LMFDB - Embedded newform 475.2.a.d.1.1

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,-4] 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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	\(q-2.80194 q^{2} -0.554958 q^{3} +5.85086 q^{4} +1.55496 q^{6} +3.04892 q^{7} -10.7899 q^{8} -2.69202 q^{9} -2.93900 q^{11} -3.24698 q^{12} -3.24698 q^{13} -8.54288 q^{14} +18.5308 q^{16} +2.15883 q^{17} +7.54288 q^{18} -1.00000 q^{19} -1.69202 q^{21} +8.23490 q^{22} -1.19806 q^{23} +5.98792 q^{24} +9.09783 q^{26} +3.15883 q^{27} +17.8388 q^{28} -1.77479 q^{29} -9.34481 q^{31} -30.3424 q^{32} +1.63102 q^{33} -6.04892 q^{34} -15.7506 q^{36} +1.15883 q^{37} +2.80194 q^{38} +1.80194 q^{39} +8.57002 q^{41} +4.74094 q^{42} -5.27413 q^{43} -17.1957 q^{44} +3.35690 q^{46} -2.35690 q^{47} -10.2838 q^{48} +2.29590 q^{49} -1.19806 q^{51} -18.9976 q^{52} -8.82371 q^{53} -8.85086 q^{54} -32.8974 q^{56} +0.554958 q^{57} +4.97285 q^{58} -5.70171 q^{59} -9.96077 q^{61} +26.1836 q^{62} -8.20775 q^{63} +47.9560 q^{64} -4.57002 q^{66} -4.98254 q^{67} +12.6310 q^{68} +0.664874 q^{69} +2.70171 q^{71} +29.0465 q^{72} -13.7778 q^{73} -3.24698 q^{74} -5.85086 q^{76} -8.96077 q^{77} -5.04892 q^{78} +5.66487 q^{79} +6.32304 q^{81} -24.0127 q^{82} +3.00969 q^{83} -9.89977 q^{84} +14.7778 q^{86} +0.984935 q^{87} +31.7114 q^{88} -10.2838 q^{89} -9.89977 q^{91} -7.00969 q^{92} +5.18598 q^{93} +6.60388 q^{94} +16.8388 q^{96} -3.24698 q^{97} -6.43296 q^{98} +7.91185 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27}+ \cdots + 20 q^{99}+O(q^{100}) $ 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 + q^11 - 5 * q^12 - 5 * q^13 - 7 * q^14 + 18 * q^16 - 2 * q^17 + 4 * q^18 - 3 * q^19 + q^22 - 8 * q^23 - q^24 + 9 * q^26 + q^27 + 21 * q^28 - 7 * q^29 - 5 * q^31 - 27 * q^32 - 10 * q^33 - 9 * q^34 - 11 * q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^41 - 5 * q^43 - 15 * q^44 + 6 * q^46 - 3 * q^47 + 2 * q^48 - 7 * q^49 - 8 * q^51 - 16 * q^52 - 19 * q^53 - 13 * q^54 - 35 * q^56 + 2 * q^57 + 21 * q^58 + 10 * q^59 - 17 * q^61 + 23 * q^62 - 7 * q^63 + 49 * q^64 + 11 * q^66 + q^67 + 23 * q^68 + 3 * q^69 - 19 * q^71 + 37 * q^72 + q^73 - 5 * q^74 - 4 * q^76 - 14 * q^77 - 6 * q^78 + 18 * q^79 - q^81 - 6 * q^82 - 13 * q^83 - 7 * q^84 + 2 * q^86 + 28 * q^87 + 46 * q^88 + 2 * q^89 - 7 * q^91 + q^92 + q^93 + 11 * q^94 + 18 * q^96 - 5 * q^97 + 20 * 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 $

$2$	−2.80194	−1.98127	−0.990635	−	0.136540i	$-0.956402\pi$
$2$	−2.80194	−1.98127	−0.990635	+	0.136540i	$0.956402\pi$
$3$	−0.554958	−0.320405	−0.160203	−	0.987084i	$-0.551215\pi$
$3$	−0.554958	−0.320405	−0.160203	+	0.987084i	$0.551215\pi$
$4$	5.85086	2.92543
$5$	0	0
$5$	0	0
$6$	1.55496	0.634809
$7$	3.04892	1.15238	0.576191	−	0.817315i	$-0.304538\pi$
$7$	3.04892	1.15238	0.576191	+	0.817315i	$0.304538\pi$
$8$	−10.7899	−3.81479
$9$	−2.69202	−0.897340
$10$	0	0
$11$	−2.93900	−0.886142	−0.443071	−	0.896486i	$-0.646111\pi$
$11$	−2.93900	−0.886142	−0.443071	+	0.896486i	$0.646111\pi$
$12$	−3.24698	−0.937322
$13$	−3.24698	−0.900550	−0.450275	−	0.892890i	$-0.648674\pi$
$13$	−3.24698	−0.900550	−0.450275	+	0.892890i	$0.648674\pi$
$14$	−8.54288	−2.28318
$15$	0	0
$16$	18.5308	4.63270
$17$	2.15883	0.523594	0.261797	−	0.965123i	$-0.415685\pi$
$17$	2.15883	0.523594	0.261797	+	0.965123i	$0.415685\pi$
$18$	7.54288	1.77787
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.69202	−0.369229
$22$	8.23490	1.75569
$23$	−1.19806	−0.249813	−0.124907	−	0.992169i	$-0.539863\pi$
$23$	−1.19806	−0.249813	−0.124907	+	0.992169i	$0.539863\pi$
$24$	5.98792	1.22228
$25$	0	0
$26$	9.09783	1.78423
$27$	3.15883	0.607918
$28$	17.8388	3.37121
$29$	−1.77479	−0.329570	−0.164785	−	0.986329i	$-0.552693\pi$
$29$	−1.77479	−0.329570	−0.164785	+	0.986329i	$0.552693\pi$
$30$	0	0
$31$	−9.34481	−1.67838	−0.839189	−	0.543840i	$-0.816970\pi$
$31$	−9.34481	−1.67838	−0.839189	+	0.543840i	$0.816970\pi$
$32$	−30.3424	−5.36383
$33$	1.63102	0.283925
$34$	−6.04892	−1.03738
$35$	0	0
$36$	−15.7506	−2.62510
$37$	1.15883	0.190511	0.0952555	−	0.995453i	$-0.469633\pi$
$37$	1.15883	0.190511	0.0952555	+	0.995453i	$0.469633\pi$
$38$	2.80194	0.454534
$39$	1.80194	0.288541
$40$	0	0
$41$	8.57002	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$41$	8.57002	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$42$	4.74094	0.731543
$43$	−5.27413	−0.804297	−0.402148	−	0.915575i	$-0.631736\pi$
$43$	−5.27413	−0.804297	−0.402148	+	0.915575i	$0.631736\pi$
$44$	−17.1957	−2.59234
$45$	0	0
$46$	3.35690	0.494947
$47$	−2.35690	−0.343789	−0.171894	−	0.985115i	$-0.554989\pi$
$47$	−2.35690	−0.343789	−0.171894	+	0.985115i	$0.554989\pi$
$48$	−10.2838	−1.48434
$49$	2.29590	0.327985
$50$	0	0
$51$	−1.19806	−0.167762
$52$	−18.9976	−2.63449
$53$	−8.82371	−1.21203	−0.606015	−	0.795453i	$-0.707233\pi$
$53$	−8.82371	−1.21203	−0.606015	+	0.795453i	$0.707233\pi$
$54$	−8.85086	−1.20445
$55$	0	0
$56$	−32.8974	−4.39610
$57$	0.554958	0.0735060
$58$	4.97285	0.652968
$59$	−5.70171	−0.742299	−0.371150	−	0.928573i	$-0.621036\pi$
$59$	−5.70171	−0.742299	−0.371150	+	0.928573i	$0.621036\pi$
$60$	0	0
$61$	−9.96077	−1.27535	−0.637673	−	0.770307i	$-0.720103\pi$
$61$	−9.96077	−1.27535	−0.637673	+	0.770307i	$0.720103\pi$
$62$	26.1836	3.32532
$63$	−8.20775	−1.03408
$64$	47.9560	5.99450
$65$	0	0
$66$	−4.57002	−0.562531
$67$	−4.98254	−0.608714	−0.304357	−	0.952558i	$-0.598442\pi$
$67$	−4.98254	−0.608714	−0.304357	+	0.952558i	$0.598442\pi$
$68$	12.6310	1.53174
$69$	0.664874	0.0800415
$70$	0	0
$71$	2.70171	0.320634	0.160317	−	0.987066i	$-0.448748\pi$
$71$	2.70171	0.320634	0.160317	+	0.987066i	$0.448748\pi$
$72$	29.0465	3.42317
$73$	−13.7778	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$73$	−13.7778	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$74$	−3.24698	−0.377454
$75$	0	0
$76$	−5.85086	−0.671139
$77$	−8.96077	−1.02117
$78$	−5.04892	−0.571677
$79$	5.66487	0.637348	0.318674	−	0.947864i	$-0.396762\pi$
$79$	5.66487	0.637348	0.318674	+	0.947864i	$0.396762\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	−24.0127	−2.65176
$83$	3.00969	0.330356	0.165178	−	0.986264i	$-0.447180\pi$
$83$	3.00969	0.330356	0.165178	+	0.986264i	$0.447180\pi$
$84$	−9.89977	−1.08015
$85$	0	0
$86$	14.7778	1.59353
$87$	0.984935	0.105596
$88$	31.7114	3.38045
$89$	−10.2838	−1.09008	−0.545041	−	0.838409i	$-0.683486\pi$
$89$	−10.2838	−1.09008	−0.545041	+	0.838409i	$0.683486\pi$
$90$	0	0
$91$	−9.89977	−1.03778
$92$	−7.00969	−0.730811
$93$	5.18598	0.537761
$94$	6.60388	0.681138
$95$	0	0
$96$	16.8388	1.71860
$97$	−3.24698	−0.329681	−0.164840	−	0.986320i	$-0.552711\pi$
$97$	−3.24698	−0.329681	−0.164840	+	0.986320i	$0.552711\pi$
$98$	−6.43296	−0.649827
$99$	7.91185	0.795171
$100$	0	0
$101$	5.09246	0.506718	0.253359	−	0.967372i	$-0.418465\pi$
$101$	5.09246	0.506718	0.253359	+	0.967372i	$0.418465\pi$
$102$	3.35690	0.332382
$103$	14.3110	1.41010	0.705051	−	0.709157i	$-0.250924\pi$
$103$	14.3110	1.41010	0.705051	+	0.709157i	$0.250924\pi$
$104$	35.0344	3.43541
$105$	0	0
$106$	24.7235	2.40136
$107$	8.18598	0.791369	0.395684	−	0.918387i	$-0.370507\pi$
$107$	8.18598	0.791369	0.395684	+	0.918387i	$0.370507\pi$
$108$	18.4819	1.77842
$109$	−11.3274	−1.08496	−0.542482	−	0.840067i	$-0.682515\pi$
$109$	−11.3274	−1.08496	−0.542482	+	0.840067i	$0.682515\pi$
$110$	0	0
$111$	−0.643104	−0.0610407
$112$	56.4989	5.33864
$113$	−3.43535	−0.323171	−0.161585	−	0.986859i	$-0.551661\pi$
$113$	−3.43535	−0.323171	−0.161585	+	0.986859i	$0.551661\pi$
$114$	−1.55496	−0.145635
$115$	0	0
$116$	−10.3840	−0.964134
$117$	8.74094	0.808100
$118$	15.9758	1.47069
$119$	6.58211	0.603381
$120$	0	0
$121$	−2.36227	−0.214752
$122$	27.9095	2.52680
$123$	−4.75600	−0.428834
$124$	−54.6752	−4.90997
$125$	0	0
$126$	22.9976	2.04879
$127$	−1.46144	−0.129681	−0.0648407	−	0.997896i	$-0.520654\pi$
$127$	−1.46144	−0.129681	−0.0648407	+	0.997896i	$0.520654\pi$
$128$	−73.6848	−6.51288
$129$	2.92692	0.257701
$130$	0	0
$131$	−13.2295	−1.15587	−0.577934	−	0.816083i	$-0.696141\pi$
$131$	−13.2295	−1.15587	−0.577934	+	0.816083i	$0.696141\pi$
$132$	9.54288	0.830601
$133$	−3.04892	−0.264375
$134$	13.9608	1.20603
$135$	0	0
$136$	−23.2935	−1.99740
$137$	−16.1739	−1.38183	−0.690915	−	0.722936i	$-0.742792\pi$
$137$	−16.1739	−1.38183	−0.690915	+	0.722936i	$0.742792\pi$
$138$	−1.86294	−0.158584
$139$	13.0978	1.11094	0.555472	−	0.831535i	$-0.312538\pi$
$139$	13.0978	1.11094	0.555472	+	0.831535i	$0.312538\pi$
$140$	0	0
$141$	1.30798	0.110152
$142$	−7.57002	−0.635262
$143$	9.54288	0.798015
$144$	−49.8853	−4.15711
$145$	0	0
$146$	38.6045	3.19493
$147$	−1.27413	−0.105088
$148$	6.78017	0.557326
$149$	11.0640	0.906397	0.453198	−	0.891410i	$-0.350283\pi$
$149$	11.0640	0.906397	0.453198	+	0.891410i	$0.350283\pi$
$150$	0	0
$151$	10.4426	0.849811	0.424905	−	0.905238i	$-0.360307\pi$
$151$	10.4426	0.849811	0.424905	+	0.905238i	$0.360307\pi$
$152$	10.7899	0.875173
$153$	−5.81163	−0.469842
$154$	25.1075	2.02322
$155$	0	0
$156$	10.5429	0.844106
$157$	−2.48427	−0.198266	−0.0991332	−	0.995074i	$-0.531607\pi$
$157$	−2.48427	−0.198266	−0.0991332	+	0.995074i	$0.531607\pi$
$158$	−15.8726	−1.26276
$159$	4.89679	0.388341
$160$	0	0
$161$	−3.65279	−0.287880
$162$	−17.7168	−1.39196
$163$	−3.16852	−0.248178	−0.124089	−	0.992271i	$-0.539601\pi$
$163$	−3.16852	−0.248178	−0.124089	+	0.992271i	$0.539601\pi$
$164$	50.1420	3.91543
$165$	0	0
$166$	−8.43296	−0.654525
$167$	−4.74632	−0.367281	−0.183640	−	0.982993i	$-0.558788\pi$
$167$	−4.74632	−0.367281	−0.183640	+	0.982993i	$0.558788\pi$
$168$	18.2567	1.40853
$169$	−2.45712	−0.189009
$170$	0	0
$171$	2.69202	0.205864
$172$	−30.8582	−2.35291
$173$	−3.96316	−0.301314	−0.150657	−	0.988586i	$-0.548139\pi$
$173$	−3.96316	−0.301314	−0.150657	+	0.988586i	$0.548139\pi$
$174$	−2.75973	−0.209214
$175$	0	0
$176$	−54.4620	−4.10523
$177$	3.16421	0.237837
$178$	28.8146	2.15975
$179$	−14.0858	−1.05282	−0.526409	−	0.850231i	$-0.676462\pi$
$179$	−14.0858	−1.05282	−0.526409	+	0.850231i	$0.676462\pi$
$180$	0	0
$181$	−12.2513	−0.910631	−0.455316	−	0.890330i	$-0.650474\pi$
$181$	−12.2513	−0.910631	−0.455316	+	0.890330i	$0.650474\pi$
$182$	27.7385	2.05612
$183$	5.52781	0.408628
$184$	12.9269	0.952985
$185$	0	0
$186$	−14.5308	−1.06545
$187$	−6.34481	−0.463979
$188$	−13.7899	−1.00573
$189$	9.63102	0.700554
$190$	0	0
$191$	20.1468	1.45777	0.728884	−	0.684637i	$-0.240039\pi$
$191$	20.1468	1.45777	0.728884	+	0.684637i	$0.240039\pi$
$192$	−26.6136	−1.92067
$193$	9.52781	0.685827	0.342913	−	0.939367i	$-0.388586\pi$
$193$	9.52781	0.685827	0.342913	+	0.939367i	$0.388586\pi$
$194$	9.09783	0.653186
$195$	0	0
$196$	13.4330	0.959497
$197$	−4.96316	−0.353611	−0.176805	−	0.984246i	$-0.556576\pi$
$197$	−4.96316	−0.353611	−0.176805	+	0.984246i	$0.556576\pi$
$198$	−22.1685	−1.57545
$199$	5.42221	0.384370	0.192185	−	0.981359i	$-0.438443\pi$
$199$	5.42221	0.384370	0.192185	+	0.981359i	$0.438443\pi$
$200$	0	0
$201$	2.76510	0.195035
$202$	−14.2687	−1.00395
$203$	−5.41119	−0.379791
$204$	−7.00969	−0.490776
$205$	0	0
$206$	−40.0984	−2.79379
$207$	3.22521	0.224168
$208$	−60.1691	−4.17198
$209$	2.93900	0.203295
$210$	0	0
$211$	15.9638	1.09899	0.549495	−	0.835497i	$-0.314820\pi$
$211$	15.9638	1.09899	0.549495	+	0.835497i	$0.314820\pi$
$212$	−51.6262	−3.54570
$213$	−1.49934	−0.102733
$214$	−22.9366	−1.56791
$215$	0	0
$216$	−34.0834	−2.31908
$217$	−28.4916	−1.93413
$218$	31.7385	2.14961
$219$	7.64609	0.516675
$220$	0	0
$221$	−7.00969	−0.471523
$222$	1.80194	0.120938
$223$	23.2150	1.55459	0.777297	−	0.629134i	$-0.216590\pi$
$223$	23.2150	1.55459	0.777297	+	0.629134i	$0.216590\pi$
$224$	−92.5115	−6.18119
$225$	0	0
$226$	9.62565	0.640288
$227$	17.4795	1.16015	0.580077	−	0.814562i	$-0.303022\pi$
$227$	17.4795	1.16015	0.580077	+	0.814562i	$0.303022\pi$
$228$	3.24698	0.215036
$229$	−2.32544	−0.153669	−0.0768346	−	0.997044i	$-0.524481\pi$
$229$	−2.32544	−0.153669	−0.0768346	+	0.997044i	$0.524481\pi$
$230$	0	0
$231$	4.97285	0.327190
$232$	19.1497	1.25724
$233$	18.9342	1.24042	0.620211	−	0.784435i	$-0.287047\pi$
$233$	18.9342	1.24042	0.620211	+	0.784435i	$0.287047\pi$
$234$	−24.4916	−1.60106
$235$	0	0
$236$	−33.3599	−2.17154
$237$	−3.14377	−0.204210
$238$	−18.4426	−1.19546
$239$	11.1685	0.722432	0.361216	−	0.932482i	$-0.382362\pi$
$239$	11.1685	0.722432	0.361216	+	0.932482i	$0.382362\pi$
$240$	0	0
$241$	7.42998	0.478607	0.239303	−	0.970945i	$-0.423081\pi$
$241$	7.42998	0.478607	0.239303	+	0.970945i	$0.423081\pi$
$242$	6.61894	0.425482
$243$	−12.9855	−0.833022
$244$	−58.2790	−3.73093
$245$	0	0
$246$	13.3260	0.849637
$247$	3.24698	0.206600
$248$	100.829	6.40266
$249$	−1.67025	−0.105848
$250$	0	0
$251$	−14.7409	−0.930440	−0.465220	−	0.885195i	$-0.654025\pi$
$251$	−14.7409	−0.930440	−0.465220	+	0.885195i	$0.654025\pi$
$252$	−48.0224	−3.02512
$253$	3.52111	0.221370
$254$	4.09485	0.256934
$255$	0	0
$256$	110.548	6.90927
$257$	−3.34721	−0.208793	−0.104397	−	0.994536i	$-0.533291\pi$
$257$	−3.34721	−0.208793	−0.104397	+	0.994536i	$0.533291\pi$
$258$	−8.20105	−0.510575
$259$	3.53319	0.219542
$260$	0	0
$261$	4.77777	0.295737
$262$	37.0683	2.29009
$263$	16.6853	1.02886	0.514430	−	0.857532i	$-0.328003\pi$
$263$	16.6853	1.02886	0.514430	+	0.857532i	$0.328003\pi$
$264$	−17.5985	−1.08311
$265$	0	0
$266$	8.54288	0.523797
$267$	5.70709	0.349268
$268$	−29.1521	−1.78075
$269$	15.5961	0.950911	0.475456	−	0.879740i	$-0.342283\pi$
$269$	15.5961	0.950911	0.475456	+	0.879740i	$0.342283\pi$
$270$	0	0
$271$	−13.2131	−0.802640	−0.401320	−	0.915938i	$-0.631449\pi$
$271$	−13.2131	−0.802640	−0.401320	+	0.915938i	$0.631449\pi$
$272$	40.0049	2.42565
$273$	5.49396	0.332510
$274$	45.3183	2.73778
$275$	0	0
$276$	3.89008	0.234156
$277$	−12.9758	−0.779642	−0.389821	−	0.920891i	$-0.627463\pi$
$277$	−12.9758	−0.779642	−0.389821	+	0.920891i	$0.627463\pi$
$278$	−36.6993	−2.20108
$279$	25.1564	1.50608
$280$	0	0
$281$	24.8901	1.48482	0.742409	−	0.669947i	$-0.233683\pi$
$281$	24.8901	1.48482	0.742409	+	0.669947i	$0.233683\pi$
$282$	−3.66487	−0.218240
$283$	13.5821	0.807372	0.403686	−	0.914898i	$-0.367729\pi$
$283$	13.5821	0.807372	0.403686	+	0.914898i	$0.367729\pi$
$284$	15.8073	0.937992
$285$	0	0
$286$	−26.7385	−1.58108
$287$	26.1293	1.54236
$288$	81.6824	4.81318
$289$	−12.3394	−0.725849
$290$	0	0
$291$	1.80194	0.105631
$292$	−80.6118	−4.71745
$293$	−1.27652	−0.0745751	−0.0372875	−	0.999305i	$-0.511872\pi$
$293$	−1.27652	−0.0745751	−0.0372875	+	0.999305i	$0.511872\pi$
$294$	3.57002	0.208208
$295$	0	0
$296$	−12.5036	−0.726760
$297$	−9.28382	−0.538702
$298$	−31.0006	−1.79582
$299$	3.89008	0.224969
$300$	0	0
$301$	−16.0804	−0.926857
$302$	−29.2597	−1.68370
$303$	−2.82610	−0.162355
$304$	−18.5308	−1.06281
$305$	0	0
$306$	16.2838	0.930884
$307$	32.2295	1.83944	0.919718	−	0.392580i	$-0.128417\pi$
$307$	32.2295	1.83944	0.919718	+	0.392580i	$0.128417\pi$
$308$	−52.4282	−2.98737
$309$	−7.94198	−0.451804
$310$	0	0
$311$	12.4983	0.708712	0.354356	−	0.935111i	$-0.384700\pi$
$311$	12.4983	0.708712	0.354356	+	0.935111i	$0.384700\pi$
$312$	−19.4426	−1.10072
$313$	20.9390	1.18354	0.591771	−	0.806106i	$-0.298429\pi$
$313$	20.9390	1.18354	0.591771	+	0.806106i	$0.298429\pi$
$314$	6.96077	0.392819
$315$	0	0
$316$	33.1444	1.86452
$317$	−15.8562	−0.890575	−0.445287	−	0.895388i	$-0.646898\pi$
$317$	−15.8562	−0.890575	−0.445287	+	0.895388i	$0.646898\pi$
$318$	−13.7205	−0.769407
$319$	5.21611	0.292046
$320$	0	0
$321$	−4.54288	−0.253559
$322$	10.2349	0.570369
$323$	−2.15883	−0.120121
$324$	36.9952	2.05529
$325$	0	0
$326$	8.87800	0.491707
$327$	6.28621	0.347628
$328$	−92.4693	−5.10576
$329$	−7.18598	−0.396176
$330$	0	0
$331$	−13.2349	−0.727456	−0.363728	−	0.931505i	$-0.618496\pi$
$331$	−13.2349	−0.727456	−0.363728	+	0.931505i	$0.618496\pi$
$332$	17.6093	0.966433
$333$	−3.11960	−0.170953
$334$	13.2989	0.727682
$335$	0	0
$336$	−31.3545	−1.71053
$337$	26.0780	1.42056	0.710279	−	0.703920i	$-0.248569\pi$
$337$	26.0780	1.42056	0.710279	+	0.703920i	$0.248569\pi$
$338$	6.88471	0.374479
$339$	1.90648	0.103546
$340$	0	0
$341$	27.4644	1.48728
$342$	−7.54288	−0.407872
$343$	−14.3424	−0.774418
$344$	56.9071	3.06822
$345$	0	0
$346$	11.1045	0.596984
$347$	−8.98254	−0.482208	−0.241104	−	0.970499i	$-0.577510\pi$
$347$	−8.98254	−0.482208	−0.241104	+	0.970499i	$0.577510\pi$
$348$	5.76271	0.308914
$349$	0.599564	0.0320939	0.0160470	−	0.999871i	$-0.494892\pi$
$349$	0.599564	0.0320939	0.0160470	+	0.999871i	$0.494892\pi$
$350$	0	0
$351$	−10.2567	−0.547460
$352$	89.1764	4.75312
$353$	−31.8896	−1.69731	−0.848656	−	0.528945i	$-0.822588\pi$
$353$	−31.8896	−1.69731	−0.848656	+	0.528945i	$0.822588\pi$
$354$	−8.86592	−0.471218
$355$	0	0
$356$	−60.1691	−3.18896
$357$	−3.65279	−0.193326
$358$	39.4674	2.08592
$359$	−18.3763	−0.969863	−0.484931	−	0.874552i	$-0.661155\pi$
$359$	−18.3763	−0.969863	−0.484931	+	0.874552i	$0.661155\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	34.3274	1.80421
$363$	1.31096	0.0688077
$364$	−57.9221	−3.03594
$365$	0	0
$366$	−15.4886	−0.809601
$367$	14.6377	0.764083	0.382042	−	0.924145i	$-0.375221\pi$
$367$	14.6377	0.764083	0.382042	+	0.924145i	$0.375221\pi$
$368$	−22.2010	−1.15731
$369$	−23.0707	−1.20101
$370$	0	0
$371$	−26.9028	−1.39672
$372$	30.3424	1.57318
$373$	1.02715	0.0531837	0.0265918	−	0.999646i	$-0.491535\pi$
$373$	1.02715	0.0531837	0.0265918	+	0.999646i	$0.491535\pi$
$374$	17.7778	0.919267
$375$	0	0
$376$	25.4306	1.31148
$377$	5.76271	0.296795
$378$	−26.9855	−1.38799
$379$	−22.5284	−1.15721	−0.578603	−	0.815609i	$-0.696402\pi$
$379$	−22.5284	−1.15721	−0.578603	+	0.815609i	$0.696402\pi$
$380$	0	0
$381$	0.811035	0.0415506
$382$	−56.4499	−2.88823
$383$	−32.6698	−1.66935	−0.834674	−	0.550745i	$-0.814344\pi$
$383$	−32.6698	−1.66935	−0.834674	+	0.550745i	$0.814344\pi$
$384$	40.8920	2.08676
$385$	0	0
$386$	−26.6963	−1.35881
$387$	14.1981	0.721728
$388$	−18.9976	−0.964457
$389$	−8.16421	−0.413942	−0.206971	−	0.978347i	$-0.566361\pi$
$389$	−8.16421	−0.413942	−0.206971	+	0.978347i	$0.566361\pi$
$390$	0	0
$391$	−2.58642	−0.130801
$392$	−24.7724	−1.25120
$393$	7.34183	0.370346
$394$	13.9065	0.700598
$395$	0	0
$396$	46.2911	2.32622
$397$	−37.5478	−1.88447	−0.942235	−	0.334954i	$-0.891279\pi$
$397$	−37.5478	−1.88447	−0.942235	+	0.334954i	$0.891279\pi$
$398$	−15.1927	−0.761541
$399$	1.69202	0.0847070
$400$	0	0
$401$	32.2519	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$401$	32.2519	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$402$	−7.74764	−0.386417
$403$	30.3424	1.51146
$404$	29.7952	1.48237
$405$	0	0
$406$	15.1618	0.752468
$407$	−3.40581	−0.168820
$408$	12.9269	0.639978
$409$	−31.5459	−1.55984	−0.779921	−	0.625878i	$-0.784741\pi$
$409$	−31.5459	−1.55984	−0.779921	+	0.625878i	$0.784741\pi$
$410$	0	0
$411$	8.97584	0.442745
$412$	83.7314	4.12515
$413$	−17.3840	−0.855413
$414$	−9.03684	−0.444136
$415$	0	0
$416$	98.5212	4.83040
$417$	−7.26875	−0.355952
$418$	−8.23490	−0.402782
$419$	−14.6866	−0.717490	−0.358745	−	0.933436i	$-0.616795\pi$
$419$	−14.6866	−0.717490	−0.358745	+	0.933436i	$0.616795\pi$
$420$	0	0
$421$	15.1142	0.736622	0.368311	−	0.929703i	$-0.379936\pi$
$421$	15.1142	0.736622	0.368311	+	0.929703i	$0.379936\pi$
$422$	−44.7294	−2.17740
$423$	6.34481	0.308495
$424$	95.2065	4.62364
$425$	0	0
$426$	4.20105	0.203541
$427$	−30.3696	−1.46969
$428$	47.8950	2.31509
$429$	−5.29590	−0.255688
$430$	0	0
$431$	1.67696	0.0807761	0.0403881	−	0.999184i	$-0.487141\pi$
$431$	1.67696	0.0807761	0.0403881	+	0.999184i	$0.487141\pi$
$432$	58.5357	2.81630
$433$	13.1545	0.632166	0.316083	−	0.948732i	$-0.397632\pi$
$433$	13.1545	0.632166	0.316083	+	0.948732i	$0.397632\pi$
$434$	79.8316	3.83204
$435$	0	0
$436$	−66.2747	−3.17398
$437$	1.19806	0.0573111
$438$	−21.4239	−1.02367
$439$	6.45580	0.308118	0.154059	−	0.988062i	$-0.450765\pi$
$439$	6.45580	0.308118	0.154059	+	0.988062i	$0.450765\pi$
$440$	0	0
$441$	−6.18060	−0.294314
$442$	19.6407	0.934213
$443$	25.3709	1.20541	0.602704	−	0.797965i	$-0.294090\pi$
$443$	25.3709	1.20541	0.602704	+	0.797965i	$0.294090\pi$
$444$	−3.76271	−0.178570
$445$	0	0
$446$	−65.0471	−3.08007
$447$	−6.14005	−0.290414
$448$	146.214	6.90795
$449$	−3.59956	−0.169874	−0.0849370	−	0.996386i	$-0.527069\pi$
$449$	−3.59956	−0.169874	−0.0849370	+	0.996386i	$0.527069\pi$
$450$	0	0
$451$	−25.1873	−1.18602
$452$	−20.0998	−0.945413
$453$	−5.79523	−0.272284
$454$	−48.9764	−2.29858
$455$	0	0
$456$	−5.98792	−0.280410
$457$	12.2784	0.574361	0.287181	−	0.957876i	$-0.407282\pi$
$457$	12.2784	0.574361	0.287181	+	0.957876i	$0.407282\pi$
$458$	6.51573	0.304460
$459$	6.81940	0.318302
$460$	0	0
$461$	39.4935	1.83939	0.919697	−	0.392628i	$-0.128434\pi$
$461$	39.4935	1.83939	0.919697	+	0.392628i	$0.128434\pi$
$462$	−13.9336	−0.648251
$463$	−18.3991	−0.855079	−0.427540	−	0.903997i	$-0.640620\pi$
$463$	−18.3991	−0.855079	−0.427540	+	0.903997i	$0.640620\pi$
$464$	−32.8883	−1.52680
$465$	0	0
$466$	−53.0525	−2.45761
$467$	36.2282	1.67644	0.838220	−	0.545332i	$-0.183596\pi$
$467$	36.2282	1.67644	0.838220	+	0.545332i	$0.183596\pi$
$468$	51.1420	2.36404
$469$	−15.1914	−0.701472
$470$	0	0
$471$	1.37867	0.0635256
$472$	61.5206	2.83172
$473$	15.5007	0.712721
$474$	8.80864	0.404594
$475$	0	0
$476$	38.5109	1.76515
$477$	23.7536	1.08760
$478$	−31.2935	−1.43133
$479$	−34.3129	−1.56780	−0.783898	−	0.620890i	$-0.786771\pi$
$479$	−34.3129	−1.56780	−0.783898	+	0.620890i	$0.786771\pi$
$480$	0	0
$481$	−3.76271	−0.171565
$482$	−20.8183	−0.948249
$483$	2.02715	0.0922384
$484$	−13.8213	−0.628242
$485$	0	0
$486$	36.3846	1.65044
$487$	24.4722	1.10894	0.554470	−	0.832203i	$-0.312921\pi$
$487$	24.4722	1.10894	0.554470	+	0.832203i	$0.312921\pi$
$488$	107.475	4.86518
$489$	1.75840	0.0795175
$490$	0	0
$491$	18.3067	0.826168	0.413084	−	0.910693i	$-0.364452\pi$
$491$	18.3067	0.826168	0.413084	+	0.910693i	$0.364452\pi$
$492$	−27.8267	−1.25452
$493$	−3.83148	−0.172561
$494$	−9.09783	−0.409331
$495$	0	0
$496$	−173.167	−7.77542
$497$	8.23729	0.369493
$498$	4.67994	0.209713
$499$	−25.3424	−1.13448	−0.567241	−	0.823552i	$-0.691989\pi$
$499$	−25.3424	−1.13448	−0.567241	+	0.823552i	$0.691989\pi$
$500$	0	0
$501$	2.63401	0.117679
$502$	41.3032	1.84345
$503$	−14.1105	−0.629156	−0.314578	−	0.949232i	$-0.601863\pi$
$503$	−14.1105	−0.629156	−0.314578	+	0.949232i	$0.601863\pi$
$504$	88.5605	3.94480
$505$	0	0
$506$	−9.86592	−0.438594
$507$	1.36360	0.0605596
$508$	−8.55065	−0.379374
$509$	−17.8019	−0.789057	−0.394529	−	0.918884i	$-0.629092\pi$
$509$	−17.8019	−0.789057	−0.394529	+	0.918884i	$0.629092\pi$
$510$	0	0
$511$	−42.0073	−1.85829
$512$	−162.380	−7.17625
$513$	−3.15883	−0.139466
$514$	9.37867	0.413675
$515$	0	0
$516$	17.1250	0.753885
$517$	6.92692	0.304646
$518$	−9.89977	−0.434971
$519$	2.19939	0.0965425
$520$	0	0
$521$	13.3948	0.586837	0.293418	−	0.955984i	$-0.405207\pi$
$521$	13.3948	0.586837	0.293418	+	0.955984i	$0.405207\pi$
$522$	−13.3870	−0.585934
$523$	1.46921	0.0642439	0.0321219	−	0.999484i	$-0.489774\pi$
$523$	1.46921	0.0642439	0.0321219	+	0.999484i	$0.489774\pi$
$524$	−77.4040	−3.38141
$525$	0	0
$526$	−46.7512	−2.03845
$527$	−20.1739	−0.878789
$528$	30.2241	1.31534
$529$	−21.5646	−0.937593
$530$	0	0
$531$	15.3491	0.666095
$532$	−17.8388	−0.773409
$533$	−27.8267	−1.20531
$534$	−15.9909	−0.691994
$535$	0	0
$536$	53.7609	2.32212
$537$	7.81700	0.337329
$538$	−43.6993	−1.88401
$539$	−6.74764	−0.290642
$540$	0	0
$541$	−42.7144	−1.83643	−0.918217	−	0.396077i	$-0.870371\pi$
$541$	−42.7144	−1.83643	−0.918217	+	0.396077i	$0.870371\pi$
$542$	37.0224	1.59025
$543$	6.79895	0.291771
$544$	−65.5042	−2.80847
$545$	0	0
$546$	−15.3937	−0.658791
$547$	−10.6866	−0.456928	−0.228464	−	0.973552i	$-0.573370\pi$
$547$	−10.6866	−0.456928	−0.228464	+	0.973552i	$0.573370\pi$
$548$	−94.6311	−4.04244
$549$	26.8146	1.14442
$550$	0	0
$551$	1.77479	0.0756086
$552$	−7.17390	−0.305341
$553$	17.2717	0.734469
$554$	36.3575	1.54468
$555$	0	0
$556$	76.6335	3.24999
$557$	34.2083	1.44945	0.724727	−	0.689036i	$-0.241966\pi$
$557$	34.2083	1.44945	0.724727	+	0.689036i	$0.241966\pi$
$558$	−70.4868	−2.98394
$559$	17.1250	0.724310
$560$	0	0
$561$	3.52111	0.148661
$562$	−69.7405	−2.94182
$563$	5.46788	0.230444	0.115222	−	0.993340i	$-0.463242\pi$
$563$	5.46788	0.230444	0.115222	+	0.993340i	$0.463242\pi$
$564$	7.65279	0.322241
$565$	0	0
$566$	−38.0562	−1.59962
$567$	19.2784	0.809618
$568$	−29.1511	−1.22315
$569$	−17.4819	−0.732878	−0.366439	−	0.930442i	$-0.619423\pi$
$569$	−17.4819	−0.732878	−0.366439	+	0.930442i	$0.619423\pi$
$570$	0	0
$571$	−7.21877	−0.302096	−0.151048	−	0.988526i	$-0.548265\pi$
$571$	−7.21877	−0.302096	−0.151048	+	0.988526i	$0.548265\pi$
$572$	55.8340	2.33454
$573$	−11.1806	−0.467076
$574$	−73.2127	−3.05584
$575$	0	0
$576$	−129.099	−5.37911
$577$	−1.80625	−0.0751952	−0.0375976	−	0.999293i	$-0.511971\pi$
$577$	−1.80625	−0.0751952	−0.0375976	+	0.999293i	$0.511971\pi$
$578$	34.5743	1.43810
$579$	−5.28754	−0.219743
$580$	0	0
$581$	9.17629	0.380697
$582$	−5.04892	−0.209284
$583$	25.9329	1.07403
$584$	148.660	6.15160
$585$	0	0
$586$	3.57673	0.147753
$587$	−2.39075	−0.0986767	−0.0493384	−	0.998782i	$-0.515711\pi$
$587$	−2.39075	−0.0986767	−0.0493384	+	0.998782i	$0.515711\pi$
$588$	−7.45473	−0.307428
$589$	9.34481	0.385046
$590$	0	0
$591$	2.75435	0.113299
$592$	21.4741	0.882580
$593$	42.2650	1.73562	0.867808	−	0.496899i	$-0.165528\pi$
$593$	42.2650	1.73562	0.867808	+	0.496899i	$0.165528\pi$
$594$	26.0127	1.06731
$595$	0	0
$596$	64.7338	2.65160
$597$	−3.00910	−0.123154
$598$	−10.8998	−0.445725
$599$	8.95838	0.366029	0.183015	−	0.983110i	$-0.441414\pi$
$599$	8.95838	0.366029	0.183015	+	0.983110i	$0.441414\pi$
$600$	0	0
$601$	−19.8998	−0.811729	−0.405864	−	0.913933i	$-0.633029\pi$
$601$	−19.8998	−0.811729	−0.405864	+	0.913933i	$0.633029\pi$
$602$	45.0562	1.83635
$603$	13.4131	0.546224
$604$	61.0984	2.48606
$605$	0	0
$606$	7.91856	0.321669
$607$	−26.0084	−1.05565	−0.527823	−	0.849354i	$-0.676992\pi$
$607$	−26.0084	−1.05565	−0.527823	+	0.849354i	$0.676992\pi$
$608$	30.3424	1.23055
$609$	3.00298	0.121687
$610$	0	0
$611$	7.65279	0.309599
$612$	−34.0030	−1.37449
$613$	−41.3763	−1.67117	−0.835586	−	0.549360i	$-0.814872\pi$
$613$	−41.3763	−1.67117	−0.835586	+	0.549360i	$0.814872\pi$
$614$	−90.3051	−3.64442
$615$	0	0
$616$	96.6854	3.89557
$617$	−43.9845	−1.77075	−0.885374	−	0.464880i	$-0.846098\pi$
$617$	−43.9845	−1.77075	−0.885374	+	0.464880i	$0.846098\pi$
$618$	22.2529	0.895145
$619$	23.4553	0.942749	0.471374	−	0.881933i	$-0.343758\pi$
$619$	23.4553	0.942749	0.471374	+	0.881933i	$0.343758\pi$
$620$	0	0
$621$	−3.78448	−0.151866
$622$	−35.0194	−1.40415
$623$	−31.3545	−1.25619
$624$	33.3913	1.33672
$625$	0	0
$626$	−58.6698	−2.34492
$627$	−1.63102	−0.0651368
$628$	−14.5351	−0.580014
$629$	2.50173	0.0997505
$630$	0	0
$631$	−34.8877	−1.38886	−0.694429	−	0.719562i	$-0.744343\pi$
$631$	−34.8877	−1.38886	−0.694429	+	0.719562i	$0.744343\pi$
$632$	−61.1232	−2.43135
$633$	−8.85922	−0.352122
$634$	44.4282	1.76447
$635$	0	0
$636$	28.6504	1.13606
$637$	−7.45473	−0.295367
$638$	−14.6152	−0.578622
$639$	−7.27306	−0.287718
$640$	0	0
$641$	−38.5314	−1.52190	−0.760949	−	0.648812i	$-0.775266\pi$
$641$	−38.5314	−1.52190	−0.760949	+	0.648812i	$0.775266\pi$
$642$	12.7289	0.502368
$643$	18.6353	0.734906	0.367453	−	0.930042i	$-0.380230\pi$
$643$	18.6353	0.734906	0.367453	+	0.930042i	$0.380230\pi$
$644$	−21.3720	−0.842173
$645$	0	0
$646$	6.04892	0.237991
$647$	−19.2282	−0.755938	−0.377969	−	0.925818i	$-0.623377\pi$
$647$	−19.2282	−0.755938	−0.377969	+	0.925818i	$0.623377\pi$
$648$	−68.2247	−2.68012
$649$	16.7573	0.657783
$650$	0	0
$651$	15.8116	0.619706
$652$	−18.5386	−0.726026
$653$	3.98553	0.155966	0.0779828	−	0.996955i	$-0.475152\pi$
$653$	3.98553	0.155966	0.0779828	+	0.996955i	$0.475152\pi$
$654$	−17.6136	−0.688745
$655$	0	0
$656$	158.809	6.20046
$657$	37.0901	1.44702
$658$	20.1347	0.784931
$659$	33.5719	1.30778	0.653889	−	0.756591i	$-0.273136\pi$
$659$	33.5719	1.30778	0.653889	+	0.756591i	$0.273136\pi$
$660$	0	0
$661$	36.9909	1.43878	0.719390	−	0.694607i	$-0.244422\pi$
$661$	36.9909	1.43878	0.719390	+	0.694607i	$0.244422\pi$
$662$	37.0834	1.44129
$663$	3.89008	0.151078
$664$	−32.4741	−1.26024
$665$	0	0
$666$	8.74094	0.338704
$667$	2.12631	0.0823310
$668$	−27.7700	−1.07445
$669$	−12.8834	−0.498100
$670$	0	0
$671$	29.2747	1.13014
$672$	51.3400	1.98048
$673$	−12.0747	−0.465447	−0.232723	−	0.972543i	$-0.574764\pi$
$673$	−12.0747	−0.465447	−0.232723	+	0.972543i	$0.574764\pi$
$674$	−73.0689	−2.81451
$675$	0	0
$676$	−14.3763	−0.552934
$677$	−41.9135	−1.61087	−0.805434	−	0.592686i	$-0.798067\pi$
$677$	−41.9135	−1.61087	−0.805434	+	0.592686i	$0.798067\pi$
$678$	−5.34183	−0.205152
$679$	−9.89977	−0.379918
$680$	0	0
$681$	−9.70038	−0.371719
$682$	−76.9536	−2.94671
$683$	−9.17092	−0.350915	−0.175458	−	0.984487i	$-0.556141\pi$
$683$	−9.17092	−0.350915	−0.175458	+	0.984487i	$0.556141\pi$
$684$	15.7506	0.602240
$685$	0	0
$686$	40.1866	1.53433
$687$	1.29052	0.0492364
$688$	−97.7338	−3.72606
$689$	28.6504	1.09149
$690$	0	0
$691$	−0.111244	−0.00423193	−0.00211596	−	0.999998i	$-0.500674\pi$
$691$	−0.111244	−0.00423193	−0.00211596	+	0.999998i	$0.500674\pi$
$692$	−23.1879	−0.881472
$693$	24.1226	0.916341
$694$	25.1685	0.955384
$695$	0	0
$696$	−10.6273	−0.402827
$697$	18.5013	0.700785
$698$	−1.67994	−0.0635867
$699$	−10.5077	−0.397438
$700$	0	0
$701$	−27.7832	−1.04936	−0.524678	−	0.851301i	$-0.675814\pi$
$701$	−27.7832	−1.04936	−0.524678	+	0.851301i	$0.675814\pi$
$702$	28.7385	1.08467
$703$	−1.15883	−0.0437062
$704$	−140.943	−5.31198
$705$	0	0
$706$	89.3527	3.36283
$707$	15.5265	0.583933
$708$	18.5133	0.695774
$709$	37.4668	1.40710	0.703548	−	0.710648i	$-0.251598\pi$
$709$	37.4668	1.40710	0.703548	+	0.710648i	$0.251598\pi$
$710$	0	0
$711$	−15.2500	−0.571918
$712$	110.961	4.15844
$713$	11.1957	0.419281
$714$	10.2349	0.383031
$715$	0	0
$716$	−82.4137	−3.07994
$717$	−6.19806	−0.231471
$718$	51.4892	1.92156
$719$	−21.0513	−0.785081	−0.392541	−	0.919735i	$-0.628404\pi$
$719$	−21.0513	−0.785081	−0.392541	+	0.919735i	$0.628404\pi$
$720$	0	0
$721$	43.6329	1.62498
$722$	−2.80194	−0.104277
$723$	−4.12333	−0.153348
$724$	−71.6805	−2.66399
$725$	0	0
$726$	−3.67324	−0.136327
$727$	44.7023	1.65792	0.828958	−	0.559310i	$-0.188934\pi$
$727$	44.7023	1.65792	0.828958	+	0.559310i	$0.188934\pi$
$728$	106.817	3.95891
$729$	−11.7627	−0.435656
$730$	0	0
$731$	−11.3860	−0.421125
$732$	32.3424	1.19541
$733$	−25.0301	−0.924509	−0.462254	−	0.886747i	$-0.652959\pi$
$733$	−25.0301	−0.924509	−0.462254	+	0.886747i	$0.652959\pi$
$734$	−41.0140	−1.51385
$735$	0	0
$736$	36.3521	1.33996
$737$	14.6437	0.539407
$738$	64.6426	2.37953
$739$	−23.9982	−0.882788	−0.441394	−	0.897313i	$-0.645516\pi$
$739$	−23.9982	−0.882788	−0.441394	+	0.897313i	$0.645516\pi$
$740$	0	0
$741$	−1.80194	−0.0661958
$742$	75.3798	2.76728
$743$	−10.4601	−0.383744	−0.191872	−	0.981420i	$-0.561456\pi$
$743$	−10.4601	−0.383744	−0.191872	+	0.981420i	$0.561456\pi$
$744$	−55.9560	−2.05145
$745$	0	0
$746$	−2.87800	−0.105371
$747$	−8.10215	−0.296442
$748$	−37.1226	−1.35734
$749$	24.9584	0.911959
$750$	0	0
$751$	−29.9420	−1.09260	−0.546299	−	0.837590i	$-0.683964\pi$
$751$	−29.9420	−1.09260	−0.546299	+	0.837590i	$0.683964\pi$
$752$	−43.6752	−1.59267
$753$	8.18060	0.298118
$754$	−16.1468	−0.588030
$755$	0	0
$756$	56.3497	2.04942
$757$	−27.2174	−0.989235	−0.494617	−	0.869111i	$-0.664692\pi$
$757$	−27.2174	−0.989235	−0.494617	+	0.869111i	$0.664692\pi$
$758$	63.1232	2.29274
$759$	−1.95407	−0.0709281
$760$	0	0
$761$	18.6813	0.677195	0.338598	−	0.940931i	$-0.390047\pi$
$761$	18.6813	0.677195	0.338598	+	0.940931i	$0.390047\pi$
$762$	−2.27247	−0.0823229
$763$	−34.5362	−1.25029
$764$	117.876	4.26459
$765$	0	0
$766$	91.5387	3.30743
$767$	18.5133	0.668478
$768$	−61.3497	−2.21377
$769$	6.34780	0.228907	0.114454	−	0.993429i	$-0.463488\pi$
$769$	6.34780	0.228907	0.114454	+	0.993429i	$0.463488\pi$
$770$	0	0
$771$	1.85756	0.0668984
$772$	55.7458	2.00634
$773$	−49.7851	−1.79064	−0.895322	−	0.445419i	$-0.853055\pi$
$773$	−49.7851	−1.79064	−0.895322	+	0.445419i	$0.853055\pi$
$774$	−39.7821	−1.42994
$775$	0	0
$776$	35.0344	1.25766
$777$	−1.96077	−0.0703423
$778$	22.8756	0.820130
$779$	−8.57002	−0.307053
$780$	0	0
$781$	−7.94033	−0.284127
$782$	7.24698	0.259151
$783$	−5.60627	−0.200352
$784$	42.5448	1.51946
$785$	0	0
$786$	−20.5714	−0.733756
$787$	9.19865	0.327897	0.163948	−	0.986469i	$-0.447577\pi$
$787$	9.19865	0.327897	0.163948	+	0.986469i	$0.447577\pi$
$788$	−29.0388	−1.03446
$789$	−9.25965	−0.329652
$790$	0	0
$791$	−10.4741	−0.372416
$792$	−85.3678	−3.03341
$793$	32.3424	1.14851
$794$	105.207	3.73364
$795$	0	0
$796$	31.7245	1.12445
$797$	−13.5084	−0.478493	−0.239247	−	0.970959i	$-0.576900\pi$
$797$	−13.5084	−0.478493	−0.239247	+	0.970959i	$0.576900\pi$
$798$	−4.74094	−0.167827
$799$	−5.08815	−0.180006
$800$	0	0
$801$	27.6843	0.978175
$802$	−90.3678	−3.19100
$803$	40.4929	1.42896
$804$	16.1782	0.570562
$805$	0	0
$806$	−85.0176	−2.99462
$807$	−8.65519	−0.304677
$808$	−54.9469	−1.93302
$809$	31.9560	1.12351	0.561756	−	0.827303i	$-0.310126\pi$
$809$	31.9560	1.12351	0.561756	+	0.827303i	$0.310126\pi$
$810$	0	0
$811$	40.7012	1.42921	0.714607	−	0.699526i	$-0.246606\pi$
$811$	40.7012	1.42921	0.714607	+	0.699526i	$0.246606\pi$
$812$	−31.6601	−1.11105
$813$	7.33273	0.257170
$814$	9.54288	0.334478
$815$	0	0
$816$	−22.2010	−0.777192
$817$	5.27413	0.184518
$818$	88.3895	3.09047
$819$	26.6504	0.931240
$820$	0	0
$821$	−13.4300	−0.468709	−0.234355	−	0.972151i	$-0.575298\pi$
$821$	−13.4300	−0.468709	−0.234355	+	0.972151i	$0.575298\pi$
$822$	−25.1497	−0.877198
$823$	26.7275	0.931663	0.465832	−	0.884873i	$-0.345755\pi$
$823$	26.7275	0.931663	0.465832	+	0.884873i	$0.345755\pi$
$824$	−154.413	−5.37924
$825$	0	0
$826$	48.7090	1.69480
$827$	−33.3435	−1.15947	−0.579733	−	0.814806i	$-0.696843\pi$
$827$	−33.3435	−1.15947	−0.579733	+	0.814806i	$0.696843\pi$
$828$	18.8702	0.655786
$829$	12.2849	0.426672	0.213336	−	0.976979i	$-0.431567\pi$
$829$	12.2849	0.426672	0.213336	+	0.976979i	$0.431567\pi$
$830$	0	0
$831$	7.20105	0.249802
$832$	−155.712	−5.39835
$833$	4.95646	0.171731
$834$	20.3666	0.705237
$835$	0	0
$836$	17.1957	0.594725
$837$	−29.5187	−1.02032
$838$	41.1511	1.42154
$839$	36.9922	1.27711	0.638557	−	0.769575i	$-0.279532\pi$
$839$	36.9922	1.27711	0.638557	+	0.769575i	$0.279532\pi$
$840$	0	0
$841$	−25.8501	−0.891383
$842$	−42.3491	−1.45945
$843$	−13.8130	−0.475743
$844$	93.4016	3.21502
$845$	0	0
$846$	−17.7778	−0.611212
$847$	−7.20237	−0.247477
$848$	−163.510	−5.61497
$849$	−7.53750	−0.258686
$850$	0	0
$851$	−1.38835	−0.0475922
$852$	−8.77240	−0.300537
$853$	−7.16959	−0.245482	−0.122741	−	0.992439i	$-0.539168\pi$
$853$	−7.16959	−0.245482	−0.122741	+	0.992439i	$0.539168\pi$
$854$	85.0936	2.91184
$855$	0	0
$856$	−88.3256	−3.01891
$857$	−2.55124	−0.0871486	−0.0435743	−	0.999050i	$-0.513875\pi$
$857$	−2.55124	−0.0871486	−0.0435743	+	0.999050i	$0.513875\pi$
$858$	14.8388	0.506587
$859$	53.9493	1.84073	0.920363	−	0.391065i	$-0.127893\pi$
$859$	53.9493	1.84073	0.920363	+	0.391065i	$0.127893\pi$
$860$	0	0
$861$	−14.5007	−0.494181
$862$	−4.69873	−0.160039
$863$	−21.4698	−0.730840	−0.365420	−	0.930843i	$-0.619075\pi$
$863$	−21.4698	−0.730840	−0.365420	+	0.930843i	$0.619075\pi$
$864$	−95.8467	−3.26077
$865$	0	0
$866$	−36.8582	−1.25249
$867$	6.84787	0.232566
$868$	−166.700	−5.65817
$869$	−16.6491	−0.564781
$870$	0	0
$871$	16.1782	0.548178
$872$	122.221	4.13891
$873$	8.74094	0.295836
$874$	−3.35690	−0.113549
$875$	0	0
$876$	44.7362	1.51149
$877$	38.9638	1.31571	0.657856	−	0.753144i	$-0.271463\pi$
$877$	38.9638	1.31571	0.657856	+	0.753144i	$0.271463\pi$
$878$	−18.0887	−0.610465
$879$	0.708415	0.0238942
$880$	0	0
$881$	−13.4373	−0.452713	−0.226357	−	0.974045i	$-0.572681\pi$
$881$	−13.4373	−0.452713	−0.226357	+	0.974045i	$0.572681\pi$
$882$	17.3177	0.583116
$883$	27.8799	0.938234	0.469117	−	0.883136i	$-0.344572\pi$
$883$	27.8799	0.938234	0.469117	+	0.883136i	$0.344572\pi$
$884$	−41.0127	−1.37941
$885$	0	0
$886$	−71.0877	−2.38824
$887$	28.9135	0.970821	0.485410	−	0.874286i	$-0.338670\pi$
$887$	28.9135	0.970821	0.485410	+	0.874286i	$0.338670\pi$
$888$	6.93900	0.232858
$889$	−4.45580	−0.149443
$890$	0	0
$891$	−18.5834	−0.622568
$892$	135.828	4.54785
$893$	2.35690	0.0788705
$894$	17.2040	0.575389
$895$	0	0
$896$	−224.659	−7.50533
$897$	−2.15883	−0.0720814
$898$	10.0858	0.336566
$899$	16.5851	0.553144
$900$	0	0
$901$	−19.0489	−0.634611
$902$	70.5733	2.34983
$903$	8.92394	0.296970
$904$	37.0670	1.23283
$905$	0	0
$906$	16.2379	0.539467
$907$	−30.7127	−1.01980	−0.509900	−	0.860234i	$-0.670317\pi$
$907$	−30.7127	−1.01980	−0.509900	+	0.860234i	$0.670317\pi$
$908$	102.270	3.39395
$909$	−13.7090	−0.454699
$910$	0	0
$911$	39.3690	1.30435	0.652176	−	0.758067i	$-0.273856\pi$
$911$	39.3690	1.30435	0.652176	+	0.758067i	$0.273856\pi$
$912$	10.2838	0.340531
$913$	−8.84548	−0.292743
$914$	−34.4034	−1.13796
$915$	0	0
$916$	−13.6058	−0.449548
$917$	−40.3357	−1.33200
$918$	−19.1075	−0.630642
$919$	−7.87023	−0.259615	−0.129808	−	0.991539i	$-0.541436\pi$
$919$	−7.87023	−0.259615	−0.129808	+	0.991539i	$0.541436\pi$
$920$	0	0
$921$	−17.8860	−0.589365
$922$	−110.658	−3.64434
$923$	−8.77240	−0.288747
$924$	29.0954	0.957170
$925$	0	0
$926$	51.5532	1.69414
$927$	−38.5254	−1.26534
$928$	53.8514	1.76776
$929$	39.2924	1.28914	0.644572	−	0.764544i	$-0.277036\pi$
$929$	39.2924	1.28914	0.644572	+	0.764544i	$0.277036\pi$
$930$	0	0
$931$	−2.29590	−0.0752450
$932$	110.781	3.62876
$933$	−6.93602	−0.227075
$934$	−101.509	−3.32148
$935$	0	0
$936$	−94.3135	−3.08273
$937$	26.7554	0.874061	0.437031	−	0.899447i	$-0.356030\pi$
$937$	26.7554	0.874061	0.437031	+	0.899447i	$0.356030\pi$
$938$	42.5652	1.38980
$939$	−11.6203	−0.379213
$940$	0	0
$941$	−41.9191	−1.36653	−0.683263	−	0.730173i	$-0.739440\pi$
$941$	−41.9191	−1.36653	−0.683263	+	0.730173i	$0.739440\pi$
$942$	−3.86294	−0.125861
$943$	−10.2674	−0.334353
$944$	−105.657	−3.43885
$945$	0	0
$946$	−43.4319	−1.41209
$947$	−55.4868	−1.80308	−0.901539	−	0.432698i	$-0.857562\pi$
$947$	−55.4868	−1.80308	−0.901539	+	0.432698i	$0.857562\pi$
$948$	−18.3937	−0.597401
$949$	44.7362	1.45220
$950$	0	0
$951$	8.79954	0.285345
$952$	−71.0200	−2.30177
$953$	−8.48081	−0.274720	−0.137360	−	0.990521i	$-0.543862\pi$
$953$	−8.48081	−0.274720	−0.137360	+	0.990521i	$0.543862\pi$
$954$	−66.5561	−2.15483
$955$	0	0
$956$	65.3454	2.11342
$957$	−2.89472	−0.0935731
$958$	96.1426	3.10623
$959$	−49.3129	−1.59240
$960$	0	0
$961$	56.3256	1.81695
$962$	10.5429	0.339916
$963$	−22.0368	−0.710127
$964$	43.4717	1.40013
$965$	0	0
$966$	−5.67994	−0.182749
$967$	7.63102	0.245397	0.122699	−	0.992444i	$-0.460845\pi$
$967$	7.63102	0.245397	0.122699	+	0.992444i	$0.460845\pi$
$968$	25.4886	0.819234
$969$	1.19806	0.0384873
$970$	0	0
$971$	−19.3599	−0.621288	−0.310644	−	0.950526i	$-0.600545\pi$
$971$	−19.3599	−0.621288	−0.310644	+	0.950526i	$0.600545\pi$
$972$	−75.9764	−2.43695
$973$	39.9342	1.28023
$974$	−68.5695	−2.19711
$975$	0	0
$976$	−184.581	−5.90829
$977$	−55.8883	−1.78802	−0.894012	−	0.448042i	$-0.852121\pi$
$977$	−55.8883	−1.78802	−0.894012	+	0.448042i	$0.852121\pi$
$978$	−4.92692	−0.157546
$979$	30.2241	0.965968
$980$	0	0
$981$	30.4935	0.973582
$982$	−51.2941	−1.63686
$983$	4.58450	0.146223	0.0731114	−	0.997324i	$-0.476707\pi$
$983$	4.58450	0.146223	0.0731114	+	0.997324i	$0.476707\pi$
$984$	51.3166	1.63591
$985$	0	0
$986$	10.7356	0.341890
$987$	3.98792	0.126937
$988$	18.9976	0.604394
$989$	6.31873	0.200924
$990$	0	0
$991$	27.5666	0.875681	0.437840	−	0.899053i	$-0.355743\pi$
$991$	27.5666	0.875681	0.437840	+	0.899053i	$0.355743\pi$
$992$	283.544	9.00254
$993$	7.34481	0.233081
$994$	−23.0804	−0.732065
$995$	0	0
$996$	−9.77240	−0.309650
$997$	−15.9119	−0.503933	−0.251967	−	0.967736i	$-0.581077\pi$
$997$	−15.9119	−0.503933	−0.251967	+	0.967736i	$0.581077\pi$
$998$	71.0079	2.24772
$999$	3.66056	0.115815

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.d.1.1	✓	3
3.2	odd	2		4275.2.a.bn.1.3		3
4.3	odd	2		7600.2.a.bw.1.2		3
5.2	odd	4		475.2.b.c.324.1		6
5.3	odd	4		475.2.b.c.324.6		6
5.4	even	2		475.2.a.h.1.3	yes	3
15.14	odd	2		4275.2.a.z.1.1		3
19.18	odd	2		9025.2.a.be.1.3		3
20.19	odd	2		7600.2.a.bn.1.2		3
95.94	odd	2		9025.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.1	✓	3	1.1	even	1	trivial
475.2.a.h.1.3	yes	3	5.4	even	2
475.2.b.c.324.1		6	5.2	odd	4
475.2.b.c.324.6		6	5.3	odd	4
4275.2.a.z.1.1		3	15.14	odd	2
4275.2.a.bn.1.3		3	3.2	odd	2
7600.2.a.bn.1.2		3	20.19	odd	2
7600.2.a.bw.1.2		3	4.3	odd	2
9025.2.a.w.1.1		3	95.94	odd	2
9025.2.a.be.1.3		3	19.18	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)

\(f(q)\)	\(=\)	\(q-2.80194 q^{2} -0.554958 q^{3} +5.85086 q^{4} +1.55496 q^{6} +3.04892 q^{7} -10.7899 q^{8} -2.69202 q^{9} -2.93900 q^{11} -3.24698 q^{12} -3.24698 q^{13} -8.54288 q^{14} +18.5308 q^{16} +2.15883 q^{17} +7.54288 q^{18} -1.00000 q^{19} -1.69202 q^{21} +8.23490 q^{22} -1.19806 q^{23} +5.98792 q^{24} +9.09783 q^{26} +3.15883 q^{27} +17.8388 q^{28} -1.77479 q^{29} -9.34481 q^{31} -30.3424 q^{32} +1.63102 q^{33} -6.04892 q^{34} -15.7506 q^{36} +1.15883 q^{37} +2.80194 q^{38} +1.80194 q^{39} +8.57002 q^{41} +4.74094 q^{42} -5.27413 q^{43} -17.1957 q^{44} +3.35690 q^{46} -2.35690 q^{47} -10.2838 q^{48} +2.29590 q^{49} -1.19806 q^{51} -18.9976 q^{52} -8.82371 q^{53} -8.85086 q^{54} -32.8974 q^{56} +0.554958 q^{57} +4.97285 q^{58} -5.70171 q^{59} -9.96077 q^{61} +26.1836 q^{62} -8.20775 q^{63} +47.9560 q^{64} -4.57002 q^{66} -4.98254 q^{67} +12.6310 q^{68} +0.664874 q^{69} +2.70171 q^{71} +29.0465 q^{72} -13.7778 q^{73} -3.24698 q^{74} -5.85086 q^{76} -8.96077 q^{77} -5.04892 q^{78} +5.66487 q^{79} +6.32304 q^{81} -24.0127 q^{82} +3.00969 q^{83} -9.89977 q^{84} +14.7778 q^{86} +0.984935 q^{87} +31.7114 q^{88} -10.2838 q^{89} -9.89977 q^{91} -7.00969 q^{92} +5.18598 q^{93} +6.60388 q^{94} +16.8388 q^{96} -3.24698 q^{97} -6.43296 q^{98} +7.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27}+ \cdots + 20 q^{99}+O(q^{100}) \) 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 + q^11 - 5 * q^12 - 5 * q^13 - 7 * q^14 + 18 * q^16 - 2 * q^17 + 4 * q^18 - 3 * q^19 + q^22 - 8 * q^23 - q^24 + 9 * q^26 + q^27 + 21 * q^28 - 7 * q^29 - 5 * q^31 - 27 * q^32 - 10 * q^33 - 9 * q^34 - 11 * q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^41 - 5 * q^43 - 15 * q^44 + 6 * q^46 - 3 * q^47 + 2 * q^48 - 7 * q^49 - 8 * q^51 - 16 * q^52 - 19 * q^53 - 13 * q^54 - 35 * q^56 + 2 * q^57 + 21 * q^58 + 10 * q^59 - 17 * q^61 + 23 * q^62 - 7 * q^63 + 49 * q^64 + 11 * q^66 + q^67 + 23 * q^68 + 3 * q^69 - 19 * q^71 + 37 * q^72 + q^73 - 5 * q^74 - 4 * q^76 - 14 * q^77 - 6 * q^78 + 18 * q^79 - q^81 - 6 * q^82 - 13 * q^83 - 7 * q^84 + 2 * q^86 + 28 * q^87 + 46 * q^88 + 2 * q^89 - 7 * q^91 + q^92 + q^93 + 11 * q^94 + 18 * q^96 - 5 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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