LMFDB - Embedded newform 1850.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1791440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} + 13x - 4 $ x^5 - 9x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.332924$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +0.332924 q^{3} +1.00000 q^{4} -0.332924 q^{6} -3.51336 q^{7} -1.00000 q^{8} -2.88916 q^{9} -0.290044 q^{11} +0.332924 q^{12} +7.12558 q^{13} +3.51336 q^{14} +1.00000 q^{16} -6.17921 q^{17} +2.88916 q^{18} +5.83553 q^{19} -1.16968 q^{21} +0.290044 q^{22} -6.45973 q^{23} -0.332924 q^{24} -7.12558 q^{26} -1.96065 q^{27} -3.51336 q^{28} -1.18043 q^{29} +9.77838 q^{31} -1.00000 q^{32} -0.0965628 q^{33} +6.17921 q^{34} -2.88916 q^{36} -1.00000 q^{37} -5.83553 q^{38} +2.37228 q^{39} +1.64077 q^{41} +1.16968 q^{42} -5.34889 q^{43} -0.290044 q^{44} +6.45973 q^{46} +5.69256 q^{47} +0.332924 q^{48} +5.34367 q^{49} -2.05721 q^{51} +7.12558 q^{52} +9.32034 q^{53} +1.96065 q^{54} +3.51336 q^{56} +1.94279 q^{57} +1.18043 q^{58} +5.94279 q^{59} -1.27699 q^{61} -9.77838 q^{62} +10.1507 q^{63} +1.00000 q^{64} +0.0965628 q^{66} +6.04334 q^{67} -6.17921 q^{68} -2.15060 q^{69} +13.4995 q^{71} +2.88916 q^{72} +1.71348 q^{73} +1.00000 q^{74} +5.83553 q^{76} +1.01903 q^{77} -2.37228 q^{78} +8.25422 q^{79} +8.01474 q^{81} -1.64077 q^{82} +2.41807 q^{83} -1.16968 q^{84} +5.34889 q^{86} -0.392995 q^{87} +0.290044 q^{88} -10.2797 q^{89} -25.0347 q^{91} -6.45973 q^{92} +3.25546 q^{93} -5.69256 q^{94} -0.332924 q^{96} +15.1272 q^{97} -5.34367 q^{98} +0.837984 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31}+ \cdots - 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.332924	0.192214	0.0961070	−	0.995371i	$-0.469361\pi$
$3$	0.332924	0.192214	0.0961070	+	0.995371i	$0.469361\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.332924	−0.135916
$7$	−3.51336	−1.32792	−0.663962	−	0.747766i	$-0.731126\pi$
$7$	−3.51336	−1.32792	−0.663962	+	0.747766i	$0.731126\pi$
$8$	−1.00000	−0.353553
$9$	−2.88916	−0.963054
$10$	0	0
$11$	−0.290044	−0.0874516	−0.0437258	−	0.999044i	$-0.513923\pi$
$11$	−0.290044	−0.0874516	−0.0437258	+	0.999044i	$0.513923\pi$
$12$	0.332924	0.0961070
$13$	7.12558	1.97628	0.988140	−	0.153559i	$-0.0490734\pi$
$13$	7.12558	1.97628	0.988140	+	0.153559i	$0.0490734\pi$
$14$	3.51336	0.938984
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.17921	−1.49868	−0.749339	−	0.662187i	$-0.769628\pi$
$17$	−6.17921	−1.49868	−0.749339	+	0.662187i	$0.769628\pi$
$18$	2.88916	0.680982
$19$	5.83553	1.33876	0.669381	−	0.742919i	$-0.266559\pi$
$19$	5.83553	1.33876	0.669381	+	0.742919i	$0.266559\pi$
$20$	0	0
$21$	−1.16968	−0.255246
$22$	0.290044	0.0618376
$23$	−6.45973	−1.34695	−0.673473	−	0.739212i	$-0.735198\pi$
$23$	−6.45973	−1.34695	−0.673473	+	0.739212i	$0.735198\pi$
$24$	−0.332924	−0.0679579
$25$	0	0
$26$	−7.12558	−1.39744
$27$	−1.96065	−0.377326
$28$	−3.51336	−0.663962
$29$	−1.18043	−0.219201	−0.109600	−	0.993976i	$-0.534957\pi$
$29$	−1.18043	−0.219201	−0.109600	+	0.993976i	$0.534957\pi$
$30$	0	0
$31$	9.77838	1.75625	0.878124	−	0.478433i	$-0.158795\pi$
$31$	9.77838	1.75625	0.878124	+	0.478433i	$0.158795\pi$
$32$	−1.00000	−0.176777
$33$	−0.0965628	−0.0168094
$34$	6.17921	1.05972
$35$	0	0
$36$	−2.88916	−0.481527
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−5.83553	−0.946648
$39$	2.37228	0.379869
$40$	0	0
$41$	1.64077	0.256245	0.128123	−	0.991758i	$-0.459105\pi$
$41$	1.64077	0.256245	0.128123	+	0.991758i	$0.459105\pi$
$42$	1.16968	0.180486
$43$	−5.34889	−0.815698	−0.407849	−	0.913049i	$-0.633721\pi$
$43$	−5.34889	−0.815698	−0.407849	+	0.913049i	$0.633721\pi$
$44$	−0.290044	−0.0437258
$45$	0	0
$46$	6.45973	0.952435
$47$	5.69256	0.830346	0.415173	−	0.909743i	$-0.363721\pi$
$47$	5.69256	0.830346	0.415173	+	0.909743i	$0.363721\pi$
$48$	0.332924	0.0480535
$49$	5.34367	0.763382
$50$	0	0
$51$	−2.05721	−0.288067
$52$	7.12558	0.988140
$53$	9.32034	1.28025	0.640123	−	0.768272i	$-0.278883\pi$
$53$	9.32034	1.28025	0.640123	+	0.768272i	$0.278883\pi$
$54$	1.96065	0.266810
$55$	0	0
$56$	3.51336	0.469492
$57$	1.94279	0.257329
$58$	1.18043	0.154998
$59$	5.94279	0.773686	0.386843	−	0.922146i	$-0.373566\pi$
$59$	5.94279	0.773686	0.386843	+	0.922146i	$0.373566\pi$
$60$	0	0
$61$	−1.27699	−0.163502	−0.0817512	−	0.996653i	$-0.526051\pi$
$61$	−1.27699	−0.163502	−0.0817512	+	0.996653i	$0.526051\pi$
$62$	−9.77838	−1.24185
$63$	10.1507	1.27886
$64$	1.00000	0.125000
$65$	0	0
$66$	0.0965628	0.0118861
$67$	6.04334	0.738312	0.369156	−	0.929368i	$-0.379647\pi$
$67$	6.04334	0.738312	0.369156	+	0.929368i	$0.379647\pi$
$68$	−6.17921	−0.749339
$69$	−2.15060	−0.258902
$70$	0	0
$71$	13.4995	1.60210	0.801050	−	0.598597i	$-0.204275\pi$
$71$	13.4995	1.60210	0.801050	+	0.598597i	$0.204275\pi$
$72$	2.88916	0.340491
$73$	1.71348	0.200548	0.100274	−	0.994960i	$-0.468028\pi$
$73$	1.71348	0.200548	0.100274	+	0.994960i	$0.468028\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	5.83553	0.669381
$77$	1.01903	0.116129
$78$	−2.37228	−0.268608
$79$	8.25422	0.928672	0.464336	−	0.885659i	$-0.346293\pi$
$79$	8.25422	0.928672	0.464336	+	0.885659i	$0.346293\pi$
$80$	0	0
$81$	8.01474	0.890526
$82$	−1.64077	−0.181193
$83$	2.41807	0.265418	0.132709	−	0.991155i	$-0.457632\pi$
$83$	2.41807	0.265418	0.132709	+	0.991155i	$0.457632\pi$
$84$	−1.16968	−0.127623
$85$	0	0
$86$	5.34889	0.576785
$87$	−0.392995	−0.0421335
$88$	0.290044	0.0309188
$89$	−10.2797	−1.08965	−0.544823	−	0.838551i	$-0.683403\pi$
$89$	−10.2797	−1.08965	−0.544823	+	0.838551i	$0.683403\pi$
$90$	0	0
$91$	−25.0347	−2.62435
$92$	−6.45973	−0.673473
$93$	3.25546	0.337576
$94$	−5.69256	−0.587143
$95$	0	0
$96$	−0.332924	−0.0339790
$97$	15.1272	1.53594	0.767968	−	0.640489i	$-0.221268\pi$
$97$	15.1272	1.53594	0.767968	+	0.640489i	$0.221268\pi$
$98$	−5.34367	−0.539793
$99$	0.837984	0.0842206
$100$	0	0
$101$	2.63730	0.262421	0.131210	−	0.991355i	$-0.458114\pi$
$101$	2.63730	0.262421	0.131210	+	0.991355i	$0.458114\pi$
$102$	2.05721	0.203694
$103$	−1.72469	−0.169939	−0.0849695	−	0.996384i	$-0.527079\pi$
$103$	−1.72469	−0.169939	−0.0849695	+	0.996384i	$0.527079\pi$
$104$	−7.12558	−0.698720
$105$	0	0
$106$	−9.32034	−0.905271
$107$	10.0279	0.969438	0.484719	−	0.874670i	$-0.338922\pi$
$107$	10.0279	0.969438	0.484719	+	0.874670i	$0.338922\pi$
$108$	−1.96065	−0.188663
$109$	4.87361	0.466807	0.233403	−	0.972380i	$-0.425014\pi$
$109$	4.87361	0.466807	0.233403	+	0.972380i	$0.425014\pi$
$110$	0	0
$111$	−0.332924	−0.0315998
$112$	−3.51336	−0.331981
$113$	−15.9290	−1.49847	−0.749236	−	0.662303i	$-0.769579\pi$
$113$	−15.9290	−1.49847	−0.749236	+	0.662303i	$0.769579\pi$
$114$	−1.94279	−0.181959
$115$	0	0
$116$	−1.18043	−0.109600
$117$	−20.5869	−1.90326
$118$	−5.94279	−0.547078
$119$	21.7098	1.99013
$120$	0	0
$121$	−10.9159	−0.992352
$122$	1.27699	0.115614
$123$	0.546253	0.0492540
$124$	9.77838	0.878124
$125$	0	0
$126$	−10.1507	−0.904292
$127$	1.31265	0.116479	0.0582395	−	0.998303i	$-0.481451\pi$
$127$	1.31265	0.116479	0.0582395	+	0.998303i	$0.481451\pi$
$128$	−1.00000	−0.0883883
$129$	−1.78078	−0.156789
$130$	0	0
$131$	2.41562	0.211054	0.105527	−	0.994416i	$-0.466347\pi$
$131$	2.41562	0.211054	0.105527	+	0.994416i	$0.466347\pi$
$132$	−0.0965628	−0.00840471
$133$	−20.5023	−1.77778
$134$	−6.04334	−0.522065
$135$	0	0
$136$	6.17921	0.529862
$137$	6.87366	0.587256	0.293628	−	0.955920i	$-0.405137\pi$
$137$	6.87366	0.587256	0.293628	+	0.955920i	$0.405137\pi$
$138$	2.15060	0.183071
$139$	−9.80616	−0.831748	−0.415874	−	0.909422i	$-0.636524\pi$
$139$	−9.80616	−0.831748	−0.415874	+	0.909422i	$0.636524\pi$
$140$	0	0
$141$	1.89519	0.159604
$142$	−13.4995	−1.13286
$143$	−2.06673	−0.172829
$144$	−2.88916	−0.240763
$145$	0	0
$146$	−1.71348	−0.141809
$147$	1.77904	0.146733
$148$	−1.00000	−0.0821995
$149$	−9.07687	−0.743606	−0.371803	−	0.928312i	$-0.621260\pi$
$149$	−9.07687	−0.743606	−0.371803	+	0.928312i	$0.621260\pi$
$150$	0	0
$151$	20.1964	1.64356	0.821780	−	0.569805i	$-0.192981\pi$
$151$	20.1964	1.64356	0.821780	+	0.569805i	$0.192981\pi$
$152$	−5.83553	−0.473324
$153$	17.8527	1.44331
$154$	−1.01903	−0.0821157
$155$	0	0
$156$	2.37228	0.189934
$157$	−17.2677	−1.37811	−0.689057	−	0.724707i	$-0.741975\pi$
$157$	−17.2677	−1.37811	−0.689057	+	0.724707i	$0.741975\pi$
$158$	−8.25422	−0.656670
$159$	3.10297	0.246081
$160$	0	0
$161$	22.6953	1.78864
$162$	−8.01474	−0.629697
$163$	16.6901	1.30727	0.653634	−	0.756810i	$-0.273243\pi$
$163$	16.6901	1.30727	0.653634	+	0.756810i	$0.273243\pi$
$164$	1.64077	0.128123
$165$	0	0
$166$	−2.41807	−0.187679
$167$	1.47354	0.114026	0.0570130	−	0.998373i	$-0.481842\pi$
$167$	1.47354	0.114026	0.0570130	+	0.998373i	$0.481842\pi$
$168$	1.16968	0.0902430
$169$	37.7738	2.90568
$170$	0	0
$171$	−16.8598	−1.28930
$172$	−5.34889	−0.407849
$173$	13.8432	1.05248	0.526240	−	0.850336i	$-0.323601\pi$
$173$	13.8432	1.05248	0.526240	+	0.850336i	$0.323601\pi$
$174$	0.392995	0.0297929
$175$	0	0
$176$	−0.290044	−0.0218629
$177$	1.97850	0.148713
$178$	10.2797	0.770496
$179$	21.7817	1.62804	0.814020	−	0.580836i	$-0.197274\pi$
$179$	21.7817	1.62804	0.814020	+	0.580836i	$0.197274\pi$
$180$	0	0
$181$	2.03100	0.150963	0.0754817	−	0.997147i	$-0.475951\pi$
$181$	2.03100	0.150963	0.0754817	+	0.997147i	$0.475951\pi$
$182$	25.0347	1.85569
$183$	−0.425143	−0.0314275
$184$	6.45973	0.476217
$185$	0	0
$186$	−3.25546	−0.238702
$187$	1.79224	0.131062
$188$	5.69256	0.415173
$189$	6.88845	0.501061
$190$	0	0
$191$	−8.44668	−0.611180	−0.305590	−	0.952163i	$-0.598854\pi$
$191$	−8.44668	−0.611180	−0.305590	+	0.952163i	$0.598854\pi$
$192$	0.332924	0.0240268
$193$	3.64159	0.262127	0.131064	−	0.991374i	$-0.458161\pi$
$193$	3.64159	0.262127	0.131064	+	0.991374i	$0.458161\pi$
$194$	−15.1272	−1.08607
$195$	0	0
$196$	5.34367	0.381691
$197$	10.1939	0.726288	0.363144	−	0.931733i	$-0.381703\pi$
$197$	10.1939	0.726288	0.363144	+	0.931733i	$0.381703\pi$
$198$	−0.837984	−0.0595530
$199$	−5.25299	−0.372375	−0.186187	−	0.982514i	$-0.559613\pi$
$199$	−5.25299	−0.372375	−0.186187	+	0.982514i	$0.559613\pi$
$200$	0	0
$201$	2.01198	0.141914
$202$	−2.63730	−0.185560
$203$	4.14728	0.291082
$204$	−2.05721	−0.144033
$205$	0	0
$206$	1.72469	0.120165
$207$	18.6632	1.29718
$208$	7.12558	0.494070
$209$	−1.69256	−0.117077
$210$	0	0
$211$	−4.81998	−0.331821	−0.165910	−	0.986141i	$-0.553056\pi$
$211$	−4.81998	−0.331821	−0.165910	+	0.986141i	$0.553056\pi$
$212$	9.32034	0.640123
$213$	4.49433	0.307946
$214$	−10.0279	−0.685496
$215$	0	0
$216$	1.96065	0.133405
$217$	−34.3549	−2.33216
$218$	−4.87361	−0.330082
$219$	0.570460	0.0385481
$220$	0	0
$221$	−44.0304	−2.96180
$222$	0.332924	0.0223444
$223$	3.70392	0.248033	0.124017	−	0.992280i	$-0.460422\pi$
$223$	3.70392	0.248033	0.124017	+	0.992280i	$0.460422\pi$
$224$	3.51336	0.234746
$225$	0	0
$226$	15.9290	1.05958
$227$	6.31512	0.419149	0.209575	−	0.977793i	$-0.432792\pi$
$227$	6.31512	0.419149	0.209575	+	0.977793i	$0.432792\pi$
$228$	1.94279	0.128664
$229$	−20.5548	−1.35830	−0.679150	−	0.734000i	$-0.737651\pi$
$229$	−20.5548	−1.35830	−0.679150	+	0.734000i	$0.737651\pi$
$230$	0	0
$231$	0.339260	0.0223216
$232$	1.18043	0.0774992
$233$	21.4393	1.40453	0.702267	−	0.711914i	$-0.252171\pi$
$233$	21.4393	1.40453	0.702267	+	0.711914i	$0.252171\pi$
$234$	20.5869	1.34581
$235$	0	0
$236$	5.94279	0.386843
$237$	2.74803	0.178504
$238$	−21.7098	−1.40723
$239$	−20.7479	−1.34207	−0.671034	−	0.741426i	$-0.734150\pi$
$239$	−20.7479	−1.34207	−0.671034	+	0.741426i	$0.734150\pi$
$240$	0	0
$241$	10.7731	0.693957	0.346978	−	0.937873i	$-0.387208\pi$
$241$	10.7731	0.693957	0.346978	+	0.937873i	$0.387208\pi$
$242$	10.9159	0.701699
$243$	8.55024	0.548498
$244$	−1.27699	−0.0817512
$245$	0	0
$246$	−0.546253	−0.0348278
$247$	41.5815	2.64577
$248$	−9.77838	−0.620927
$249$	0.805036	0.0510171
$250$	0	0
$251$	−4.46812	−0.282025	−0.141013	−	0.990008i	$-0.545036\pi$
$251$	−4.46812	−0.282025	−0.141013	+	0.990008i	$0.545036\pi$
$252$	10.1507	0.639431
$253$	1.87361	0.117793
$254$	−1.31265	−0.0823631
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.24594	−0.0777194	−0.0388597	−	0.999245i	$-0.512373\pi$
$257$	−1.24594	−0.0777194	−0.0388597	+	0.999245i	$0.512373\pi$
$258$	1.78078	0.110866
$259$	3.51336	0.218309
$260$	0	0
$261$	3.41046	0.211102
$262$	−2.41562	−0.149237
$263$	16.3230	1.00652	0.503259	−	0.864135i	$-0.332134\pi$
$263$	16.3230	1.00652	0.503259	+	0.864135i	$0.332134\pi$
$264$	0.0965628	0.00594303
$265$	0	0
$266$	20.5023	1.25708
$267$	−3.42236	−0.209445
$268$	6.04334	0.369156
$269$	−18.5763	−1.13262	−0.566309	−	0.824193i	$-0.691629\pi$
$269$	−18.5763	−1.13262	−0.566309	+	0.824193i	$0.691629\pi$
$270$	0	0
$271$	16.4181	0.997327	0.498663	−	0.866796i	$-0.333824\pi$
$271$	16.4181	0.997327	0.498663	+	0.866796i	$0.333824\pi$
$272$	−6.17921	−0.374669
$273$	−8.33466	−0.504437
$274$	−6.87366	−0.415253
$275$	0	0
$276$	−2.15060	−0.129451
$277$	15.6104	0.937937	0.468968	−	0.883215i	$-0.344626\pi$
$277$	15.6104	0.937937	0.468968	+	0.883215i	$0.344626\pi$
$278$	9.80616	0.588134
$279$	−28.2513	−1.69136
$280$	0	0
$281$	−21.8665	−1.30445	−0.652224	−	0.758026i	$-0.726164\pi$
$281$	−21.8665	−1.30445	−0.652224	+	0.758026i	$0.726164\pi$
$282$	−1.89519	−0.112857
$283$	−2.24778	−0.133616	−0.0668082	−	0.997766i	$-0.521282\pi$
$283$	−2.24778	−0.133616	−0.0668082	+	0.997766i	$0.521282\pi$
$284$	13.4995	0.801050
$285$	0	0
$286$	2.06673	0.122208
$287$	−5.76461	−0.340274
$288$	2.88916	0.170245
$289$	21.1826	1.24603
$290$	0	0
$291$	5.03622	0.295228
$292$	1.71348	0.100274
$293$	−26.8794	−1.57031	−0.785157	−	0.619297i	$-0.787418\pi$
$293$	−26.8794	−1.57031	−0.785157	+	0.619297i	$0.787418\pi$
$294$	−1.77904	−0.103756
$295$	0	0
$296$	1.00000	0.0581238
$297$	0.568674	0.0329978
$298$	9.07687	0.525809
$299$	−46.0293	−2.66194
$300$	0	0
$301$	18.7926	1.08318
$302$	−20.1964	−1.16217
$303$	0.878021	0.0504410
$304$	5.83553	0.334691
$305$	0	0
$306$	−17.8527	−1.02057
$307$	−23.9979	−1.36963	−0.684815	−	0.728717i	$-0.740117\pi$
$307$	−23.9979	−1.36963	−0.684815	+	0.728717i	$0.740117\pi$
$308$	1.01903	0.0580645
$309$	−0.574192	−0.0326647
$310$	0	0
$311$	−5.07563	−0.287812	−0.143906	−	0.989591i	$-0.545966\pi$
$311$	−5.07563	−0.287812	−0.143906	+	0.989591i	$0.545966\pi$
$312$	−2.37228	−0.134304
$313$	8.96950	0.506986	0.253493	−	0.967337i	$-0.418420\pi$
$313$	8.96950	0.506986	0.253493	+	0.967337i	$0.418420\pi$
$314$	17.2677	0.974474
$315$	0	0
$316$	8.25422	0.464336
$317$	22.0181	1.23666	0.618330	−	0.785918i	$-0.287809\pi$
$317$	22.0181	1.23666	0.618330	+	0.785918i	$0.287809\pi$
$318$	−3.10297	−0.174006
$319$	0.342377	0.0191695
$320$	0	0
$321$	3.33855	0.186339
$322$	−22.6953	−1.26476
$323$	−36.0589	−2.00637
$324$	8.01474	0.445263
$325$	0	0
$326$	−16.6901	−0.924379
$327$	1.62254	0.0897268
$328$	−1.64077	−0.0905964
$329$	−20.0000	−1.10264
$330$	0	0
$331$	−10.7134	−0.588864	−0.294432	−	0.955672i	$-0.595130\pi$
$331$	−10.7134	−0.588864	−0.294432	+	0.955672i	$0.595130\pi$
$332$	2.41807	0.132709
$333$	2.88916	0.158325
$334$	−1.47354	−0.0806286
$335$	0	0
$336$	−1.16968	−0.0638114
$337$	−1.34097	−0.0730471	−0.0365235	−	0.999333i	$-0.511628\pi$
$337$	−1.34097	−0.0730471	−0.0365235	+	0.999333i	$0.511628\pi$
$338$	−37.7738	−2.05463
$339$	−5.30315	−0.288027
$340$	0	0
$341$	−2.83616	−0.153587
$342$	16.8598	0.911673
$343$	5.81926	0.314211
$344$	5.34889	0.288393
$345$	0	0
$346$	−13.8432	−0.744216
$347$	−0.883744	−0.0474419	−0.0237209	−	0.999719i	$-0.507551\pi$
$347$	−0.883744	−0.0474419	−0.0237209	+	0.999719i	$0.507551\pi$
$348$	−0.392995	−0.0210667
$349$	−9.00521	−0.482038	−0.241019	−	0.970520i	$-0.577482\pi$
$349$	−9.00521	−0.482038	−0.241019	+	0.970520i	$0.577482\pi$
$350$	0	0
$351$	−13.9707	−0.745702
$352$	0.290044	0.0154594
$353$	−6.79030	−0.361411	−0.180706	−	0.983537i	$-0.557838\pi$
$353$	−6.79030	−0.361411	−0.180706	+	0.983537i	$0.557838\pi$
$354$	−1.97850	−0.105156
$355$	0	0
$356$	−10.2797	−0.544823
$357$	7.22771	0.382531
$358$	−21.7817	−1.15120
$359$	−10.2263	−0.539722	−0.269861	−	0.962899i	$-0.586978\pi$
$359$	−10.2263	−0.539722	−0.269861	+	0.962899i	$0.586978\pi$
$360$	0	0
$361$	15.0534	0.792286
$362$	−2.03100	−0.106747
$363$	−3.63416	−0.190744
$364$	−25.0347	−1.31217
$365$	0	0
$366$	0.425143	0.0222226
$367$	−17.1583	−0.895657	−0.447829	−	0.894119i	$-0.647802\pi$
$367$	−17.1583	−0.895657	−0.447829	+	0.894119i	$0.647802\pi$
$368$	−6.45973	−0.336737
$369$	−4.74045	−0.246778
$370$	0	0
$371$	−32.7457	−1.70007
$372$	3.25546	0.168788
$373$	−17.1601	−0.888515	−0.444257	−	0.895899i	$-0.646532\pi$
$373$	−17.1601	−0.888515	−0.444257	+	0.895899i	$0.646532\pi$
$374$	−1.79224	−0.0926747
$375$	0	0
$376$	−5.69256	−0.293571
$377$	−8.41126	−0.433202
$378$	−6.88845	−0.354304
$379$	−27.5435	−1.41482	−0.707408	−	0.706805i	$-0.750136\pi$
$379$	−27.5435	−1.41482	−0.707408	+	0.706805i	$0.750136\pi$
$380$	0	0
$381$	0.437014	0.0223889
$382$	8.44668	0.432170
$383$	3.49954	0.178818	0.0894091	−	0.995995i	$-0.471502\pi$
$383$	3.49954	0.178818	0.0894091	+	0.995995i	$0.471502\pi$
$384$	−0.332924	−0.0169895
$385$	0	0
$386$	−3.64159	−0.185352
$387$	15.4538	0.785561
$388$	15.1272	0.767968
$389$	−13.0015	−0.659203	−0.329602	−	0.944120i	$-0.606914\pi$
$389$	−13.0015	−0.659203	−0.329602	+	0.944120i	$0.606914\pi$
$390$	0	0
$391$	39.9160	2.01864
$392$	−5.34367	−0.269896
$393$	0.804219	0.0405675
$394$	−10.1939	−0.513563
$395$	0	0
$396$	0.837984	0.0421103
$397$	−26.6658	−1.33832	−0.669160	−	0.743118i	$-0.733346\pi$
$397$	−26.6658	−1.33832	−0.669160	+	0.743118i	$0.733346\pi$
$398$	5.25299	0.263309
$399$	−6.82572	−0.341713
$400$	0	0
$401$	20.3546	1.01646	0.508231	−	0.861221i	$-0.330300\pi$
$401$	20.3546	1.01646	0.508231	+	0.861221i	$0.330300\pi$
$402$	−2.01198	−0.100348
$403$	69.6766	3.47084
$404$	2.63730	0.131210
$405$	0	0
$406$	−4.14728	−0.205826
$407$	0.290044	0.0143770
$408$	2.05721	0.101847
$409$	29.9603	1.48144	0.740720	−	0.671814i	$-0.234485\pi$
$409$	29.9603	1.48144	0.740720	+	0.671814i	$0.234485\pi$
$410$	0	0
$411$	2.28841	0.112879
$412$	−1.72469	−0.0849695
$413$	−20.8791	−1.02740
$414$	−18.6632	−0.917246
$415$	0	0
$416$	−7.12558	−0.349360
$417$	−3.26471	−0.159874
$418$	1.69256	0.0827859
$419$	23.3149	1.13901	0.569504	−	0.821988i	$-0.307135\pi$
$419$	23.3149	1.13901	0.569504	+	0.821988i	$0.307135\pi$
$420$	0	0
$421$	24.0289	1.17110	0.585548	−	0.810638i	$-0.300880\pi$
$421$	24.0289	1.17110	0.585548	+	0.810638i	$0.300880\pi$
$422$	4.81998	0.234633
$423$	−16.4467	−0.799667
$424$	−9.32034	−0.452636
$425$	0	0
$426$	−4.49433	−0.217751
$427$	4.48654	0.217119
$428$	10.0279	0.484719
$429$	−0.688066	−0.0332201
$430$	0	0
$431$	−19.2917	−0.929247	−0.464624	−	0.885508i	$-0.653810\pi$
$431$	−19.2917	−0.929247	−0.464624	+	0.885508i	$0.653810\pi$
$432$	−1.96065	−0.0943316
$433$	−12.5816	−0.604634	−0.302317	−	0.953207i	$-0.597760\pi$
$433$	−12.5816	−0.604634	−0.302317	+	0.953207i	$0.597760\pi$
$434$	34.3549	1.64909
$435$	0	0
$436$	4.87361	0.233403
$437$	−37.6959	−1.80324
$438$	−0.570460	−0.0272576
$439$	14.0428	0.670225	0.335113	−	0.942178i	$-0.391226\pi$
$439$	14.0428	0.670225	0.335113	+	0.942178i	$0.391226\pi$
$440$	0	0
$441$	−15.4387	−0.735178
$442$	44.0304	2.09431
$443$	−24.5696	−1.16734	−0.583668	−	0.811992i	$-0.698383\pi$
$443$	−24.5696	−1.16734	−0.583668	+	0.811992i	$0.698383\pi$
$444$	−0.332924	−0.0157999
$445$	0	0
$446$	−3.70392	−0.175386
$447$	−3.02191	−0.142932
$448$	−3.51336	−0.165990
$449$	−12.7574	−0.602061	−0.301030	−	0.953615i	$-0.597331\pi$
$449$	−12.7574	−0.602061	−0.301030	+	0.953615i	$0.597331\pi$
$450$	0	0
$451$	−0.475896	−0.0224091
$452$	−15.9290	−0.749236
$453$	6.72387	0.315915
$454$	−6.31512	−0.296383
$455$	0	0
$456$	−1.94279	−0.0909795
$457$	3.34889	0.156654	0.0783272	−	0.996928i	$-0.475042\pi$
$457$	3.34889	0.156654	0.0783272	+	0.996928i	$0.475042\pi$
$458$	20.5548	0.960463
$459$	12.1152	0.565491
$460$	0	0
$461$	−0.859501	−0.0400310	−0.0200155	−	0.999800i	$-0.506372\pi$
$461$	−0.859501	−0.0400310	−0.0200155	+	0.999800i	$0.506372\pi$
$462$	−0.339260	−0.0157838
$463$	13.8711	0.644643	0.322321	−	0.946630i	$-0.395537\pi$
$463$	13.8711	0.644643	0.322321	+	0.946630i	$0.395537\pi$
$464$	−1.18043	−0.0548002
$465$	0	0
$466$	−21.4393	−0.993155
$467$	40.9281	1.89392	0.946962	−	0.321344i	$-0.104135\pi$
$467$	40.9281	1.89392	0.946962	+	0.321344i	$0.104135\pi$
$468$	−20.5869	−0.951632
$469$	−21.2324	−0.980422
$470$	0	0
$471$	−5.74885	−0.264893
$472$	−5.94279	−0.273539
$473$	1.55141	0.0713341
$474$	−2.74803	−0.126221
$475$	0	0
$476$	21.7098	0.995065
$477$	−26.9280	−1.23295
$478$	20.7479	0.948986
$479$	8.58990	0.392483	0.196241	−	0.980556i	$-0.437126\pi$
$479$	8.58990	0.392483	0.196241	+	0.980556i	$0.437126\pi$
$480$	0	0
$481$	−7.12558	−0.324898
$482$	−10.7731	−0.490702
$483$	7.55583	0.343802
$484$	−10.9159	−0.496176
$485$	0	0
$486$	−8.55024	−0.387847
$487$	25.8389	1.17087	0.585436	−	0.810718i	$-0.300923\pi$
$487$	25.8389	1.17087	0.585436	+	0.810718i	$0.300923\pi$
$488$	1.27699	0.0578068
$489$	5.55654	0.251275
$490$	0	0
$491$	11.4864	0.518376	0.259188	−	0.965827i	$-0.416545\pi$
$491$	11.4864	0.518376	0.259188	+	0.965827i	$0.416545\pi$
$492$	0.546253	0.0246270
$493$	7.29413	0.328511
$494$	−41.5815	−1.87084
$495$	0	0
$496$	9.77838	0.439062
$497$	−47.4287	−2.12747
$498$	−0.805036	−0.0360745
$499$	−9.74599	−0.436290	−0.218145	−	0.975916i	$-0.570001\pi$
$499$	−9.74599	−0.436290	−0.218145	+	0.975916i	$0.570001\pi$
$500$	0	0
$501$	0.490578	0.0219174
$502$	4.46812	0.199422
$503$	17.4803	0.779408	0.389704	−	0.920940i	$-0.372577\pi$
$503$	17.4803	0.779408	0.389704	+	0.920940i	$0.372577\pi$
$504$	−10.1507	−0.452146
$505$	0	0
$506$	−1.87361	−0.0832920
$507$	12.5758	0.558512
$508$	1.31265	0.0582395
$509$	−16.3273	−0.723695	−0.361847	−	0.932237i	$-0.617854\pi$
$509$	−16.3273	−0.723695	−0.361847	+	0.932237i	$0.617854\pi$
$510$	0	0
$511$	−6.02007	−0.266312
$512$	−1.00000	−0.0441942
$513$	−11.4414	−0.505151
$514$	1.24594	0.0549559
$515$	0	0
$516$	−1.78078	−0.0783943
$517$	−1.65109	−0.0726151
$518$	−3.51336	−0.154368
$519$	4.60875	0.202301
$520$	0	0
$521$	−4.78335	−0.209562	−0.104781	−	0.994495i	$-0.533414\pi$
$521$	−4.78335	−0.209562	−0.104781	+	0.994495i	$0.533414\pi$
$522$	−3.41046	−0.149272
$523$	−24.1224	−1.05480	−0.527399	−	0.849618i	$-0.676833\pi$
$523$	−24.1224	−1.05480	−0.527399	+	0.849618i	$0.676833\pi$
$524$	2.41562	0.105527
$525$	0	0
$526$	−16.3230	−0.711716
$527$	−60.4226	−2.63205
$528$	−0.0965628	−0.00420236
$529$	18.7281	0.814264
$530$	0	0
$531$	−17.1697	−0.745101
$532$	−20.5023	−0.888888
$533$	11.6914	0.506412
$534$	3.42236	0.148100
$535$	0	0
$536$	−6.04334	−0.261033
$537$	7.25166	0.312932
$538$	18.5763	0.800881
$539$	−1.54990	−0.0667590
$540$	0	0
$541$	4.11745	0.177023	0.0885115	−	0.996075i	$-0.471789\pi$
$541$	4.11745	0.177023	0.0885115	+	0.996075i	$0.471789\pi$
$542$	−16.4181	−0.705217
$543$	0.676171	0.0290173
$544$	6.17921	0.264931
$545$	0	0
$546$	8.33466	0.356691
$547$	9.22742	0.394536	0.197268	−	0.980350i	$-0.436793\pi$
$547$	9.22742	0.394536	0.197268	+	0.980350i	$0.436793\pi$
$548$	6.87366	0.293628
$549$	3.68944	0.157462
$550$	0	0
$551$	−6.88845	−0.293458
$552$	2.15060	0.0915357
$553$	−29.0000	−1.23321
$554$	−15.6104	−0.663222
$555$	0	0
$556$	−9.80616	−0.415874
$557$	−2.25125	−0.0953885	−0.0476943	−	0.998862i	$-0.515187\pi$
$557$	−2.25125	−0.0953885	−0.0476943	+	0.998862i	$0.515187\pi$
$558$	28.2513	1.19597
$559$	−38.1139	−1.61205
$560$	0	0
$561$	0.596681	0.0251919
$562$	21.8665	0.922384
$563$	−5.79316	−0.244153	−0.122076	−	0.992521i	$-0.538955\pi$
$563$	−5.79316	−0.244153	−0.122076	+	0.992521i	$0.538955\pi$
$564$	1.89519	0.0798020
$565$	0	0
$566$	2.24778	0.0944811
$567$	−28.1586	−1.18255
$568$	−13.4995	−0.566428
$569$	−34.4807	−1.44551	−0.722753	−	0.691106i	$-0.757124\pi$
$569$	−34.4807	−1.44551	−0.722753	+	0.691106i	$0.757124\pi$
$570$	0	0
$571$	−19.2152	−0.804133	−0.402066	−	0.915611i	$-0.631708\pi$
$571$	−19.2152	−0.804133	−0.402066	+	0.915611i	$0.631708\pi$
$572$	−2.06673	−0.0864144
$573$	−2.81211	−0.117477
$574$	5.76461	0.240610
$575$	0	0
$576$	−2.88916	−0.120382
$577$	−6.33354	−0.263669	−0.131834	−	0.991272i	$-0.542087\pi$
$577$	−6.33354	−0.263669	−0.131834	+	0.991272i	$0.542087\pi$
$578$	−21.1826	−0.881079
$579$	1.21237	0.0503845
$580$	0	0
$581$	−8.49555	−0.352455
$582$	−5.03622	−0.208758
$583$	−2.70331	−0.111960
$584$	−1.71348	−0.0709044
$585$	0	0
$586$	26.8794	1.11038
$587$	21.9099	0.904320	0.452160	−	0.891937i	$-0.350654\pi$
$587$	21.9099	0.904320	0.452160	+	0.891937i	$0.350654\pi$
$588$	1.77904	0.0733664
$589$	57.0620	2.35120
$590$	0	0
$591$	3.39381	0.139603
$592$	−1.00000	−0.0410997
$593$	9.40482	0.386210	0.193105	−	0.981178i	$-0.438144\pi$
$593$	9.40482	0.386210	0.193105	+	0.981178i	$0.438144\pi$
$594$	−0.568674	−0.0233330
$595$	0	0
$596$	−9.07687	−0.371803
$597$	−1.74885	−0.0715756
$598$	46.0293	1.88228
$599$	16.4095	0.670474	0.335237	−	0.942134i	$-0.391184\pi$
$599$	16.4095	0.670474	0.335237	+	0.942134i	$0.391184\pi$
$600$	0	0
$601$	−7.01392	−0.286104	−0.143052	−	0.989715i	$-0.545692\pi$
$601$	−7.01392	−0.286104	−0.143052	+	0.989715i	$0.545692\pi$
$602$	−18.7926	−0.765927
$603$	−17.4602	−0.711034
$604$	20.1964	0.821780
$605$	0	0
$606$	−0.878021	−0.0356672
$607$	−16.5621	−0.672234	−0.336117	−	0.941820i	$-0.609114\pi$
$607$	−16.5621	−0.672234	−0.336117	+	0.941820i	$0.609114\pi$
$608$	−5.83553	−0.236662
$609$	1.38073	0.0559500
$610$	0	0
$611$	40.5628	1.64099
$612$	17.8527	0.721653
$613$	−23.9370	−0.966804	−0.483402	−	0.875398i	$-0.660599\pi$
$613$	−23.9370	−0.966804	−0.483402	+	0.875398i	$0.660599\pi$
$614$	23.9979	0.968475
$615$	0	0
$616$	−1.01903	−0.0410578
$617$	13.9489	0.561563	0.280781	−	0.959772i	$-0.409406\pi$
$617$	13.9489	0.561563	0.280781	+	0.959772i	$0.409406\pi$
$618$	0.574192	0.0230974
$619$	21.6942	0.871964	0.435982	−	0.899955i	$-0.356401\pi$
$619$	21.6942	0.871964	0.435982	+	0.899955i	$0.356401\pi$
$620$	0	0
$621$	12.6652	0.508238
$622$	5.07563	0.203514
$623$	36.1163	1.44697
$624$	2.37228	0.0949671
$625$	0	0
$626$	−8.96950	−0.358493
$627$	−0.563495	−0.0225038
$628$	−17.2677	−0.689057
$629$	6.17921	0.246381
$630$	0	0
$631$	−32.4902	−1.29342	−0.646708	−	0.762738i	$-0.723855\pi$
$631$	−32.4902	−1.29342	−0.646708	+	0.762738i	$0.723855\pi$
$632$	−8.25422	−0.328335
$633$	−1.60469	−0.0637806
$634$	−22.0181	−0.874451
$635$	0	0
$636$	3.10297	0.123041
$637$	38.0768	1.50866
$638$	−0.342377	−0.0135549
$639$	−39.0024	−1.54291
$640$	0	0
$641$	−28.1621	−1.11234	−0.556168	−	0.831070i	$-0.687729\pi$
$641$	−28.1621	−1.11234	−0.556168	+	0.831070i	$0.687729\pi$
$642$	−3.33855	−0.131762
$643$	−8.32452	−0.328287	−0.164144	−	0.986436i	$-0.552486\pi$
$643$	−8.32452	−0.328287	−0.164144	+	0.986436i	$0.552486\pi$
$644$	22.6953	0.894321
$645$	0	0
$646$	36.0589	1.41872
$647$	−1.75827	−0.0691249	−0.0345624	−	0.999403i	$-0.511004\pi$
$647$	−1.75827	−0.0691249	−0.0345624	+	0.999403i	$0.511004\pi$
$648$	−8.01474	−0.314849
$649$	−1.72367	−0.0676600
$650$	0	0
$651$	−11.4376	−0.448275
$652$	16.6901	0.653634
$653$	−24.7416	−0.968213	−0.484106	−	0.875009i	$-0.660855\pi$
$653$	−24.7416	−0.968213	−0.484106	+	0.875009i	$0.660855\pi$
$654$	−1.62254	−0.0634464
$655$	0	0
$656$	1.64077	0.0640613
$657$	−4.95053	−0.193138
$658$	20.0000	0.779681
$659$	18.2640	0.711466	0.355733	−	0.934588i	$-0.384231\pi$
$659$	18.2640	0.711466	0.355733	+	0.934588i	$0.384231\pi$
$660$	0	0
$661$	2.61857	0.101851	0.0509253	−	0.998702i	$-0.483783\pi$
$661$	2.61857	0.101851	0.0509253	+	0.998702i	$0.483783\pi$
$662$	10.7134	0.416390
$663$	−14.6588	−0.569300
$664$	−2.41807	−0.0938394
$665$	0	0
$666$	−2.88916	−0.111953
$667$	7.62527	0.295252
$668$	1.47354	0.0570130
$669$	1.23313	0.0476754
$670$	0	0
$671$	0.370385	0.0142986
$672$	1.16968	0.0451215
$673$	48.4181	1.86638	0.933190	−	0.359384i	$-0.117013\pi$
$673$	48.4181	1.86638	0.933190	+	0.359384i	$0.117013\pi$
$674$	1.34097	0.0516521
$675$	0	0
$676$	37.7738	1.45284
$677$	7.49862	0.288195	0.144098	−	0.989563i	$-0.453972\pi$
$677$	7.49862	0.288195	0.144098	+	0.989563i	$0.453972\pi$
$678$	5.30315	0.203666
$679$	−53.1473	−2.03961
$680$	0	0
$681$	2.10246	0.0805664
$682$	2.83616	0.108602
$683$	33.2987	1.27414	0.637070	−	0.770806i	$-0.280146\pi$
$683$	33.2987	1.27414	0.637070	+	0.770806i	$0.280146\pi$
$684$	−16.8598	−0.644650
$685$	0	0
$686$	−5.81926	−0.222181
$687$	−6.84320	−0.261084
$688$	−5.34889	−0.203924
$689$	66.4128	2.53012
$690$	0	0
$691$	38.3695	1.45964	0.729822	−	0.683638i	$-0.239603\pi$
$691$	38.3695	1.45964	0.729822	+	0.683638i	$0.239603\pi$
$692$	13.8432	0.526240
$693$	−2.94414	−0.111839
$694$	0.883744	0.0335465
$695$	0	0
$696$	0.392995	0.0148964
$697$	−10.1387	−0.384029
$698$	9.00521	0.340852
$699$	7.13766	0.269971
$700$	0	0
$701$	−40.6339	−1.53472	−0.767360	−	0.641217i	$-0.778430\pi$
$701$	−40.6339	−1.53472	−0.767360	+	0.641217i	$0.778430\pi$
$702$	13.9707	0.527291
$703$	−5.83553	−0.220091
$704$	−0.290044	−0.0109315
$705$	0	0
$706$	6.79030	0.255556
$707$	−9.26577	−0.348475
$708$	1.97850	0.0743566
$709$	11.1957	0.420464	0.210232	−	0.977651i	$-0.432578\pi$
$709$	11.1957	0.420464	0.210232	+	0.977651i	$0.432578\pi$
$710$	0	0
$711$	−23.8478	−0.894361
$712$	10.2797	0.385248
$713$	−63.1656	−2.36557
$714$	−7.22771	−0.270490
$715$	0	0
$716$	21.7817	0.814020
$717$	−6.90748	−0.257964
$718$	10.2263	0.381641
$719$	30.2226	1.12711	0.563556	−	0.826078i	$-0.309433\pi$
$719$	30.2226	1.12711	0.563556	+	0.826078i	$0.309433\pi$
$720$	0	0
$721$	6.05946	0.225666
$722$	−15.0534	−0.560231
$723$	3.58663	0.133388
$724$	2.03100	0.0754817
$725$	0	0
$726$	3.63416	0.134876
$727$	19.1911	0.711758	0.355879	−	0.934532i	$-0.384182\pi$
$727$	19.1911	0.711758	0.355879	+	0.934532i	$0.384182\pi$
$728$	25.0347	0.927847
$729$	−21.1976	−0.785097
$730$	0	0
$731$	33.0519	1.22247
$732$	−0.425143	−0.0157137
$733$	−5.39567	−0.199294	−0.0996468	−	0.995023i	$-0.531771\pi$
$733$	−5.39567	−0.199294	−0.0996468	+	0.995023i	$0.531771\pi$
$734$	17.1583	0.633325
$735$	0	0
$736$	6.45973	0.238109
$737$	−1.75284	−0.0645665
$738$	4.74045	0.174498
$739$	24.7127	0.909072	0.454536	−	0.890728i	$-0.349805\pi$
$739$	24.7127	0.909072	0.454536	+	0.890728i	$0.349805\pi$
$740$	0	0
$741$	13.8435	0.508554
$742$	32.7457	1.20213
$743$	−45.4537	−1.66753	−0.833767	−	0.552116i	$-0.813820\pi$
$743$	−45.4537	−1.66753	−0.833767	+	0.552116i	$0.813820\pi$
$744$	−3.25546	−0.119351
$745$	0	0
$746$	17.1601	0.628275
$747$	−6.98620	−0.255612
$748$	1.79224	0.0655309
$749$	−35.2317	−1.28734
$750$	0	0
$751$	49.6360	1.81124	0.905621	−	0.424088i	$-0.139405\pi$
$751$	49.6360	1.81124	0.905621	+	0.424088i	$0.139405\pi$
$752$	5.69256	0.207586
$753$	−1.48755	−0.0542093
$754$	8.41126	0.306320
$755$	0	0
$756$	6.88845	0.250530
$757$	27.9481	1.01579	0.507896	−	0.861419i	$-0.330424\pi$
$757$	27.9481	1.01579	0.507896	+	0.861419i	$0.330424\pi$
$758$	27.5435	1.00043
$759$	0.623769	0.0226414
$760$	0	0
$761$	−53.4744	−1.93844	−0.969222	−	0.246188i	$-0.920822\pi$
$761$	−53.4744	−1.93844	−0.969222	+	0.246188i	$0.920822\pi$
$762$	−0.437014	−0.0158313
$763$	−17.1227	−0.619884
$764$	−8.44668	−0.305590
$765$	0	0
$766$	−3.49954	−0.126444
$767$	42.3458	1.52902
$768$	0.332924	0.0120134
$769$	−8.46659	−0.305313	−0.152657	−	0.988279i	$-0.548783\pi$
$769$	−8.46659	−0.305313	−0.152657	+	0.988279i	$0.548783\pi$
$770$	0	0
$771$	−0.414803	−0.0149388
$772$	3.64159	0.131064
$773$	0.530866	0.0190939	0.00954697	−	0.999954i	$-0.496961\pi$
$773$	0.530866	0.0190939	0.00954697	+	0.999954i	$0.496961\pi$
$774$	−15.4538	−0.555475
$775$	0	0
$776$	−15.1272	−0.543035
$777$	1.16968	0.0419621
$778$	13.0015	0.466127
$779$	9.57477	0.343052
$780$	0	0
$781$	−3.91546	−0.140106
$782$	−39.9160	−1.42739
$783$	2.31441	0.0827102
$784$	5.34367	0.190846
$785$	0	0
$786$	−0.804219	−0.0286855
$787$	2.16917	0.0773227	0.0386613	−	0.999252i	$-0.487691\pi$
$787$	2.16917	0.0773227	0.0386613	+	0.999252i	$0.487691\pi$
$788$	10.1939	0.363144
$789$	5.43432	0.193467
$790$	0	0
$791$	55.9642	1.98986
$792$	−0.837984	−0.0297765
$793$	−9.09932	−0.323126
$794$	26.6658	0.946336
$795$	0	0
$796$	−5.25299	−0.186187
$797$	−26.2510	−0.929857	−0.464928	−	0.885348i	$-0.653920\pi$
$797$	−26.2510	−0.929857	−0.464928	+	0.885348i	$0.653920\pi$
$798$	6.82572	0.241628
$799$	−35.1755	−1.24442
$800$	0	0
$801$	29.6997	1.04939
$802$	−20.3546	−0.718747
$803$	−0.496986	−0.0175382
$804$	2.01198	0.0709569
$805$	0	0
$806$	−69.6766	−2.45425
$807$	−6.18451	−0.217705
$808$	−2.63730	−0.0927798
$809$	6.67443	0.234661	0.117330	−	0.993093i	$-0.462566\pi$
$809$	6.67443	0.234661	0.117330	+	0.993093i	$0.462566\pi$
$810$	0	0
$811$	−8.24768	−0.289615	−0.144808	−	0.989460i	$-0.546256\pi$
$811$	−8.24768	−0.289615	−0.144808	+	0.989460i	$0.546256\pi$
$812$	4.14728	0.145541
$813$	5.46598	0.191700
$814$	−0.290044	−0.0101660
$815$	0	0
$816$	−2.05721	−0.0720167
$817$	−31.2136	−1.09203
$818$	−29.9603	−1.04754
$819$	72.3292	2.52739
$820$	0	0
$821$	−9.61631	−0.335611	−0.167806	−	0.985820i	$-0.553668\pi$
$821$	−9.61631	−0.335611	−0.167806	+	0.985820i	$0.553668\pi$
$822$	−2.28841	−0.0798174
$823$	−32.7309	−1.14093	−0.570464	−	0.821322i	$-0.693237\pi$
$823$	−32.7309	−1.14093	−0.570464	+	0.821322i	$0.693237\pi$
$824$	1.72469	0.0600825
$825$	0	0
$826$	20.8791	0.726478
$827$	39.9956	1.39078	0.695391	−	0.718631i	$-0.255231\pi$
$827$	39.9956	1.39078	0.695391	+	0.718631i	$0.255231\pi$
$828$	18.6632	0.648591
$829$	36.8620	1.28027	0.640134	−	0.768263i	$-0.278879\pi$
$829$	36.8620	1.28027	0.640134	+	0.768263i	$0.278879\pi$
$830$	0	0
$831$	5.19708	0.180285
$832$	7.12558	0.247035
$833$	−33.0197	−1.14406
$834$	3.26471	0.113048
$835$	0	0
$836$	−1.69256	−0.0585385
$837$	−19.1719	−0.662679
$838$	−23.3149	−0.805400
$839$	36.7629	1.26919	0.634597	−	0.772843i	$-0.281166\pi$
$839$	36.7629	1.26919	0.634597	+	0.772843i	$0.281166\pi$
$840$	0	0
$841$	−27.6066	−0.951951
$842$	−24.0289	−0.828089
$843$	−7.27991	−0.250733
$844$	−4.81998	−0.165910
$845$	0	0
$846$	16.4467	0.565450
$847$	38.3514	1.31777
$848$	9.32034	0.320062
$849$	−0.748340	−0.0256830
$850$	0	0
$851$	6.45973	0.221437
$852$	4.49433	0.153973
$853$	−15.9547	−0.546278	−0.273139	−	0.961975i	$-0.588062\pi$
$853$	−15.9547	−0.546278	−0.273139	+	0.961975i	$0.588062\pi$
$854$	−4.48654	−0.153526
$855$	0	0
$856$	−10.0279	−0.342748
$857$	10.2157	0.348963	0.174481	−	0.984660i	$-0.444175\pi$
$857$	10.2157	0.348963	0.174481	+	0.984660i	$0.444175\pi$
$858$	0.688066	0.0234902
$859$	7.41899	0.253133	0.126566	−	0.991958i	$-0.459604\pi$
$859$	7.41899	0.253133	0.126566	+	0.991958i	$0.459604\pi$
$860$	0	0
$861$	−1.91918	−0.0654055
$862$	19.2917	0.657077
$863$	−9.35839	−0.318563	−0.159282	−	0.987233i	$-0.550918\pi$
$863$	−9.35839	−0.318563	−0.159282	+	0.987233i	$0.550918\pi$
$864$	1.96065	0.0667025
$865$	0	0
$866$	12.5816	0.427541
$867$	7.05220	0.239505
$868$	−34.3549	−1.16608
$869$	−2.39409	−0.0812139
$870$	0	0
$871$	43.0623	1.45911
$872$	−4.87361	−0.165041
$873$	−43.7050	−1.47919
$874$	37.6959	1.27508
$875$	0	0
$876$	0.570460	0.0192741
$877$	13.1486	0.443997	0.221998	−	0.975047i	$-0.428742\pi$
$877$	13.1486	0.443997	0.221998	+	0.975047i	$0.428742\pi$
$878$	−14.0428	−0.473921
$879$	−8.94882	−0.301836
$880$	0	0
$881$	26.7781	0.902178	0.451089	−	0.892479i	$-0.351036\pi$
$881$	26.7781	0.902178	0.451089	+	0.892479i	$0.351036\pi$
$882$	15.4387	0.519849
$883$	16.7245	0.562824	0.281412	−	0.959587i	$-0.409197\pi$
$883$	16.7245	0.562824	0.281412	+	0.959587i	$0.409197\pi$
$884$	−44.0304	−1.48090
$885$	0	0
$886$	24.5696	0.825432
$887$	−18.1469	−0.609312	−0.304656	−	0.952462i	$-0.598542\pi$
$887$	−18.1469	−0.609312	−0.304656	+	0.952462i	$0.598542\pi$
$888$	0.332924	0.0111722
$889$	−4.61181	−0.154675
$890$	0	0
$891$	−2.32463	−0.0778780
$892$	3.70392	0.124017
$893$	33.2191	1.11164
$894$	3.02191	0.101068
$895$	0	0
$896$	3.51336	0.117373
$897$	−15.3243	−0.511663
$898$	12.7574	0.425721
$899$	−11.5427	−0.384971
$900$	0	0
$901$	−57.5923	−1.91868
$902$	0.475896	0.0158456
$903$	6.25650	0.208203
$904$	15.9290	0.529790
$905$	0	0
$906$	−6.72387	−0.223386
$907$	−50.1481	−1.66514	−0.832570	−	0.553920i	$-0.813131\pi$
$907$	−50.1481	−1.66514	−0.832570	+	0.553920i	$0.813131\pi$
$908$	6.31512	0.209575
$909$	−7.61958	−0.252725
$910$	0	0
$911$	2.64691	0.0876960	0.0438480	−	0.999038i	$-0.486038\pi$
$911$	2.64691	0.0876960	0.0438480	+	0.999038i	$0.486038\pi$
$912$	1.94279	0.0643322
$913$	−0.701348	−0.0232112
$914$	−3.34889	−0.110771
$915$	0	0
$916$	−20.5548	−0.679150
$917$	−8.48693	−0.280263
$918$	−12.1152	−0.399862
$919$	−39.5369	−1.30420	−0.652101	−	0.758132i	$-0.726112\pi$
$919$	−39.5369	−1.30420	−0.652101	+	0.758132i	$0.726112\pi$
$920$	0	0
$921$	−7.98947	−0.263262
$922$	0.859501	0.0283062
$923$	96.1920	3.16620
$924$	0.339260	0.0111608
$925$	0	0
$926$	−13.8711	−0.455831
$927$	4.98292	0.163660
$928$	1.18043	0.0387496
$929$	−39.2999	−1.28939	−0.644693	−	0.764441i	$-0.723015\pi$
$929$	−39.2999	−1.28939	−0.644693	+	0.764441i	$0.723015\pi$
$930$	0	0
$931$	31.1832	1.02199
$932$	21.4393	0.702267
$933$	−1.68980	−0.0553216
$934$	−40.9281	−1.33921
$935$	0	0
$936$	20.5869	0.672905
$937$	27.7584	0.906826	0.453413	−	0.891301i	$-0.350206\pi$
$937$	27.7584	0.906826	0.453413	+	0.891301i	$0.350206\pi$
$938$	21.2324	0.693263
$939$	2.98617	0.0974499
$940$	0	0
$941$	48.2212	1.57197	0.785983	−	0.618249i	$-0.212158\pi$
$941$	48.2212	1.57197	0.785983	+	0.618249i	$0.212158\pi$
$942$	5.74885	0.187308
$943$	−10.5989	−0.345149
$944$	5.94279	0.193421
$945$	0	0
$946$	−1.55141	−0.0504408
$947$	26.5111	0.861495	0.430748	−	0.902472i	$-0.358250\pi$
$947$	26.5111	0.861495	0.430748	+	0.902472i	$0.358250\pi$
$948$	2.74803	0.0892519
$949$	12.2095	0.396339
$950$	0	0
$951$	7.33037	0.237703
$952$	−21.7098	−0.703617
$953$	31.8487	1.03168	0.515840	−	0.856685i	$-0.327480\pi$
$953$	31.8487	1.03168	0.515840	+	0.856685i	$0.327480\pi$
$954$	26.9280	0.871825
$955$	0	0
$956$	−20.7479	−0.671034
$957$	0.113986	0.00368464
$958$	−8.58990	−0.277527
$959$	−24.1496	−0.779832
$960$	0	0
$961$	64.6166	2.08441
$962$	7.12558	0.229738
$963$	−28.9723	−0.933620
$964$	10.7731	0.346978
$965$	0	0
$966$	−7.55583	−0.243105
$967$	−19.4699	−0.626109	−0.313054	−	0.949735i	$-0.601352\pi$
$967$	−19.4699	−0.626109	−0.313054	+	0.949735i	$0.601352\pi$
$968$	10.9159	0.350849
$969$	−12.0049	−0.385653
$970$	0	0
$971$	−21.8178	−0.700168	−0.350084	−	0.936718i	$-0.613847\pi$
$971$	−21.8178	−0.700168	−0.350084	+	0.936718i	$0.613847\pi$
$972$	8.55024	0.274249
$973$	34.4525	1.10450
$974$	−25.8389	−0.827932
$975$	0	0
$976$	−1.27699	−0.0408756
$977$	41.8705	1.33956	0.669779	−	0.742561i	$-0.266389\pi$
$977$	41.8705	1.33956	0.669779	+	0.742561i	$0.266389\pi$
$978$	−5.55654	−0.177679
$979$	2.98157	0.0952913
$980$	0	0
$981$	−14.0806	−0.449560
$982$	−11.4864	−0.366547
$983$	37.0175	1.18067	0.590337	−	0.807157i	$-0.298995\pi$
$983$	37.0175	1.18067	0.590337	+	0.807157i	$0.298995\pi$
$984$	−0.546253	−0.0174139
$985$	0	0
$986$	−7.29413	−0.232292
$987$	−6.65849	−0.211942
$988$	41.5815	1.32288
$989$	34.5524	1.09870
$990$	0	0
$991$	−28.4749	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$991$	−28.4749	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$992$	−9.77838	−0.310464
$993$	−3.56677	−0.113188
$994$	47.4287	1.50435
$995$	0	0
$996$	0.805036	0.0255085
$997$	14.2442	0.451118	0.225559	−	0.974229i	$-0.427579\pi$
$997$	14.2442	0.451118	0.225559	+	0.974229i	$0.427579\pi$
$998$	9.74599	0.308504
$999$	1.96065	0.0620321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bd.1.3		5
5.2	odd	4		370.2.b.d.149.3	✓	10
5.3	odd	4		370.2.b.d.149.8	yes	10
5.4	even	2		1850.2.a.be.1.3		5
15.2	even	4		3330.2.d.p.1999.9		10
15.8	even	4		3330.2.d.p.1999.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.d.149.3	✓	10	5.2	odd	4
370.2.b.d.149.8	yes	10	5.3	odd	4
1850.2.a.bd.1.3		5	1.1	even	1	trivial
1850.2.a.be.1.3		5	5.4	even	2
3330.2.d.p.1999.4		10	15.8	even	4
3330.2.d.p.1999.9		10	15.2	even	4

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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.332924 q^{3} +1.00000 q^{4} -0.332924 q^{6} -3.51336 q^{7} -1.00000 q^{8} -2.88916 q^{9} -0.290044 q^{11} +0.332924 q^{12} +7.12558 q^{13} +3.51336 q^{14} +1.00000 q^{16} -6.17921 q^{17} +2.88916 q^{18} +5.83553 q^{19} -1.16968 q^{21} +0.290044 q^{22} -6.45973 q^{23} -0.332924 q^{24} -7.12558 q^{26} -1.96065 q^{27} -3.51336 q^{28} -1.18043 q^{29} +9.77838 q^{31} -1.00000 q^{32} -0.0965628 q^{33} +6.17921 q^{34} -2.88916 q^{36} -1.00000 q^{37} -5.83553 q^{38} +2.37228 q^{39} +1.64077 q^{41} +1.16968 q^{42} -5.34889 q^{43} -0.290044 q^{44} +6.45973 q^{46} +5.69256 q^{47} +0.332924 q^{48} +5.34367 q^{49} -2.05721 q^{51} +7.12558 q^{52} +9.32034 q^{53} +1.96065 q^{54} +3.51336 q^{56} +1.94279 q^{57} +1.18043 q^{58} +5.94279 q^{59} -1.27699 q^{61} -9.77838 q^{62} +10.1507 q^{63} +1.00000 q^{64} +0.0965628 q^{66} +6.04334 q^{67} -6.17921 q^{68} -2.15060 q^{69} +13.4995 q^{71} +2.88916 q^{72} +1.71348 q^{73} +1.00000 q^{74} +5.83553 q^{76} +1.01903 q^{77} -2.37228 q^{78} +8.25422 q^{79} +8.01474 q^{81} -1.64077 q^{82} +2.41807 q^{83} -1.16968 q^{84} +5.34889 q^{86} -0.392995 q^{87} +0.290044 q^{88} -10.2797 q^{89} -25.0347 q^{91} -6.45973 q^{92} +3.25546 q^{93} -5.69256 q^{94} -0.332924 q^{96} +15.1272 q^{97} -5.34367 q^{98} +0.837984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31}+ \cdots - 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

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Groups

Database

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