LMFDB - Embedded newform 855.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
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	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} +3.35026 q^{7} -2.67513 q^{8} -1.48119 q^{10} -0.962389 q^{11} +6.15633 q^{13} +4.96239 q^{14} -4.35026 q^{16} +6.31265 q^{17} -1.00000 q^{19} -0.193937 q^{20} -1.42548 q^{22} +4.96239 q^{23} +1.00000 q^{25} +9.11871 q^{26} +0.649738 q^{28} +3.61213 q^{29} -5.92478 q^{31} -1.09332 q^{32} +9.35026 q^{34} -3.35026 q^{35} +10.1563 q^{37} -1.48119 q^{38} +2.67513 q^{40} -6.31265 q^{41} -4.12601 q^{43} -0.186642 q^{44} +7.35026 q^{46} -3.35026 q^{47} +4.22425 q^{49} +1.48119 q^{50} +1.19394 q^{52} -1.84367 q^{53} +0.962389 q^{55} -8.96239 q^{56} +5.35026 q^{58} +6.38787 q^{59} -11.2750 q^{61} -8.77575 q^{62} +7.08110 q^{64} -6.15633 q^{65} -6.73084 q^{67} +1.22425 q^{68} -4.96239 q^{70} +0.775746 q^{71} +0.387873 q^{73} +15.0435 q^{74} -0.193937 q^{76} -3.22425 q^{77} -0.836381 q^{79} +4.35026 q^{80} -9.35026 q^{82} +7.03761 q^{83} -6.31265 q^{85} -6.11142 q^{86} +2.57452 q^{88} -7.08840 q^{89} +20.6253 q^{91} +0.962389 q^{92} -4.96239 q^{94} +1.00000 q^{95} +10.9927 q^{97} +6.25694 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} + 8 q^{11} + 8 q^{13} + 4 q^{14} - 3 q^{16} - 2 q^{17} - 3 q^{19} - q^{20} - 16 q^{22} + 4 q^{23} + 3 q^{25} + 6 q^{26} + 12 q^{28} + 10 q^{29} + 4 q^{31}+ \cdots + 15 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 + 8 * q^11 + 8 * q^13 + 4 * q^14 - 3 * q^16 - 2 * q^17 - 3 * q^19 - q^20 - 16 * q^22 + 4 * q^23 + 3 * q^25 + 6 * q^26 + 12 * q^28 + 10 * q^29 + 4 * q^31 + 3 * q^32 + 18 * q^34 + 20 * q^37 + q^38 + 3 * q^40 + 2 * q^41 - 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - q^50 + 4 * q^52 - 16 * q^53 - 8 * q^55 - 16 * q^56 + 6 * q^58 + 20 * q^59 - 2 * q^61 - 28 * q^62 - 11 * q^64 - 8 * q^65 + 2 * q^67 + 2 * q^68 - 4 * q^70 + 4 * q^71 + 2 * q^73 + 2 * q^74 - q^76 - 8 * q^77 + 3 * q^80 - 18 * q^82 + 32 * q^83 + 2 * q^85 + 16 * q^86 - 4 * q^88 - 2 * q^89 + 20 * q^91 - 8 * q^92 - 4 * q^94 + 3 * q^95 + 20 * q^97 + 15 * q^98

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	0
$3$	0	0
$4$	0.193937	0.0969683
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.35026	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$7$	3.35026	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$8$	−2.67513	−0.945802
$9$	0	0
$10$	−1.48119	−0.468395
$11$	−0.962389	−0.290171	−0.145086	−	0.989419i	$-0.546346\pi$
$11$	−0.962389	−0.290171	−0.145086	+	0.989419i	$0.546346\pi$
$12$	0	0
$13$	6.15633	1.70746	0.853729	−	0.520718i	$-0.174336\pi$
$13$	6.15633	1.70746	0.853729	+	0.520718i	$0.174336\pi$
$14$	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	$0.222481\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.193937	−0.0433655
$21$	0	0
$22$	−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	$0.326913\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.11871	1.78833
$27$	0	0
$28$	0.649738	0.122789
$29$	3.61213	0.670755	0.335378	−	0.942084i	$-0.391136\pi$
$29$	3.61213	0.670755	0.335378	+	0.942084i	$0.391136\pi$
$30$	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	0
$34$	9.35026	1.60356
$35$	−3.35026	−0.566298
$36$	0	0
$37$	10.1563	1.66969	0.834845	−	0.550485i	$-0.185557\pi$
$37$	10.1563	1.66969	0.834845	+	0.550485i	$0.185557\pi$
$38$	−1.48119	−0.240281
$39$	0	0
$40$	2.67513	0.422975
$41$	−6.31265	−0.985870	−0.492935	−	0.870066i	$-0.664076\pi$
$41$	−6.31265	−0.985870	−0.492935	+	0.870066i	$0.664076\pi$
$42$	0	0
$43$	−4.12601	−0.629210	−0.314605	−	0.949223i	$-0.601872\pi$
$43$	−4.12601	−0.629210	−0.314605	+	0.949223i	$0.601872\pi$
$44$	−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	$0.578572\pi$
$48$	0	0
$49$	4.22425	0.603465
$50$	1.48119	0.209473
$51$	0	0
$52$	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	$0.540414\pi$
$54$	0	0
$55$	0.962389	0.129768
$56$	−8.96239	−1.19765
$57$	0	0
$58$	5.35026	0.702524
$59$	6.38787	0.831630	0.415815	−	0.909449i	$-0.363496\pi$
$59$	6.38787	0.831630	0.415815	+	0.909449i	$0.363496\pi$
$60$	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
$63$	0	0
$64$	7.08110	0.885138
$65$	−6.15633	−0.763598
$66$	0	0
$67$	−6.73084	−0.822303	−0.411152	−	0.911567i	$-0.634873\pi$
$67$	−6.73084	−0.822303	−0.411152	+	0.911567i	$0.634873\pi$
$68$	1.22425	0.148463
$69$	0	0
$70$	−4.96239	−0.593119
$71$	0.775746	0.0920641	0.0460321	−	0.998940i	$-0.485342\pi$
$71$	0.775746	0.0920641	0.0460321	+	0.998940i	$0.485342\pi$
$72$	0	0
$73$	0.387873	0.0453971	0.0226986	−	0.999742i	$-0.492774\pi$
$73$	0.387873	0.0453971	0.0226986	+	0.999742i	$0.492774\pi$
$74$	15.0435	1.74877
$75$	0	0
$76$	−0.193937	−0.0222460
$77$	−3.22425	−0.367438
$78$	0	0
$79$	−0.836381	−0.0941002	−0.0470501	−	0.998893i	$-0.514982\pi$
$79$	−0.836381	−0.0941002	−0.0470501	+	0.998893i	$0.514982\pi$
$80$	4.35026	0.486374
$81$	0	0
$82$	−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	0
$85$	−6.31265	−0.684703
$86$	−6.11142	−0.659011
$87$	0	0
$88$	2.57452	0.274444
$89$	−7.08840	−0.751369	−0.375684	−	0.926748i	$-0.622592\pi$
$89$	−7.08840	−0.751369	−0.375684	+	0.926748i	$0.622592\pi$
$90$	0	0
$91$	20.6253	2.16212
$92$	0.962389	0.100336
$93$	0	0
$94$	−4.96239	−0.511831
$95$	1.00000	0.102598
$96$	0	0
$97$	10.9927	1.11614	0.558070	−	0.829794i	$-0.311542\pi$
$97$	10.9927	1.11614	0.558070	+	0.829794i	$0.311542\pi$
$98$	6.25694	0.632046
$99$	0	0
$100$	0.193937	0.0193937
$101$	2.64974	0.263659	0.131829	−	0.991272i	$-0.457915\pi$
$101$	2.64974	0.263659	0.131829	+	0.991272i	$0.457915\pi$
$102$	0	0
$103$	−10.7308	−1.05734	−0.528671	−	0.848827i	$-0.677309\pi$
$103$	−10.7308	−1.05734	−0.528671	+	0.848827i	$0.677309\pi$
$104$	−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	0
$109$	−2.77575	−0.265868	−0.132934	−	0.991125i	$-0.542440\pi$
$109$	−2.77575	−0.265868	−0.132934	+	0.991125i	$0.542440\pi$
$110$	1.42548	0.135915
$111$	0	0
$112$	−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	$0.606681\pi$
$114$	0	0
$115$	−4.96239	−0.462745
$116$	0.700523	0.0650420
$117$	0	0
$118$	9.46168	0.871018
$119$	21.1490	1.93873
$120$	0	0
$121$	−10.0738	−0.915801
$122$	−16.7005	−1.51199
$123$	0	0
$124$	−1.14903	−0.103186
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.4314	1.19184	0.595920	−	0.803043i	$-0.296787\pi$
$127$	13.4314	1.19184	0.595920	+	0.803043i	$0.296787\pi$
$128$	12.6751	1.12033
$129$	0	0
$130$	−9.11871	−0.799764
$131$	−20.6253	−1.80204	−0.901020	−	0.433777i	$-0.857181\pi$
$131$	−20.6253	−1.80204	−0.901020	+	0.433777i	$0.857181\pi$
$132$	0	0
$133$	−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	$0.832365\pi$
$138$	0	0
$139$	−17.5877	−1.49177	−0.745884	−	0.666076i	$-0.767973\pi$
$139$	−17.5877	−1.49177	−0.745884	+	0.666076i	$0.767973\pi$
$140$	−0.649738	−0.0549129
$141$	0	0
$142$	1.14903	0.0964245
$143$	−5.92478	−0.495455
$144$	0	0
$145$	−3.61213	−0.299971
$146$	0.574515	0.0475472
$147$	0	0
$148$	1.96968	0.161907
$149$	7.42548	0.608319	0.304160	−	0.952621i	$-0.401624\pi$
$149$	7.42548	0.608319	0.304160	+	0.952621i	$0.401624\pi$
$150$	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
$153$	0	0
$154$	−4.77575	−0.384841
$155$	5.92478	0.475890
$156$	0	0
$157$	4.38787	0.350190	0.175095	−	0.984552i	$-0.443977\pi$
$157$	4.38787	0.350190	0.175095	+	0.984552i	$0.443977\pi$
$158$	−1.23884	−0.0985570
$159$	0	0
$160$	1.09332	0.0864346
$161$	16.6253	1.31026
$162$	0	0
$163$	−0.649738	−0.0508914	−0.0254457	−	0.999676i	$-0.508100\pi$
$163$	−0.649738	−0.0508914	−0.0254457	+	0.999676i	$0.508100\pi$
$164$	−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	$0.297490\pi$
$168$	0	0
$169$	24.9003	1.91541
$170$	−9.35026	−0.717132
$171$	0	0
$172$	−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	$0.539362\pi$
$174$	0	0
$175$	3.35026	0.253256
$176$	4.18664	0.315580
$177$	0	0
$178$	−10.4993	−0.786955
$179$	−15.0132	−1.12214	−0.561069	−	0.827769i	$-0.689610\pi$
$179$	−15.0132	−1.12214	−0.561069	+	0.827769i	$0.689610\pi$
$180$	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
$183$	0	0
$184$	−13.2750	−0.978649
$185$	−10.1563	−0.746708
$186$	0	0
$187$	−6.07522	−0.444264
$188$	−0.649738	−0.0473870
$189$	0	0
$190$	1.48119	0.107457
$191$	21.7743	1.57554	0.787768	−	0.615972i	$-0.211237\pi$
$191$	21.7743	1.57554	0.787768	+	0.615972i	$0.211237\pi$
$192$	0	0
$193$	12.5442	0.902951	0.451476	−	0.892283i	$-0.350898\pi$
$193$	12.5442	0.902951	0.451476	+	0.892283i	$0.350898\pi$
$194$	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	$0.838850\pi$
$198$	0	0
$199$	23.0738	1.63566	0.817829	−	0.575461i	$-0.195177\pi$
$199$	23.0738	1.63566	0.817829	+	0.575461i	$0.195177\pi$
$200$	−2.67513	−0.189160
$201$	0	0
$202$	3.92478	0.276146
$203$	12.1016	0.849364
$204$	0	0
$205$	6.31265	0.440895
$206$	−15.8945	−1.10742
$207$	0	0
$208$	−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	0
$214$	−7.11871	−0.486625
$215$	4.12601	0.281391
$216$	0	0
$217$	−19.8496	−1.34748
$218$	−4.11142	−0.278460
$219$	0	0
$220$	0.186642	0.0125834
$221$	38.8627	2.61419
$222$	0	0
$223$	−0.0303172	−0.00203019	−0.00101509	−	0.999999i	$-0.500323\pi$
$223$	−0.0303172	−0.00203019	−0.00101509	+	0.999999i	$0.500323\pi$
$224$	−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	0
$229$	1.87399	0.123837	0.0619184	−	0.998081i	$-0.480278\pi$
$229$	1.87399	0.123837	0.0619184	+	0.998081i	$0.480278\pi$
$230$	−7.35026	−0.484662
$231$	0	0
$232$	−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	$0.381001\pi$
$234$	0	0
$235$	3.35026	0.218547
$236$	1.23884	0.0806418
$237$	0	0
$238$	31.3258	2.03055
$239$	−9.29948	−0.601533	−0.300767	−	0.953698i	$-0.597243\pi$
$239$	−9.29948	−0.601533	−0.300767	+	0.953698i	$0.597243\pi$
$240$	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
$243$	0	0
$244$	−2.18664	−0.139985
$245$	−4.22425	−0.269878
$246$	0	0
$247$	−6.15633	−0.391718
$248$	15.8496	1.00645
$249$	0	0
$250$	−1.48119	−0.0936790
$251$	−24.1016	−1.52128	−0.760639	−	0.649175i	$-0.775114\pi$
$251$	−24.1016	−1.52128	−0.760639	+	0.649175i	$0.775114\pi$
$252$	0	0
$253$	−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	$0.363628\pi$
$258$	0	0
$259$	34.0263	2.11429
$260$	−1.19394	−0.0740448
$261$	0	0
$262$	−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	$0.630867\pi$
$264$	0	0
$265$	1.84367	0.113256
$266$	−4.96239	−0.304264
$267$	0	0
$268$	−1.30536	−0.0797373
$269$	11.4010	0.695134	0.347567	−	0.937655i	$-0.387008\pi$
$269$	11.4010	0.695134	0.347567	+	0.937655i	$0.387008\pi$
$270$	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
$273$	0	0
$274$	−29.9756	−1.81089
$275$	−0.962389	−0.0580342
$276$	0	0
$277$	−29.7889	−1.78984	−0.894921	−	0.446224i	$-0.852769\pi$
$277$	−29.7889	−1.78984	−0.894921	+	0.446224i	$0.852769\pi$
$278$	−26.0508	−1.56242
$279$	0	0
$280$	8.96239	0.535605
$281$	11.6121	0.692721	0.346361	−	0.938101i	$-0.387417\pi$
$281$	11.6121	0.692721	0.346361	+	0.938101i	$0.387417\pi$
$282$	0	0
$283$	2.26187	0.134454	0.0672270	−	0.997738i	$-0.478585\pi$
$283$	2.26187	0.134454	0.0672270	+	0.997738i	$0.478585\pi$
$284$	0.150446	0.00892730
$285$	0	0
$286$	−8.77575	−0.518921
$287$	−21.1490	−1.24839
$288$	0	0
$289$	22.8496	1.34409
$290$	−5.35026	−0.314178
$291$	0	0
$292$	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	$0.517151\pi$
$294$	0	0
$295$	−6.38787	−0.371916
$296$	−27.1695	−1.57920
$297$	0	0
$298$	10.9986	0.637131
$299$	30.5501	1.76676
$300$	0	0
$301$	−13.8232	−0.796756
$302$	−2.38787	−0.137407
$303$	0	0
$304$	4.35026	0.249505
$305$	11.2750	0.645607
$306$	0	0
$307$	26.2071	1.49572	0.747859	−	0.663857i	$-0.231082\pi$
$307$	26.2071	1.49572	0.747859	+	0.663857i	$0.231082\pi$
$308$	−0.625301	−0.0356298
$309$	0	0
$310$	8.77575	0.498429
$311$	−6.51388	−0.369368	−0.184684	−	0.982798i	$-0.559126\pi$
$311$	−6.51388	−0.369368	−0.184684	+	0.982798i	$0.559126\pi$
$312$	0	0
$313$	16.0752	0.908625	0.454313	−	0.890842i	$-0.349885\pi$
$313$	16.0752	0.908625	0.454313	+	0.890842i	$0.349885\pi$
$314$	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	$0.448889\pi$
$318$	0	0
$319$	−3.47627	−0.194634
$320$	−7.08110	−0.395846
$321$	0	0
$322$	24.6253	1.37231
$323$	−6.31265	−0.351245
$324$	0	0
$325$	6.15633	0.341491
$326$	−0.962389	−0.0533018
$327$	0	0
$328$	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
$333$	0	0
$334$	22.7454	1.24457
$335$	6.73084	0.367745
$336$	0	0
$337$	3.76845	0.205281	0.102640	−	0.994719i	$-0.467271\pi$
$337$	3.76845	0.205281	0.102640	+	0.994719i	$0.467271\pi$
$338$	36.8822	2.00613
$339$	0	0
$340$	−1.22425	−0.0663945
$341$	5.70194	0.308777
$342$	0	0
$343$	−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	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	4.96239	0.265251
$351$	0	0
$352$	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	$0.547074\pi$
$354$	0	0
$355$	−0.775746	−0.0411723
$356$	−1.37470	−0.0728589
$357$	0	0
$358$	−22.2374	−1.17528
$359$	10.3634	0.546961	0.273481	−	0.961878i	$-0.411825\pi$
$359$	10.3634	0.546961	0.273481	+	0.961878i	$0.411825\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	13.6629	0.718107
$363$	0	0
$364$	4.00000	0.209657
$365$	−0.387873	−0.0203022
$366$	0	0
$367$	3.35026	0.174882	0.0874411	−	0.996170i	$-0.472131\pi$
$367$	3.35026	0.174882	0.0874411	+	0.996170i	$0.472131\pi$
$368$	−21.5877	−1.12534
$369$	0	0
$370$	−15.0435	−0.782074
$371$	−6.17679	−0.320683
$372$	0	0
$373$	−12.6048	−0.652653	−0.326327	−	0.945257i	$-0.605811\pi$
$373$	−12.6048	−0.652653	−0.326327	+	0.945257i	$0.605811\pi$
$374$	−8.99859	−0.465306
$375$	0	0
$376$	8.96239	0.462200
$377$	22.2374	1.14529
$378$	0	0
$379$	−37.2506	−1.91343	−0.956717	−	0.291018i	$-0.906006\pi$
$379$	−37.2506	−1.91343	−0.956717	+	0.291018i	$0.906006\pi$
$380$	0.193937	0.00994874
$381$	0	0
$382$	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	$0.788744\pi$
$384$	0	0
$385$	3.22425	0.164323
$386$	18.5804	0.945717
$387$	0	0
$388$	2.13189	0.108230
$389$	1.37470	0.0697000	0.0348500	−	0.999393i	$-0.488905\pi$
$389$	1.37470	0.0697000	0.0348500	+	0.999393i	$0.488905\pi$
$390$	0	0
$391$	31.3258	1.58422
$392$	−11.3004	−0.570758
$393$	0	0
$394$	−36.3634	−1.83196
$395$	0.836381	0.0420829
$396$	0	0
$397$	−9.38646	−0.471093	−0.235546	−	0.971863i	$-0.575688\pi$
$397$	−9.38646	−0.471093	−0.235546	+	0.971863i	$0.575688\pi$
$398$	34.1768	1.71313
$399$	0	0
$400$	−4.35026	−0.217513
$401$	−14.1016	−0.704199	−0.352099	−	0.935963i	$-0.614532\pi$
$401$	−14.1016	−0.704199	−0.352099	+	0.935963i	$0.614532\pi$
$402$	0	0
$403$	−36.4749	−1.81694
$404$	0.513881	0.0255665
$405$	0	0
$406$	17.9248	0.889592
$407$	−9.77433	−0.484496
$408$	0	0
$409$	35.1490	1.73801	0.869004	−	0.494805i	$-0.164761\pi$
$409$	35.1490	1.73801	0.869004	+	0.494805i	$0.164761\pi$
$410$	9.35026	0.461777
$411$	0	0
$412$	−2.08110	−0.102529
$413$	21.4010	1.05308
$414$	0	0
$415$	−7.03761	−0.345463
$416$	−6.73084	−0.330007
$417$	0	0
$418$	1.42548	0.0697227
$419$	−9.02776	−0.441035	−0.220518	−	0.975383i	$-0.570775\pi$
$419$	−9.02776	−0.441035	−0.220518	+	0.975383i	$0.570775\pi$
$420$	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
$423$	0	0
$424$	4.93207	0.239523
$425$	6.31265	0.306209
$426$	0	0
$427$	−37.7743	−1.82803
$428$	−0.932071	−0.0450534
$429$	0	0
$430$	6.11142	0.294719
$431$	−16.3127	−0.785753	−0.392876	−	0.919591i	$-0.628520\pi$
$431$	−16.3127	−0.785753	−0.392876	+	0.919591i	$0.628520\pi$
$432$	0	0
$433$	11.1432	0.535506	0.267753	−	0.963488i	$-0.413719\pi$
$433$	11.1432	0.535506	0.267753	+	0.963488i	$0.413719\pi$
$434$	−29.4010	−1.41130
$435$	0	0
$436$	−0.538319	−0.0257808
$437$	−4.96239	−0.237383
$438$	0	0
$439$	27.3865	1.30708	0.653542	−	0.756890i	$-0.273282\pi$
$439$	27.3865	1.30708	0.653542	+	0.756890i	$0.273282\pi$
$440$	−2.57452	−0.122735
$441$	0	0
$442$	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	$0.653418\pi$
$444$	0	0
$445$	7.08840	0.336022
$446$	−0.0449056	−0.00212634
$447$	0	0
$448$	23.7235	1.12083
$449$	22.1016	1.04304	0.521519	−	0.853240i	$-0.325366\pi$
$449$	22.1016	1.04304	0.521519	+	0.853240i	$0.325366\pi$
$450$	0	0
$451$	6.07522	0.286071
$452$	−1.35614	−0.0637875
$453$	0	0
$454$	−7.11871	−0.334098
$455$	−20.6253	−0.966929
$456$	0	0
$457$	−17.8496	−0.834967	−0.417483	−	0.908685i	$-0.637088\pi$
$457$	−17.8496	−0.834967	−0.417483	+	0.908685i	$0.637088\pi$
$458$	2.77575	0.129702
$459$	0	0
$460$	−0.962389	−0.0448716
$461$	35.2506	1.64178	0.820892	−	0.571083i	$-0.193477\pi$
$461$	35.2506	1.64178	0.820892	+	0.571083i	$0.193477\pi$
$462$	0	0
$463$	−26.3634	−1.22521	−0.612606	−	0.790388i	$-0.709879\pi$
$463$	−26.3634	−1.22521	−0.612606	+	0.790388i	$0.709879\pi$
$464$	−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	$0.449818\pi$
$468$	0	0
$469$	−22.5501	−1.04127
$470$	4.96239	0.228898
$471$	0	0
$472$	−17.0884	−0.786557
$473$	3.97082	0.182579
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	4.10157	0.187995
$477$	0	0
$478$	−13.7743	−0.630023
$479$	−12.7104	−0.580752	−0.290376	−	0.956913i	$-0.593780\pi$
$479$	−12.7104	−0.580752	−0.290376	+	0.956913i	$0.593780\pi$
$480$	0	0
$481$	62.5256	2.85092
$482$	−3.42548	−0.156027
$483$	0	0
$484$	−1.95368	−0.0888036
$485$	−10.9927	−0.499153
$486$	0	0
$487$	−15.7586	−0.714090	−0.357045	−	0.934087i	$-0.616216\pi$
$487$	−15.7586	−0.714090	−0.357045	+	0.934087i	$0.616216\pi$
$488$	30.1622	1.36538
$489$	0	0
$490$	−6.25694	−0.282660
$491$	14.5501	0.656636	0.328318	−	0.944567i	$-0.393518\pi$
$491$	14.5501	0.656636	0.328318	+	0.944567i	$0.393518\pi$
$492$	0	0
$493$	22.8021	1.02695
$494$	−9.11871	−0.410270
$495$	0	0
$496$	25.7743	1.15730
$497$	2.59895	0.116579
$498$	0	0
$499$	−5.48612	−0.245592	−0.122796	−	0.992432i	$-0.539186\pi$
$499$	−5.48612	−0.245592	−0.122796	+	0.992432i	$0.539186\pi$
$500$	−0.193937	−0.00867311
$501$	0	0
$502$	−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	$0.195455\pi$
$504$	0	0
$505$	−2.64974	−0.117912
$506$	−7.07381	−0.314469
$507$	0	0
$508$	2.60483	0.115571
$509$	−39.1900	−1.73706	−0.868532	−	0.495632i	$-0.834936\pi$
$509$	−39.1900	−1.73706	−0.868532	+	0.495632i	$0.834936\pi$
$510$	0	0
$511$	1.29948	0.0574855
$512$	−18.5188	−0.818423
$513$	0	0
$514$	19.7294	0.870228
$515$	10.7308	0.472857
$516$	0	0
$517$	3.22425	0.141803
$518$	50.3996	2.21443
$519$	0	0
$520$	16.4690	0.722212
$521$	17.7283	0.776690	0.388345	−	0.921514i	$-0.373047\pi$
$521$	17.7283	0.776690	0.388345	+	0.921514i	$0.373047\pi$
$522$	0	0
$523$	−40.7572	−1.78219	−0.891094	−	0.453819i	$-0.850061\pi$
$523$	−40.7572	−1.78219	−0.891094	+	0.453819i	$0.850061\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−19.1998	−0.837152
$527$	−37.4010	−1.62922
$528$	0	0
$529$	1.62530	0.0706652
$530$	2.73084	0.118620
$531$	0	0
$532$	−0.649738	−0.0281697
$533$	−38.8627	−1.68333
$534$	0	0
$535$	4.80606	0.207784
$536$	18.0059	0.777736
$537$	0	0
$538$	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
$543$	0	0
$544$	−6.90175	−0.295910
$545$	2.77575	0.118900
$546$	0	0
$547$	8.55405	0.365745	0.182872	−	0.983137i	$-0.441461\pi$
$547$	8.55405	0.365745	0.182872	+	0.983137i	$0.441461\pi$
$548$	−3.92478	−0.167658
$549$	0	0
$550$	−1.42548	−0.0607829
$551$	−3.61213	−0.153882
$552$	0	0
$553$	−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	$0.471386\pi$
$558$	0	0
$559$	−25.4010	−1.07435
$560$	14.5745	0.615886
$561$	0	0
$562$	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	0
$565$	6.99271	0.294185
$566$	3.35026	0.140822
$567$	0	0
$568$	−2.07522	−0.0870744
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	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
$573$	0	0
$574$	−31.3258	−1.30751
$575$	4.96239	0.206946
$576$	0	0
$577$	30.1016	1.25314	0.626572	−	0.779363i	$-0.284457\pi$
$577$	30.1016	1.25314	0.626572	+	0.779363i	$0.284457\pi$
$578$	33.8446	1.40775
$579$	0	0
$580$	−0.700523	−0.0290877
$581$	23.5778	0.978174
$582$	0	0
$583$	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	0
$589$	5.92478	0.244126
$590$	−9.46168	−0.389531
$591$	0	0
$592$	−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	$0.748949\pi$
$594$	0	0
$595$	−21.1490	−0.867026
$596$	1.44007	0.0589877
$597$	0	0
$598$	45.2506	1.85043
$599$	−14.6107	−0.596978	−0.298489	−	0.954413i	$-0.596483\pi$
$599$	−14.6107	−0.596978	−0.298489	+	0.954413i	$0.596483\pi$
$600$	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
$603$	0	0
$604$	−0.312650	−0.0127216
$605$	10.0738	0.409559
$606$	0	0
$607$	−8.80606	−0.357427	−0.178714	−	0.983901i	$-0.557194\pi$
$607$	−8.80606	−0.357427	−0.178714	+	0.983901i	$0.557194\pi$
$608$	1.09332	0.0443400
$609$	0	0
$610$	16.7005	0.676184
$611$	−20.6253	−0.834410
$612$	0	0
$613$	10.4142	0.420626	0.210313	−	0.977634i	$-0.432552\pi$
$613$	10.4142	0.420626	0.210313	+	0.977634i	$0.432552\pi$
$614$	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	$0.386884\pi$
$618$	0	0
$619$	−10.6351	−0.427463	−0.213731	−	0.976892i	$-0.568562\pi$
$619$	−10.6351	−0.427463	−0.213731	+	0.976892i	$0.568562\pi$
$620$	1.14903	0.0461462
$621$	0	0
$622$	−9.64832	−0.386863
$623$	−23.7480	−0.951443
$624$	0	0
$625$	1.00000	0.0400000
$626$	23.8105	0.951660
$627$	0	0
$628$	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
$633$	0	0
$634$	8.43278	0.334908
$635$	−13.4314	−0.533007
$636$	0	0
$637$	26.0059	1.03039
$638$	−5.14903	−0.203852
$639$	0	0
$640$	−12.6751	−0.501029
$641$	16.7612	0.662026	0.331013	−	0.943626i	$-0.392610\pi$
$641$	16.7612	0.662026	0.331013	+	0.943626i	$0.392610\pi$
$642$	0	0
$643$	5.73813	0.226290	0.113145	−	0.993578i	$-0.463908\pi$
$643$	5.73813	0.226290	0.113145	+	0.993578i	$0.463908\pi$
$644$	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	$0.238809\pi$
$648$	0	0
$649$	−6.14762	−0.241315
$650$	9.11871	0.357665
$651$	0	0
$652$	−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	$0.573923\pi$
$654$	0	0
$655$	20.6253	0.805897
$656$	27.4617	1.07220
$657$	0	0
$658$	−16.6253	−0.648122
$659$	−1.23884	−0.0482584	−0.0241292	−	0.999709i	$-0.507681\pi$
$659$	−1.23884	−0.0482584	−0.0241292	+	0.999709i	$0.507681\pi$
$660$	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
$663$	0	0
$664$	−18.8265	−0.730611
$665$	3.35026	0.129918
$666$	0	0
$667$	17.9248	0.694050
$668$	2.97812	0.115227
$669$	0	0
$670$	9.96968	0.385162
$671$	10.8510	0.418897
$672$	0	0
$673$	39.9307	1.53921	0.769607	−	0.638518i	$-0.220452\pi$
$673$	39.9307	1.53921	0.769607	+	0.638518i	$0.220452\pi$
$674$	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	$0.481313\pi$
$678$	0	0
$679$	36.8284	1.41335
$680$	16.8872	0.647593
$681$	0	0
$682$	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	0
$685$	20.2374	0.773232
$686$	−13.7743	−0.525906
$687$	0	0
$688$	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
$693$	0	0
$694$	33.1246	1.25739
$695$	17.5877	0.667139
$696$	0	0
$697$	−39.8496	−1.50941
$698$	−14.8119	−0.560640
$699$	0	0
$700$	0.649738	0.0245578
$701$	25.0494	0.946102	0.473051	−	0.881035i	$-0.343153\pi$
$701$	25.0494	0.946102	0.473051	+	0.881035i	$0.343153\pi$
$702$	0	0
$703$	−10.1563	−0.383053
$704$	−6.81477	−0.256841
$705$	0	0
$706$	−8.20123	−0.308657
$707$	8.87732	0.333866
$708$	0	0
$709$	−41.6991	−1.56604	−0.783021	−	0.621995i	$-0.786323\pi$
$709$	−41.6991	−1.56604	−0.783021	+	0.621995i	$0.786323\pi$
$710$	−1.14903	−0.0431224
$711$	0	0
$712$	18.9624	0.710646
$713$	−29.4010	−1.10108
$714$	0	0
$715$	5.92478	0.221574
$716$	−2.91160	−0.108812
$717$	0	0
$718$	15.3503	0.572867
$719$	−30.6351	−1.14250	−0.571249	−	0.820777i	$-0.693541\pi$
$719$	−30.6351	−1.14250	−0.571249	+	0.820777i	$0.693541\pi$
$720$	0	0
$721$	−35.9511	−1.33889
$722$	1.48119	0.0551243
$723$	0	0
$724$	1.78892	0.0664847
$725$	3.61213	0.134151
$726$	0	0
$727$	7.50071	0.278186	0.139093	−	0.990279i	$-0.455581\pi$
$727$	7.50071	0.278186	0.139093	+	0.990279i	$0.455581\pi$
$728$	−55.1754	−2.04494
$729$	0	0
$730$	−0.574515	−0.0212638
$731$	−26.0460	−0.963348
$732$	0	0
$733$	9.84955	0.363802	0.181901	−	0.983317i	$-0.441775\pi$
$733$	9.84955	0.363802	0.181901	+	0.983317i	$0.441775\pi$
$734$	4.96239	0.183165
$735$	0	0
$736$	−5.42548	−0.199986
$737$	6.47768	0.238609
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−1.96968	−0.0724070
$741$	0	0
$742$	−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	$0.406906\pi$
$744$	0	0
$745$	−7.42548	−0.272049
$746$	−18.6702	−0.683565
$747$	0	0
$748$	−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
$753$	0	0
$754$	32.9380	1.19953
$755$	1.61213	0.0586713
$756$	0	0
$757$	−15.7743	−0.573328	−0.286664	−	0.958031i	$-0.592546\pi$
$757$	−15.7743	−0.573328	−0.286664	+	0.958031i	$0.592546\pi$
$758$	−55.1754	−2.00406
$759$	0	0
$760$	−2.67513	−0.0970372
$761$	−43.2262	−1.56695	−0.783474	−	0.621425i	$-0.786554\pi$
$761$	−43.2262	−1.56695	−0.783474	+	0.621425i	$0.786554\pi$
$762$	0	0
$763$	−9.29948	−0.336664
$764$	4.22284	0.152777
$765$	0	0
$766$	−45.6688	−1.65008
$767$	39.3258	1.41997
$768$	0	0
$769$	−19.1246	−0.689650	−0.344825	−	0.938667i	$-0.612062\pi$
$769$	−19.1246	−0.689650	−0.344825	+	0.938667i	$0.612062\pi$
$770$	4.77575	0.172106
$771$	0	0
$772$	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	$0.654126\pi$
$774$	0	0
$775$	−5.92478	−0.212824
$776$	−29.4069	−1.05565
$777$	0	0
$778$	2.03620	0.0730012
$779$	6.31265	0.226174
$780$	0	0
$781$	−0.746569	−0.0267144
$782$	46.3996	1.65925
$783$	0	0
$784$	−18.3766	−0.656307
$785$	−4.38787	−0.156610
$786$	0	0
$787$	19.9814	0.712261	0.356131	−	0.934436i	$-0.384096\pi$
$787$	19.9814	0.712261	0.356131	+	0.934436i	$0.384096\pi$
$788$	−4.76116	−0.169609
$789$	0	0
$790$	1.23884	0.0440760
$791$	−23.4274	−0.832982
$792$	0	0
$793$	−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	$0.669372\pi$
$798$	0	0
$799$	−21.1490	−0.748199
$800$	−1.09332	−0.0386547
$801$	0	0
$802$	−20.8872	−0.737551
$803$	−0.373285	−0.0131729
$804$	0	0
$805$	−16.6253	−0.585965
$806$	−54.0263	−1.90300
$807$	0	0
$808$	−7.08840	−0.249369
$809$	17.2243	0.605573	0.302786	−	0.953058i	$-0.402083\pi$
$809$	17.2243	0.605573	0.302786	+	0.953058i	$0.402083\pi$
$810$	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
$813$	0	0
$814$	−14.4777	−0.507443
$815$	0.649738	0.0227593
$816$	0	0
$817$	4.12601	0.144351
$818$	52.0625	1.82032
$819$	0	0
$820$	1.22425	0.0427528
$821$	−39.2506	−1.36986	−0.684928	−	0.728611i	$-0.740166\pi$
$821$	−39.2506	−1.36986	−0.684928	+	0.728611i	$0.740166\pi$
$822$	0	0
$823$	−45.5271	−1.58697	−0.793487	−	0.608588i	$-0.791736\pi$
$823$	−45.5271	−1.58697	−0.793487	+	0.608588i	$0.791736\pi$
$824$	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	$0.595916\pi$
$828$	0	0
$829$	1.69911	0.0590125	0.0295062	−	0.999565i	$-0.490607\pi$
$829$	1.69911	0.0590125	0.0295062	+	0.999565i	$0.490607\pi$
$830$	−10.4241	−0.361825
$831$	0	0
$832$	43.5936	1.51134
$833$	26.6662	0.923930
$834$	0	0
$835$	−15.3561	−0.531421
$836$	0.186642	0.00645516
$837$	0	0
$838$	−13.3719	−0.461924
$839$	50.5910	1.74660	0.873298	−	0.487187i	$-0.161977\pi$
$839$	50.5910	1.74660	0.873298	+	0.487187i	$0.161977\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	22.5285	0.776382
$843$	0	0
$844$	−4.06063	−0.139773
$845$	−24.9003	−0.856598
$846$	0	0
$847$	−33.7499	−1.15966
$848$	8.02047	0.275424
$849$	0	0
$850$	9.35026	0.320711
$851$	50.3996	1.72768
$852$	0	0
$853$	−22.5237	−0.771198	−0.385599	−	0.922667i	$-0.626005\pi$
$853$	−22.5237	−0.771198	−0.385599	+	0.922667i	$0.626005\pi$
$854$	−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	$0.367833\pi$
$858$	0	0
$859$	15.1754	0.517777	0.258889	−	0.965907i	$-0.416644\pi$
$859$	15.1754	0.517777	0.258889	+	0.965907i	$0.416644\pi$
$860$	0.800184	0.0272860
$861$	0	0
$862$	−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	$0.671244\pi$
$864$	0	0
$865$	3.24472	0.110324
$866$	16.5052	0.560869
$867$	0	0
$868$	−3.84955	−0.130662
$869$	0.804923	0.0273051
$870$	0	0
$871$	−41.4372	−1.40405
$872$	7.42548	0.251459
$873$	0	0
$874$	−7.35026	−0.248626
$875$	−3.35026	−0.113260
$876$	0	0
$877$	−5.53102	−0.186769	−0.0933847	−	0.995630i	$-0.529769\pi$
$877$	−5.53102	−0.186769	−0.0933847	+	0.995630i	$0.529769\pi$
$878$	40.5647	1.36899
$879$	0	0
$880$	−4.18664	−0.141132
$881$	−20.8265	−0.701664	−0.350832	−	0.936438i	$-0.614101\pi$
$881$	−20.8265	−0.701664	−0.350832	+	0.936438i	$0.614101\pi$
$882$	0	0
$883$	43.1509	1.45214	0.726072	−	0.687618i	$-0.241344\pi$
$883$	43.1509	1.45214	0.726072	+	0.687618i	$0.241344\pi$
$884$	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	$0.231083\pi$
$888$	0	0
$889$	44.9986	1.50920
$890$	10.4993	0.351937
$891$	0	0
$892$	−0.00587961	−0.000196864 0
$893$	3.35026	0.112112
$894$	0	0
$895$	15.0132	0.501835
$896$	42.4650	1.41866
$897$	0	0
$898$	32.7367	1.09244
$899$	−21.4010	−0.713765
$900$	0	0
$901$	−11.6385	−0.387734
$902$	8.99859	0.299620
$903$	0	0
$904$	18.7064	0.622166
$905$	−9.22425	−0.306625
$906$	0	0
$907$	53.7558	1.78493	0.892466	−	0.451115i	$-0.148974\pi$
$907$	53.7558	1.78493	0.892466	+	0.451115i	$0.148974\pi$
$908$	−0.932071	−0.0309319
$909$	0	0
$910$	−30.5501	−1.01273
$911$	2.28630	0.0757486	0.0378743	−	0.999283i	$-0.487941\pi$
$911$	2.28630	0.0757486	0.0378743	+	0.999283i	$0.487941\pi$
$912$	0	0
$913$	−6.77292	−0.224151
$914$	−26.4387	−0.874513
$915$	0	0
$916$	0.363436	0.0120082
$917$	−69.1002	−2.28189
$918$	0	0
$919$	−34.8510	−1.14963	−0.574814	−	0.818284i	$-0.694925\pi$
$919$	−34.8510	−1.14963	−0.574814	+	0.818284i	$0.694925\pi$
$920$	13.2750	0.437665
$921$	0	0
$922$	52.2130	1.71954
$923$	4.77575	0.157196
$924$	0	0
$925$	10.1563	0.333938
$926$	−39.0494	−1.28324
$927$	0	0
$928$	−3.94921	−0.129639
$929$	−41.6991	−1.36810	−0.684052	−	0.729434i	$-0.739784\pi$
$929$	−41.6991	−1.36810	−0.684052	+	0.729434i	$0.739784\pi$
$930$	0	0
$931$	−4.22425	−0.138444
$932$	2.16220	0.0708254
$933$	0	0
$934$	10.0508	0.328872
$935$	6.07522	0.198681
$936$	0	0
$937$	21.9102	0.715775	0.357887	−	0.933765i	$-0.383497\pi$
$937$	21.9102	0.715775	0.357887	+	0.933765i	$0.383497\pi$
$938$	−33.4010	−1.09058
$939$	0	0
$940$	0.649738	0.0211921
$941$	3.55149	0.115775	0.0578877	−	0.998323i	$-0.481563\pi$
$941$	3.55149	0.115775	0.0578877	+	0.998323i	$0.481563\pi$
$942$	0	0
$943$	−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	0
$949$	2.38787	0.0775136
$950$	−1.48119	−0.0480563
$951$	0	0
$952$	−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	$0.379207\pi$
$954$	0	0
$955$	−21.7743	−0.704601
$956$	−1.80351	−0.0583296
$957$	0	0
$958$	−18.8265	−0.608258
$959$	−67.8007	−2.18940
$960$	0	0
$961$	4.10299	0.132354
$962$	92.6126	2.98595
$963$	0	0
$964$	−0.448507	−0.0144455
$965$	−12.5442	−0.403812
$966$	0	0
$967$	27.6629	0.889579	0.444790	−	0.895635i	$-0.353278\pi$
$967$	27.6629	0.889579	0.444790	+	0.895635i	$0.353278\pi$
$968$	26.9488	0.866166
$969$	0	0
$970$	−16.2823	−0.522794
$971$	42.1768	1.35352	0.676759	−	0.736205i	$-0.263384\pi$
$971$	42.1768	1.35352	0.676759	+	0.736205i	$0.263384\pi$
$972$	0	0
$973$	−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	0
$979$	6.82179	0.218025
$980$	−0.819237	−0.0261696
$981$	0	0
$982$	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	$0.756831\pi$
$984$	0	0
$985$	24.5501	0.782231
$986$	33.7743	1.07559
$987$	0	0
$988$	−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
$993$	0	0
$994$	3.84955	0.122100
$995$	−23.0738	−0.731489
$996$	0	0
$997$	13.6873	0.433483	0.216741	−	0.976229i	$-0.430457\pi$
$997$	13.6873	0.433483	0.216741	+	0.976229i	$0.430457\pi$
$998$	−8.12601	−0.257224
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.i.1.3		3
3.2	odd	2		95.2.a.a.1.1	✓	3
5.4	even	2		4275.2.a.bk.1.1		3
12.11	even	2		1520.2.a.p.1.2		3
15.2	even	4		475.2.b.d.324.2		6
15.8	even	4		475.2.b.d.324.5		6
15.14	odd	2		475.2.a.f.1.3		3
21.20	even	2		4655.2.a.u.1.1		3
24.5	odd	2		6080.2.a.bo.1.2		3
24.11	even	2		6080.2.a.by.1.2		3
57.56	even	2		1805.2.a.f.1.3		3
60.59	even	2		7600.2.a.bx.1.2		3
285.284	even	2		9025.2.a.bb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.a.1.1	✓	3	3.2	odd	2
475.2.a.f.1.3		3	15.14	odd	2
475.2.b.d.324.2		6	15.2	even	4
475.2.b.d.324.5		6	15.8	even	4
855.2.a.i.1.3		3	1.1	even	1	trivial
1520.2.a.p.1.2		3	12.11	even	2
1805.2.a.f.1.3		3	57.56	even	2
4275.2.a.bk.1.1		3	5.4	even	2
4655.2.a.u.1.1		3	21.20	even	2
6080.2.a.bo.1.2		3	24.5	odd	2
6080.2.a.by.1.2		3	24.11	even	2
7600.2.a.bx.1.2		3	60.59	even	2
9025.2.a.bb.1.1		3	285.284	even	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} +3.35026 q^{7} -2.67513 q^{8} -1.48119 q^{10} -0.962389 q^{11} +6.15633 q^{13} +4.96239 q^{14} -4.35026 q^{16} +6.31265 q^{17} -1.00000 q^{19} -0.193937 q^{20} -1.42548 q^{22} +4.96239 q^{23} +1.00000 q^{25} +9.11871 q^{26} +0.649738 q^{28} +3.61213 q^{29} -5.92478 q^{31} -1.09332 q^{32} +9.35026 q^{34} -3.35026 q^{35} +10.1563 q^{37} -1.48119 q^{38} +2.67513 q^{40} -6.31265 q^{41} -4.12601 q^{43} -0.186642 q^{44} +7.35026 q^{46} -3.35026 q^{47} +4.22425 q^{49} +1.48119 q^{50} +1.19394 q^{52} -1.84367 q^{53} +0.962389 q^{55} -8.96239 q^{56} +5.35026 q^{58} +6.38787 q^{59} -11.2750 q^{61} -8.77575 q^{62} +7.08110 q^{64} -6.15633 q^{65} -6.73084 q^{67} +1.22425 q^{68} -4.96239 q^{70} +0.775746 q^{71} +0.387873 q^{73} +15.0435 q^{74} -0.193937 q^{76} -3.22425 q^{77} -0.836381 q^{79} +4.35026 q^{80} -9.35026 q^{82} +7.03761 q^{83} -6.31265 q^{85} -6.11142 q^{86} +2.57452 q^{88} -7.08840 q^{89} +20.6253 q^{91} +0.962389 q^{92} -4.96239 q^{94} +1.00000 q^{95} +10.9927 q^{97} +6.25694 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 3 q^{5} - 3 q^{8} + q^{10} + 8 q^{11} + 8 q^{13} + 4 q^{14} - 3 q^{16} - 2 q^{17} - 3 q^{19} - q^{20} - 16 q^{22} + 4 q^{23} + 3 q^{25} + 6 q^{26} + 12 q^{28} + 10 q^{29} + 4 q^{31}+ \cdots + 15 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 3 * q^5 - 3 * q^8 + q^10 + 8 * q^11 + 8 * q^13 + 4 * q^14 - 3 * q^16 - 2 * q^17 - 3 * q^19 - q^20 - 16 * q^22 + 4 * q^23 + 3 * q^25 + 6 * q^26 + 12 * q^28 + 10 * q^29 + 4 * q^31 + 3 * q^32 + 18 * q^34 + 20 * q^37 + q^38 + 3 * q^40 + 2 * q^41 - 4 * q^43 + 12 * q^44 + 12 * q^46 + 11 * q^49 - q^50 + 4 * q^52 - 16 * q^53 - 8 * q^55 - 16 * q^56 + 6 * q^58 + 20 * q^59 - 2 * q^61 - 28 * q^62 - 11 * q^64 - 8 * q^65 + 2 * q^67 + 2 * q^68 - 4 * q^70 + 4 * q^71 + 2 * q^73 + 2 * q^74 - q^76 - 8 * q^77 + 3 * q^80 - 18 * q^82 + 32 * q^83 + 2 * q^85 + 16 * q^86 - 4 * q^88 - 2 * q^89 + 20 * q^91 - 8 * q^92 - 4 * q^94 + 3 * q^95 + 20 * q^97 + 15 * q^98

Introduction

L-functions

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