LMFDB - Embedded newform 575.2.b.f.24.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [575,2,Mod(24,575)] 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("575.24"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	575.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$4.59139811622$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 25x^{12} + 248x^{10} + 1239x^{8} + 3259x^{6} + 4248x^{4} + 2149x^{2} + 81 $ x^14 + 25x^12 + 248x^10 + 1239x^8 + 3259x^6 + 4248x^4 + 2149x^2 + 81
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.8
Root			$0.202227i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.2.b.f.24.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.202227i q^{2} +2.69619i q^{3} +1.95910 q^{4} -0.545243 q^{6} -2.81698i q^{7} +0.800639i q^{8} -4.26945 q^{9} -1.84786 q^{11} +5.28212i q^{12} +6.49089i q^{13} +0.569670 q^{14} +3.75630 q^{16} +7.06930i q^{17} -0.863400i q^{18} +0.252319 q^{19} +7.59513 q^{21} -0.373689i q^{22} +1.00000i q^{23} -2.15868 q^{24} -1.31263 q^{26} -3.42269i q^{27} -5.51876i q^{28} -4.12351 q^{29} +3.54811 q^{31} +2.36090i q^{32} -4.98220i q^{33} -1.42961 q^{34} -8.36430 q^{36} -7.91248i q^{37} +0.0510258i q^{38} -17.5007 q^{39} +6.53833 q^{41} +1.53594i q^{42} -4.93835i q^{43} -3.62016 q^{44} -0.202227 q^{46} -0.851917i q^{47} +10.1277i q^{48} -0.935388 q^{49} -19.0602 q^{51} +12.7163i q^{52} -12.5831i q^{53} +0.692161 q^{54} +2.25538 q^{56} +0.680301i q^{57} -0.833886i q^{58} +10.3616 q^{59} +1.01207 q^{61} +0.717524i q^{62} +12.0270i q^{63} +7.03516 q^{64} +1.00754 q^{66} -3.37930i q^{67} +13.8495i q^{68} -2.69619 q^{69} -0.851917 q^{71} -3.41829i q^{72} -9.75748i q^{73} +1.60012 q^{74} +0.494319 q^{76} +5.20540i q^{77} -3.53911i q^{78} +16.5320 q^{79} -3.58013 q^{81} +1.32223i q^{82} +0.696772i q^{83} +14.8796 q^{84} +0.998669 q^{86} -11.1178i q^{87} -1.47947i q^{88} -13.1835 q^{89} +18.2847 q^{91} +1.95910i q^{92} +9.56638i q^{93} +0.172281 q^{94} -6.36545 q^{96} +5.48684i q^{97} -0.189161i q^{98} +7.88937 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 22 q^{4} + 10 q^{6} - 30 q^{9} - 2 q^{11} - 14 q^{14} + 14 q^{16} - 30 q^{19} + 4 q^{21} - 36 q^{24} - 40 q^{26} - 6 q^{29} + 28 q^{31} - 40 q^{34} + 16 q^{39} + 38 q^{41} + 6 q^{44} + 2 q^{46} - 80 q^{49}+ \cdots + 106 q^{99}+O(q^{100}) $ 14 * q - 22 * q^4 + 10 * q^6 - 30 * q^9 - 2 * q^11 - 14 * q^14 + 14 * q^16 - 30 * q^19 + 4 * q^21 - 36 * q^24 - 40 * q^26 - 6 * q^29 + 28 * q^31 - 40 * q^34 + 16 * q^39 + 38 * q^41 + 6 * q^44 + 2 * q^46 - 80 * q^49 + 4 * q^51 + 2 * q^54 - 18 * q^56 + 32 * q^59 + 80 * q^61 + 8 * q^64 - 108 * q^66 - 28 * q^71 + 36 * q^74 + 70 * q^76 + 2 * q^79 + 94 * q^81 + 120 * q^84 - 70 * q^86 - 32 * q^89 + 50 * q^91 - 14 * q^94 - 38 * q^96 + 106 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.202227i	0.142996i	0.997441	+	0.0714981i	$0.0227780\pi$
$2$	0.202227i	0.142996i	−0.997441	+	0.0714981i	$0.977222\pi$
$3$	2.69619i	1.55665i	0.627863	+	0.778324i	$0.283930\pi$
$3$	2.69619i	1.55665i	−0.627863	+	0.778324i	$0.716070\pi$
$4$	1.95910	0.979552
$5$	0	0
$5$	0	0
$6$	−0.545243	−0.222595
$7$	− 2.81698i	− 1.06472i	−0.846518	−	0.532360i	$-0.821305\pi$
$7$	− 2.81698i	− 1.06472i	0.846518	−	0.532360i	$-0.178695\pi$
$8$	0.800639i	0.283069i
$9$	−4.26945	−1.42315
$10$	0	0
$11$	−1.84786	−0.557152	−0.278576	−	0.960414i	$-0.589862\pi$
$11$	−1.84786	−0.557152	−0.278576	+	0.960414i	$0.589862\pi$
$12$	5.28212i	1.52482i
$13$	6.49089i	1.80025i	0.435633	+	0.900124i	$0.356525\pi$
$13$	6.49089i	1.80025i	−0.435633	+	0.900124i	$0.643475\pi$
$14$	0.569670	0.152251
$15$	0	0
$16$	3.75630	0.939074
$17$	7.06930i	1.71456i	0.514853	+	0.857279i	$0.327847\pi$
$17$	7.06930i	1.71456i	−0.514853	+	0.857279i	$0.672153\pi$
$18$	− 0.863400i	− 0.203505i
$19$	0.252319	0.0578860	0.0289430	−	0.999581i	$-0.490786\pi$
$19$	0.252319	0.0578860	0.0289430	+	0.999581i	$0.490786\pi$
$20$	0	0
$21$	7.59513	1.65739
$22$	− 0.373689i	− 0.0796707i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	−2.15868	−0.440638
$25$	0	0
$26$	−1.31263	−0.257429
$27$	− 3.42269i	− 0.658696i
$28$	− 5.51876i	− 1.04295i
$29$	−4.12351	−0.765717	−0.382858	−	0.923807i	$-0.625060\pi$
$29$	−4.12351	−0.765717	−0.382858	+	0.923807i	$0.625060\pi$
$30$	0	0
$31$	3.54811	0.637259	0.318630	−	0.947879i	$-0.396777\pi$
$31$	3.54811	0.637259	0.318630	+	0.947879i	$0.396777\pi$
$32$	2.36090i	0.417353i
$33$	− 4.98220i	− 0.867289i
$34$	−1.42961	−0.245175
$35$	0	0
$36$	−8.36430	−1.39405
$37$	− 7.91248i	− 1.30080i	−0.759591	−	0.650402i	$-0.774601\pi$
$37$	− 7.91248i	− 1.30080i	0.759591	−	0.650402i	$-0.225399\pi$
$38$	0.0510258i	0.00827748i
$39$	−17.5007	−2.80235
$40$	0	0
$41$	6.53833	1.02111	0.510557	−	0.859844i	$-0.329439\pi$
$41$	6.53833	1.02111	0.510557	+	0.859844i	$0.329439\pi$
$42$	1.53594i	0.237001i
$43$	− 4.93835i	− 0.753092i	−0.926398	−	0.376546i	$-0.877112\pi$
$43$	− 4.93835i	− 0.753092i	0.926398	−	0.376546i	$-0.122888\pi$
$44$	−3.62016	−0.545760
$45$	0	0
$46$	−0.202227	−0.0298168
$47$	− 0.851917i	− 0.124265i	−0.998068	−	0.0621324i	$-0.980210\pi$
$47$	− 0.851917i	− 0.124265i	0.998068	−	0.0621324i	$-0.0197901\pi$
$48$	10.1277i	1.46181i
$49$	−0.935388	−0.133627
$50$	0	0
$51$	−19.0602	−2.66896
$52$	12.7163i	1.76344i
$53$	− 12.5831i	− 1.72842i	−0.503135	−	0.864208i	$-0.667820\pi$
$53$	− 12.5831i	− 1.72842i	0.503135	−	0.864208i	$-0.332180\pi$
$54$	0.692161	0.0941911
$55$	0	0
$56$	2.25538	0.301388
$57$	0.680301i	0.0901080i
$58$	− 0.833886i	− 0.109495i
$59$	10.3616	1.34897	0.674484	−	0.738290i	$-0.264366\pi$
$59$	10.3616	1.34897	0.674484	+	0.738290i	$0.264366\pi$
$60$	0	0
$61$	1.01207	0.129582	0.0647911	−	0.997899i	$-0.479362\pi$
$61$	1.01207	0.129582	0.0647911	+	0.997899i	$0.479362\pi$
$62$	0.717524i	0.0911257i
$63$	12.0270i	1.51526i
$64$	7.03516	0.879394
$65$	0	0
$66$	1.00754	0.124019
$67$	− 3.37930i	− 0.412848i	−0.978463	−	0.206424i	$-0.933817\pi$
$67$	− 3.37930i	− 0.412848i	0.978463	−	0.206424i	$-0.0661826\pi$
$68$	13.8495i	1.67950i
$69$	−2.69619	−0.324583
$70$	0	0
$71$	−0.851917	−0.101104	−0.0505520	−	0.998721i	$-0.516098\pi$
$71$	−0.851917	−0.101104	−0.0505520	+	0.998721i	$0.516098\pi$
$72$	− 3.41829i	− 0.402849i
$73$	− 9.75748i	− 1.14203i	−0.820941	−	0.571013i	$-0.806551\pi$
$73$	− 9.75748i	− 1.14203i	0.820941	−	0.571013i	$-0.193449\pi$
$74$	1.60012	0.186010
$75$	0	0
$76$	0.494319	0.0567023
$77$	5.20540i	0.593211i
$78$	− 3.53911i	− 0.400726i
$79$	16.5320	1.86000	0.929999	−	0.367561i	$-0.119807\pi$
$79$	16.5320	1.86000	0.929999	+	0.367561i	$0.119807\pi$
$80$	0	0
$81$	−3.58013	−0.397793
$82$	1.32223i	0.146016i
$83$	0.696772i	0.0764806i	0.999269	+	0.0382403i	$0.0121752\pi$
$83$	0.696772i	0.0764806i	−0.999269	+	0.0382403i	$0.987825\pi$
$84$	14.8796	1.62350
$85$	0	0
$86$	0.998669	0.107689
$87$	− 11.1178i	− 1.19195i
$88$	− 1.47947i	− 0.157712i
$89$	−13.1835	−1.39745	−0.698724	−	0.715392i	$-0.746248\pi$
$89$	−13.1835	−1.39745	−0.698724	+	0.715392i	$0.746248\pi$
$90$	0	0
$91$	18.2847	1.91676
$92$	1.95910i	0.204251i
$93$	9.56638i	0.991988i
$94$	0.172281	0.0177694
$95$	0	0
$96$	−6.36545	−0.649671
$97$	5.48684i	0.557104i	0.960421	+	0.278552i	$0.0898544\pi$
$97$	5.48684i	0.557104i	−0.960421	+	0.278552i	$0.910146\pi$
$98$	− 0.189161i	− 0.0191081i
$99$	7.88937	0.792911
$100$	0	0
$101$	−9.56925	−0.952176	−0.476088	−	0.879398i	$-0.657946\pi$
$101$	−9.56925	−0.952176	−0.476088	+	0.879398i	$0.657946\pi$
$102$	− 3.85449i	− 0.381651i
$103$	− 9.10123i	− 0.896770i	−0.893840	−	0.448385i	$-0.851999\pi$
$103$	− 9.10123i	− 0.896770i	0.893840	−	0.448385i	$-0.148001\pi$
$104$	−5.19686	−0.509594
$105$	0	0
$106$	2.54464	0.247157
$107$	− 11.7093i	− 1.13198i	−0.824411	−	0.565991i	$-0.808494\pi$
$107$	− 11.7093i	− 1.13198i	0.824411	−	0.565991i	$-0.191506\pi$
$108$	− 6.70540i	− 0.645228i
$109$	−1.46844	−0.140651	−0.0703257	−	0.997524i	$-0.522404\pi$
$109$	−1.46844	−0.140651	−0.0703257	+	0.997524i	$0.522404\pi$
$110$	0	0
$111$	21.3336	2.02489
$112$	− 10.5814i	− 0.999850i
$113$	1.62939i	0.153280i	0.997059	+	0.0766401i	$0.0244193\pi$
$113$	1.62939i	0.153280i	−0.997059	+	0.0766401i	$0.975581\pi$
$114$	−0.137575	−0.0128851
$115$	0	0
$116$	−8.07839	−0.750060
$117$	− 27.7125i	− 2.56203i
$118$	2.09540i	0.192897i
$119$	19.9141	1.82552
$120$	0	0
$121$	−7.58540	−0.689581
$122$	0.204668i	0.0185298i
$123$	17.6286i	1.58952i
$124$	6.95112	0.624229
$125$	0	0
$126$	−2.43218	−0.216676
$127$	8.59258i	0.762468i	0.924479	+	0.381234i	$0.124501\pi$
$127$	8.59258i	0.762468i	−0.924479	+	0.381234i	$0.875499\pi$
$128$	6.14451i	0.543103i
$129$	13.3147	1.17230
$130$	0	0
$131$	2.93777	0.256674	0.128337	−	0.991731i	$-0.459036\pi$
$131$	2.93777	0.256674	0.128337	+	0.991731i	$0.459036\pi$
$132$	− 9.76064i	− 0.849555i
$133$	− 0.710778i	− 0.0616323i
$134$	0.683387	0.0590357
$135$	0	0
$136$	−5.65996	−0.485337
$137$	21.9366i	1.87417i	0.349101	+	0.937085i	$0.386487\pi$
$137$	21.9366i	1.87417i	−0.349101	+	0.937085i	$0.613513\pi$
$138$	− 0.545243i	− 0.0464142i
$139$	−15.3324	−1.30048	−0.650239	−	0.759730i	$-0.725331\pi$
$139$	−15.3324	−1.30048	−0.650239	+	0.759730i	$0.725331\pi$
$140$	0	0
$141$	2.29693	0.193437
$142$	− 0.172281i	− 0.0144575i
$143$	− 11.9943i	− 1.00301i
$144$	−16.0373	−1.33644
$145$	0	0
$146$	1.97323	0.163305
$147$	− 2.52199i	− 0.208010i
$148$	− 15.5014i	− 1.27420i
$149$	15.0218	1.23063	0.615316	−	0.788281i	$-0.289028\pi$
$149$	15.0218	1.23063	0.615316	+	0.788281i	$0.289028\pi$
$150$	0	0
$151$	17.5294	1.42652	0.713262	−	0.700897i	$-0.247217\pi$
$151$	17.5294	1.42652	0.713262	+	0.700897i	$0.247217\pi$
$152$	0.202016i	0.0163857i
$153$	− 30.1820i	− 2.44007i
$154$	−1.05267	−0.0848269
$155$	0	0
$156$	−34.2857	−2.74505
$157$	− 7.10315i	− 0.566893i	−0.958988	−	0.283446i	$-0.908522\pi$
$157$	− 7.10315i	− 0.566893i	0.958988	−	0.283446i	$-0.0914778\pi$
$158$	3.34323i	0.265973i
$159$	33.9263	2.69053
$160$	0	0
$161$	2.81698	0.222009
$162$	− 0.724001i	− 0.0568829i
$163$	− 14.5905i	− 1.14282i	−0.820666	−	0.571408i	$-0.806397\pi$
$163$	− 14.5905i	− 1.14282i	0.820666	−	0.571408i	$-0.193603\pi$
$164$	12.8093	1.00024
$165$	0	0
$166$	−0.140906	−0.0109364
$167$	9.45066i	0.731314i	0.930750	+	0.365657i	$0.119156\pi$
$167$	9.45066i	0.731314i	−0.930750	+	0.365657i	$0.880844\pi$
$168$	6.08095i	0.469156i
$169$	−29.1316	−2.24090
$170$	0	0
$171$	−1.07726	−0.0823805
$172$	− 9.67475i	− 0.737692i
$173$	− 23.3055i	− 1.77188i	−0.463799	−	0.885940i	$-0.653514\pi$
$173$	− 23.3055i	− 1.77188i	0.463799	−	0.885940i	$-0.346486\pi$
$174$	2.24832	0.170445
$175$	0	0
$176$	−6.94113	−0.523207
$177$	27.9369i	2.09987i
$178$	− 2.66606i	− 0.199830i
$179$	−23.5562	−1.76068	−0.880338	−	0.474347i	$-0.842684\pi$
$179$	−23.5562	−1.76068	−0.880338	+	0.474347i	$0.842684\pi$
$180$	0	0
$181$	0.505059	0.0375407	0.0187704	−	0.999824i	$-0.494025\pi$
$181$	0.505059	0.0375407	0.0187704	+	0.999824i	$0.494025\pi$
$182$	3.69767i	0.274089i
$183$	2.72874i	0.201714i
$184$	−0.800639	−0.0590239
$185$	0	0
$186$	−1.93458	−0.141851
$187$	− 13.0631i	− 0.955269i
$188$	− 1.66899i	− 0.121724i
$189$	−9.64165	−0.701327
$190$	0	0
$191$	2.51313	0.181844	0.0909219	−	0.995858i	$-0.471019\pi$
$191$	2.51313	0.181844	0.0909219	+	0.995858i	$0.471019\pi$
$192$	18.9681i	1.36891i
$193$	− 4.17037i	− 0.300190i	−0.988672	−	0.150095i	$-0.952042\pi$
$193$	− 4.17037i	− 0.300190i	0.988672	−	0.150095i	$-0.0479579\pi$
$194$	−1.10959	−0.0796638
$195$	0	0
$196$	−1.83252	−0.130894
$197$	6.60102i	0.470303i	0.971959	+	0.235151i	$0.0755586\pi$
$197$	6.60102i	0.470303i	−0.971959	+	0.235151i	$0.924441\pi$
$198$	1.59545i	0.113383i
$199$	9.91175	0.702626	0.351313	−	0.936258i	$-0.385735\pi$
$199$	9.91175	0.702626	0.351313	+	0.936258i	$0.385735\pi$
$200$	0	0
$201$	9.11125	0.642658
$202$	− 1.93516i	− 0.136158i
$203$	11.6159i	0.815273i
$204$	−37.3409	−2.61439
$205$	0	0
$206$	1.84052	0.128235
$207$	− 4.26945i	− 0.296747i
$208$	24.3817i	1.69057i
$209$	−0.466251	−0.0322513
$210$	0	0
$211$	8.33546	0.573837	0.286918	−	0.957955i	$-0.407369\pi$
$211$	8.33546	0.573837	0.286918	+	0.957955i	$0.407369\pi$
$212$	− 24.6515i	− 1.69307i
$213$	− 2.29693i	− 0.157383i
$214$	2.36794	0.161869
$215$	0	0
$216$	2.74034	0.186456
$217$	− 9.99496i	− 0.678502i
$218$	− 0.296959i	− 0.0201126i
$219$	26.3080	1.77773
$220$	0	0
$221$	−45.8860	−3.08663
$222$	4.31423i	0.289552i
$223$	13.4909i	0.903420i	0.892165	+	0.451710i	$0.149186\pi$
$223$	13.4909i	0.903420i	−0.892165	+	0.451710i	$0.850814\pi$
$224$	6.65062	0.444363
$225$	0	0
$226$	−0.329507	−0.0219185
$227$	5.77415i	0.383244i	0.981469	+	0.191622i	$0.0613747\pi$
$227$	5.77415i	0.383244i	−0.981469	+	0.191622i	$0.938625\pi$
$228$	1.33278i	0.0882655i
$229$	−22.3181	−1.47482	−0.737412	−	0.675444i	$-0.763952\pi$
$229$	−22.3181	−1.47482	−0.737412	+	0.675444i	$0.763952\pi$
$230$	0	0
$231$	−14.0348	−0.923420
$232$	− 3.30144i	− 0.216750i
$233$	21.8227i	1.42965i	0.699303	+	0.714825i	$0.253494\pi$
$233$	21.8227i	1.42965i	−0.699303	+	0.714825i	$0.746506\pi$
$234$	5.60423	0.366360
$235$	0	0
$236$	20.2995	1.32138
$237$	44.5735i	2.89536i
$238$	4.02717i	0.261043i
$239$	26.7115	1.72782	0.863912	−	0.503644i	$-0.168007\pi$
$239$	26.7115	1.72782	0.863912	+	0.503644i	$0.168007\pi$
$240$	0	0
$241$	15.1565	0.976318	0.488159	−	0.872755i	$-0.337669\pi$
$241$	15.1565	0.976318	0.488159	+	0.872755i	$0.337669\pi$
$242$	− 1.53397i	− 0.0986076i
$243$	− 19.9208i	− 1.27792i
$244$	1.98275	0.126933
$245$	0	0
$246$	−3.56498	−0.227295
$247$	1.63778i	0.104209i
$248$	2.84075i	0.180388i
$249$	−1.87863	−0.119053
$250$	0	0
$251$	−13.2026	−0.833340	−0.416670	−	0.909058i	$-0.636803\pi$
$251$	−13.2026	−0.833340	−0.416670	+	0.909058i	$0.636803\pi$
$252$	23.5621i	1.48427i
$253$	− 1.84786i	− 0.116174i
$254$	−1.73765	−0.109030
$255$	0	0
$256$	12.8277	0.801733
$257$	− 13.9450i	− 0.869866i	−0.900463	−	0.434933i	$-0.856772\pi$
$257$	− 13.9450i	− 0.869866i	0.900463	−	0.434933i	$-0.143228\pi$
$258$	2.69260i	0.167634i
$259$	−22.2893	−1.38499
$260$	0	0
$261$	17.6051	1.08973
$262$	0.594097i	0.0367035i
$263$	21.7883i	1.34352i	0.740767	+	0.671762i	$0.234462\pi$
$263$	21.7883i	1.34352i	−0.740767	+	0.671762i	$0.765538\pi$
$264$	3.98894	0.245502
$265$	0	0
$266$	0.143739	0.00881319
$267$	− 35.5452i	− 2.17533i
$268$	− 6.62041i	− 0.404406i
$269$	−10.2596	−0.625540	−0.312770	−	0.949829i	$-0.601257\pi$
$269$	−10.2596	−0.625540	−0.312770	+	0.949829i	$0.601257\pi$
$270$	0	0
$271$	4.11888	0.250204	0.125102	−	0.992144i	$-0.460074\pi$
$271$	4.11888	0.250204	0.125102	+	0.992144i	$0.460074\pi$
$272$	26.5544i	1.61010i
$273$	49.2991i	2.98372i
$274$	−4.43618	−0.267999
$275$	0	0
$276$	−5.28212	−0.317946
$277$	− 9.62736i	− 0.578452i	−0.957261	−	0.289226i	$-0.906602\pi$
$277$	− 9.62736i	− 0.578452i	0.957261	−	0.289226i	$-0.0933980\pi$
$278$	− 3.10063i	− 0.185964i
$279$	−15.1485	−0.906916
$280$	0	0
$281$	16.9191	1.00931	0.504654	−	0.863322i	$-0.331620\pi$
$281$	16.9191	1.00931	0.504654	+	0.863322i	$0.331620\pi$
$282$	0.464502i	0.0276607i
$283$	− 7.51375i	− 0.446646i	−0.974744	−	0.223323i	$-0.928309\pi$
$283$	− 7.51375i	− 0.446646i	0.974744	−	0.223323i	$-0.0716905\pi$
$284$	−1.66899	−0.0990366
$285$	0	0
$286$	2.42557	0.143427
$287$	− 18.4183i	− 1.08720i
$288$	− 10.0798i	− 0.593956i
$289$	−32.9750	−1.93971
$290$	0	0
$291$	−14.7936	−0.867214
$292$	− 19.1159i	− 1.11867i
$293$	15.7923i	0.922596i	0.887245	+	0.461298i	$0.152616\pi$
$293$	15.7923i	0.922596i	−0.887245	+	0.461298i	$0.847384\pi$
$294$	0.510014	0.0297446
$295$	0	0
$296$	6.33503	0.368216
$297$	6.32466i	0.366994i
$298$	3.03781i	0.175976i
$299$	−6.49089	−0.375378
$300$	0	0
$301$	−13.9112	−0.801831
$302$	3.54493i	0.203988i
$303$	− 25.8005i	− 1.48220i
$304$	0.947785	0.0543592
$305$	0	0
$306$	6.10363	0.348921
$307$	25.0861i	1.43174i	0.698234	+	0.715870i	$0.253970\pi$
$307$	25.0861i	1.43174i	−0.698234	+	0.715870i	$0.746030\pi$
$308$	10.1979i	0.581081i
$309$	24.5387	1.39596
$310$	0	0
$311$	18.2599	1.03542	0.517711	−	0.855555i	$-0.326784\pi$
$311$	18.2599	1.03542	0.517711	+	0.855555i	$0.326784\pi$
$312$	− 14.0117i	− 0.793258i
$313$	11.3712i	0.642738i	0.946954	+	0.321369i	$0.104143\pi$
$313$	11.3712i	0.642738i	−0.946954	+	0.321369i	$0.895857\pi$
$314$	1.43645	0.0810636
$315$	0	0
$316$	32.3880	1.82197
$317$	− 15.7315i	− 0.883570i	−0.897121	−	0.441785i	$-0.854345\pi$
$317$	− 15.7315i	− 0.883570i	0.897121	−	0.441785i	$-0.145655\pi$
$318$	6.86083i	0.384736i
$319$	7.61969	0.426621
$320$	0	0
$321$	31.5706	1.76210
$322$	0.569670i	0.0317465i
$323$	1.78372i	0.0992488i
$324$	−7.01386	−0.389659
$325$	0	0
$326$	2.95059	0.163418
$327$	− 3.95921i	− 0.218945i
$328$	5.23484i	0.289045i
$329$	−2.39984	−0.132307
$330$	0	0
$331$	12.5991	0.692507	0.346253	−	0.938141i	$-0.387454\pi$
$331$	12.5991	0.692507	0.346253	+	0.938141i	$0.387454\pi$
$332$	1.36505i	0.0749168i
$333$	33.7819i	1.85124i
$334$	−1.91118	−0.104575
$335$	0	0
$336$	28.5295	1.55641
$337$	16.5473i	0.901388i	0.892679	+	0.450694i	$0.148823\pi$
$337$	16.5473i	0.901388i	−0.892679	+	0.450694i	$0.851177\pi$
$338$	− 5.89121i	− 0.320440i
$339$	−4.39315	−0.238603
$340$	0	0
$341$	−6.55643	−0.355050
$342$	− 0.217852i	− 0.0117801i
$343$	− 17.0839i	− 0.922444i
$344$	3.95384	0.213177
$345$	0	0
$346$	4.71300	0.253372
$347$	− 8.43461i	− 0.452794i	−0.974035	−	0.226397i	$-0.927305\pi$
$347$	− 8.43461i	− 0.452794i	0.974035	−	0.226397i	$-0.0726946\pi$
$348$	− 21.7809i	− 1.16758i
$349$	13.2201	0.707658	0.353829	−	0.935310i	$-0.384879\pi$
$349$	13.2201	0.707658	0.353829	+	0.935310i	$0.384879\pi$
$350$	0	0
$351$	22.2163	1.18582
$352$	− 4.36263i	− 0.232529i
$353$	− 1.28663i	− 0.0684803i	−0.999414	−	0.0342401i	$-0.989099\pi$
$353$	− 1.28663i	− 0.0684803i	0.999414	−	0.0342401i	$-0.0109011\pi$
$354$	−5.64960	−0.300273
$355$	0	0
$356$	−25.8278	−1.36887
$357$	53.6922i	2.84169i
$358$	− 4.76371i	− 0.251770i
$359$	−14.7493	−0.778438	−0.389219	−	0.921145i	$-0.627255\pi$
$359$	−14.7493	−0.778438	−0.389219	+	0.921145i	$0.627255\pi$
$360$	0	0
$361$	−18.9363	−0.996649
$362$	0.102137i	0.00536818i
$363$	− 20.4517i	− 1.07344i
$364$	35.8217	1.87757
$365$	0	0
$366$	−0.551825	−0.0288443
$367$	25.9403i	1.35407i	0.735950	+	0.677036i	$0.236736\pi$
$367$	25.9403i	1.35407i	−0.735950	+	0.677036i	$0.763264\pi$
$368$	3.75630i	0.195811i
$369$	−27.9151	−1.45320
$370$	0	0
$371$	−35.4462	−1.84028
$372$	18.7415i	0.971704i
$373$	13.1177i	0.679209i	0.940568	+	0.339604i	$0.110293\pi$
$373$	13.1177i	0.679209i	−0.940568	+	0.339604i	$0.889707\pi$
$374$	2.64172	0.136600
$375$	0	0
$376$	0.682078	0.0351755
$377$	− 26.7653i	− 1.37848i
$378$	− 1.94980i	− 0.100287i
$379$	−9.63674	−0.495006	−0.247503	−	0.968887i	$-0.579610\pi$
$379$	−9.63674	−0.495006	−0.247503	+	0.968887i	$0.579610\pi$
$380$	0	0
$381$	−23.1672	−1.18689
$382$	0.508224i	0.0260030i
$383$	− 16.1595i	− 0.825711i	−0.910797	−	0.412856i	$-0.864531\pi$
$383$	− 16.1595i	− 0.825711i	0.910797	−	0.412856i	$-0.135469\pi$
$384$	−16.5668	−0.845419
$385$	0	0
$386$	0.843362	0.0429260
$387$	21.0841i	1.07176i
$388$	10.7493i	0.545712i
$389$	−27.6271	−1.40075	−0.700374	−	0.713776i	$-0.746983\pi$
$389$	−27.6271	−1.40075	−0.700374	+	0.713776i	$0.746983\pi$
$390$	0	0
$391$	−7.06930	−0.357510
$392$	− 0.748908i	− 0.0378256i
$393$	7.92080i	0.399551i
$394$	−1.33491	−0.0672516
$395$	0	0
$396$	15.4561	0.776698
$397$	− 10.7707i	− 0.540565i	−0.962781	−	0.270282i	$-0.912883\pi$
$397$	− 10.7707i	− 0.540565i	0.962781	−	0.270282i	$-0.0871170\pi$
$398$	2.00443i	0.100473i
$399$	1.91639	0.0959397
$400$	0	0
$401$	−31.5635	−1.57621	−0.788104	−	0.615542i	$-0.788937\pi$
$401$	−31.5635	−1.57621	−0.788104	+	0.615542i	$0.788937\pi$
$402$	1.84254i	0.0918977i
$403$	23.0304i	1.14723i
$404$	−18.7472	−0.932706
$405$	0	0
$406$	−2.34904	−0.116581
$407$	14.6212i	0.724745i
$408$	− 15.2603i	− 0.755499i
$409$	−15.6426	−0.773479	−0.386740	−	0.922189i	$-0.626399\pi$
$409$	−15.6426	−0.773479	−0.386740	+	0.922189i	$0.626399\pi$
$410$	0	0
$411$	−59.1453	−2.91742
$412$	− 17.8303i	− 0.878433i
$413$	− 29.1885i	− 1.43627i
$414$	0.863400	0.0424338
$415$	0	0
$416$	−15.3244	−0.751338
$417$	− 41.3391i	− 2.02439i
$418$	− 0.0942887i	− 0.00461181i
$419$	11.1799	0.546175	0.273088	−	0.961989i	$-0.411955\pi$
$419$	11.1799	0.546175	0.273088	+	0.961989i	$0.411955\pi$
$420$	0	0
$421$	0.661200	0.0322249	0.0161125	−	0.999870i	$-0.494871\pi$
$421$	0.661200	0.0322249	0.0161125	+	0.999870i	$0.494871\pi$
$422$	1.68566i	0.0820565i
$423$	3.63722i	0.176848i
$424$	10.0745	0.489260
$425$	0	0
$426$	0.464502	0.0225052
$427$	− 2.85098i	− 0.137969i
$428$	− 22.9398i	− 1.10884i
$429$	32.3389	1.56134
$430$	0	0
$431$	−15.2276	−0.733488	−0.366744	−	0.930322i	$-0.619527\pi$
$431$	−15.2276	−0.733488	−0.366744	+	0.930322i	$0.619527\pi$
$432$	− 12.8566i	− 0.618565i
$433$	− 22.2662i	− 1.07005i	−0.844838	−	0.535023i	$-0.820303\pi$
$433$	− 22.2662i	− 1.07005i	0.844838	−	0.535023i	$-0.179697\pi$
$434$	2.02125	0.0970233
$435$	0	0
$436$	−2.87684	−0.137775
$437$	0.252319i	0.0120701i
$438$	5.32020i	0.254209i
$439$	−21.1491	−1.00939	−0.504695	−	0.863298i	$-0.668395\pi$
$439$	−21.1491	−1.00939	−0.504695	+	0.863298i	$0.668395\pi$
$440$	0	0
$441$	3.99359	0.190171
$442$	− 9.27941i	− 0.441376i
$443$	− 9.94721i	− 0.472606i	−0.971679	−	0.236303i	$-0.924064\pi$
$443$	− 9.94721i	− 0.472606i	0.971679	−	0.236303i	$-0.0759358\pi$
$444$	41.7947	1.98349
$445$	0	0
$446$	−2.72824	−0.129186
$447$	40.5016i	1.91566i
$448$	− 19.8179i	− 0.936308i
$449$	8.32870	0.393056	0.196528	−	0.980498i	$-0.437033\pi$
$449$	8.32870	0.393056	0.196528	+	0.980498i	$0.437033\pi$
$450$	0	0
$451$	−12.0819	−0.568916
$452$	3.19215i	0.150146i
$453$	47.2627i	2.22060i
$454$	−1.16769	−0.0548024
$455$	0	0
$456$	−0.544675	−0.0255067
$457$	− 33.2599i	− 1.55583i	−0.628369	−	0.777916i	$-0.716277\pi$
$457$	− 33.2599i	− 1.55583i	0.628369	−	0.777916i	$-0.283723\pi$
$458$	− 4.51333i	− 0.210894i
$459$	24.1960	1.12937
$460$	0	0
$461$	14.8481	0.691544	0.345772	−	0.938318i	$-0.387617\pi$
$461$	14.8481	0.691544	0.345772	+	0.938318i	$0.387617\pi$
$462$	− 2.83821i	− 0.132046i
$463$	15.6422i	0.726956i	0.931603	+	0.363478i	$0.118411\pi$
$463$	15.6422i	0.726956i	−0.931603	+	0.363478i	$0.881589\pi$
$464$	−15.4891	−0.719065
$465$	0	0
$466$	−4.41314	−0.204435
$467$	− 5.26987i	− 0.243861i	−0.992539	−	0.121930i	$-0.961092\pi$
$467$	− 5.26987i	− 0.243861i	0.992539	−	0.121930i	$-0.0389084\pi$
$468$	− 54.2918i	− 2.50964i
$469$	−9.51944	−0.439567
$470$	0	0
$471$	19.1514	0.882452
$472$	8.29591i	0.381850i
$473$	9.12541i	0.419587i
$474$	−9.01398	−0.414026
$475$	0	0
$476$	39.0138	1.78819
$477$	53.7228i	2.45980i
$478$	5.40179i	0.247072i
$479$	−0.404422	−0.0184785	−0.00923927	−	0.999957i	$-0.502941\pi$
$479$	−0.404422	−0.0184785	−0.00923927	+	0.999957i	$0.502941\pi$
$480$	0	0
$481$	51.3590	2.34177
$482$	3.06506i	0.139610i
$483$	7.59513i	0.345590i
$484$	−14.8606	−0.675481
$485$	0	0
$486$	4.02853	0.182738
$487$	− 17.0409i	− 0.772198i	−0.922457	−	0.386099i	$-0.873822\pi$
$487$	− 17.0409i	− 0.772198i	0.922457	−	0.386099i	$-0.126178\pi$
$488$	0.810302i	0.0366807i
$489$	39.3388	1.77896
$490$	0	0
$491$	−9.42821	−0.425489	−0.212745	−	0.977108i	$-0.568240\pi$
$491$	−9.42821	−0.425489	−0.212745	+	0.977108i	$0.568240\pi$
$492$	34.5362i	1.55701i
$493$	− 29.1503i	− 1.31287i
$494$	−0.331203	−0.0149015
$495$	0	0
$496$	13.3278	0.598434
$497$	2.39984i	0.107647i
$498$	− 0.379910i	− 0.0170242i
$499$	25.7959	1.15478	0.577391	−	0.816468i	$-0.304071\pi$
$499$	25.7959	1.15478	0.577391	+	0.816468i	$0.304071\pi$
$500$	0	0
$501$	−25.4808	−1.13840
$502$	− 2.66992i	− 0.119165i
$503$	− 21.6111i	− 0.963589i	−0.876284	−	0.481795i	$-0.839985\pi$
$503$	− 21.6111i	− 0.963589i	0.876284	−	0.481795i	$-0.160015\pi$
$504$	−9.62926	−0.428921
$505$	0	0
$506$	0.373689	0.0166125
$507$	− 78.5445i	− 3.48828i
$508$	16.8338i	0.746877i
$509$	6.13182	0.271788	0.135894	−	0.990723i	$-0.456609\pi$
$509$	6.13182	0.271788	0.135894	+	0.990723i	$0.456609\pi$
$510$	0	0
$511$	−27.4866	−1.21594
$512$	14.8831i	0.657748i
$513$	− 0.863609i	− 0.0381293i
$514$	2.82006	0.124388
$515$	0	0
$516$	26.0850	1.14833
$517$	1.57423i	0.0692345i
$518$	− 4.50750i	− 0.198048i
$519$	62.8360	2.75819
$520$	0	0
$521$	22.4021	0.981454	0.490727	−	0.871313i	$-0.336731\pi$
$521$	22.4021	0.981454	0.490727	+	0.871313i	$0.336731\pi$
$522$	3.56024i	0.155827i
$523$	− 8.25468i	− 0.360952i	−0.983579	−	0.180476i	$-0.942236\pi$
$523$	− 8.25468i	− 0.360952i	0.983579	−	0.180476i	$-0.0577638\pi$
$524$	5.75540	0.251426
$525$	0	0
$526$	−4.40619	−0.192119
$527$	25.0827i	1.09262i
$528$	− 18.7146i	− 0.814449i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−44.2384	−1.91978
$532$	− 1.39249i	− 0.0603720i
$533$	42.4395i	1.83826i
$534$	7.18821	0.311064
$535$	0	0
$536$	2.70560	0.116864
$537$	− 63.5121i	− 2.74075i
$538$	− 2.07477i	− 0.0894498i
$539$	1.72847	0.0744505
$540$	0	0
$541$	24.9079	1.07087	0.535437	−	0.844575i	$-0.320147\pi$
$541$	24.9079	1.07087	0.535437	+	0.844575i	$0.320147\pi$
$542$	0.832950i	0.0357782i
$543$	1.36174i	0.0584376i
$544$	−16.6899	−0.715575
$545$	0	0
$546$	−9.96962	−0.426660
$547$	− 19.8049i	− 0.846795i	−0.905944	−	0.423398i	$-0.860837\pi$
$547$	− 19.8049i	− 0.846795i	0.905944	−	0.423398i	$-0.139163\pi$
$548$	42.9761i	1.83585i
$549$	−4.32098	−0.184415
$550$	0	0
$551$	−1.04044	−0.0443243
$552$	− 2.15868i	− 0.0918793i
$553$	− 46.5704i	− 1.98038i
$554$	1.94692	0.0827165
$555$	0	0
$556$	−30.0378	−1.27389
$557$	− 36.6380i	− 1.55240i	−0.630486	−	0.776201i	$-0.717144\pi$
$557$	− 36.6380i	− 1.55240i	0.630486	−	0.776201i	$-0.282856\pi$
$558$	− 3.06344i	− 0.129686i
$559$	32.0543	1.35575
$560$	0	0
$561$	35.2207	1.48702
$562$	3.42150i	0.144327i
$563$	− 19.0783i	− 0.804054i	−0.915628	−	0.402027i	$-0.868306\pi$
$563$	− 19.0783i	− 0.804054i	0.915628	−	0.402027i	$-0.131694\pi$
$564$	4.49993	0.189481
$565$	0	0
$566$	1.51949	0.0638687
$567$	10.0852i	0.423537i
$568$	− 0.682078i	− 0.0286194i
$569$	−4.50086	−0.188686	−0.0943429	−	0.995540i	$-0.530075\pi$
$569$	−4.50086	−0.188686	−0.0943429	+	0.995540i	$0.530075\pi$
$570$	0	0
$571$	1.25913	0.0526930	0.0263465	−	0.999653i	$-0.491613\pi$
$571$	1.25913	0.0526930	0.0263465	+	0.999653i	$0.491613\pi$
$572$	− 23.4981i	− 0.982503i
$573$	6.77589i	0.283067i
$574$	3.72469	0.155466
$575$	0	0
$576$	−30.0363	−1.25151
$577$	1.33814i	0.0557077i	0.999612	+	0.0278539i	$0.00886730\pi$
$577$	1.33814i	0.0557077i	−0.999612	+	0.0278539i	$0.991133\pi$
$578$	− 6.66845i	− 0.277371i
$579$	11.2441	0.467289
$580$	0	0
$581$	1.96279	0.0814304
$582$	− 2.99166i	− 0.124008i
$583$	23.2518i	0.962990i
$584$	7.81221	0.323272
$585$	0	0
$586$	−3.19363	−0.131928
$587$	− 22.4997i	− 0.928664i	−0.885661	−	0.464332i	$-0.846295\pi$
$587$	− 22.4997i	− 0.928664i	0.885661	−	0.464332i	$-0.153705\pi$
$588$	− 4.94083i	− 0.203757i
$589$	0.895256	0.0368884
$590$	0	0
$591$	−17.7976	−0.732096
$592$	− 29.7216i	− 1.22155i
$593$	30.8988i	1.26886i	0.772979	+	0.634432i	$0.218766\pi$
$593$	30.8988i	1.26886i	−0.772979	+	0.634432i	$0.781234\pi$
$594$	−1.27902	−0.0524788
$595$	0	0
$596$	29.4292	1.20547
$597$	26.7240i	1.09374i
$598$	− 1.31263i	− 0.0536776i
$599$	7.34413	0.300073	0.150036	−	0.988680i	$-0.452061\pi$
$599$	7.34413	0.300073	0.150036	+	0.988680i	$0.452061\pi$
$600$	0	0
$601$	−5.10253	−0.208136	−0.104068	−	0.994570i	$-0.533186\pi$
$601$	−5.10253	−0.208136	−0.104068	+	0.994570i	$0.533186\pi$
$602$	− 2.81323i	− 0.114659i
$603$	14.4278i	0.587544i
$604$	34.3420	1.39735
$605$	0	0
$606$	5.21757	0.211949
$607$	21.4965i	0.872518i	0.899821	+	0.436259i	$0.143697\pi$
$607$	21.4965i	0.872518i	−0.899821	+	0.436259i	$0.856303\pi$
$608$	0.595701i	0.0241589i
$609$	−31.3186	−1.26909
$610$	0	0
$611$	5.52970	0.223708
$612$	− 59.1298i	− 2.39018i
$613$	44.4369i	1.79479i	0.441227	+	0.897396i	$0.354543\pi$
$613$	44.4369i	1.79479i	−0.441227	+	0.897396i	$0.645457\pi$
$614$	−5.07310	−0.204733
$615$	0	0
$616$	−4.16765	−0.167919
$617$	25.0662i	1.00913i	0.863375	+	0.504564i	$0.168347\pi$
$617$	25.0662i	1.00913i	−0.863375	+	0.504564i	$0.831653\pi$
$618$	4.96238i	0.199616i
$619$	11.6002	0.466252	0.233126	−	0.972447i	$-0.425105\pi$
$619$	11.6002	0.466252	0.233126	+	0.972447i	$0.425105\pi$
$620$	0	0
$621$	3.42269	0.137348
$622$	3.69265i	0.148062i
$623$	37.1377i	1.48789i
$624$	−65.7378	−2.63162
$625$	0	0
$626$	−2.29957	−0.0919091
$627$	− 1.25710i	− 0.0502039i
$628$	− 13.9158i	− 0.555301i
$629$	55.9357	2.23030
$630$	0	0
$631$	−9.17437	−0.365226	−0.182613	−	0.983185i	$-0.558456\pi$
$631$	−9.17437	−0.365226	−0.182613	+	0.983185i	$0.558456\pi$
$632$	13.2362i	0.526507i
$633$	22.4740i	0.893261i
$634$	3.18134	0.126347
$635$	0	0
$636$	66.4652	2.63552
$637$	− 6.07150i	− 0.240562i
$638$	1.54091i	0.0610052i
$639$	3.63722	0.143886
$640$	0	0
$641$	3.98400	0.157359	0.0786793	−	0.996900i	$-0.474930\pi$
$641$	3.98400	0.157359	0.0786793	+	0.996900i	$0.474930\pi$
$642$	6.38443i	0.251973i
$643$	− 13.3005i	− 0.524520i	−0.964997	−	0.262260i	$-0.915532\pi$
$643$	− 13.3005i	− 0.524520i	0.964997	−	0.262260i	$-0.0844678\pi$
$644$	5.51876	0.217470
$645$	0	0
$646$	−0.360717	−0.0141922
$647$	3.02871i	0.119071i	0.998226	+	0.0595354i	$0.0189619\pi$
$647$	3.02871i	0.119071i	−0.998226	+	0.0595354i	$0.981038\pi$
$648$	− 2.86639i	− 0.112603i
$649$	−19.1469	−0.751580
$650$	0	0
$651$	26.9483	1.05619
$652$	− 28.5843i	− 1.11945i
$653$	− 24.8569i	− 0.972727i	−0.873757	−	0.486363i	$-0.838323\pi$
$653$	− 24.8569i	− 0.972727i	0.873757	−	0.486363i	$-0.161677\pi$
$654$	0.800660	0.0313083
$655$	0	0
$656$	24.5599	0.958903
$657$	41.6591i	1.62528i
$658$	− 0.485312i	− 0.0189194i
$659$	−30.2901	−1.17994	−0.589968	−	0.807427i	$-0.700860\pi$
$659$	−30.2901	−1.17994	−0.589968	+	0.807427i	$0.700860\pi$
$660$	0	0
$661$	−41.7574	−1.62418	−0.812088	−	0.583535i	$-0.801669\pi$
$661$	−41.7574	−1.62418	−0.812088	+	0.583535i	$0.801669\pi$
$662$	2.54787i	0.0990259i
$663$	− 123.718i	− 4.80479i
$664$	−0.557862	−0.0216493
$665$	0	0
$666$	−6.83163	−0.264720
$667$	− 4.12351i	− 0.159663i
$668$	18.5148i	0.716360i
$669$	−36.3742	−1.40631
$670$	0	0
$671$	−1.87017	−0.0721971
$672$	17.9314i	0.691717i
$673$	18.1446i	0.699424i	0.936857	+	0.349712i	$0.113721\pi$
$673$	18.1446i	0.699424i	−0.936857	+	0.349712i	$0.886279\pi$
$674$	−3.34631	−0.128895
$675$	0	0
$676$	−57.0719	−2.19507
$677$	16.4647i	0.632790i	0.948627	+	0.316395i	$0.102473\pi$
$677$	16.4647i	0.632790i	−0.948627	+	0.316395i	$0.897527\pi$
$678$	− 0.888415i	− 0.0341194i
$679$	15.4563	0.593159
$680$	0	0
$681$	−15.5682	−0.596575
$682$	− 1.32589i	− 0.0507709i
$683$	− 46.7050i	− 1.78712i	−0.448948	−	0.893558i	$-0.648201\pi$
$683$	− 46.7050i	− 1.78712i	0.448948	−	0.893558i	$-0.351799\pi$
$684$	−2.11047	−0.0806959
$685$	0	0
$686$	3.45483	0.131906
$687$	− 60.1740i	− 2.29578i
$688$	− 18.5499i	− 0.707209i
$689$	81.6752	3.11158
$690$	0	0
$691$	−11.5366	−0.438872	−0.219436	−	0.975627i	$-0.570422\pi$
$691$	−11.5366	−0.438872	−0.219436	+	0.975627i	$0.570422\pi$
$692$	− 45.6578i	− 1.73565i
$693$	− 22.2242i	− 0.844228i
$694$	1.70571	0.0647478
$695$	0	0
$696$	8.90132	0.337404
$697$	46.2214i	1.75076i
$698$	2.67347i	0.101192i
$699$	−58.8381	−2.22546
$700$	0	0
$701$	31.8694	1.20369	0.601845	−	0.798613i	$-0.294432\pi$
$701$	31.8694	1.20369	0.601845	+	0.798613i	$0.294432\pi$
$702$	4.49274i	0.169567i
$703$	− 1.99647i	− 0.0752982i
$704$	−13.0000	−0.489957
$705$	0	0
$706$	0.260191	0.00979243
$707$	26.9564i	1.01380i
$708$	54.7313i	2.05693i
$709$	−5.06884	−0.190364	−0.0951822	−	0.995460i	$-0.530343\pi$
$709$	−5.06884	−0.190364	−0.0951822	+	0.995460i	$0.530343\pi$
$710$	0	0
$711$	−70.5827	−2.64706
$712$	− 10.5552i	− 0.395573i
$713$	3.54811i	0.132878i
$714$	−10.8580	−0.406352
$715$	0	0
$716$	−46.1491	−1.72467
$717$	72.0193i	2.68961i
$718$	− 2.98271i	− 0.111314i
$719$	−6.53346	−0.243657	−0.121828	−	0.992551i	$-0.538876\pi$
$719$	−6.53346	−0.243657	−0.121828	+	0.992551i	$0.538876\pi$
$720$	0	0
$721$	−25.6380	−0.954809
$722$	− 3.82944i	− 0.142517i
$723$	40.8649i	1.51978i
$724$	0.989462	0.0367731
$725$	0	0
$726$	4.13589	0.153497
$727$	46.2193i	1.71418i	0.515169	+	0.857089i	$0.327729\pi$
$727$	46.2193i	1.71418i	−0.515169	+	0.857089i	$0.672271\pi$
$728$	14.6395i	0.542574i
$729$	42.9699	1.59148
$730$	0	0
$731$	34.9107	1.29122
$732$	5.34588i	0.197589i
$733$	− 20.2561i	− 0.748177i	−0.927393	−	0.374089i	$-0.877956\pi$
$733$	− 20.2561i	− 0.748177i	0.927393	−	0.374089i	$-0.122044\pi$
$734$	−5.24583	−0.193627
$735$	0	0
$736$	−2.36090	−0.0870240
$737$	6.24450i	0.230019i
$738$	− 5.64519i	− 0.207802i
$739$	−0.454209	−0.0167083	−0.00835417	−	0.999965i	$-0.502659\pi$
$739$	−0.454209	−0.0167083	−0.00835417	+	0.999965i	$0.502659\pi$
$740$	0	0
$741$	−4.41576	−0.162217
$742$	− 7.16820i	− 0.263153i
$743$	− 17.9525i	− 0.658614i	−0.944223	−	0.329307i	$-0.893185\pi$
$743$	− 17.9525i	− 0.658614i	0.944223	−	0.329307i	$-0.106815\pi$
$744$	−7.65922	−0.280801
$745$	0	0
$746$	−2.65276	−0.0971243
$747$	− 2.97483i	− 0.108843i
$748$	− 25.5920i	− 0.935736i
$749$	−32.9849	−1.20524
$750$	0	0
$751$	14.8866	0.543219	0.271610	−	0.962408i	$-0.412444\pi$
$751$	14.8866	0.543219	0.271610	+	0.962408i	$0.412444\pi$
$752$	− 3.20005i	− 0.116694i
$753$	− 35.5967i	− 1.29722i
$754$	5.41266	0.197118
$755$	0	0
$756$	−18.8890	−0.686986
$757$	31.7028i	1.15226i	0.817358	+	0.576129i	$0.195438\pi$
$757$	31.7028i	1.15226i	−0.817358	+	0.576129i	$0.804562\pi$
$758$	− 1.94881i	− 0.0707840i
$759$	4.98220	0.180842
$760$	0	0
$761$	−6.80194	−0.246570	−0.123285	−	0.992371i	$-0.539343\pi$
$761$	−6.80194	−0.246570	−0.123285	+	0.992371i	$0.539343\pi$
$762$	− 4.68505i	− 0.169721i
$763$	4.13658i	0.149754i
$764$	4.92349	0.178126
$765$	0	0
$766$	3.26789	0.118074
$767$	67.2561i	2.42848i
$768$	34.5860i	1.24802i
$769$	−19.2067	−0.692612	−0.346306	−	0.938122i	$-0.612564\pi$
$769$	−19.2067	−0.692612	−0.346306	+	0.938122i	$0.612564\pi$
$770$	0	0
$771$	37.5985	1.35408
$772$	− 8.17019i	− 0.294051i
$773$	27.6037i	0.992835i	0.868084	+	0.496418i	$0.165351\pi$
$773$	27.6037i	0.992835i	−0.868084	+	0.496418i	$0.834649\pi$
$774$	−4.26377	−0.153258
$775$	0	0
$776$	−4.39297	−0.157699
$777$	− 60.0962i	− 2.15594i
$778$	− 5.58694i	− 0.200302i
$779$	1.64974	0.0591082
$780$	0	0
$781$	1.57423	0.0563303
$782$	− 1.42961i	− 0.0511226i
$783$	14.1135i	0.504375i
$784$	−3.51360	−0.125486
$785$	0	0
$786$	−1.60180	−0.0571344
$787$	35.6271i	1.26997i	0.772525	+	0.634984i	$0.218993\pi$
$787$	35.6271i	1.26997i	−0.772525	+	0.634984i	$0.781007\pi$
$788$	12.9321i	0.460686i
$789$	−58.7455	−2.09139
$790$	0	0
$791$	4.58997	0.163200
$792$	6.31653i	0.224448i
$793$	6.56923i	0.233280i
$794$	2.17812	0.0772987
$795$	0	0
$796$	19.4182	0.688258
$797$	− 9.85579i	− 0.349110i	−0.984647	−	0.174555i	$-0.944151\pi$
$797$	− 9.85579i	− 0.349110i	0.984647	−	0.174555i	$-0.0558487\pi$
$798$	0.387547i	0.0137190i
$799$	6.02246	0.213059
$800$	0	0
$801$	56.2863	1.98878
$802$	− 6.38301i	− 0.225392i
$803$	18.0305i	0.636282i
$804$	17.8499	0.629517
$805$	0	0
$806$	−4.65737	−0.164049
$807$	− 27.6619i	− 0.973745i
$808$	− 7.66151i	− 0.269531i
$809$	−35.4171	−1.24520	−0.622599	−	0.782541i	$-0.713923\pi$
$809$	−35.4171	−1.24520	−0.622599	+	0.782541i	$0.713923\pi$
$810$	0	0
$811$	−23.6023	−0.828790	−0.414395	−	0.910097i	$-0.636007\pi$
$811$	−23.6023	−0.828790	−0.414395	+	0.910097i	$0.636007\pi$
$812$	22.7567i	0.798603i
$813$	11.1053i	0.389480i
$814$	−2.95680	−0.103636
$815$	0	0
$816$	−71.5958	−2.50635
$817$	− 1.24604i	− 0.0435934i
$818$	− 3.16337i	− 0.110605i
$819$	−78.0657	−2.72784
$820$	0	0
$821$	−21.1904	−0.739548	−0.369774	−	0.929122i	$-0.620565\pi$
$821$	−21.1904	−0.739548	−0.369774	+	0.929122i	$0.620565\pi$
$822$	− 11.9608i	− 0.417180i
$823$	28.8463i	1.00552i	0.864427	+	0.502759i	$0.167682\pi$
$823$	28.8463i	1.00552i	−0.864427	+	0.502759i	$0.832318\pi$
$824$	7.28679	0.253847
$825$	0	0
$826$	5.90271	0.205381
$827$	− 7.85313i	− 0.273080i	−0.990635	−	0.136540i	$-0.956402\pi$
$827$	− 7.85313i	− 0.273080i	0.990635	−	0.136540i	$-0.0435982\pi$
$828$	− 8.36430i	− 0.290680i
$829$	7.58148	0.263316	0.131658	−	0.991295i	$-0.457970\pi$
$829$	7.58148	0.263316	0.131658	+	0.991295i	$0.457970\pi$
$830$	0	0
$831$	25.9572	0.900446
$832$	45.6644i	1.58313i
$833$	− 6.61254i	− 0.229111i
$834$	8.35990	0.289480
$835$	0	0
$836$	−0.913435	−0.0315918
$837$	− 12.1441i	− 0.419760i
$838$	2.26089i	0.0781010i
$839$	−33.2424	−1.14765	−0.573827	−	0.818977i	$-0.694542\pi$
$839$	−33.2424	−1.14765	−0.573827	+	0.818977i	$0.694542\pi$
$840$	0	0
$841$	−11.9967	−0.413678
$842$	0.133713i	0.00460804i
$843$	45.6171i	1.57114i
$844$	16.3300	0.562103
$845$	0	0
$846$	−0.735545	−0.0252886
$847$	21.3679i	0.734211i
$848$	− 47.2657i	− 1.62311i
$849$	20.2585	0.695271
$850$	0	0
$851$	7.91248	0.271236
$852$	− 4.49993i	− 0.154165i
$853$	21.1798i	0.725183i	0.931948	+	0.362591i	$0.118108\pi$
$853$	21.1798i	0.725183i	−0.931948	+	0.362591i	$0.881892\pi$
$854$	0.576546	0.0197290
$855$	0	0
$856$	9.37493	0.320429
$857$	13.2340i	0.452066i	0.974120	+	0.226033i	$0.0725758\pi$
$857$	13.2340i	0.452066i	−0.974120	+	0.226033i	$0.927424\pi$
$858$	6.53980i	0.223265i
$859$	31.6863	1.08112	0.540561	−	0.841305i	$-0.318212\pi$
$859$	31.6863	1.08112	0.540561	+	0.841305i	$0.318212\pi$
$860$	0	0
$861$	49.6594	1.69239
$862$	− 3.07944i	− 0.104886i
$863$	− 1.12760i	− 0.0383840i	−0.999816	−	0.0191920i	$-0.993891\pi$
$863$	− 1.12760i	− 0.0383840i	0.999816	−	0.0191920i	$-0.00610938\pi$
$864$	8.08063	0.274909
$865$	0	0
$866$	4.50283	0.153013
$867$	− 88.9070i	− 3.01944i
$868$	− 19.5812i	− 0.664628i
$869$	−30.5490	−1.03630
$870$	0	0
$871$	21.9347	0.743228
$872$	− 1.17569i	− 0.0398140i
$873$	− 23.4258i	− 0.792843i
$874$	−0.0510258	−0.00172597
$875$	0	0
$876$	51.5402	1.74138
$877$	30.9499i	1.04510i	0.852608	+	0.522552i	$0.175020\pi$
$877$	30.9499i	1.04510i	−0.852608	+	0.522552i	$0.824980\pi$
$878$	− 4.27692i	− 0.144339i
$879$	−42.5791	−1.43616
$880$	0	0
$881$	45.9333	1.54753	0.773765	−	0.633472i	$-0.218371\pi$
$881$	45.9333	1.54753	0.773765	+	0.633472i	$0.218371\pi$
$882$	0.807614i	0.0271938i
$883$	− 14.0404i	− 0.472496i	−0.971693	−	0.236248i	$-0.924082\pi$
$883$	− 14.0404i	− 0.472496i	0.971693	−	0.236248i	$-0.0759177\pi$
$884$	−89.8955	−3.02351
$885$	0	0
$886$	2.01160	0.0675809
$887$	− 33.4833i	− 1.12426i	−0.827049	−	0.562129i	$-0.809982\pi$
$887$	− 33.4833i	− 1.12426i	0.827049	−	0.562129i	$-0.190018\pi$
$888$	17.0805i	0.573183i
$889$	24.2051	0.811814
$890$	0	0
$891$	6.61560	0.221631
$892$	26.4302i	0.884947i
$893$	− 0.214955i	− 0.00719319i
$894$	−8.19053	−0.273932
$895$	0	0
$896$	17.3090	0.578252
$897$	− 17.5007i	− 0.584331i
$898$	1.68429i	0.0562055i
$899$	−14.6307	−0.487960
$900$	0	0
$901$	88.9534	2.96347
$902$	− 2.44330i	− 0.0813529i
$903$	− 37.5074i	− 1.24817i
$904$	−1.30455	−0.0433888
$905$	0	0
$906$	−9.55780	−0.317537
$907$	22.0732i	0.732929i	0.930432	+	0.366464i	$0.119432\pi$
$907$	22.0732i	0.732929i	−0.930432	+	0.366464i	$0.880568\pi$
$908$	11.3122i	0.375407i
$909$	40.8555	1.35509
$910$	0	0
$911$	−29.9624	−0.992699	−0.496349	−	0.868123i	$-0.665327\pi$
$911$	−29.9624	−0.992699	−0.496349	+	0.868123i	$0.665327\pi$
$912$	2.55541i	0.0846181i
$913$	− 1.28754i	− 0.0426113i
$914$	6.72605	0.222478
$915$	0	0
$916$	−43.7235	−1.44467
$917$	− 8.27565i	− 0.273286i
$918$	4.89309i	0.161496i
$919$	−27.8980	−0.920269	−0.460134	−	0.887849i	$-0.652199\pi$
$919$	−27.8980	−0.920269	−0.460134	+	0.887849i	$0.652199\pi$
$920$	0	0
$921$	−67.6370	−2.22871
$922$	3.00269i	0.0988883i
$923$	− 5.52970i	− 0.182012i
$924$	−27.4956	−0.904538
$925$	0	0
$926$	−3.16328	−0.103952
$927$	38.8573i	1.27624i
$928$	− 9.73521i	− 0.319574i
$929$	25.6063	0.840115	0.420057	−	0.907498i	$-0.362010\pi$
$929$	25.6063	0.840115	0.420057	+	0.907498i	$0.362010\pi$
$930$	0	0
$931$	−0.236016	−0.00773512
$932$	42.7529i	1.40042i
$933$	49.2322i	1.61179i
$934$	1.06571	0.0348711
$935$	0	0
$936$	22.1877	0.725229
$937$	− 23.1946i	− 0.757736i	−0.925451	−	0.378868i	$-0.876313\pi$
$937$	− 23.1946i	− 0.757736i	0.925451	−	0.378868i	$-0.123687\pi$
$938$	− 1.92509i	− 0.0628564i
$939$	−30.6589	−1.00052
$940$	0	0
$941$	8.04530	0.262269	0.131135	−	0.991365i	$-0.458138\pi$
$941$	8.04530	0.262269	0.131135	+	0.991365i	$0.458138\pi$
$942$	3.87294i	0.126187i
$943$	6.53833i	0.212917i
$944$	38.9213	1.26678
$945$	0	0
$946$	−1.84541	−0.0599993
$947$	− 42.2271i	− 1.37220i	−0.727509	−	0.686098i	$-0.759322\pi$
$947$	− 42.2271i	− 1.37220i	0.727509	−	0.686098i	$-0.240678\pi$
$948$	87.3242i	2.83616i
$949$	63.3347	2.05593
$950$	0	0
$951$	42.4152	1.37541
$952$	15.9440i	0.516748i
$953$	− 13.3175i	− 0.431396i	−0.976460	−	0.215698i	$-0.930797\pi$
$953$	− 13.3175i	− 0.431396i	0.976460	−	0.215698i	$-0.0692027\pi$
$954$	−10.8642	−0.351742
$955$	0	0
$956$	52.3306	1.69249
$957$	20.5442i	0.664098i
$958$	− 0.0817852i	− 0.00264236i
$959$	61.7950	1.99546
$960$	0	0
$961$	−18.4109	−0.593901
$962$	10.3862i	0.334864i
$963$	49.9924i	1.61098i
$964$	29.6932	0.956355
$965$	0	0
$966$	−1.53594	−0.0494181
$967$	− 42.9969i	− 1.38269i	−0.722526	−	0.691344i	$-0.757019\pi$
$967$	− 42.9969i	− 1.38269i	0.722526	−	0.691344i	$-0.242981\pi$
$968$	− 6.07316i	− 0.195199i
$969$	−4.80925	−0.154495
$970$	0	0
$971$	−9.01250	−0.289225	−0.144612	−	0.989488i	$-0.546194\pi$
$971$	−9.01250	−0.289225	−0.144612	+	0.989488i	$0.546194\pi$
$972$	− 39.0269i	− 1.25179i
$973$	43.1911i	1.38464i
$974$	3.44614	0.110421
$975$	0	0
$976$	3.80164	0.121687
$977$	− 41.1074i	− 1.31514i	−0.753392	−	0.657571i	$-0.771584\pi$
$977$	− 41.1074i	− 1.31514i	0.753392	−	0.657571i	$-0.228416\pi$
$978$	7.95537i	0.254385i
$979$	24.3613	0.778591
$980$	0	0
$981$	6.26945	0.200168
$982$	− 1.90664i	− 0.0608434i
$983$	− 59.8861i	− 1.91007i	−0.296493	−	0.955035i	$-0.595817\pi$
$983$	− 59.8861i	− 1.91007i	0.296493	−	0.955035i	$-0.404183\pi$
$984$	−14.1141	−0.449942
$985$	0	0
$986$	5.89499	0.187735
$987$	− 6.47042i	− 0.205956i
$988$	3.20857i	0.102078i
$989$	4.93835	0.157030
$990$	0	0
$991$	20.8275	0.661607	0.330804	−	0.943700i	$-0.392680\pi$
$991$	20.8275	0.661607	0.330804	+	0.943700i	$0.392680\pi$
$992$	8.37674i	0.265962i
$993$	33.9695i	1.07799i
$994$	−0.485312	−0.0153932
$995$	0	0
$996$	−3.68043	−0.116619
$997$	12.7634i	0.404222i	0.979363	+	0.202111i	$0.0647801\pi$
$997$	12.7634i	0.404222i	−0.979363	+	0.202111i	$0.935220\pi$
$998$	5.21662i	0.165129i
$999$	−27.0819	−0.856834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.2.b.f.24.8		14
5.2	odd	4		575.2.a.l.1.4	yes	7
5.3	odd	4		575.2.a.k.1.4	✓	7
5.4	even	2	inner	575.2.b.f.24.7		14
15.2	even	4		5175.2.a.cb.1.4		7
15.8	even	4		5175.2.a.cg.1.4		7
20.3	even	4		9200.2.a.da.1.6		7
20.7	even	4		9200.2.a.db.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.2.a.k.1.4	✓	7	5.3	odd	4
575.2.a.l.1.4	yes	7	5.2	odd	4
575.2.b.f.24.7		14	5.4	even	2	inner
575.2.b.f.24.8		14	1.1	even	1	trivial
5175.2.a.cb.1.4		7	15.2	even	4
5175.2.a.cg.1.4		7	15.8	even	4
9200.2.a.da.1.6		7	20.3	even	4
9200.2.a.db.1.2		7	20.7	even	4

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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.202227i q^{2} +2.69619i q^{3} +1.95910 q^{4} -0.545243 q^{6} -2.81698i q^{7} +0.800639i q^{8} -4.26945 q^{9} -1.84786 q^{11} +5.28212i q^{12} +6.49089i q^{13} +0.569670 q^{14} +3.75630 q^{16} +7.06930i q^{17} -0.863400i q^{18} +0.252319 q^{19} +7.59513 q^{21} -0.373689i q^{22} +1.00000i q^{23} -2.15868 q^{24} -1.31263 q^{26} -3.42269i q^{27} -5.51876i q^{28} -4.12351 q^{29} +3.54811 q^{31} +2.36090i q^{32} -4.98220i q^{33} -1.42961 q^{34} -8.36430 q^{36} -7.91248i q^{37} +0.0510258i q^{38} -17.5007 q^{39} +6.53833 q^{41} +1.53594i q^{42} -4.93835i q^{43} -3.62016 q^{44} -0.202227 q^{46} -0.851917i q^{47} +10.1277i q^{48} -0.935388 q^{49} -19.0602 q^{51} +12.7163i q^{52} -12.5831i q^{53} +0.692161 q^{54} +2.25538 q^{56} +0.680301i q^{57} -0.833886i q^{58} +10.3616 q^{59} +1.01207 q^{61} +0.717524i q^{62} +12.0270i q^{63} +7.03516 q^{64} +1.00754 q^{66} -3.37930i q^{67} +13.8495i q^{68} -2.69619 q^{69} -0.851917 q^{71} -3.41829i q^{72} -9.75748i q^{73} +1.60012 q^{74} +0.494319 q^{76} +5.20540i q^{77} -3.53911i q^{78} +16.5320 q^{79} -3.58013 q^{81} +1.32223i q^{82} +0.696772i q^{83} +14.8796 q^{84} +0.998669 q^{86} -11.1178i q^{87} -1.47947i q^{88} -13.1835 q^{89} +18.2847 q^{91} +1.95910i q^{92} +9.56638i q^{93} +0.172281 q^{94} -6.36545 q^{96} +5.48684i q^{97} -0.189161i q^{98} +7.88937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 22 q^{4} + 10 q^{6} - 30 q^{9} - 2 q^{11} - 14 q^{14} + 14 q^{16} - 30 q^{19} + 4 q^{21} - 36 q^{24} - 40 q^{26} - 6 q^{29} + 28 q^{31} - 40 q^{34} + 16 q^{39} + 38 q^{41} + 6 q^{44} + 2 q^{46} - 80 q^{49}+ \cdots + 106 q^{99}+O(q^{100}) \) 14 * q - 22 * q^4 + 10 * q^6 - 30 * q^9 - 2 * q^11 - 14 * q^14 + 14 * q^16 - 30 * q^19 + 4 * q^21 - 36 * q^24 - 40 * q^26 - 6 * q^29 + 28 * q^31 - 40 * q^34 + 16 * q^39 + 38 * q^41 + 6 * q^44 + 2 * q^46 - 80 * q^49 + 4 * q^51 + 2 * q^54 - 18 * q^56 + 32 * q^59 + 80 * q^61 + 8 * q^64 - 108 * q^66 - 28 * q^71 + 36 * q^74 + 70 * q^76 + 2 * q^79 + 94 * q^81 + 120 * q^84 - 70 * q^86 - 32 * q^89 + 50 * q^91 - 14 * q^94 - 38 * q^96 + 106 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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