LMFDB - Embedded newform 3775.2.a.s.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3775,2,Mod(1,3775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3775 = 5^{2} \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3775.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:	$30.1435267630$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 755)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	3775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.834752 q^{2} -2.74035 q^{3} -1.30319 q^{4} +2.28751 q^{6} +3.57510 q^{7} +2.75734 q^{8} +4.50951 q^{9} +O(q^{10})$	\(q-0.834752 q^{2} -2.74035 q^{3} -1.30319 q^{4} +2.28751 q^{6} +3.57510 q^{7} +2.75734 q^{8} +4.50951 q^{9} +1.26414 q^{11} +3.57119 q^{12} -1.32452 q^{13} -2.98432 q^{14} +0.304682 q^{16} +1.38627 q^{17} -3.76433 q^{18} -3.60701 q^{19} -9.79703 q^{21} -1.05524 q^{22} -0.986357 q^{23} -7.55608 q^{24} +1.10564 q^{26} -4.13660 q^{27} -4.65903 q^{28} -3.68422 q^{29} -5.24190 q^{31} -5.76902 q^{32} -3.46419 q^{33} -1.15719 q^{34} -5.87675 q^{36} +8.41329 q^{37} +3.01096 q^{38} +3.62964 q^{39} -7.84474 q^{41} +8.17808 q^{42} +4.34313 q^{43} -1.64742 q^{44} +0.823364 q^{46} -8.23064 q^{47} -0.834935 q^{48} +5.78135 q^{49} -3.79887 q^{51} +1.72610 q^{52} +3.02719 q^{53} +3.45303 q^{54} +9.85778 q^{56} +9.88447 q^{57} +3.07541 q^{58} -4.89815 q^{59} -6.63187 q^{61} +4.37568 q^{62} +16.1220 q^{63} +4.20634 q^{64} +2.89174 q^{66} +13.9879 q^{67} -1.80657 q^{68} +2.70296 q^{69} +13.9957 q^{71} +12.4343 q^{72} -7.00313 q^{73} -7.02301 q^{74} +4.70062 q^{76} +4.51944 q^{77} -3.02985 q^{78} -6.88193 q^{79} -2.19282 q^{81} +6.54841 q^{82} +5.33248 q^{83} +12.7674 q^{84} -3.62544 q^{86} +10.0960 q^{87} +3.48567 q^{88} +3.46816 q^{89} -4.73529 q^{91} +1.28541 q^{92} +14.3646 q^{93} +6.87054 q^{94} +15.8091 q^{96} -15.1864 q^{97} -4.82599 q^{98} +5.70067 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 22 q^{4} - 16 q^{6} + 8 q^{9}+O(q^{10}) $ 22 * q + 22 * q^4 - 16 * q^6 + 8 * q^9	$ 22 q + 22 q^{4} - 16 q^{6} + 8 q^{9} - 16 q^{11} - 50 q^{14} + 22 q^{16} - 6 q^{19} - 58 q^{21} - 20 q^{24} - 8 q^{26} - 10 q^{29} - 40 q^{31} - 24 q^{34} - 22 q^{36} - 6 q^{39} - 70 q^{41} - 56 q^{44} - 8 q^{46} - 46 q^{49} + 2 q^{54} - 46 q^{56} - 78 q^{59} - 12 q^{61} - 56 q^{64} - 32 q^{66} + 26 q^{69} - 74 q^{71} + 18 q^{74} - 74 q^{76} - 42 q^{79} - 50 q^{81} + 8 q^{84} - 76 q^{86} - 118 q^{89} + 14 q^{91} + 132 q^{94} - 54 q^{96} - 40 q^{99}+O(q^{100}) $ 22 * q + 22 * q^4 - 16 * q^6 + 8 * q^9 - 16 * q^11 - 50 * q^14 + 22 * q^16 - 6 * q^19 - 58 * q^21 - 20 * q^24 - 8 * q^26 - 10 * q^29 - 40 * q^31 - 24 * q^34 - 22 * q^36 - 6 * q^39 - 70 * q^41 - 56 * q^44 - 8 * q^46 - 46 * q^49 + 2 * q^54 - 46 * q^56 - 78 * q^59 - 12 * q^61 - 56 * q^64 - 32 * q^66 + 26 * q^69 - 74 * q^71 + 18 * q^74 - 74 * q^76 - 42 * q^79 - 50 * q^81 + 8 * q^84 - 76 * q^86 - 118 * q^89 + 14 * q^91 + 132 * q^94 - 54 * q^96 - 40 * q^99

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 $

$2$	−0.834752	−0.590259	−0.295129	−	0.955457i	$-0.595363\pi$
$2$	−0.834752	−0.590259	−0.295129	+	0.955457i	$0.595363\pi$
$3$	−2.74035	−1.58214	−0.791071	−	0.611725i	$-0.790476\pi$
$3$	−2.74035	−1.58214	−0.791071	+	0.611725i	$0.790476\pi$
$4$	−1.30319	−0.651595
$5$	0	0
$5$	0	0
$6$	2.28751	0.933873
$7$	3.57510	1.35126	0.675631	−	0.737240i	$-0.263871\pi$
$7$	3.57510	1.35126	0.675631	+	0.737240i	$0.263871\pi$
$8$	2.75734	0.974868
$9$	4.50951	1.50317
$10$	0	0
$11$	1.26414	0.381153	0.190577	−	0.981672i	$-0.438964\pi$
$11$	1.26414	0.381153	0.190577	+	0.981672i	$0.438964\pi$
$12$	3.57119	1.03092
$13$	−1.32452	−0.367355	−0.183678	−	0.982987i	$-0.558800\pi$
$13$	−1.32452	−0.367355	−0.183678	+	0.982987i	$0.558800\pi$
$14$	−2.98432	−0.797594
$15$	0	0
$16$	0.304682	0.0761705
$17$	1.38627	0.336220	0.168110	−	0.985768i	$-0.446234\pi$
$17$	1.38627	0.336220	0.168110	+	0.985768i	$0.446234\pi$
$18$	−3.76433	−0.887260
$19$	−3.60701	−0.827505	−0.413752	−	0.910389i	$-0.635782\pi$
$19$	−3.60701	−0.827505	−0.413752	+	0.910389i	$0.635782\pi$
$20$	0	0
$21$	−9.79703	−2.13789
$22$	−1.05524	−0.224979
$23$	−0.986357	−0.205670	−0.102835	−	0.994698i	$-0.532791\pi$
$23$	−0.986357	−0.205670	−0.102835	+	0.994698i	$0.532791\pi$
$24$	−7.55608	−1.54238
$25$	0	0
$26$	1.10564	0.216835
$27$	−4.13660	−0.796089
$28$	−4.65903	−0.880475
$29$	−3.68422	−0.684142	−0.342071	−	0.939674i	$-0.611128\pi$
$29$	−3.68422	−0.684142	−0.342071	+	0.939674i	$0.611128\pi$
$30$	0	0
$31$	−5.24190	−0.941473	−0.470736	−	0.882274i	$-0.656012\pi$
$31$	−5.24190	−0.941473	−0.470736	+	0.882274i	$0.656012\pi$
$32$	−5.76902	−1.01983
$33$	−3.46419	−0.603038
$34$	−1.15719	−0.198457
$35$	0	0
$36$	−5.87675	−0.979459
$37$	8.41329	1.38314	0.691568	−	0.722311i	$-0.256920\pi$
$37$	8.41329	1.38314	0.691568	+	0.722311i	$0.256920\pi$
$38$	3.01096	0.488442
$39$	3.62964	0.581208
$40$	0	0
$41$	−7.84474	−1.22514	−0.612571	−	0.790416i	$-0.709865\pi$
$41$	−7.84474	−1.22514	−0.612571	+	0.790416i	$0.709865\pi$
$42$	8.17808	1.26191
$43$	4.34313	0.662321	0.331161	−	0.943574i	$-0.392560\pi$
$43$	4.34313	0.662321	0.331161	+	0.943574i	$0.392560\pi$
$44$	−1.64742	−0.248357
$45$	0	0
$46$	0.823364	0.121398
$47$	−8.23064	−1.20056	−0.600281	−	0.799789i	$-0.704945\pi$
$47$	−8.23064	−1.20056	−0.600281	+	0.799789i	$0.704945\pi$
$48$	−0.834935	−0.120512
$49$	5.78135	0.825907
$50$	0	0
$51$	−3.79887	−0.531948
$52$	1.72610	0.239367
$53$	3.02719	0.415817	0.207908	−	0.978148i	$-0.433334\pi$
$53$	3.02719	0.415817	0.207908	+	0.978148i	$0.433334\pi$
$54$	3.45303	0.469898
$55$	0	0
$56$	9.85778	1.31730
$57$	9.88447	1.30923
$58$	3.07541	0.403821
$59$	−4.89815	−0.637684	−0.318842	−	0.947808i	$-0.603294\pi$
$59$	−4.89815	−0.637684	−0.318842	+	0.947808i	$0.603294\pi$
$60$	0	0
$61$	−6.63187	−0.849123	−0.424562	−	0.905399i	$-0.639572\pi$
$61$	−6.63187	−0.849123	−0.424562	+	0.905399i	$0.639572\pi$
$62$	4.37568	0.555712
$63$	16.1220	2.03118
$64$	4.20634	0.525792
$65$	0	0
$66$	2.89174	0.355949
$67$	13.9879	1.70889	0.854446	−	0.519541i	$-0.173897\pi$
$67$	13.9879	1.70889	0.854446	+	0.519541i	$0.173897\pi$
$68$	−1.80657	−0.219079
$69$	2.70296	0.325399
$70$	0	0
$71$	13.9957	1.66099	0.830493	−	0.557030i	$-0.188059\pi$
$71$	13.9957	1.66099	0.830493	+	0.557030i	$0.188059\pi$
$72$	12.4343	1.46539
$73$	−7.00313	−0.819655	−0.409827	−	0.912163i	$-0.634411\pi$
$73$	−7.00313	−0.819655	−0.409827	+	0.912163i	$0.634411\pi$
$74$	−7.02301	−0.816408
$75$	0	0
$76$	4.70062	0.539198
$77$	4.51944	0.515038
$78$	−3.02985	−0.343063
$79$	−6.88193	−0.774278	−0.387139	−	0.922021i	$-0.626537\pi$
$79$	−6.88193	−0.774278	−0.387139	+	0.922021i	$0.626537\pi$
$80$	0	0
$81$	−2.19282	−0.243647
$82$	6.54841	0.723151
$83$	5.33248	0.585315	0.292658	−	0.956217i	$-0.405460\pi$
$83$	5.33248	0.585315	0.292658	+	0.956217i	$0.405460\pi$
$84$	12.7674	1.39304
$85$	0	0
$86$	−3.62544	−0.390941
$87$	10.0960	1.08241
$88$	3.48567	0.371574
$89$	3.46816	0.367625	0.183812	−	0.982961i	$-0.441156\pi$
$89$	3.46816	0.367625	0.183812	+	0.982961i	$0.441156\pi$
$90$	0	0
$91$	−4.73529	−0.496393
$92$	1.28541	0.134013
$93$	14.3646	1.48954
$94$	6.87054	0.708642
$95$	0	0
$96$	15.8091	1.61351
$97$	−15.1864	−1.54194	−0.770972	−	0.636870i	$-0.780229\pi$
$97$	−15.1864	−1.54194	−0.770972	+	0.636870i	$0.780229\pi$
$98$	−4.82599	−0.487499
$99$	5.70067	0.572939
$100$	0	0
$101$	8.07990	0.803980	0.401990	−	0.915644i	$-0.368319\pi$
$101$	8.07990	0.803980	0.401990	+	0.915644i	$0.368319\pi$
$102$	3.17111	0.313987
$103$	10.8270	1.06681	0.533406	−	0.845859i	$-0.320912\pi$
$103$	10.8270	1.06681	0.533406	+	0.845859i	$0.320912\pi$
$104$	−3.65215	−0.358123
$105$	0	0
$106$	−2.52695	−0.245440
$107$	−6.80114	−0.657491	−0.328745	−	0.944419i	$-0.606626\pi$
$107$	−6.80114	−0.657491	−0.328745	+	0.944419i	$0.606626\pi$
$108$	5.39077	0.518727
$109$	7.45058	0.713636	0.356818	−	0.934174i	$-0.383862\pi$
$109$	7.45058	0.713636	0.356818	+	0.934174i	$0.383862\pi$
$110$	0	0
$111$	−23.0554	−2.18832
$112$	1.08927	0.102926
$113$	9.11915	0.857857	0.428929	−	0.903338i	$-0.358891\pi$
$113$	9.11915	0.857857	0.428929	+	0.903338i	$0.358891\pi$
$114$	−8.25108	−0.772784
$115$	0	0
$116$	4.80124	0.445784
$117$	−5.97293	−0.552198
$118$	4.08874	0.376399
$119$	4.95606	0.454321
$120$	0	0
$121$	−9.40194	−0.854722
$122$	5.53596	0.501202
$123$	21.4973	1.93835
$124$	6.83119	0.613459
$125$	0	0
$126$	−13.4578	−1.19892
$127$	−17.6485	−1.56605	−0.783026	−	0.621989i	$-0.786325\pi$
$127$	−17.6485	−1.56605	−0.783026	+	0.621989i	$0.786325\pi$
$128$	8.02679	0.709475
$129$	−11.9017	−1.04789
$130$	0	0
$131$	−16.6591	−1.45551	−0.727757	−	0.685835i	$-0.759437\pi$
$131$	−16.6591	−1.45551	−0.727757	+	0.685835i	$0.759437\pi$
$132$	4.51450	0.392937
$133$	−12.8954	−1.11818
$134$	−11.6764	−1.00869
$135$	0	0
$136$	3.82243	0.327770
$137$	18.0037	1.53816	0.769078	−	0.639155i	$-0.220716\pi$
$137$	18.0037	1.53816	0.769078	+	0.639155i	$0.220716\pi$
$138$	−2.25630	−0.192069
$139$	6.50021	0.551340	0.275670	−	0.961252i	$-0.411100\pi$
$139$	6.50021	0.551340	0.275670	+	0.961252i	$0.411100\pi$
$140$	0	0
$141$	22.5548	1.89946
$142$	−11.6829	−0.980411
$143$	−1.67438	−0.140019
$144$	1.37397	0.114497
$145$	0	0
$146$	5.84588	0.483808
$147$	−15.8429	−1.30670
$148$	−10.9641	−0.901244
$149$	−3.50122	−0.286832	−0.143416	−	0.989663i	$-0.545809\pi$
$149$	−3.50122	−0.286832	−0.143416	+	0.989663i	$0.545809\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−9.94576	−0.806708
$153$	6.25141	0.505397
$154$	−3.77261	−0.304005
$155$	0	0
$156$	−4.73011	−0.378712
$157$	−8.24609	−0.658110	−0.329055	−	0.944311i	$-0.606730\pi$
$157$	−8.24609	−0.658110	−0.329055	+	0.944311i	$0.606730\pi$
$158$	5.74470	0.457024
$159$	−8.29557	−0.657881
$160$	0	0
$161$	−3.52633	−0.277914
$162$	1.83046	0.143815
$163$	13.8200	1.08246	0.541232	−	0.840873i	$-0.317958\pi$
$163$	13.8200	1.08246	0.541232	+	0.840873i	$0.317958\pi$
$164$	10.2232	0.798296
$165$	0	0
$166$	−4.45130	−0.345488
$167$	−14.8455	−1.14878	−0.574389	−	0.818583i	$-0.694760\pi$
$167$	−14.8455	−1.14878	−0.574389	+	0.818583i	$0.694760\pi$
$168$	−27.0138	−2.08416
$169$	−11.2457	−0.865050
$170$	0	0
$171$	−16.2659	−1.24388
$172$	−5.65992	−0.431565
$173$	−14.5493	−1.10616	−0.553080	−	0.833128i	$-0.686547\pi$
$173$	−14.5493	−1.10616	−0.553080	+	0.833128i	$0.686547\pi$
$174$	−8.42769	−0.638902
$175$	0	0
$176$	0.385161	0.0290326
$177$	13.4226	1.00891
$178$	−2.89506	−0.216994
$179$	−6.44243	−0.481530	−0.240765	−	0.970583i	$-0.577398\pi$
$179$	−6.44243	−0.481530	−0.240765	+	0.970583i	$0.577398\pi$
$180$	0	0
$181$	−14.9764	−1.11318	−0.556592	−	0.830786i	$-0.687891\pi$
$181$	−14.9764	−1.11318	−0.556592	+	0.830786i	$0.687891\pi$
$182$	3.95279	0.293000
$183$	18.1736	1.34343
$184$	−2.71973	−0.200501
$185$	0	0
$186$	−11.9909	−0.879216
$187$	1.75244	0.128151
$188$	10.7261	0.782280
$189$	−14.7888	−1.07572
$190$	0	0
$191$	3.66057	0.264870	0.132435	−	0.991192i	$-0.457720\pi$
$191$	3.66057	0.264870	0.132435	+	0.991192i	$0.457720\pi$
$192$	−11.5268	−0.831877
$193$	20.9420	1.50744	0.753721	−	0.657195i	$-0.228257\pi$
$193$	20.9420	1.50744	0.753721	+	0.657195i	$0.228257\pi$
$194$	12.6769	0.910145
$195$	0	0
$196$	−7.53419	−0.538157
$197$	−0.106413	−0.00758161	−0.00379080	−	0.999993i	$-0.501207\pi$
$197$	−0.106413	−0.00758161	−0.00379080	+	0.999993i	$0.501207\pi$
$198$	−4.75864	−0.338182
$199$	−15.0240	−1.06502	−0.532512	−	0.846423i	$-0.678752\pi$
$199$	−15.0240	−1.06502	−0.532512	+	0.846423i	$0.678752\pi$
$200$	0	0
$201$	−38.3317	−2.70371
$202$	−6.74471	−0.474556
$203$	−13.1715	−0.924455
$204$	4.95065	0.346615
$205$	0	0
$206$	−9.03782	−0.629695
$207$	−4.44799	−0.309157
$208$	−0.403557	−0.0279816
$209$	−4.55977	−0.315406
$210$	0	0
$211$	22.2232	1.52990	0.764952	−	0.644087i	$-0.222762\pi$
$211$	22.2232	1.52990	0.764952	+	0.644087i	$0.222762\pi$
$212$	−3.94501	−0.270944
$213$	−38.3531	−2.62791
$214$	5.67726	0.388090
$215$	0	0
$216$	−11.4060	−0.776081
$217$	−18.7403	−1.27218
$218$	−6.21938	−0.421230
$219$	19.1910	1.29681
$220$	0	0
$221$	−1.83614	−0.123512
$222$	19.2455	1.29167
$223$	16.4695	1.10288	0.551440	−	0.834214i	$-0.314078\pi$
$223$	16.4695	1.10288	0.551440	+	0.834214i	$0.314078\pi$
$224$	−20.6248	−1.37805
$225$	0	0
$226$	−7.61222	−0.506358
$227$	4.62175	0.306757	0.153378	−	0.988168i	$-0.450985\pi$
$227$	4.62175	0.306757	0.153378	+	0.988168i	$0.450985\pi$
$228$	−12.8813	−0.853087
$229$	4.65221	0.307426	0.153713	−	0.988116i	$-0.450877\pi$
$229$	4.65221	0.307426	0.153713	+	0.988116i	$0.450877\pi$
$230$	0	0
$231$	−12.3848	−0.814862
$232$	−10.1587	−0.666948
$233$	−17.7252	−1.16122	−0.580608	−	0.814183i	$-0.697185\pi$
$233$	−17.7252	−1.16122	−0.580608	+	0.814183i	$0.697185\pi$
$234$	4.98592	0.325940
$235$	0	0
$236$	6.38321	0.415512
$237$	18.8589	1.22502
$238$	−4.13708	−0.268167
$239$	−0.516206	−0.0333906	−0.0166953	−	0.999861i	$-0.505315\pi$
$239$	−0.516206	−0.0333906	−0.0166953	+	0.999861i	$0.505315\pi$
$240$	0	0
$241$	14.1515	0.911578	0.455789	−	0.890088i	$-0.349357\pi$
$241$	14.1515	0.911578	0.455789	+	0.890088i	$0.349357\pi$
$242$	7.84829	0.504507
$243$	18.4189	1.18157
$244$	8.64258	0.553284
$245$	0	0
$246$	−17.9449	−1.14413
$247$	4.77755	0.303988
$248$	−14.4537	−0.917812
$249$	−14.6129	−0.926052
$250$	0	0
$251$	−1.56027	−0.0984834	−0.0492417	−	0.998787i	$-0.515680\pi$
$251$	−1.56027	−0.0984834	−0.0492417	+	0.998787i	$0.515680\pi$
$252$	−21.0100	−1.32350
$253$	−1.24690	−0.0783917
$254$	14.7321	0.924376
$255$	0	0
$256$	−15.1131	−0.944566
$257$	−29.3509	−1.83086	−0.915429	−	0.402479i	$-0.868149\pi$
$257$	−29.3509	−1.83086	−0.915429	+	0.402479i	$0.868149\pi$
$258$	9.93496	0.618524
$259$	30.0784	1.86898
$260$	0	0
$261$	−16.6140	−1.02838
$262$	13.9062	0.859130
$263$	−25.7122	−1.58548	−0.792742	−	0.609557i	$-0.791347\pi$
$263$	−25.7122	−1.58548	−0.792742	+	0.609557i	$0.791347\pi$
$264$	−9.55196	−0.587883
$265$	0	0
$266$	10.7645	0.660013
$267$	−9.50398	−0.581634
$268$	−18.2289	−1.11350
$269$	20.7638	1.26599	0.632995	−	0.774156i	$-0.281826\pi$
$269$	20.7638	1.26599	0.632995	+	0.774156i	$0.281826\pi$
$270$	0	0
$271$	−30.6812	−1.86375	−0.931875	−	0.362780i	$-0.881828\pi$
$271$	−30.6812	−1.86375	−0.931875	+	0.362780i	$0.881828\pi$
$272$	0.422372	0.0256101
$273$	12.9763	0.785364
$274$	−15.0286	−0.907910
$275$	0	0
$276$	−3.52247	−0.212028
$277$	−12.0036	−0.721228	−0.360614	−	0.932715i	$-0.617433\pi$
$277$	−12.0036	−0.721228	−0.360614	+	0.932715i	$0.617433\pi$
$278$	−5.42606	−0.325433
$279$	−23.6384	−1.41520
$280$	0	0
$281$	−15.8347	−0.944617	−0.472309	−	0.881433i	$-0.656579\pi$
$281$	−15.8347	−0.944617	−0.472309	+	0.881433i	$0.656579\pi$
$282$	−18.8277	−1.12117
$283$	−22.5023	−1.33763	−0.668813	−	0.743431i	$-0.733197\pi$
$283$	−22.5023	−1.33763	−0.668813	+	0.743431i	$0.733197\pi$
$284$	−18.2391	−1.08229
$285$	0	0
$286$	1.39769	0.0826472
$287$	−28.0457	−1.65549
$288$	−26.0155	−1.53298
$289$	−15.0783	−0.886956
$290$	0	0
$291$	41.6160	2.43957
$292$	9.12641	0.534083
$293$	6.90587	0.403445	0.201723	−	0.979443i	$-0.435346\pi$
$293$	6.90587	0.403445	0.201723	+	0.979443i	$0.435346\pi$
$294$	13.2249	0.771292
$295$	0	0
$296$	23.1983	1.34838
$297$	−5.22925	−0.303432
$298$	2.92265	0.169305
$299$	1.30645	0.0755539
$300$	0	0
$301$	15.5271	0.894969
$302$	−0.834752	−0.0480346
$303$	−22.1418	−1.27201
$304$	−1.09899	−0.0630314
$305$	0	0
$306$	−5.21838	−0.298315
$307$	−7.43377	−0.424268	−0.212134	−	0.977241i	$-0.568041\pi$
$307$	−7.43377	−0.424268	−0.212134	+	0.977241i	$0.568041\pi$
$308$	−5.88968	−0.335596
$309$	−29.6696	−1.68785
$310$	0	0
$311$	12.4728	0.707270	0.353635	−	0.935384i	$-0.384946\pi$
$311$	12.4728	0.707270	0.353635	+	0.935384i	$0.384946\pi$
$312$	10.0082	0.566601
$313$	0.466321	0.0263580	0.0131790	−	0.999913i	$-0.495805\pi$
$313$	0.466321	0.0263580	0.0131790	+	0.999913i	$0.495805\pi$
$314$	6.88344	0.388455
$315$	0	0
$316$	8.96846	0.504515
$317$	−23.0241	−1.29316	−0.646580	−	0.762846i	$-0.723802\pi$
$317$	−23.0241	−1.29316	−0.646580	+	0.762846i	$0.723802\pi$
$318$	6.92474	0.388320
$319$	−4.65738	−0.260763
$320$	0	0
$321$	18.6375	1.04024
$322$	2.94361	0.164041
$323$	−5.00030	−0.278224
$324$	2.85766	0.158759
$325$	0	0
$326$	−11.5363	−0.638934
$327$	−20.4172	−1.12907
$328$	−21.6306	−1.19435
$329$	−29.4254	−1.62227
$330$	0	0
$331$	27.5187	1.51256	0.756282	−	0.654246i	$-0.227014\pi$
$331$	27.5187	1.51256	0.756282	+	0.654246i	$0.227014\pi$
$332$	−6.94923	−0.381388
$333$	37.9399	2.07909
$334$	12.3923	0.678076
$335$	0	0
$336$	−2.98498	−0.162844
$337$	6.80776	0.370842	0.185421	−	0.982659i	$-0.440635\pi$
$337$	6.80776	0.370842	0.185421	+	0.982659i	$0.440635\pi$
$338$	9.38733	0.510603
$339$	−24.9896	−1.35725
$340$	0	0
$341$	−6.62651	−0.358845
$342$	13.5780	0.734212
$343$	−4.35680	−0.235245
$344$	11.9755	0.645676
$345$	0	0
$346$	12.1450	0.652920
$347$	−3.51899	−0.188909	−0.0944547	−	0.995529i	$-0.530111\pi$
$347$	−3.51899	−0.188909	−0.0944547	+	0.995529i	$0.530111\pi$
$348$	−13.1571	−0.705293
$349$	5.78715	0.309779	0.154890	−	0.987932i	$-0.450498\pi$
$349$	5.78715	0.309779	0.154890	+	0.987932i	$0.450498\pi$
$350$	0	0
$351$	5.47900	0.292447
$352$	−7.29286	−0.388711
$353$	10.2709	0.546664	0.273332	−	0.961920i	$-0.411874\pi$
$353$	10.2709	0.546664	0.273332	+	0.961920i	$0.411874\pi$
$354$	−11.2046	−0.595516
$355$	0	0
$356$	−4.51968	−0.239542
$357$	−13.5813	−0.718801
$358$	5.37783	0.284227
$359$	21.0426	1.11059	0.555293	−	0.831655i	$-0.312606\pi$
$359$	21.0426	1.11059	0.555293	+	0.831655i	$0.312606\pi$
$360$	0	0
$361$	−5.98948	−0.315236
$362$	12.5015	0.657067
$363$	25.7646	1.35229
$364$	6.17098	0.323447
$365$	0	0
$366$	−15.1705	−0.792973
$367$	36.3327	1.89655	0.948275	−	0.317451i	$-0.102827\pi$
$367$	36.3327	1.89655	0.948275	+	0.317451i	$0.102827\pi$
$368$	−0.300525	−0.0156660
$369$	−35.3760	−1.84160
$370$	0	0
$371$	10.8225	0.561877
$372$	−18.7198	−0.970579
$373$	−19.9586	−1.03342	−0.516709	−	0.856161i	$-0.672843\pi$
$373$	−19.9586	−1.03342	−0.516709	+	0.856161i	$0.672843\pi$
$374$	−1.46286	−0.0756425
$375$	0	0
$376$	−22.6947	−1.17039
$377$	4.87982	0.251323
$378$	12.3449	0.634955
$379$	12.7374	0.654274	0.327137	−	0.944977i	$-0.393916\pi$
$379$	12.7374	0.654274	0.327137	+	0.944977i	$0.393916\pi$
$380$	0	0
$381$	48.3631	2.47772
$382$	−3.05567	−0.156342
$383$	−30.8698	−1.57737	−0.788687	−	0.614795i	$-0.789239\pi$
$383$	−30.8698	−1.57737	−0.788687	+	0.614795i	$0.789239\pi$
$384$	−21.9962	−1.12249
$385$	0	0
$386$	−17.4814	−0.889780
$387$	19.5854	0.995582
$388$	19.7907	1.00472
$389$	8.34635	0.423177	0.211588	−	0.977359i	$-0.432136\pi$
$389$	8.34635	0.423177	0.211588	+	0.977359i	$0.432136\pi$
$390$	0	0
$391$	−1.36736	−0.0691503
$392$	15.9412	0.805150
$393$	45.6518	2.30283
$394$	0.0888284	0.00447511
$395$	0	0
$396$	−7.42905	−0.373324
$397$	32.6803	1.64018	0.820090	−	0.572235i	$-0.193923\pi$
$397$	32.6803	1.64018	0.820090	+	0.572235i	$0.193923\pi$
$398$	12.5413	0.628639
$399$	35.3380	1.76911
$400$	0	0
$401$	−13.7155	−0.684918	−0.342459	−	0.939533i	$-0.611260\pi$
$401$	−13.7155	−0.684918	−0.342459	+	0.939533i	$0.611260\pi$
$402$	31.9974	1.59589
$403$	6.94299	0.345855
$404$	−10.5296	−0.523869
$405$	0	0
$406$	10.9949	0.545668
$407$	10.6356	0.527187
$408$	−10.4748	−0.518579
$409$	20.6661	1.02187	0.510937	−	0.859618i	$-0.329299\pi$
$409$	20.6661	1.02187	0.510937	+	0.859618i	$0.329299\pi$
$410$	0	0
$411$	−49.3363	−2.43358
$412$	−14.1096	−0.695129
$413$	−17.5114	−0.861678
$414$	3.71297	0.182483
$415$	0	0
$416$	7.64117	0.374639
$417$	−17.8128	−0.872298
$418$	3.80628	0.186171
$419$	−28.8169	−1.40780	−0.703898	−	0.710301i	$-0.748559\pi$
$419$	−28.8169	−1.40780	−0.703898	+	0.710301i	$0.748559\pi$
$420$	0	0
$421$	−8.81290	−0.429515	−0.214757	−	0.976667i	$-0.568896\pi$
$421$	−8.81290	−0.429515	−0.214757	+	0.976667i	$0.568896\pi$
$422$	−18.5508	−0.903039
$423$	−37.1162	−1.80465
$424$	8.34701	0.405367
$425$	0	0
$426$	32.0153	1.55115
$427$	−23.7096	−1.14739
$428$	8.86317	0.428418
$429$	4.58838	0.221529
$430$	0	0
$431$	14.5511	0.700904	0.350452	−	0.936581i	$-0.386028\pi$
$431$	14.5511	0.700904	0.350452	+	0.936581i	$0.386028\pi$
$432$	−1.26035	−0.0606384
$433$	−0.0241996	−0.00116296	−0.000581480	−	1.00000i	$-0.500185\pi$
$433$	−0.0241996	−0.00116296	−0.000581480		1.00000i	$0.500185\pi$
$434$	15.6435	0.750913
$435$	0	0
$436$	−9.70951	−0.465001
$437$	3.55780	0.170193
$438$	−16.0197	−0.765453
$439$	26.1675	1.24891	0.624453	−	0.781063i	$-0.285322\pi$
$439$	26.1675	1.24891	0.624453	+	0.781063i	$0.285322\pi$
$440$	0	0
$441$	26.0711	1.24148
$442$	1.53272	0.0729042
$443$	−32.8766	−1.56202	−0.781008	−	0.624521i	$-0.785294\pi$
$443$	−32.8766	−1.56202	−0.781008	+	0.624521i	$0.785294\pi$
$444$	30.0455	1.42590
$445$	0	0
$446$	−13.7480	−0.650985
$447$	9.59458	0.453808
$448$	15.0381	0.710482
$449$	−31.6661	−1.49441	−0.747207	−	0.664592i	$-0.768606\pi$
$449$	−31.6661	−1.49441	−0.747207	+	0.664592i	$0.768606\pi$
$450$	0	0
$451$	−9.91686	−0.466967
$452$	−11.8840	−0.558975
$453$	−2.74035	−0.128753
$454$	−3.85802	−0.181066
$455$	0	0
$456$	27.2549	1.27633
$457$	1.71831	0.0803791	0.0401895	−	0.999192i	$-0.487204\pi$
$457$	1.71831	0.0803791	0.0401895	+	0.999192i	$0.487204\pi$
$458$	−3.88344	−0.181461
$459$	−5.73445	−0.267661
$460$	0	0
$461$	0.811206	0.0377817	0.0188908	−	0.999822i	$-0.493987\pi$
$461$	0.811206	0.0377817	0.0188908	+	0.999822i	$0.493987\pi$
$462$	10.3383	0.480979
$463$	−27.9365	−1.29832	−0.649161	−	0.760651i	$-0.724880\pi$
$463$	−27.9365	−1.29832	−0.649161	+	0.760651i	$0.724880\pi$
$464$	−1.12251	−0.0521114
$465$	0	0
$466$	14.7961	0.685418
$467$	−17.3652	−0.803565	−0.401783	−	0.915735i	$-0.631609\pi$
$467$	−17.3652	−0.803565	−0.401783	+	0.915735i	$0.631609\pi$
$468$	7.78387	0.359809
$469$	50.0081	2.30916
$470$	0	0
$471$	22.5972	1.04122
$472$	−13.5059	−0.621658
$473$	5.49033	0.252446
$474$	−15.7425	−0.723077
$475$	0	0
$476$	−6.45869	−0.296033
$477$	13.6512	0.625044
$478$	0.430904	0.0197091
$479$	24.1514	1.10350	0.551752	−	0.834008i	$-0.313959\pi$
$479$	24.1514	1.10350	0.551752	+	0.834008i	$0.313959\pi$
$480$	0	0
$481$	−11.1436	−0.508102
$482$	−11.8130	−0.538067
$483$	9.66337	0.439698
$484$	12.2525	0.556933
$485$	0	0
$486$	−15.3752	−0.697433
$487$	40.3289	1.82748	0.913740	−	0.406301i	$-0.133181\pi$
$487$	40.3289	1.82748	0.913740	+	0.406301i	$0.133181\pi$
$488$	−18.2863	−0.827783
$489$	−37.8716	−1.71261
$490$	0	0
$491$	19.7951	0.893341	0.446671	−	0.894699i	$-0.352610\pi$
$491$	19.7951	0.893341	0.446671	+	0.894699i	$0.352610\pi$
$492$	−28.0151	−1.26302
$493$	−5.10733	−0.230022
$494$	−3.98807	−0.179432
$495$	0	0
$496$	−1.59711	−0.0717124
$497$	50.0361	2.24442
$498$	12.1981	0.546610
$499$	−12.8065	−0.573299	−0.286650	−	0.958036i	$-0.592542\pi$
$499$	−12.8065	−0.573299	−0.286650	+	0.958036i	$0.592542\pi$
$500$	0	0
$501$	40.6818	1.81753
$502$	1.30244	0.0581307
$503$	−16.2146	−0.722974	−0.361487	−	0.932377i	$-0.617731\pi$
$503$	−16.2146	−0.722974	−0.361487	+	0.932377i	$0.617731\pi$
$504$	44.4538	1.98013
$505$	0	0
$506$	1.04085	0.0462714
$507$	30.8170	1.36863
$508$	22.9993	1.02043
$509$	−12.3936	−0.549337	−0.274669	−	0.961539i	$-0.588568\pi$
$509$	−12.3936	−0.549337	−0.274669	+	0.961539i	$0.588568\pi$
$510$	0	0
$511$	−25.0369	−1.10757
$512$	−3.43794	−0.151937
$513$	14.9207	0.658767
$514$	24.5007	1.08068
$515$	0	0
$516$	15.5102	0.682797
$517$	−10.4047	−0.457598
$518$	−25.1080	−1.10318
$519$	39.8700	1.75010
$520$	0	0
$521$	−35.9616	−1.57551	−0.787753	−	0.615992i	$-0.788755\pi$
$521$	−35.9616	−1.57551	−0.787753	+	0.615992i	$0.788755\pi$
$522$	13.8686	0.607012
$523$	−16.8647	−0.737442	−0.368721	−	0.929540i	$-0.620204\pi$
$523$	−16.8647	−0.737442	−0.368721	+	0.929540i	$0.620204\pi$
$524$	21.7100	0.948406
$525$	0	0
$526$	21.4633	0.935846
$527$	−7.26670	−0.316542
$528$	−1.05548	−0.0459337
$529$	−22.0271	−0.957700
$530$	0	0
$531$	−22.0883	−0.958549
$532$	16.8052	0.728597
$533$	10.3905	0.450062
$534$	7.93347	0.343315
$535$	0	0
$536$	38.5694	1.66594
$537$	17.6545	0.761848
$538$	−17.3326	−0.747261
$539$	7.30845	0.314797
$540$	0	0
$541$	17.9708	0.772625	0.386313	−	0.922368i	$-0.373749\pi$
$541$	17.9708	0.772625	0.386313	+	0.922368i	$0.373749\pi$
$542$	25.6112	1.10009
$543$	41.0405	1.76122
$544$	−7.99743	−0.342887
$545$	0	0
$546$	−10.8320	−0.463568
$547$	40.3004	1.72312	0.861561	−	0.507654i	$-0.169487\pi$
$547$	40.3004	1.72312	0.861561	+	0.507654i	$0.169487\pi$
$548$	−23.4622	−1.00225
$549$	−29.9065	−1.27638
$550$	0	0
$551$	13.2890	0.566131
$552$	7.45300	0.317221
$553$	−24.6036	−1.04625
$554$	10.0201	0.425711
$555$	0	0
$556$	−8.47100	−0.359251
$557$	−27.7934	−1.17764	−0.588821	−	0.808263i	$-0.700408\pi$
$557$	−27.7934	−1.17764	−0.588821	+	0.808263i	$0.700408\pi$
$558$	19.7322	0.835331
$559$	−5.75256	−0.243307
$560$	0	0
$561$	−4.80231	−0.202754
$562$	13.2180	0.557569
$563$	6.20396	0.261466	0.130733	−	0.991418i	$-0.458267\pi$
$563$	6.20396	0.261466	0.130733	+	0.991418i	$0.458267\pi$
$564$	−29.3932	−1.23768
$565$	0	0
$566$	18.7839	0.789545
$567$	−7.83956	−0.329230
$568$	38.5910	1.61924
$569$	−30.1629	−1.26449	−0.632247	−	0.774767i	$-0.717867\pi$
$569$	−30.1629	−1.26449	−0.632247	+	0.774767i	$0.717867\pi$
$570$	0	0
$571$	−30.2435	−1.26565	−0.632825	−	0.774295i	$-0.718105\pi$
$571$	−30.2435	−1.26565	−0.632825	+	0.774295i	$0.718105\pi$
$572$	2.18203	0.0912354
$573$	−10.0312	−0.419061
$574$	23.4112	0.977165
$575$	0	0
$576$	18.9685	0.790356
$577$	−2.42981	−0.101154	−0.0505772	−	0.998720i	$-0.516106\pi$
$577$	−2.42981	−0.101154	−0.0505772	+	0.998720i	$0.516106\pi$
$578$	12.5866	0.523533
$579$	−57.3885	−2.38499
$580$	0	0
$581$	19.0641	0.790914
$582$	−34.7390	−1.43998
$583$	3.82680	0.158490
$584$	−19.3100	−0.799055
$585$	0	0
$586$	−5.76468	−0.238137
$587$	30.6017	1.26307	0.631533	−	0.775349i	$-0.282426\pi$
$587$	30.6017	1.26307	0.631533	+	0.775349i	$0.282426\pi$
$588$	20.6463	0.851440
$589$	18.9076	0.779073
$590$	0	0
$591$	0.291609	0.0119952
$592$	2.56338	0.105354
$593$	−16.9989	−0.698063	−0.349032	−	0.937111i	$-0.613489\pi$
$593$	−16.9989	−0.698063	−0.349032	+	0.937111i	$0.613489\pi$
$594$	4.36512	0.179103
$595$	0	0
$596$	4.56276	0.186898
$597$	41.1710	1.68502
$598$	−1.09056	−0.0445963
$599$	25.7140	1.05065	0.525323	−	0.850903i	$-0.323945\pi$
$599$	25.7140	1.05065	0.525323	+	0.850903i	$0.323945\pi$
$600$	0	0
$601$	−12.2166	−0.498325	−0.249163	−	0.968462i	$-0.580155\pi$
$601$	−12.2166	−0.498325	−0.249163	+	0.968462i	$0.580155\pi$
$602$	−12.9613	−0.528263
$603$	63.0785	2.56876
$604$	−1.30319	−0.0530260
$605$	0	0
$606$	18.4829	0.750815
$607$	36.2152	1.46993	0.734965	−	0.678105i	$-0.237198\pi$
$607$	36.2152	1.46993	0.734965	+	0.678105i	$0.237198\pi$
$608$	20.8089	0.843913
$609$	36.0944	1.46262
$610$	0	0
$611$	10.9016	0.441033
$612$	−8.14677	−0.329314
$613$	5.17043	0.208832	0.104416	−	0.994534i	$-0.466703\pi$
$613$	5.17043	0.208832	0.104416	+	0.994534i	$0.466703\pi$
$614$	6.20536	0.250428
$615$	0	0
$616$	12.4616	0.502094
$617$	−23.2959	−0.937858	−0.468929	−	0.883236i	$-0.655360\pi$
$617$	−23.2959	−0.937858	−0.468929	+	0.883236i	$0.655360\pi$
$618$	24.7668	0.996266
$619$	−8.91303	−0.358245	−0.179122	−	0.983827i	$-0.557326\pi$
$619$	−8.91303	−0.358245	−0.179122	+	0.983827i	$0.557326\pi$
$620$	0	0
$621$	4.08016	0.163731
$622$	−10.4117	−0.417472
$623$	12.3990	0.496757
$624$	1.10589	0.0442709
$625$	0	0
$626$	−0.389263	−0.0155581
$627$	12.4954	0.499017
$628$	10.7462	0.428821
$629$	11.6631	0.465038
$630$	0	0
$631$	−32.4167	−1.29049	−0.645244	−	0.763977i	$-0.723244\pi$
$631$	−32.4167	−1.29049	−0.645244	+	0.763977i	$0.723244\pi$
$632$	−18.9758	−0.754819
$633$	−60.8992	−2.42053
$634$	19.2194	0.763299
$635$	0	0
$636$	10.8107	0.428672
$637$	−7.65750	−0.303401
$638$	3.88775	0.153918
$639$	63.1139	2.49675
$640$	0	0
$641$	−25.9056	−1.02321	−0.511605	−	0.859221i	$-0.670949\pi$
$641$	−25.9056	−1.02321	−0.511605	+	0.859221i	$0.670949\pi$
$642$	−15.5577	−0.614013
$643$	0.238497	0.00940541	0.00470271	−	0.999989i	$-0.498503\pi$
$643$	0.238497	0.00940541	0.00470271	+	0.999989i	$0.498503\pi$
$644$	4.59547	0.181087
$645$	0	0
$646$	4.17401	0.164224
$647$	−17.4366	−0.685505	−0.342753	−	0.939426i	$-0.611359\pi$
$647$	−17.4366	−0.685505	−0.342753	+	0.939426i	$0.611359\pi$
$648$	−6.04636	−0.237523
$649$	−6.19195	−0.243055
$650$	0	0
$651$	51.3550	2.01276
$652$	−18.0101	−0.705328
$653$	−10.0573	−0.393574	−0.196787	−	0.980446i	$-0.563051\pi$
$653$	−10.0573	−0.393574	−0.196787	+	0.980446i	$0.563051\pi$
$654$	17.0433	0.666445
$655$	0	0
$656$	−2.39015	−0.0933196
$657$	−31.5807	−1.23208
$658$	24.5629	0.957560
$659$	−13.1959	−0.514039	−0.257019	−	0.966406i	$-0.582740\pi$
$659$	−13.1959	−0.514039	−0.257019	+	0.966406i	$0.582740\pi$
$660$	0	0
$661$	−41.4831	−1.61351	−0.806753	−	0.590889i	$-0.798777\pi$
$661$	−41.4831	−1.61351	−0.806753	+	0.590889i	$0.798777\pi$
$662$	−22.9713	−0.892804
$663$	5.03167	0.195414
$664$	14.7035	0.570605
$665$	0	0
$666$	−31.6704	−1.22720
$667$	3.63396	0.140707
$668$	19.3465	0.748538
$669$	−45.1322	−1.74491
$670$	0	0
$671$	−8.38362	−0.323646
$672$	56.5192	2.18028
$673$	−33.5197	−1.29209	−0.646045	−	0.763300i	$-0.723578\pi$
$673$	−33.5197	−1.29209	−0.646045	+	0.763300i	$0.723578\pi$
$674$	−5.68279	−0.218893
$675$	0	0
$676$	14.6552	0.563662
$677$	41.6564	1.60098	0.800492	−	0.599343i	$-0.204571\pi$
$677$	41.6564	1.60098	0.800492	+	0.599343i	$0.204571\pi$
$678$	20.8602	0.801129
$679$	−54.2928	−2.08357
$680$	0	0
$681$	−12.6652	−0.485332
$682$	5.53149	0.211812
$683$	25.4285	0.972994	0.486497	−	0.873682i	$-0.338274\pi$
$683$	25.4285	0.972994	0.486497	+	0.873682i	$0.338274\pi$
$684$	21.1975	0.810507
$685$	0	0
$686$	3.63685	0.138856
$687$	−12.7487	−0.486392
$688$	1.32327	0.0504493
$689$	−4.00957	−0.152753
$690$	0	0
$691$	30.9463	1.17725	0.588627	−	0.808405i	$-0.299669\pi$
$691$	30.9463	1.17725	0.588627	+	0.808405i	$0.299669\pi$
$692$	18.9604	0.720768
$693$	20.3805	0.774190
$694$	2.93748	0.111505
$695$	0	0
$696$	27.8383	1.05521
$697$	−10.8749	−0.411918
$698$	−4.83083	−0.182850
$699$	48.5733	1.83721
$700$	0	0
$701$	−11.3162	−0.427406	−0.213703	−	0.976899i	$-0.568552\pi$
$701$	−11.3162	−0.427406	−0.213703	+	0.976899i	$0.568552\pi$
$702$	−4.57360	−0.172620
$703$	−30.3468	−1.14455
$704$	5.31741	0.200407
$705$	0	0
$706$	−8.57363	−0.322673
$707$	28.8865	1.08639
$708$	−17.4922	−0.657398
$709$	−47.4593	−1.78237	−0.891186	−	0.453637i	$-0.850126\pi$
$709$	−47.4593	−1.78237	−0.891186	+	0.453637i	$0.850126\pi$
$710$	0	0
$711$	−31.0342	−1.16387
$712$	9.56292	0.358386
$713$	5.17039	0.193632
$714$	11.3370	0.424278
$715$	0	0
$716$	8.39571	0.313762
$717$	1.41459	0.0528287
$718$	−17.5654	−0.655533
$719$	−41.5330	−1.54892	−0.774459	−	0.632624i	$-0.781978\pi$
$719$	−41.5330	−1.54892	−0.774459	+	0.632624i	$0.781978\pi$
$720$	0	0
$721$	38.7075	1.44154
$722$	4.99973	0.186071
$723$	−38.7800	−1.44225
$724$	19.5170	0.725345
$725$	0	0
$726$	−21.5071	−0.798202
$727$	−9.89900	−0.367134	−0.183567	−	0.983007i	$-0.558764\pi$
$727$	−9.89900	−0.367134	−0.183567	+	0.983007i	$0.558764\pi$
$728$	−13.0568	−0.483918
$729$	−43.8957	−1.62577
$730$	0	0
$731$	6.02076	0.222686
$732$	−23.6837	−0.875374
$733$	−3.39915	−0.125550	−0.0627752	−	0.998028i	$-0.519995\pi$
$733$	−3.39915	−0.125550	−0.0627752	+	0.998028i	$0.519995\pi$
$734$	−30.3287	−1.11945
$735$	0	0
$736$	5.69032	0.209748
$737$	17.6827	0.651349
$738$	29.5301	1.08702
$739$	6.06663	0.223164	0.111582	−	0.993755i	$-0.464408\pi$
$739$	6.06663	0.223164	0.111582	+	0.993755i	$0.464408\pi$
$740$	0	0
$741$	−13.0922	−0.480952
$742$	−9.03412	−0.331653
$743$	−13.9343	−0.511201	−0.255600	−	0.966783i	$-0.582273\pi$
$743$	−13.9343	−0.511201	−0.255600	+	0.966783i	$0.582273\pi$
$744$	39.6082	1.45211
$745$	0	0
$746$	16.6605	0.609984
$747$	24.0469	0.879830
$748$	−2.28377	−0.0835028
$749$	−24.3148	−0.888442
$750$	0	0
$751$	−20.1723	−0.736098	−0.368049	−	0.929806i	$-0.619974\pi$
$751$	−20.1723	−0.736098	−0.368049	+	0.929806i	$0.619974\pi$
$752$	−2.50773	−0.0914473
$753$	4.27569	0.155815
$754$	−4.07343	−0.148346
$755$	0	0
$756$	19.2726	0.700936
$757$	−15.0487	−0.546955	−0.273478	−	0.961878i	$-0.588174\pi$
$757$	−15.0487	−0.546955	−0.273478	+	0.961878i	$0.588174\pi$
$758$	−10.6325	−0.386191
$759$	3.41693	0.124027
$760$	0	0
$761$	−52.1497	−1.89042	−0.945212	−	0.326456i	$-0.894146\pi$
$761$	−52.1497	−1.89042	−0.945212	+	0.326456i	$0.894146\pi$
$762$	−40.3712	−1.46249
$763$	26.6366	0.964309
$764$	−4.77042	−0.172588
$765$	0	0
$766$	25.7686	0.931059
$767$	6.48768	0.234257
$768$	41.4150	1.49444
$769$	8.31650	0.299901	0.149950	−	0.988694i	$-0.452089\pi$
$769$	8.31650	0.299901	0.149950	+	0.988694i	$0.452089\pi$
$770$	0	0
$771$	80.4317	2.89668
$772$	−27.2915	−0.982241
$773$	16.4904	0.593118	0.296559	−	0.955015i	$-0.404161\pi$
$773$	16.4904	0.593118	0.296559	+	0.955015i	$0.404161\pi$
$774$	−16.3490	−0.587651
$775$	0	0
$776$	−41.8741	−1.50319
$777$	−82.4252	−2.95699
$778$	−6.96713	−0.249784
$779$	28.2960	1.01381
$780$	0	0
$781$	17.6926	0.633090
$782$	1.14141	0.0408166
$783$	15.2401	0.544638
$784$	1.76147	0.0629097
$785$	0	0
$786$	−38.1079	−1.35927
$787$	−15.3217	−0.546158	−0.273079	−	0.961992i	$-0.588042\pi$
$787$	−15.3217	−0.546158	−0.273079	+	0.961992i	$0.588042\pi$
$788$	0.138676	0.00494014
$789$	70.4605	2.50846
$790$	0	0
$791$	32.6019	1.15919
$792$	15.7187	0.558540
$793$	8.78403	0.311930
$794$	−27.2800	−0.968130
$795$	0	0
$796$	19.5791	0.693964
$797$	−31.0655	−1.10040	−0.550198	−	0.835034i	$-0.685448\pi$
$797$	−31.0655	−1.10040	−0.550198	+	0.835034i	$0.685448\pi$
$798$	−29.4984	−1.04423
$799$	−11.4099	−0.403653
$800$	0	0
$801$	15.6397	0.552603
$802$	11.4490	0.404278
$803$	−8.85296	−0.312414
$804$	49.9534	1.76172
$805$	0	0
$806$	−5.79567	−0.204144
$807$	−56.9000	−2.00297
$808$	22.2791	0.783775
$809$	7.56561	0.265993	0.132996	−	0.991117i	$-0.457540\pi$
$809$	7.56561	0.265993	0.132996	+	0.991117i	$0.457540\pi$
$810$	0	0
$811$	33.6387	1.18121	0.590607	−	0.806959i	$-0.298888\pi$
$811$	33.6387	1.18121	0.590607	+	0.806959i	$0.298888\pi$
$812$	17.1649	0.602370
$813$	84.0772	2.94872
$814$	−8.87808	−0.311177
$815$	0	0
$816$	−1.15745	−0.0405187
$817$	−15.6657	−0.548074
$818$	−17.2511	−0.603170
$819$	−21.3538	−0.746164
$820$	0	0
$821$	−40.5961	−1.41681	−0.708406	−	0.705805i	$-0.750585\pi$
$821$	−40.5961	−1.41681	−0.708406	+	0.705805i	$0.750585\pi$
$822$	41.1836	1.43644
$823$	−50.9129	−1.77471	−0.887356	−	0.461085i	$-0.847460\pi$
$823$	−50.9129	−1.77471	−0.887356	+	0.461085i	$0.847460\pi$
$824$	29.8536	1.04000
$825$	0	0
$826$	14.6176	0.508613
$827$	−14.1454	−0.491885	−0.245942	−	0.969284i	$-0.579097\pi$
$827$	−14.1454	−0.491885	−0.245942	+	0.969284i	$0.579097\pi$
$828$	5.79658	0.201445
$829$	11.1554	0.387444	0.193722	−	0.981056i	$-0.437944\pi$
$829$	11.1554	0.387444	0.193722	+	0.981056i	$0.437944\pi$
$830$	0	0
$831$	32.8941	1.14109
$832$	−5.57137	−0.193152
$833$	8.01452	0.277687
$834$	14.8693	0.514882
$835$	0	0
$836$	5.94225	0.205517
$837$	21.6836	0.749496
$838$	24.0549	0.830964
$839$	−39.4960	−1.36355	−0.681777	−	0.731560i	$-0.738792\pi$
$839$	−39.4960	−1.36355	−0.681777	+	0.731560i	$0.738792\pi$
$840$	0	0
$841$	−15.4265	−0.531949
$842$	7.35659	0.253525
$843$	43.3925	1.49452
$844$	−28.9610	−0.996878
$845$	0	0
$846$	30.9828	1.06521
$847$	−33.6129	−1.15495
$848$	0.922331	0.0316730
$849$	61.6643	2.11631
$850$	0	0
$851$	−8.29851	−0.284469
$852$	49.9814	1.71233
$853$	15.0348	0.514781	0.257390	−	0.966307i	$-0.417137\pi$
$853$	15.0348	0.514781	0.257390	+	0.966307i	$0.417137\pi$
$854$	19.7916	0.677255
$855$	0	0
$856$	−18.7531	−0.640967
$857$	−14.6551	−0.500610	−0.250305	−	0.968167i	$-0.580531\pi$
$857$	−14.6551	−0.500610	−0.250305	+	0.968167i	$0.580531\pi$
$858$	−3.83016	−0.130760
$859$	−51.8840	−1.77026	−0.885130	−	0.465344i	$-0.845931\pi$
$859$	−51.8840	−1.77026	−0.885130	+	0.465344i	$0.845931\pi$
$860$	0	0
$861$	76.8551	2.61921
$862$	−12.1466	−0.413715
$863$	56.2800	1.91579	0.957897	−	0.287112i	$-0.0926952\pi$
$863$	56.2800	1.91579	0.957897	+	0.287112i	$0.0926952\pi$
$864$	23.8641	0.811874
$865$	0	0
$866$	0.0202007	0.000686447 0
$867$	41.3197	1.40329
$868$	24.4222	0.828943
$869$	−8.69974	−0.295118
$870$	0	0
$871$	−18.5272	−0.627770
$872$	20.5438	0.695701
$873$	−68.4832	−2.31781
$874$	−2.96988	−0.100458
$875$	0	0
$876$	−25.0096	−0.844994
$877$	−21.3168	−0.719818	−0.359909	−	0.932987i	$-0.617192\pi$
$877$	−21.3168	−0.719818	−0.359909	+	0.932987i	$0.617192\pi$
$878$	−21.8433	−0.737177
$879$	−18.9245	−0.638307
$880$	0	0
$881$	40.9599	1.37997	0.689987	−	0.723822i	$-0.257616\pi$
$881$	40.9599	1.37997	0.689987	+	0.723822i	$0.257616\pi$
$882$	−21.7629	−0.732794
$883$	22.1909	0.746784	0.373392	−	0.927674i	$-0.378195\pi$
$883$	22.1909	0.746784	0.373392	+	0.927674i	$0.378195\pi$
$884$	2.39284	0.0804799
$885$	0	0
$886$	27.4438	0.921993
$887$	42.6018	1.43043	0.715214	−	0.698906i	$-0.246329\pi$
$887$	42.6018	1.43043	0.715214	+	0.698906i	$0.246329\pi$
$888$	−63.5715	−2.13332
$889$	−63.0952	−2.11615
$890$	0	0
$891$	−2.77204	−0.0928668
$892$	−21.4629	−0.718631
$893$	29.6880	0.993471
$894$	−8.00909	−0.267864
$895$	0	0
$896$	28.6966	0.958686
$897$	−3.58012	−0.119537
$898$	26.4333	0.882091
$899$	19.3123	0.644101
$900$	0	0
$901$	4.19651	0.139806
$902$	8.27812	0.275631
$903$	−42.5498	−1.41597
$904$	25.1446	0.836297
$905$	0	0
$906$	2.28751	0.0759975
$907$	7.80742	0.259241	0.129621	−	0.991564i	$-0.458624\pi$
$907$	7.80742	0.259241	0.129621	+	0.991564i	$0.458624\pi$
$908$	−6.02302	−0.199881
$909$	36.4364	1.20852
$910$	0	0
$911$	−10.4869	−0.347446	−0.173723	−	0.984795i	$-0.555580\pi$
$911$	−10.4869	−0.347446	−0.173723	+	0.984795i	$0.555580\pi$
$912$	3.01162	0.0997246
$913$	6.74101	0.223095
$914$	−1.43436	−0.0474444
$915$	0	0
$916$	−6.06271	−0.200317
$917$	−59.5581	−1.96678
$918$	4.78684	0.157989
$919$	21.0406	0.694066	0.347033	−	0.937853i	$-0.387189\pi$
$919$	21.0406	0.694066	0.347033	+	0.937853i	$0.387189\pi$
$920$	0	0
$921$	20.3711	0.671252
$922$	−0.677156	−0.0223009
$923$	−18.5376	−0.610172
$924$	16.1398	0.530960
$925$	0	0
$926$	23.3201	0.766345
$927$	48.8243	1.60360
$928$	21.2543	0.697708
$929$	−1.12032	−0.0367564	−0.0183782	−	0.999831i	$-0.505850\pi$
$929$	−1.12032	−0.0367564	−0.0183782	+	0.999831i	$0.505850\pi$
$930$	0	0
$931$	−20.8534	−0.683442
$932$	23.0993	0.756643
$933$	−34.1799	−1.11900
$934$	14.4956	0.474311
$935$	0	0
$936$	−16.4694	−0.538320
$937$	41.1765	1.34518	0.672589	−	0.740016i	$-0.265182\pi$
$937$	41.1765	1.34518	0.672589	+	0.740016i	$0.265182\pi$
$938$	−41.7443	−1.36300
$939$	−1.27788	−0.0417021
$940$	0	0
$941$	−49.1613	−1.60261	−0.801306	−	0.598254i	$-0.795861\pi$
$941$	−49.1613	−1.60261	−0.801306	+	0.598254i	$0.795861\pi$
$942$	−18.8630	−0.614590
$943$	7.73771	0.251975
$944$	−1.49238	−0.0485727
$945$	0	0
$946$	−4.58307	−0.149008
$947$	−14.2599	−0.463385	−0.231692	−	0.972789i	$-0.574426\pi$
$947$	−14.2599	−0.463385	−0.231692	+	0.972789i	$0.574426\pi$
$948$	−24.5767	−0.798214
$949$	9.27578	0.301104
$950$	0	0
$951$	63.0940	2.04596
$952$	13.6656	0.442903
$953$	−17.8905	−0.579531	−0.289766	−	0.957098i	$-0.593577\pi$
$953$	−17.8905	−0.579531	−0.289766	+	0.957098i	$0.593577\pi$
$954$	−11.3953	−0.368938
$955$	0	0
$956$	0.672714	0.0217571
$957$	12.7628	0.412564
$958$	−20.1604	−0.651353
$959$	64.3649	2.07845
$960$	0	0
$961$	−3.52249	−0.113629
$962$	9.30210	0.299912
$963$	−30.6698	−0.988321
$964$	−18.4421	−0.593979
$965$	0	0
$966$	−8.06651	−0.259536
$967$	24.9973	0.803858	0.401929	−	0.915671i	$-0.368340\pi$
$967$	24.9973	0.803858	0.401929	+	0.915671i	$0.368340\pi$
$968$	−25.9244	−0.833241
$969$	13.7026	0.440190
$970$	0	0
$971$	−49.6417	−1.59308	−0.796538	−	0.604588i	$-0.793338\pi$
$971$	−49.6417	−1.59308	−0.796538	+	0.604588i	$0.793338\pi$
$972$	−24.0033	−0.769906
$973$	23.2389	0.745005
$974$	−33.6647	−1.07869
$975$	0	0
$976$	−2.02061	−0.0646781
$977$	−21.1768	−0.677505	−0.338752	−	0.940876i	$-0.610005\pi$
$977$	−21.1768	−0.677505	−0.338752	+	0.940876i	$0.610005\pi$
$978$	31.6134	1.01088
$979$	4.38425	0.140121
$980$	0	0
$981$	33.5985	1.07272
$982$	−16.5240	−0.527302
$983$	5.41296	0.172647	0.0863234	−	0.996267i	$-0.472488\pi$
$983$	5.41296	0.172647	0.0863234	+	0.996267i	$0.472488\pi$
$984$	59.2755	1.88963
$985$	0	0
$986$	4.26335	0.135773
$987$	80.6358	2.56666
$988$	−6.22605	−0.198077
$989$	−4.28388	−0.136219
$990$	0	0
$991$	−15.1226	−0.480384	−0.240192	−	0.970725i	$-0.577210\pi$
$991$	−15.1226	−0.480384	−0.240192	+	0.970725i	$0.577210\pi$
$992$	30.2406	0.960141
$993$	−75.4108	−2.39309
$994$	−41.7677	−1.32479
$995$	0	0
$996$	19.0433	0.603411
$997$	30.9845	0.981290	0.490645	−	0.871360i	$-0.336761\pi$
$997$	30.9845	0.981290	0.490645	+	0.871360i	$0.336761\pi$
$998$	10.6903	0.338395
$999$	−34.8024	−1.10110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3775.2.a.s.1.8		22
5.2	odd	4		755.2.b.c.454.8	✓	22
5.3	odd	4		755.2.b.c.454.15	yes	22
5.4	even	2	inner	3775.2.a.s.1.15		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.b.c.454.8	✓	22	5.2	odd	4
755.2.b.c.454.15	yes	22	5.3	odd	4
3775.2.a.s.1.8		22	1.1	even	1	trivial
3775.2.a.s.1.15		22	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 3775 = 5^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3775.a (trivial)

\(f(q)\)	\(=\)	\(q-0.834752 q^{2} -2.74035 q^{3} -1.30319 q^{4} +2.28751 q^{6} +3.57510 q^{7} +2.75734 q^{8} +4.50951 q^{9} +O(q^{10})\)	\(q-0.834752 q^{2} -2.74035 q^{3} -1.30319 q^{4} +2.28751 q^{6} +3.57510 q^{7} +2.75734 q^{8} +4.50951 q^{9} +1.26414 q^{11} +3.57119 q^{12} -1.32452 q^{13} -2.98432 q^{14} +0.304682 q^{16} +1.38627 q^{17} -3.76433 q^{18} -3.60701 q^{19} -9.79703 q^{21} -1.05524 q^{22} -0.986357 q^{23} -7.55608 q^{24} +1.10564 q^{26} -4.13660 q^{27} -4.65903 q^{28} -3.68422 q^{29} -5.24190 q^{31} -5.76902 q^{32} -3.46419 q^{33} -1.15719 q^{34} -5.87675 q^{36} +8.41329 q^{37} +3.01096 q^{38} +3.62964 q^{39} -7.84474 q^{41} +8.17808 q^{42} +4.34313 q^{43} -1.64742 q^{44} +0.823364 q^{46} -8.23064 q^{47} -0.834935 q^{48} +5.78135 q^{49} -3.79887 q^{51} +1.72610 q^{52} +3.02719 q^{53} +3.45303 q^{54} +9.85778 q^{56} +9.88447 q^{57} +3.07541 q^{58} -4.89815 q^{59} -6.63187 q^{61} +4.37568 q^{62} +16.1220 q^{63} +4.20634 q^{64} +2.89174 q^{66} +13.9879 q^{67} -1.80657 q^{68} +2.70296 q^{69} +13.9957 q^{71} +12.4343 q^{72} -7.00313 q^{73} -7.02301 q^{74} +4.70062 q^{76} +4.51944 q^{77} -3.02985 q^{78} -6.88193 q^{79} -2.19282 q^{81} +6.54841 q^{82} +5.33248 q^{83} +12.7674 q^{84} -3.62544 q^{86} +10.0960 q^{87} +3.48567 q^{88} +3.46816 q^{89} -4.73529 q^{91} +1.28541 q^{92} +14.3646 q^{93} +6.87054 q^{94} +15.8091 q^{96} -15.1864 q^{97} -4.82599 q^{98} +5.70067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 22 q^{4} - 16 q^{6} + 8 q^{9}+O(q^{10}) \) 22 * q + 22 * q^4 - 16 * q^6 + 8 * q^9	\( 22 q + 22 q^{4} - 16 q^{6} + 8 q^{9} - 16 q^{11} - 50 q^{14} + 22 q^{16} - 6 q^{19} - 58 q^{21} - 20 q^{24} - 8 q^{26} - 10 q^{29} - 40 q^{31} - 24 q^{34} - 22 q^{36} - 6 q^{39} - 70 q^{41} - 56 q^{44} - 8 q^{46} - 46 q^{49} + 2 q^{54} - 46 q^{56} - 78 q^{59} - 12 q^{61} - 56 q^{64} - 32 q^{66} + 26 q^{69} - 74 q^{71} + 18 q^{74} - 74 q^{76} - 42 q^{79} - 50 q^{81} + 8 q^{84} - 76 q^{86} - 118 q^{89} + 14 q^{91} + 132 q^{94} - 54 q^{96} - 40 q^{99}+O(q^{100}) \) 22 * q + 22 * q^4 - 16 * q^6 + 8 * q^9 - 16 * q^11 - 50 * q^14 + 22 * q^16 - 6 * q^19 - 58 * q^21 - 20 * q^24 - 8 * q^26 - 10 * q^29 - 40 * q^31 - 24 * q^34 - 22 * q^36 - 6 * q^39 - 70 * q^41 - 56 * q^44 - 8 * q^46 - 46 * q^49 + 2 * q^54 - 46 * q^56 - 78 * q^59 - 12 * q^61 - 56 * q^64 - 32 * q^66 + 26 * q^69 - 74 * q^71 + 18 * q^74 - 74 * q^76 - 42 * q^79 - 50 * q^81 + 8 * q^84 - 76 * q^86 - 118 * q^89 + 14 * q^91 + 132 * q^94 - 54 * q^96 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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