LMFDB - Embedded newform 475.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$3.79289409601$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ 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)

$f(q)$	$=$	\(q-2.17009 q^{2} +1.70928 q^{3} +2.70928 q^{4} -3.70928 q^{6} -1.07838 q^{7} -1.53919 q^{8} -0.0783777 q^{9} -6.34017 q^{11} +4.63090 q^{12} -1.36910 q^{13} +2.34017 q^{14} -2.07838 q^{16} -3.26180 q^{17} +0.170086 q^{18} -1.00000 q^{19} -1.84324 q^{21} +13.7587 q^{22} -2.34017 q^{23} -2.63090 q^{24} +2.97107 q^{26} -5.26180 q^{27} -2.92162 q^{28} +1.41855 q^{29} +8.68035 q^{31} +7.58864 q^{32} -10.8371 q^{33} +7.07838 q^{34} -0.212347 q^{36} -5.36910 q^{37} +2.17009 q^{38} -2.34017 q^{39} -3.26180 q^{41} +4.00000 q^{42} +11.9155 q^{43} -17.1773 q^{44} +5.07838 q^{46} -1.07838 q^{47} -3.55252 q^{48} -5.83710 q^{49} -5.57531 q^{51} -3.70928 q^{52} -6.63090 q^{53} +11.4186 q^{54} +1.65983 q^{56} -1.70928 q^{57} -3.07838 q^{58} -11.4186 q^{59} +5.60197 q^{61} -18.8371 q^{62} +0.0845208 q^{63} -12.3112 q^{64} +23.5174 q^{66} -10.3896 q^{67} -8.83710 q^{68} -4.00000 q^{69} -10.8371 q^{71} +0.120638 q^{72} -5.41855 q^{73} +11.6514 q^{74} -2.70928 q^{76} +6.83710 q^{77} +5.07838 q^{78} +14.2557 q^{79} -8.75872 q^{81} +7.07838 q^{82} +14.3402 q^{83} -4.99386 q^{84} -25.8576 q^{86} +2.42469 q^{87} +9.75872 q^{88} +7.57531 q^{89} +1.47641 q^{91} -6.34017 q^{92} +14.8371 q^{93} +2.34017 q^{94} +12.9711 q^{96} +8.88655 q^{97} +12.6670 q^{98} +0.496928 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9} - 8 q^{11} + 10 q^{12} - 8 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} - 5 q^{18} - 3 q^{19} - 12 q^{21} + 16 q^{22} + 4 q^{23} - 4 q^{24} - 6 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) $ 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 3 * q^8 + 3 * q^9 - 8 * q^11 + 10 * q^12 - 8 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 - 5 * q^18 - 3 * q^19 - 12 * q^21 + 16 * q^22 + 4 * q^23 - 4 * q^24 - 6 * q^26 - 8 * q^27 - 12 * q^28 - 10 * q^29 + 4 * q^31 + 3 * q^32 - 4 * q^33 + 18 * q^34 - 11 * q^36 - 20 * q^37 + q^38 + 4 * q^39 - 2 * q^41 + 12 * q^42 + 4 * q^43 - 12 * q^44 + 12 * q^46 - 10 * q^48 + 11 * q^49 + 4 * q^51 - 4 * q^52 - 16 * q^53 + 20 * q^54 + 16 * q^56 + 2 * q^57 - 6 * q^58 - 20 * q^59 - 2 * q^61 - 28 * q^62 + 32 * q^63 - 11 * q^64 + 20 * q^66 - 2 * q^67 + 2 * q^68 - 12 * q^69 - 4 * q^71 + 13 * q^72 - 2 * q^73 - 2 * q^74 - q^76 - 8 * q^77 + 12 * q^78 - q^81 + 18 * q^82 + 32 * q^83 + 20 * q^84 - 16 * q^86 + 28 * q^87 + 4 * q^88 + 2 * q^89 + 20 * q^91 - 8 * q^92 + 16 * q^93 - 4 * q^94 + 24 * q^96 - 20 * q^97 + 15 * q^98 - 16 * 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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	2.70928	1.35464
$5$	0	0
$5$	0	0
$6$	−3.70928	−1.51431
$7$	−1.07838	−0.407588	−0.203794	−	0.979014i	$-0.565327\pi$
$7$	−1.07838	−0.407588	−0.203794	+	0.979014i	$0.565327\pi$
$8$	−1.53919	−0.544185
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	−6.34017	−1.91163	−0.955817	−	0.293962i	$-0.905026\pi$
$11$	−6.34017	−1.91163	−0.955817	+	0.293962i	$0.905026\pi$
$12$	4.63090	1.33682
$13$	−1.36910	−0.379721	−0.189860	−	0.981811i	$-0.560804\pi$
$13$	−1.36910	−0.379721	−0.189860	+	0.981811i	$0.560804\pi$
$14$	2.34017	0.625438
$15$	0	0
$16$	−2.07838	−0.519594
$17$	−3.26180	−0.791102	−0.395551	−	0.918444i	$-0.629446\pi$
$17$	−3.26180	−0.791102	−0.395551	+	0.918444i	$0.629446\pi$
$18$	0.170086	0.0400898
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.84324	−0.402229
$22$	13.7587	2.93337
$23$	−2.34017	−0.487960	−0.243980	−	0.969780i	$-0.578453\pi$
$23$	−2.34017	−0.487960	−0.243980	+	0.969780i	$0.578453\pi$
$24$	−2.63090	−0.537030
$25$	0	0
$26$	2.97107	0.582675
$27$	−5.26180	−1.01263
$28$	−2.92162	−0.552135
$29$	1.41855	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$29$	1.41855	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$30$	0	0
$31$	8.68035	1.55904	0.779518	−	0.626380i	$-0.215464\pi$
$31$	8.68035	1.55904	0.779518	+	0.626380i	$0.215464\pi$
$32$	7.58864	1.34149
$33$	−10.8371	−1.88650
$34$	7.07838	1.21393
$35$	0	0
$36$	−0.212347	−0.0353911
$37$	−5.36910	−0.882675	−0.441337	−	0.897341i	$-0.645496\pi$
$37$	−5.36910	−0.882675	−0.441337	+	0.897341i	$0.645496\pi$
$38$	2.17009	0.352035
$39$	−2.34017	−0.374728
$40$	0	0
$41$	−3.26180	−0.509407	−0.254703	−	0.967019i	$-0.581978\pi$
$41$	−3.26180	−0.509407	−0.254703	+	0.967019i	$0.581978\pi$
$42$	4.00000	0.617213
$43$	11.9155	1.81709	0.908547	−	0.417783i	$-0.137193\pi$
$43$	11.9155	1.81709	0.908547	+	0.417783i	$0.137193\pi$
$44$	−17.1773	−2.58957
$45$	0	0
$46$	5.07838	0.748766
$47$	−1.07838	−0.157298	−0.0786488	−	0.996902i	$-0.525061\pi$
$47$	−1.07838	−0.157298	−0.0786488	+	0.996902i	$0.525061\pi$
$48$	−3.55252	−0.512762
$49$	−5.83710	−0.833872
$50$	0	0
$51$	−5.57531	−0.780699
$52$	−3.70928	−0.514384
$53$	−6.63090	−0.910824	−0.455412	−	0.890281i	$-0.650508\pi$
$53$	−6.63090	−0.910824	−0.455412	+	0.890281i	$0.650508\pi$
$54$	11.4186	1.55387
$55$	0	0
$56$	1.65983	0.221804
$57$	−1.70928	−0.226399
$58$	−3.07838	−0.404211
$59$	−11.4186	−1.48657	−0.743284	−	0.668976i	$-0.766733\pi$
$59$	−11.4186	−1.48657	−0.743284	+	0.668976i	$0.766733\pi$
$60$	0	0
$61$	5.60197	0.717259	0.358629	−	0.933480i	$-0.383244\pi$
$61$	5.60197	0.717259	0.358629	+	0.933480i	$0.383244\pi$
$62$	−18.8371	−2.39231
$63$	0.0845208	0.0106486
$64$	−12.3112	−1.53891
$65$	0	0
$66$	23.5174	2.89480
$67$	−10.3896	−1.26929	−0.634647	−	0.772802i	$-0.718855\pi$
$67$	−10.3896	−1.26929	−0.634647	+	0.772802i	$0.718855\pi$
$68$	−8.83710	−1.07166
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−10.8371	−1.28613	−0.643064	−	0.765813i	$-0.722337\pi$
$71$	−10.8371	−1.28613	−0.643064	+	0.765813i	$0.722337\pi$
$72$	0.120638	0.0142173
$73$	−5.41855	−0.634193	−0.317097	−	0.948393i	$-0.602708\pi$
$73$	−5.41855	−0.634193	−0.317097	+	0.948393i	$0.602708\pi$
$74$	11.6514	1.35445
$75$	0	0
$76$	−2.70928	−0.310775
$77$	6.83710	0.779160
$78$	5.07838	0.575013
$79$	14.2557	1.60389	0.801943	−	0.597400i	$-0.203800\pi$
$79$	14.2557	1.60389	0.801943	+	0.597400i	$0.203800\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	7.07838	0.781676
$83$	14.3402	1.57404	0.787019	−	0.616928i	$-0.211623\pi$
$83$	14.3402	1.57404	0.787019	+	0.616928i	$0.211623\pi$
$84$	−4.99386	−0.544874
$85$	0	0
$86$	−25.8576	−2.78830
$87$	2.42469	0.259954
$88$	9.75872	1.04028
$89$	7.57531	0.802981	0.401490	−	0.915863i	$-0.368492\pi$
$89$	7.57531	0.802981	0.401490	+	0.915863i	$0.368492\pi$
$90$	0	0
$91$	1.47641	0.154770
$92$	−6.34017	−0.661009
$93$	14.8371	1.53854
$94$	2.34017	0.241370
$95$	0	0
$96$	12.9711	1.32385
$97$	8.88655	0.902292	0.451146	−	0.892450i	$-0.351015\pi$
$97$	8.88655	0.902292	0.451146	+	0.892450i	$0.351015\pi$
$98$	12.6670	1.27956
$99$	0.496928	0.0499432
$100$	0	0
$101$	−4.92162	−0.489720	−0.244860	−	0.969558i	$-0.578742\pi$
$101$	−4.92162	−0.489720	−0.244860	+	0.969558i	$0.578742\pi$
$102$	12.0989	1.19797
$103$	−6.38962	−0.629588	−0.314794	−	0.949160i	$-0.601935\pi$
$103$	−6.38962	−0.629588	−0.314794	+	0.949160i	$0.601935\pi$
$104$	2.10731	0.206638
$105$	0	0
$106$	14.3896	1.39764
$107$	−2.29072	−0.221453	−0.110726	−	0.993851i	$-0.535318\pi$
$107$	−2.29072	−0.221453	−0.110726	+	0.993851i	$0.535318\pi$
$108$	−14.2557	−1.37175
$109$	−12.8371	−1.22957	−0.614786	−	0.788694i	$-0.710757\pi$
$109$	−12.8371	−1.22957	−0.614786	+	0.788694i	$0.710757\pi$
$110$	0	0
$111$	−9.17727	−0.871068
$112$	2.24128	0.211781
$113$	12.8865	1.21226	0.606132	−	0.795364i	$-0.292720\pi$
$113$	12.8865	1.21226	0.606132	+	0.795364i	$0.292720\pi$
$114$	3.70928	0.347405
$115$	0	0
$116$	3.84324	0.356836
$117$	0.107307	0.00992055
$118$	24.7792	2.28111
$119$	3.51745	0.322444
$120$	0	0
$121$	29.1978	2.65434
$122$	−12.1568	−1.10062
$123$	−5.57531	−0.502708
$124$	23.5174	2.11193
$125$	0	0
$126$	−0.183417	−0.0163401
$127$	8.23287	0.730549	0.365274	−	0.930900i	$-0.380975\pi$
$127$	8.23287	0.730549	0.365274	+	0.930900i	$0.380975\pi$
$128$	11.5392	1.01993
$129$	20.3668	1.79320
$130$	0	0
$131$	1.47641	0.128995	0.0644973	−	0.997918i	$-0.479456\pi$
$131$	1.47641	0.128995	0.0644973	+	0.997918i	$0.479456\pi$
$132$	−29.3607	−2.55552
$133$	1.07838	0.0935072
$134$	22.5464	1.94771
$135$	0	0
$136$	5.02052	0.430506
$137$	3.94214	0.336800	0.168400	−	0.985719i	$-0.446140\pi$
$137$	3.94214	0.336800	0.168400	+	0.985719i	$0.446140\pi$
$138$	8.68035	0.738920
$139$	8.86376	0.751815	0.375907	−	0.926657i	$-0.377331\pi$
$139$	8.86376	0.751815	0.375907	+	0.926657i	$0.377331\pi$
$140$	0	0
$141$	−1.84324	−0.155229
$142$	23.5174	1.97354
$143$	8.68035	0.725887
$144$	0.162899	0.0135749
$145$	0	0
$146$	11.7587	0.973159
$147$	−9.97721	−0.822907
$148$	−14.5464	−1.19570
$149$	−19.7587	−1.61870	−0.809349	−	0.587328i	$-0.800180\pi$
$149$	−19.7587	−1.61870	−0.809349	+	0.587328i	$0.800180\pi$
$150$	0	0
$151$	3.41855	0.278198	0.139099	−	0.990279i	$-0.455579\pi$
$151$	3.41855	0.278198	0.139099	+	0.990279i	$0.455579\pi$
$152$	1.53919	0.124845
$153$	0.255652	0.0206683
$154$	−14.8371	−1.19561
$155$	0	0
$156$	−6.34017	−0.507620
$157$	−9.41855	−0.751682	−0.375841	−	0.926684i	$-0.622646\pi$
$157$	−9.41855	−0.751682	−0.375841	+	0.926684i	$0.622646\pi$
$158$	−30.9360	−2.46114
$159$	−11.3340	−0.898847
$160$	0	0
$161$	2.52359	0.198887
$162$	19.0072	1.49335
$163$	2.92162	0.228839	0.114420	−	0.993433i	$-0.463499\pi$
$163$	2.92162	0.228839	0.114420	+	0.993433i	$0.463499\pi$
$164$	−8.83710	−0.690062
$165$	0	0
$166$	−31.1194	−2.41534
$167$	−20.9132	−1.61831	−0.809156	−	0.587593i	$-0.800076\pi$
$167$	−20.9132	−1.61831	−0.809156	+	0.587593i	$0.800076\pi$
$168$	2.83710	0.218887
$169$	−11.1256	−0.855812
$170$	0	0
$171$	0.0783777	0.00599370
$172$	32.2823	2.46150
$173$	1.05559	0.0802551	0.0401276	−	0.999195i	$-0.487224\pi$
$173$	1.05559	0.0802551	0.0401276	+	0.999195i	$0.487224\pi$
$174$	−5.26180	−0.398896
$175$	0	0
$176$	13.1773	0.993274
$177$	−19.5174	−1.46702
$178$	−16.4391	−1.23216
$179$	0.894960	0.0668925	0.0334462	−	0.999441i	$-0.489352\pi$
$179$	0.894960	0.0668925	0.0334462	+	0.999441i	$0.489352\pi$
$180$	0	0
$181$	−0.837101	−0.0622213	−0.0311106	−	0.999516i	$-0.509904\pi$
$181$	−0.837101	−0.0622213	−0.0311106	+	0.999516i	$0.509904\pi$
$182$	−3.20394	−0.237492
$183$	9.57531	0.707827
$184$	3.60197	0.265541
$185$	0	0
$186$	−32.1978	−2.36086
$187$	20.6803	1.51230
$188$	−2.92162	−0.213081
$189$	5.67420	0.412738
$190$	0	0
$191$	22.0410	1.59483	0.797417	−	0.603429i	$-0.206199\pi$
$191$	22.0410	1.59483	0.797417	+	0.603429i	$0.206199\pi$
$192$	−21.0433	−1.51867
$193$	−12.7877	−0.920475	−0.460238	−	0.887796i	$-0.652236\pi$
$193$	−12.7877	−0.920475	−0.460238	+	0.887796i	$0.652236\pi$
$194$	−19.2846	−1.38455
$195$	0	0
$196$	−15.8143	−1.12959
$197$	9.20394	0.655753	0.327877	−	0.944721i	$-0.393667\pi$
$197$	9.20394	0.655753	0.327877	+	0.944721i	$0.393667\pi$
$198$	−1.07838	−0.0766370
$199$	−16.1978	−1.14823	−0.574116	−	0.818774i	$-0.694654\pi$
$199$	−16.1978	−1.14823	−0.574116	+	0.818774i	$0.694654\pi$
$200$	0	0
$201$	−17.7587	−1.25260
$202$	10.6803	0.751467
$203$	−1.52973	−0.107366
$204$	−15.1050	−1.05756
$205$	0	0
$206$	13.8660	0.966092
$207$	0.183417	0.0127484
$208$	2.84551	0.197301
$209$	6.34017	0.438559
$210$	0	0
$211$	7.78539	0.535968	0.267984	−	0.963423i	$-0.413643\pi$
$211$	7.78539	0.535968	0.267984	+	0.963423i	$0.413643\pi$
$212$	−17.9649	−1.23384
$213$	−18.5236	−1.26922
$214$	4.97107	0.339815
$215$	0	0
$216$	8.09890	0.551060
$217$	−9.36069	−0.635445
$218$	27.8576	1.88676
$219$	−9.26180	−0.625854
$220$	0	0
$221$	4.46573	0.300398
$222$	19.9155	1.33664
$223$	−12.5464	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$223$	−12.5464	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$224$	−8.18342	−0.546778
$225$	0	0
$226$	−27.9649	−1.86020
$227$	−2.29072	−0.152041	−0.0760204	−	0.997106i	$-0.524221\pi$
$227$	−2.29072	−0.152041	−0.0760204	+	0.997106i	$0.524221\pi$
$228$	−4.63090	−0.306689
$229$	−5.91548	−0.390906	−0.195453	−	0.980713i	$-0.562618\pi$
$229$	−5.91548	−0.390906	−0.195453	+	0.980713i	$0.562618\pi$
$230$	0	0
$231$	11.6865	0.768915
$232$	−2.18342	−0.143348
$233$	−13.5174	−0.885557	−0.442779	−	0.896631i	$-0.646007\pi$
$233$	−13.5174	−0.885557	−0.442779	+	0.896631i	$0.646007\pi$
$234$	−0.232866	−0.0152229
$235$	0	0
$236$	−30.9360	−2.01376
$237$	24.3668	1.58280
$238$	−7.63317	−0.494785
$239$	13.8432	0.895445	0.447723	−	0.894173i	$-0.352235\pi$
$239$	13.8432	0.895445	0.447723	+	0.894173i	$0.352235\pi$
$240$	0	0
$241$	7.26180	0.467773	0.233887	−	0.972264i	$-0.424856\pi$
$241$	7.26180	0.467773	0.233887	+	0.972264i	$0.424856\pi$
$242$	−63.3617	−4.07305
$243$	0.814315	0.0522383
$244$	15.1773	0.971625
$245$	0	0
$246$	12.0989	0.771397
$247$	1.36910	0.0871139
$248$	−13.3607	−0.848405
$249$	24.5113	1.55334
$250$	0	0
$251$	10.4703	0.660877	0.330439	−	0.943827i	$-0.392803\pi$
$251$	10.4703	0.660877	0.330439	+	0.943827i	$0.392803\pi$
$252$	0.228990	0.0144250
$253$	14.8371	0.932801
$254$	−17.8660	−1.12101
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	23.6248	1.47367	0.736836	−	0.676072i	$-0.236319\pi$
$257$	23.6248	1.47367	0.736836	+	0.676072i	$0.236319\pi$
$258$	−44.1978	−2.75163
$259$	5.78992	0.359768
$260$	0	0
$261$	−0.111183	−0.00688204
$262$	−3.20394	−0.197940
$263$	−5.65983	−0.349000	−0.174500	−	0.984657i	$-0.555831\pi$
$263$	−5.65983	−0.349000	−0.174500	+	0.984657i	$0.555831\pi$
$264$	16.6803	1.02660
$265$	0	0
$266$	−2.34017	−0.143485
$267$	12.9483	0.792422
$268$	−28.1483	−1.71943
$269$	−2.31351	−0.141057	−0.0705286	−	0.997510i	$-0.522469\pi$
$269$	−2.31351	−0.141057	−0.0705286	+	0.997510i	$0.522469\pi$
$270$	0	0
$271$	−19.7009	−1.19674	−0.598371	−	0.801219i	$-0.704185\pi$
$271$	−19.7009	−1.19674	−0.598371	+	0.801219i	$0.704185\pi$
$272$	6.77924	0.411052
$273$	2.52359	0.152735
$274$	−8.55479	−0.516814
$275$	0	0
$276$	−10.8371	−0.652317
$277$	25.7321	1.54609	0.773045	−	0.634351i	$-0.218733\pi$
$277$	25.7321	1.54609	0.773045	+	0.634351i	$0.218733\pi$
$278$	−19.2351	−1.15365
$279$	−0.680346	−0.0407312
$280$	0	0
$281$	−6.58145	−0.392616	−0.196308	−	0.980542i	$-0.562895\pi$
$281$	−6.58145	−0.392616	−0.196308	+	0.980542i	$0.562895\pi$
$282$	4.00000	0.238197
$283$	0.496928	0.0295393	0.0147697	−	0.999891i	$-0.495298\pi$
$283$	0.496928	0.0295393	0.0147697	+	0.999891i	$0.495298\pi$
$284$	−29.3607	−1.74224
$285$	0	0
$286$	−18.8371	−1.11386
$287$	3.51745	0.207628
$288$	−0.594780	−0.0350478
$289$	−6.36069	−0.374158
$290$	0	0
$291$	15.1896	0.890428
$292$	−14.6803	−0.859102
$293$	−6.63090	−0.387381	−0.193691	−	0.981063i	$-0.562046\pi$
$293$	−6.63090	−0.387381	−0.193691	+	0.981063i	$0.562046\pi$
$294$	21.6514	1.26274
$295$	0	0
$296$	8.26406	0.480339
$297$	33.3607	1.93578
$298$	42.8781	2.48386
$299$	3.20394	0.185288
$300$	0	0
$301$	−12.8494	−0.740626
$302$	−7.41855	−0.426890
$303$	−8.41241	−0.483280
$304$	2.07838	0.119203
$305$	0	0
$306$	−0.554787	−0.0317151
$307$	−14.6042	−0.833508	−0.416754	−	0.909019i	$-0.636832\pi$
$307$	−14.6042	−0.833508	−0.416754	+	0.909019i	$0.636832\pi$
$308$	18.5236	1.05548
$309$	−10.9216	−0.621309
$310$	0	0
$311$	19.3340	1.09633	0.548166	−	0.836369i	$-0.315326\pi$
$311$	19.3340	1.09633	0.548166	+	0.836369i	$0.315326\pi$
$312$	3.60197	0.203921
$313$	−30.6803	−1.73416	−0.867078	−	0.498173i	$-0.834005\pi$
$313$	−30.6803	−1.73416	−0.867078	+	0.498173i	$0.834005\pi$
$314$	20.4391	1.15344
$315$	0	0
$316$	38.6225	2.17268
$317$	−18.7298	−1.05197	−0.525985	−	0.850494i	$-0.676303\pi$
$317$	−18.7298	−1.05197	−0.525985	+	0.850494i	$0.676303\pi$
$318$	24.5958	1.37927
$319$	−8.99386	−0.503559
$320$	0	0
$321$	−3.91548	−0.218541
$322$	−5.47641	−0.305188
$323$	3.26180	0.181491
$324$	−23.7298	−1.31832
$325$	0	0
$326$	−6.34017	−0.351150
$327$	−21.9421	−1.21340
$328$	5.02052	0.277212
$329$	1.16290	0.0641127
$330$	0	0
$331$	−2.73820	−0.150505	−0.0752527	−	0.997164i	$-0.523976\pi$
$331$	−2.73820	−0.150505	−0.0752527	+	0.997164i	$0.523976\pi$
$332$	38.8515	2.13225
$333$	0.420818	0.0230607
$334$	45.3835	2.48327
$335$	0	0
$336$	3.83096	0.208996
$337$	6.04945	0.329534	0.164767	−	0.986332i	$-0.447313\pi$
$337$	6.04945	0.329534	0.164767	+	0.986332i	$0.447313\pi$
$338$	24.1434	1.31323
$339$	22.0267	1.19632
$340$	0	0
$341$	−55.0349	−2.98031
$342$	−0.170086	−0.00919722
$343$	13.8432	0.747465
$344$	−18.3402	−0.988836
$345$	0	0
$346$	−2.29072	−0.123150
$347$	5.97334	0.320666	0.160333	−	0.987063i	$-0.448743\pi$
$347$	5.97334	0.320666	0.160333	+	0.987063i	$0.448743\pi$
$348$	6.56916	0.352144
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	7.20394	0.384518
$352$	−48.1133	−2.56445
$353$	14.0989	0.750409	0.375204	−	0.926942i	$-0.377573\pi$
$353$	14.0989	0.750409	0.375204	+	0.926942i	$0.377573\pi$
$354$	42.3545	2.25112
$355$	0	0
$356$	20.5236	1.08775
$357$	6.01229	0.318204
$358$	−1.94214	−0.102645
$359$	6.02666	0.318075	0.159038	−	0.987273i	$-0.449161\pi$
$359$	6.02666	0.318075	0.159038	+	0.987273i	$0.449161\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.81658	0.0954775
$363$	49.9071	2.61944
$364$	4.00000	0.209657
$365$	0	0
$366$	−20.7792	−1.08615
$367$	−1.07838	−0.0562909	−0.0281454	−	0.999604i	$-0.508960\pi$
$367$	−1.07838	−0.0562909	−0.0281454	+	0.999604i	$0.508960\pi$
$368$	4.86376	0.253541
$369$	0.255652	0.0133087
$370$	0	0
$371$	7.15061	0.371241
$372$	40.1978	2.08416
$373$	−12.3051	−0.637134	−0.318567	−	0.947900i	$-0.603202\pi$
$373$	−12.3051	−0.637134	−0.318567	+	0.947900i	$0.603202\pi$
$374$	−44.8781	−2.32059
$375$	0	0
$376$	1.65983	0.0855990
$377$	−1.94214	−0.100025
$378$	−12.3135	−0.633339
$379$	1.04718	0.0537901	0.0268950	−	0.999638i	$-0.491438\pi$
$379$	1.04718	0.0537901	0.0268950	+	0.999638i	$0.491438\pi$
$380$	0	0
$381$	14.0722	0.720942
$382$	−47.8310	−2.44724
$383$	−0.0806452	−0.00412078	−0.00206039	−	0.999998i	$-0.500656\pi$
$383$	−0.0806452	−0.00412078	−0.00206039	+	0.999998i	$0.500656\pi$
$384$	19.7237	1.00652
$385$	0	0
$386$	27.7503	1.41245
$387$	−0.933908	−0.0474732
$388$	24.0761	1.22228
$389$	−20.5236	−1.04059	−0.520294	−	0.853987i	$-0.674178\pi$
$389$	−20.5236	−1.04059	−0.520294	+	0.853987i	$0.674178\pi$
$390$	0	0
$391$	7.63317	0.386026
$392$	8.98440	0.453781
$393$	2.52359	0.127298
$394$	−19.9733	−1.00624
$395$	0	0
$396$	1.34632	0.0676549
$397$	−39.4596	−1.98042	−0.990210	−	0.139586i	$-0.955423\pi$
$397$	−39.4596	−1.98042	−0.990210	+	0.139586i	$0.955423\pi$
$398$	35.1506	1.76194
$399$	1.84324	0.0922776
$400$	0	0
$401$	0.470266	0.0234840	0.0117420	−	0.999931i	$-0.496262\pi$
$401$	0.470266	0.0234840	0.0117420	+	0.999931i	$0.496262\pi$
$402$	38.5380	1.92210
$403$	−11.8843	−0.591998
$404$	−13.3340	−0.663393
$405$	0	0
$406$	3.31965	0.164752
$407$	34.0410	1.68735
$408$	8.58145	0.424845
$409$	10.4826	0.518329	0.259164	−	0.965833i	$-0.416553\pi$
$409$	10.4826	0.518329	0.259164	+	0.965833i	$0.416553\pi$
$410$	0	0
$411$	6.73820	0.332371
$412$	−17.3112	−0.852864
$413$	12.3135	0.605908
$414$	−0.398032	−0.0195622
$415$	0	0
$416$	−10.3896	−0.509393
$417$	15.1506	0.741929
$418$	−13.7587	−0.672961
$419$	34.6681	1.69365	0.846823	−	0.531875i	$-0.178512\pi$
$419$	34.6681	1.69365	0.846823	+	0.531875i	$0.178512\pi$
$420$	0	0
$421$	−34.6102	−1.68680	−0.843399	−	0.537288i	$-0.819449\pi$
$421$	−34.6102	−1.68680	−0.843399	+	0.537288i	$0.819449\pi$
$422$	−16.8950	−0.822434
$423$	0.0845208	0.00410954
$424$	10.2062	0.495657
$425$	0	0
$426$	40.1978	1.94759
$427$	−6.04104	−0.292346
$428$	−6.20620	−0.299988
$429$	14.8371	0.716342
$430$	0	0
$431$	6.73820	0.324568	0.162284	−	0.986744i	$-0.448114\pi$
$431$	6.73820	0.324568	0.162284	+	0.986744i	$0.448114\pi$
$432$	10.9360	0.526158
$433$	−20.4741	−0.983924	−0.491962	−	0.870617i	$-0.663720\pi$
$433$	−20.4741	−0.983924	−0.491962	+	0.870617i	$0.663720\pi$
$434$	20.3135	0.975080
$435$	0	0
$436$	−34.7792	−1.66562
$437$	2.34017	0.111946
$438$	20.0989	0.960362
$439$	−21.4596	−1.02421	−0.512105	−	0.858923i	$-0.671134\pi$
$439$	−21.4596	−1.02421	−0.512105	+	0.858923i	$0.671134\pi$
$440$	0	0
$441$	0.457499	0.0217857
$442$	−9.69102	−0.460955
$443$	21.5441	1.02359	0.511796	−	0.859107i	$-0.328980\pi$
$443$	21.5441	1.02359	0.511796	+	0.859107i	$0.328980\pi$
$444$	−24.8638	−1.17998
$445$	0	0
$446$	27.2267	1.28922
$447$	−33.7731	−1.59741
$448$	13.2762	0.627240
$449$	−8.47027	−0.399737	−0.199868	−	0.979823i	$-0.564051\pi$
$449$	−8.47027	−0.399737	−0.199868	+	0.979823i	$0.564051\pi$
$450$	0	0
$451$	20.6803	0.973799
$452$	34.9132	1.64218
$453$	5.84324	0.274540
$454$	4.97107	0.233304
$455$	0	0
$456$	2.63090	0.123203
$457$	−11.3607	−0.531431	−0.265715	−	0.964052i	$-0.585608\pi$
$457$	−11.3607	−0.531431	−0.265715	+	0.964052i	$0.585608\pi$
$458$	12.8371	0.599838
$459$	17.1629	0.801096
$460$	0	0
$461$	3.04718	0.141921	0.0709607	−	0.997479i	$-0.477394\pi$
$461$	3.04718	0.141921	0.0709607	+	0.997479i	$0.477394\pi$
$462$	−25.3607	−1.17989
$463$	9.97334	0.463500	0.231750	−	0.972775i	$-0.425555\pi$
$463$	9.97334	0.463500	0.231750	+	0.972775i	$0.425555\pi$
$464$	−2.94828	−0.136871
$465$	0	0
$466$	29.3340	1.35887
$467$	−1.49079	−0.0689853	−0.0344927	−	0.999405i	$-0.510982\pi$
$467$	−1.49079	−0.0689853	−0.0344927	+	0.999405i	$0.510982\pi$
$468$	0.290725	0.0134388
$469$	11.2039	0.517350
$470$	0	0
$471$	−16.0989	−0.741798
$472$	17.5753	0.808969
$473$	−75.5462	−3.47362
$474$	−52.8781	−2.42877
$475$	0	0
$476$	9.52973	0.436795
$477$	0.519715	0.0237961
$478$	−30.0410	−1.37405
$479$	−10.1711	−0.464731	−0.232365	−	0.972629i	$-0.574647\pi$
$479$	−10.1711	−0.464731	−0.232365	+	0.972629i	$0.574647\pi$
$480$	0	0
$481$	7.35085	0.335170
$482$	−15.7587	−0.717790
$483$	4.31351	0.196272
$484$	79.1049	3.59568
$485$	0	0
$486$	−1.76713	−0.0801588
$487$	24.2784	1.10016	0.550081	−	0.835112i	$-0.314597\pi$
$487$	24.2784	1.10016	0.550081	+	0.835112i	$0.314597\pi$
$488$	−8.62249	−0.390322
$489$	4.99386	0.225830
$490$	0	0
$491$	19.2039	0.866662	0.433331	−	0.901235i	$-0.357338\pi$
$491$	19.2039	0.866662	0.433331	+	0.901235i	$0.357338\pi$
$492$	−15.1050	−0.680988
$493$	−4.62702	−0.208391
$494$	−2.97107	−0.133675
$495$	0	0
$496$	−18.0410	−0.810067
$497$	11.6865	0.524211
$498$	−53.1917	−2.38357
$499$	7.33403	0.328316	0.164158	−	0.986434i	$-0.447509\pi$
$499$	7.33403	0.328316	0.164158	+	0.986434i	$0.447509\pi$
$500$	0	0
$501$	−35.7464	−1.59703
$502$	−22.7214	−1.01410
$503$	−29.0616	−1.29579	−0.647895	−	0.761729i	$-0.724351\pi$
$503$	−29.0616	−1.29579	−0.647895	+	0.761729i	$0.724351\pi$
$504$	−0.130094	−0.00579483
$505$	0	0
$506$	−32.1978	−1.43137
$507$	−19.0166	−0.844559
$508$	22.3051	0.989629
$509$	26.0456	1.15445	0.577225	−	0.816585i	$-0.304136\pi$
$509$	26.0456	1.15445	0.577225	+	0.816585i	$0.304136\pi$
$510$	0	0
$511$	5.84324	0.258490
$512$	−22.1701	−0.979789
$513$	5.26180	0.232314
$514$	−51.2678	−2.26132
$515$	0	0
$516$	55.1794	2.42914
$517$	6.83710	0.300695
$518$	−12.5646	−0.552058
$519$	1.80430	0.0791998
$520$	0	0
$521$	−38.8248	−1.70095	−0.850473	−	0.526019i	$-0.823684\pi$
$521$	−38.8248	−1.70095	−0.850473	+	0.526019i	$0.823684\pi$
$522$	0.241276	0.0105604
$523$	−4.59970	−0.201131	−0.100565	−	0.994930i	$-0.532065\pi$
$523$	−4.59970	−0.201131	−0.100565	+	0.994930i	$0.532065\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	12.2823	0.535534
$527$	−28.3135	−1.23336
$528$	22.5236	0.980213
$529$	−17.5236	−0.761895
$530$	0	0
$531$	0.894960	0.0388380
$532$	2.92162	0.126668
$533$	4.46573	0.193432
$534$	−28.0989	−1.21596
$535$	0	0
$536$	15.9916	0.690731
$537$	1.52973	0.0660129
$538$	5.02052	0.216450
$539$	37.0082	1.59406
$540$	0	0
$541$	−12.1256	−0.521318	−0.260659	−	0.965431i	$-0.583940\pi$
$541$	−12.1256	−0.521318	−0.260659	+	0.965431i	$0.583940\pi$
$542$	42.7526	1.83638
$543$	−1.43084	−0.0614031
$544$	−24.7526	−1.06126
$545$	0	0
$546$	−5.47641	−0.234369
$547$	9.54023	0.407911	0.203955	−	0.978980i	$-0.434620\pi$
$547$	9.54023	0.407911	0.203955	+	0.978980i	$0.434620\pi$
$548$	10.6803	0.456242
$549$	−0.439070	−0.0187390
$550$	0	0
$551$	−1.41855	−0.0604323
$552$	6.15676	0.262049
$553$	−15.3730	−0.653726
$554$	−55.8408	−2.37245
$555$	0	0
$556$	24.0144	1.01844
$557$	−19.9421	−0.844976	−0.422488	−	0.906369i	$-0.638843\pi$
$557$	−19.9421	−0.844976	−0.422488	+	0.906369i	$0.638843\pi$
$558$	1.47641	0.0625014
$559$	−16.3135	−0.689988
$560$	0	0
$561$	35.3484	1.49241
$562$	14.2823	0.602463
$563$	−23.5525	−0.992620	−0.496310	−	0.868145i	$-0.665312\pi$
$563$	−23.5525	−0.992620	−0.496310	+	0.868145i	$0.665312\pi$
$564$	−4.99386	−0.210279
$565$	0	0
$566$	−1.07838	−0.0453276
$567$	9.44521	0.396662
$568$	16.6803	0.699892
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−23.5031	−0.983573	−0.491786	−	0.870716i	$-0.663656\pi$
$571$	−23.5031	−0.983573	−0.491786	+	0.870716i	$0.663656\pi$
$572$	23.5174	0.983314
$573$	37.6742	1.57386
$574$	−7.63317	−0.318602
$575$	0	0
$576$	0.964928	0.0402053
$577$	−16.4703	−0.685666	−0.342833	−	0.939396i	$-0.611387\pi$
$577$	−16.4703	−0.685666	−0.342833	+	0.939396i	$0.611387\pi$
$578$	13.8033	0.574140
$579$	−21.8576	−0.908372
$580$	0	0
$581$	−15.4641	−0.641560
$582$	−32.9627	−1.36635
$583$	42.0410	1.74116
$584$	8.34017	0.345119
$585$	0	0
$586$	14.3896	0.594430
$587$	28.8104	1.18913	0.594567	−	0.804046i	$-0.297323\pi$
$587$	28.8104	1.18913	0.594567	+	0.804046i	$0.297323\pi$
$588$	−27.0310	−1.11474
$589$	−8.68035	−0.357667
$590$	0	0
$591$	15.7321	0.647131
$592$	11.1590	0.458633
$593$	43.2450	1.77586	0.887929	−	0.459980i	$-0.152144\pi$
$593$	43.2450	1.77586	0.887929	+	0.459980i	$0.152144\pi$
$594$	−72.3956	−2.97043
$595$	0	0
$596$	−53.5318	−2.19275
$597$	−27.6865	−1.13313
$598$	−6.95282	−0.284322
$599$	−44.2967	−1.80991	−0.904957	−	0.425503i	$-0.860097\pi$
$599$	−44.2967	−1.80991	−0.904957	+	0.425503i	$0.860097\pi$
$600$	0	0
$601$	−24.3090	−0.991584	−0.495792	−	0.868441i	$-0.665122\pi$
$601$	−24.3090	−0.991584	−0.495792	+	0.868441i	$0.665122\pi$
$602$	27.8843	1.13648
$603$	0.814315	0.0331615
$604$	9.26180	0.376857
$605$	0	0
$606$	18.2557	0.741585
$607$	6.29072	0.255333	0.127666	−	0.991817i	$-0.459251\pi$
$607$	6.29072	0.255333	0.127666	+	0.991817i	$0.459251\pi$
$608$	−7.58864	−0.307760
$609$	−2.61474	−0.105954
$610$	0	0
$611$	1.47641	0.0597291
$612$	0.692632	0.0279980
$613$	12.7915	0.516645	0.258322	−	0.966059i	$-0.416830\pi$
$613$	12.7915	0.516645	0.258322	+	0.966059i	$0.416830\pi$
$614$	31.6925	1.27900
$615$	0	0
$616$	−10.5236	−0.424008
$617$	−17.9299	−0.721829	−0.360914	−	0.932599i	$-0.617535\pi$
$617$	−17.9299	−0.721829	−0.360914	+	0.932599i	$0.617535\pi$
$618$	23.7009	0.953389
$619$	26.8515	1.07925	0.539626	−	0.841905i	$-0.318566\pi$
$619$	26.8515	1.07925	0.539626	+	0.841905i	$0.318566\pi$
$620$	0	0
$621$	12.3135	0.494124
$622$	−41.9565	−1.68230
$623$	−8.16904	−0.327286
$624$	4.86376	0.194706
$625$	0	0
$626$	66.5790	2.66103
$627$	10.8371	0.432792
$628$	−25.5174	−1.01826
$629$	17.5129	0.698286
$630$	0	0
$631$	35.5318	1.41450	0.707250	−	0.706964i	$-0.249936\pi$
$631$	35.5318	1.41450	0.707250	+	0.706964i	$0.249936\pi$
$632$	−21.9421	−0.872812
$633$	13.3074	0.528920
$634$	40.6453	1.61423
$635$	0	0
$636$	−30.7070	−1.21761
$637$	7.99159	0.316638
$638$	19.5174	0.772703
$639$	0.849388	0.0336013
$640$	0	0
$641$	12.9360	0.510941	0.255471	−	0.966817i	$-0.417770\pi$
$641$	12.9360	0.510941	0.255471	+	0.966817i	$0.417770\pi$
$642$	8.49693	0.335347
$643$	−8.49693	−0.335086	−0.167543	−	0.985865i	$-0.553583\pi$
$643$	−8.49693	−0.335086	−0.167543	+	0.985865i	$0.553583\pi$
$644$	6.83710	0.269420
$645$	0	0
$646$	−7.07838	−0.278495
$647$	45.4908	1.78843	0.894214	−	0.447640i	$-0.147736\pi$
$647$	45.4908	1.78843	0.894214	+	0.447640i	$0.147736\pi$
$648$	13.4813	0.529597
$649$	72.3956	2.84178
$650$	0	0
$651$	−16.0000	−0.627089
$652$	7.91548	0.309994
$653$	−35.9421	−1.40652	−0.703262	−	0.710930i	$-0.748274\pi$
$653$	−35.9421	−1.40652	−0.703262	+	0.710930i	$0.748274\pi$
$654$	47.6163	1.86195
$655$	0	0
$656$	6.77924	0.264685
$657$	0.424694	0.0165689
$658$	−2.52359	−0.0983798
$659$	30.9360	1.20510	0.602548	−	0.798083i	$-0.294152\pi$
$659$	30.9360	1.20510	0.602548	+	0.798083i	$0.294152\pi$
$660$	0	0
$661$	10.0989	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$661$	10.0989	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$662$	5.94214	0.230948
$663$	7.63317	0.296448
$664$	−22.0722	−0.856569
$665$	0	0
$666$	−0.913212	−0.0353862
$667$	−3.31965	−0.128538
$668$	−56.6596	−2.19223
$669$	−21.4452	−0.829120
$670$	0	0
$671$	−35.5174	−1.37114
$672$	−13.9877	−0.539588
$673$	8.67194	0.334279	0.167139	−	0.985933i	$-0.446547\pi$
$673$	8.67194	0.334279	0.167139	+	0.985933i	$0.446547\pi$
$674$	−13.1278	−0.505665
$675$	0	0
$676$	−30.1422	−1.15932
$677$	−41.9793	−1.61340	−0.806698	−	0.590964i	$-0.798747\pi$
$677$	−41.9793	−1.61340	−0.806698	+	0.590964i	$0.798747\pi$
$678$	−47.7998	−1.83574
$679$	−9.58306	−0.367764
$680$	0	0
$681$	−3.91548	−0.150041
$682$	119.430	4.57323
$683$	−46.3896	−1.77505	−0.887525	−	0.460760i	$-0.847577\pi$
$683$	−46.3896	−1.77505	−0.887525	+	0.460760i	$0.847577\pi$
$684$	0.212347	0.00811929
$685$	0	0
$686$	−30.0410	−1.14697
$687$	−10.1112	−0.385766
$688$	−24.7649	−0.944152
$689$	9.07838	0.345859
$690$	0	0
$691$	−34.8515	−1.32581	−0.662906	−	0.748702i	$-0.730677\pi$
$691$	−34.8515	−1.32581	−0.662906	+	0.748702i	$0.730677\pi$
$692$	2.85989	0.108717
$693$	−0.535877	−0.0203563
$694$	−12.9627	−0.492056
$695$	0	0
$696$	−3.73206	−0.141463
$697$	10.6393	0.402993
$698$	21.7009	0.821390
$699$	−23.1050	−0.873913
$700$	0	0
$701$	35.6430	1.34622	0.673109	−	0.739543i	$-0.264959\pi$
$701$	35.6430	1.34622	0.673109	+	0.739543i	$0.264959\pi$
$702$	−15.6332	−0.590036
$703$	5.36910	0.202500
$704$	78.0554	2.94182
$705$	0	0
$706$	−30.5958	−1.15149
$707$	5.30737	0.199604
$708$	−52.8781	−1.98728
$709$	16.7214	0.627985	0.313992	−	0.949426i	$-0.398333\pi$
$709$	16.7214	0.627985	0.313992	+	0.949426i	$0.398333\pi$
$710$	0	0
$711$	−1.11733	−0.0419030
$712$	−11.6598	−0.436970
$713$	−20.3135	−0.760747
$714$	−13.0472	−0.488278
$715$	0	0
$716$	2.42469	0.0906151
$717$	23.6619	0.883670
$718$	−13.0784	−0.488081
$719$	−6.85148	−0.255517	−0.127758	−	0.991805i	$-0.540778\pi$
$719$	−6.85148	−0.255517	−0.127758	+	0.991805i	$0.540778\pi$
$720$	0	0
$721$	6.89043	0.256613
$722$	−2.17009	−0.0807623
$723$	12.4124	0.461622
$724$	−2.26794	−0.0842873
$725$	0	0
$726$	−108.303	−4.01949
$727$	−34.4391	−1.27727	−0.638637	−	0.769508i	$-0.720502\pi$
$727$	−34.4391	−1.27727	−0.638637	+	0.769508i	$0.720502\pi$
$728$	−2.27247	−0.0842235
$729$	27.6681	1.02474
$730$	0	0
$731$	−38.8659	−1.43751
$732$	25.9421	0.958849
$733$	19.3607	0.715103	0.357552	−	0.933893i	$-0.383612\pi$
$733$	19.3607	0.715103	0.357552	+	0.933893i	$0.383612\pi$
$734$	2.34017	0.0863774
$735$	0	0
$736$	−17.7587	−0.654595
$737$	65.8720	2.42643
$738$	−0.554787	−0.0204220
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	2.34017	0.0859684
$742$	−15.5174	−0.569663
$743$	12.7154	0.466483	0.233242	−	0.972419i	$-0.425067\pi$
$743$	12.7154	0.466483	0.233242	+	0.972419i	$0.425067\pi$
$744$	−22.8371	−0.837249
$745$	0	0
$746$	26.7031	0.977671
$747$	−1.12395	−0.0411232
$748$	56.0288	2.04861
$749$	2.47027	0.0902616
$750$	0	0
$751$	3.15836	0.115250	0.0576252	−	0.998338i	$-0.481647\pi$
$751$	3.15836	0.115250	0.0576252	+	0.998338i	$0.481647\pi$
$752$	2.24128	0.0817309
$753$	17.8966	0.652187
$754$	4.21461	0.153487
$755$	0	0
$756$	15.3730	0.559110
$757$	−28.0410	−1.01917	−0.509584	−	0.860421i	$-0.670201\pi$
$757$	−28.0410	−1.01917	−0.509584	+	0.860421i	$0.670201\pi$
$758$	−2.27247	−0.0825399
$759$	25.3607	0.920535
$760$	0	0
$761$	−16.4924	−0.597849	−0.298924	−	0.954277i	$-0.596628\pi$
$761$	−16.4924	−0.597849	−0.298924	+	0.954277i	$0.596628\pi$
$762$	−30.5380	−1.10627
$763$	13.8432	0.501159
$764$	59.7152	2.16042
$765$	0	0
$766$	0.175007	0.00632326
$767$	15.6332	0.564481
$768$	−0.715418	−0.0258154
$769$	26.9627	0.972298	0.486149	−	0.873876i	$-0.338401\pi$
$769$	26.9627	0.972298	0.486149	+	0.873876i	$0.338401\pi$
$770$	0	0
$771$	40.3812	1.45429
$772$	−34.6453	−1.24691
$773$	−42.1939	−1.51761	−0.758805	−	0.651318i	$-0.774216\pi$
$773$	−42.1939	−1.51761	−0.758805	+	0.651318i	$0.774216\pi$
$774$	2.02666	0.0728469
$775$	0	0
$776$	−13.6781	−0.491014
$777$	9.89657	0.355037
$778$	44.5380	1.59676
$779$	3.26180	0.116866
$780$	0	0
$781$	68.7091	2.45860
$782$	−16.5646	−0.592350
$783$	−7.46412	−0.266746
$784$	12.1317	0.433275
$785$	0	0
$786$	−5.47641	−0.195337
$787$	35.4368	1.26319	0.631593	−	0.775300i	$-0.282401\pi$
$787$	35.4368	1.26319	0.631593	+	0.775300i	$0.282401\pi$
$788$	24.9360	0.888308
$789$	−9.67420	−0.344411
$790$	0	0
$791$	−13.8966	−0.494105
$792$	−0.764867	−0.0271784
$793$	−7.66967	−0.272358
$794$	85.6307	3.03892
$795$	0	0
$796$	−43.8843	−1.55544
$797$	−15.2579	−0.540463	−0.270232	−	0.962795i	$-0.587100\pi$
$797$	−15.2579	−0.540463	−0.270232	+	0.962795i	$0.587100\pi$
$798$	−4.00000	−0.141598
$799$	3.51745	0.124438
$800$	0	0
$801$	−0.593735	−0.0209786
$802$	−1.02052	−0.0360358
$803$	34.3545	1.21235
$804$	−48.1133	−1.69682
$805$	0	0
$806$	25.7899	0.908411
$807$	−3.95443	−0.139202
$808$	7.57531	0.266498
$809$	−7.16290	−0.251834	−0.125917	−	0.992041i	$-0.540187\pi$
$809$	−7.16290	−0.251834	−0.125917	+	0.992041i	$0.540187\pi$
$810$	0	0
$811$	−50.3545	−1.76819	−0.884094	−	0.467310i	$-0.845223\pi$
$811$	−50.3545	−1.76819	−0.884094	+	0.467310i	$0.845223\pi$
$812$	−4.14447	−0.145442
$813$	−33.6742	−1.18101
$814$	−73.8720	−2.58921
$815$	0	0
$816$	11.5876	0.405647
$817$	−11.9155	−0.416870
$818$	−22.7480	−0.795367
$819$	−0.115718	−0.00404350
$820$	0	0
$821$	0.952819	0.0332536	0.0166268	−	0.999862i	$-0.494707\pi$
$821$	0.952819	0.0332536	0.0166268	+	0.999862i	$0.494707\pi$
$822$	−14.6225	−0.510018
$823$	44.2290	1.54173	0.770863	−	0.637001i	$-0.219825\pi$
$823$	44.2290	1.54173	0.770863	+	0.637001i	$0.219825\pi$
$824$	9.83483	0.342613
$825$	0	0
$826$	−26.7214	−0.929756
$827$	37.8615	1.31657	0.658287	−	0.752767i	$-0.271281\pi$
$827$	37.8615	1.31657	0.658287	+	0.752767i	$0.271281\pi$
$828$	0.496928	0.0172695
$829$	−56.7214	−1.97002	−0.985008	−	0.172511i	$-0.944812\pi$
$829$	−56.7214	−1.97002	−0.985008	+	0.172511i	$0.944812\pi$
$830$	0	0
$831$	43.9832	1.52576
$832$	16.8554	0.584354
$833$	19.0394	0.659677
$834$	−32.8781	−1.13848
$835$	0	0
$836$	17.1773	0.594088
$837$	−45.6742	−1.57873
$838$	−75.2327	−2.59887
$839$	−28.3591	−0.979064	−0.489532	−	0.871985i	$-0.662832\pi$
$839$	−28.3591	−0.979064	−0.489532	+	0.871985i	$0.662832\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	75.1071	2.58836
$843$	−11.2495	−0.387454
$844$	21.0928	0.726043
$845$	0	0
$846$	−0.183417	−0.00630602
$847$	−31.4863	−1.08188
$848$	13.7815	0.473259
$849$	0.849388	0.0291509
$850$	0	0
$851$	12.5646	0.430710
$852$	−50.1855	−1.71933
$853$	17.0061	0.582279	0.291140	−	0.956681i	$-0.405966\pi$
$853$	17.0061	0.582279	0.291140	+	0.956681i	$0.405966\pi$
$854$	13.1096	0.448600
$855$	0	0
$856$	3.52586	0.120511
$857$	−15.4101	−0.526400	−0.263200	−	0.964741i	$-0.584778\pi$
$857$	−15.4101	−0.526400	−0.263200	+	0.964741i	$0.584778\pi$
$858$	−32.1978	−1.09921
$859$	−37.7275	−1.28725	−0.643623	−	0.765342i	$-0.722570\pi$
$859$	−37.7275	−1.28725	−0.643623	+	0.765342i	$0.722570\pi$
$860$	0	0
$861$	6.01229	0.204898
$862$	−14.6225	−0.498044
$863$	−32.1340	−1.09385	−0.546927	−	0.837181i	$-0.684202\pi$
$863$	−32.1340	−1.09385	−0.546927	+	0.837181i	$0.684202\pi$
$864$	−39.9299	−1.35844
$865$	0	0
$866$	44.4307	1.50982
$867$	−10.8722	−0.369238
$868$	−25.3607	−0.860798
$869$	−90.3833	−3.06604
$870$	0	0
$871$	14.2245	0.481977
$872$	19.7587	0.669115
$873$	−0.696508	−0.0235732
$874$	−5.07838	−0.171779
$875$	0	0
$876$	−25.0928	−0.847806
$877$	19.8927	0.671729	0.335864	−	0.941910i	$-0.390972\pi$
$877$	19.8927	0.671729	0.335864	+	0.941910i	$0.390972\pi$
$878$	46.5692	1.57163
$879$	−11.3340	−0.382287
$880$	0	0
$881$	24.0722	0.811014	0.405507	−	0.914092i	$-0.367095\pi$
$881$	24.0722	0.811014	0.405507	+	0.914092i	$0.367095\pi$
$882$	−0.992812	−0.0334297
$883$	31.1727	1.04905	0.524523	−	0.851396i	$-0.324244\pi$
$883$	31.1727	1.04905	0.524523	+	0.851396i	$0.324244\pi$
$884$	12.0989	0.406930
$885$	0	0
$886$	−46.7526	−1.57068
$887$	−4.86764	−0.163439	−0.0817197	−	0.996655i	$-0.526041\pi$
$887$	−4.86764	−0.163439	−0.0817197	+	0.996655i	$0.526041\pi$
$888$	14.1256	0.474023
$889$	−8.87814	−0.297763
$890$	0	0
$891$	55.5318	1.86039
$892$	−33.9916	−1.13812
$893$	1.07838	0.0360865
$894$	73.2905	2.45120
$895$	0	0
$896$	−12.4436	−0.415712
$897$	5.47641	0.182852
$898$	18.3812	0.613389
$899$	12.3135	0.410679
$900$	0	0
$901$	21.6286	0.720554
$902$	−44.8781	−1.49428
$903$	−21.9631	−0.730888
$904$	−19.8348	−0.659697
$905$	0	0
$906$	−12.6803	−0.421276
$907$	45.4778	1.51007	0.755033	−	0.655686i	$-0.227621\pi$
$907$	45.4778	1.51007	0.755033	+	0.655686i	$0.227621\pi$
$908$	−6.20620	−0.205960
$909$	0.385746	0.0127944
$910$	0	0
$911$	−20.9483	−0.694048	−0.347024	−	0.937856i	$-0.612808\pi$
$911$	−20.9483	−0.694048	−0.347024	+	0.937856i	$0.612808\pi$
$912$	3.55252	0.117636
$913$	−90.9192	−3.00899
$914$	24.6537	0.815471
$915$	0	0
$916$	−16.0267	−0.529536
$917$	−1.59213	−0.0525767
$918$	−37.2450	−1.22927
$919$	−59.5174	−1.96330	−0.981650	−	0.190693i	$-0.938926\pi$
$919$	−59.5174	−1.96330	−0.981650	+	0.190693i	$0.938926\pi$
$920$	0	0
$921$	−24.9627	−0.822548
$922$	−6.61265	−0.217776
$923$	14.8371	0.488369
$924$	31.6619	1.04160
$925$	0	0
$926$	−21.6430	−0.711233
$927$	0.500804	0.0164486
$928$	10.7649	0.353374
$929$	−16.7214	−0.548611	−0.274305	−	0.961643i	$-0.588448\pi$
$929$	−16.7214	−0.548611	−0.274305	+	0.961643i	$0.588448\pi$
$930$	0	0
$931$	5.83710	0.191303
$932$	−36.6225	−1.19961
$933$	33.0472	1.08192
$934$	3.23513	0.105857
$935$	0	0
$936$	−0.165166	−0.00539862
$937$	32.4534	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$937$	32.4534	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$938$	−24.3135	−0.793864
$939$	−52.4412	−1.71135
$940$	0	0
$941$	−23.6742	−0.771757	−0.385878	−	0.922550i	$-0.626102\pi$
$941$	−23.6742	−0.771757	−0.385878	+	0.922550i	$0.626102\pi$
$942$	34.9360	1.13828
$943$	7.63317	0.248570
$944$	23.7321	0.772413
$945$	0	0
$946$	163.942	5.33021
$947$	−21.9733	−0.714038	−0.357019	−	0.934097i	$-0.616207\pi$
$947$	−21.9733	−0.714038	−0.357019	+	0.934097i	$0.616207\pi$
$948$	66.0165	2.14412
$949$	7.41855	0.240816
$950$	0	0
$951$	−32.0144	−1.03814
$952$	−5.41402	−0.175469
$953$	53.2990	1.72652	0.863261	−	0.504757i	$-0.168418\pi$
$953$	53.2990	1.72652	0.863261	+	0.504757i	$0.168418\pi$
$954$	−1.12783	−0.0365147
$955$	0	0
$956$	37.5052	1.21300
$957$	−15.3730	−0.496938
$958$	22.0722	0.713122
$959$	−4.25112	−0.137276
$960$	0	0
$961$	44.3484	1.43059
$962$	−15.9520	−0.514313
$963$	0.179542	0.00578565
$964$	19.6742	0.633663
$965$	0	0
$966$	−9.36069	−0.301175
$967$	−15.8166	−0.508627	−0.254314	−	0.967122i	$-0.581850\pi$
$967$	−15.8166	−0.508627	−0.254314	+	0.967122i	$0.581850\pi$
$968$	−44.9409	−1.44446
$969$	5.57531	0.179105
$970$	0	0
$971$	−43.1506	−1.38477	−0.692385	−	0.721529i	$-0.743440\pi$
$971$	−43.1506	−1.38477	−0.692385	+	0.721529i	$0.743440\pi$
$972$	2.20620	0.0707640
$973$	−9.55849	−0.306431
$974$	−52.6863	−1.68818
$975$	0	0
$976$	−11.6430	−0.372684
$977$	8.47414	0.271112	0.135556	−	0.990770i	$-0.456718\pi$
$977$	8.47414	0.271112	0.135556	+	0.990770i	$0.456718\pi$
$978$	−10.8371	−0.346532
$979$	−48.0288	−1.53501
$980$	0	0
$981$	1.00614	0.0321237
$982$	−41.6742	−1.32988
$983$	5.59356	0.178407	0.0892034	−	0.996013i	$-0.471568\pi$
$983$	5.59356	0.178407	0.0892034	+	0.996013i	$0.471568\pi$
$984$	8.58145	0.273567
$985$	0	0
$986$	10.0410	0.319772
$987$	1.98771	0.0632696
$988$	3.70928	0.118008
$989$	−27.8843	−0.886669
$990$	0	0
$991$	32.8950	1.04494	0.522471	−	0.852657i	$-0.325010\pi$
$991$	32.8950	1.04494	0.522471	+	0.852657i	$0.325010\pi$
$992$	65.8720	2.09144
$993$	−4.68035	−0.148526
$994$	−25.3607	−0.804392
$995$	0	0
$996$	66.4079	2.10421
$997$	−23.2618	−0.736708	−0.368354	−	0.929686i	$-0.620079\pi$
$997$	−23.2618	−0.736708	−0.368354	+	0.929686i	$0.620079\pi$
$998$	−15.9155	−0.503796
$999$	28.2511	0.893826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.f.1.1		3
3.2	odd	2		4275.2.a.bk.1.3		3
4.3	odd	2		7600.2.a.bx.1.1		3
5.2	odd	4		475.2.b.d.324.1		6
5.3	odd	4		475.2.b.d.324.6		6
5.4	even	2		95.2.a.a.1.3	✓	3
15.14	odd	2		855.2.a.i.1.1		3
19.18	odd	2		9025.2.a.bb.1.3		3
20.19	odd	2		1520.2.a.p.1.3		3
35.34	odd	2		4655.2.a.u.1.3		3
40.19	odd	2		6080.2.a.by.1.1		3
40.29	even	2		6080.2.a.bo.1.3		3
95.94	odd	2		1805.2.a.f.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.3	✓	3	5.4	even	2
475.2.a.f.1.1		3	1.1	even	1	trivial
475.2.b.d.324.1		6	5.2	odd	4
475.2.b.d.324.6		6	5.3	odd	4
855.2.a.i.1.1		3	15.14	odd	2
1520.2.a.p.1.3		3	20.19	odd	2
1805.2.a.f.1.1		3	95.94	odd	2
4275.2.a.bk.1.3		3	3.2	odd	2
4655.2.a.u.1.3		3	35.34	odd	2
6080.2.a.bo.1.3		3	40.29	even	2
6080.2.a.by.1.1		3	40.19	odd	2
7600.2.a.bx.1.1		3	4.3	odd	2
9025.2.a.bb.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +1.70928 q^{3} +2.70928 q^{4} -3.70928 q^{6} -1.07838 q^{7} -1.53919 q^{8} -0.0783777 q^{9} -6.34017 q^{11} +4.63090 q^{12} -1.36910 q^{13} +2.34017 q^{14} -2.07838 q^{16} -3.26180 q^{17} +0.170086 q^{18} -1.00000 q^{19} -1.84324 q^{21} +13.7587 q^{22} -2.34017 q^{23} -2.63090 q^{24} +2.97107 q^{26} -5.26180 q^{27} -2.92162 q^{28} +1.41855 q^{29} +8.68035 q^{31} +7.58864 q^{32} -10.8371 q^{33} +7.07838 q^{34} -0.212347 q^{36} -5.36910 q^{37} +2.17009 q^{38} -2.34017 q^{39} -3.26180 q^{41} +4.00000 q^{42} +11.9155 q^{43} -17.1773 q^{44} +5.07838 q^{46} -1.07838 q^{47} -3.55252 q^{48} -5.83710 q^{49} -5.57531 q^{51} -3.70928 q^{52} -6.63090 q^{53} +11.4186 q^{54} +1.65983 q^{56} -1.70928 q^{57} -3.07838 q^{58} -11.4186 q^{59} +5.60197 q^{61} -18.8371 q^{62} +0.0845208 q^{63} -12.3112 q^{64} +23.5174 q^{66} -10.3896 q^{67} -8.83710 q^{68} -4.00000 q^{69} -10.8371 q^{71} +0.120638 q^{72} -5.41855 q^{73} +11.6514 q^{74} -2.70928 q^{76} +6.83710 q^{77} +5.07838 q^{78} +14.2557 q^{79} -8.75872 q^{81} +7.07838 q^{82} +14.3402 q^{83} -4.99386 q^{84} -25.8576 q^{86} +2.42469 q^{87} +9.75872 q^{88} +7.57531 q^{89} +1.47641 q^{91} -6.34017 q^{92} +14.8371 q^{93} +2.34017 q^{94} +12.9711 q^{96} +8.88655 q^{97} +12.6670 q^{98} +0.496928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9} - 8 q^{11} + 10 q^{12} - 8 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} - 5 q^{18} - 3 q^{19} - 12 q^{21} + 16 q^{22} + 4 q^{23} - 4 q^{24} - 6 q^{26}+ \cdots - 16 q^{99}+O(q^{100}) \) 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 3 * q^8 + 3 * q^9 - 8 * q^11 + 10 * q^12 - 8 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 - 5 * q^18 - 3 * q^19 - 12 * q^21 + 16 * q^22 + 4 * q^23 - 4 * q^24 - 6 * q^26 - 8 * q^27 - 12 * q^28 - 10 * q^29 + 4 * q^31 + 3 * q^32 - 4 * q^33 + 18 * q^34 - 11 * q^36 - 20 * q^37 + q^38 + 4 * q^39 - 2 * q^41 + 12 * q^42 + 4 * q^43 - 12 * q^44 + 12 * q^46 - 10 * q^48 + 11 * q^49 + 4 * q^51 - 4 * q^52 - 16 * q^53 + 20 * q^54 + 16 * q^56 + 2 * q^57 - 6 * q^58 - 20 * q^59 - 2 * q^61 - 28 * q^62 + 32 * q^63 - 11 * q^64 + 20 * q^66 - 2 * q^67 + 2 * q^68 - 12 * q^69 - 4 * q^71 + 13 * q^72 - 2 * q^73 - 2 * q^74 - q^76 - 8 * q^77 + 12 * q^78 - q^81 + 18 * q^82 + 32 * q^83 + 20 * q^84 - 16 * q^86 + 28 * q^87 + 4 * q^88 + 2 * q^89 + 20 * q^91 - 8 * q^92 + 16 * q^93 - 4 * q^94 + 24 * q^96 - 20 * q^97 + 15 * q^98 - 16 * q^99

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