LMFDB - Embedded newform 475.2.a.f.1.3

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			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.19394 q^{6} -3.35026 q^{7} -2.67513 q^{8} -2.35026 q^{9} +0.962389 q^{11} -0.156325 q^{12} -6.15633 q^{13} -4.96239 q^{14} -4.35026 q^{16} +6.31265 q^{17} -3.48119 q^{18} -1.00000 q^{19} +2.70052 q^{21} +1.42548 q^{22} +4.96239 q^{23} +2.15633 q^{24} -9.11871 q^{26} +4.31265 q^{27} -0.649738 q^{28} -3.61213 q^{29} -5.92478 q^{31} -1.09332 q^{32} -0.775746 q^{33} +9.35026 q^{34} -0.455802 q^{36} -10.1563 q^{37} -1.48119 q^{38} +4.96239 q^{39} +6.31265 q^{41} +4.00000 q^{42} +4.12601 q^{43} +0.186642 q^{44} +7.35026 q^{46} -3.35026 q^{47} +3.50659 q^{48} +4.22425 q^{49} -5.08840 q^{51} -1.19394 q^{52} -1.84367 q^{53} +6.38787 q^{54} +8.96239 q^{56} +0.806063 q^{57} -5.35026 q^{58} -6.38787 q^{59} -11.2750 q^{61} -8.77575 q^{62} +7.87399 q^{63} +7.08110 q^{64} -1.14903 q^{66} +6.73084 q^{67} +1.22425 q^{68} -4.00000 q^{69} -0.775746 q^{71} +6.28726 q^{72} -0.387873 q^{73} -15.0435 q^{74} -0.193937 q^{76} -3.22425 q^{77} +7.35026 q^{78} -0.836381 q^{79} +3.57452 q^{81} +9.35026 q^{82} +7.03761 q^{83} +0.523730 q^{84} +6.11142 q^{86} +2.91160 q^{87} -2.57452 q^{88} +7.08840 q^{89} +20.6253 q^{91} +0.962389 q^{92} +4.77575 q^{93} -4.96239 q^{94} +0.881286 q^{96} -10.9927 q^{97} +6.25694 q^{98} -2.26187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9} - 8 q^{11} + 10 q^{12} - 8 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} - 5 q^{18} - 3 q^{19} - 12 q^{21} + 16 q^{22} + 4 q^{23} - 4 q^{24} - 6 q^{26}+ \cdots - 16 q^{99}+O(q^{100})$ 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	−0.806063	−0.465381	−0.232690	−	0.972551i	$-0.574753\pi$
$3$	−0.806063	−0.465381	−0.232690	+	0.972551i	$0.574753\pi$
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	−1.19394	−0.487423
$7$	−3.35026	−1.26628	−0.633140	−	0.774037i	$-0.718234\pi$
$7$	−3.35026	−1.26628	−0.633140	+	0.774037i	$0.718234\pi$
$8$	−2.67513	−0.945802
$9$	−2.35026	−0.783421
$10$	0	0
$11$	0.962389	0.290171	0.145086	−	0.989419i	$-0.453654\pi$
$11$	0.962389	0.290171	0.145086	+	0.989419i	$0.453654\pi$
$12$	−0.156325	−0.0451272
$13$	−6.15633	−1.70746	−0.853729	−	0.520718i	$-0.825664\pi$
$13$	−6.15633	−1.70746	−0.853729	+	0.520718i	$0.825664\pi$
$14$	−4.96239	−1.32625
$15$	0	0
$16$	−4.35026	−1.08757
$17$	6.31265	1.53104	0.765521	−	0.643411i	$-0.222481\pi$
$17$	6.31265	1.53104	0.765521	+	0.643411i	$0.222481\pi$
$18$	−3.48119	−0.820525
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.70052	0.589303
$22$	1.42548	0.303914
$23$	4.96239	1.03473	0.517365	−	0.855765i	$-0.326913\pi$
$23$	4.96239	1.03473	0.517365	+	0.855765i	$0.326913\pi$
$24$	2.15633	0.440158
$25$	0	0
$26$	−9.11871	−1.78833
$27$	4.31265	0.829970
$28$	−0.649738	−0.122789
$29$	−3.61213	−0.670755	−0.335378	−	0.942084i	$-0.608864\pi$
$29$	−3.61213	−0.670755	−0.335378	+	0.942084i	$0.608864\pi$
$30$	0	0
$31$	−5.92478	−1.06412	−0.532061	−	0.846706i	$-0.678582\pi$
$31$	−5.92478	−1.06412	−0.532061	+	0.846706i	$0.678582\pi$
$32$	−1.09332	−0.193274
$33$	−0.775746	−0.135040
$34$	9.35026	1.60356
$35$	0	0
$36$	−0.455802	−0.0759669
$37$	−10.1563	−1.66969	−0.834845	−	0.550485i	$-0.814443\pi$
$37$	−10.1563	−1.66969	−0.834845	+	0.550485i	$0.814443\pi$
$38$	−1.48119	−0.240281
$39$	4.96239	0.794618
$40$	0	0
$41$	6.31265	0.985870	0.492935	−	0.870066i	$-0.335924\pi$
$41$	6.31265	0.985870	0.492935	+	0.870066i	$0.335924\pi$
$42$	4.00000	0.617213
$43$	4.12601	0.629210	0.314605	−	0.949223i	$-0.398128\pi$
$43$	4.12601	0.629210	0.314605	+	0.949223i	$0.398128\pi$
$44$	0.186642	0.0281374
$45$	0	0
$46$	7.35026	1.08374
$47$	−3.35026	−0.488686	−0.244343	−	0.969689i	$-0.578572\pi$
$47$	−3.35026	−0.488686	−0.244343	+	0.969689i	$0.578572\pi$
$48$	3.50659	0.506132
$49$	4.22425	0.603465
$50$	0	0
$51$	−5.08840	−0.712518
$52$	−1.19394	−0.165569
$53$	−1.84367	−0.253248	−0.126624	−	0.991951i	$-0.540414\pi$
$53$	−1.84367	−0.253248	−0.126624	+	0.991951i	$0.540414\pi$
$54$	6.38787	0.869279
$55$	0	0
$56$	8.96239	1.19765
$57$	0.806063	0.106766
$58$	−5.35026	−0.702524
$59$	−6.38787	−0.831630	−0.415815	−	0.909449i	$-0.636504\pi$
$59$	−6.38787	−0.831630	−0.415815	+	0.909449i	$0.636504\pi$
$60$	0	0
$61$	−11.2750	−1.44362	−0.721810	−	0.692091i	$-0.756690\pi$
$61$	−11.2750	−1.44362	−0.721810	+	0.692091i	$0.756690\pi$
$62$	−8.77575	−1.11452
$63$	7.87399	0.992030
$64$	7.08110	0.885138
$65$	0	0
$66$	−1.14903	−0.141436
$67$	6.73084	0.822303	0.411152	−	0.911567i	$-0.365127\pi$
$67$	6.73084	0.822303	0.411152	+	0.911567i	$0.365127\pi$
$68$	1.22425	0.148463
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−0.775746	−0.0920641	−0.0460321	−	0.998940i	$-0.514658\pi$
$71$	−0.775746	−0.0920641	−0.0460321	+	0.998940i	$0.514658\pi$
$72$	6.28726	0.740960
$73$	−0.387873	−0.0453971	−0.0226986	−	0.999742i	$-0.507226\pi$
$73$	−0.387873	−0.0453971	−0.0226986	+	0.999742i	$0.507226\pi$
$74$	−15.0435	−1.74877
$75$	0	0
$76$	−0.193937	−0.0222460
$77$	−3.22425	−0.367438
$78$	7.35026	0.832253
$79$	−0.836381	−0.0941002	−0.0470501	−	0.998893i	$-0.514982\pi$
$79$	−0.836381	−0.0941002	−0.0470501	+	0.998893i	$0.514982\pi$
$80$	0	0
$81$	3.57452	0.397168
$82$	9.35026	1.03256
$83$	7.03761	0.772478	0.386239	−	0.922399i	$-0.373774\pi$
$83$	7.03761	0.772478	0.386239	+	0.922399i	$0.373774\pi$
$84$	0.523730	0.0571437
$85$	0	0
$86$	6.11142	0.659011
$87$	2.91160	0.312157
$88$	−2.57452	−0.274444
$89$	7.08840	0.751369	0.375684	−	0.926748i	$-0.377408\pi$
$89$	7.08840	0.751369	0.375684	+	0.926748i	$0.377408\pi$
$90$	0	0
$91$	20.6253	2.16212
$92$	0.962389	0.100336
$93$	4.77575	0.495222
$94$	−4.96239	−0.511831
$95$	0	0
$96$	0.881286	0.0899459
$97$	−10.9927	−1.11614	−0.558070	−	0.829794i	$-0.688458\pi$
$97$	−10.9927	−1.11614	−0.558070	+	0.829794i	$0.688458\pi$
$98$	6.25694	0.632046
$99$	−2.26187	−0.227326
$100$	0	0
$101$	−2.64974	−0.263659	−0.131829	−	0.991272i	$-0.542085\pi$
$101$	−2.64974	−0.263659	−0.131829	+	0.991272i	$0.542085\pi$
$102$	−7.53690	−0.746265
$103$	10.7308	1.05734	0.528671	−	0.848827i	$-0.322691\pi$
$103$	10.7308	1.05734	0.528671	+	0.848827i	$0.322691\pi$
$104$	16.4690	1.61492
$105$	0	0
$106$	−2.73084	−0.265243
$107$	−4.80606	−0.464620	−0.232310	−	0.972642i	$-0.574628\pi$
$107$	−4.80606	−0.464620	−0.232310	+	0.972642i	$0.574628\pi$
$108$	0.836381	0.0804808
$109$	−2.77575	−0.265868	−0.132934	−	0.991125i	$-0.542440\pi$
$109$	−2.77575	−0.265868	−0.132934	+	0.991125i	$0.542440\pi$
$110$	0	0
$111$	8.18664	0.777042
$112$	14.5745	1.37716
$113$	−6.99271	−0.657818	−0.328909	−	0.944362i	$-0.606681\pi$
$113$	−6.99271	−0.657818	−0.328909	+	0.944362i	$0.606681\pi$
$114$	1.19394	0.111822
$115$	0	0
$116$	−0.700523	−0.0650420
$117$	14.4690	1.33766
$118$	−9.46168	−0.871018
$119$	−21.1490	−1.93873
$120$	0	0
$121$	−10.0738	−0.915801
$122$	−16.7005	−1.51199
$123$	−5.08840	−0.458805
$124$	−1.14903	−0.103186
$125$	0	0
$126$	11.6629	1.03901
$127$	−13.4314	−1.19184	−0.595920	−	0.803043i	$-0.703213\pi$
$127$	−13.4314	−1.19184	−0.595920	+	0.803043i	$0.703213\pi$
$128$	12.6751	1.12033
$129$	−3.32582	−0.292822
$130$	0	0
$131$	20.6253	1.80204	0.901020	−	0.433777i	$-0.142819\pi$
$131$	20.6253	1.80204	0.901020	+	0.433777i	$0.142819\pi$
$132$	−0.150446	−0.0130946
$133$	3.35026	0.290505
$134$	9.96968	0.861249
$135$	0	0
$136$	−16.8872	−1.44806
$137$	−20.2374	−1.72900	−0.864500	−	0.502633i	$-0.832365\pi$
$137$	−20.2374	−1.72900	−0.864500	+	0.502633i	$0.832365\pi$
$138$	−5.92478	−0.504351
$139$	−17.5877	−1.49177	−0.745884	−	0.666076i	$-0.767973\pi$
$139$	−17.5877	−1.49177	−0.745884	+	0.666076i	$0.767973\pi$
$140$	0	0
$141$	2.70052	0.227425
$142$	−1.14903	−0.0964245
$143$	−5.92478	−0.495455
$144$	10.2243	0.852021
$145$	0	0
$146$	−0.574515	−0.0475472
$147$	−3.40502	−0.280841
$148$	−1.96968	−0.161907
$149$	−7.42548	−0.608319	−0.304160	−	0.952621i	$-0.598376\pi$
$149$	−7.42548	−0.608319	−0.304160	+	0.952621i	$0.598376\pi$
$150$	0	0
$151$	−1.61213	−0.131193	−0.0655965	−	0.997846i	$-0.520895\pi$
$151$	−1.61213	−0.131193	−0.0655965	+	0.997846i	$0.520895\pi$
$152$	2.67513	0.216982
$153$	−14.8364	−1.19945
$154$	−4.77575	−0.384841
$155$	0	0
$156$	0.962389	0.0770528
$157$	−4.38787	−0.350190	−0.175095	−	0.984552i	$-0.556023\pi$
$157$	−4.38787	−0.350190	−0.175095	+	0.984552i	$0.556023\pi$
$158$	−1.23884	−0.0985570
$159$	1.48612	0.117857
$160$	0	0
$161$	−16.6253	−1.31026
$162$	5.29455	0.415979
$163$	0.649738	0.0508914	0.0254457	−	0.999676i	$-0.491900\pi$
$163$	0.649738	0.0508914	0.0254457	+	0.999676i	$0.491900\pi$
$164$	1.22425	0.0955982
$165$	0	0
$166$	10.4241	0.809065
$167$	15.3561	1.18829	0.594147	−	0.804357i	$-0.297490\pi$
$167$	15.3561	1.18829	0.594147	+	0.804357i	$0.297490\pi$
$168$	−7.22425	−0.557363
$169$	24.9003	1.91541
$170$	0	0
$171$	2.35026	0.179729
$172$	0.800184	0.0610134
$173$	−3.24472	−0.246692	−0.123346	−	0.992364i	$-0.539362\pi$
$173$	−3.24472	−0.246692	−0.123346	+	0.992364i	$0.539362\pi$
$174$	4.31265	0.326941
$175$	0	0
$176$	−4.18664	−0.315580
$177$	5.14903	0.387025
$178$	10.4993	0.786955
$179$	15.0132	1.12214	0.561069	−	0.827769i	$-0.310390\pi$
$179$	15.0132	1.12214	0.561069	+	0.827769i	$0.310390\pi$
$180$	0	0
$181$	9.22425	0.685633	0.342817	−	0.939402i	$-0.388619\pi$
$181$	9.22425	0.685633	0.342817	+	0.939402i	$0.388619\pi$
$182$	30.5501	2.26452
$183$	9.08840	0.671834
$184$	−13.2750	−0.978649
$185$	0	0
$186$	7.07381	0.518677
$187$	6.07522	0.444264
$188$	−0.649738	−0.0473870
$189$	−14.4485	−1.05097
$190$	0	0
$191$	−21.7743	−1.57554	−0.787768	−	0.615972i	$-0.788763\pi$
$191$	−21.7743	−1.57554	−0.787768	+	0.615972i	$0.788763\pi$
$192$	−5.70782	−0.411926
$193$	−12.5442	−0.902951	−0.451476	−	0.892283i	$-0.649102\pi$
$193$	−12.5442	−0.902951	−0.451476	+	0.892283i	$0.649102\pi$
$194$	−16.2823	−1.16900
$195$	0	0
$196$	0.819237	0.0585169
$197$	−24.5501	−1.74912	−0.874560	−	0.484917i	$-0.838850\pi$
$197$	−24.5501	−1.74912	−0.874560	+	0.484917i	$0.838850\pi$
$198$	−3.35026	−0.238093
$199$	23.0738	1.63566	0.817829	−	0.575461i	$-0.195177\pi$
$199$	23.0738	1.63566	0.817829	+	0.575461i	$0.195177\pi$
$200$	0	0
$201$	−5.42548	−0.382684
$202$	−3.92478	−0.276146
$203$	12.1016	0.849364
$204$	−0.986826	−0.0690917
$205$	0	0
$206$	15.8945	1.10742
$207$	−11.6629	−0.810628
$208$	26.7816	1.85697
$209$	−0.962389	−0.0665698
$210$	0	0
$211$	−20.9380	−1.44143	−0.720714	−	0.693233i	$-0.756186\pi$
$211$	−20.9380	−1.44143	−0.720714	+	0.693233i	$0.756186\pi$
$212$	−0.357556	−0.0245570
$213$	0.625301	0.0428449
$214$	−7.11871	−0.486625
$215$	0	0
$216$	−11.5369	−0.784987
$217$	19.8496	1.34748
$218$	−4.11142	−0.278460
$219$	0.312650	0.0211270
$220$	0	0
$221$	−38.8627	−2.61419
$222$	12.1260	0.813844
$223$	0.0303172	0.00203019	0.00101509	−	0.999999i	$-0.499677\pi$
$223$	0.0303172	0.00203019	0.00101509	+	0.999999i	$0.499677\pi$
$224$	3.66291	0.244739
$225$	0	0
$226$	−10.3576	−0.688974
$227$	−4.80606	−0.318990	−0.159495	−	0.987199i	$-0.550987\pi$
$227$	−4.80606	−0.318990	−0.159495	+	0.987199i	$0.550987\pi$
$228$	0.156325	0.0103529
$229$	1.87399	0.123837	0.0619184	−	0.998081i	$-0.480278\pi$
$229$	1.87399	0.123837	0.0619184	+	0.998081i	$0.480278\pi$
$230$	0	0
$231$	2.59895	0.170999
$232$	9.66291	0.634401
$233$	11.1490	0.730397	0.365199	−	0.930930i	$-0.381001\pi$
$233$	11.1490	0.730397	0.365199	+	0.930930i	$0.381001\pi$
$234$	21.4314	1.40101
$235$	0	0
$236$	−1.23884	−0.0806418
$237$	0.674176	0.0437924
$238$	−31.3258	−2.03055
$239$	9.29948	0.601533	0.300767	−	0.953698i	$-0.402757\pi$
$239$	9.29948	0.601533	0.300767	+	0.953698i	$0.402757\pi$
$240$	0	0
$241$	−2.31265	−0.148971	−0.0744855	−	0.997222i	$-0.523731\pi$
$241$	−2.31265	−0.148971	−0.0744855	+	0.997222i	$0.523731\pi$
$242$	−14.9213	−0.959175
$243$	−15.8192	−1.01480
$244$	−2.18664	−0.139985
$245$	0	0
$246$	−7.53690	−0.480535
$247$	6.15633	0.391718
$248$	15.8496	1.00645
$249$	−5.67276	−0.359497
$250$	0	0
$251$	24.1016	1.52128	0.760639	−	0.649175i	$-0.224886\pi$
$251$	24.1016	1.52128	0.760639	+	0.649175i	$0.224886\pi$
$252$	1.52705	0.0961954
$253$	4.77575	0.300249
$254$	−19.8945	−1.24829
$255$	0	0
$256$	4.61213	0.288258
$257$	13.3199	0.830875	0.415438	−	0.909622i	$-0.363628\pi$
$257$	13.3199	0.830875	0.415438	+	0.909622i	$0.363628\pi$
$258$	−4.92619	−0.306691
$259$	34.0263	2.11429
$260$	0	0
$261$	8.48944	0.525483
$262$	30.5501	1.88739
$263$	−12.9624	−0.799295	−0.399648	−	0.916669i	$-0.630867\pi$
$263$	−12.9624	−0.799295	−0.399648	+	0.916669i	$0.630867\pi$
$264$	2.07522	0.127721
$265$	0	0
$266$	4.96239	0.304264
$267$	−5.71370	−0.349673
$268$	1.30536	0.0797373
$269$	−11.4010	−0.695134	−0.347567	−	0.937655i	$-0.612992\pi$
$269$	−11.4010	−0.695134	−0.347567	+	0.937655i	$0.612992\pi$
$270$	0	0
$271$	16.8119	1.02125	0.510626	−	0.859803i	$-0.329414\pi$
$271$	16.8119	1.02125	0.510626	+	0.859803i	$0.329414\pi$
$272$	−27.4617	−1.66511
$273$	−16.6253	−1.00621
$274$	−29.9756	−1.81089
$275$	0	0
$276$	−0.775746	−0.0466944
$277$	29.7889	1.78984	0.894921	−	0.446224i	$-0.147231\pi$
$277$	29.7889	1.78984	0.894921	+	0.446224i	$0.147231\pi$
$278$	−26.0508	−1.56242
$279$	13.9248	0.833655
$280$	0	0
$281$	−11.6121	−0.692721	−0.346361	−	0.938101i	$-0.612583\pi$
$281$	−11.6121	−0.692721	−0.346361	+	0.938101i	$0.612583\pi$
$282$	4.00000	0.238197
$283$	−2.26187	−0.134454	−0.0672270	−	0.997738i	$-0.521415\pi$
$283$	−2.26187	−0.134454	−0.0672270	+	0.997738i	$0.521415\pi$
$284$	−0.150446	−0.00892730
$285$	0	0
$286$	−8.77575	−0.518921
$287$	−21.1490	−1.24839
$288$	2.56959	0.151415
$289$	22.8496	1.34409
$290$	0	0
$291$	8.86082	0.519430
$292$	−0.0752228	−0.00440208
$293$	−1.84367	−0.107709	−0.0538543	−	0.998549i	$-0.517151\pi$
$293$	−1.84367	−0.107709	−0.0538543	+	0.998549i	$0.517151\pi$
$294$	−5.04349	−0.294142
$295$	0	0
$296$	27.1695	1.57920
$297$	4.15045	0.240833
$298$	−10.9986	−0.637131
$299$	−30.5501	−1.76676
$300$	0	0
$301$	−13.8232	−0.796756
$302$	−2.38787	−0.137407
$303$	2.13586	0.122702
$304$	4.35026	0.249505
$305$	0	0
$306$	−21.9756	−1.25626
$307$	−26.2071	−1.49572	−0.747859	−	0.663857i	$-0.768918\pi$
$307$	−26.2071	−1.49572	−0.747859	+	0.663857i	$0.768918\pi$
$308$	−0.625301	−0.0356298
$309$	−8.64974	−0.492066
$310$	0	0
$311$	6.51388	0.369368	0.184684	−	0.982798i	$-0.440874\pi$
$311$	6.51388	0.369368	0.184684	+	0.982798i	$0.440874\pi$
$312$	−13.2750	−0.751551
$313$	−16.0752	−0.908625	−0.454313	−	0.890842i	$-0.650115\pi$
$313$	−16.0752	−0.908625	−0.454313	+	0.890842i	$0.650115\pi$
$314$	−6.49929	−0.366776
$315$	0	0
$316$	−0.162205	−0.00912473
$317$	5.69323	0.319764	0.159882	−	0.987136i	$-0.448889\pi$
$317$	5.69323	0.319764	0.159882	+	0.987136i	$0.448889\pi$
$318$	2.20123	0.123439
$319$	−3.47627	−0.194634
$320$	0	0
$321$	3.87399	0.216225
$322$	−24.6253	−1.37231
$323$	−6.31265	−0.351245
$324$	0.693229	0.0385127
$325$	0	0
$326$	0.962389	0.0533018
$327$	2.23743	0.123730
$328$	−16.8872	−0.932438
$329$	11.2243	0.618813
$330$	0	0
$331$	−12.3127	−0.676764	−0.338382	−	0.941009i	$-0.609880\pi$
$331$	−12.3127	−0.676764	−0.338382	+	0.941009i	$0.609880\pi$
$332$	1.36485	0.0749059
$333$	23.8700	1.30807
$334$	22.7454	1.24457
$335$	0	0
$336$	−11.7480	−0.640905
$337$	−3.76845	−0.205281	−0.102640	−	0.994719i	$-0.532729\pi$
$337$	−3.76845	−0.205281	−0.102640	+	0.994719i	$0.532729\pi$
$338$	36.8822	2.00613
$339$	5.63656	0.306136
$340$	0	0
$341$	−5.70194	−0.308777
$342$	3.48119	0.188241
$343$	9.29948	0.502125
$344$	−11.0376	−0.595108
$345$	0	0
$346$	−4.80606	−0.258376
$347$	22.3634	1.20053	0.600266	−	0.799800i	$-0.295061\pi$
$347$	22.3634	1.20053	0.600266	+	0.799800i	$0.295061\pi$
$348$	0.564666	0.0302693
$349$	−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
$351$	−26.5501	−1.41714
$352$	−1.05220	−0.0560824
$353$	−5.53690	−0.294700	−0.147350	−	0.989084i	$-0.547074\pi$
$353$	−5.53690	−0.294700	−0.147350	+	0.989084i	$0.547074\pi$
$354$	7.62672	0.405355
$355$	0	0
$356$	1.37470	0.0728589
$357$	17.0475	0.902247
$358$	22.2374	1.17528
$359$	−10.3634	−0.546961	−0.273481	−	0.961878i	$-0.588175\pi$
$359$	−10.3634	−0.546961	−0.273481	+	0.961878i	$0.588175\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	13.6629	0.718107
$363$	8.12013	0.426196
$364$	4.00000	0.209657
$365$	0	0
$366$	13.4617	0.703653
$367$	−3.35026	−0.174882	−0.0874411	−	0.996170i	$-0.527869\pi$
$367$	−3.35026	−0.174882	−0.0874411	+	0.996170i	$0.527869\pi$
$368$	−21.5877	−1.12534
$369$	−14.8364	−0.772351
$370$	0	0
$371$	6.17679	0.320683
$372$	0.926192	0.0480208
$373$	12.6048	0.652653	0.326327	−	0.945257i	$-0.394189\pi$
$373$	12.6048	0.652653	0.326327	+	0.945257i	$0.394189\pi$
$374$	8.99859	0.465306
$375$	0	0
$376$	8.96239	0.462200
$377$	22.2374	1.14529
$378$	−21.4010	−1.10075
$379$	−37.2506	−1.91343	−0.956717	−	0.291018i	$-0.906006\pi$
$379$	−37.2506	−1.91343	−0.956717	+	0.291018i	$0.906006\pi$
$380$	0	0
$381$	10.8265	0.554660
$382$	−32.2520	−1.65016
$383$	−30.8324	−1.57546	−0.787731	−	0.616019i	$-0.788744\pi$
$383$	−30.8324	−1.57546	−0.787731	+	0.616019i	$0.788744\pi$
$384$	−10.2170	−0.521382
$385$	0	0
$386$	−18.5804	−0.945717
$387$	−9.69720	−0.492936
$388$	−2.13189	−0.108230
$389$	−1.37470	−0.0697000	−0.0348500	−	0.999393i	$-0.511095\pi$
$389$	−1.37470	−0.0697000	−0.0348500	+	0.999393i	$0.511095\pi$
$390$	0	0
$391$	31.3258	1.58422
$392$	−11.3004	−0.570758
$393$	−16.6253	−0.838635
$394$	−36.3634	−1.83196
$395$	0	0
$396$	−0.438658	−0.0220434
$397$	9.38646	0.471093	0.235546	−	0.971863i	$-0.424312\pi$
$397$	9.38646	0.471093	0.235546	+	0.971863i	$0.424312\pi$
$398$	34.1768	1.71313
$399$	−2.70052	−0.135195
$400$	0	0
$401$	14.1016	0.704199	0.352099	−	0.935963i	$-0.385468\pi$
$401$	14.1016	0.704199	0.352099	+	0.935963i	$0.385468\pi$
$402$	−8.03620	−0.400809
$403$	36.4749	1.81694
$404$	−0.513881	−0.0255665
$405$	0	0
$406$	17.9248	0.889592
$407$	−9.77433	−0.484496
$408$	13.6121	0.673901
$409$	35.1490	1.73801	0.869004	−	0.494805i	$-0.164761\pi$
$409$	35.1490	1.73801	0.869004	+	0.494805i	$0.164761\pi$
$410$	0	0
$411$	16.3127	0.804644
$412$	2.08110	0.102529
$413$	21.4010	1.05308
$414$	−17.2750	−0.849022
$415$	0	0
$416$	6.73084	0.330007
$417$	14.1768	0.694241
$418$	−1.42548	−0.0697227
$419$	9.02776	0.441035	0.220518	−	0.975383i	$-0.429225\pi$
$419$	9.02776	0.441035	0.220518	+	0.975383i	$0.429225\pi$
$420$	0	0
$421$	15.2097	0.741274	0.370637	−	0.928778i	$-0.379139\pi$
$421$	15.2097	0.741274	0.370637	+	0.928778i	$0.379139\pi$
$422$	−31.0132	−1.50970
$423$	7.87399	0.382847
$424$	4.93207	0.239523
$425$	0	0
$426$	0.926192	0.0448741
$427$	37.7743	1.82803
$428$	−0.932071	−0.0450534
$429$	4.77575	0.230575
$430$	0	0
$431$	16.3127	0.785753	0.392876	−	0.919591i	$-0.371480\pi$
$431$	16.3127	0.785753	0.392876	+	0.919591i	$0.371480\pi$
$432$	−18.7612	−0.902647
$433$	−11.1432	−0.535506	−0.267753	−	0.963488i	$-0.586281\pi$
$433$	−11.1432	−0.535506	−0.267753	+	0.963488i	$0.586281\pi$
$434$	29.4010	1.41130
$435$	0	0
$436$	−0.538319	−0.0257808
$437$	−4.96239	−0.237383
$438$	0.463096	0.0221276
$439$	27.3865	1.30708	0.653542	−	0.756890i	$-0.273282\pi$
$439$	27.3865	1.30708	0.653542	+	0.756890i	$0.273282\pi$
$440$	0	0
$441$	−9.92810	−0.472767
$442$	−57.5633	−2.73800
$443$	−19.5125	−0.927065	−0.463533	−	0.886080i	$-0.653418\pi$
$443$	−19.5125	−0.927065	−0.463533	+	0.886080i	$0.653418\pi$
$444$	1.58769	0.0753484
$445$	0	0
$446$	0.0449056	0.00212634
$447$	5.98541	0.283100
$448$	−23.7235	−1.12083
$449$	−22.1016	−1.04304	−0.521519	−	0.853240i	$-0.674634\pi$
$449$	−22.1016	−1.04304	−0.521519	+	0.853240i	$0.674634\pi$
$450$	0	0
$451$	6.07522	0.286071
$452$	−1.35614	−0.0637875
$453$	1.29948	0.0610547
$454$	−7.11871	−0.334098
$455$	0	0
$456$	−2.15633	−0.100979
$457$	17.8496	0.834967	0.417483	−	0.908685i	$-0.362912\pi$
$457$	17.8496	0.834967	0.417483	+	0.908685i	$0.362912\pi$
$458$	2.77575	0.129702
$459$	27.2243	1.27072
$460$	0	0
$461$	−35.2506	−1.64178	−0.820892	−	0.571083i	$-0.806523\pi$
$461$	−35.2506	−1.64178	−0.820892	+	0.571083i	$0.806523\pi$
$462$	3.84955	0.179097
$463$	26.3634	1.22521	0.612606	−	0.790388i	$-0.290121\pi$
$463$	26.3634	1.22521	0.612606	+	0.790388i	$0.290121\pi$
$464$	15.7137	0.729490
$465$	0	0
$466$	16.5139	0.764991
$467$	6.78560	0.314000	0.157000	−	0.987599i	$-0.449818\pi$
$467$	6.78560	0.314000	0.157000	+	0.987599i	$0.449818\pi$
$468$	2.80606	0.129710
$469$	−22.5501	−1.04127
$470$	0	0
$471$	3.53690	0.162972
$472$	17.0884	0.786557
$473$	3.97082	0.182579
$474$	0.998585	0.0458665
$475$	0	0
$476$	−4.10157	−0.187995
$477$	4.33312	0.198400
$478$	13.7743	0.630023
$479$	12.7104	0.580752	0.290376	−	0.956913i	$-0.406220\pi$
$479$	12.7104	0.580752	0.290376	+	0.956913i	$0.406220\pi$
$480$	0	0
$481$	62.5256	2.85092
$482$	−3.42548	−0.156027
$483$	13.4010	0.609769
$484$	−1.95368	−0.0888036
$485$	0	0
$486$	−23.4314	−1.06287
$487$	15.7586	0.714090	0.357045	−	0.934087i	$-0.383784\pi$
$487$	15.7586	0.714090	0.357045	+	0.934087i	$0.383784\pi$
$488$	30.1622	1.36538
$489$	−0.523730	−0.0236839
$490$	0	0
$491$	−14.5501	−0.656636	−0.328318	−	0.944567i	$-0.606482\pi$
$491$	−14.5501	−0.656636	−0.328318	+	0.944567i	$0.606482\pi$
$492$	−0.986826	−0.0444896
$493$	−22.8021	−1.02695
$494$	9.11871	0.410270
$495$	0	0
$496$	25.7743	1.15730
$497$	2.59895	0.116579
$498$	−8.40246	−0.376523
$499$	−5.48612	−0.245592	−0.122796	−	0.992432i	$-0.539186\pi$
$499$	−5.48612	−0.245592	−0.122796	+	0.992432i	$0.539186\pi$
$500$	0	0
$501$	−12.3780	−0.553009
$502$	35.6991	1.59333
$503$	36.6615	1.63466	0.817328	−	0.576173i	$-0.195455\pi$
$503$	36.6615	1.63466	0.817328	+	0.576173i	$0.195455\pi$
$504$	−21.0640	−0.938263
$505$	0	0
$506$	7.07381	0.314469
$507$	−20.0713	−0.891396
$508$	−2.60483	−0.115571
$509$	39.1900	1.73706	0.868532	−	0.495632i	$-0.165064\pi$
$509$	39.1900	1.73706	0.868532	+	0.495632i	$0.165064\pi$
$510$	0	0
$511$	1.29948	0.0574855
$512$	−18.5188	−0.818423
$513$	−4.31265	−0.190408
$514$	19.7294	0.870228
$515$	0	0
$516$	−0.644999	−0.0283945
$517$	−3.22425	−0.141803
$518$	50.3996	2.21443
$519$	2.61545	0.114806
$520$	0	0
$521$	−17.7283	−0.776690	−0.388345	−	0.921514i	$-0.626953\pi$
$521$	−17.7283	−0.776690	−0.388345	+	0.921514i	$0.626953\pi$
$522$	12.5745	0.550372
$523$	40.7572	1.78219	0.891094	−	0.453819i	$-0.149939\pi$
$523$	40.7572	1.78219	0.891094	+	0.453819i	$0.149939\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−19.1998	−0.837152
$527$	−37.4010	−1.62922
$528$	3.37470	0.146865
$529$	1.62530	0.0706652
$530$	0	0
$531$	15.0132	0.651516
$532$	0.649738	0.0281697
$533$	−38.8627	−1.68333
$534$	−8.46310	−0.366234
$535$	0	0
$536$	−18.0059	−0.777736
$537$	−12.1016	−0.522221
$538$	−16.8872	−0.728057
$539$	4.06537	0.175108
$540$	0	0
$541$	23.9003	1.02756	0.513778	−	0.857923i	$-0.328246\pi$
$541$	23.9003	1.02756	0.513778	+	0.857923i	$0.328246\pi$
$542$	24.9018	1.06962
$543$	−7.43533	−0.319081
$544$	−6.90175	−0.295910
$545$	0	0
$546$	−24.6253	−1.05387
$547$	−8.55405	−0.365745	−0.182872	−	0.983137i	$-0.558539\pi$
$547$	−8.55405	−0.365745	−0.182872	+	0.983137i	$0.558539\pi$
$548$	−3.92478	−0.167658
$549$	26.4993	1.13096
$550$	0	0
$551$	3.61213	0.153882
$552$	10.7005	0.455445
$553$	2.80209	0.119157
$554$	44.1232	1.87461
$555$	0	0
$556$	−3.41090	−0.144654
$557$	4.23743	0.179546	0.0897728	−	0.995962i	$-0.471386\pi$
$557$	4.23743	0.179546	0.0897728	+	0.995962i	$0.471386\pi$
$558$	20.6253	0.873139
$559$	−25.4010	−1.07435
$560$	0	0
$561$	−4.89701	−0.206752
$562$	−17.1998	−0.725530
$563$	−16.4934	−0.695114	−0.347557	−	0.937659i	$-0.612989\pi$
$563$	−16.4934	−0.695114	−0.347557	+	0.937659i	$0.612989\pi$
$564$	0.523730	0.0220530
$565$	0	0
$566$	−3.35026	−0.140822
$567$	−11.9756	−0.502926
$568$	2.07522	0.0870744
$569$	−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
$571$	−26.2619	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$571$	−26.2619	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$572$	−1.14903	−0.0480434
$573$	17.5515	0.733224
$574$	−31.3258	−1.30751
$575$	0	0
$576$	−16.6424	−0.693435
$577$	−30.1016	−1.25314	−0.626572	−	0.779363i	$-0.715543\pi$
$577$	−30.1016	−1.25314	−0.626572	+	0.779363i	$0.715543\pi$
$578$	33.8446	1.40775
$579$	10.1114	0.420216
$580$	0	0
$581$	−23.5778	−0.978174
$582$	13.1246	0.544032
$583$	−1.77433	−0.0734853
$584$	1.03761	0.0429367
$585$	0	0
$586$	−2.73084	−0.112810
$587$	35.1392	1.45035	0.725175	−	0.688565i	$-0.241759\pi$
$587$	35.1392	1.45035	0.725175	+	0.688565i	$0.241759\pi$
$588$	−0.660357	−0.0272327
$589$	5.92478	0.244126
$590$	0	0
$591$	19.7889	0.814007
$592$	44.1827	1.81590
$593$	−34.3244	−1.40953	−0.704767	−	0.709439i	$-0.748949\pi$
$593$	−34.3244	−1.40953	−0.704767	+	0.709439i	$0.748949\pi$
$594$	6.14762	0.252240
$595$	0	0
$596$	−1.44007	−0.0589877
$597$	−18.5990	−0.761204
$598$	−45.2506	−1.85043
$599$	14.6107	0.596978	0.298489	−	0.954413i	$-0.403517\pi$
$599$	14.6107	0.596978	0.298489	+	0.954413i	$0.403517\pi$
$600$	0	0
$601$	23.5633	0.961165	0.480583	−	0.876949i	$-0.340425\pi$
$601$	23.5633	0.961165	0.480583	+	0.876949i	$0.340425\pi$
$602$	−20.4749	−0.834493
$603$	−15.8192	−0.644209
$604$	−0.312650	−0.0127216
$605$	0	0
$606$	3.16362	0.128513
$607$	8.80606	0.357427	0.178714	−	0.983901i	$-0.442806\pi$
$607$	8.80606	0.357427	0.178714	+	0.983901i	$0.442806\pi$
$608$	1.09332	0.0443400
$609$	−9.75463	−0.395278
$610$	0	0
$611$	20.6253	0.834410
$612$	−2.87732	−0.116309
$613$	−10.4142	−0.420626	−0.210313	−	0.977634i	$-0.567448\pi$
$613$	−10.4142	−0.420626	−0.210313	+	0.977634i	$0.567448\pi$
$614$	−38.8178	−1.56656
$615$	0	0
$616$	8.62530	0.347523
$617$	17.2849	0.695863	0.347932	−	0.937520i	$-0.386884\pi$
$617$	17.2849	0.695863	0.347932	+	0.937520i	$0.386884\pi$
$618$	−12.8119	−0.515372
$619$	−10.6351	−0.427463	−0.213731	−	0.976892i	$-0.568562\pi$
$619$	−10.6351	−0.427463	−0.213731	+	0.976892i	$0.568562\pi$
$620$	0	0
$621$	21.4010	0.858794
$622$	9.64832	0.386863
$623$	−23.7480	−0.951443
$624$	−21.5877	−0.864199
$625$	0	0
$626$	−23.8105	−0.951660
$627$	0.775746	0.0309803
$628$	−0.850969	−0.0339574
$629$	−64.1133	−2.55637
$630$	0	0
$631$	−16.5599	−0.659240	−0.329620	−	0.944114i	$-0.606921\pi$
$631$	−16.5599	−0.659240	−0.329620	+	0.944114i	$0.606921\pi$
$632$	2.23743	0.0890001
$633$	16.8773	0.670813
$634$	8.43278	0.334908
$635$	0	0
$636$	0.288213	0.0114284
$637$	−26.0059	−1.03039
$638$	−5.14903	−0.203852
$639$	1.82321	0.0721249
$640$	0	0
$641$	−16.7612	−0.662026	−0.331013	−	0.943626i	$-0.607390\pi$
$641$	−16.7612	−0.662026	−0.331013	+	0.943626i	$0.607390\pi$
$642$	5.73813	0.226466
$643$	−5.73813	−0.226290	−0.113145	−	0.993578i	$-0.536092\pi$
$643$	−5.73813	−0.226290	−0.113145	+	0.993578i	$0.536092\pi$
$644$	−3.22425	−0.127053
$645$	0	0
$646$	−9.35026	−0.367881
$647$	37.2144	1.46305	0.731525	−	0.681815i	$-0.238809\pi$
$647$	37.2144	1.46305	0.731525	+	0.681815i	$0.238809\pi$
$648$	−9.56230	−0.375642
$649$	−6.14762	−0.241315
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0.126008	0.00493485
$653$	−11.7626	−0.460305	−0.230153	−	0.973155i	$-0.573923\pi$
$653$	−11.7626	−0.460305	−0.230153	+	0.973155i	$0.573923\pi$
$654$	3.31406	0.129590
$655$	0	0
$656$	−27.4617	−1.07220
$657$	0.911603	0.0355650
$658$	16.6253	0.648122
$659$	1.23884	0.0482584	0.0241292	−	0.999709i	$-0.492319\pi$
$659$	1.23884	0.0482584	0.0241292	+	0.999709i	$0.492319\pi$
$660$	0	0
$661$	−9.53690	−0.370943	−0.185471	−	0.982650i	$-0.559381\pi$
$661$	−9.53690	−0.370943	−0.185471	+	0.982650i	$0.559381\pi$
$662$	−18.2374	−0.708818
$663$	31.3258	1.21659
$664$	−18.8265	−0.730611
$665$	0	0
$666$	35.3561	1.37002
$667$	−17.9248	−0.694050
$668$	2.97812	0.115227
$669$	−0.0244376	−0.000944811 0
$670$	0	0
$671$	−10.8510	−0.418897
$672$	−2.95254	−0.113897
$673$	−39.9307	−1.53921	−0.769607	−	0.638518i	$-0.779548\pi$
$673$	−39.9307	−1.53921	−0.769607	+	0.638518i	$0.779548\pi$
$674$	−5.58181	−0.215003
$675$	0	0
$676$	4.82909	0.185734
$677$	3.05334	0.117349	0.0586747	−	0.998277i	$-0.481313\pi$
$677$	3.05334	0.117349	0.0586747	+	0.998277i	$0.481313\pi$
$678$	8.34885	0.320636
$679$	36.8284	1.41335
$680$	0	0
$681$	3.87399	0.148452
$682$	−8.44568	−0.323402
$683$	−29.2692	−1.11995	−0.559977	−	0.828508i	$-0.689190\pi$
$683$	−29.2692	−1.11995	−0.559977	+	0.828508i	$0.689190\pi$
$684$	0.455802	0.0174280
$685$	0	0
$686$	13.7743	0.525906
$687$	−1.51056	−0.0576313
$688$	−17.9492	−0.684307
$689$	11.3503	0.432411
$690$	0	0
$691$	2.63515	0.100246	0.0501229	−	0.998743i	$-0.484039\pi$
$691$	2.63515	0.100246	0.0501229	+	0.998743i	$0.484039\pi$
$692$	−0.629270	−0.0239213
$693$	7.57784	0.287858
$694$	33.1246	1.25739
$695$	0	0
$696$	−7.78892	−0.295238
$697$	39.8496	1.50941
$698$	−14.8119	−0.560640
$699$	−8.98683	−0.339913
$700$	0	0
$701$	−25.0494	−0.946102	−0.473051	−	0.881035i	$-0.656847\pi$
$701$	−25.0494	−0.946102	−0.473051	+	0.881035i	$0.656847\pi$
$702$	−39.3258	−1.48426
$703$	10.1563	0.383053
$704$	6.81477	0.256841
$705$	0	0
$706$	−8.20123	−0.308657
$707$	8.87732	0.333866
$708$	0.998585	0.0375291
$709$	−41.6991	−1.56604	−0.783021	−	0.621995i	$-0.786323\pi$
$709$	−41.6991	−1.56604	−0.783021	+	0.621995i	$0.786323\pi$
$710$	0	0
$711$	1.96571	0.0737200
$712$	−18.9624	−0.710646
$713$	−29.4010	−1.10108
$714$	25.2506	0.944980
$715$	0	0
$716$	2.91160	0.108812
$717$	−7.49597	−0.279942
$718$	−15.3503	−0.572867
$719$	30.6351	1.14250	0.571249	−	0.820777i	$-0.306459\pi$
$719$	30.6351	1.14250	0.571249	+	0.820777i	$0.306459\pi$
$720$	0	0
$721$	−35.9511	−1.33889
$722$	1.48119	0.0551243
$723$	1.86414	0.0693282
$724$	1.78892	0.0664847
$725$	0	0
$726$	12.0275	0.446382
$727$	−7.50071	−0.278186	−0.139093	−	0.990279i	$-0.544419\pi$
$727$	−7.50071	−0.278186	−0.139093	+	0.990279i	$0.544419\pi$
$728$	−55.1754	−2.04494
$729$	2.02776	0.0751023
$730$	0	0
$731$	26.0460	0.963348
$732$	1.76257	0.0651466
$733$	−9.84955	−0.363802	−0.181901	−	0.983317i	$-0.558225\pi$
$733$	−9.84955	−0.363802	−0.181901	+	0.983317i	$0.558225\pi$
$734$	−4.96239	−0.183165
$735$	0	0
$736$	−5.42548	−0.199986
$737$	6.47768	0.238609
$738$	−21.9756	−0.808932
$739$	−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
$741$	−4.96239	−0.182298
$742$	9.14903	0.335871
$743$	15.7177	0.576625	0.288313	−	0.957536i	$-0.406906\pi$
$743$	15.7177	0.576625	0.288313	+	0.957536i	$0.406906\pi$
$744$	−12.7757	−0.468382
$745$	0	0
$746$	18.6702	0.683565
$747$	−16.5402	−0.605175
$748$	1.17821	0.0430795
$749$	16.1016	0.588339
$750$	0	0
$751$	−43.7400	−1.59610	−0.798048	−	0.602593i	$-0.794134\pi$
$751$	−43.7400	−1.59610	−0.798048	+	0.602593i	$0.794134\pi$
$752$	14.5745	0.531478
$753$	−19.4274	−0.707974
$754$	32.9380	1.19953
$755$	0	0
$756$	−2.80209	−0.101911
$757$	15.7743	0.573328	0.286664	−	0.958031i	$-0.407454\pi$
$757$	15.7743	0.573328	0.286664	+	0.958031i	$0.407454\pi$
$758$	−55.1754	−2.00406
$759$	−3.84955	−0.139730
$760$	0	0
$761$	43.2262	1.56695	0.783474	−	0.621425i	$-0.213446\pi$
$761$	43.2262	1.56695	0.783474	+	0.621425i	$0.213446\pi$
$762$	16.0362	0.580930
$763$	9.29948	0.336664
$764$	−4.22284	−0.152777
$765$	0	0
$766$	−45.6688	−1.65008
$767$	39.3258	1.41997
$768$	−3.71767	−0.134150
$769$	−19.1246	−0.689650	−0.344825	−	0.938667i	$-0.612062\pi$
$769$	−19.1246	−0.689650	−0.344825	+	0.938667i	$0.612062\pi$
$770$	0	0
$771$	−10.7367	−0.386674
$772$	−2.43278	−0.0875576
$773$	−25.8846	−0.931005	−0.465502	−	0.885047i	$-0.654126\pi$
$773$	−25.8846	−0.931005	−0.465502	+	0.885047i	$0.654126\pi$
$774$	−14.3634	−0.516283
$775$	0	0
$776$	29.4069	1.05565
$777$	−27.4274	−0.983952
$778$	−2.03620	−0.0730012
$779$	−6.31265	−0.226174
$780$	0	0
$781$	−0.746569	−0.0267144
$782$	46.3996	1.65925
$783$	−15.5778	−0.556707
$784$	−18.3766	−0.656307
$785$	0	0
$786$	−24.6253	−0.878355
$787$	−19.9814	−0.712261	−0.356131	−	0.934436i	$-0.615904\pi$
$787$	−19.9814	−0.712261	−0.356131	+	0.934436i	$0.615904\pi$
$788$	−4.76116	−0.169609
$789$	10.4485	0.371977
$790$	0	0
$791$	23.4274	0.832982
$792$	6.05079	0.215005
$793$	69.4128	2.46492
$794$	13.9032	0.493405
$795$	0	0
$796$	4.47486	0.158607
$797$	−28.6458	−1.01469	−0.507343	−	0.861744i	$-0.669372\pi$
$797$	−28.6458	−1.01469	−0.507343	+	0.861744i	$0.669372\pi$
$798$	−4.00000	−0.141598
$799$	−21.1490	−0.748199
$800$	0	0
$801$	−16.6596	−0.588638
$802$	20.8872	0.737551
$803$	−0.373285	−0.0131729
$804$	−1.05220	−0.0371082
$805$	0	0
$806$	54.0263	1.90300
$807$	9.18997	0.323502
$808$	7.08840	0.249369
$809$	−17.2243	−0.605573	−0.302786	−	0.953058i	$-0.597917\pi$
$809$	−17.2243	−0.605573	−0.302786	+	0.953058i	$0.597917\pi$
$810$	0	0
$811$	−15.6267	−0.548728	−0.274364	−	0.961626i	$-0.588467\pi$
$811$	−15.6267	−0.548728	−0.274364	+	0.961626i	$0.588467\pi$
$812$	2.34694	0.0823613
$813$	−13.5515	−0.475272
$814$	−14.4777	−0.507443
$815$	0	0
$816$	22.1359	0.774910
$817$	−4.12601	−0.144351
$818$	52.0625	1.82032
$819$	−48.4749	−1.69385
$820$	0	0
$821$	39.2506	1.36986	0.684928	−	0.728611i	$-0.259834\pi$
$821$	39.2506	1.36986	0.684928	+	0.728611i	$0.259834\pi$
$822$	24.1622	0.842754
$823$	45.5271	1.58697	0.793487	−	0.608588i	$-0.208264\pi$
$823$	45.5271	1.58697	0.793487	+	0.608588i	$0.208264\pi$
$824$	−28.7064	−1.00003
$825$	0	0
$826$	31.6991	1.10295
$827$	−17.0698	−0.593576	−0.296788	−	0.954943i	$-0.595916\pi$
$827$	−17.0698	−0.593576	−0.296788	+	0.954943i	$0.595916\pi$
$828$	−2.26187	−0.0786052
$829$	1.69911	0.0590125	0.0295062	−	0.999565i	$-0.490607\pi$
$829$	1.69911	0.0590125	0.0295062	+	0.999565i	$0.490607\pi$
$830$	0	0
$831$	−24.0118	−0.832959
$832$	−43.5936	−1.51134
$833$	26.6662	0.923930
$834$	20.9986	0.727122
$835$	0	0
$836$	−0.186642	−0.00645516
$837$	−25.5515	−0.883189
$838$	13.3719	0.461924
$839$	−50.5910	−1.74660	−0.873298	−	0.487187i	$-0.838023\pi$
$839$	−50.5910	−1.74660	−0.873298	+	0.487187i	$0.838023\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	22.5285	0.776382
$843$	9.36011	0.322379
$844$	−4.06063	−0.139773
$845$	0	0
$846$	11.6629	0.400979
$847$	33.7499	1.15966
$848$	8.02047	0.275424
$849$	1.82321	0.0625723
$850$	0	0
$851$	−50.3996	−1.72768
$852$	0.121269	0.00415460
$853$	22.5237	0.771198	0.385599	−	0.922667i	$-0.373995\pi$
$853$	22.5237	0.771198	0.385599	+	0.922667i	$0.373995\pi$
$854$	55.9511	1.91461
$855$	0	0
$856$	12.8568	0.439438
$857$	23.6180	0.806776	0.403388	−	0.915029i	$-0.367833\pi$
$857$	23.6180	0.806776	0.403388	+	0.915029i	$0.367833\pi$
$858$	7.07381	0.241496
$859$	15.1754	0.517777	0.258889	−	0.965907i	$-0.416644\pi$
$859$	15.1754	0.517777	0.258889	+	0.965907i	$0.416644\pi$
$860$	0	0
$861$	17.0475	0.580976
$862$	24.1622	0.822968
$863$	−30.1055	−1.02480	−0.512402	−	0.858746i	$-0.671244\pi$
$863$	−30.1055	−1.02480	−0.512402	+	0.858746i	$0.671244\pi$
$864$	−4.71511	−0.160411
$865$	0	0
$866$	−16.5052	−0.560869
$867$	−18.4182	−0.625515
$868$	3.84955	0.130662
$869$	−0.804923	−0.0273051
$870$	0	0
$871$	−41.4372	−1.40405
$872$	7.42548	0.251459
$873$	25.8357	0.874407
$874$	−7.35026	−0.248626
$875$	0	0
$876$	0.0606343	0.00204864
$877$	5.53102	0.186769	0.0933847	−	0.995630i	$-0.470231\pi$
$877$	5.53102	0.186769	0.0933847	+	0.995630i	$0.470231\pi$
$878$	40.5647	1.36899
$879$	1.48612	0.0501255
$880$	0	0
$881$	20.8265	0.701664	0.350832	−	0.936438i	$-0.385899\pi$
$881$	20.8265	0.701664	0.350832	+	0.936438i	$0.385899\pi$
$882$	−14.7054	−0.495158
$883$	−43.1509	−1.45214	−0.726072	−	0.687618i	$-0.758656\pi$
$883$	−43.1509	−1.45214	−0.726072	+	0.687618i	$0.758656\pi$
$884$	−7.53690	−0.253494
$885$	0	0
$886$	−28.9018	−0.970973
$887$	44.5461	1.49571	0.747856	−	0.663861i	$-0.231083\pi$
$887$	44.5461	1.49571	0.747856	+	0.663861i	$0.231083\pi$
$888$	−21.9003	−0.734927
$889$	44.9986	1.50920
$890$	0	0
$891$	3.44007	0.115247
$892$	0.00587961	0.000196864 0
$893$	3.35026	0.112112
$894$	8.86556	0.296509
$895$	0	0
$896$	−42.4650	−1.41866
$897$	24.6253	0.822215
$898$	−32.7367	−1.09244
$899$	21.4010	0.713765
$900$	0	0
$901$	−11.6385	−0.387734
$902$	8.99859	0.299620
$903$	11.1424	0.370795
$904$	18.7064	0.622166
$905$	0	0
$906$	1.92478	0.0639464
$907$	−53.7558	−1.78493	−0.892466	−	0.451115i	$-0.851026\pi$
$907$	−53.7558	−1.78493	−0.892466	+	0.451115i	$0.851026\pi$
$908$	−0.932071	−0.0309319
$909$	6.22758	0.206556
$910$	0	0
$911$	−2.28630	−0.0757486	−0.0378743	−	0.999283i	$-0.512059\pi$
$911$	−2.28630	−0.0757486	−0.0378743	+	0.999283i	$0.512059\pi$
$912$	−3.50659	−0.116115
$913$	6.77292	0.224151
$914$	26.4387	0.874513
$915$	0	0
$916$	0.363436	0.0120082
$917$	−69.1002	−2.28189
$918$	40.3244	1.33090
$919$	−34.8510	−1.14963	−0.574814	−	0.818284i	$-0.694925\pi$
$919$	−34.8510	−1.14963	−0.574814	+	0.818284i	$0.694925\pi$
$920$	0	0
$921$	21.1246	0.696079
$922$	−52.2130	−1.71954
$923$	4.77575	0.157196
$924$	0.504032	0.0165814
$925$	0	0
$926$	39.0494	1.28324
$927$	−25.2203	−0.828343
$928$	3.94921	0.129639
$929$	41.6991	1.36810	0.684052	−	0.729434i	$-0.260216\pi$
$929$	41.6991	1.36810	0.684052	+	0.729434i	$0.260216\pi$
$930$	0	0
$931$	−4.22425	−0.138444
$932$	2.16220	0.0708254
$933$	−5.25060	−0.171897
$934$	10.0508	0.328872
$935$	0	0
$936$	−38.7064	−1.26516
$937$	−21.9102	−0.715775	−0.357887	−	0.933765i	$-0.616503\pi$
$937$	−21.9102	−0.715775	−0.357887	+	0.933765i	$0.616503\pi$
$938$	−33.4010	−1.09058
$939$	12.9576	0.422857
$940$	0	0
$941$	−3.55149	−0.115775	−0.0578877	−	0.998323i	$-0.518437\pi$
$941$	−3.55149	−0.115775	−0.0578877	+	0.998323i	$0.518437\pi$
$942$	5.23884	0.170691
$943$	31.3258	1.02011
$944$	27.7889	0.904452
$945$	0	0
$946$	5.88156	0.191226
$947$	−38.3634	−1.24664	−0.623322	−	0.781965i	$-0.714217\pi$
$947$	−38.3634	−1.24664	−0.623322	+	0.781965i	$0.714217\pi$
$948$	0.130747	0.00424648
$949$	2.38787	0.0775136
$950$	0	0
$951$	−4.58910	−0.148812
$952$	56.5764	1.83365
$953$	22.8714	0.740879	0.370439	−	0.928857i	$-0.379207\pi$
$953$	22.8714	0.740879	0.370439	+	0.928857i	$0.379207\pi$
$954$	6.41819	0.207797
$955$	0	0
$956$	1.80351	0.0583296
$957$	2.80209	0.0905788
$958$	18.8265	0.608258
$959$	67.8007	2.18940
$960$	0	0
$961$	4.10299	0.132354
$962$	92.6126	2.98595
$963$	11.2955	0.363993
$964$	−0.448507	−0.0144455
$965$	0	0
$966$	19.8496	0.638649
$967$	−27.6629	−0.889579	−0.444790	−	0.895635i	$-0.646722\pi$
$967$	−27.6629	−0.889579	−0.444790	+	0.895635i	$0.646722\pi$
$968$	26.9488	0.866166
$969$	5.08840	0.163463
$970$	0	0
$971$	−42.1768	−1.35352	−0.676759	−	0.736205i	$-0.736616\pi$
$971$	−42.1768	−1.35352	−0.676759	+	0.736205i	$0.736616\pi$
$972$	−3.06793	−0.0984039
$973$	58.9234	1.88900
$974$	23.3416	0.747912
$975$	0	0
$976$	49.0494	1.57003
$977$	−0.856849	−0.0274130	−0.0137065	−	0.999906i	$-0.504363\pi$
$977$	−0.856849	−0.0274130	−0.0137065	+	0.999906i	$0.504363\pi$
$978$	−0.775746	−0.0248056
$979$	6.82179	0.218025
$980$	0	0
$981$	6.52373	0.208287
$982$	−21.5515	−0.687736
$983$	−45.2809	−1.44424	−0.722119	−	0.691769i	$-0.756831\pi$
$983$	−45.2809	−1.44424	−0.722119	+	0.691769i	$0.756831\pi$
$984$	13.6121	0.433939
$985$	0	0
$986$	−33.7743	−1.07559
$987$	−9.04746	−0.287984
$988$	1.19394	0.0379842
$989$	20.4749	0.651063
$990$	0	0
$991$	47.0132	1.49342	0.746711	−	0.665148i	$-0.231632\pi$
$991$	47.0132	1.49342	0.746711	+	0.665148i	$0.231632\pi$
$992$	6.47768	0.205667
$993$	9.92478	0.314953
$994$	3.84955	0.122100
$995$	0	0
$996$	−1.10016	−0.0348598
$997$	−13.6873	−0.433483	−0.216741	−	0.976229i	$-0.569543\pi$
$997$	−13.6873	−0.433483	−0.216741	+	0.976229i	$0.569543\pi$
$998$	−8.12601	−0.257224
$999$	−43.8007	−1.38579

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.f.1.3		3
3.2	odd	2		4275.2.a.bk.1.1		3
4.3	odd	2		7600.2.a.bx.1.2		3
5.2	odd	4		475.2.b.d.324.5		6
5.3	odd	4		475.2.b.d.324.2		6
5.4	even	2		95.2.a.a.1.1	✓	3
15.14	odd	2		855.2.a.i.1.3		3
19.18	odd	2		9025.2.a.bb.1.1		3
20.19	odd	2		1520.2.a.p.1.2		3
35.34	odd	2		4655.2.a.u.1.1		3
40.19	odd	2		6080.2.a.by.1.2		3
40.29	even	2		6080.2.a.bo.1.2		3
95.94	odd	2		1805.2.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.1	✓	3	5.4	even	2
475.2.a.f.1.3		3	1.1	even	1	trivial
475.2.b.d.324.2		6	5.3	odd	4
475.2.b.d.324.5		6	5.2	odd	4
855.2.a.i.1.3		3	15.14	odd	2
1520.2.a.p.1.2		3	20.19	odd	2
1805.2.a.f.1.3		3	95.94	odd	2
4275.2.a.bk.1.1		3	3.2	odd	2
4655.2.a.u.1.1		3	35.34	odd	2
6080.2.a.bo.1.2		3	40.29	even	2
6080.2.a.by.1.2		3	40.19	odd	2
7600.2.a.bx.1.2		3	4.3	odd	2
9025.2.a.bb.1.1		3	19.18	odd	2

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Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	475.2.a.f.1.3

Level	$475$
Weight	$2$
Character	475.1
Self dual	yes
Analytic conductor	$3.793$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$