
$622$	−51.2954	−2.05676
$623$	10.6737	0.427634
$624$	−93.6432	−3.74873
$625$	0	0
$626$	−53.6468	−2.14416
$627$	2.99450	0.119589
$628$	35.7213	1.42544
$629$	−10.7824	−0.429924
$630$	0	0
$631$	−31.9210	−1.27076	−0.635378	−	0.772201i	$-0.719156\pi$
$631$	−31.9210	−1.27076	−0.635378	+	0.772201i	$0.719156\pi$
$632$	−69.9752	−2.78346
$633$	−27.5331	−1.09434
$634$	−85.1716	−3.38259
$635$	0	0
$636$	−135.994	−5.39250
$637$	−20.7604	−0.822558
$638$	−8.68403	−0.343804
$639$	63.3528	2.50620
$640$	0	0
$641$	−18.8432	−0.744262	−0.372131	−	0.928180i	$-0.621373\pi$
$641$	−18.8432	−0.744262	−0.372131	+	0.928180i	$0.621373\pi$
$642$	63.0144	2.48698
$643$	15.4713	0.610129	0.305064	−	0.952332i	$-0.401322\pi$
$643$	15.4713	0.610129	0.305064	+	0.952332i	$0.401322\pi$
$644$	24.6302	0.970565
$645$	0	0
$646$	5.46957	0.215197
$647$	−8.22385	−0.323313	−0.161656	−	0.986847i	$-0.551684\pi$
$647$	−8.22385	−0.323313	−0.161656	+	0.986847i	$0.551684\pi$
$648$	61.1176	2.40093
$649$	−0.737428	−0.0289466
$650$	0	0
$651$	−15.3800	−0.602790
$652$	80.3951	3.14852
$653$	−9.29054	−0.363567	−0.181783	−	0.983339i	$-0.558187\pi$
$653$	−9.29054	−0.363567	−0.181783	+	0.983339i	$0.558187\pi$
$654$	−41.1156	−1.60775
$655$	0	0
$656$	−73.4013	−2.86584
$657$	65.6796	2.56240
$658$	3.96914	0.154733
$659$	31.5012	1.22711	0.613556	−	0.789651i	$-0.289738\pi$
$659$	31.5012	1.22711	0.613556	+	0.789651i	$0.289738\pi$
$660$	0	0
$661$	1.08128	0.0420569	0.0210285	−	0.999779i	$-0.493306\pi$
$661$	1.08128	0.0420569	0.0210285	+	0.999779i	$0.493306\pi$
$662$	−51.5289	−2.00273
$663$	22.5518	0.875840
$664$	−98.4295	−3.81980
$665$	0	0
$666$	78.9362	3.05871
$667$	−16.1396	−0.624927
$668$	32.8237	1.26999
$669$	−48.9391	−1.89210
$670$	0	0
$671$	−5.70238	−0.220138
$672$	−28.1938	−1.08760
$673$	−43.8982	−1.69215	−0.846075	−	0.533064i	$-0.821041\pi$
$673$	−43.8982	−1.69215	−0.846075	+	0.533064i	$0.821041\pi$
$674$	51.7800	1.99449
$675$	0	0
$676$	−1.30897	−0.0503449
$677$	10.1422	0.389797	0.194898	−	0.980823i	$-0.437562\pi$
$677$	10.1422	0.389797	0.194898	+	0.980823i	$0.437562\pi$
$678$	−139.750	−5.36705
$679$	9.66341	0.370848
$680$	0	0
$681$	−20.8548	−0.799157
$682$	12.2504	0.469090
$683$	−0.666978	−0.0255212	−0.0127606	−	0.999919i	$-0.504062\pi$
$683$	−0.666978	−0.0255212	−0.0127606	+	0.999919i	$0.504062\pi$
$684$	−28.1075	−1.07472
$685$	0	0
$686$	−36.0693	−1.37713
$687$	−20.0027	−0.763150
$688$	69.7321	2.65851
$689$	34.3883	1.31009
$690$	0	0
$691$	23.8707	0.908085	0.454042	−	0.890980i	$-0.349981\pi$
$691$	23.8707	0.908085	0.454042	+	0.890980i	$0.349981\pi$
$692$	120.803	4.59225
$693$	6.48066	0.246180
$694$	30.1391	1.14406
$695$	0	0
$696$	70.4834	2.67167
$697$	17.6770	0.669565
$698$	−88.0001	−3.33085
$699$	13.3588	0.505276
$700$	0	0
$701$	42.6513	1.61092	0.805460	−	0.592651i	$-0.201919\pi$
$701$	42.6513	1.61092	0.805460	+	0.592651i	$0.201919\pi$
$702$	−82.0931	−3.09840
$703$	−5.10670	−0.192603
$704$	4.92184	0.185499
$705$	0	0
$706$	−81.0061	−3.04871
$707$	−7.78001	−0.292597
$708$	10.4018	0.390922
$709$	−29.4487	−1.10597	−0.552985	−	0.833191i	$-0.686511\pi$
$709$	−29.4487	−1.10597	−0.552985	+	0.833191i	$0.686511\pi$
$710$	0	0
$711$	−59.4686	−2.23025
$712$	69.0040	2.58603
$713$	22.7677	0.852658
$714$	17.7884	0.665713
$715$	0	0
$716$	−73.8812	−2.76107
$717$	−9.60039	−0.358533
$718$	−23.3836	−0.872668
$719$	−38.5175	−1.43646	−0.718230	−	0.695805i	$-0.755048\pi$
$719$	−38.5175	−1.43646	−0.718230	+	0.695805i	$0.755048\pi$
$720$	0	0
$721$	−6.95810	−0.259133
$722$	2.59045	0.0964067
$723$	55.5799	2.06704
$724$	−48.2236	−1.79222
$725$	0	0
$726$	−7.75712	−0.287894
$727$	13.9379	0.516929	0.258464	−	0.966021i	$-0.416784\pi$
$727$	13.9379	0.516929	0.258464	+	0.966021i	$0.416784\pi$
$728$	27.1991	1.00807
$729$	−27.8764	−1.03246
$730$	0	0
$731$	−16.7934	−0.621126
$732$	80.4346	2.97295
$733$	−23.2536	−0.858891	−0.429445	−	0.903093i	$-0.641291\pi$
$733$	−23.2536	−0.858891	−0.429445	+	0.903093i	$0.641291\pi$
$734$	28.0404	1.03499
$735$	0	0
$736$	41.7366	1.53843
$737$	−2.12759	−0.0783709
$738$	−129.410	−4.76365
$739$	28.2385	1.03877	0.519385	−	0.854540i	$-0.326161\pi$
$739$	28.2385	1.03877	0.519385	+	0.854540i	$0.326161\pi$
$740$	0	0
$741$	10.6808	0.392370
$742$	27.1247	0.995780
$743$	24.5547	0.900823	0.450412	−	0.892821i	$-0.351277\pi$
$743$	24.5547	0.900823	0.450412	+	0.892821i	$0.351277\pi$
$744$	−99.4294	−3.64526
$745$	0	0
$746$	56.2321	2.05880
$747$	−83.6505	−3.06061
$748$	−9.94579	−0.363654
$749$	−8.82263	−0.322372
$750$	0	0
$751$	−39.8070	−1.45258	−0.726290	−	0.687389i	$-0.758757\pi$
$751$	−39.8070	−1.45258	−0.726290	+	0.687389i	$0.758757\pi$
$752$	12.3690	0.451050
$753$	−6.29811	−0.229516
$754$	−30.9742	−1.12802
$755$	0	0
$756$	−45.4540	−1.65315
$757$	−13.9769	−0.508000	−0.254000	−	0.967204i	$-0.581746\pi$
$757$	−13.9769	−0.508000	−0.254000	+	0.967204i	$0.581746\pi$
$758$	52.3535	1.90156
$759$	−14.4169	−0.523300
$760$	0	0
$761$	30.8110	1.11690	0.558449	−	0.829539i	$-0.311397\pi$
$761$	30.8110	1.11690	0.558449	+	0.829539i	$0.311397\pi$
$762$	−30.9658	−1.12177
$763$	5.75657	0.208402
$764$	52.0024	1.88138
$765$	0	0
$766$	90.4970	3.26979
$767$	−2.63026	−0.0949733
$768$	65.0685	2.34796
$769$	41.5445	1.49813	0.749066	−	0.662496i	$-0.230503\pi$
$769$	41.5445	1.49813	0.749066	+	0.662496i	$0.230503\pi$
$770$	0	0
$771$	59.3173	2.13626
$772$	72.5896	2.61256
$773$	16.7752	0.603363	0.301681	−	0.953409i	$-0.402452\pi$
$773$	16.7752	0.603363	0.301681	+	0.953409i	$0.402452\pi$
$774$	122.941	4.41903
$775$	0	0
$776$	62.4724	2.24263
$777$	−16.6082	−0.595817
$778$	−83.5057	−2.99382
$779$	8.37205	0.299960
$780$	0	0
$781$	10.6171	0.379909
$782$	−26.3330	−0.941665
$783$	29.7849	1.06443
$784$	−51.0303	−1.82251
$785$	0	0
$786$	−92.5237	−3.30021
$787$	45.4889	1.62150	0.810752	−	0.585390i	$-0.199059\pi$
$787$	45.4889	1.62150	0.810752	+	0.585390i	$0.199059\pi$
$788$	74.2063	2.64349
$789$	39.2856	1.39860
$790$	0	0
$791$	19.5663	0.695697
$792$	41.8964	1.48872
$793$	−20.3393	−0.722269
$794$	−55.9731	−1.98641
$795$	0	0
$796$	41.3303	1.46491
$797$	−1.55392	−0.0550428	−0.0275214	−	0.999621i	$-0.508761\pi$
$797$	−1.55392	−0.0550428	−0.0275214	+	0.999621i	$0.508761\pi$
$798$	8.42480	0.298235
$799$	−2.97878	−0.105382
$800$	0	0
$801$	58.6432	2.07206
$802$	85.4373	3.01690
$803$	11.0070	0.388430
$804$	30.0107	1.05840
$805$	0	0
$806$	43.6947	1.53908
$807$	71.3639	2.51213
$808$	−50.2965	−1.76943
$809$	10.4343	0.366849	0.183425	−	0.983034i	$-0.441282\pi$
$809$	10.4343	0.366849	0.183425	+	0.983034i	$0.441282\pi$
$810$	0	0
$811$	47.3513	1.66273	0.831364	−	0.555729i	$-0.187561\pi$
$811$	47.3513	1.66273	0.831364	+	0.555729i	$0.187561\pi$
$812$	−17.1501	−0.601850
$813$	−33.2800	−1.16718
$814$	13.2287	0.463664
$815$	0	0
$816$	55.4337	1.94057
$817$	−7.95355	−0.278259
$818$	21.0511	0.736036
$819$	23.1152	0.807712
$820$	0	0
$821$	−15.8887	−0.554521	−0.277260	−	0.960795i	$-0.589426\pi$
$821$	−15.8887	−0.554521	−0.277260	+	0.960795i	$0.589426\pi$
$822$	169.680	5.91827
$823$	−15.6533	−0.545640	−0.272820	−	0.962065i	$-0.587956\pi$
$823$	−15.6533	−0.545640	−0.272820	+	0.962065i	$0.587956\pi$
$824$	−44.9831	−1.56706
$825$	0	0
$826$	−2.07469	−0.0721878
$827$	−42.8097	−1.48864	−0.744319	−	0.667824i	$-0.767226\pi$
$827$	−42.8097	−1.48864	−0.744319	+	0.667824i	$0.767226\pi$
$828$	135.322	4.70277
$829$	−27.2954	−0.948007	−0.474004	−	0.880523i	$-0.657192\pi$
$829$	−27.2954	−0.948007	−0.474004	+	0.880523i	$0.657192\pi$
$830$	0	0
$831$	48.8268	1.69378
$832$	17.5553	0.608619
$833$	12.2895	0.425805
$834$	−90.4107	−3.13066
$835$	0	0
$836$	−4.71045	−0.162914
$837$	−42.0169	−1.45232
$838$	77.1163	2.66394
$839$	−4.39518	−0.151739	−0.0758693	−	0.997118i	$-0.524173\pi$
$839$	−4.39518	−0.151739	−0.0758693	+	0.997118i	$0.524173\pi$
$840$	0	0
$841$	−17.7620	−0.612481
$842$	−7.65608	−0.263846
$843$	−16.1760	−0.557132
$844$	43.3104	1.49080
$845$	0	0
$846$	21.8071	0.749743
$847$	1.08607	0.0373179
$848$	84.5285	2.90272
$849$	−10.5645	−0.362571
$850$	0	0
$851$	24.5859	0.842795
$852$	−149.759	−5.13065
$853$	0.792470	0.0271337	0.0135668	−	0.999908i	$-0.495681\pi$
$853$	0.792470	0.0271337	0.0135668	+	0.999908i	$0.495681\pi$
$854$	−16.0432	−0.548986
$855$	0	0
$856$	−57.0369	−1.94948
$857$	35.8171	1.22349	0.611744	−	0.791056i	$-0.290468\pi$
$857$	35.8171	1.22349	0.611744	+	0.791056i	$0.290468\pi$
$858$	−27.6682	−0.944576
$859$	−45.8663	−1.56494	−0.782470	−	0.622689i	$-0.786040\pi$
$859$	−45.8663	−1.56494	−0.782470	+	0.622689i	$0.786040\pi$
$860$	0	0
$861$	27.2280	0.927927
$862$	−40.7250	−1.38710
$863$	1.85336	0.0630892	0.0315446	−	0.999502i	$-0.489957\pi$
$863$	1.85336	0.0630892	0.0315446	+	0.999502i	$0.489957\pi$
$864$	−77.0231	−2.62038
$865$	0	0
$866$	−44.3837	−1.50822
$867$	37.5566	1.27549
$868$	24.1932	0.821172
$869$	−9.96615	−0.338078
$870$	0	0
$871$	−7.58872	−0.257134
$872$	37.2154	1.26027
$873$	53.0924	1.79690
$874$	−12.4716	−0.421859
$875$	0	0
$876$	−155.259	−5.24572
$877$	28.7139	0.969598	0.484799	−	0.874626i	$-0.338893\pi$
$877$	28.7139	0.969598	0.484799	+	0.874626i	$0.338893\pi$
$878$	15.8824	0.536005
$879$	92.7943	3.12987
$880$	0	0
$881$	14.9317	0.503061	0.251531	−	0.967849i	$-0.419066\pi$
$881$	14.9317	0.503061	0.251531	+	0.967849i	$0.419066\pi$
$882$	−89.9689	−3.02941
$883$	−28.3711	−0.954765	−0.477382	−	0.878696i	$-0.658414\pi$
$883$	−28.3711	−0.954765	−0.477382	+	0.878696i	$0.658414\pi$
$884$	−35.4747	−1.19314
$885$	0	0
$886$	−43.8316	−1.47255
$887$	2.80865	0.0943053	0.0471527	−	0.998888i	$-0.484985\pi$
$887$	2.80865	0.0943053	0.0471527	+	0.998888i	$0.484985\pi$
$888$	−107.370	−3.60309
$889$	4.33550	0.145408
$890$	0	0
$891$	8.70462	0.291616
$892$	76.9828	2.57757
$893$	−1.41079	−0.0472102
$894$	−75.0483	−2.50999
$895$	0	0
$896$	−4.98316	−0.166476
$897$	−51.4223	−1.71694
$898$	−27.4031	−0.914453
$899$	−15.8533	−0.528736
$900$	0	0
$901$	−20.3567	−0.678181
$902$	−21.6874	−0.722112
$903$	−25.8669	−0.860797
$904$	126.493	4.20709
$905$	0	0
$906$	−113.349	−3.76578
$907$	47.3996	1.57388	0.786939	−	0.617031i	$-0.211665\pi$
$907$	47.3996	1.57388	0.786939	+	0.617031i	$0.211665\pi$
$908$	32.8052	1.08868
$909$	−42.7446	−1.41775
$910$	0	0
$911$	−21.0979	−0.699005	−0.349502	−	0.936935i	$-0.613649\pi$
$911$	−21.0979	−0.699005	−0.349502	+	0.936935i	$0.613649\pi$
$912$	26.2541	0.869359
$913$	−14.0187	−0.463952
$914$	−0.536312	−0.0177396
$915$	0	0
$916$	31.4648	1.03963
$917$	12.9542	0.427786
$918$	48.5964	1.60392
$919$	19.3565	0.638512	0.319256	−	0.947668i	$-0.396567\pi$
$919$	19.3565	0.638512	0.319256	+	0.947668i	$0.396567\pi$
$920$	0	0
$921$	95.0906	3.13334
$922$	−63.9463	−2.10596
$923$	37.8691	1.24648
$924$	−15.3195	−0.503976
$925$	0	0
$926$	83.6892	2.75020
$927$	−38.2290	−1.25560
$928$	−29.0613	−0.953984
$929$	−9.90582	−0.324999	−0.162500	−	0.986709i	$-0.551956\pi$
$929$	−9.90582	−0.324999	−0.162500	+	0.986709i	$0.551956\pi$
$930$	0	0
$931$	5.82045	0.190757
$932$	−21.0138	−0.688329
$933$	59.2963	1.94127
$934$	−52.6924	−1.72415
$935$	0	0
$936$	149.436	4.88448
$937$	41.1901	1.34562	0.672811	−	0.739814i	$-0.265087\pi$
$937$	41.1901	1.34562	0.672811	+	0.739814i	$0.265087\pi$
$938$	−5.98581	−0.195444
$939$	62.0145	2.02377
$940$	0	0
$941$	−46.3419	−1.51070	−0.755351	−	0.655320i	$-0.772534\pi$
$941$	−46.3419	−1.51070	−0.755351	+	0.655320i	$0.772534\pi$
$942$	−58.8256	−1.91664
$943$	−40.3068	−1.31257
$944$	−6.46534	−0.210429
$945$	0	0
$946$	20.6033	0.669871
$947$	−24.4233	−0.793650	−0.396825	−	0.917894i	$-0.629888\pi$
$947$	−24.4233	−0.793650	−0.396825	+	0.917894i	$0.629888\pi$
$948$	140.577	4.56573
$949$	39.2599	1.27443
$950$	0	0
$951$	98.4564	3.19267
$952$	−16.1010	−0.521836
$953$	10.5765	0.342607	0.171303	−	0.985218i	$-0.445202\pi$
$953$	10.5765	0.342607	0.171303	+	0.985218i	$0.445202\pi$
$954$	149.028	4.82495
$955$	0	0
$956$	15.1017	0.488424
$957$	10.0385	0.324500
$958$	17.5018	0.565458
$959$	−23.7568	−0.767148
$960$	0	0
$961$	−8.63618	−0.278586
$962$	47.1841	1.52128
$963$	−48.4730	−1.56202
$964$	−87.4288	−2.81589
$965$	0	0
$966$	−40.5608	−1.30502
$967$	−16.1131	−0.518161	−0.259081	−	0.965856i	$-0.583420\pi$
$967$	−16.1131	−0.518161	−0.259081	+	0.965856i	$0.583420\pi$
$968$	7.02129	0.225673
$969$	−6.32269	−0.203114
$970$	0	0
$971$	−57.4053	−1.84222	−0.921111	−	0.389299i	$-0.872717\pi$
$971$	−57.4053	−1.84222	−0.921111	+	0.389299i	$0.872717\pi$
$972$	2.77259	0.0889308
$973$	12.6584	0.405808
$974$	57.6825	1.84827
$975$	0	0
$976$	−49.9951	−1.60030
$977$	12.2856	0.393050	0.196525	−	0.980499i	$-0.437034\pi$
$977$	12.2856	0.393050	0.196525	+	0.980499i	$0.437034\pi$
$978$	−132.394	−4.23349
$979$	9.82782	0.314099
$980$	0	0
$981$	31.6276	1.00979
$982$	97.1538	3.10030
$983$	−2.24299	−0.0715404	−0.0357702	−	0.999360i	$-0.511388\pi$
$983$	−2.24299	−0.0715404	−0.0357702	+	0.999360i	$0.511388\pi$
$984$	176.025	5.61146
$985$	0	0
$986$	18.3357	0.583929
$987$	−4.58823	−0.146045
$988$	−16.8013	−0.534519
$989$	38.2920	1.21761
$990$	0	0
$991$	31.9640	1.01537	0.507685	−	0.861543i	$-0.330502\pi$
$991$	31.9640	1.01537	0.507685	+	0.861543i	$0.330502\pi$
$992$	40.9961	1.30163
$993$	59.5662	1.89028
$994$	29.8703	0.947429
$995$	0	0
$996$	197.740	6.26565
$997$	10.1521	0.321519	0.160759	−	0.986994i	$-0.448606\pi$
$997$	10.1521	0.321519	0.160759	+	0.986994i	$0.448606\pi$
$998$	−46.9683	−1.48676
$999$	−45.3723	−1.43552

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.x.1.13	yes	15
5.4	even	2		5225.2.a.s.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.s.1.3	✓	15	5.4	even	2
5225.2.a.x.1.13	yes	15	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)

\(f(q)\)	\(=\)	\(q+2.59045 q^{2} -2.99450 q^{3} +4.71045 q^{4} -7.75712 q^{6} +1.08607 q^{7} +7.02129 q^{8} +5.96706 q^{9} +1.00000 q^{11} -14.1055 q^{12} +3.56681 q^{13} +2.81342 q^{14} +8.76742 q^{16} -2.11143 q^{17} +15.4574 q^{18} -1.00000 q^{19} -3.25225 q^{21} +2.59045 q^{22} +4.81445 q^{23} -21.0253 q^{24} +9.23965 q^{26} -8.88487 q^{27} +5.11589 q^{28} -3.35232 q^{29} +4.72904 q^{31} +8.66902 q^{32} -2.99450 q^{33} -5.46957 q^{34} +28.1075 q^{36} +5.10670 q^{37} -2.59045 q^{38} -10.6808 q^{39} -8.37205 q^{41} -8.42480 q^{42} +7.95355 q^{43} +4.71045 q^{44} +12.4716 q^{46} +1.41079 q^{47} -26.2541 q^{48} -5.82045 q^{49} +6.32269 q^{51} +16.8013 q^{52} +9.64120 q^{53} -23.0158 q^{54} +7.62562 q^{56} +2.99450 q^{57} -8.68403 q^{58} -0.737428 q^{59} -5.70238 q^{61} +12.2504 q^{62} +6.48066 q^{63} +4.92184 q^{64} -7.75712 q^{66} -2.12759 q^{67} -9.94579 q^{68} -14.4169 q^{69} +10.6171 q^{71} +41.8964 q^{72} +11.0070 q^{73} +13.2287 q^{74} -4.71045 q^{76} +1.08607 q^{77} -27.6682 q^{78} -9.96615 q^{79} +8.70462 q^{81} -21.6874 q^{82} -14.0187 q^{83} -15.3195 q^{84} +20.6033 q^{86} +10.0385 q^{87} +7.02129 q^{88} +9.82782 q^{89} +3.87381 q^{91} +22.6782 q^{92} -14.1611 q^{93} +3.65458 q^{94} -25.9594 q^{96} +8.89758 q^{97} -15.0776 q^{98} +5.96706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 15 * q^7 + 15 * q^8 + 19 * q^9 + 15 * q^11 + 9 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 11 * q^17 + 16 * q^18 - 15 * q^19 + 5 * q^22 + 10 * q^23 + 17 * q^24 - 17 * q^26 + 13 * q^27 + 30 * q^28 + 5 * q^29 + 6 * q^31 + 40 * q^32 + 4 * q^33 + 17 * q^34 + 28 * q^36 + 13 * q^37 - 5 * q^38 - 22 * q^39 + 30 * q^42 + 36 * q^43 + 17 * q^44 + 13 * q^46 - 6 * q^47 + 14 * q^48 + 16 * q^49 + 4 * q^51 + 50 * q^52 + 9 * q^53 + 9 * q^54 - 18 * q^56 - 4 * q^57 + 2 * q^58 - 7 * q^59 - 2 * q^61 + 11 * q^62 + 39 * q^63 + 17 * q^64 - q^66 + 35 * q^67 - 18 * q^68 - 9 * q^69 + 13 * q^71 + 68 * q^72 + 2 * q^73 + 13 * q^74 - 17 * q^76 + 15 * q^77 - 10 * q^78 + 6 * q^79 + 11 * q^81 + 14 * q^82 + 30 * q^83 - 6 * q^84 - 25 * q^86 + 19 * q^87 + 15 * q^88 + 55 * q^89 + 26 * q^91 - 18 * q^92 + 14 * q^93 - 22 * q^94 - 17 * q^96 + 28 * q^97 - 22 * q^98 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more