LMFDB - Embedded newform 2835.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2835,2,Mod(1,2835)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2835, 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("2835.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2835 = 3^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2835.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:	$22.6375889730$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75080$ of defining polynomial
Character	$\chi$	$=$	2835.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75080 q^{2} +1.06530 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.63647 q^{8} +O(q^{10})$	\(q-1.75080 q^{2} +1.06530 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.63647 q^{8} +1.75080 q^{10} +6.11007 q^{11} -5.81610 q^{13} -1.75080 q^{14} -4.99574 q^{16} -4.73907 q^{17} +3.68123 q^{19} -1.06530 q^{20} -10.6975 q^{22} +1.00000 q^{23} +1.00000 q^{25} +10.1828 q^{26} +1.06530 q^{28} +1.67803 q^{29} -6.95844 q^{31} +5.47360 q^{32} +8.29717 q^{34} -1.00000 q^{35} -4.49413 q^{37} -6.44511 q^{38} -1.63647 q^{40} +3.97471 q^{41} -11.3270 q^{43} +6.50907 q^{44} -1.75080 q^{46} -4.17537 q^{47} +1.00000 q^{49} -1.75080 q^{50} -6.19591 q^{52} +8.30597 q^{53} -6.11007 q^{55} +1.63647 q^{56} -2.93790 q^{58} -0.837128 q^{59} -9.25667 q^{61} +12.1828 q^{62} +0.408295 q^{64} +5.81610 q^{65} +4.51787 q^{67} -5.04854 q^{68} +1.75080 q^{70} -10.3667 q^{71} +12.2346 q^{73} +7.86833 q^{74} +3.92163 q^{76} +6.11007 q^{77} +8.83286 q^{79} +4.99574 q^{80} -6.95892 q^{82} +6.34144 q^{83} +4.73907 q^{85} +19.8313 q^{86} +9.99894 q^{88} +9.66496 q^{89} -5.81610 q^{91} +1.06530 q^{92} +7.31024 q^{94} -3.68123 q^{95} -8.17354 q^{97} -1.75080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 4 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + q^4 - 4 * q^5 + 4 * q^7 + 3 * q^8	$ 4 q + q^{2} + q^{4} - 4 q^{5} + 4 q^{7} + 3 q^{8} - q^{10} + q^{11} - 12 q^{13} + q^{14} - q^{16} - 5 q^{17} - 9 q^{19} - q^{20} - 17 q^{22} + 4 q^{23} + 4 q^{25} + q^{26} + q^{28} + 15 q^{29} - 16 q^{31} + 2 q^{32} - 11 q^{34} - 4 q^{35} - 15 q^{37} - 24 q^{38} - 3 q^{40} + 2 q^{41} - 7 q^{43} - 3 q^{44} + q^{46} + 10 q^{47} + 4 q^{49} + q^{50} - 15 q^{52} - q^{55} + 3 q^{56} + 17 q^{58} + 13 q^{59} - 32 q^{61} + 9 q^{62} - 15 q^{64} + 12 q^{65} - 41 q^{68} - q^{70} - 13 q^{71} - 2 q^{73} - 13 q^{74} - 11 q^{76} + q^{77} + q^{80} + 6 q^{82} - 17 q^{83} + 5 q^{85} + 15 q^{86} - 11 q^{88} + 17 q^{89} - 12 q^{91} + q^{92} + 15 q^{94} + 9 q^{95} - 4 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 - 4 * q^5 + 4 * q^7 + 3 * q^8 - q^10 + q^11 - 12 * q^13 + q^14 - q^16 - 5 * q^17 - 9 * q^19 - q^20 - 17 * q^22 + 4 * q^23 + 4 * q^25 + q^26 + q^28 + 15 * q^29 - 16 * q^31 + 2 * q^32 - 11 * q^34 - 4 * q^35 - 15 * q^37 - 24 * q^38 - 3 * q^40 + 2 * q^41 - 7 * q^43 - 3 * q^44 + q^46 + 10 * q^47 + 4 * q^49 + q^50 - 15 * q^52 - q^55 + 3 * q^56 + 17 * q^58 + 13 * q^59 - 32 * q^61 + 9 * q^62 - 15 * q^64 + 12 * q^65 - 41 * q^68 - q^70 - 13 * q^71 - 2 * q^73 - 13 * q^74 - 11 * q^76 + q^77 + q^80 + 6 * q^82 - 17 * q^83 + 5 * q^85 + 15 * q^86 - 11 * q^88 + 17 * q^89 - 12 * q^91 + q^92 + 15 * q^94 + 9 * q^95 - 4 * 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 $

$2$	−1.75080	−1.23800	−0.619001	−	0.785390i	$-0.712462\pi$
$2$	−1.75080	−1.23800	−0.619001	+	0.785390i	$0.712462\pi$
$3$	0	0
$3$	0	0
$4$	1.06530	0.532651
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.63647	0.578579
$9$	0	0
$10$	1.75080	0.553652
$11$	6.11007	1.84225	0.921127	−	0.389262i	$-0.127270\pi$
$11$	6.11007	1.84225	0.921127	+	0.389262i	$0.127270\pi$
$12$	0	0
$13$	−5.81610	−1.61310	−0.806548	−	0.591168i	$-0.798667\pi$
$13$	−5.81610	−1.61310	−0.806548	+	0.591168i	$0.798667\pi$
$14$	−1.75080	−0.467921
$15$	0	0
$16$	−4.99574	−1.24893
$17$	−4.73907	−1.14939	−0.574697	−	0.818367i	$-0.694880\pi$
$17$	−4.73907	−1.14939	−0.574697	+	0.818367i	$0.694880\pi$
$18$	0	0
$19$	3.68123	0.844533	0.422266	−	0.906472i	$-0.361235\pi$
$19$	3.68123	0.844533	0.422266	+	0.906472i	$0.361235\pi$
$20$	−1.06530	−0.238209
$21$	0	0
$22$	−10.6975	−2.28072
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	10.1828	1.99702
$27$	0	0
$28$	1.06530	0.201323
$29$	1.67803	0.311603	0.155801	−	0.987788i	$-0.450204\pi$
$29$	1.67803	0.311603	0.155801	+	0.987788i	$0.450204\pi$
$30$	0	0
$31$	−6.95844	−1.24977	−0.624886	−	0.780716i	$-0.714855\pi$
$31$	−6.95844	−1.24977	−0.624886	+	0.780716i	$0.714855\pi$
$32$	5.47360	0.967604
$33$	0	0
$34$	8.29717	1.42295
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−4.49413	−0.738831	−0.369416	−	0.929264i	$-0.620442\pi$
$37$	−4.49413	−0.738831	−0.369416	+	0.929264i	$0.620442\pi$
$38$	−6.44511	−1.04553
$39$	0	0
$40$	−1.63647	−0.258749
$41$	3.97471	0.620745	0.310373	−	0.950615i	$-0.399546\pi$
$41$	3.97471	0.620745	0.310373	+	0.950615i	$0.399546\pi$
$42$	0	0
$43$	−11.3270	−1.72735	−0.863676	−	0.504048i	$-0.831843\pi$
$43$	−11.3270	−1.72735	−0.863676	+	0.504048i	$0.831843\pi$
$44$	6.50907	0.981279
$45$	0	0
$46$	−1.75080	−0.258141
$47$	−4.17537	−0.609040	−0.304520	−	0.952506i	$-0.598496\pi$
$47$	−4.17537	−0.609040	−0.304520	+	0.952506i	$0.598496\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.75080	−0.247601
$51$	0	0
$52$	−6.19591	−0.859218
$53$	8.30597	1.14091	0.570457	−	0.821328i	$-0.306766\pi$
$53$	8.30597	1.14091	0.570457	+	0.821328i	$0.306766\pi$
$54$	0	0
$55$	−6.11007	−0.823881
$56$	1.63647	0.218682
$57$	0	0
$58$	−2.93790	−0.385765
$59$	−0.837128	−0.108985	−0.0544924	−	0.998514i	$-0.517354\pi$
$59$	−0.837128	−0.108985	−0.0544924	+	0.998514i	$0.517354\pi$
$60$	0	0
$61$	−9.25667	−1.18519	−0.592597	−	0.805499i	$-0.701897\pi$
$61$	−9.25667	−1.18519	−0.592597	+	0.805499i	$0.701897\pi$
$62$	12.1828	1.54722
$63$	0	0
$64$	0.408295	0.0510369
$65$	5.81610	0.721399
$66$	0	0
$67$	4.51787	0.551946	0.275973	−	0.961165i	$-0.411000\pi$
$67$	4.51787	0.551946	0.275973	+	0.961165i	$0.411000\pi$
$68$	−5.04854	−0.612226
$69$	0	0
$70$	1.75080	0.209261
$71$	−10.3667	−1.23030	−0.615152	−	0.788408i	$-0.710906\pi$
$71$	−10.3667	−1.23030	−0.615152	+	0.788408i	$0.710906\pi$
$72$	0	0
$73$	12.2346	1.43195	0.715975	−	0.698126i	$-0.245983\pi$
$73$	12.2346	1.43195	0.715975	+	0.698126i	$0.245983\pi$
$74$	7.86833	0.914675
$75$	0	0
$76$	3.92163	0.449841
$77$	6.11007	0.696307
$78$	0	0
$79$	8.83286	0.993775	0.496887	−	0.867815i	$-0.334476\pi$
$79$	8.83286	0.993775	0.496887	+	0.867815i	$0.334476\pi$
$80$	4.99574	0.558540
$81$	0	0
$82$	−6.95892	−0.768485
$83$	6.34144	0.696064	0.348032	−	0.937483i	$-0.386850\pi$
$83$	6.34144	0.696064	0.348032	+	0.937483i	$0.386850\pi$
$84$	0	0
$85$	4.73907	0.514024
$86$	19.8313	2.13847
$87$	0	0
$88$	9.99894	1.06589
$89$	9.66496	1.02448	0.512242	−	0.858841i	$-0.328815\pi$
$89$	9.66496	1.02448	0.512242	+	0.858841i	$0.328815\pi$
$90$	0	0
$91$	−5.81610	−0.609693
$92$	1.06530	0.111065
$93$	0	0
$94$	7.31024	0.753993
$95$	−3.68123	−0.377687
$96$	0	0
$97$	−8.17354	−0.829897	−0.414949	−	0.909845i	$-0.636200\pi$
$97$	−8.17354	−0.829897	−0.414949	+	0.909845i	$0.636200\pi$
$98$	−1.75080	−0.176858
$99$	0	0
$100$	1.06530	0.106530
$101$	−12.1320	−1.20718	−0.603588	−	0.797296i	$-0.706263\pi$
$101$	−12.1320	−1.20718	−0.603588	+	0.797296i	$0.706263\pi$
$102$	0	0
$103$	−1.39580	−0.137532	−0.0687661	−	0.997633i	$-0.521906\pi$
$103$	−1.39580	−0.137532	−0.0687661	+	0.997633i	$0.521906\pi$
$104$	−9.51787	−0.933304
$105$	0	0
$106$	−14.5421	−1.41245
$107$	0.484840	0.0468713	0.0234356	−	0.999725i	$-0.492540\pi$
$107$	0.484840	0.0468713	0.0234356	+	0.999725i	$0.492540\pi$
$108$	0	0
$109$	−10.7345	−1.02818	−0.514091	−	0.857736i	$-0.671870\pi$
$109$	−10.7345	−1.02818	−0.514091	+	0.857736i	$0.671870\pi$
$110$	10.6975	1.01997
$111$	0	0
$112$	−4.99574	−0.472053
$113$	1.11860	0.105229	0.0526143	−	0.998615i	$-0.483245\pi$
$113$	1.11860	0.105229	0.0526143	+	0.998615i	$0.483245\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	1.78761	0.165976
$117$	0	0
$118$	1.46564	0.134923
$119$	−4.73907	−0.434430
$120$	0	0
$121$	26.3329	2.39390
$122$	16.2066	1.46727
$123$	0	0
$124$	−7.41284	−0.665692
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.7567	−1.04324	−0.521620	−	0.853178i	$-0.674672\pi$
$127$	−11.7567	−1.04324	−0.521620	+	0.853178i	$0.674672\pi$
$128$	−11.6620	−1.03079
$129$	0	0
$130$	−10.1828	−0.893094
$131$	0.970168	0.0847639	0.0423820	−	0.999101i	$-0.486505\pi$
$131$	0.970168	0.0847639	0.0423820	+	0.999101i	$0.486505\pi$
$132$	0	0
$133$	3.68123	0.319203
$134$	−7.90990	−0.683311
$135$	0	0
$136$	−7.75534	−0.665015
$137$	−10.7671	−0.919893	−0.459947	−	0.887947i	$-0.652131\pi$
$137$	−10.7671	−0.919893	−0.459947	+	0.887947i	$0.652131\pi$
$138$	0	0
$139$	−10.8254	−0.918198	−0.459099	−	0.888385i	$-0.651828\pi$
$139$	−10.8254	−0.918198	−0.459099	+	0.888385i	$0.651828\pi$
$140$	−1.06530	−0.0900345
$141$	0	0
$142$	18.1501	1.52312
$143$	−35.5368	−2.97173
$144$	0	0
$145$	−1.67803	−0.139353
$146$	−21.4203	−1.77276
$147$	0	0
$148$	−4.78761	−0.393539
$149$	13.8150	1.13177	0.565886	−	0.824483i	$-0.308534\pi$
$149$	13.8150	1.13177	0.565886	+	0.824483i	$0.308534\pi$
$150$	0	0
$151$	10.8401	0.882156	0.441078	−	0.897469i	$-0.354596\pi$
$151$	10.8401	0.882156	0.441078	+	0.897469i	$0.354596\pi$
$152$	6.02423	0.488629
$153$	0	0
$154$	−10.6975	−0.862030
$155$	6.95844	0.558915
$156$	0	0
$157$	−11.0701	−0.883486	−0.441743	−	0.897142i	$-0.645640\pi$
$157$	−11.0701	−0.883486	−0.441743	+	0.897142i	$0.645640\pi$
$158$	−15.4646	−1.23030
$159$	0	0
$160$	−5.47360	−0.432726
$161$	1.00000	0.0788110
$162$	0	0
$163$	11.2066	0.877767	0.438883	−	0.898544i	$-0.355374\pi$
$163$	11.2066	0.877767	0.438883	+	0.898544i	$0.355374\pi$
$164$	4.23427	0.330641
$165$	0	0
$166$	−11.1026	−0.861729
$167$	−23.6399	−1.82931	−0.914657	−	0.404231i	$-0.867539\pi$
$167$	−23.6399	−1.82931	−0.914657	+	0.404231i	$0.867539\pi$
$168$	0	0
$169$	20.8270	1.60208
$170$	−8.29717	−0.636364
$171$	0	0
$172$	−12.0667	−0.920075
$173$	3.08904	0.234855	0.117428	−	0.993081i	$-0.462535\pi$
$173$	3.08904	0.234855	0.117428	+	0.993081i	$0.462535\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−30.5243	−2.30085
$177$	0	0
$178$	−16.9214	−1.26831
$179$	−22.9526	−1.71556	−0.857780	−	0.514017i	$-0.828157\pi$
$179$	−22.9526	−1.71556	−0.857780	+	0.514017i	$0.828157\pi$
$180$	0	0
$181$	−4.10028	−0.304772	−0.152386	−	0.988321i	$-0.548696\pi$
$181$	−4.10028	−0.304772	−0.152386	+	0.988321i	$0.548696\pi$
$182$	10.1828	0.754802
$183$	0	0
$184$	1.63647	0.120642
$185$	4.49413	0.330415
$186$	0	0
$187$	−28.9560	−2.11747
$188$	−4.44803	−0.324406
$189$	0	0
$190$	6.44511	0.467577
$191$	8.73481	0.632028	0.316014	−	0.948754i	$-0.397655\pi$
$191$	8.73481	0.632028	0.316014	+	0.948754i	$0.397655\pi$
$192$	0	0
$193$	−16.1828	−1.16487	−0.582433	−	0.812879i	$-0.697899\pi$
$193$	−16.1828	−1.16487	−0.582433	+	0.812879i	$0.697899\pi$
$194$	14.3102	1.02742
$195$	0	0
$196$	1.06530	0.0760930
$197$	0.418652	0.0298278	0.0149139	−	0.999889i	$-0.495253\pi$
$197$	0.418652	0.0298278	0.0149139	+	0.999889i	$0.495253\pi$
$198$	0	0
$199$	−7.48667	−0.530716	−0.265358	−	0.964150i	$-0.585490\pi$
$199$	−7.48667	−0.530716	−0.265358	+	0.964150i	$0.585490\pi$
$200$	1.63647	0.115716
$201$	0	0
$202$	21.2407	1.49449
$203$	1.67803	0.117775
$204$	0	0
$205$	−3.97471	−0.277606
$206$	2.44376	0.170265
$207$	0	0
$208$	29.0557	2.01465
$209$	22.4926	1.55584
$210$	0	0
$211$	6.25938	0.430913	0.215457	−	0.976513i	$-0.430876\pi$
$211$	6.25938	0.430913	0.215457	+	0.976513i	$0.430876\pi$
$212$	8.84837	0.607709
$213$	0	0
$214$	−0.848858	−0.0580267
$215$	11.3270	0.772495
$216$	0	0
$217$	−6.95844	−0.472369
$218$	18.7940	1.27289
$219$	0	0
$220$	−6.50907	−0.438841
$221$	27.5629	1.85408
$222$	0	0
$223$	−23.2414	−1.55636	−0.778181	−	0.628040i	$-0.783857\pi$
$223$	−23.2414	−1.55636	−0.778181	+	0.628040i	$0.783857\pi$
$224$	5.47360	0.365720
$225$	0	0
$226$	−1.95844	−0.130273
$227$	−19.7684	−1.31208	−0.656039	−	0.754727i	$-0.727769\pi$
$227$	−19.7684	−1.31208	−0.656039	+	0.754727i	$0.727769\pi$
$228$	0	0
$229$	−0.0103574	−0.000684438 0	−0.000342219	−	1.00000i	$-0.500109\pi$
$229$	−0.0103574	−0.000684438 0	−0.000342219		1.00000i	$0.500109\pi$
$230$	1.75080	0.115444
$231$	0	0
$232$	2.74605	0.180287
$233$	6.48609	0.424918	0.212459	−	0.977170i	$-0.431853\pi$
$233$	6.48609	0.424918	0.212459	+	0.977170i	$0.431853\pi$
$234$	0	0
$235$	4.17537	0.272371
$236$	−0.891794	−0.0580508
$237$	0	0
$238$	8.29717	0.537825
$239$	−15.7986	−1.02192	−0.510962	−	0.859603i	$-0.670711\pi$
$239$	−15.7986	−1.02192	−0.510962	+	0.859603i	$0.670711\pi$
$240$	0	0
$241$	−6.41799	−0.413419	−0.206709	−	0.978402i	$-0.566275\pi$
$241$	−6.41799	−0.413419	−0.206709	+	0.978402i	$0.566275\pi$
$242$	−46.1037	−2.96366
$243$	0	0
$244$	−9.86115	−0.631295
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−21.4104	−1.36231
$248$	−11.3873	−0.723092
$249$	0	0
$250$	1.75080	0.110730
$251$	−4.15861	−0.262489	−0.131245	−	0.991350i	$-0.541897\pi$
$251$	−4.15861	−0.262489	−0.131245	+	0.991350i	$0.541897\pi$
$252$	0	0
$253$	6.11007	0.384137
$254$	20.5837	1.29153
$255$	0	0
$256$	19.6013	1.22508
$257$	−11.7481	−0.732825	−0.366413	−	0.930452i	$-0.619414\pi$
$257$	−11.7481	−0.732825	−0.366413	+	0.930452i	$0.619414\pi$
$258$	0	0
$259$	−4.49413	−0.279252
$260$	6.19591	0.384254
$261$	0	0
$262$	−1.69857	−0.104938
$263$	12.3985	0.764525	0.382263	−	0.924054i	$-0.375145\pi$
$263$	12.3985	0.764525	0.382263	+	0.924054i	$0.375145\pi$
$264$	0	0
$265$	−8.30597	−0.510232
$266$	−6.44511	−0.395175
$267$	0	0
$268$	4.81290	0.293995
$269$	18.8491	1.14925	0.574626	−	0.818416i	$-0.305148\pi$
$269$	18.8491	1.14925	0.574626	+	0.818416i	$0.305148\pi$
$270$	0	0
$271$	−21.0866	−1.28092	−0.640461	−	0.767991i	$-0.721257\pi$
$271$	−21.0866	−1.28092	−0.640461	+	0.767991i	$0.721257\pi$
$272$	23.6751	1.43552
$273$	0	0
$274$	18.8510	1.13883
$275$	6.11007	0.368451
$276$	0	0
$277$	−18.4973	−1.11140	−0.555699	−	0.831384i	$-0.687549\pi$
$277$	−18.4973	−1.11140	−0.555699	+	0.831384i	$0.687549\pi$
$278$	18.9531	1.13673
$279$	0	0
$280$	−1.63647	−0.0977978
$281$	−17.6284	−1.05162	−0.525812	−	0.850601i	$-0.676238\pi$
$281$	−17.6284	−1.05162	−0.525812	+	0.850601i	$0.676238\pi$
$282$	0	0
$283$	−3.79188	−0.225404	−0.112702	−	0.993629i	$-0.535950\pi$
$283$	−3.79188	−0.225404	−0.112702	+	0.993629i	$0.535950\pi$
$284$	−11.0437	−0.655323
$285$	0	0
$286$	62.2178	3.67902
$287$	3.97471	0.234620
$288$	0	0
$289$	5.45878	0.321105
$290$	2.93790	0.172519
$291$	0	0
$292$	13.0335	0.762729
$293$	8.07460	0.471723	0.235861	−	0.971787i	$-0.424209\pi$
$293$	8.07460	0.471723	0.235861	+	0.971787i	$0.424209\pi$
$294$	0	0
$295$	0.837128	0.0487395
$296$	−7.35451	−0.427472
$297$	0	0
$298$	−24.1874	−1.40114
$299$	−5.81610	−0.336354
$300$	0	0
$301$	−11.3270	−0.652877
$302$	−18.9789	−1.09211
$303$	0	0
$304$	−18.3905	−1.05477
$305$	9.25667	0.530035
$306$	0	0
$307$	−1.03255	−0.0589305	−0.0294653	−	0.999566i	$-0.509380\pi$
$307$	−1.03255	−0.0589305	−0.0294653	+	0.999566i	$0.509380\pi$
$308$	6.50907	0.370889
$309$	0	0
$310$	−12.1828	−0.691939
$311$	−8.28942	−0.470050	−0.235025	−	0.971989i	$-0.575517\pi$
$311$	−8.28942	−0.470050	−0.235025	+	0.971989i	$0.575517\pi$
$312$	0	0
$313$	20.0693	1.13438	0.567191	−	0.823586i	$-0.308030\pi$
$313$	20.0693	1.13438	0.567191	+	0.823586i	$0.308030\pi$
$314$	19.3815	1.09376
$315$	0	0
$316$	9.40967	0.529335
$317$	−20.7642	−1.16623	−0.583116	−	0.812389i	$-0.698167\pi$
$317$	−20.7642	−1.16623	−0.583116	+	0.812389i	$0.698167\pi$
$318$	0	0
$319$	10.2529	0.574052
$320$	−0.408295	−0.0228244
$321$	0	0
$322$	−1.75080	−0.0975683
$323$	−17.4456	−0.970701
$324$	0	0
$325$	−5.81610	−0.322619
$326$	−19.6205	−1.08668
$327$	0	0
$328$	6.50449	0.359151
$329$	−4.17537	−0.230196
$330$	0	0
$331$	−11.1632	−0.613582	−0.306791	−	0.951777i	$-0.599255\pi$
$331$	−11.1632	−0.613582	−0.306791	+	0.951777i	$0.599255\pi$
$332$	6.75555	0.370759
$333$	0	0
$334$	41.3888	2.26470
$335$	−4.51787	−0.246838
$336$	0	0
$337$	6.07249	0.330790	0.165395	−	0.986227i	$-0.447110\pi$
$337$	6.07249	0.330790	0.165395	+	0.986227i	$0.447110\pi$
$338$	−36.4640	−1.98338
$339$	0	0
$340$	5.04854	0.273796
$341$	−42.5165	−2.30240
$342$	0	0
$343$	1.00000	0.0539949
$344$	−18.5363	−0.999410
$345$	0	0
$346$	−5.40830	−0.290752
$347$	−19.5051	−1.04709	−0.523544	−	0.851999i	$-0.675390\pi$
$347$	−19.5051	−1.04709	−0.523544	+	0.851999i	$0.675390\pi$
$348$	0	0
$349$	−12.5882	−0.673831	−0.336916	−	0.941535i	$-0.609384\pi$
$349$	−12.5882	−0.673831	−0.336916	+	0.941535i	$0.609384\pi$
$350$	−1.75080	−0.0935842
$351$	0	0
$352$	33.4440	1.78257
$353$	−16.4171	−0.873796	−0.436898	−	0.899511i	$-0.643923\pi$
$353$	−16.4171	−0.873796	−0.436898	+	0.899511i	$0.643923\pi$
$354$	0	0
$355$	10.3667	0.550209
$356$	10.2961	0.545692
$357$	0	0
$358$	40.1855	2.12387
$359$	13.2324	0.698381	0.349191	−	0.937052i	$-0.386457\pi$
$359$	13.2324	0.698381	0.349191	+	0.937052i	$0.386457\pi$
$360$	0	0
$361$	−5.44852	−0.286764
$362$	7.17878	0.377308
$363$	0	0
$364$	−6.19591	−0.324754
$365$	−12.2346	−0.640387
$366$	0	0
$367$	−20.2724	−1.05821	−0.529104	−	0.848557i	$-0.677472\pi$
$367$	−20.2724	−1.05821	−0.529104	+	0.848557i	$0.677472\pi$
$368$	−4.99574	−0.260421
$369$	0	0
$370$	−7.86833	−0.409055
$371$	8.30597	0.431225
$372$	0	0
$373$	30.2507	1.56632	0.783162	−	0.621818i	$-0.213606\pi$
$373$	30.2507	1.56632	0.783162	+	0.621818i	$0.213606\pi$
$374$	50.6962	2.62144
$375$	0	0
$376$	−6.83286	−0.352378
$377$	−9.75961	−0.502645
$378$	0	0
$379$	−0.698602	−0.0358848	−0.0179424	−	0.999839i	$-0.505712\pi$
$379$	−0.698602	−0.0358848	−0.0179424	+	0.999839i	$0.505712\pi$
$380$	−3.92163	−0.201175
$381$	0	0
$382$	−15.2929	−0.782453
$383$	−6.09513	−0.311447	−0.155723	−	0.987801i	$-0.549771\pi$
$383$	−6.09513	−0.311447	−0.155723	+	0.987801i	$0.549771\pi$
$384$	0	0
$385$	−6.11007	−0.311398
$386$	28.3329	1.44211
$387$	0	0
$388$	−8.70729	−0.442046
$389$	−13.5589	−0.687462	−0.343731	−	0.939068i	$-0.611691\pi$
$389$	−13.5589	−0.687462	−0.343731	+	0.939068i	$0.611691\pi$
$390$	0	0
$391$	−4.73907	−0.239665
$392$	1.63647	0.0826542
$393$	0	0
$394$	−0.732977	−0.0369268
$395$	−8.83286	−0.444430
$396$	0	0
$397$	−16.4109	−0.823639	−0.411819	−	0.911265i	$-0.635107\pi$
$397$	−16.4109	−0.823639	−0.411819	+	0.911265i	$0.635107\pi$
$398$	13.1077	0.657028
$399$	0	0
$400$	−4.99574	−0.249787
$401$	3.50083	0.174823	0.0874117	−	0.996172i	$-0.472140\pi$
$401$	3.50083	0.174823	0.0874117	+	0.996172i	$0.472140\pi$
$402$	0	0
$403$	40.4710	2.01600
$404$	−12.9242	−0.643004
$405$	0	0
$406$	−2.93790	−0.145806
$407$	−27.4595	−1.36112
$408$	0	0
$409$	6.75717	0.334121	0.167060	−	0.985947i	$-0.446573\pi$
$409$	6.75717	0.334121	0.167060	+	0.985947i	$0.446573\pi$
$410$	6.95892	0.343677
$411$	0	0
$412$	−1.48695	−0.0732566
$413$	−0.837128	−0.0411924
$414$	0	0
$415$	−6.34144	−0.311289
$416$	−31.8350	−1.56084
$417$	0	0
$418$	−39.3800	−1.92614
$419$	30.2284	1.47675	0.738376	−	0.674389i	$-0.235593\pi$
$419$	30.2284	1.47675	0.738376	+	0.674389i	$0.235593\pi$
$420$	0	0
$421$	−1.30122	−0.0634176	−0.0317088	−	0.999497i	$-0.510095\pi$
$421$	−1.30122	−0.0634176	−0.0317088	+	0.999497i	$0.510095\pi$
$422$	−10.9589	−0.533472
$423$	0	0
$424$	13.5925	0.660109
$425$	−4.73907	−0.229879
$426$	0	0
$427$	−9.25667	−0.447961
$428$	0.516501	0.0249660
$429$	0	0
$430$	−19.8313	−0.956351
$431$	6.84759	0.329837	0.164918	−	0.986307i	$-0.447264\pi$
$431$	6.84759	0.329837	0.164918	+	0.986307i	$0.447264\pi$
$432$	0	0
$433$	−5.73858	−0.275779	−0.137889	−	0.990448i	$-0.544032\pi$
$433$	−5.73858	−0.275779	−0.137889	+	0.990448i	$0.544032\pi$
$434$	12.1828	0.584795
$435$	0	0
$436$	−11.4355	−0.547662
$437$	3.68123	0.176097
$438$	0	0
$439$	12.3396	0.588938	0.294469	−	0.955661i	$-0.404857\pi$
$439$	12.3396	0.588938	0.294469	+	0.955661i	$0.404857\pi$
$440$	−9.99894	−0.476681
$441$	0	0
$442$	−48.2572	−2.29536
$443$	−10.3393	−0.491235	−0.245618	−	0.969367i	$-0.578991\pi$
$443$	−10.3393	−0.491235	−0.245618	+	0.969367i	$0.578991\pi$
$444$	0	0
$445$	−9.66496	−0.458163
$446$	40.6911	1.92678
$447$	0	0
$448$	0.408295	0.0192901
$449$	−24.4414	−1.15346	−0.576731	−	0.816934i	$-0.695672\pi$
$449$	−24.4414	−1.15346	−0.576731	+	0.816934i	$0.695672\pi$
$450$	0	0
$451$	24.2857	1.14357
$452$	1.19164	0.0560501
$453$	0	0
$454$	34.6106	1.62436
$455$	5.81610	0.272663
$456$	0	0
$457$	−39.9555	−1.86904	−0.934520	−	0.355910i	$-0.884171\pi$
$457$	−39.9555	−1.86904	−0.934520	+	0.355910i	$0.884171\pi$
$458$	0.0181338	0.000847337 0
$459$	0	0
$460$	−1.06530	−0.0496700
$461$	15.1655	0.706326	0.353163	−	0.935562i	$-0.385106\pi$
$461$	15.1655	0.706326	0.353163	+	0.935562i	$0.385106\pi$
$462$	0	0
$463$	10.4286	0.484656	0.242328	−	0.970194i	$-0.422089\pi$
$463$	10.4286	0.484656	0.242328	+	0.970194i	$0.422089\pi$
$464$	−8.38301	−0.389171
$465$	0	0
$466$	−11.3559	−0.526050
$467$	27.5899	1.27671	0.638353	−	0.769744i	$-0.279616\pi$
$467$	27.5899	1.27671	0.638353	+	0.769744i	$0.279616\pi$
$468$	0	0
$469$	4.51787	0.208616
$470$	−7.31024	−0.337196
$471$	0	0
$472$	−1.36993	−0.0630563
$473$	−69.2087	−3.18222
$474$	0	0
$475$	3.68123	0.168907
$476$	−5.04854	−0.231400
$477$	0	0
$478$	27.6601	1.26514
$479$	−21.4888	−0.981849	−0.490924	−	0.871202i	$-0.663341\pi$
$479$	−21.4888	−0.981849	−0.490924	+	0.871202i	$0.663341\pi$
$480$	0	0
$481$	26.1383	1.19181
$482$	11.2366	0.511814
$483$	0	0
$484$	28.0525	1.27511
$485$	8.17354	0.371141
$486$	0	0
$487$	11.6178	0.526451	0.263225	−	0.964734i	$-0.415214\pi$
$487$	11.6178	0.526451	0.263225	+	0.964734i	$0.415214\pi$
$488$	−15.1483	−0.685729
$489$	0	0
$490$	1.75080	0.0790931
$491$	−34.7403	−1.56781	−0.783903	−	0.620883i	$-0.786774\pi$
$491$	−34.7403	−1.56781	−0.783903	+	0.620883i	$0.786774\pi$
$492$	0	0
$493$	−7.95231	−0.358154
$494$	37.4854	1.68655
$495$	0	0
$496$	34.7625	1.56088
$497$	−10.3667	−0.465011
$498$	0	0
$499$	−3.03709	−0.135959	−0.0679794	−	0.997687i	$-0.521655\pi$
$499$	−3.03709	−0.135959	−0.0679794	+	0.997687i	$0.521655\pi$
$500$	−1.06530	−0.0476418
$501$	0	0
$502$	7.28089	0.324962
$503$	3.40995	0.152042	0.0760210	−	0.997106i	$-0.475778\pi$
$503$	3.40995	0.152042	0.0760210	+	0.997106i	$0.475778\pi$
$504$	0	0
$505$	12.1320	0.539866
$506$	−10.6975	−0.475562
$507$	0	0
$508$	−12.5245	−0.555683
$509$	31.3417	1.38919	0.694597	−	0.719399i	$-0.255583\pi$
$509$	31.3417	1.38919	0.694597	+	0.719399i	$0.255583\pi$
$510$	0	0
$511$	12.2346	0.541226
$512$	−10.9939	−0.485867
$513$	0	0
$514$	20.5686	0.907240
$515$	1.39580	0.0615062
$516$	0	0
$517$	−25.5118	−1.12201
$518$	7.86833	0.345715
$519$	0	0
$520$	9.51787	0.417386
$521$	11.9819	0.524936	0.262468	−	0.964941i	$-0.415464\pi$
$521$	11.9819	0.524936	0.262468	+	0.964941i	$0.415464\pi$
$522$	0	0
$523$	−23.4259	−1.02434	−0.512172	−	0.858883i	$-0.671159\pi$
$523$	−23.4259	−1.02434	−0.512172	+	0.858883i	$0.671159\pi$
$524$	1.03352	0.0451496
$525$	0	0
$526$	−21.7073	−0.946484
$527$	32.9765	1.43648
$528$	0	0
$529$	−22.0000	−0.956522
$530$	14.5421	0.631669
$531$	0	0
$532$	3.92163	0.170024
$533$	−23.1173	−1.00132
$534$	0	0
$535$	−0.484840	−0.0209615
$536$	7.39336	0.319345
$537$	0	0
$538$	−33.0011	−1.42278
$539$	6.11007	0.263179
$540$	0	0
$541$	7.62446	0.327801	0.163901	−	0.986477i	$-0.447592\pi$
$541$	7.62446	0.327801	0.163901	+	0.986477i	$0.447592\pi$
$542$	36.9185	1.58579
$543$	0	0
$544$	−25.9398	−1.11216
$545$	10.7345	0.459817
$546$	0	0
$547$	41.3585	1.76836	0.884180	−	0.467145i	$-0.154718\pi$
$547$	41.3585	1.76836	0.884180	+	0.467145i	$0.154718\pi$
$548$	−11.4702	−0.489982
$549$	0	0
$550$	−10.6975	−0.456143
$551$	6.17723	0.263159
$552$	0	0
$553$	8.83286	0.375612
$554$	32.3851	1.37591
$555$	0	0
$556$	−11.5323	−0.489079
$557$	10.2775	0.435472	0.217736	−	0.976008i	$-0.430133\pi$
$557$	10.2775	0.435472	0.217736	+	0.976008i	$0.430133\pi$
$558$	0	0
$559$	65.8790	2.78638
$560$	4.99574	0.211108
$561$	0	0
$562$	30.8639	1.30191
$563$	8.98992	0.378880	0.189440	−	0.981892i	$-0.439333\pi$
$563$	8.98992	0.378880	0.189440	+	0.981892i	$0.439333\pi$
$564$	0	0
$565$	−1.11860	−0.0470597
$566$	6.63882	0.279050
$567$	0	0
$568$	−16.9648	−0.711829
$569$	−7.94223	−0.332956	−0.166478	−	0.986045i	$-0.553239\pi$
$569$	−7.94223	−0.332956	−0.166478	+	0.986045i	$0.553239\pi$
$570$	0	0
$571$	39.1724	1.63931	0.819656	−	0.572856i	$-0.194165\pi$
$571$	39.1724	1.63931	0.819656	+	0.572856i	$0.194165\pi$
$572$	−37.8574	−1.58290
$573$	0	0
$574$	−6.95892	−0.290460
$575$	1.00000	0.0417029
$576$	0	0
$577$	5.80815	0.241796	0.120898	−	0.992665i	$-0.461423\pi$
$577$	5.80815	0.241796	0.120898	+	0.992665i	$0.461423\pi$
$578$	−9.55724	−0.397529
$579$	0	0
$580$	−1.78761	−0.0742265
$581$	6.34144	0.263087
$582$	0	0
$583$	50.7500	2.10185
$584$	20.0215	0.828496
$585$	0	0
$586$	−14.1370	−0.583994
$587$	−2.00028	−0.0825603	−0.0412802	−	0.999148i	$-0.513144\pi$
$587$	−2.00028	−0.0825603	−0.0412802	+	0.999148i	$0.513144\pi$
$588$	0	0
$589$	−25.6156	−1.05547
$590$	−1.46564	−0.0603396
$591$	0	0
$592$	22.4515	0.922751
$593$	−18.6458	−0.765690	−0.382845	−	0.923813i	$-0.625056\pi$
$593$	−18.6458	−0.765690	−0.382845	+	0.923813i	$0.625056\pi$
$594$	0	0
$595$	4.73907	0.194283
$596$	14.7172	0.602840
$597$	0	0
$598$	10.1828	0.416407
$599$	−28.0399	−1.14568	−0.572840	−	0.819667i	$-0.694158\pi$
$599$	−28.0399	−1.14568	−0.572840	+	0.819667i	$0.694158\pi$
$600$	0	0
$601$	29.9063	1.21990	0.609952	−	0.792438i	$-0.291189\pi$
$601$	29.9063	1.21990	0.609952	+	0.792438i	$0.291189\pi$
$602$	19.8313	0.808264
$603$	0	0
$604$	11.5480	0.469882
$605$	−26.3329	−1.07059
$606$	0	0
$607$	15.2527	0.619087	0.309544	−	0.950885i	$-0.399824\pi$
$607$	15.2527	0.619087	0.309544	+	0.950885i	$0.399824\pi$
$608$	20.1496	0.817174
$609$	0	0
$610$	−16.2066	−0.656185
$611$	24.2844	0.982441
$612$	0	0
$613$	21.4518	0.866431	0.433215	−	0.901290i	$-0.357379\pi$
$613$	21.4518	0.866431	0.433215	+	0.901290i	$0.357379\pi$
$614$	1.80778	0.0729562
$615$	0	0
$616$	9.99894	0.402869
$617$	31.6000	1.27217	0.636085	−	0.771619i	$-0.280553\pi$
$617$	31.6000	1.27217	0.636085	+	0.771619i	$0.280553\pi$
$618$	0	0
$619$	18.7450	0.753425	0.376712	−	0.926330i	$-0.377055\pi$
$619$	18.7450	0.753425	0.376712	+	0.926330i	$0.377055\pi$
$620$	7.41284	0.297707
$621$	0	0
$622$	14.5131	0.581923
$623$	9.66496	0.387218
$624$	0	0
$625$	1.00000	0.0400000
$626$	−35.1373	−1.40437
$627$	0	0
$628$	−11.7930	−0.470590
$629$	21.2980	0.849208
$630$	0	0
$631$	−34.9978	−1.39324	−0.696619	−	0.717441i	$-0.745313\pi$
$631$	−34.9978	−1.39324	−0.696619	+	0.717441i	$0.745313\pi$
$632$	14.4547	0.574978
$633$	0	0
$634$	36.3539	1.44380
$635$	11.7567	0.466551
$636$	0	0
$637$	−5.81610	−0.230442
$638$	−17.9508	−0.710677
$639$	0	0
$640$	11.6620	0.460983
$641$	41.2830	1.63058	0.815291	−	0.579052i	$-0.196577\pi$
$641$	41.2830	1.63058	0.815291	+	0.579052i	$0.196577\pi$
$642$	0	0
$643$	5.26182	0.207506	0.103753	−	0.994603i	$-0.466915\pi$
$643$	5.26182	0.207506	0.103753	+	0.994603i	$0.466915\pi$
$644$	1.06530	0.0419788
$645$	0	0
$646$	30.5438	1.20173
$647$	4.79934	0.188682	0.0943408	−	0.995540i	$-0.469926\pi$
$647$	4.79934	0.188682	0.0943408	+	0.995540i	$0.469926\pi$
$648$	0	0
$649$	−5.11491	−0.200778
$650$	10.1828	0.399404
$651$	0	0
$652$	11.9384	0.467543
$653$	−14.4765	−0.566509	−0.283254	−	0.959045i	$-0.591414\pi$
$653$	−14.4765	−0.566509	−0.283254	+	0.959045i	$0.591414\pi$
$654$	0	0
$655$	−0.970168	−0.0379076
$656$	−19.8566	−0.775270
$657$	0	0
$658$	7.31024	0.284983
$659$	28.0171	1.09139	0.545696	−	0.837983i	$-0.316265\pi$
$659$	28.0171	1.09139	0.545696	+	0.837983i	$0.316265\pi$
$660$	0	0
$661$	−48.4337	−1.88385	−0.941926	−	0.335820i	$-0.890986\pi$
$661$	−48.4337	−1.88385	−0.941926	+	0.335820i	$0.890986\pi$
$662$	19.5444	0.759617
$663$	0	0
$664$	10.3776	0.402728
$665$	−3.68123	−0.142752
$666$	0	0
$667$	1.67803	0.0649737
$668$	−25.1837	−0.974386
$669$	0	0
$670$	7.90990	0.305586
$671$	−56.5588	−2.18343
$672$	0	0
$673$	−35.9400	−1.38539	−0.692693	−	0.721232i	$-0.743576\pi$
$673$	−35.9400	−1.38539	−0.692693	+	0.721232i	$0.743576\pi$
$674$	−10.6317	−0.409518
$675$	0	0
$676$	22.1871	0.853350
$677$	28.4439	1.09319	0.546593	−	0.837398i	$-0.315924\pi$
$677$	28.4439	1.09319	0.546593	+	0.837398i	$0.315924\pi$
$678$	0	0
$679$	−8.17354	−0.313672
$680$	7.75534	0.297404
$681$	0	0
$682$	74.4379	2.85038
$683$	41.1926	1.57619	0.788096	−	0.615553i	$-0.211067\pi$
$683$	41.1926	1.57619	0.788096	+	0.615553i	$0.211067\pi$
$684$	0	0
$685$	10.7671	0.411389
$686$	−1.75080	−0.0668459
$687$	0	0
$688$	56.5867	2.15735
$689$	−48.3084	−1.84040
$690$	0	0
$691$	−6.56690	−0.249817	−0.124908	−	0.992168i	$-0.539864\pi$
$691$	−6.56690	−0.249817	−0.124908	+	0.992168i	$0.539864\pi$
$692$	3.29076	0.125096
$693$	0	0
$694$	34.1495	1.29630
$695$	10.8254	0.410631
$696$	0	0
$697$	−18.8364	−0.713481
$698$	22.0394	0.834205
$699$	0	0
$700$	1.06530	0.0402646
$701$	2.61012	0.0985828	0.0492914	−	0.998784i	$-0.484304\pi$
$701$	2.61012	0.0985828	0.0492914	+	0.998784i	$0.484304\pi$
$702$	0	0
$703$	−16.5440	−0.623967
$704$	2.49471	0.0940229
$705$	0	0
$706$	28.7431	1.08176
$707$	−12.1320	−0.456270
$708$	0	0
$709$	−12.9056	−0.484681	−0.242341	−	0.970191i	$-0.577915\pi$
$709$	−12.9056	−0.484681	−0.242341	+	0.970191i	$0.577915\pi$
$710$	−18.1501	−0.681160
$711$	0	0
$712$	15.8164	0.592745
$713$	−6.95844	−0.260596
$714$	0	0
$715$	35.5368	1.32900
$716$	−24.4515	−0.913795
$717$	0	0
$718$	−23.1674	−0.864598
$719$	−8.13828	−0.303507	−0.151753	−	0.988418i	$-0.548492\pi$
$719$	−8.13828	−0.303507	−0.151753	+	0.988418i	$0.548492\pi$
$720$	0	0
$721$	−1.39580	−0.0519823
$722$	9.53927	0.355015
$723$	0	0
$724$	−4.36804	−0.162337
$725$	1.67803	0.0623206
$726$	0	0
$727$	28.5247	1.05792	0.528962	−	0.848646i	$-0.322581\pi$
$727$	28.5247	1.05792	0.528962	+	0.848646i	$0.322581\pi$
$728$	−9.51787	−0.352756
$729$	0	0
$730$	21.4203	0.792801
$731$	53.6794	1.98541
$732$	0	0
$733$	−37.8081	−1.39647	−0.698236	−	0.715867i	$-0.746032\pi$
$733$	−37.8081	−1.39647	−0.698236	+	0.715867i	$0.746032\pi$
$734$	35.4929	1.31007
$735$	0	0
$736$	5.47360	0.201759
$737$	27.6045	1.01683
$738$	0	0
$739$	−39.0949	−1.43813	−0.719065	−	0.694943i	$-0.755430\pi$
$739$	−39.0949	−1.43813	−0.719065	+	0.694943i	$0.755430\pi$
$740$	4.78761	0.175996
$741$	0	0
$742$	−14.5421	−0.533857
$743$	−9.30917	−0.341520	−0.170760	−	0.985313i	$-0.554622\pi$
$743$	−9.30917	−0.341520	−0.170760	+	0.985313i	$0.554622\pi$
$744$	0	0
$745$	−13.8150	−0.506144
$746$	−52.9630	−1.93911
$747$	0	0
$748$	−30.8469	−1.12788
$749$	0.484840	0.0177157
$750$	0	0
$751$	−17.7751	−0.648624	−0.324312	−	0.945950i	$-0.605133\pi$
$751$	−17.7751	−0.648624	−0.324312	+	0.945950i	$0.605133\pi$
$752$	20.8590	0.760651
$753$	0	0
$754$	17.0871	0.622276
$755$	−10.8401	−0.394512
$756$	0	0
$757$	−11.2184	−0.407739	−0.203870	−	0.978998i	$-0.565352\pi$
$757$	−11.2184	−0.407739	−0.203870	+	0.978998i	$0.565352\pi$
$758$	1.22311	0.0444254
$759$	0	0
$760$	−6.02423	−0.218522
$761$	42.1917	1.52945	0.764723	−	0.644359i	$-0.222876\pi$
$761$	42.1917	1.52945	0.764723	+	0.644359i	$0.222876\pi$
$762$	0	0
$763$	−10.7345	−0.388616
$764$	9.30521	0.336651
$765$	0	0
$766$	10.6714	0.385572
$767$	4.86882	0.175803
$768$	0	0
$769$	3.44182	0.124115	0.0620575	−	0.998073i	$-0.480234\pi$
$769$	3.44182	0.124115	0.0620575	+	0.998073i	$0.480234\pi$
$770$	10.6975	0.385511
$771$	0	0
$772$	−17.2396	−0.620467
$773$	−20.9848	−0.754770	−0.377385	−	0.926056i	$-0.623177\pi$
$773$	−20.9848	−0.754770	−0.377385	+	0.926056i	$0.623177\pi$
$774$	0	0
$775$	−6.95844	−0.249954
$776$	−13.3757	−0.480161
$777$	0	0
$778$	23.7389	0.851079
$779$	14.6318	0.524240
$780$	0	0
$781$	−63.3414	−2.26653
$782$	8.29717	0.296706
$783$	0	0
$784$	−4.99574	−0.178419
$785$	11.0701	0.395107
$786$	0	0
$787$	−26.7801	−0.954606	−0.477303	−	0.878739i	$-0.658386\pi$
$787$	−26.7801	−0.954606	−0.477303	+	0.878739i	$0.658386\pi$
$788$	0.445991	0.0158878
$789$	0	0
$790$	15.4646	0.550205
$791$	1.11860	0.0397727
$792$	0	0
$793$	53.8377	1.91183
$794$	28.7322	1.01967
$795$	0	0
$796$	−7.97556	−0.282686
$797$	−49.6335	−1.75811	−0.879054	−	0.476722i	$-0.841825\pi$
$797$	−49.6335	−1.75811	−0.879054	+	0.476722i	$0.841825\pi$
$798$	0	0
$799$	19.7874	0.700027
$800$	5.47360	0.193521
$801$	0	0
$802$	−6.12926	−0.216432
$803$	74.7541	2.63801
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	−70.8566	−2.49582
$807$	0	0
$808$	−19.8536	−0.698448
$809$	55.4296	1.94880	0.974401	−	0.224818i	$-0.0721789\pi$
$809$	55.4296	1.94880	0.974401	+	0.224818i	$0.0721789\pi$
$810$	0	0
$811$	−51.0851	−1.79384	−0.896920	−	0.442193i	$-0.854200\pi$
$811$	−51.0851	−1.79384	−0.896920	+	0.442193i	$0.854200\pi$
$812$	1.78761	0.0627329
$813$	0	0
$814$	48.0760	1.68506
$815$	−11.2066	−0.392549
$816$	0	0
$817$	−41.6973	−1.45880
$818$	−11.8305	−0.413642
$819$	0	0
$820$	−4.23427	−0.147867
$821$	−5.11433	−0.178491	−0.0892457	−	0.996010i	$-0.528446\pi$
$821$	−5.11433	−0.178491	−0.0892457	+	0.996010i	$0.528446\pi$
$822$	0	0
$823$	−37.2614	−1.29885	−0.649425	−	0.760425i	$-0.724991\pi$
$823$	−37.2614	−1.29885	−0.649425	+	0.760425i	$0.724991\pi$
$824$	−2.28418	−0.0795732
$825$	0	0
$826$	1.46564	0.0509963
$827$	−23.9366	−0.832356	−0.416178	−	0.909283i	$-0.636631\pi$
$827$	−23.9366	−0.832356	−0.416178	+	0.909283i	$0.636631\pi$
$828$	0	0
$829$	−11.3556	−0.394397	−0.197198	−	0.980364i	$-0.563184\pi$
$829$	−11.3556	−0.394397	−0.197198	+	0.980364i	$0.563184\pi$
$830$	11.1026	0.385377
$831$	0	0
$832$	−2.37469	−0.0823274
$833$	−4.73907	−0.164199
$834$	0	0
$835$	23.6399	0.818094
$836$	23.9614	0.828722
$837$	0	0
$838$	−52.9238	−1.82822
$839$	37.9278	1.30941	0.654707	−	0.755883i	$-0.272792\pi$
$839$	37.9278	1.30941	0.654707	+	0.755883i	$0.272792\pi$
$840$	0	0
$841$	−26.1842	−0.902904
$842$	2.27818	0.0785112
$843$	0	0
$844$	6.66813	0.229527
$845$	−20.8270	−0.716472
$846$	0	0
$847$	26.3329	0.904810
$848$	−41.4944	−1.42493
$849$	0	0
$850$	8.29717	0.284590
$851$	−4.49413	−0.154057
$852$	0	0
$853$	39.0387	1.33666	0.668329	−	0.743866i	$-0.267010\pi$
$853$	39.0387	1.33666	0.668329	+	0.743866i	$0.267010\pi$
$854$	16.2066	0.554578
$855$	0	0
$856$	0.793426	0.0271187
$857$	−23.7175	−0.810173	−0.405087	−	0.914278i	$-0.632759\pi$
$857$	−23.7175	−0.810173	−0.405087	+	0.914278i	$0.632759\pi$
$858$	0	0
$859$	45.7085	1.55955	0.779777	−	0.626058i	$-0.215333\pi$
$859$	45.7085	1.55955	0.779777	+	0.626058i	$0.215333\pi$
$860$	12.0667	0.411470
$861$	0	0
$862$	−11.9888	−0.408339
$863$	5.54957	0.188909	0.0944547	−	0.995529i	$-0.469889\pi$
$863$	5.54957	0.188909	0.0944547	+	0.995529i	$0.469889\pi$
$864$	0	0
$865$	−3.08904	−0.105031
$866$	10.0471	0.341415
$867$	0	0
$868$	−7.41284	−0.251608
$869$	53.9694	1.83079
$870$	0	0
$871$	−26.2764	−0.890342
$872$	−17.5667	−0.594884
$873$	0	0
$874$	−6.44511	−0.218009
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	21.3177	0.719848	0.359924	−	0.932982i	$-0.382803\pi$
$877$	21.3177	0.719848	0.359924	+	0.932982i	$0.382803\pi$
$878$	−21.6042	−0.729106
$879$	0	0
$880$	30.5243	1.02897
$881$	−8.12987	−0.273902	−0.136951	−	0.990578i	$-0.543730\pi$
$881$	−8.12987	−0.273902	−0.136951	+	0.990578i	$0.543730\pi$
$882$	0	0
$883$	−4.24853	−0.142975	−0.0714873	−	0.997442i	$-0.522775\pi$
$883$	−4.24853	−0.142975	−0.0714873	+	0.997442i	$0.522775\pi$
$884$	29.3628	0.987579
$885$	0	0
$886$	18.1021	0.608151
$887$	51.5497	1.73087	0.865435	−	0.501021i	$-0.167042\pi$
$887$	51.5497	1.73087	0.865435	+	0.501021i	$0.167042\pi$
$888$	0	0
$889$	−11.7567	−0.394308
$890$	16.9214	0.567207
$891$	0	0
$892$	−24.7592	−0.828998
$893$	−15.3705	−0.514354
$894$	0	0
$895$	22.9526	0.767222
$896$	−11.6620	−0.389601
$897$	0	0
$898$	42.7921	1.42799
$899$	−11.6765	−0.389432
$900$	0	0
$901$	−39.3626	−1.31136
$902$	−42.5195	−1.41574
$903$	0	0
$904$	1.83055	0.0608831
$905$	4.10028	0.136298
$906$	0	0
$907$	−2.14550	−0.0712403	−0.0356202	−	0.999365i	$-0.511341\pi$
$907$	−2.14550	−0.0712403	−0.0356202	+	0.999365i	$0.511341\pi$
$908$	−21.0594	−0.698880
$909$	0	0
$910$	−10.1828	−0.337558
$911$	−34.3396	−1.13772	−0.568860	−	0.822434i	$-0.692615\pi$
$911$	−34.3396	−1.13772	−0.568860	+	0.822434i	$0.692615\pi$
$912$	0	0
$913$	38.7466	1.28233
$914$	69.9542	2.31388
$915$	0	0
$916$	−0.0110338	−0.000364567 0
$917$	0.970168	0.0320378
$918$	0	0
$919$	2.02313	0.0667370	0.0333685	−	0.999443i	$-0.489377\pi$
$919$	2.02313	0.0667370	0.0333685	+	0.999443i	$0.489377\pi$
$920$	−1.63647	−0.0539528
$921$	0	0
$922$	−26.5517	−0.874434
$923$	60.2940	1.98460
$924$	0	0
$925$	−4.49413	−0.147766
$926$	−18.2583	−0.600006
$927$	0	0
$928$	9.18487	0.301508
$929$	11.6930	0.383634	0.191817	−	0.981431i	$-0.438562\pi$
$929$	11.6930	0.383634	0.191817	+	0.981431i	$0.438562\pi$
$930$	0	0
$931$	3.68123	0.120648
$932$	6.90965	0.226333
$933$	0	0
$934$	−48.3043	−1.58057
$935$	28.9560	0.946964
$936$	0	0
$937$	13.5123	0.441427	0.220713	−	0.975339i	$-0.429161\pi$
$937$	13.5123	0.441427	0.220713	+	0.975339i	$0.429161\pi$
$938$	−7.90990	−0.258267
$939$	0	0
$940$	4.44803	0.145079
$941$	5.58384	0.182028	0.0910140	−	0.995850i	$-0.470989\pi$
$941$	5.58384	0.182028	0.0910140	+	0.995850i	$0.470989\pi$
$942$	0	0
$943$	3.97471	0.129434
$944$	4.18207	0.136115
$945$	0	0
$946$	121.171	3.93960
$947$	25.2107	0.819236	0.409618	−	0.912257i	$-0.365662\pi$
$947$	25.2107	0.819236	0.409618	+	0.912257i	$0.365662\pi$
$948$	0	0
$949$	−71.1576	−2.30987
$950$	−6.44511	−0.209107
$951$	0	0
$952$	−7.75534	−0.251352
$953$	−32.3742	−1.04870	−0.524351	−	0.851502i	$-0.675692\pi$
$953$	−32.3742	−1.04870	−0.524351	+	0.851502i	$0.675692\pi$
$954$	0	0
$955$	−8.73481	−0.282652
$956$	−16.8302	−0.544329
$957$	0	0
$958$	37.6226	1.21553
$959$	−10.7671	−0.347687
$960$	0	0
$961$	17.4198	0.561931
$962$	−45.7630	−1.47546
$963$	0	0
$964$	−6.83710	−0.220208
$965$	16.1828	0.520944
$966$	0	0
$967$	27.4153	0.881616	0.440808	−	0.897601i	$-0.354692\pi$
$967$	27.4153	0.881616	0.440808	+	0.897601i	$0.354692\pi$
$968$	43.0930	1.38506
$969$	0	0
$970$	−14.3102	−0.459474
$971$	−16.6145	−0.533185	−0.266592	−	0.963809i	$-0.585898\pi$
$971$	−16.6145	−0.533185	−0.266592	+	0.963809i	$0.585898\pi$
$972$	0	0
$973$	−10.8254	−0.347046
$974$	−20.3404	−0.651748
$975$	0	0
$976$	46.2439	1.48023
$977$	61.0578	1.95341	0.976707	−	0.214578i	$-0.0688376\pi$
$977$	61.0578	1.95341	0.976707	+	0.214578i	$0.0688376\pi$
$978$	0	0
$979$	59.0536	1.88736
$980$	−1.06530	−0.0340298
$981$	0	0
$982$	60.8233	1.94095
$983$	−7.02047	−0.223918	−0.111959	−	0.993713i	$-0.535713\pi$
$983$	−7.02047	−0.223918	−0.111959	+	0.993713i	$0.535713\pi$
$984$	0	0
$985$	−0.418652	−0.0133394
$986$	13.9229	0.443396
$987$	0	0
$988$	−22.8086	−0.725638
$989$	−11.3270	−0.360178
$990$	0	0
$991$	60.0698	1.90818	0.954090	−	0.299522i	$-0.0968270\pi$
$991$	60.0698	1.90818	0.954090	+	0.299522i	$0.0968270\pi$
$992$	−38.0877	−1.20929
$993$	0	0
$994$	18.1501	0.575686
$995$	7.48667	0.237343
$996$	0	0
$997$	14.8900	0.471572	0.235786	−	0.971805i	$-0.424234\pi$
$997$	14.8900	0.471572	0.235786	+	0.971805i	$0.424234\pi$
$998$	5.31734	0.168317
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2835.2.a.p.1.1		4
3.2	odd	2		2835.2.a.m.1.4		4
9.2	odd	6		315.2.i.c.211.1	yes	8
9.4	even	3		945.2.i.d.316.4		8
9.5	odd	6		315.2.i.c.106.1	✓	8
9.7	even	3		945.2.i.d.631.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.i.c.106.1	✓	8	9.5	odd	6
315.2.i.c.211.1	yes	8	9.2	odd	6
945.2.i.d.316.4		8	9.4	even	3
945.2.i.d.631.4		8	9.7	even	3
2835.2.a.m.1.4		4	3.2	odd	2
2835.2.a.p.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2835 = 3^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2835.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75080 q^{2} +1.06530 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.63647 q^{8} +O(q^{10})\)	\(q-1.75080 q^{2} +1.06530 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.63647 q^{8} +1.75080 q^{10} +6.11007 q^{11} -5.81610 q^{13} -1.75080 q^{14} -4.99574 q^{16} -4.73907 q^{17} +3.68123 q^{19} -1.06530 q^{20} -10.6975 q^{22} +1.00000 q^{23} +1.00000 q^{25} +10.1828 q^{26} +1.06530 q^{28} +1.67803 q^{29} -6.95844 q^{31} +5.47360 q^{32} +8.29717 q^{34} -1.00000 q^{35} -4.49413 q^{37} -6.44511 q^{38} -1.63647 q^{40} +3.97471 q^{41} -11.3270 q^{43} +6.50907 q^{44} -1.75080 q^{46} -4.17537 q^{47} +1.00000 q^{49} -1.75080 q^{50} -6.19591 q^{52} +8.30597 q^{53} -6.11007 q^{55} +1.63647 q^{56} -2.93790 q^{58} -0.837128 q^{59} -9.25667 q^{61} +12.1828 q^{62} +0.408295 q^{64} +5.81610 q^{65} +4.51787 q^{67} -5.04854 q^{68} +1.75080 q^{70} -10.3667 q^{71} +12.2346 q^{73} +7.86833 q^{74} +3.92163 q^{76} +6.11007 q^{77} +8.83286 q^{79} +4.99574 q^{80} -6.95892 q^{82} +6.34144 q^{83} +4.73907 q^{85} +19.8313 q^{86} +9.99894 q^{88} +9.66496 q^{89} -5.81610 q^{91} +1.06530 q^{92} +7.31024 q^{94} -3.68123 q^{95} -8.17354 q^{97} -1.75080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 4 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + q^4 - 4 * q^5 + 4 * q^7 + 3 * q^8	\( 4 q + q^{2} + q^{4} - 4 q^{5} + 4 q^{7} + 3 q^{8} - q^{10} + q^{11} - 12 q^{13} + q^{14} - q^{16} - 5 q^{17} - 9 q^{19} - q^{20} - 17 q^{22} + 4 q^{23} + 4 q^{25} + q^{26} + q^{28} + 15 q^{29} - 16 q^{31} + 2 q^{32} - 11 q^{34} - 4 q^{35} - 15 q^{37} - 24 q^{38} - 3 q^{40} + 2 q^{41} - 7 q^{43} - 3 q^{44} + q^{46} + 10 q^{47} + 4 q^{49} + q^{50} - 15 q^{52} - q^{55} + 3 q^{56} + 17 q^{58} + 13 q^{59} - 32 q^{61} + 9 q^{62} - 15 q^{64} + 12 q^{65} - 41 q^{68} - q^{70} - 13 q^{71} - 2 q^{73} - 13 q^{74} - 11 q^{76} + q^{77} + q^{80} + 6 q^{82} - 17 q^{83} + 5 q^{85} + 15 q^{86} - 11 q^{88} + 17 q^{89} - 12 q^{91} + q^{92} + 15 q^{94} + 9 q^{95} - 4 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 - 4 * q^5 + 4 * q^7 + 3 * q^8 - q^10 + q^11 - 12 * q^13 + q^14 - q^16 - 5 * q^17 - 9 * q^19 - q^20 - 17 * q^22 + 4 * q^23 + 4 * q^25 + q^26 + q^28 + 15 * q^29 - 16 * q^31 + 2 * q^32 - 11 * q^34 - 4 * q^35 - 15 * q^37 - 24 * q^38 - 3 * q^40 + 2 * q^41 - 7 * q^43 - 3 * q^44 + q^46 + 10 * q^47 + 4 * q^49 + q^50 - 15 * q^52 - q^55 + 3 * q^56 + 17 * q^58 + 13 * q^59 - 32 * q^61 + 9 * q^62 - 15 * q^64 + 12 * q^65 - 41 * q^68 - q^70 - 13 * q^71 - 2 * q^73 - 13 * q^74 - 11 * q^76 + q^77 + q^80 + 6 * q^82 - 17 * q^83 + 5 * q^85 + 15 * q^86 - 11 * q^88 + 17 * q^89 - 12 * q^91 + q^92 + 15 * q^94 + 9 * q^95 - 4 * q^97 + q^98

Introduction

L-functions

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