LMFDB - Embedded newform 855.2.a.k.1.2

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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ 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)

$f(q)$	$=$	$q+0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -3.59210 q^{7} -1.21432 q^{8} +O(q^{10})$	\(q+0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -3.59210 q^{7} -1.21432 q^{8} -0.311108 q^{10} -3.11753 q^{11} +4.64296 q^{13} -1.11753 q^{14} +3.42864 q^{16} +7.52543 q^{17} +1.00000 q^{19} +1.90321 q^{20} -0.969888 q^{22} +5.52543 q^{23} +1.00000 q^{25} +1.44446 q^{26} +6.83654 q^{28} +8.16839 q^{29} -3.95407 q^{31} +3.49532 q^{32} +2.34122 q^{34} +3.59210 q^{35} -8.83654 q^{37} +0.311108 q^{38} +1.21432 q^{40} +7.87310 q^{41} -5.45875 q^{43} +5.93332 q^{44} +1.71900 q^{46} -9.13828 q^{47} +5.90321 q^{49} +0.311108 q^{50} -8.83654 q^{52} +5.03657 q^{53} +3.11753 q^{55} +4.36196 q^{56} +2.54125 q^{58} -6.42864 q^{59} -0.280996 q^{61} -1.23014 q^{62} -5.76986 q^{64} -4.64296 q^{65} +3.67307 q^{67} -14.3225 q^{68} +1.11753 q^{70} +3.37778 q^{71} +3.18421 q^{73} -2.74912 q^{74} -1.90321 q^{76} +11.1985 q^{77} +12.9906 q^{79} -3.42864 q^{80} +2.44938 q^{82} +8.57628 q^{83} -7.52543 q^{85} -1.69826 q^{86} +3.78568 q^{88} +17.7397 q^{89} -16.6780 q^{91} -10.5161 q^{92} -2.84299 q^{94} -1.00000 q^{95} -7.26517 q^{97} +1.83654 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8	$ 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8} - q^{10} + 4 q^{11} - 6 q^{13} + 10 q^{14} - 3 q^{16} + 16 q^{17} + 3 q^{19} - q^{20} + 4 q^{22} + 10 q^{23} + 3 q^{25} + 4 q^{26} + 14 q^{28} - 2 q^{29} + 8 q^{31} - 3 q^{32} + 14 q^{34} + 4 q^{35} - 20 q^{37} + q^{38} - 3 q^{40} + 10 q^{41} - 10 q^{43} + 18 q^{44} + 12 q^{46} + 6 q^{47} + 11 q^{49} + q^{50} - 20 q^{52} + 8 q^{53} - 4 q^{55} + 14 q^{58} - 6 q^{59} + 6 q^{61} - 10 q^{62} - 11 q^{64} + 6 q^{65} - 2 q^{67} + 4 q^{68} - 10 q^{70} + 10 q^{71} - 4 q^{73} - 22 q^{74} + q^{76} + 14 q^{77} + 12 q^{79} + 3 q^{80} - 26 q^{82} + 6 q^{83} - 16 q^{85} - 32 q^{86} + 18 q^{88} + 40 q^{89} - 4 q^{91} + 2 q^{92} + 12 q^{94} - 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100}) $ 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8 - q^10 + 4 * q^11 - 6 * q^13 + 10 * q^14 - 3 * q^16 + 16 * q^17 + 3 * q^19 - q^20 + 4 * q^22 + 10 * q^23 + 3 * q^25 + 4 * q^26 + 14 * q^28 - 2 * q^29 + 8 * q^31 - 3 * q^32 + 14 * q^34 + 4 * q^35 - 20 * q^37 + q^38 - 3 * q^40 + 10 * q^41 - 10 * q^43 + 18 * q^44 + 12 * q^46 + 6 * q^47 + 11 * q^49 + q^50 - 20 * q^52 + 8 * q^53 - 4 * q^55 + 14 * q^58 - 6 * q^59 + 6 * q^61 - 10 * q^62 - 11 * q^64 + 6 * q^65 - 2 * q^67 + 4 * q^68 - 10 * q^70 + 10 * q^71 - 4 * q^73 - 22 * q^74 + q^76 + 14 * q^77 + 12 * q^79 + 3 * q^80 - 26 * q^82 + 6 * q^83 - 16 * q^85 - 32 * q^86 + 18 * q^88 + 40 * q^89 - 4 * q^91 + 2 * q^92 + 12 * q^94 - 3 * q^95 - 2 * q^97 - q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	0	0
$3$	0	0
$4$	−1.90321	−0.951606
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.59210	−1.35769	−0.678844	−	0.734283i	$-0.737519\pi$
$7$	−3.59210	−1.35769	−0.678844	+	0.734283i	$0.737519\pi$
$8$	−1.21432	−0.429327
$9$	0	0
$10$	−0.311108	−0.0983809
$11$	−3.11753	−0.939971	−0.469986	−	0.882674i	$-0.655741\pi$
$11$	−3.11753	−0.939971	−0.469986	+	0.882674i	$0.655741\pi$
$12$	0	0
$13$	4.64296	1.28773	0.643863	−	0.765141i	$-0.277331\pi$
$13$	4.64296	1.28773	0.643863	+	0.765141i	$0.277331\pi$
$14$	−1.11753	−0.298673
$15$	0	0
$16$	3.42864	0.857160
$17$	7.52543	1.82518	0.912592	−	0.408871i	$-0.134077\pi$
$17$	7.52543	1.82518	0.912592	+	0.408871i	$0.134077\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.90321	0.425571
$21$	0	0
$22$	−0.969888	−0.206781
$23$	5.52543	1.15213	0.576066	−	0.817403i	$-0.304587\pi$
$23$	5.52543	1.15213	0.576066	+	0.817403i	$0.304587\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.44446	0.283282
$27$	0	0
$28$	6.83654	1.29198
$29$	8.16839	1.51683	0.758416	−	0.651771i	$-0.225974\pi$
$29$	8.16839	1.51683	0.758416	+	0.651771i	$0.225974\pi$
$30$	0	0
$31$	−3.95407	−0.710171	−0.355086	−	0.934834i	$-0.615548\pi$
$31$	−3.95407	−0.710171	−0.355086	+	0.934834i	$0.615548\pi$
$32$	3.49532	0.617890
$33$	0	0
$34$	2.34122	0.401516
$35$	3.59210	0.607176
$36$	0	0
$37$	−8.83654	−1.45272	−0.726359	−	0.687316i	$-0.758789\pi$
$37$	−8.83654	−1.45272	−0.726359	+	0.687316i	$0.758789\pi$
$38$	0.311108	0.0504684
$39$	0	0
$40$	1.21432	0.192001
$41$	7.87310	1.22957	0.614786	−	0.788694i	$-0.289242\pi$
$41$	7.87310	1.22957	0.614786	+	0.788694i	$0.289242\pi$
$42$	0	0
$43$	−5.45875	−0.832452	−0.416226	−	0.909261i	$-0.636647\pi$
$43$	−5.45875	−0.832452	−0.416226	+	0.909261i	$0.636647\pi$
$44$	5.93332	0.894482
$45$	0	0
$46$	1.71900	0.253453
$47$	−9.13828	−1.33295	−0.666477	−	0.745525i	$-0.732199\pi$
$47$	−9.13828	−1.33295	−0.666477	+	0.745525i	$0.732199\pi$
$48$	0	0
$49$	5.90321	0.843316
$50$	0.311108	0.0439973
$51$	0	0
$52$	−8.83654	−1.22541
$53$	5.03657	0.691825	0.345913	−	0.938267i	$-0.387569\pi$
$53$	5.03657	0.691825	0.345913	+	0.938267i	$0.387569\pi$
$54$	0	0
$55$	3.11753	0.420368
$56$	4.36196	0.582892
$57$	0	0
$58$	2.54125	0.333682
$59$	−6.42864	−0.836938	−0.418469	−	0.908231i	$-0.637433\pi$
$59$	−6.42864	−0.836938	−0.418469	+	0.908231i	$0.637433\pi$
$60$	0	0
$61$	−0.280996	−0.0359779	−0.0179889	−	0.999838i	$-0.505726\pi$
$61$	−0.280996	−0.0359779	−0.0179889	+	0.999838i	$0.505726\pi$
$62$	−1.23014	−0.156228
$63$	0	0
$64$	−5.76986	−0.721232
$65$	−4.64296	−0.575888
$66$	0	0
$67$	3.67307	0.448737	0.224369	−	0.974504i	$-0.427968\pi$
$67$	3.67307	0.448737	0.224369	+	0.974504i	$0.427968\pi$
$68$	−14.3225	−1.73686
$69$	0	0
$70$	1.11753	0.133571
$71$	3.37778	0.400869	0.200435	−	0.979707i	$-0.435765\pi$
$71$	3.37778	0.400869	0.200435	+	0.979707i	$0.435765\pi$
$72$	0	0
$73$	3.18421	0.372683	0.186342	−	0.982485i	$-0.440337\pi$
$73$	3.18421	0.372683	0.186342	+	0.982485i	$0.440337\pi$
$74$	−2.74912	−0.319578
$75$	0	0
$76$	−1.90321	−0.218313
$77$	11.1985	1.27619
$78$	0	0
$79$	12.9906	1.46156	0.730780	−	0.682613i	$-0.239156\pi$
$79$	12.9906	1.46156	0.730780	+	0.682613i	$0.239156\pi$
$80$	−3.42864	−0.383334
$81$	0	0
$82$	2.44938	0.270489
$83$	8.57628	0.941369	0.470685	−	0.882302i	$-0.344007\pi$
$83$	8.57628	0.941369	0.470685	+	0.882302i	$0.344007\pi$
$84$	0	0
$85$	−7.52543	−0.816247
$86$	−1.69826	−0.183128
$87$	0	0
$88$	3.78568	0.403555
$89$	17.7397	1.88041	0.940205	−	0.340610i	$-0.110634\pi$
$89$	17.7397	1.88041	0.940205	+	0.340610i	$0.110634\pi$
$90$	0	0
$91$	−16.6780	−1.74833
$92$	−10.5161	−1.09638
$93$	0	0
$94$	−2.84299	−0.293232
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−7.26517	−0.737667	−0.368833	−	0.929496i	$-0.620243\pi$
$97$	−7.26517	−0.737667	−0.368833	+	0.929496i	$0.620243\pi$
$98$	1.83654	0.185518
$99$	0	0
$100$	−1.90321	−0.190321
$101$	−10.7239	−1.06707	−0.533535	−	0.845778i	$-0.679137\pi$
$101$	−10.7239	−1.06707	−0.533535	+	0.845778i	$0.679137\pi$
$102$	0	0
$103$	5.24443	0.516749	0.258375	−	0.966045i	$-0.416813\pi$
$103$	5.24443	0.516749	0.258375	+	0.966045i	$0.416813\pi$
$104$	−5.63804	−0.552855
$105$	0	0
$106$	1.56691	0.152192
$107$	6.23506	0.602766	0.301383	−	0.953503i	$-0.402552\pi$
$107$	6.23506	0.602766	0.301383	+	0.953503i	$0.402552\pi$
$108$	0	0
$109$	9.74620	0.933517	0.466758	−	0.884385i	$-0.345422\pi$
$109$	9.74620	0.933517	0.466758	+	0.884385i	$0.345422\pi$
$110$	0.969888	0.0924752
$111$	0	0
$112$	−12.3160	−1.16376
$113$	17.8938	1.68331	0.841656	−	0.540015i	$-0.181581\pi$
$113$	17.8938	1.68331	0.841656	+	0.540015i	$0.181581\pi$
$114$	0	0
$115$	−5.52543	−0.515249
$116$	−15.5462	−1.44343
$117$	0	0
$118$	−2.00000	−0.184115
$119$	−27.0321	−2.47803
$120$	0	0
$121$	−1.28100	−0.116454
$122$	−0.0874201	−0.00791465
$123$	0	0
$124$	7.52543	0.675803
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.86665	−0.520581	−0.260290	−	0.965530i	$-0.583818\pi$
$127$	−5.86665	−0.520581	−0.260290	+	0.965530i	$0.583818\pi$
$128$	−8.78568	−0.776552
$129$	0	0
$130$	−1.44446	−0.126688
$131$	9.25088	0.808254	0.404127	−	0.914703i	$-0.367575\pi$
$131$	9.25088	0.808254	0.404127	+	0.914703i	$0.367575\pi$
$132$	0	0
$133$	−3.59210	−0.311475
$134$	1.14272	0.0987161
$135$	0	0
$136$	−9.13828	−0.783601
$137$	−9.09234	−0.776811	−0.388406	−	0.921489i	$-0.626974\pi$
$137$	−9.09234	−0.776811	−0.388406	+	0.921489i	$0.626974\pi$
$138$	0	0
$139$	19.4795	1.65223	0.826115	−	0.563502i	$-0.190546\pi$
$139$	19.4795	1.65223	0.826115	+	0.563502i	$0.190546\pi$
$140$	−6.83654	−0.577793
$141$	0	0
$142$	1.05086	0.0881858
$143$	−14.4746	−1.21042
$144$	0	0
$145$	−8.16839	−0.678348
$146$	0.990632	0.0819853
$147$	0	0
$148$	16.8178	1.38241
$149$	16.8573	1.38100	0.690501	−	0.723331i	$-0.257390\pi$
$149$	16.8573	1.38100	0.690501	+	0.723331i	$0.257390\pi$
$150$	0	0
$151$	22.4844	1.82976	0.914878	−	0.403731i	$-0.132287\pi$
$151$	22.4844	1.82976	0.914878	+	0.403731i	$0.132287\pi$
$152$	−1.21432	−0.0984943
$153$	0	0
$154$	3.48394	0.280744
$155$	3.95407	0.317598
$156$	0	0
$157$	−0.387152	−0.0308981	−0.0154491	−	0.999881i	$-0.504918\pi$
$157$	−0.387152	−0.0308981	−0.0154491	+	0.999881i	$0.504918\pi$
$158$	4.04149	0.321523
$159$	0	0
$160$	−3.49532	−0.276329
$161$	−19.8479	−1.56423
$162$	0	0
$163$	−17.3067	−1.35556	−0.677781	−	0.735264i	$-0.737058\pi$
$163$	−17.3067	−1.35556	−0.677781	+	0.735264i	$0.737058\pi$
$164$	−14.9842	−1.17007
$165$	0	0
$166$	2.66815	0.207088
$167$	6.81135	0.527078	0.263539	−	0.964649i	$-0.415110\pi$
$167$	6.81135	0.527078	0.263539	+	0.964649i	$0.415110\pi$
$168$	0	0
$169$	8.55707	0.658236
$170$	−2.34122	−0.179563
$171$	0	0
$172$	10.3892	0.792166
$173$	−14.8430	−1.12849	−0.564246	−	0.825607i	$-0.690833\pi$
$173$	−14.8430	−1.12849	−0.564246	+	0.825607i	$0.690833\pi$
$174$	0	0
$175$	−3.59210	−0.271538
$176$	−10.6889	−0.805706
$177$	0	0
$178$	5.51897	0.413665
$179$	6.72393	0.502570	0.251285	−	0.967913i	$-0.419147\pi$
$179$	6.72393	0.502570	0.251285	+	0.967913i	$0.419147\pi$
$180$	0	0
$181$	−4.32693	−0.321618	−0.160809	−	0.986986i	$-0.551410\pi$
$181$	−4.32693	−0.321618	−0.160809	+	0.986986i	$0.551410\pi$
$182$	−5.18865	−0.384609
$183$	0	0
$184$	−6.70964	−0.494641
$185$	8.83654	0.649675
$186$	0	0
$187$	−23.4608	−1.71562
$188$	17.3921	1.26845
$189$	0	0
$190$	−0.311108	−0.0225701
$191$	−17.3526	−1.25559	−0.627795	−	0.778379i	$-0.716042\pi$
$191$	−17.3526	−1.25559	−0.627795	+	0.778379i	$0.716042\pi$
$192$	0	0
$193$	7.56046	0.544214	0.272107	−	0.962267i	$-0.412280\pi$
$193$	7.56046	0.544214	0.272107	+	0.962267i	$0.412280\pi$
$194$	−2.26025	−0.162277
$195$	0	0
$196$	−11.2351	−0.802505
$197$	12.9447	0.922272	0.461136	−	0.887330i	$-0.347442\pi$
$197$	12.9447	0.922272	0.461136	+	0.887330i	$0.347442\pi$
$198$	0	0
$199$	−0.133353	−0.00945315	−0.00472658	−	0.999989i	$-0.501505\pi$
$199$	−0.133353	−0.00945315	−0.00472658	+	0.999989i	$0.501505\pi$
$200$	−1.21432	−0.0858654
$201$	0	0
$202$	−3.33630	−0.234741
$203$	−29.3417	−2.05938
$204$	0	0
$205$	−7.87310	−0.549881
$206$	1.63158	0.113678
$207$	0	0
$208$	15.9190	1.10379
$209$	−3.11753	−0.215644
$210$	0	0
$211$	28.3368	1.95078	0.975392	−	0.220478i	$-0.0707617\pi$
$211$	28.3368	1.95078	0.975392	+	0.220478i	$0.0707617\pi$
$212$	−9.58565	−0.658345
$213$	0	0
$214$	1.93978	0.132600
$215$	5.45875	0.372284
$216$	0	0
$217$	14.2034	0.964191
$218$	3.03212	0.205361
$219$	0	0
$220$	−5.93332	−0.400025
$221$	34.9403	2.35034
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−12.5555	−0.838902
$225$	0	0
$226$	5.56691	0.370306
$227$	0.709636	0.0471002	0.0235501	−	0.999723i	$-0.492503\pi$
$227$	0.709636	0.0471002	0.0235501	+	0.999723i	$0.492503\pi$
$228$	0	0
$229$	−26.4242	−1.74616	−0.873080	−	0.487577i	$-0.837881\pi$
$229$	−26.4242	−1.74616	−0.873080	+	0.487577i	$0.837881\pi$
$230$	−1.71900	−0.113348
$231$	0	0
$232$	−9.91903	−0.651216
$233$	−17.5210	−1.14784	−0.573919	−	0.818912i	$-0.694578\pi$
$233$	−17.5210	−1.14784	−0.573919	+	0.818912i	$0.694578\pi$
$234$	0	0
$235$	9.13828	0.596115
$236$	12.2351	0.796435
$237$	0	0
$238$	−8.40990	−0.545133
$239$	−2.49532	−0.161409	−0.0807043	−	0.996738i	$-0.525717\pi$
$239$	−2.49532	−0.161409	−0.0807043	+	0.996738i	$0.525717\pi$
$240$	0	0
$241$	−2.85728	−0.184054	−0.0920268	−	0.995757i	$-0.529335\pi$
$241$	−2.85728	−0.184054	−0.0920268	+	0.995757i	$0.529335\pi$
$242$	−0.398528	−0.0256183
$243$	0	0
$244$	0.534795	0.0342368
$245$	−5.90321	−0.377142
$246$	0	0
$247$	4.64296	0.295424
$248$	4.80150	0.304896
$249$	0	0
$250$	−0.311108	−0.0196762
$251$	14.9240	0.941992	0.470996	−	0.882135i	$-0.343895\pi$
$251$	14.9240	0.941992	0.470996	+	0.882135i	$0.343895\pi$
$252$	0	0
$253$	−17.2257	−1.08297
$254$	−1.82516	−0.114521
$255$	0	0
$256$	8.80642	0.550401
$257$	−23.4750	−1.46433	−0.732167	−	0.681126i	$-0.761491\pi$
$257$	−23.4750	−1.46433	−0.732167	+	0.681126i	$0.761491\pi$
$258$	0	0
$259$	31.7418	1.97234
$260$	8.83654	0.548019
$261$	0	0
$262$	2.87802	0.177805
$263$	16.3225	1.00649	0.503244	−	0.864145i	$-0.332140\pi$
$263$	16.3225	1.00649	0.503244	+	0.864145i	$0.332140\pi$
$264$	0	0
$265$	−5.03657	−0.309394
$266$	−1.11753	−0.0685203
$267$	0	0
$268$	−6.99063	−0.427021
$269$	−15.5877	−0.950396	−0.475198	−	0.879879i	$-0.657624\pi$
$269$	−15.5877	−0.950396	−0.475198	+	0.879879i	$0.657624\pi$
$270$	0	0
$271$	20.3082	1.23363	0.616817	−	0.787106i	$-0.288422\pi$
$271$	20.3082	1.23363	0.616817	+	0.787106i	$0.288422\pi$
$272$	25.8020	1.56447
$273$	0	0
$274$	−2.82870	−0.170888
$275$	−3.11753	−0.187994
$276$	0	0
$277$	−29.1338	−1.75048	−0.875241	−	0.483687i	$-0.839297\pi$
$277$	−29.1338	−1.75048	−0.875241	+	0.483687i	$0.839297\pi$
$278$	6.06022	0.363468
$279$	0	0
$280$	−4.36196	−0.260677
$281$	−18.3017	−1.09179	−0.545895	−	0.837854i	$-0.683810\pi$
$281$	−18.3017	−1.09179	−0.545895	+	0.837854i	$0.683810\pi$
$282$	0	0
$283$	−9.39853	−0.558684	−0.279342	−	0.960192i	$-0.590116\pi$
$283$	−9.39853	−0.558684	−0.279342	+	0.960192i	$0.590116\pi$
$284$	−6.42864	−0.381470
$285$	0	0
$286$	−4.50315	−0.266277
$287$	−28.2810	−1.66937
$288$	0	0
$289$	39.6321	2.33130
$290$	−2.54125	−0.149227
$291$	0	0
$292$	−6.06022	−0.354648
$293$	−0.811346	−0.0473993	−0.0236997	−	0.999719i	$-0.507545\pi$
$293$	−0.811346	−0.0473993	−0.0236997	+	0.999719i	$0.507545\pi$
$294$	0	0
$295$	6.42864	0.374290
$296$	10.7304	0.623691
$297$	0	0
$298$	5.24443	0.303802
$299$	25.6543	1.48363
$300$	0	0
$301$	19.6084	1.13021
$302$	6.99508	0.402521
$303$	0	0
$304$	3.42864	0.196646
$305$	0.280996	0.0160898
$306$	0	0
$307$	−20.6035	−1.17590	−0.587951	−	0.808896i	$-0.700065\pi$
$307$	−20.6035	−1.17590	−0.587951	+	0.808896i	$0.700065\pi$
$308$	−21.3131	−1.21443
$309$	0	0
$310$	1.23014	0.0698673
$311$	−4.43509	−0.251491	−0.125746	−	0.992063i	$-0.540132\pi$
$311$	−4.43509	−0.251491	−0.125746	+	0.992063i	$0.540132\pi$
$312$	0	0
$313$	−12.2953	−0.694971	−0.347485	−	0.937685i	$-0.612964\pi$
$313$	−12.2953	−0.694971	−0.347485	+	0.937685i	$0.612964\pi$
$314$	−0.120446	−0.00679717
$315$	0	0
$316$	−24.7239	−1.39083
$317$	−0.220773	−0.0123999	−0.00619993	−	0.999981i	$-0.501974\pi$
$317$	−0.220773	−0.0123999	−0.00619993	+	0.999981i	$0.501974\pi$
$318$	0	0
$319$	−25.4652	−1.42578
$320$	5.76986	0.322545
$321$	0	0
$322$	−6.17484	−0.344110
$323$	7.52543	0.418726
$324$	0	0
$325$	4.64296	0.257545
$326$	−5.38424	−0.298205
$327$	0	0
$328$	−9.56046	−0.527888
$329$	32.8256	1.80974
$330$	0	0
$331$	10.6681	0.586374	0.293187	−	0.956055i	$-0.405284\pi$
$331$	10.6681	0.586374	0.293187	+	0.956055i	$0.405284\pi$
$332$	−16.3225	−0.895813
$333$	0	0
$334$	2.11906	0.115950
$335$	−3.67307	−0.200681
$336$	0	0
$337$	−8.02074	−0.436918	−0.218459	−	0.975846i	$-0.570103\pi$
$337$	−8.02074	−0.436918	−0.218459	+	0.975846i	$0.570103\pi$
$338$	2.66217	0.144803
$339$	0	0
$340$	14.3225	0.776746
$341$	12.3269	0.667541
$342$	0	0
$343$	3.93978	0.212728
$344$	6.62867	0.357394
$345$	0	0
$346$	−4.61777	−0.248253
$347$	−7.06515	−0.379277	−0.189638	−	0.981854i	$-0.560732\pi$
$347$	−7.06515	−0.379277	−0.189638	+	0.981854i	$0.560732\pi$
$348$	0	0
$349$	19.6731	1.05308	0.526538	−	0.850152i	$-0.323490\pi$
$349$	19.6731	1.05308	0.526538	+	0.850152i	$0.323490\pi$
$350$	−1.11753	−0.0597346
$351$	0	0
$352$	−10.8968	−0.580799
$353$	5.61285	0.298742	0.149371	−	0.988781i	$-0.452275\pi$
$353$	5.61285	0.298742	0.149371	+	0.988781i	$0.452275\pi$
$354$	0	0
$355$	−3.37778	−0.179274
$356$	−33.7625	−1.78941
$357$	0	0
$358$	2.09187	0.110559
$359$	−10.0065	−0.528120	−0.264060	−	0.964506i	$-0.585062\pi$
$359$	−10.0065	−0.528120	−0.264060	+	0.964506i	$0.585062\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.34614	−0.0707516
$363$	0	0
$364$	31.7418	1.66372
$365$	−3.18421	−0.166669
$366$	0	0
$367$	8.25581	0.430950	0.215475	−	0.976509i	$-0.430870\pi$
$367$	8.25581	0.430950	0.215475	+	0.976509i	$0.430870\pi$
$368$	18.9447	0.987561
$369$	0	0
$370$	2.74912	0.142920
$371$	−18.0919	−0.939283
$372$	0	0
$373$	6.71609	0.347746	0.173873	−	0.984768i	$-0.444372\pi$
$373$	6.71609	0.347746	0.173873	+	0.984768i	$0.444372\pi$
$374$	−7.29883	−0.377413
$375$	0	0
$376$	11.0968	0.572273
$377$	37.9255	1.95326
$378$	0	0
$379$	−35.7003	−1.83380	−0.916900	−	0.399117i	$-0.869317\pi$
$379$	−35.7003	−1.83380	−0.916900	+	0.399117i	$0.869317\pi$
$380$	1.90321	0.0976327
$381$	0	0
$382$	−5.39853	−0.276213
$383$	−1.53972	−0.0786759	−0.0393379	−	0.999226i	$-0.512525\pi$
$383$	−1.53972	−0.0786759	−0.0393379	+	0.999226i	$0.512525\pi$
$384$	0	0
$385$	−11.1985	−0.570728
$386$	2.35212	0.119720
$387$	0	0
$388$	13.8272	0.701968
$389$	−15.5111	−0.786446	−0.393223	−	0.919443i	$-0.628640\pi$
$389$	−15.5111	−0.786446	−0.393223	+	0.919443i	$0.628640\pi$
$390$	0	0
$391$	41.5812	2.10285
$392$	−7.16839	−0.362058
$393$	0	0
$394$	4.02720	0.202887
$395$	−12.9906	−0.653630
$396$	0	0
$397$	−4.78415	−0.240110	−0.120055	−	0.992767i	$-0.538307\pi$
$397$	−4.78415	−0.240110	−0.120055	+	0.992767i	$0.538307\pi$
$398$	−0.0414872	−0.00207956
$399$	0	0
$400$	3.42864	0.171432
$401$	−3.57781	−0.178668	−0.0893338	−	0.996002i	$-0.528474\pi$
$401$	−3.57781	−0.178668	−0.0893338	+	0.996002i	$0.528474\pi$
$402$	0	0
$403$	−18.3586	−0.914506
$404$	20.4099	1.01543
$405$	0	0
$406$	−9.12843	−0.453036
$407$	27.5482	1.36551
$408$	0	0
$409$	−16.8385	−0.832612	−0.416306	−	0.909224i	$-0.636676\pi$
$409$	−16.8385	−0.832612	−0.416306	+	0.909224i	$0.636676\pi$
$410$	−2.44938	−0.120966
$411$	0	0
$412$	−9.98126	−0.491742
$413$	23.0923	1.13630
$414$	0	0
$415$	−8.57628	−0.420993
$416$	16.2286	0.795673
$417$	0	0
$418$	−0.969888	−0.0474388
$419$	13.4445	0.656805	0.328402	−	0.944538i	$-0.393490\pi$
$419$	13.4445	0.656805	0.328402	+	0.944538i	$0.393490\pi$
$420$	0	0
$421$	−1.74620	−0.0851046	−0.0425523	−	0.999094i	$-0.513549\pi$
$421$	−1.74620	−0.0851046	−0.0425523	+	0.999094i	$0.513549\pi$
$422$	8.81579	0.429146
$423$	0	0
$424$	−6.11600	−0.297019
$425$	7.52543	0.365037
$426$	0	0
$427$	1.00937	0.0488467
$428$	−11.8666	−0.573596
$429$	0	0
$430$	1.69826	0.0818974
$431$	22.1432	1.06660	0.533300	−	0.845926i	$-0.320952\pi$
$431$	22.1432	1.06660	0.533300	+	0.845926i	$0.320952\pi$
$432$	0	0
$433$	36.3892	1.74875	0.874376	−	0.485250i	$-0.161271\pi$
$433$	36.3892	1.74875	0.874376	+	0.485250i	$0.161271\pi$
$434$	4.41880	0.212109
$435$	0	0
$436$	−18.5491	−0.888340
$437$	5.52543	0.264317
$438$	0	0
$439$	−1.63158	−0.0778712	−0.0389356	−	0.999242i	$-0.512397\pi$
$439$	−1.63158	−0.0778712	−0.0389356	+	0.999242i	$0.512397\pi$
$440$	−3.78568	−0.180475
$441$	0	0
$442$	10.8702	0.517042
$443$	−5.68736	−0.270215	−0.135107	−	0.990831i	$-0.543138\pi$
$443$	−5.68736	−0.270215	−0.135107	+	0.990831i	$0.543138\pi$
$444$	0	0
$445$	−17.7397	−0.840945
$446$	−1.24443	−0.0589255
$447$	0	0
$448$	20.7259	0.979208
$449$	35.0988	1.65641	0.828207	−	0.560422i	$-0.189361\pi$
$449$	35.0988	1.65641	0.828207	+	0.560422i	$0.189361\pi$
$450$	0	0
$451$	−24.5446	−1.15576
$452$	−34.0558	−1.60185
$453$	0	0
$454$	0.220773	0.0103614
$455$	16.6780	0.781876
$456$	0	0
$457$	24.6450	1.15284	0.576421	−	0.817153i	$-0.304449\pi$
$457$	24.6450	1.15284	0.576421	+	0.817153i	$0.304449\pi$
$458$	−8.22077	−0.384132
$459$	0	0
$460$	10.5161	0.490314
$461$	20.4701	0.953389	0.476694	−	0.879069i	$-0.341835\pi$
$461$	20.4701	0.953389	0.476694	+	0.879069i	$0.341835\pi$
$462$	0	0
$463$	−26.6113	−1.23673	−0.618366	−	0.785890i	$-0.712205\pi$
$463$	−26.6113	−1.23673	−0.618366	+	0.785890i	$0.712205\pi$
$464$	28.0065	1.30017
$465$	0	0
$466$	−5.45091	−0.252509
$467$	35.9353	1.66289	0.831444	−	0.555608i	$-0.187514\pi$
$467$	35.9353	1.66289	0.831444	+	0.555608i	$0.187514\pi$
$468$	0	0
$469$	−13.1941	−0.609245
$470$	2.84299	0.131137
$471$	0	0
$472$	7.80642	0.359320
$473$	17.0178	0.782481
$474$	0	0
$475$	1.00000	0.0458831
$476$	51.4479	2.35811
$477$	0	0
$478$	−0.776312	−0.0355077
$479$	−17.1461	−0.783426	−0.391713	−	0.920087i	$-0.628117\pi$
$479$	−17.1461	−0.783426	−0.391713	+	0.920087i	$0.628117\pi$
$480$	0	0
$481$	−41.0277	−1.87070
$482$	−0.888922	−0.0404893
$483$	0	0
$484$	2.43801	0.110819
$485$	7.26517	0.329895
$486$	0	0
$487$	−23.5941	−1.06915	−0.534576	−	0.845121i	$-0.679529\pi$
$487$	−23.5941	−1.06915	−0.534576	+	0.845121i	$0.679529\pi$
$488$	0.341219	0.0154463
$489$	0	0
$490$	−1.83654	−0.0829662
$491$	33.4859	1.51120	0.755600	−	0.655033i	$-0.227345\pi$
$491$	33.4859	1.51120	0.755600	+	0.655033i	$0.227345\pi$
$492$	0	0
$493$	61.4706	2.76850
$494$	1.44446	0.0649894
$495$	0	0
$496$	−13.5571	−0.608730
$497$	−12.1334	−0.544255
$498$	0	0
$499$	−22.7052	−1.01642	−0.508212	−	0.861232i	$-0.669693\pi$
$499$	−22.7052	−1.01642	−0.508212	+	0.861232i	$0.669693\pi$
$500$	1.90321	0.0851142
$501$	0	0
$502$	4.64296	0.207225
$503$	7.07805	0.315595	0.157797	−	0.987472i	$-0.449561\pi$
$503$	7.07805	0.315595	0.157797	+	0.987472i	$0.449561\pi$
$504$	0	0
$505$	10.7239	0.477208
$506$	−5.35905	−0.238239
$507$	0	0
$508$	11.1655	0.495388
$509$	−25.9748	−1.15131	−0.575657	−	0.817692i	$-0.695253\pi$
$509$	−25.9748	−1.15131	−0.575657	+	0.817692i	$0.695253\pi$
$510$	0	0
$511$	−11.4380	−0.505988
$512$	20.3111	0.897633
$513$	0	0
$514$	−7.30327	−0.322133
$515$	−5.24443	−0.231097
$516$	0	0
$517$	28.4889	1.25294
$518$	9.87511	0.433887
$519$	0	0
$520$	5.63804	0.247244
$521$	12.7491	0.558549	0.279274	−	0.960211i	$-0.409906\pi$
$521$	12.7491	0.558549	0.279274	+	0.960211i	$0.409906\pi$
$522$	0	0
$523$	31.4479	1.37512	0.687559	−	0.726128i	$-0.258682\pi$
$523$	31.4479	1.37512	0.687559	+	0.726128i	$0.258682\pi$
$524$	−17.6064	−0.769139
$525$	0	0
$526$	5.07805	0.221414
$527$	−29.7560	−1.29619
$528$	0	0
$529$	7.53035	0.327407
$530$	−1.56691	−0.0680624
$531$	0	0
$532$	6.83654	0.296401
$533$	36.5545	1.58335
$534$	0	0
$535$	−6.23506	−0.269565
$536$	−4.46028	−0.192655
$537$	0	0
$538$	−4.84944	−0.209074
$539$	−18.4035	−0.792693
$540$	0	0
$541$	20.4889	0.880885	0.440443	−	0.897781i	$-0.354822\pi$
$541$	20.4889	0.880885	0.440443	+	0.897781i	$0.354822\pi$
$542$	6.31804	0.271383
$543$	0	0
$544$	26.3037	1.12776
$545$	−9.74620	−0.417481
$546$	0	0
$547$	14.2065	0.607425	0.303713	−	0.952764i	$-0.401774\pi$
$547$	14.2065	0.607425	0.303713	+	0.952764i	$0.401774\pi$
$548$	17.3047	0.739218
$549$	0	0
$550$	−0.969888	−0.0413562
$551$	8.16839	0.347985
$552$	0	0
$553$	−46.6637	−1.98434
$554$	−9.06376	−0.385082
$555$	0	0
$556$	−37.0736	−1.57227
$557$	−24.5303	−1.03938	−0.519692	−	0.854354i	$-0.673953\pi$
$557$	−24.5303	−1.03938	−0.519692	+	0.854354i	$0.673953\pi$
$558$	0	0
$559$	−25.3448	−1.07197
$560$	12.3160	0.520447
$561$	0	0
$562$	−5.69381	−0.240179
$563$	−20.2306	−0.852619	−0.426309	−	0.904577i	$-0.640187\pi$
$563$	−20.2306	−0.852619	−0.426309	+	0.904577i	$0.640187\pi$
$564$	0	0
$565$	−17.8938	−0.752800
$566$	−2.92396	−0.122903
$567$	0	0
$568$	−4.10171	−0.172104
$569$	8.64740	0.362518	0.181259	−	0.983435i	$-0.441983\pi$
$569$	8.64740	0.362518	0.181259	+	0.983435i	$0.441983\pi$
$570$	0	0
$571$	9.33630	0.390712	0.195356	−	0.980732i	$-0.437414\pi$
$571$	9.33630	0.390712	0.195356	+	0.980732i	$0.437414\pi$
$572$	27.5482	1.15185
$573$	0	0
$574$	−8.79844	−0.367240
$575$	5.52543	0.230426
$576$	0	0
$577$	26.5718	1.10620	0.553100	−	0.833115i	$-0.313445\pi$
$577$	26.5718	1.10620	0.553100	+	0.833115i	$0.313445\pi$
$578$	12.3298	0.512854
$579$	0	0
$580$	15.5462	0.645520
$581$	−30.8069	−1.27809
$582$	0	0
$583$	−15.7017	−0.650296
$584$	−3.86665	−0.160003
$585$	0	0
$586$	−0.252416	−0.0104272
$587$	−20.8845	−0.861995	−0.430997	−	0.902353i	$-0.641838\pi$
$587$	−20.8845	−0.861995	−0.430997	+	0.902353i	$0.641838\pi$
$588$	0	0
$589$	−3.95407	−0.162924
$590$	2.00000	0.0823387
$591$	0	0
$592$	−30.2973	−1.24521
$593$	14.4286	0.592513	0.296257	−	0.955108i	$-0.404262\pi$
$593$	14.4286	0.592513	0.296257	+	0.955108i	$0.404262\pi$
$594$	0	0
$595$	27.0321	1.10821
$596$	−32.0830	−1.31417
$597$	0	0
$598$	7.98126	0.326378
$599$	−46.3497	−1.89380	−0.946898	−	0.321533i	$-0.895802\pi$
$599$	−46.3497	−1.89380	−0.946898	+	0.321533i	$0.895802\pi$
$600$	0	0
$601$	−18.0415	−0.735928	−0.367964	−	0.929840i	$-0.619945\pi$
$601$	−18.0415	−0.735928	−0.367964	+	0.929840i	$0.619945\pi$
$602$	6.10033	0.248631
$603$	0	0
$604$	−42.7926	−1.74121
$605$	1.28100	0.0520799
$606$	0	0
$607$	4.87601	0.197911	0.0989557	−	0.995092i	$-0.468450\pi$
$607$	4.87601	0.197911	0.0989557	+	0.995092i	$0.468450\pi$
$608$	3.49532	0.141754
$609$	0	0
$610$	0.0874201	0.00353954
$611$	−42.4286	−1.71648
$612$	0	0
$613$	−17.8064	−0.719195	−0.359597	−	0.933108i	$-0.617086\pi$
$613$	−17.8064	−0.719195	−0.359597	+	0.933108i	$0.617086\pi$
$614$	−6.40990	−0.258683
$615$	0	0
$616$	−13.5986	−0.547901
$617$	−0.595019	−0.0239545	−0.0119773	−	0.999928i	$-0.503813\pi$
$617$	−0.595019	−0.0239545	−0.0119773	+	0.999928i	$0.503813\pi$
$618$	0	0
$619$	20.3783	0.819071	0.409536	−	0.912294i	$-0.365691\pi$
$619$	20.3783	0.819071	0.409536	+	0.912294i	$0.365691\pi$
$620$	−7.52543	−0.302228
$621$	0	0
$622$	−1.37979	−0.0553246
$623$	−63.7230	−2.55301
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.82516	−0.152884
$627$	0	0
$628$	0.736833	0.0294028
$629$	−66.4987	−2.65148
$630$	0	0
$631$	−2.50177	−0.0995939	−0.0497969	−	0.998759i	$-0.515857\pi$
$631$	−2.50177	−0.0995939	−0.0497969	+	0.998759i	$0.515857\pi$
$632$	−15.7748	−0.627487
$633$	0	0
$634$	−0.0686843	−0.00272780
$635$	5.86665	0.232811
$636$	0	0
$637$	27.4084	1.08596
$638$	−7.92242	−0.313652
$639$	0	0
$640$	8.78568	0.347285
$641$	33.6479	1.32901	0.664506	−	0.747283i	$-0.268642\pi$
$641$	33.6479	1.32901	0.664506	+	0.747283i	$0.268642\pi$
$642$	0	0
$643$	18.2242	0.718691	0.359345	−	0.933205i	$-0.383000\pi$
$643$	18.2242	0.718691	0.359345	+	0.933205i	$0.383000\pi$
$644$	37.7748	1.48853
$645$	0	0
$646$	2.34122	0.0921140
$647$	−26.7195	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$647$	−26.7195	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$648$	0	0
$649$	20.0415	0.786697
$650$	1.44446	0.0566564
$651$	0	0
$652$	32.9382	1.28996
$653$	−43.9037	−1.71808	−0.859042	−	0.511905i	$-0.828940\pi$
$653$	−43.9037	−1.71808	−0.859042	+	0.511905i	$0.828940\pi$
$654$	0	0
$655$	−9.25088	−0.361462
$656$	26.9940	1.05394
$657$	0	0
$658$	10.2123	0.398117
$659$	15.3274	0.597071	0.298535	−	0.954399i	$-0.403502\pi$
$659$	15.3274	0.597071	0.298535	+	0.954399i	$0.403502\pi$
$660$	0	0
$661$	23.1526	0.900530	0.450265	−	0.892895i	$-0.351329\pi$
$661$	23.1526	0.900530	0.450265	+	0.892895i	$0.351329\pi$
$662$	3.31894	0.128994
$663$	0	0
$664$	−10.4143	−0.404155
$665$	3.59210	0.139296
$666$	0	0
$667$	45.1338	1.74759
$668$	−12.9634	−0.501570
$669$	0	0
$670$	−1.14272	−0.0441472
$671$	0.876015	0.0338182
$672$	0	0
$673$	3.97926	0.153389	0.0766945	−	0.997055i	$-0.475563\pi$
$673$	3.97926	0.153389	0.0766945	+	0.997055i	$0.475563\pi$
$674$	−2.49532	−0.0961160
$675$	0	0
$676$	−16.2859	−0.626381
$677$	21.4652	0.824975	0.412487	−	0.910963i	$-0.364660\pi$
$677$	21.4652	0.824975	0.412487	+	0.910963i	$0.364660\pi$
$678$	0	0
$679$	26.0973	1.00152
$680$	9.13828	0.350437
$681$	0	0
$682$	3.83500	0.146850
$683$	−16.5205	−0.632140	−0.316070	−	0.948736i	$-0.602363\pi$
$683$	−16.5205	−0.632140	−0.316070	+	0.948736i	$0.602363\pi$
$684$	0	0
$685$	9.09234	0.347401
$686$	1.22570	0.0467973
$687$	0	0
$688$	−18.7161	−0.713544
$689$	23.3846	0.890881
$690$	0	0
$691$	−31.2543	−1.18897	−0.594484	−	0.804107i	$-0.702644\pi$
$691$	−31.2543	−1.18897	−0.594484	+	0.804107i	$0.702644\pi$
$692$	28.2494	1.07388
$693$	0	0
$694$	−2.19802	−0.0834358
$695$	−19.4795	−0.738899
$696$	0	0
$697$	59.2484	2.24419
$698$	6.12045	0.231662
$699$	0	0
$700$	6.83654	0.258397
$701$	−11.3176	−0.427458	−0.213729	−	0.976893i	$-0.568561\pi$
$701$	−11.3176	−0.427458	−0.213729	+	0.976893i	$0.568561\pi$
$702$	0	0
$703$	−8.83654	−0.333276
$704$	17.9877	0.677938
$705$	0	0
$706$	1.74620	0.0657191
$707$	38.5215	1.44875
$708$	0	0
$709$	17.3131	0.650208	0.325104	−	0.945678i	$-0.394601\pi$
$709$	17.3131	0.650208	0.325104	+	0.945678i	$0.394601\pi$
$710$	−1.05086	−0.0394379
$711$	0	0
$712$	−21.5417	−0.807310
$713$	−21.8479	−0.818211
$714$	0	0
$715$	14.4746	0.541318
$716$	−12.7971	−0.478248
$717$	0	0
$718$	−3.11309	−0.116179
$719$	21.1175	0.787551	0.393776	−	0.919207i	$-0.371169\pi$
$719$	21.1175	0.787551	0.393776	+	0.919207i	$0.371169\pi$
$720$	0	0
$721$	−18.8385	−0.701584
$722$	0.311108	0.0115782
$723$	0	0
$724$	8.23506	0.306054
$725$	8.16839	0.303366
$726$	0	0
$727$	19.1220	0.709195	0.354597	−	0.935019i	$-0.384618\pi$
$727$	19.1220	0.709195	0.354597	+	0.935019i	$0.384618\pi$
$728$	20.2524	0.750604
$729$	0	0
$730$	−0.990632	−0.0366649
$731$	−41.0794	−1.51938
$732$	0	0
$733$	−27.1842	−1.00407	−0.502036	−	0.864847i	$-0.667415\pi$
$733$	−27.1842	−1.00407	−0.502036	+	0.864847i	$0.667415\pi$
$734$	2.56845	0.0948031
$735$	0	0
$736$	19.3131	0.711891
$737$	−11.4509	−0.421800
$738$	0	0
$739$	−25.9684	−0.955261	−0.477631	−	0.878561i	$-0.658504\pi$
$739$	−25.9684	−0.955261	−0.477631	+	0.878561i	$0.658504\pi$
$740$	−16.8178	−0.618235
$741$	0	0
$742$	−5.62852	−0.206630
$743$	−0.866170	−0.0317767	−0.0158884	−	0.999874i	$-0.505058\pi$
$743$	−0.866170	−0.0317767	−0.0158884	+	0.999874i	$0.505058\pi$
$744$	0	0
$745$	−16.8573	−0.617603
$746$	2.08943	0.0764994
$747$	0	0
$748$	44.6508	1.63259
$749$	−22.3970	−0.818368
$750$	0	0
$751$	−6.01429	−0.219465	−0.109732	−	0.993961i	$-0.534999\pi$
$751$	−6.01429	−0.219465	−0.109732	+	0.993961i	$0.534999\pi$
$752$	−31.3319	−1.14256
$753$	0	0
$754$	11.7989	0.429691
$755$	−22.4844	−0.818292
$756$	0	0
$757$	−43.7560	−1.59034	−0.795170	−	0.606386i	$-0.792619\pi$
$757$	−43.7560	−1.59034	−0.795170	+	0.606386i	$0.792619\pi$
$758$	−11.1066	−0.403411
$759$	0	0
$760$	1.21432	0.0440480
$761$	7.68598	0.278616	0.139308	−	0.990249i	$-0.455512\pi$
$761$	7.68598	0.278616	0.139308	+	0.990249i	$0.455512\pi$
$762$	0	0
$763$	−35.0094	−1.26742
$764$	33.0257	1.19483
$765$	0	0
$766$	−0.479018	−0.0173076
$767$	−29.8479	−1.07775
$768$	0	0
$769$	−31.2444	−1.12670	−0.563352	−	0.826217i	$-0.690488\pi$
$769$	−31.2444	−1.12670	−0.563352	+	0.826217i	$0.690488\pi$
$770$	−3.48394	−0.125552
$771$	0	0
$772$	−14.3892	−0.517877
$773$	38.0687	1.36924	0.684618	−	0.728902i	$-0.259969\pi$
$773$	38.0687	1.36924	0.684618	+	0.728902i	$0.259969\pi$
$774$	0	0
$775$	−3.95407	−0.142034
$776$	8.82225	0.316700
$777$	0	0
$778$	−4.82564	−0.173007
$779$	7.87310	0.282083
$780$	0	0
$781$	−10.5303	−0.376806
$782$	12.9362	0.462599
$783$	0	0
$784$	20.2400	0.722857
$785$	0.387152	0.0138181
$786$	0	0
$787$	19.5111	0.695497	0.347748	−	0.937588i	$-0.386946\pi$
$787$	19.5111	0.695497	0.347748	+	0.937588i	$0.386946\pi$
$788$	−24.6365	−0.877639
$789$	0	0
$790$	−4.04149	−0.143790
$791$	−64.2766	−2.28541
$792$	0	0
$793$	−1.30465	−0.0463296
$794$	−1.48839	−0.0528208
$795$	0	0
$796$	0.253799	0.00899567
$797$	−13.4938	−0.477974	−0.238987	−	0.971023i	$-0.576815\pi$
$797$	−13.4938	−0.477974	−0.238987	+	0.971023i	$0.576815\pi$
$798$	0	0
$799$	−68.7694	−2.43289
$800$	3.49532	0.123578
$801$	0	0
$802$	−1.11309	−0.0393044
$803$	−9.92687	−0.350312
$804$	0	0
$805$	19.8479	0.699547
$806$	−5.71150	−0.201179
$807$	0	0
$808$	13.0223	0.458122
$809$	39.8163	1.39987	0.699933	−	0.714209i	$-0.253213\pi$
$809$	39.8163	1.39987	0.699933	+	0.714209i	$0.253213\pi$
$810$	0	0
$811$	35.5496	1.24831	0.624157	−	0.781299i	$-0.285443\pi$
$811$	35.5496	1.24831	0.624157	+	0.781299i	$0.285443\pi$
$812$	55.8435	1.95972
$813$	0	0
$814$	8.57045	0.300394
$815$	17.3067	0.606226
$816$	0	0
$817$	−5.45875	−0.190978
$818$	−5.23860	−0.183163
$819$	0	0
$820$	14.9842	0.523270
$821$	15.9684	0.557300	0.278650	−	0.960393i	$-0.410113\pi$
$821$	15.9684	0.557300	0.278650	+	0.960393i	$0.410113\pi$
$822$	0	0
$823$	−31.4400	−1.09593	−0.547965	−	0.836501i	$-0.684597\pi$
$823$	−31.4400	−1.09593	−0.547965	+	0.836501i	$0.684597\pi$
$824$	−6.36842	−0.221854
$825$	0	0
$826$	7.18421	0.249971
$827$	−53.0593	−1.84505	−0.922527	−	0.385933i	$-0.873879\pi$
$827$	−53.0593	−1.84505	−0.922527	+	0.385933i	$0.873879\pi$
$828$	0	0
$829$	−31.1526	−1.08197	−0.540987	−	0.841031i	$-0.681949\pi$
$829$	−31.1526	−1.08197	−0.540987	+	0.841031i	$0.681949\pi$
$830$	−2.66815	−0.0926128
$831$	0	0
$832$	−26.7892	−0.928749
$833$	44.4242	1.53921
$834$	0	0
$835$	−6.81135	−0.235716
$836$	5.93332	0.205208
$837$	0	0
$838$	4.18268	0.144488
$839$	23.6642	0.816978	0.408489	−	0.912763i	$-0.366056\pi$
$839$	23.6642	0.816978	0.408489	+	0.912763i	$0.366056\pi$
$840$	0	0
$841$	37.7225	1.30078
$842$	−0.543257	−0.0187219
$843$	0	0
$844$	−53.9309	−1.85638
$845$	−8.55707	−0.294372
$846$	0	0
$847$	4.60147	0.158108
$848$	17.2686	0.593005
$849$	0	0
$850$	2.34122	0.0803032
$851$	−48.8256	−1.67372
$852$	0	0
$853$	−39.7560	−1.36122	−0.680611	−	0.732645i	$-0.738285\pi$
$853$	−39.7560	−1.36122	−0.680611	+	0.732645i	$0.738285\pi$
$854$	0.314022	0.0107456
$855$	0	0
$856$	−7.57136	−0.258784
$857$	−42.3640	−1.44713	−0.723563	−	0.690259i	$-0.757497\pi$
$857$	−42.3640	−1.44713	−0.723563	+	0.690259i	$0.757497\pi$
$858$	0	0
$859$	11.0638	0.377491	0.188745	−	0.982026i	$-0.439558\pi$
$859$	11.0638	0.377491	0.188745	+	0.982026i	$0.439558\pi$
$860$	−10.3892	−0.354267
$861$	0	0
$862$	6.88892	0.234638
$863$	−29.7649	−1.01321	−0.506605	−	0.862178i	$-0.669100\pi$
$863$	−29.7649	−1.01321	−0.506605	+	0.862178i	$0.669100\pi$
$864$	0	0
$865$	14.8430	0.504677
$866$	11.3210	0.384702
$867$	0	0
$868$	−27.0321	−0.917530
$869$	−40.4987	−1.37382
$870$	0	0
$871$	17.0539	0.577850
$872$	−11.8350	−0.400784
$873$	0	0
$874$	1.71900	0.0581462
$875$	3.59210	0.121435
$876$	0	0
$877$	−3.24644	−0.109624	−0.0548122	−	0.998497i	$-0.517456\pi$
$877$	−3.24644	−0.109624	−0.0548122	+	0.998497i	$0.517456\pi$
$878$	−0.507598	−0.0171306
$879$	0	0
$880$	10.6889	0.360322
$881$	30.3269	1.02174	0.510870	−	0.859658i	$-0.329323\pi$
$881$	30.3269	1.02174	0.510870	+	0.859658i	$0.329323\pi$
$882$	0	0
$883$	−4.92840	−0.165854	−0.0829270	−	0.996556i	$-0.526427\pi$
$883$	−4.92840	−0.165854	−0.0829270	+	0.996556i	$0.526427\pi$
$884$	−66.4987	−2.23659
$885$	0	0
$886$	−1.76938	−0.0594436
$887$	−8.08297	−0.271400	−0.135700	−	0.990750i	$-0.543328\pi$
$887$	−8.08297	−0.271400	−0.135700	+	0.990750i	$0.543328\pi$
$888$	0	0
$889$	21.0736	0.706786
$890$	−5.51897	−0.184996
$891$	0	0
$892$	7.61285	0.254897
$893$	−9.13828	−0.305801
$894$	0	0
$895$	−6.72393	−0.224756
$896$	31.5591	1.05431
$897$	0	0
$898$	10.9195	0.364389
$899$	−32.2983	−1.07721
$900$	0	0
$901$	37.9023	1.26271
$902$	−7.63603	−0.254252
$903$	0	0
$904$	−21.7288	−0.722691
$905$	4.32693	0.143832
$906$	0	0
$907$	13.1427	0.436397	0.218198	−	0.975904i	$-0.429982\pi$
$907$	13.1427	0.436397	0.218198	+	0.975904i	$0.429982\pi$
$908$	−1.35059	−0.0448208
$909$	0	0
$910$	5.18865	0.172002
$911$	−6.83854	−0.226571	−0.113286	−	0.993562i	$-0.536137\pi$
$911$	−6.83854	−0.226571	−0.113286	+	0.993562i	$0.536137\pi$
$912$	0	0
$913$	−26.7368	−0.884860
$914$	7.66724	0.253610
$915$	0	0
$916$	50.2908	1.66166
$917$	−33.2301	−1.09736
$918$	0	0
$919$	−33.9398	−1.11957	−0.559785	−	0.828638i	$-0.689116\pi$
$919$	−33.9398	−1.11957	−0.559785	+	0.828638i	$0.689116\pi$
$920$	6.70964	0.221210
$921$	0	0
$922$	6.36842	0.209733
$923$	15.6829	0.516209
$924$	0	0
$925$	−8.83654	−0.290543
$926$	−8.27899	−0.272064
$927$	0	0
$928$	28.5511	0.937236
$929$	41.5714	1.36391	0.681956	−	0.731393i	$-0.261130\pi$
$929$	41.5714	1.36391	0.681956	+	0.731393i	$0.261130\pi$
$930$	0	0
$931$	5.90321	0.193470
$932$	33.3461	1.09229
$933$	0	0
$934$	11.1798	0.365813
$935$	23.4608	0.767249
$936$	0	0
$937$	−26.2538	−0.857674	−0.428837	−	0.903382i	$-0.641077\pi$
$937$	−26.2538	−0.857674	−0.428837	+	0.903382i	$0.641077\pi$
$938$	−4.10477	−0.134026
$939$	0	0
$940$	−17.3921	−0.567267
$941$	−45.8642	−1.49513	−0.747565	−	0.664188i	$-0.768777\pi$
$941$	−45.8642	−1.49513	−0.747565	+	0.664188i	$0.768777\pi$
$942$	0	0
$943$	43.5022	1.41663
$944$	−22.0415	−0.717389
$945$	0	0
$946$	5.29438	0.172135
$947$	19.6271	0.637796	0.318898	−	0.947789i	$-0.396687\pi$
$947$	19.6271	0.637796	0.318898	+	0.947789i	$0.396687\pi$
$948$	0	0
$949$	14.7841	0.479914
$950$	0.311108	0.0100937
$951$	0	0
$952$	32.8256	1.06388
$953$	−16.2079	−0.525024	−0.262512	−	0.964929i	$-0.584551\pi$
$953$	−16.2079	−0.525024	−0.262512	+	0.964929i	$0.584551\pi$
$954$	0	0
$955$	17.3526	0.561517
$956$	4.74912	0.153597
$957$	0	0
$958$	−5.33429	−0.172343
$959$	32.6606	1.05467
$960$	0	0
$961$	−15.3654	−0.495657
$962$	−12.7640	−0.411529
$963$	0	0
$964$	5.43801	0.175146
$965$	−7.56046	−0.243380
$966$	0	0
$967$	39.8499	1.28149	0.640743	−	0.767755i	$-0.278626\pi$
$967$	39.8499	1.28149	0.640743	+	0.767755i	$0.278626\pi$
$968$	1.55554	0.0499969
$969$	0	0
$970$	2.26025	0.0725723
$971$	−19.3907	−0.622277	−0.311138	−	0.950365i	$-0.600710\pi$
$971$	−19.3907	−0.622277	−0.311138	+	0.950365i	$0.600710\pi$
$972$	0	0
$973$	−69.9724	−2.24321
$974$	−7.34031	−0.235199
$975$	0	0
$976$	−0.963435	−0.0308388
$977$	10.7096	0.342632	0.171316	−	0.985216i	$-0.445198\pi$
$977$	10.7096	0.342632	0.171316	+	0.985216i	$0.445198\pi$
$978$	0	0
$979$	−55.3042	−1.76753
$980$	11.2351	0.358891
$981$	0	0
$982$	10.4177	0.332443
$983$	47.9639	1.52981	0.764906	−	0.644142i	$-0.222786\pi$
$983$	47.9639	1.52981	0.764906	+	0.644142i	$0.222786\pi$
$984$	0	0
$985$	−12.9447	−0.412452
$986$	19.1240	0.609032
$987$	0	0
$988$	−8.83654	−0.281128
$989$	−30.1619	−0.959094
$990$	0	0
$991$	3.81627	0.121228	0.0606139	−	0.998161i	$-0.480694\pi$
$991$	3.81627	0.121228	0.0606139	+	0.998161i	$0.480694\pi$
$992$	−13.8207	−0.438808
$993$	0	0
$994$	−3.77478	−0.119729
$995$	0.133353	0.00422758
$996$	0	0
$997$	37.8894	1.19997	0.599985	−	0.800012i	$-0.295173\pi$
$997$	37.8894	1.19997	0.599985	+	0.800012i	$0.295173\pi$
$998$	−7.06376	−0.223599
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.k.1.2	yes	3
3.2	odd	2		855.2.a.j.1.2	✓	3
5.4	even	2		4275.2.a.bc.1.2		3
15.14	odd	2		4275.2.a.bl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.a.j.1.2	✓	3	3.2	odd	2
855.2.a.k.1.2	yes	3	1.1	even	1	trivial
4275.2.a.bc.1.2		3	5.4	even	2
4275.2.a.bl.1.2		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -3.59210 q^{7} -1.21432 q^{8} +O(q^{10})\)	\(q+0.311108 q^{2} -1.90321 q^{4} -1.00000 q^{5} -3.59210 q^{7} -1.21432 q^{8} -0.311108 q^{10} -3.11753 q^{11} +4.64296 q^{13} -1.11753 q^{14} +3.42864 q^{16} +7.52543 q^{17} +1.00000 q^{19} +1.90321 q^{20} -0.969888 q^{22} +5.52543 q^{23} +1.00000 q^{25} +1.44446 q^{26} +6.83654 q^{28} +8.16839 q^{29} -3.95407 q^{31} +3.49532 q^{32} +2.34122 q^{34} +3.59210 q^{35} -8.83654 q^{37} +0.311108 q^{38} +1.21432 q^{40} +7.87310 q^{41} -5.45875 q^{43} +5.93332 q^{44} +1.71900 q^{46} -9.13828 q^{47} +5.90321 q^{49} +0.311108 q^{50} -8.83654 q^{52} +5.03657 q^{53} +3.11753 q^{55} +4.36196 q^{56} +2.54125 q^{58} -6.42864 q^{59} -0.280996 q^{61} -1.23014 q^{62} -5.76986 q^{64} -4.64296 q^{65} +3.67307 q^{67} -14.3225 q^{68} +1.11753 q^{70} +3.37778 q^{71} +3.18421 q^{73} -2.74912 q^{74} -1.90321 q^{76} +11.1985 q^{77} +12.9906 q^{79} -3.42864 q^{80} +2.44938 q^{82} +8.57628 q^{83} -7.52543 q^{85} -1.69826 q^{86} +3.78568 q^{88} +17.7397 q^{89} -16.6780 q^{91} -10.5161 q^{92} -2.84299 q^{94} -1.00000 q^{95} -7.26517 q^{97} +1.83654 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8	\( 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8} - q^{10} + 4 q^{11} - 6 q^{13} + 10 q^{14} - 3 q^{16} + 16 q^{17} + 3 q^{19} - q^{20} + 4 q^{22} + 10 q^{23} + 3 q^{25} + 4 q^{26} + 14 q^{28} - 2 q^{29} + 8 q^{31} - 3 q^{32} + 14 q^{34} + 4 q^{35} - 20 q^{37} + q^{38} - 3 q^{40} + 10 q^{41} - 10 q^{43} + 18 q^{44} + 12 q^{46} + 6 q^{47} + 11 q^{49} + q^{50} - 20 q^{52} + 8 q^{53} - 4 q^{55} + 14 q^{58} - 6 q^{59} + 6 q^{61} - 10 q^{62} - 11 q^{64} + 6 q^{65} - 2 q^{67} + 4 q^{68} - 10 q^{70} + 10 q^{71} - 4 q^{73} - 22 q^{74} + q^{76} + 14 q^{77} + 12 q^{79} + 3 q^{80} - 26 q^{82} + 6 q^{83} - 16 q^{85} - 32 q^{86} + 18 q^{88} + 40 q^{89} - 4 q^{91} + 2 q^{92} + 12 q^{94} - 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100}) \) 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8 - q^10 + 4 * q^11 - 6 * q^13 + 10 * q^14 - 3 * q^16 + 16 * q^17 + 3 * q^19 - q^20 + 4 * q^22 + 10 * q^23 + 3 * q^25 + 4 * q^26 + 14 * q^28 - 2 * q^29 + 8 * q^31 - 3 * q^32 + 14 * q^34 + 4 * q^35 - 20 * q^37 + q^38 - 3 * q^40 + 10 * q^41 - 10 * q^43 + 18 * q^44 + 12 * q^46 + 6 * q^47 + 11 * q^49 + q^50 - 20 * q^52 + 8 * q^53 - 4 * q^55 + 14 * q^58 - 6 * q^59 + 6 * q^61 - 10 * q^62 - 11 * q^64 + 6 * q^65 - 2 * q^67 + 4 * q^68 - 10 * q^70 + 10 * q^71 - 4 * q^73 - 22 * q^74 + q^76 + 14 * q^77 + 12 * q^79 + 3 * q^80 - 26 * q^82 + 6 * q^83 - 16 * q^85 - 32 * q^86 + 18 * q^88 + 40 * q^89 - 4 * q^91 + 2 * q^92 + 12 * q^94 - 3 * q^95 - 2 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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