LMFDB - Embedded newform 475.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,2,Mod(1,475)] 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("475.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.273891$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.27389 q^{2} -1.65109 q^{3} -0.377203 q^{4} -2.10331 q^{6} +3.65109 q^{7} -3.02830 q^{8} -0.273891 q^{9} +2.65109 q^{11} +0.622797 q^{12} +6.13161 q^{13} +4.65109 q^{14} -3.10331 q^{16} +2.34891 q^{17} -0.348907 q^{18} +1.00000 q^{19} -6.02830 q^{21} +3.37720 q^{22} +5.48052 q^{23} +5.00000 q^{24} +7.81100 q^{26} +5.40550 q^{27} -1.37720 q^{28} +0.651093 q^{29} -6.67939 q^{31} +2.10331 q^{32} -4.37720 q^{33} +2.99225 q^{34} +0.103312 q^{36} -8.70769 q^{37} +1.27389 q^{38} -10.1239 q^{39} +1.93273 q^{41} -7.67939 q^{42} -2.65884 q^{43} -1.00000 q^{44} +6.98158 q^{46} +3.71836 q^{47} +5.12386 q^{48} +6.33048 q^{49} -3.87826 q^{51} -2.31286 q^{52} +13.7544 q^{53} +6.88601 q^{54} -11.0566 q^{56} -1.65109 q^{57} +0.829422 q^{58} -7.84997 q^{59} -1.92498 q^{61} -8.50881 q^{62} -1.00000 q^{63} +8.88601 q^{64} -5.57608 q^{66} -4.44447 q^{67} -0.886014 q^{68} -9.04884 q^{69} +3.54778 q^{71} +0.829422 q^{72} -2.48052 q^{73} -11.0926 q^{74} -0.377203 q^{76} +9.67939 q^{77} -12.8967 q^{78} -15.1599 q^{79} -8.10331 q^{81} +2.46209 q^{82} -14.7282 q^{83} +2.27389 q^{84} -3.38708 q^{86} -1.07502 q^{87} -8.02830 q^{88} -5.06727 q^{89} +22.3871 q^{91} -2.06727 q^{92} +11.0283 q^{93} +4.73678 q^{94} -3.47277 q^{96} +3.22717 q^{97} +8.06434 q^{98} -0.726109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} + 7 q^{14} - 6 q^{16} + 14 q^{17} - 8 q^{18} + 3 q^{19} - 6 q^{21} + 5 q^{22} + 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 + 4 * q^7 + 3 * q^8 + q^9 + q^11 + 7 * q^12 + 3 * q^13 + 7 * q^14 - 6 * q^16 + 14 * q^17 - 8 * q^18 + 3 * q^19 - 6 * q^21 + 5 * q^22 + 8 * q^23 + 15 * q^24 - 11 * q^26 - q^27 + q^28 - 5 * q^29 - q^31 + 3 * q^32 - 8 * q^33 + 5 * q^34 - 3 * q^36 + 5 * q^37 + 2 * q^38 - 11 * q^39 + q^41 - 4 * q^42 - 5 * q^43 - 3 * q^44 - 12 * q^46 + 9 * q^47 - 4 * q^48 - 7 * q^49 + 18 * q^51 - 22 * q^52 + 31 * q^53 - 5 * q^54 - 9 * q^56 + 2 * q^57 + q^58 - 6 * q^59 + 3 * q^61 - 5 * q^62 - 3 * q^63 + q^64 - q^66 - 13 * q^67 + 23 * q^68 + q^69 + 7 * q^71 + q^72 + q^73 - q^74 + 4 * q^76 + 10 * q^77 - 42 * q^78 - 18 * q^79 - 21 * q^81 - 34 * q^82 + 3 * q^83 + 5 * q^84 + 40 * q^86 - 12 * q^87 - 12 * q^88 - 20 * q^89 + 17 * q^91 - 11 * q^92 + 21 * q^93 + 45 * q^94 + 2 * q^96 - 13 * q^97 + 4 * 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 $

$2$	1.27389	0.900777	0.450388	−	0.892833i	$-0.351286\pi$
$2$	1.27389	0.900777	0.450388	+	0.892833i	$0.351286\pi$
$3$	−1.65109	−0.953259	−0.476630	−	0.879104i	$-0.658142\pi$
$3$	−1.65109	−0.953259	−0.476630	+	0.879104i	$0.658142\pi$
$4$	−0.377203	−0.188601
$5$	0	0
$5$	0	0
$6$	−2.10331	−0.858674
$7$	3.65109	1.37998	0.689992	−	0.723817i	$-0.257614\pi$
$7$	3.65109	1.37998	0.689992	+	0.723817i	$0.257614\pi$
$8$	−3.02830	−1.07066
$9$	−0.273891	−0.0912969
$10$	0	0
$11$	2.65109	0.799335	0.399667	−	0.916660i	$-0.369126\pi$
$11$	2.65109	0.799335	0.399667	+	0.916660i	$0.369126\pi$
$12$	0.622797	0.179786
$13$	6.13161	1.70060	0.850301	−	0.526297i	$-0.176420\pi$
$13$	6.13161	1.70060	0.850301	+	0.526297i	$0.176420\pi$
$14$	4.65109	1.24306
$15$	0	0
$16$	−3.10331	−0.775828
$17$	2.34891	0.569694	0.284847	−	0.958573i	$-0.408057\pi$
$17$	2.34891	0.569694	0.284847	+	0.958573i	$0.408057\pi$
$18$	−0.348907	−0.0822381
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.02830	−1.31548
$22$	3.37720	0.720022
$23$	5.48052	1.14277	0.571383	−	0.820683i	$-0.306407\pi$
$23$	5.48052	1.14277	0.571383	+	0.820683i	$0.306407\pi$
$24$	5.00000	1.02062
$25$	0	0
$26$	7.81100	1.53186
$27$	5.40550	1.04029
$28$	−1.37720	−0.260267
$29$	0.651093	0.120905	0.0604525	−	0.998171i	$-0.480746\pi$
$29$	0.651093	0.120905	0.0604525	+	0.998171i	$0.480746\pi$
$30$	0	0
$31$	−6.67939	−1.19965	−0.599827	−	0.800130i	$-0.704764\pi$
$31$	−6.67939	−1.19965	−0.599827	+	0.800130i	$0.704764\pi$
$32$	2.10331	0.371817
$33$	−4.37720	−0.761973
$34$	2.99225	0.513167
$35$	0	0
$36$	0.103312	0.0172187
$37$	−8.70769	−1.43153	−0.715767	−	0.698339i	$-0.753923\pi$
$37$	−8.70769	−1.43153	−0.715767	+	0.698339i	$0.753923\pi$
$38$	1.27389	0.206652
$39$	−10.1239	−1.62111
$40$	0	0
$41$	1.93273	0.301842	0.150921	−	0.988546i	$-0.451776\pi$
$41$	1.93273	0.301842	0.150921	+	0.988546i	$0.451776\pi$
$42$	−7.67939	−1.18496
$43$	−2.65884	−0.405470	−0.202735	−	0.979234i	$-0.564983\pi$
$43$	−2.65884	−0.405470	−0.202735	+	0.979234i	$0.564983\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	6.98158	1.02938
$47$	3.71836	0.542378	0.271189	−	0.962526i	$-0.412583\pi$
$47$	3.71836	0.542378	0.271189	+	0.962526i	$0.412583\pi$
$48$	5.12386	0.739565
$49$	6.33048	0.904355
$50$	0	0
$51$	−3.87826	−0.543066
$52$	−2.31286	−0.320736
$53$	13.7544	1.88931	0.944656	−	0.328061i	$-0.106395\pi$
$53$	13.7544	1.88931	0.944656	+	0.328061i	$0.106395\pi$
$54$	6.88601	0.937068
$55$	0	0
$56$	−11.0566	−1.47750
$57$	−1.65109	−0.218693
$58$	0.829422	0.108908
$59$	−7.84997	−1.02198	−0.510989	−	0.859587i	$-0.670721\pi$
$59$	−7.84997	−1.02198	−0.510989	+	0.859587i	$0.670721\pi$
$60$	0	0
$61$	−1.92498	−0.246469	−0.123234	−	0.992378i	$-0.539327\pi$
$61$	−1.92498	−0.246469	−0.123234	+	0.992378i	$0.539327\pi$
$62$	−8.50881	−1.08062
$63$	−1.00000	−0.125988
$64$	8.88601	1.11075
$65$	0	0
$66$	−5.57608	−0.686368
$67$	−4.44447	−0.542978	−0.271489	−	0.962442i	$-0.587516\pi$
$67$	−4.44447	−0.542978	−0.271489	+	0.962442i	$0.587516\pi$
$68$	−0.886014	−0.107445
$69$	−9.04884	−1.08935
$70$	0	0
$71$	3.54778	0.421044	0.210522	−	0.977589i	$-0.432484\pi$
$71$	3.54778	0.421044	0.210522	+	0.977589i	$0.432484\pi$
$72$	0.829422	0.0977483
$73$	−2.48052	−0.290322	−0.145161	−	0.989408i	$-0.546370\pi$
$73$	−2.48052	−0.290322	−0.145161	+	0.989408i	$0.546370\pi$
$74$	−11.0926	−1.28949
$75$	0	0
$76$	−0.377203	−0.0432681
$77$	9.67939	1.10307
$78$	−12.8967	−1.46026
$79$	−15.1599	−1.70562	−0.852811	−	0.522219i	$-0.825104\pi$
$79$	−15.1599	−1.70562	−0.852811	+	0.522219i	$0.825104\pi$
$80$	0	0
$81$	−8.10331	−0.900368
$82$	2.46209	0.271893
$83$	−14.7282	−1.61663	−0.808317	−	0.588748i	$-0.799621\pi$
$83$	−14.7282	−1.61663	−0.808317	+	0.588748i	$0.799621\pi$
$84$	2.27389	0.248102
$85$	0	0
$86$	−3.38708	−0.365238
$87$	−1.07502	−0.115254
$88$	−8.02830	−0.855819
$89$	−5.06727	−0.537129	−0.268565	−	0.963262i	$-0.586549\pi$
$89$	−5.06727	−0.537129	−0.268565	+	0.963262i	$0.586549\pi$
$90$	0	0
$91$	22.3871	2.34680
$92$	−2.06727	−0.215527
$93$	11.0283	1.14358
$94$	4.73678	0.488562
$95$	0	0
$96$	−3.47277	−0.354438
$97$	3.22717	0.327670	0.163835	−	0.986488i	$-0.447614\pi$
$97$	3.22717	0.327670	0.163835	+	0.986488i	$0.447614\pi$
$98$	8.06434	0.814622
$99$	−0.726109	−0.0729767
$100$	0	0
$101$	16.8032	1.67199	0.835993	−	0.548740i	$-0.184892\pi$
$101$	16.8032	1.67199	0.835993	+	0.548740i	$0.184892\pi$
$102$	−4.94048	−0.489181
$103$	−9.85772	−0.971310	−0.485655	−	0.874151i	$-0.661419\pi$
$103$	−9.85772	−0.971310	−0.485655	+	0.874151i	$0.661419\pi$
$104$	−18.5683	−1.82077
$105$	0	0
$106$	17.5216	1.70185
$107$	−5.13936	−0.496841	−0.248420	−	0.968652i	$-0.579911\pi$
$107$	−5.13936	−0.496841	−0.248420	+	0.968652i	$0.579911\pi$
$108$	−2.03897	−0.196200
$109$	13.9738	1.33845	0.669225	−	0.743060i	$-0.266626\pi$
$109$	13.9738	1.33845	0.669225	+	0.743060i	$0.266626\pi$
$110$	0	0
$111$	14.3772	1.36462
$112$	−11.3305	−1.07063
$113$	−0.527235	−0.0495981	−0.0247990	−	0.999692i	$-0.507895\pi$
$113$	−0.527235	−0.0495981	−0.0247990	+	0.999692i	$0.507895\pi$
$114$	−2.10331	−0.196993
$115$	0	0
$116$	−0.245594	−0.0228029
$117$	−1.67939	−0.155260
$118$	−10.0000	−0.920575
$119$	8.57608	0.786168
$120$	0	0
$121$	−3.97170	−0.361064
$122$	−2.45222	−0.222013
$123$	−3.19112	−0.287734
$124$	2.51948	0.226256
$125$	0	0
$126$	−1.27389	−0.113487
$127$	−11.8217	−1.04900	−0.524502	−	0.851409i	$-0.675748\pi$
$127$	−11.8217	−1.04900	−0.524502	+	0.851409i	$0.675748\pi$
$128$	7.11319	0.628723
$129$	4.39000	0.386518
$130$	0	0
$131$	8.12386	0.709785	0.354892	−	0.934907i	$-0.384517\pi$
$131$	8.12386	0.709785	0.354892	+	0.934907i	$0.384517\pi$
$132$	1.65109	0.143709
$133$	3.65109	0.316590
$134$	−5.66177	−0.489102
$135$	0	0
$136$	−7.11319	−0.609951
$137$	17.5761	1.50163	0.750813	−	0.660515i	$-0.229662\pi$
$137$	17.5761	1.50163	0.750813	+	0.660515i	$0.229662\pi$
$138$	−11.5272	−0.981263
$139$	18.4154	1.56197	0.780986	−	0.624549i	$-0.214717\pi$
$139$	18.4154	1.56197	0.780986	+	0.624549i	$0.214717\pi$
$140$	0	0
$141$	−6.13936	−0.517027
$142$	4.51948	0.379267
$143$	16.2555	1.35935
$144$	0.849968	0.0708307
$145$	0	0
$146$	−3.15990	−0.261516
$147$	−10.4522	−0.862084
$148$	3.28456	0.269989
$149$	−17.2915	−1.41658	−0.708288	−	0.705924i	$-0.750532\pi$
$149$	−17.2915	−1.41658	−0.708288	+	0.705924i	$0.750532\pi$
$150$	0	0
$151$	2.58383	0.210269	0.105134	−	0.994458i	$-0.466473\pi$
$151$	2.58383	0.210269	0.105134	+	0.994458i	$0.466473\pi$
$152$	−3.02830	−0.245627
$153$	−0.643343	−0.0520112
$154$	12.3305	0.993619
$155$	0	0
$156$	3.81875	0.305745
$157$	−10.9738	−0.875807	−0.437903	−	0.899022i	$-0.644279\pi$
$157$	−10.9738	−0.875807	−0.437903	+	0.899022i	$0.644279\pi$
$158$	−19.3121	−1.53638
$159$	−22.7098	−1.80100
$160$	0	0
$161$	20.0099	1.57700
$162$	−10.3227	−0.811031
$163$	22.6794	1.77639	0.888193	−	0.459470i	$-0.151961\pi$
$163$	22.6794	1.77639	0.888193	+	0.459470i	$0.151961\pi$
$164$	−0.729033	−0.0569279
$165$	0	0
$166$	−18.7622	−1.45623
$167$	−5.16283	−0.399512	−0.199756	−	0.979846i	$-0.564015\pi$
$167$	−5.16283	−0.399512	−0.199756	+	0.979846i	$0.564015\pi$
$168$	18.2555	1.40844
$169$	24.5966	1.89205
$170$	0	0
$171$	−0.273891	−0.0209449
$172$	1.00292	0.0764722
$173$	−17.2165	−1.30895	−0.654473	−	0.756085i	$-0.727109\pi$
$173$	−17.2165	−1.30895	−0.654473	+	0.756085i	$0.727109\pi$
$174$	−1.36945	−0.103818
$175$	0	0
$176$	−8.22717	−0.620146
$177$	12.9610	0.974211
$178$	−6.45514	−0.483833
$179$	10.6999	0.799751	0.399875	−	0.916570i	$-0.369053\pi$
$179$	10.6999	0.799751	0.399875	+	0.916570i	$0.369053\pi$
$180$	0	0
$181$	−16.7720	−1.24666	−0.623328	−	0.781961i	$-0.714220\pi$
$181$	−16.7720	−1.24666	−0.623328	+	0.781961i	$0.714220\pi$
$182$	28.5187	2.11395
$183$	3.17833	0.234949
$184$	−16.5966	−1.22352
$185$	0	0
$186$	14.0488	1.03011
$187$	6.22717	0.455376
$188$	−1.40258	−0.102293
$189$	19.7360	1.43558
$190$	0	0
$191$	−13.8598	−1.00286	−0.501431	−	0.865197i	$-0.667193\pi$
$191$	−13.8598	−1.00286	−0.501431	+	0.865197i	$0.667193\pi$
$192$	−14.6716	−1.05883
$193$	21.0694	1.51661	0.758304	−	0.651901i	$-0.226028\pi$
$193$	21.0694	1.51661	0.758304	+	0.651901i	$0.226028\pi$
$194$	4.11106	0.295157
$195$	0	0
$196$	−2.38788	−0.170563
$197$	21.2555	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$197$	21.2555	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$198$	−0.924984	−0.0657357
$199$	−7.69006	−0.545134	−0.272567	−	0.962137i	$-0.587873\pi$
$199$	−7.69006	−0.545134	−0.272567	+	0.962137i	$0.587873\pi$
$200$	0	0
$201$	7.33823	0.517599
$202$	21.4055	1.50609
$203$	2.37720	0.166847
$204$	1.46289	0.102423
$205$	0	0
$206$	−12.5577	−0.874933
$207$	−1.50106	−0.104331
$208$	−19.0283	−1.31937
$209$	2.65109	0.183380
$210$	0	0
$211$	−1.69781	−0.116882	−0.0584411	−	0.998291i	$-0.518613\pi$
$211$	−1.69781	−0.116882	−0.0584411	+	0.998291i	$0.518613\pi$
$212$	−5.18820	−0.356327
$213$	−5.85772	−0.401364
$214$	−6.54698	−0.447542
$215$	0	0
$216$	−16.3695	−1.11380
$217$	−24.3871	−1.65550
$218$	17.8011	1.20564
$219$	4.09556	0.276752
$220$	0	0
$221$	14.4026	0.968822
$222$	18.3150	1.22922
$223$	−25.2632	−1.69175	−0.845875	−	0.533381i	$-0.820921\pi$
$223$	−25.2632	−1.69175	−0.845875	+	0.533381i	$0.820921\pi$
$224$	7.67939	0.513101
$225$	0	0
$226$	−0.671640	−0.0446768
$227$	−16.8217	−1.11649	−0.558247	−	0.829675i	$-0.688526\pi$
$227$	−16.8217	−1.11649	−0.558247	+	0.829675i	$0.688526\pi$
$228$	0.622797	0.0412457
$229$	−14.6249	−0.966442	−0.483221	−	0.875498i	$-0.660533\pi$
$229$	−14.6249	−0.966442	−0.483221	+	0.875498i	$0.660533\pi$
$230$	0	0
$231$	−15.9816	−1.05151
$232$	−1.97170	−0.129449
$233$	8.91431	0.583996	0.291998	−	0.956419i	$-0.405680\pi$
$233$	8.91431	0.583996	0.291998	+	0.956419i	$0.405680\pi$
$234$	−2.13936	−0.139854
$235$	0	0
$236$	2.96103	0.192747
$237$	25.0304	1.62590
$238$	10.9250	0.708162
$239$	−24.6015	−1.59134	−0.795668	−	0.605733i	$-0.792880\pi$
$239$	−24.6015	−1.59134	−0.795668	+	0.605733i	$0.792880\pi$
$240$	0	0
$241$	−14.6150	−0.941438	−0.470719	−	0.882283i	$-0.656005\pi$
$241$	−14.6150	−0.941438	−0.470719	+	0.882283i	$0.656005\pi$
$242$	−5.05952	−0.325238
$243$	−2.83717	−0.182005
$244$	0.726109	0.0464844
$245$	0	0
$246$	−4.06514	−0.259184
$247$	6.13161	0.390145
$248$	20.2272	1.28443
$249$	24.3177	1.54107
$250$	0	0
$251$	−13.3382	−0.841902	−0.420951	−	0.907083i	$-0.638304\pi$
$251$	−13.3382	−0.841902	−0.420951	+	0.907083i	$0.638304\pi$
$252$	0.377203	0.0237615
$253$	14.5294	0.913453
$254$	−15.0595	−0.944918
$255$	0	0
$256$	−8.71061	−0.544413
$257$	−3.35103	−0.209031	−0.104516	−	0.994523i	$-0.533329\pi$
$257$	−3.35103	−0.209031	−0.104516	+	0.994523i	$0.533329\pi$
$258$	5.59238	0.348166
$259$	−31.7926	−1.97549
$260$	0	0
$261$	−0.178328	−0.0110382
$262$	10.3489	0.639358
$263$	10.8860	0.671260	0.335630	−	0.941994i	$-0.391051\pi$
$263$	10.8860	0.671260	0.335630	+	0.941994i	$0.391051\pi$
$264$	13.2555	0.815818
$265$	0	0
$266$	4.65109	0.285177
$267$	8.36653	0.512023
$268$	1.67647	0.102406
$269$	18.4338	1.12393	0.561964	−	0.827162i	$-0.310046\pi$
$269$	18.4338	1.12393	0.561964	+	0.827162i	$0.310046\pi$
$270$	0	0
$271$	−20.4076	−1.23967	−0.619837	−	0.784730i	$-0.712801\pi$
$271$	−20.4076	−1.23967	−0.619837	+	0.784730i	$0.712801\pi$
$272$	−7.28939	−0.441984
$273$	−36.9632	−2.23711
$274$	22.3900	1.35263
$275$	0	0
$276$	3.41325	0.205453
$277$	−2.69994	−0.162223	−0.0811117	−	0.996705i	$-0.525847\pi$
$277$	−2.69994	−0.162223	−0.0811117	+	0.996705i	$0.525847\pi$
$278$	23.4592	1.40699
$279$	1.82942	0.109525
$280$	0	0
$281$	−15.2242	−0.908202	−0.454101	−	0.890950i	$-0.650040\pi$
$281$	−15.2242	−0.908202	−0.454101	+	0.890950i	$0.650040\pi$
$282$	−7.82087	−0.465726
$283$	6.18045	0.367390	0.183695	−	0.982983i	$-0.441194\pi$
$283$	6.18045	0.367390	0.183695	+	0.982983i	$0.441194\pi$
$284$	−1.33823	−0.0794095
$285$	0	0
$286$	20.7077	1.22447
$287$	7.05659	0.416537
$288$	−0.576077	−0.0339457
$289$	−11.4826	−0.675449
$290$	0	0
$291$	−5.32836	−0.312354
$292$	0.935657	0.0547552
$293$	30.6893	1.79289	0.896443	−	0.443159i	$-0.146142\pi$
$293$	30.6893	1.79289	0.896443	+	0.443159i	$0.146142\pi$
$294$	−13.3150	−0.776546
$295$	0	0
$296$	26.3695	1.53269
$297$	14.3305	0.831539
$298$	−22.0275	−1.27602
$299$	33.6044	1.94339
$300$	0	0
$301$	−9.70769	−0.559542
$302$	3.29151	0.189405
$303$	−27.7437	−1.59384
$304$	−3.10331	−0.177987
$305$	0	0
$306$	−0.819549	−0.0468505
$307$	−14.7827	−0.843693	−0.421847	−	0.906667i	$-0.638618\pi$
$307$	−14.7827	−0.843693	−0.421847	+	0.906667i	$0.638618\pi$
$308$	−3.65109	−0.208040
$309$	16.2760	0.925910
$310$	0	0
$311$	−26.9992	−1.53098	−0.765492	−	0.643445i	$-0.777504\pi$
$311$	−26.9992	−1.53098	−0.765492	+	0.643445i	$0.777504\pi$
$312$	30.6580	1.73567
$313$	−9.74666	−0.550914	−0.275457	−	0.961313i	$-0.588829\pi$
$313$	−9.74666	−0.550914	−0.275457	+	0.961313i	$0.588829\pi$
$314$	−13.9795	−0.788906
$315$	0	0
$316$	5.71836	0.321683
$317$	19.7261	1.10793	0.553964	−	0.832540i	$-0.313114\pi$
$317$	19.7261	1.10793	0.553964	+	0.832540i	$0.313114\pi$
$318$	−28.9298	−1.62230
$319$	1.72611	0.0966436
$320$	0	0
$321$	8.48556	0.473618
$322$	25.4904	1.42052
$323$	2.34891	0.130697
$324$	3.05659	0.169811
$325$	0	0
$326$	28.8911	1.60013
$327$	−23.0721	−1.27589
$328$	−5.85289	−0.323172
$329$	13.5761	0.748473
$330$	0	0
$331$	19.9426	1.09614	0.548072	−	0.836431i	$-0.315362\pi$
$331$	19.9426	1.09614	0.548072	+	0.836431i	$0.315362\pi$
$332$	5.55553	0.304899
$333$	2.38495	0.130695
$334$	−6.57688	−0.359871
$335$	0	0
$336$	18.7077	1.02059
$337$	2.05952	0.112189	0.0560945	−	0.998425i	$-0.482135\pi$
$337$	2.05952	0.112189	0.0560945	+	0.998425i	$0.482135\pi$
$338$	31.3334	1.70431
$339$	0.870514	0.0472798
$340$	0	0
$341$	−17.7077	−0.958925
$342$	−0.348907	−0.0188667
$343$	−2.44447	−0.131989
$344$	8.05177	0.434122
$345$	0	0
$346$	−21.9319	−1.17907
$347$	−10.5011	−0.563727	−0.281863	−	0.959455i	$-0.590952\pi$
$347$	−10.5011	−0.563727	−0.281863	+	0.959455i	$0.590952\pi$
$348$	0.405499	0.0217370
$349$	16.5059	0.883540	0.441770	−	0.897128i	$-0.354351\pi$
$349$	16.5059	0.883540	0.441770	+	0.897128i	$0.354351\pi$
$350$	0	0
$351$	33.1444	1.76912
$352$	5.57608	0.297206
$353$	15.8860	0.845527	0.422764	−	0.906240i	$-0.361060\pi$
$353$	15.8860	0.845527	0.422764	+	0.906240i	$0.361060\pi$
$354$	16.5109	0.877546
$355$	0	0
$356$	1.91139	0.101303
$357$	−14.1599	−0.749422
$358$	13.6305	0.720397
$359$	1.28376	0.0677544	0.0338772	−	0.999426i	$-0.489214\pi$
$359$	1.28376	0.0677544	0.0338772	+	0.999426i	$0.489214\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−21.3657	−1.12296
$363$	6.55765	0.344188
$364$	−8.44447	−0.442610
$365$	0	0
$366$	4.04884	0.211636
$367$	−19.8804	−1.03775	−0.518874	−	0.854851i	$-0.673649\pi$
$367$	−19.8804	−1.03775	−0.518874	+	0.854851i	$0.673649\pi$
$368$	−17.0078	−0.886590
$369$	−0.529358	−0.0275573
$370$	0	0
$371$	50.2186	2.60722
$372$	−4.15990	−0.215681
$373$	−16.4359	−0.851020	−0.425510	−	0.904954i	$-0.639905\pi$
$373$	−16.4359	−0.851020	−0.425510	+	0.904954i	$0.639905\pi$
$374$	7.93273	0.410192
$375$	0	0
$376$	−11.2603	−0.580705
$377$	3.99225	0.205611
$378$	25.1415	1.29314
$379$	0.338233	0.0173739	0.00868694	−	0.999962i	$-0.497235\pi$
$379$	0.338233	0.0173739	0.00868694	+	0.999962i	$0.497235\pi$
$380$	0	0
$381$	19.5187	0.999972
$382$	−17.6559	−0.903355
$383$	−13.7339	−0.701767	−0.350884	−	0.936419i	$-0.614119\pi$
$383$	−13.7339	−0.701767	−0.350884	+	0.936419i	$0.614119\pi$
$384$	−11.7445	−0.599336
$385$	0	0
$386$	26.8401	1.36612
$387$	0.728232	0.0370181
$388$	−1.21730	−0.0617989
$389$	−2.26109	−0.114642	−0.0573210	−	0.998356i	$-0.518256\pi$
$389$	−2.26109	−0.114642	−0.0573210	+	0.998356i	$0.518256\pi$
$390$	0	0
$391$	12.8732	0.651027
$392$	−19.1706	−0.968260
$393$	−13.4132	−0.676609
$394$	27.0771	1.36413
$395$	0	0
$396$	0.273891	0.0137635
$397$	−7.82942	−0.392947	−0.196474	−	0.980509i	$-0.562949\pi$
$397$	−7.82942	−0.392947	−0.196474	+	0.980509i	$0.562949\pi$
$398$	−9.79630	−0.491044
$399$	−6.02830	−0.301792
$400$	0	0
$401$	−35.0510	−1.75036	−0.875181	−	0.483796i	$-0.839258\pi$
$401$	−35.0510	−1.75036	−0.875181	+	0.483796i	$0.839258\pi$
$402$	9.34811	0.466241
$403$	−40.9554	−2.04013
$404$	−6.33823	−0.315339
$405$	0	0
$406$	3.02830	0.150292
$407$	−23.0849	−1.14428
$408$	11.7445	0.581441
$409$	−8.34811	−0.412787	−0.206394	−	0.978469i	$-0.566173\pi$
$409$	−8.34811	−0.412787	−0.206394	+	0.978469i	$0.566173\pi$
$410$	0	0
$411$	−29.0197	−1.43144
$412$	3.71836	0.183190
$413$	−28.6610	−1.41031
$414$	−1.91219	−0.0939789
$415$	0	0
$416$	12.8967	0.632312
$417$	−30.4055	−1.48896
$418$	3.37720	0.165184
$419$	−19.8705	−0.970738	−0.485369	−	0.874309i	$-0.661315\pi$
$419$	−19.8705	−0.970738	−0.485369	+	0.874309i	$0.661315\pi$
$420$	0	0
$421$	−0.400672	−0.0195276	−0.00976379	−	0.999952i	$-0.503108\pi$
$421$	−0.400672	−0.0195276	−0.00976379	+	0.999952i	$0.503108\pi$
$422$	−2.16283	−0.105285
$423$	−1.01842	−0.0495174
$424$	−41.6524	−2.02282
$425$	0	0
$426$	−7.46209	−0.361540
$427$	−7.02830	−0.340123
$428$	1.93858	0.0937048
$429$	−26.8393	−1.29581
$430$	0	0
$431$	−14.2533	−0.686559	−0.343280	−	0.939233i	$-0.611538\pi$
$431$	−14.2533	−0.686559	−0.343280	+	0.939233i	$0.611538\pi$
$432$	−16.7750	−0.807085
$433$	12.8393	0.617017	0.308509	−	0.951222i	$-0.400170\pi$
$433$	12.8393	0.617017	0.308509	+	0.951222i	$0.400170\pi$
$434$	−31.0665	−1.49124
$435$	0	0
$436$	−5.27097	−0.252434
$437$	5.48052	0.262169
$438$	5.21730	0.249292
$439$	−13.2555	−0.632649	−0.316324	−	0.948651i	$-0.602449\pi$
$439$	−13.2555	−0.632649	−0.316324	+	0.948651i	$0.602449\pi$
$440$	0	0
$441$	−1.73386	−0.0825647
$442$	18.3473	0.872692
$443$	5.72611	0.272056	0.136028	−	0.990705i	$-0.456566\pi$
$443$	5.72611	0.272056	0.136028	+	0.990705i	$0.456566\pi$
$444$	−5.42312	−0.257370
$445$	0	0
$446$	−32.1826	−1.52389
$447$	28.5499	1.35036
$448$	32.4437	1.53282
$449$	3.11399	0.146958	0.0734790	−	0.997297i	$-0.476590\pi$
$449$	3.11399	0.146958	0.0734790	+	0.997297i	$0.476590\pi$
$450$	0	0
$451$	5.12386	0.241273
$452$	0.198875	0.00935427
$453$	−4.26614	−0.200441
$454$	−21.4290	−1.00571
$455$	0	0
$456$	5.00000	0.234146
$457$	27.4535	1.28422	0.642111	−	0.766611i	$-0.278059\pi$
$457$	27.4535	1.28422	0.642111	+	0.766611i	$0.278059\pi$
$458$	−18.6305	−0.870548
$459$	12.6970	0.592646
$460$	0	0
$461$	13.9837	0.651286	0.325643	−	0.945493i	$-0.394419\pi$
$461$	13.9837	0.651286	0.325643	+	0.945493i	$0.394419\pi$
$462$	−20.3588	−0.947176
$463$	17.9992	0.836494	0.418247	−	0.908333i	$-0.362645\pi$
$463$	17.9992	0.836494	0.418247	+	0.908333i	$0.362645\pi$
$464$	−2.02055	−0.0938015
$465$	0	0
$466$	11.3559	0.526050
$467$	12.6871	0.587091	0.293545	−	0.955945i	$-0.405165\pi$
$467$	12.6871	0.587091	0.293545	+	0.955945i	$0.405165\pi$
$468$	0.633471	0.0292822
$469$	−16.2272	−0.749301
$470$	0	0
$471$	18.1188	0.834871
$472$	23.7720	1.09420
$473$	−7.04884	−0.324106
$474$	31.8860	1.46457
$475$	0	0
$476$	−3.23492	−0.148272
$477$	−3.76720	−0.172488
$478$	−31.3396	−1.43344
$479$	24.7544	1.13106	0.565529	−	0.824729i	$-0.308672\pi$
$479$	24.7544	1.13106	0.565529	+	0.824729i	$0.308672\pi$
$480$	0	0
$481$	−53.3921	−2.43447
$482$	−18.6180	−0.848025
$483$	−33.0382	−1.50329
$484$	1.49814	0.0680972
$485$	0	0
$486$	−3.61425	−0.163946
$487$	−4.08277	−0.185008	−0.0925039	−	0.995712i	$-0.529487\pi$
$487$	−4.08277	−0.185008	−0.0925039	+	0.995712i	$0.529487\pi$
$488$	5.82942	0.263886
$489$	−37.4458	−1.69336
$490$	0	0
$491$	29.2547	1.32024	0.660122	−	0.751158i	$-0.270504\pi$
$491$	29.2547	1.32024	0.660122	+	0.751158i	$0.270504\pi$
$492$	1.20370	0.0542670
$493$	1.52936	0.0688788
$494$	7.81100	0.351433
$495$	0	0
$496$	20.7282	0.930725
$497$	12.9533	0.581034
$498$	30.9781	1.38816
$499$	1.57315	0.0704240	0.0352120	−	0.999380i	$-0.488789\pi$
$499$	1.57315	0.0704240	0.0352120	+	0.999380i	$0.488789\pi$
$500$	0	0
$501$	8.52431	0.380838
$502$	−16.9914	−0.758365
$503$	20.7819	0.926619	0.463310	−	0.886196i	$-0.346662\pi$
$503$	20.7819	0.926619	0.463310	+	0.886196i	$0.346662\pi$
$504$	3.02830	0.134891
$505$	0	0
$506$	18.5088	0.822817
$507$	−40.6113	−1.80361
$508$	4.45917	0.197844
$509$	−31.8238	−1.41056	−0.705282	−	0.708926i	$-0.749180\pi$
$509$	−31.8238	−1.41056	−0.705282	+	0.708926i	$0.749180\pi$
$510$	0	0
$511$	−9.05659	−0.400640
$512$	−25.3227	−1.11912
$513$	5.40550	0.238659
$514$	−4.26884	−0.188291
$515$	0	0
$516$	−1.65592	−0.0728978
$517$	9.85772	0.433542
$518$	−40.5003	−1.77948
$519$	28.4260	1.24776
$520$	0	0
$521$	−27.4720	−1.20357	−0.601784	−	0.798659i	$-0.705543\pi$
$521$	−27.4720	−1.20357	−0.601784	+	0.798659i	$0.705543\pi$
$522$	−0.227171	−0.00994300
$523$	−10.4466	−0.456798	−0.228399	−	0.973568i	$-0.573349\pi$
$523$	−10.4466	−0.456798	−0.228399	+	0.973568i	$0.573349\pi$
$524$	−3.06434	−0.133866
$525$	0	0
$526$	13.8676	0.604656
$527$	−15.6893	−0.683435
$528$	13.5838	0.591160
$529$	7.03605	0.305915
$530$	0	0
$531$	2.15003	0.0933034
$532$	−1.37720	−0.0597093
$533$	11.8508	0.513314
$534$	10.6580	0.461219
$535$	0	0
$536$	13.4592	0.581348
$537$	−17.6666	−0.762370
$538$	23.4826	1.01241
$539$	16.7827	0.722882
$540$	0	0
$541$	13.4989	0.580365	0.290182	−	0.956971i	$-0.406284\pi$
$541$	13.4989	0.580365	0.290182	+	0.956971i	$0.406284\pi$
$542$	−25.9971	−1.11667
$543$	27.6922	1.18839
$544$	4.94048	0.211822
$545$	0	0
$546$	−47.0870	−2.01514
$547$	8.54698	0.365442	0.182721	−	0.983165i	$-0.441509\pi$
$547$	8.54698	0.365442	0.182721	+	0.983165i	$0.441509\pi$
$548$	−6.62975	−0.283209
$549$	0.527235	0.0225018
$550$	0	0
$551$	0.651093	0.0277375
$552$	27.4026	1.16633
$553$	−55.3502	−2.35373
$554$	−3.43942	−0.146127
$555$	0	0
$556$	−6.94633	−0.294590
$557$	27.2378	1.15410	0.577052	−	0.816707i	$-0.304203\pi$
$557$	27.2378	1.15410	0.577052	+	0.816707i	$0.304203\pi$
$558$	2.33048	0.0986572
$559$	−16.3030	−0.689543
$560$	0	0
$561$	−10.2816	−0.434091
$562$	−19.3940	−0.818088
$563$	−1.44235	−0.0607876	−0.0303938	−	0.999538i	$-0.509676\pi$
$563$	−1.44235	−0.0607876	−0.0303938	+	0.999538i	$0.509676\pi$
$564$	2.31578	0.0975121
$565$	0	0
$566$	7.87322	0.330936
$567$	−29.5860	−1.24249
$568$	−10.7437	−0.450797
$569$	−14.2632	−0.597945	−0.298973	−	0.954262i	$-0.596644\pi$
$569$	−14.2632	−0.597945	−0.298973	+	0.954262i	$0.596644\pi$
$570$	0	0
$571$	9.91723	0.415023	0.207512	−	0.978233i	$-0.433464\pi$
$571$	9.91723	0.415023	0.207512	+	0.978233i	$0.433464\pi$
$572$	−6.13161	−0.256375
$573$	22.8839	0.955988
$574$	8.98933	0.375207
$575$	0	0
$576$	−2.43380	−0.101408
$577$	−8.23997	−0.343034	−0.171517	−	0.985181i	$-0.554867\pi$
$577$	−8.23997	−0.343034	−0.171517	+	0.985181i	$0.554867\pi$
$578$	−14.6276	−0.608429
$579$	−34.7875	−1.44572
$580$	0	0
$581$	−53.7742	−2.23093
$582$	−6.78775	−0.281361
$583$	36.4642	1.51019
$584$	7.51173	0.310838
$585$	0	0
$586$	39.0948	1.61499
$587$	36.9554	1.52531	0.762656	−	0.646804i	$-0.223895\pi$
$587$	36.9554	1.52531	0.762656	+	0.646804i	$0.223895\pi$
$588$	3.94261	0.162590
$589$	−6.67939	−0.275219
$590$	0	0
$591$	−35.0948	−1.44361
$592$	27.0227	1.11062
$593$	−1.36170	−0.0559184	−0.0279592	−	0.999609i	$-0.508901\pi$
$593$	−1.36170	−0.0559184	−0.0279592	+	0.999609i	$0.508901\pi$
$594$	18.2555	0.749031
$595$	0	0
$596$	6.52241	0.267168
$597$	12.6970	0.519654
$598$	42.8083	1.75056
$599$	33.5753	1.37185	0.685924	−	0.727673i	$-0.259398\pi$
$599$	33.5753	1.37185	0.685924	+	0.727673i	$0.259398\pi$
$600$	0	0
$601$	12.6561	0.516255	0.258127	−	0.966111i	$-0.416895\pi$
$601$	12.6561	0.516255	0.258127	+	0.966111i	$0.416895\pi$
$602$	−12.3665	−0.504022
$603$	1.21730	0.0495722
$604$	−0.974627	−0.0396570
$605$	0	0
$606$	−35.3425	−1.43569
$607$	17.4047	0.706435	0.353217	−	0.935541i	$-0.385088\pi$
$607$	17.4047	0.706435	0.353217	+	0.935541i	$0.385088\pi$
$608$	2.10331	0.0853006
$609$	−3.92498	−0.159048
$610$	0	0
$611$	22.7995	0.922370
$612$	0.242671	0.00980939
$613$	−37.2603	−1.50493	−0.752465	−	0.658633i	$-0.771135\pi$
$613$	−37.2603	−1.50493	−0.752465	+	0.658633i	$0.771135\pi$
$614$	−18.8315	−0.759979
$615$	0	0
$616$	−29.3121	−1.18102
$617$	−18.3764	−0.739806	−0.369903	−	0.929070i	$-0.620609\pi$
$617$	−18.3764	−0.739806	−0.369903	+	0.929070i	$0.620609\pi$
$618$	20.7339	0.834038
$619$	14.5526	0.584919	0.292459	−	0.956278i	$-0.405526\pi$
$619$	14.5526	0.584919	0.292459	+	0.956278i	$0.405526\pi$
$620$	0	0
$621$	29.6249	1.18881
$622$	−34.3940	−1.37907
$623$	−18.5011	−0.741229
$624$	31.4175	1.25771
$625$	0	0
$626$	−12.4162	−0.496250
$627$	−4.37720	−0.174809
$628$	4.13936	0.165178
$629$	−20.4535	−0.815536
$630$	0	0
$631$	22.0304	0.877017	0.438509	−	0.898727i	$-0.355507\pi$
$631$	22.0304	0.877017	0.438509	+	0.898727i	$0.355507\pi$
$632$	45.9087	1.82615
$633$	2.80325	0.111419
$634$	25.1289	0.997996
$635$	0	0
$636$	8.56620	0.339672
$637$	38.8160	1.53795
$638$	2.19887	0.0870543
$639$	−0.971704	−0.0384400
$640$	0	0
$641$	−43.6765	−1.72512	−0.862558	−	0.505958i	$-0.831139\pi$
$641$	−43.6765	−1.72512	−0.862558	+	0.505958i	$0.831139\pi$
$642$	10.8097	0.426624
$643$	−25.9263	−1.02243	−0.511217	−	0.859452i	$-0.670805\pi$
$643$	−25.9263	−1.02243	−0.511217	+	0.859452i	$0.670805\pi$
$644$	−7.54778	−0.297424
$645$	0	0
$646$	2.99225	0.117728
$647$	−42.1046	−1.65530	−0.827652	−	0.561242i	$-0.810324\pi$
$647$	−42.1046	−1.65530	−0.827652	+	0.561242i	$0.810324\pi$
$648$	24.5392	0.963992
$649$	−20.8110	−0.816903
$650$	0	0
$651$	40.2653	1.57812
$652$	−8.55473	−0.335029
$653$	40.4671	1.58360	0.791801	−	0.610779i	$-0.209144\pi$
$653$	40.4671	1.58360	0.791801	+	0.610779i	$0.209144\pi$
$654$	−29.3913	−1.14929
$655$	0	0
$656$	−5.99788	−0.234178
$657$	0.679390	0.0265055
$658$	17.2944	0.674207
$659$	33.4204	1.30187	0.650937	−	0.759131i	$-0.274376\pi$
$659$	33.4204	1.30187	0.650937	+	0.759131i	$0.274376\pi$
$660$	0	0
$661$	19.4883	0.758006	0.379003	−	0.925396i	$-0.376267\pi$
$661$	19.4883	0.758006	0.379003	+	0.925396i	$0.376267\pi$
$662$	25.4047	0.987382
$663$	−23.7800	−0.923539
$664$	44.6015	1.73087
$665$	0	0
$666$	3.03817	0.117727
$667$	3.56833	0.138166
$668$	1.94743	0.0753485
$669$	41.7119	1.61268
$670$	0	0
$671$	−5.10331	−0.197011
$672$	−12.6794	−0.489118
$673$	−0.747456	−0.0288123	−0.0144062	−	0.999896i	$-0.504586\pi$
$673$	−0.747456	−0.0288123	−0.0144062	+	0.999896i	$0.504586\pi$
$674$	2.62360	0.101057
$675$	0	0
$676$	−9.27792	−0.356843
$677$	25.0275	0.961885	0.480942	−	0.876752i	$-0.340295\pi$
$677$	25.0275	0.961885	0.480942	+	0.876752i	$0.340295\pi$
$678$	1.10894	0.0425886
$679$	11.7827	0.452179
$680$	0	0
$681$	27.7742	1.06431
$682$	−22.5577	−0.863777
$683$	−36.7253	−1.40525	−0.702627	−	0.711558i	$-0.747990\pi$
$683$	−36.7253	−1.40525	−0.702627	+	0.711558i	$0.747990\pi$
$684$	0.103312	0.00395024
$685$	0	0
$686$	−3.11399	−0.118893
$687$	24.1471	0.921270
$688$	8.25122	0.314575
$689$	84.3366	3.21297
$690$	0	0
$691$	−21.3177	−0.810963	−0.405482	−	0.914103i	$-0.632896\pi$
$691$	−21.3177	−0.810963	−0.405482	+	0.914103i	$0.632896\pi$
$692$	6.49411	0.246869
$693$	−2.65109	−0.100707
$694$	−13.3772	−0.507792
$695$	0	0
$696$	3.25547	0.123398
$697$	4.53981	0.171958
$698$	21.0267	0.795872
$699$	−14.7184	−0.556699
$700$	0	0
$701$	27.1161	1.02416	0.512081	−	0.858937i	$-0.328875\pi$
$701$	27.1161	1.02416	0.512081	+	0.858937i	$0.328875\pi$
$702$	42.2223	1.59358
$703$	−8.70769	−0.328417
$704$	23.5577	0.887862
$705$	0	0
$706$	20.2370	0.761631
$707$	61.3502	2.30731
$708$	−4.88894	−0.183738
$709$	−10.1161	−0.379918	−0.189959	−	0.981792i	$-0.560836\pi$
$709$	−10.1161	−0.379918	−0.189959	+	0.981792i	$0.560836\pi$
$710$	0	0
$711$	4.15215	0.155718
$712$	15.3452	0.575085
$713$	−36.6065	−1.37092
$714$	−18.0382	−0.675062
$715$	0	0
$716$	−4.03605	−0.150834
$717$	40.6193	1.51696
$718$	1.63537	0.0610316
$719$	24.1471	0.900535	0.450268	−	0.892894i	$-0.351329\pi$
$719$	24.1471	0.900535	0.450268	+	0.892894i	$0.351329\pi$
$720$	0	0
$721$	−35.9914	−1.34039
$722$	1.27389	0.0474093
$723$	24.1308	0.897434
$724$	6.32646	0.235121
$725$	0	0
$726$	8.35373	0.310036
$727$	−19.8988	−0.738006	−0.369003	−	0.929428i	$-0.620301\pi$
$727$	−19.8988	−0.738006	−0.369003	+	0.929428i	$0.620301\pi$
$728$	−67.7947	−2.51264
$729$	28.9944	1.07387
$730$	0	0
$731$	−6.24537	−0.230994
$732$	−1.19887	−0.0443117
$733$	19.9455	0.736705	0.368352	−	0.929686i	$-0.379922\pi$
$733$	19.9455	0.736705	0.368352	+	0.929686i	$0.379922\pi$
$734$	−25.3254	−0.934779
$735$	0	0
$736$	11.5272	0.424900
$737$	−11.7827	−0.434021
$738$	−0.674344	−0.0248229
$739$	−38.2624	−1.40751	−0.703753	−	0.710445i	$-0.748494\pi$
$739$	−38.2624	−1.40751	−0.703753	+	0.710445i	$0.748494\pi$
$740$	0	0
$741$	−10.1239	−0.371909
$742$	63.9730	2.34852
$743$	−12.5547	−0.460588	−0.230294	−	0.973121i	$-0.573969\pi$
$743$	−12.5547	−0.460588	−0.230294	+	0.973121i	$0.573969\pi$
$744$	−33.3969	−1.22439
$745$	0	0
$746$	−20.9376	−0.766579
$747$	4.03392	0.147594
$748$	−2.34891	−0.0858845
$749$	−18.7643	−0.685632
$750$	0	0
$751$	23.2215	0.847366	0.423683	−	0.905810i	$-0.360737\pi$
$751$	23.2215	0.847366	0.423683	+	0.905810i	$0.360737\pi$
$752$	−11.5392	−0.420792
$753$	22.0227	0.802551
$754$	5.08569	0.185210
$755$	0	0
$756$	−7.44447	−0.270753
$757$	18.4105	0.669143	0.334571	−	0.942370i	$-0.391408\pi$
$757$	18.4105	0.669143	0.334571	+	0.942370i	$0.391408\pi$
$758$	0.430872	0.0156500
$759$	−23.9893	−0.870757
$760$	0	0
$761$	11.7982	0.427684	0.213842	−	0.976868i	$-0.431402\pi$
$761$	11.7982	0.427684	0.213842	+	0.976868i	$0.431402\pi$
$762$	24.8647	0.900752
$763$	51.0197	1.84704
$764$	5.22797	0.189141
$765$	0	0
$766$	−17.4954	−0.632136
$767$	−48.1329	−1.73798
$768$	14.3820	0.518967
$769$	41.2653	1.48807	0.744033	−	0.668143i	$-0.232910\pi$
$769$	41.2653	1.48807	0.744033	+	0.668143i	$0.232910\pi$
$770$	0	0
$771$	5.53286	0.199261
$772$	−7.94743	−0.286034
$773$	5.51736	0.198446	0.0992229	−	0.995065i	$-0.468364\pi$
$773$	5.51736	0.198446	0.0992229	+	0.995065i	$0.468364\pi$
$774$	0.927688	0.0333451
$775$	0	0
$776$	−9.77283	−0.350824
$777$	52.4925	1.88316
$778$	−2.88039	−0.103267
$779$	1.93273	0.0692474
$780$	0	0
$781$	9.40550	0.336555
$782$	16.3991	0.586430
$783$	3.51948	0.125776
$784$	−19.6455	−0.701624
$785$	0	0
$786$	−17.0870	−0.609474
$787$	16.7771	0.598038	0.299019	−	0.954247i	$-0.403341\pi$
$787$	16.7771	0.598038	0.299019	+	0.954247i	$0.403341\pi$
$788$	−8.01762	−0.285616
$789$	−17.9738	−0.639885
$790$	0	0
$791$	−1.92498	−0.0684446
$792$	2.19887	0.0781336
$793$	−11.8032	−0.419146
$794$	−9.97383	−0.353958
$795$	0	0
$796$	2.90071	0.102813
$797$	−29.4260	−1.04232	−0.521162	−	0.853458i	$-0.674501\pi$
$797$	−29.4260	−1.04232	−0.521162	+	0.853458i	$0.674501\pi$
$798$	−7.67939	−0.271847
$799$	8.73408	0.308989
$800$	0	0
$801$	1.38788	0.0490382
$802$	−44.6511	−1.57668
$803$	−6.57608	−0.232065
$804$	−2.76800	−0.0976199
$805$	0	0
$806$	−52.1727	−1.83771
$807$	−30.4359	−1.07140
$808$	−50.8852	−1.79014
$809$	5.48614	0.192883	0.0964413	−	0.995339i	$-0.469254\pi$
$809$	5.48614	0.192883	0.0964413	+	0.995339i	$0.469254\pi$
$810$	0	0
$811$	16.3927	0.575626	0.287813	−	0.957687i	$-0.407072\pi$
$811$	16.3927	0.575626	0.287813	+	0.957687i	$0.407072\pi$
$812$	−0.896688	−0.0314676
$813$	33.6949	1.18173
$814$	−29.4076	−1.03074
$815$	0	0
$816$	12.0355	0.421326
$817$	−2.65884	−0.0930212
$818$	−10.6346	−0.371829
$819$	−6.13161	−0.214256
$820$	0	0
$821$	15.1628	0.529186	0.264593	−	0.964360i	$-0.414762\pi$
$821$	15.1628	0.529186	0.264593	+	0.964360i	$0.414762\pi$
$822$	−36.9680	−1.28941
$823$	16.6094	0.578968	0.289484	−	0.957183i	$-0.406516\pi$
$823$	16.6094	0.578968	0.289484	+	0.957183i	$0.406516\pi$
$824$	29.8521	1.03995
$825$	0	0
$826$	−36.5109	−1.27038
$827$	15.2293	0.529574	0.264787	−	0.964307i	$-0.414698\pi$
$827$	15.2293	0.529574	0.264787	+	0.964307i	$0.414698\pi$
$828$	0.566205	0.0196770
$829$	47.4565	1.64823	0.824116	−	0.566422i	$-0.191673\pi$
$829$	47.4565	1.64823	0.824116	+	0.566422i	$0.191673\pi$
$830$	0	0
$831$	4.45785	0.154641
$832$	54.4856	1.88895
$833$	14.8697	0.515205
$834$	−38.7333	−1.34122
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	−36.1054	−1.24799
$838$	−25.3129	−0.874418
$839$	2.26614	0.0782359	0.0391179	−	0.999235i	$-0.487545\pi$
$839$	2.26614	0.0782359	0.0391179	+	0.999235i	$0.487545\pi$
$840$	0	0
$841$	−28.5761	−0.985382
$842$	−0.510413	−0.0175900
$843$	25.1367	0.865752
$844$	0.640420	0.0220442
$845$	0	0
$846$	−1.29736	−0.0446042
$847$	−14.5011	−0.498262
$848$	−42.6842	−1.46578
$849$	−10.2045	−0.350218
$850$	0	0
$851$	−47.7226	−1.63591
$852$	2.20955	0.0756979
$853$	51.8753	1.77618	0.888089	−	0.459672i	$-0.152033\pi$
$853$	51.8753	1.77618	0.888089	+	0.459672i	$0.152033\pi$
$854$	−8.95328	−0.306375
$855$	0	0
$856$	15.5635	0.531949
$857$	−6.17058	−0.210783	−0.105391	−	0.994431i	$-0.533610\pi$
$857$	−6.17058	−0.210783	−0.105391	+	0.994431i	$0.533610\pi$
$858$	−34.1903	−1.16724
$859$	−31.0635	−1.05987	−0.529937	−	0.848037i	$-0.677785\pi$
$859$	−31.0635	−1.05987	−0.529937	+	0.848037i	$0.677785\pi$
$860$	0	0
$861$	−11.6511	−0.397068
$862$	−18.1572	−0.618437
$863$	−3.24772	−0.110554	−0.0552768	−	0.998471i	$-0.517604\pi$
$863$	−3.24772	−0.110554	−0.0552768	+	0.998471i	$0.517604\pi$
$864$	11.3695	0.386797
$865$	0	0
$866$	16.3559	0.555795
$867$	18.9589	0.643878
$868$	9.19887	0.312230
$869$	−40.1903	−1.36336
$870$	0	0
$871$	−27.2517	−0.923390
$872$	−42.3169	−1.43303
$873$	−0.883892	−0.0299152
$874$	6.98158	0.236155
$875$	0	0
$876$	−1.54486	−0.0521959
$877$	−41.7042	−1.40825	−0.704125	−	0.710076i	$-0.748661\pi$
$877$	−41.7042	−1.40825	−0.704125	+	0.710076i	$0.748661\pi$
$878$	−16.8860	−0.569875
$879$	−50.6708	−1.70908
$880$	0	0
$881$	7.42392	0.250118	0.125059	−	0.992149i	$-0.460088\pi$
$881$	7.42392	0.250118	0.125059	+	0.992149i	$0.460088\pi$
$882$	−2.20875	−0.0743724
$883$	32.0333	1.07801	0.539004	−	0.842303i	$-0.318801\pi$
$883$	32.0333	1.07801	0.539004	+	0.842303i	$0.318801\pi$
$884$	−5.43269	−0.182721
$885$	0	0
$886$	7.29444	0.245061
$887$	20.0275	0.672457	0.336229	−	0.941780i	$-0.390848\pi$
$887$	20.0275	0.672457	0.336229	+	0.941780i	$0.390848\pi$
$888$	−43.5384	−1.46105
$889$	−43.1620	−1.44761
$890$	0	0
$891$	−21.4826	−0.719695
$892$	9.52936	0.319066
$893$	3.71836	0.124430
$894$	36.3695	1.21638
$895$	0	0
$896$	25.9709	0.867627
$897$	−55.4840	−1.85256
$898$	3.96688	0.132376
$899$	−4.34891	−0.145044
$900$	0	0
$901$	32.3078	1.07633
$902$	6.52723	0.217333
$903$	16.0283	0.533388
$904$	1.59662	0.0531029
$905$	0	0
$906$	−5.43460	−0.180552
$907$	4.94823	0.164303	0.0821517	−	0.996620i	$-0.473821\pi$
$907$	4.94823	0.164303	0.0821517	+	0.996620i	$0.473821\pi$
$908$	6.34518	0.210572
$909$	−4.60225	−0.152647
$910$	0	0
$911$	32.3014	1.07019	0.535096	−	0.844791i	$-0.320275\pi$
$911$	32.3014	1.07019	0.535096	+	0.844791i	$0.320275\pi$
$912$	5.12386	0.169668
$913$	−39.0459	−1.29223
$914$	34.9728	1.15680
$915$	0	0
$916$	5.51656	0.182272
$917$	29.6610	0.979491
$918$	16.1746	0.533841
$919$	21.3072	0.702861	0.351430	−	0.936214i	$-0.385695\pi$
$919$	21.3072	0.702861	0.351430	+	0.936214i	$0.385695\pi$
$920$	0	0
$921$	24.4076	0.804258
$922$	17.8137	0.586663
$923$	21.7536	0.716029
$924$	6.02830	0.198316
$925$	0	0
$926$	22.9290	0.753494
$927$	2.69994	0.0886775
$928$	1.36945	0.0449545
$929$	24.7848	0.813164	0.406582	−	0.913614i	$-0.366721\pi$
$929$	24.7848	0.813164	0.406582	+	0.913614i	$0.366721\pi$
$930$	0	0
$931$	6.33048	0.207473
$932$	−3.36250	−0.110142
$933$	44.5782	1.45942
$934$	16.1620	0.528838
$935$	0	0
$936$	5.08569	0.166231
$937$	−32.3687	−1.05744	−0.528719	−	0.848797i	$-0.677327\pi$
$937$	−32.3687	−1.05744	−0.528719	+	0.848797i	$0.677327\pi$
$938$	−20.6716	−0.674953
$939$	16.0926	0.525163
$940$	0	0
$941$	−1.48264	−0.0483326	−0.0241663	−	0.999708i	$-0.507693\pi$
$941$	−1.48264	−0.0483326	−0.0241663	+	0.999708i	$0.507693\pi$
$942$	23.0814	0.752032
$943$	10.5924	0.344935
$944$	24.3609	0.792880
$945$	0	0
$946$	−8.97945	−0.291947
$947$	−8.86064	−0.287932	−0.143966	−	0.989583i	$-0.545986\pi$
$947$	−8.86064	−0.287932	−0.143966	+	0.989583i	$0.545986\pi$
$948$	−9.44155	−0.306647
$949$	−15.2095	−0.493723
$950$	0	0
$951$	−32.5696	−1.05614
$952$	−25.9709	−0.841722
$953$	10.1239	0.327944	0.163972	−	0.986465i	$-0.447569\pi$
$953$	10.1239	0.327944	0.163972	+	0.986465i	$0.447569\pi$
$954$	−4.79900	−0.155373
$955$	0	0
$956$	9.27974	0.300128
$957$	−2.84997	−0.0921264
$958$	31.5344	1.01883
$959$	64.1719	2.07222
$960$	0	0
$961$	13.6142	0.439169
$962$	−68.0157	−2.19291
$963$	1.40762	0.0453600
$964$	5.51284	0.177557
$965$	0	0
$966$	−42.0870	−1.35413
$967$	41.8139	1.34465	0.672323	−	0.740258i	$-0.265297\pi$
$967$	41.8139	1.34465	0.672323	+	0.740258i	$0.265297\pi$
$968$	12.0275	0.386578
$969$	−3.87826	−0.124588
$970$	0	0
$971$	−5.53016	−0.177471	−0.0887356	−	0.996055i	$-0.528283\pi$
$971$	−5.53016	−0.177471	−0.0887356	+	0.996055i	$0.528283\pi$
$972$	1.07019	0.0343263
$973$	67.2362	2.15549
$974$	−5.20100	−0.166651
$975$	0	0
$976$	5.97383	0.191218
$977$	−31.9447	−1.02200	−0.511001	−	0.859580i	$-0.670725\pi$
$977$	−31.9447	−1.02200	−0.511001	+	0.859580i	$0.670725\pi$
$978$	−47.7018	−1.52534
$979$	−13.4338	−0.429346
$980$	0	0
$981$	−3.82730	−0.122196
$982$	37.2672	1.18925
$983$	8.82862	0.281589	0.140795	−	0.990039i	$-0.455034\pi$
$983$	8.82862	0.281589	0.140795	+	0.990039i	$0.455034\pi$
$984$	9.66367	0.308067
$985$	0	0
$986$	1.94823	0.0620444
$987$	−22.4154	−0.713489
$988$	−2.31286	−0.0735819
$989$	−14.5718	−0.463357
$990$	0	0
$991$	−43.7304	−1.38914	−0.694570	−	0.719425i	$-0.744405\pi$
$991$	−43.7304	−1.38914	−0.694570	+	0.719425i	$0.744405\pi$
$992$	−14.0488	−0.446051
$993$	−32.9271	−1.04491
$994$	16.5011	0.523382
$995$	0	0
$996$	−9.17270	−0.290648
$997$	−46.6738	−1.47817	−0.739086	−	0.673611i	$-0.764743\pi$
$997$	−46.6738	−1.47817	−0.739086	+	0.673611i	$0.764743\pi$
$998$	2.00403	0.0634363
$999$	−47.0694	−1.48921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.g.1.2	yes	3
3.2	odd	2		4275.2.a.ba.1.2		3
4.3	odd	2		7600.2.a.bh.1.3		3
5.2	odd	4		475.2.b.b.324.4		6
5.3	odd	4		475.2.b.b.324.3		6
5.4	even	2		475.2.a.e.1.2	✓	3
15.14	odd	2		4275.2.a.bm.1.2		3
19.18	odd	2		9025.2.a.y.1.2		3
20.19	odd	2		7600.2.a.cc.1.1		3
95.94	odd	2		9025.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.2	✓	3	5.4	even	2
475.2.a.g.1.2	yes	3	1.1	even	1	trivial
475.2.b.b.324.3		6	5.3	odd	4
475.2.b.b.324.4		6	5.2	odd	4
4275.2.a.ba.1.2		3	3.2	odd	2
4275.2.a.bm.1.2		3	15.14	odd	2
7600.2.a.bh.1.3		3	4.3	odd	2
7600.2.a.cc.1.1		3	20.19	odd	2
9025.2.a.y.1.2		3	19.18	odd	2
9025.2.a.bc.1.2		3	95.94	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27389 q^{2} -1.65109 q^{3} -0.377203 q^{4} -2.10331 q^{6} +3.65109 q^{7} -3.02830 q^{8} -0.273891 q^{9} +2.65109 q^{11} +0.622797 q^{12} +6.13161 q^{13} +4.65109 q^{14} -3.10331 q^{16} +2.34891 q^{17} -0.348907 q^{18} +1.00000 q^{19} -6.02830 q^{21} +3.37720 q^{22} +5.48052 q^{23} +5.00000 q^{24} +7.81100 q^{26} +5.40550 q^{27} -1.37720 q^{28} +0.651093 q^{29} -6.67939 q^{31} +2.10331 q^{32} -4.37720 q^{33} +2.99225 q^{34} +0.103312 q^{36} -8.70769 q^{37} +1.27389 q^{38} -10.1239 q^{39} +1.93273 q^{41} -7.67939 q^{42} -2.65884 q^{43} -1.00000 q^{44} +6.98158 q^{46} +3.71836 q^{47} +5.12386 q^{48} +6.33048 q^{49} -3.87826 q^{51} -2.31286 q^{52} +13.7544 q^{53} +6.88601 q^{54} -11.0566 q^{56} -1.65109 q^{57} +0.829422 q^{58} -7.84997 q^{59} -1.92498 q^{61} -8.50881 q^{62} -1.00000 q^{63} +8.88601 q^{64} -5.57608 q^{66} -4.44447 q^{67} -0.886014 q^{68} -9.04884 q^{69} +3.54778 q^{71} +0.829422 q^{72} -2.48052 q^{73} -11.0926 q^{74} -0.377203 q^{76} +9.67939 q^{77} -12.8967 q^{78} -15.1599 q^{79} -8.10331 q^{81} +2.46209 q^{82} -14.7282 q^{83} +2.27389 q^{84} -3.38708 q^{86} -1.07502 q^{87} -8.02830 q^{88} -5.06727 q^{89} +22.3871 q^{91} -2.06727 q^{92} +11.0283 q^{93} +4.73678 q^{94} -3.47277 q^{96} +3.22717 q^{97} +8.06434 q^{98} -0.726109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} + 7 q^{14} - 6 q^{16} + 14 q^{17} - 8 q^{18} + 3 q^{19} - 6 q^{21} + 5 q^{22} + 8 q^{23} + 15 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 + 4 * q^7 + 3 * q^8 + q^9 + q^11 + 7 * q^12 + 3 * q^13 + 7 * q^14 - 6 * q^16 + 14 * q^17 - 8 * q^18 + 3 * q^19 - 6 * q^21 + 5 * q^22 + 8 * q^23 + 15 * q^24 - 11 * q^26 - q^27 + q^28 - 5 * q^29 - q^31 + 3 * q^32 - 8 * q^33 + 5 * q^34 - 3 * q^36 + 5 * q^37 + 2 * q^38 - 11 * q^39 + q^41 - 4 * q^42 - 5 * q^43 - 3 * q^44 - 12 * q^46 + 9 * q^47 - 4 * q^48 - 7 * q^49 + 18 * q^51 - 22 * q^52 + 31 * q^53 - 5 * q^54 - 9 * q^56 + 2 * q^57 + q^58 - 6 * q^59 + 3 * q^61 - 5 * q^62 - 3 * q^63 + q^64 - q^66 - 13 * q^67 + 23 * q^68 + q^69 + 7 * q^71 + q^72 + q^73 - q^74 + 4 * q^76 + 10 * q^77 - 42 * q^78 - 18 * q^79 - 21 * q^81 - 34 * q^82 + 3 * q^83 + 5 * q^84 + 40 * q^86 - 12 * q^87 - 12 * q^88 - 20 * q^89 + 17 * q^91 - 11 * q^92 + 21 * q^93 + 45 * q^94 + 2 * q^96 - 13 * q^97 + 4 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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