LMFDB - Embedded newform 1850.2.a.r.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.44949 q^{3} +1.00000 q^{4} -1.44949 q^{6} -2.44949 q^{7} -1.00000 q^{8} -0.898979 q^{9} +3.44949 q^{11} +1.44949 q^{12} +0.449490 q^{13} +2.44949 q^{14} +1.00000 q^{16} -3.44949 q^{17} +0.898979 q^{18} -5.00000 q^{19} -3.55051 q^{21} -3.44949 q^{22} -2.00000 q^{23} -1.44949 q^{24} -0.449490 q^{26} -5.65153 q^{27} -2.44949 q^{28} -0.898979 q^{29} +4.44949 q^{31} -1.00000 q^{32} +5.00000 q^{33} +3.44949 q^{34} -0.898979 q^{36} +1.00000 q^{37} +5.00000 q^{38} +0.651531 q^{39} +1.00000 q^{41} +3.55051 q^{42} +1.10102 q^{43} +3.44949 q^{44} +2.00000 q^{46} -9.79796 q^{47} +1.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} +0.449490 q^{52} -6.00000 q^{53} +5.65153 q^{54} +2.44949 q^{56} -7.24745 q^{57} +0.898979 q^{58} -2.00000 q^{59} -6.44949 q^{61} -4.44949 q^{62} +2.20204 q^{63} +1.00000 q^{64} -5.00000 q^{66} -4.55051 q^{67} -3.44949 q^{68} -2.89898 q^{69} -7.55051 q^{71} +0.898979 q^{72} -12.7980 q^{73} -1.00000 q^{74} -5.00000 q^{76} -8.44949 q^{77} -0.651531 q^{78} +7.79796 q^{79} -5.49490 q^{81} -1.00000 q^{82} -3.44949 q^{83} -3.55051 q^{84} -1.10102 q^{86} -1.30306 q^{87} -3.44949 q^{88} +14.3485 q^{89} -1.10102 q^{91} -2.00000 q^{92} +6.44949 q^{93} +9.79796 q^{94} -1.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -3.10102 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * 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
$3$	1.44949	0.836863	0.418432	−	0.908248i	$-0.362580\pi$
$3$	1.44949	0.836863	0.418432	+	0.908248i	$0.362580\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.44949	−0.591752
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	−1.00000	−0.353553
$9$	−0.898979	−0.299660
$10$	0	0
$11$	3.44949	1.04006	0.520030	−	0.854148i	$-0.325921\pi$
$11$	3.44949	1.04006	0.520030	+	0.854148i	$0.325921\pi$
$12$	1.44949	0.418432
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	2.44949	0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.44949	−0.836624	−0.418312	−	0.908303i	$-0.637378\pi$
$17$	−3.44949	−0.836624	−0.418312	+	0.908303i	$0.637378\pi$
$18$	0.898979	0.211891
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−3.55051	−0.774785
$22$	−3.44949	−0.735434
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−1.44949	−0.295876
$25$	0	0
$26$	−0.449490	−0.0881522
$27$	−5.65153	−1.08764
$28$	−2.44949	−0.462910
$29$	−0.898979	−0.166936	−0.0834681	−	0.996510i	$-0.526600\pi$
$29$	−0.898979	−0.166936	−0.0834681	+	0.996510i	$0.526600\pi$
$30$	0	0
$31$	4.44949	0.799152	0.399576	−	0.916700i	$-0.369157\pi$
$31$	4.44949	0.799152	0.399576	+	0.916700i	$0.369157\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	3.44949	0.591583
$35$	0	0
$36$	−0.898979	−0.149830
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	5.00000	0.811107
$39$	0.651531	0.104328
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	3.55051	0.547856
$43$	1.10102	0.167904	0.0839520	−	0.996470i	$-0.473246\pi$
$43$	1.10102	0.167904	0.0839520	+	0.996470i	$0.473246\pi$
$44$	3.44949	0.520030
$45$	0	0
$46$	2.00000	0.294884
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	1.44949	0.209216
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0.449490	0.0623330
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	5.65153	0.769076
$55$	0	0
$56$	2.44949	0.327327
$57$	−7.24745	−0.959948
$58$	0.898979	0.118042
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−6.44949	−0.825773	−0.412886	−	0.910783i	$-0.635479\pi$
$61$	−6.44949	−0.825773	−0.412886	+	0.910783i	$0.635479\pi$
$62$	−4.44949	−0.565086
$63$	2.20204	0.277431
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	−4.55051	−0.555933	−0.277967	−	0.960591i	$-0.589660\pi$
$67$	−4.55051	−0.555933	−0.277967	+	0.960591i	$0.589660\pi$
$68$	−3.44949	−0.418312
$69$	−2.89898	−0.348996
$70$	0	0
$71$	−7.55051	−0.896081	−0.448040	−	0.894013i	$-0.647878\pi$
$71$	−7.55051	−0.896081	−0.448040	+	0.894013i	$0.647878\pi$
$72$	0.898979	0.105946
$73$	−12.7980	−1.49789	−0.748944	−	0.662633i	$-0.769439\pi$
$73$	−12.7980	−1.49789	−0.748944	+	0.662633i	$0.769439\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−8.44949	−0.962909
$78$	−0.651531	−0.0737713
$79$	7.79796	0.877339	0.438669	−	0.898648i	$-0.355450\pi$
$79$	7.79796	0.877339	0.438669	+	0.898648i	$0.355450\pi$
$80$	0	0
$81$	−5.49490	−0.610544
$82$	−1.00000	−0.110432
$83$	−3.44949	−0.378631	−0.189315	−	0.981916i	$-0.560627\pi$
$83$	−3.44949	−0.378631	−0.189315	+	0.981916i	$0.560627\pi$
$84$	−3.55051	−0.387392
$85$	0	0
$86$	−1.10102	−0.118726
$87$	−1.30306	−0.139703
$88$	−3.44949	−0.367717
$89$	14.3485	1.52093	0.760467	−	0.649376i	$-0.224970\pi$
$89$	14.3485	1.52093	0.760467	+	0.649376i	$0.224970\pi$
$90$	0	0
$91$	−1.10102	−0.115418
$92$	−2.00000	−0.208514
$93$	6.44949	0.668781
$94$	9.79796	1.01058
$95$	0	0
$96$	−1.44949	−0.147938
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	1.00000	0.101015
$99$	−3.10102	−0.311664
$100$	0	0
$101$	1.10102	0.109556	0.0547778	−	0.998499i	$-0.482555\pi$
$101$	1.10102	0.109556	0.0547778	+	0.998499i	$0.482555\pi$
$102$	5.00000	0.495074
$103$	−6.44949	−0.635487	−0.317744	−	0.948177i	$-0.602925\pi$
$103$	−6.44949	−0.635487	−0.317744	+	0.948177i	$0.602925\pi$
$104$	−0.449490	−0.0440761
$105$	0	0
$106$	6.00000	0.582772
$107$	8.55051	0.826609	0.413305	−	0.910593i	$-0.364375\pi$
$107$	8.55051	0.826609	0.413305	+	0.910593i	$0.364375\pi$
$108$	−5.65153	−0.543819
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	1.44949	0.137579
$112$	−2.44949	−0.231455
$113$	4.55051	0.428076	0.214038	−	0.976825i	$-0.431338\pi$
$113$	4.55051	0.428076	0.214038	+	0.976825i	$0.431338\pi$
$114$	7.24745	0.678786
$115$	0	0
$116$	−0.898979	−0.0834681
$117$	−0.404082	−0.0373574
$118$	2.00000	0.184115
$119$	8.44949	0.774563
$120$	0	0
$121$	0.898979	0.0817254
$122$	6.44949	0.583909
$123$	1.44949	0.130696
$124$	4.44949	0.399576
$125$	0	0
$126$	−2.20204	−0.196173
$127$	16.2474	1.44173	0.720864	−	0.693077i	$-0.243745\pi$
$127$	16.2474	1.44173	0.720864	+	0.693077i	$0.243745\pi$
$128$	−1.00000	−0.0883883
$129$	1.59592	0.140513
$130$	0	0
$131$	8.69694	0.759855	0.379928	−	0.925016i	$-0.375949\pi$
$131$	8.69694	0.759855	0.379928	+	0.925016i	$0.375949\pi$
$132$	5.00000	0.435194
$133$	12.2474	1.06199
$134$	4.55051	0.393104
$135$	0	0
$136$	3.44949	0.295791
$137$	−9.69694	−0.828465	−0.414233	−	0.910171i	$-0.635950\pi$
$137$	−9.69694	−0.828465	−0.414233	+	0.910171i	$0.635950\pi$
$138$	2.89898	0.246778
$139$	−12.5505	−1.06452	−0.532260	−	0.846581i	$-0.678657\pi$
$139$	−12.5505	−1.06452	−0.532260	+	0.846581i	$0.678657\pi$
$140$	0	0
$141$	−14.2020	−1.19603
$142$	7.55051	0.633625
$143$	1.55051	0.129660
$144$	−0.898979	−0.0749150
$145$	0	0
$146$	12.7980	1.05917
$147$	−1.44949	−0.119552
$148$	1.00000	0.0821995
$149$	−4.65153	−0.381068	−0.190534	−	0.981681i	$-0.561022\pi$
$149$	−4.65153	−0.381068	−0.190534	+	0.981681i	$0.561022\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	5.00000	0.405554
$153$	3.10102	0.250703
$154$	8.44949	0.680879
$155$	0	0
$156$	0.651531	0.0521642
$157$	−16.4495	−1.31281	−0.656406	−	0.754408i	$-0.727924\pi$
$157$	−16.4495	−1.31281	−0.656406	+	0.754408i	$0.727924\pi$
$158$	−7.79796	−0.620372
$159$	−8.69694	−0.689712
$160$	0	0
$161$	4.89898	0.386094
$162$	5.49490	0.431720
$163$	−10.1010	−0.791173	−0.395586	−	0.918429i	$-0.629459\pi$
$163$	−10.1010	−0.791173	−0.395586	+	0.918429i	$0.629459\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	3.44949	0.267732
$167$	10.4495	0.808606	0.404303	−	0.914625i	$-0.367514\pi$
$167$	10.4495	0.808606	0.404303	+	0.914625i	$0.367514\pi$
$168$	3.55051	0.273928
$169$	−12.7980	−0.984458
$170$	0	0
$171$	4.49490	0.343733
$172$	1.10102	0.0839520
$173$	6.69694	0.509159	0.254579	−	0.967052i	$-0.418063\pi$
$173$	6.69694	0.509159	0.254579	+	0.967052i	$0.418063\pi$
$174$	1.30306	0.0987848
$175$	0	0
$176$	3.44949	0.260015
$177$	−2.89898	−0.217901
$178$	−14.3485	−1.07546
$179$	−0.797959	−0.0596423	−0.0298211	−	0.999555i	$-0.509494\pi$
$179$	−0.797959	−0.0596423	−0.0298211	+	0.999555i	$0.509494\pi$
$180$	0	0
$181$	6.89898	0.512797	0.256399	−	0.966571i	$-0.417464\pi$
$181$	6.89898	0.512797	0.256399	+	0.966571i	$0.417464\pi$
$182$	1.10102	0.0816131
$183$	−9.34847	−0.691059
$184$	2.00000	0.147442
$185$	0	0
$186$	−6.44949	−0.472900
$187$	−11.8990	−0.870140
$188$	−9.79796	−0.714590
$189$	13.8434	1.00696
$190$	0	0
$191$	−7.79796	−0.564241	−0.282120	−	0.959379i	$-0.591038\pi$
$191$	−7.79796	−0.564241	−0.282120	+	0.959379i	$0.591038\pi$
$192$	1.44949	0.104608
$193$	2.55051	0.183590	0.0917949	−	0.995778i	$-0.470740\pi$
$193$	2.55051	0.183590	0.0917949	+	0.995778i	$0.470740\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	17.3485	1.23603	0.618014	−	0.786167i	$-0.287938\pi$
$197$	17.3485	1.23603	0.618014	+	0.786167i	$0.287938\pi$
$198$	3.10102	0.220380
$199$	−15.5959	−1.10557	−0.552783	−	0.833325i	$-0.686434\pi$
$199$	−15.5959	−1.10557	−0.552783	+	0.833325i	$0.686434\pi$
$200$	0	0
$201$	−6.59592	−0.465240
$202$	−1.10102	−0.0774675
$203$	2.20204	0.154553
$204$	−5.00000	−0.350070
$205$	0	0
$206$	6.44949	0.449357
$207$	1.79796	0.124967
$208$	0.449490	0.0311665
$209$	−17.2474	−1.19303
$210$	0	0
$211$	−4.55051	−0.313270	−0.156635	−	0.987657i	$-0.550065\pi$
$211$	−4.55051	−0.313270	−0.156635	+	0.987657i	$0.550065\pi$
$212$	−6.00000	−0.412082
$213$	−10.9444	−0.749897
$214$	−8.55051	−0.584501
$215$	0	0
$216$	5.65153	0.384538
$217$	−10.8990	−0.739871
$218$	−14.0000	−0.948200
$219$	−18.5505	−1.25353
$220$	0	0
$221$	−1.55051	−0.104299
$222$	−1.44949	−0.0972834
$223$	−21.7980	−1.45970	−0.729850	−	0.683608i	$-0.760410\pi$
$223$	−21.7980	−1.45970	−0.729850	+	0.683608i	$0.760410\pi$
$224$	2.44949	0.163663
$225$	0	0
$226$	−4.55051	−0.302695
$227$	16.6969	1.10821	0.554107	−	0.832445i	$-0.313060\pi$
$227$	16.6969	1.10821	0.554107	+	0.832445i	$0.313060\pi$
$228$	−7.24745	−0.479974
$229$	5.79796	0.383140	0.191570	−	0.981479i	$-0.438642\pi$
$229$	5.79796	0.383140	0.191570	+	0.981479i	$0.438642\pi$
$230$	0	0
$231$	−12.2474	−0.805823
$232$	0.898979	0.0590209
$233$	28.4949	1.86676	0.933381	−	0.358886i	$-0.116843\pi$
$233$	28.4949	1.86676	0.933381	+	0.358886i	$0.116843\pi$
$234$	0.404082	0.0264157
$235$	0	0
$236$	−2.00000	−0.130189
$237$	11.3031	0.734213
$238$	−8.44949	−0.547699
$239$	0.898979	0.0581501	0.0290751	−	0.999577i	$-0.490744\pi$
$239$	0.898979	0.0581501	0.0290751	+	0.999577i	$0.490744\pi$
$240$	0	0
$241$	−2.55051	−0.164293	−0.0821464	−	0.996620i	$-0.526178\pi$
$241$	−2.55051	−0.164293	−0.0821464	+	0.996620i	$0.526178\pi$
$242$	−0.898979	−0.0577886
$243$	8.98979	0.576696
$244$	−6.44949	−0.412886
$245$	0	0
$246$	−1.44949	−0.0924161
$247$	−2.24745	−0.143002
$248$	−4.44949	−0.282543
$249$	−5.00000	−0.316862
$250$	0	0
$251$	−30.7980	−1.94395	−0.971975	−	0.235084i	$-0.924463\pi$
$251$	−30.7980	−1.94395	−0.971975	+	0.235084i	$0.924463\pi$
$252$	2.20204	0.138716
$253$	−6.89898	−0.433735
$254$	−16.2474	−1.01946
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.89898	0.430347	0.215173	−	0.976576i	$-0.430968\pi$
$257$	6.89898	0.430347	0.215173	+	0.976576i	$0.430968\pi$
$258$	−1.59592	−0.0993575
$259$	−2.44949	−0.152204
$260$	0	0
$261$	0.808164	0.0500241
$262$	−8.69694	−0.537299
$263$	−0.202041	−0.0124584	−0.00622919	−	0.999981i	$-0.501983\pi$
$263$	−0.202041	−0.0124584	−0.00622919	+	0.999981i	$0.501983\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	−12.2474	−0.750939
$267$	20.7980	1.27281
$268$	−4.55051	−0.277967
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	30.0454	1.82513	0.912564	−	0.408933i	$-0.134099\pi$
$271$	30.0454	1.82513	0.912564	+	0.408933i	$0.134099\pi$
$272$	−3.44949	−0.209156
$273$	−1.59592	−0.0965893
$274$	9.69694	0.585813
$275$	0	0
$276$	−2.89898	−0.174498
$277$	−5.79796	−0.348366	−0.174183	−	0.984713i	$-0.555728\pi$
$277$	−5.79796	−0.348366	−0.174183	+	0.984713i	$0.555728\pi$
$278$	12.5505	0.752730
$279$	−4.00000	−0.239474
$280$	0	0
$281$	8.89898	0.530869	0.265434	−	0.964129i	$-0.414485\pi$
$281$	8.89898	0.530869	0.265434	+	0.964129i	$0.414485\pi$
$282$	14.2020	0.845719
$283$	11.6969	0.695311	0.347655	−	0.937622i	$-0.386978\pi$
$283$	11.6969	0.695311	0.347655	+	0.937622i	$0.386978\pi$
$284$	−7.55051	−0.448040
$285$	0	0
$286$	−1.55051	−0.0916836
$287$	−2.44949	−0.144589
$288$	0.898979	0.0529729
$289$	−5.10102	−0.300060
$290$	0	0
$291$	−20.2929	−1.18959
$292$	−12.7980	−0.748944
$293$	28.0454	1.63843	0.819215	−	0.573486i	$-0.194409\pi$
$293$	28.0454	1.63843	0.819215	+	0.573486i	$0.194409\pi$
$294$	1.44949	0.0845360
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	−19.4949	−1.13121
$298$	4.65153	0.269456
$299$	−0.898979	−0.0519893
$300$	0	0
$301$	−2.69694	−0.155449
$302$	14.0000	0.805609
$303$	1.59592	0.0916831
$304$	−5.00000	−0.286770
$305$	0	0
$306$	−3.10102	−0.177274
$307$	−7.65153	−0.436696	−0.218348	−	0.975871i	$-0.570067\pi$
$307$	−7.65153	−0.436696	−0.218348	+	0.975871i	$0.570067\pi$
$308$	−8.44949	−0.481454
$309$	−9.34847	−0.531816
$310$	0	0
$311$	−8.44949	−0.479127	−0.239563	−	0.970881i	$-0.577004\pi$
$311$	−8.44949	−0.479127	−0.239563	+	0.970881i	$0.577004\pi$
$312$	−0.651531	−0.0368857
$313$	−8.89898	−0.503000	−0.251500	−	0.967857i	$-0.580924\pi$
$313$	−8.89898	−0.503000	−0.251500	+	0.967857i	$0.580924\pi$
$314$	16.4495	0.928298
$315$	0	0
$316$	7.79796	0.438669
$317$	9.55051	0.536410	0.268205	−	0.963362i	$-0.413570\pi$
$317$	9.55051	0.536410	0.268205	+	0.963362i	$0.413570\pi$
$318$	8.69694	0.487700
$319$	−3.10102	−0.173624
$320$	0	0
$321$	12.3939	0.691759
$322$	−4.89898	−0.273009
$323$	17.2474	0.959674
$324$	−5.49490	−0.305272
$325$	0	0
$326$	10.1010	0.559444
$327$	20.2929	1.12220
$328$	−1.00000	−0.0552158
$329$	24.0000	1.32316
$330$	0	0
$331$	19.8990	1.09375	0.546873	−	0.837215i	$-0.315818\pi$
$331$	19.8990	1.09375	0.546873	+	0.837215i	$0.315818\pi$
$332$	−3.44949	−0.189315
$333$	−0.898979	−0.0492638
$334$	−10.4495	−0.571771
$335$	0	0
$336$	−3.55051	−0.193696
$337$	−15.6969	−0.855067	−0.427533	−	0.904000i	$-0.640617\pi$
$337$	−15.6969	−0.855067	−0.427533	+	0.904000i	$0.640617\pi$
$338$	12.7980	0.696117
$339$	6.59592	0.358241
$340$	0	0
$341$	15.3485	0.831166
$342$	−4.49490	−0.243056
$343$	19.5959	1.05808
$344$	−1.10102	−0.0593630
$345$	0	0
$346$	−6.69694	−0.360030
$347$	18.7980	1.00913	0.504564	−	0.863374i	$-0.331653\pi$
$347$	18.7980	1.00913	0.504564	+	0.863374i	$0.331653\pi$
$348$	−1.30306	−0.0698514
$349$	−12.4495	−0.666406	−0.333203	−	0.942855i	$-0.608129\pi$
$349$	−12.4495	−0.666406	−0.333203	+	0.942855i	$0.608129\pi$
$350$	0	0
$351$	−2.54031	−0.135591
$352$	−3.44949	−0.183858
$353$	−14.8990	−0.792993	−0.396496	−	0.918036i	$-0.629774\pi$
$353$	−14.8990	−0.792993	−0.396496	+	0.918036i	$0.629774\pi$
$354$	2.89898	0.154079
$355$	0	0
$356$	14.3485	0.760467
$357$	12.2474	0.648204
$358$	0.797959	0.0421734
$359$	−31.3939	−1.65691	−0.828453	−	0.560059i	$-0.810778\pi$
$359$	−31.3939	−1.65691	−0.828453	+	0.560059i	$0.810778\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−6.89898	−0.362602
$363$	1.30306	0.0683930
$364$	−1.10102	−0.0577092
$365$	0	0
$366$	9.34847	0.488652
$367$	14.2474	0.743711	0.371855	−	0.928291i	$-0.378722\pi$
$367$	14.2474	0.743711	0.371855	+	0.928291i	$0.378722\pi$
$368$	−2.00000	−0.104257
$369$	−0.898979	−0.0467990
$370$	0	0
$371$	14.6969	0.763027
$372$	6.44949	0.334390
$373$	20.0454	1.03791	0.518956	−	0.854801i	$-0.326321\pi$
$373$	20.0454	1.03791	0.518956	+	0.854801i	$0.326321\pi$
$374$	11.8990	0.615282
$375$	0	0
$376$	9.79796	0.505291
$377$	−0.404082	−0.0208113
$378$	−13.8434	−0.712026
$379$	−21.0454	−1.08103	−0.540515	−	0.841334i	$-0.681771\pi$
$379$	−21.0454	−1.08103	−0.540515	+	0.841334i	$0.681771\pi$
$380$	0	0
$381$	23.5505	1.20653
$382$	7.79796	0.398978
$383$	5.34847	0.273294	0.136647	−	0.990620i	$-0.456367\pi$
$383$	5.34847	0.273294	0.136647	+	0.990620i	$0.456367\pi$
$384$	−1.44949	−0.0739690
$385$	0	0
$386$	−2.55051	−0.129818
$387$	−0.989795	−0.0503141
$388$	−14.0000	−0.710742
$389$	13.1464	0.666550	0.333275	−	0.942830i	$-0.391846\pi$
$389$	13.1464	0.666550	0.333275	+	0.942830i	$0.391846\pi$
$390$	0	0
$391$	6.89898	0.348896
$392$	1.00000	0.0505076
$393$	12.6061	0.635895
$394$	−17.3485	−0.874003
$395$	0	0
$396$	−3.10102	−0.155832
$397$	15.3485	0.770318	0.385159	−	0.922850i	$-0.374147\pi$
$397$	15.3485	0.770318	0.385159	+	0.922850i	$0.374147\pi$
$398$	15.5959	0.781753
$399$	17.7526	0.888739
$400$	0	0
$401$	15.9444	0.796225	0.398112	−	0.917337i	$-0.369665\pi$
$401$	15.9444	0.796225	0.398112	+	0.917337i	$0.369665\pi$
$402$	6.59592	0.328974
$403$	2.00000	0.0996271
$404$	1.10102	0.0547778
$405$	0	0
$406$	−2.20204	−0.109285
$407$	3.44949	0.170985
$408$	5.00000	0.247537
$409$	−34.1464	−1.68843	−0.844216	−	0.536003i	$-0.819934\pi$
$409$	−34.1464	−1.68843	−0.844216	+	0.536003i	$0.819934\pi$
$410$	0	0
$411$	−14.0556	−0.693312
$412$	−6.44949	−0.317744
$413$	4.89898	0.241063
$414$	−1.79796	−0.0883649
$415$	0	0
$416$	−0.449490	−0.0220380
$417$	−18.1918	−0.890858
$418$	17.2474	0.843600
$419$	−21.4495	−1.04788	−0.523938	−	0.851756i	$-0.675538\pi$
$419$	−21.4495	−1.04788	−0.523938	+	0.851756i	$0.675538\pi$
$420$	0	0
$421$	32.8990	1.60340	0.801699	−	0.597728i	$-0.203930\pi$
$421$	32.8990	1.60340	0.801699	+	0.597728i	$0.203930\pi$
$422$	4.55051	0.221515
$423$	8.80816	0.428268
$424$	6.00000	0.291386
$425$	0	0
$426$	10.9444	0.530257
$427$	15.7980	0.764517
$428$	8.55051	0.413305
$429$	2.24745	0.108508
$430$	0	0
$431$	−20.6969	−0.996936	−0.498468	−	0.866908i	$-0.666104\pi$
$431$	−20.6969	−0.996936	−0.498468	+	0.866908i	$0.666104\pi$
$432$	−5.65153	−0.271909
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	10.8990	0.523168
$435$	0	0
$436$	14.0000	0.670478
$437$	10.0000	0.478365
$438$	18.5505	0.886378
$439$	−19.5959	−0.935262	−0.467631	−	0.883924i	$-0.654892\pi$
$439$	−19.5959	−0.935262	−0.467631	+	0.883924i	$0.654892\pi$
$440$	0	0
$441$	0.898979	0.0428085
$442$	1.55051	0.0737503
$443$	17.4495	0.829050	0.414525	−	0.910038i	$-0.363948\pi$
$443$	17.4495	0.829050	0.414525	+	0.910038i	$0.363948\pi$
$444$	1.44949	0.0687897
$445$	0	0
$446$	21.7980	1.03216
$447$	−6.74235	−0.318902
$448$	−2.44949	−0.115728
$449$	33.2474	1.56904	0.784522	−	0.620101i	$-0.212908\pi$
$449$	33.2474	1.56904	0.784522	+	0.620101i	$0.212908\pi$
$450$	0	0
$451$	3.44949	0.162430
$452$	4.55051	0.214038
$453$	−20.2929	−0.953442
$454$	−16.6969	−0.783626
$455$	0	0
$456$	7.24745	0.339393
$457$	−15.2474	−0.713246	−0.356623	−	0.934248i	$-0.616072\pi$
$457$	−15.2474	−0.713246	−0.356623	+	0.934248i	$0.616072\pi$
$458$	−5.79796	−0.270921
$459$	19.4949	0.909944
$460$	0	0
$461$	9.30306	0.433287	0.216643	−	0.976251i	$-0.430489\pi$
$461$	9.30306	0.433287	0.216643	+	0.976251i	$0.430489\pi$
$462$	12.2474	0.569803
$463$	−9.55051	−0.443850	−0.221925	−	0.975064i	$-0.571234\pi$
$463$	−9.55051	−0.443850	−0.221925	+	0.975064i	$0.571234\pi$
$464$	−0.898979	−0.0417341
$465$	0	0
$466$	−28.4949	−1.32000
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−0.404082	−0.0186787
$469$	11.1464	0.514694
$470$	0	0
$471$	−23.8434	−1.09864
$472$	2.00000	0.0920575
$473$	3.79796	0.174630
$474$	−11.3031	−0.519167
$475$	0	0
$476$	8.44949	0.387282
$477$	5.39388	0.246969
$478$	−0.898979	−0.0411184
$479$	6.24745	0.285453	0.142727	−	0.989762i	$-0.454413\pi$
$479$	6.24745	0.285453	0.142727	+	0.989762i	$0.454413\pi$
$480$	0	0
$481$	0.449490	0.0204950
$482$	2.55051	0.116173
$483$	7.10102	0.323108
$484$	0.898979	0.0408627
$485$	0	0
$486$	−8.98979	−0.407785
$487$	30.7423	1.39307	0.696534	−	0.717523i	$-0.254724\pi$
$487$	30.7423	1.39307	0.696534	+	0.717523i	$0.254724\pi$
$488$	6.44949	0.291955
$489$	−14.6413	−0.662104
$490$	0	0
$491$	−41.7980	−1.88632	−0.943158	−	0.332345i	$-0.892160\pi$
$491$	−41.7980	−1.88632	−0.943158	+	0.332345i	$0.892160\pi$
$492$	1.44949	0.0653480
$493$	3.10102	0.139663
$494$	2.24745	0.101117
$495$	0	0
$496$	4.44949	0.199788
$497$	18.4949	0.829610
$498$	5.00000	0.224055
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	15.1464	0.676693
$502$	30.7980	1.37458
$503$	−9.79796	−0.436869	−0.218435	−	0.975852i	$-0.570095\pi$
$503$	−9.79796	−0.436869	−0.218435	+	0.975852i	$0.570095\pi$
$504$	−2.20204	−0.0980867
$505$	0	0
$506$	6.89898	0.306697
$507$	−18.5505	−0.823857
$508$	16.2474	0.720864
$509$	29.3485	1.30085	0.650424	−	0.759571i	$-0.274591\pi$
$509$	29.3485	1.30085	0.650424	+	0.759571i	$0.274591\pi$
$510$	0	0
$511$	31.3485	1.38677
$512$	−1.00000	−0.0441942
$513$	28.2577	1.24761
$514$	−6.89898	−0.304301
$515$	0	0
$516$	1.59592	0.0702564
$517$	−33.7980	−1.48643
$518$	2.44949	0.107624
$519$	9.70714	0.426096
$520$	0	0
$521$	10.1010	0.442534	0.221267	−	0.975213i	$-0.428981\pi$
$521$	10.1010	0.442534	0.221267	+	0.975213i	$0.428981\pi$
$522$	−0.808164	−0.0353724
$523$	18.3939	0.804308	0.402154	−	0.915572i	$-0.368262\pi$
$523$	18.3939	0.804308	0.402154	+	0.915572i	$0.368262\pi$
$524$	8.69694	0.379928
$525$	0	0
$526$	0.202041	0.00880941
$527$	−15.3485	−0.668590
$528$	5.00000	0.217597
$529$	−19.0000	−0.826087
$530$	0	0
$531$	1.79796	0.0780248
$532$	12.2474	0.530994
$533$	0.449490	0.0194696
$534$	−20.7980	−0.900016
$535$	0	0
$536$	4.55051	0.196552
$537$	−1.15663	−0.0499124
$538$	−4.00000	−0.172452
$539$	−3.44949	−0.148580
$540$	0	0
$541$	25.3485	1.08982	0.544908	−	0.838496i	$-0.316565\pi$
$541$	25.3485	1.08982	0.544908	+	0.838496i	$0.316565\pi$
$542$	−30.0454	−1.29056
$543$	10.0000	0.429141
$544$	3.44949	0.147896
$545$	0	0
$546$	1.59592	0.0682990
$547$	−3.69694	−0.158070	−0.0790348	−	0.996872i	$-0.525184\pi$
$547$	−3.69694	−0.158070	−0.0790348	+	0.996872i	$0.525184\pi$
$548$	−9.69694	−0.414233
$549$	5.79796	0.247451
$550$	0	0
$551$	4.49490	0.191489
$552$	2.89898	0.123389
$553$	−19.1010	−0.812258
$554$	5.79796	0.246332
$555$	0	0
$556$	−12.5505	−0.532260
$557$	21.7980	0.923609	0.461805	−	0.886982i	$-0.347202\pi$
$557$	21.7980	0.923609	0.461805	+	0.886982i	$0.347202\pi$
$558$	4.00000	0.169334
$559$	0.494897	0.0209319
$560$	0	0
$561$	−17.2474	−0.728188
$562$	−8.89898	−0.375381
$563$	−37.5959	−1.58448	−0.792240	−	0.610210i	$-0.791085\pi$
$563$	−37.5959	−1.58448	−0.792240	+	0.610210i	$0.791085\pi$
$564$	−14.2020	−0.598014
$565$	0	0
$566$	−11.6969	−0.491659
$567$	13.4597	0.565254
$568$	7.55051	0.316812
$569$	27.0454	1.13380	0.566901	−	0.823786i	$-0.308142\pi$
$569$	27.0454	1.13380	0.566901	+	0.823786i	$0.308142\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	1.55051	0.0648301
$573$	−11.3031	−0.472192
$574$	2.44949	0.102240
$575$	0	0
$576$	−0.898979	−0.0374575
$577$	−21.0454	−0.876132	−0.438066	−	0.898943i	$-0.644336\pi$
$577$	−21.0454	−0.876132	−0.438066	+	0.898943i	$0.644336\pi$
$578$	5.10102	0.212174
$579$	3.69694	0.153640
$580$	0	0
$581$	8.44949	0.350544
$582$	20.2929	0.841166
$583$	−20.6969	−0.857180
$584$	12.7980	0.529583
$585$	0	0
$586$	−28.0454	−1.15855
$587$	6.30306	0.260155	0.130078	−	0.991504i	$-0.458477\pi$
$587$	6.30306	0.260155	0.130078	+	0.991504i	$0.458477\pi$
$588$	−1.44949	−0.0597759
$589$	−22.2474	−0.916690
$590$	0	0
$591$	25.1464	1.03439
$592$	1.00000	0.0410997
$593$	−46.3939	−1.90517	−0.952584	−	0.304275i	$-0.901586\pi$
$593$	−46.3939	−1.90517	−0.952584	+	0.304275i	$0.901586\pi$
$594$	19.4949	0.799885
$595$	0	0
$596$	−4.65153	−0.190534
$597$	−22.6061	−0.925207
$598$	0.898979	0.0367620
$599$	38.9444	1.59122	0.795612	−	0.605806i	$-0.207149\pi$
$599$	38.9444	1.59122	0.795612	+	0.605806i	$0.207149\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	2.69694	0.109919
$603$	4.09082	0.166591
$604$	−14.0000	−0.569652
$605$	0	0
$606$	−1.59592	−0.0648297
$607$	28.7423	1.16662	0.583308	−	0.812251i	$-0.301758\pi$
$607$	28.7423	1.16662	0.583308	+	0.812251i	$0.301758\pi$
$608$	5.00000	0.202777
$609$	3.19184	0.129340
$610$	0	0
$611$	−4.40408	−0.178170
$612$	3.10102	0.125351
$613$	34.4949	1.39324	0.696618	−	0.717442i	$-0.254687\pi$
$613$	34.4949	1.39324	0.696618	+	0.717442i	$0.254687\pi$
$614$	7.65153	0.308791
$615$	0	0
$616$	8.44949	0.340440
$617$	4.89898	0.197225	0.0986127	−	0.995126i	$-0.468559\pi$
$617$	4.89898	0.197225	0.0986127	+	0.995126i	$0.468559\pi$
$618$	9.34847	0.376051
$619$	−32.8990	−1.32232	−0.661161	−	0.750244i	$-0.729936\pi$
$619$	−32.8990	−1.32232	−0.661161	+	0.750244i	$0.729936\pi$
$620$	0	0
$621$	11.3031	0.453576
$622$	8.44949	0.338794
$623$	−35.1464	−1.40811
$624$	0.651531	0.0260821
$625$	0	0
$626$	8.89898	0.355675
$627$	−25.0000	−0.998404
$628$	−16.4495	−0.656406
$629$	−3.44949	−0.137540
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−7.79796	−0.310186
$633$	−6.59592	−0.262164
$634$	−9.55051	−0.379299
$635$	0	0
$636$	−8.69694	−0.344856
$637$	−0.449490	−0.0178094
$638$	3.10102	0.122771
$639$	6.78775	0.268519
$640$	0	0
$641$	21.5959	0.852987	0.426494	−	0.904491i	$-0.359749\pi$
$641$	21.5959	0.852987	0.426494	+	0.904491i	$0.359749\pi$
$642$	−12.3939	−0.489147
$643$	−32.2929	−1.27351	−0.636753	−	0.771068i	$-0.719723\pi$
$643$	−32.2929	−1.27351	−0.636753	+	0.771068i	$0.719723\pi$
$644$	4.89898	0.193047
$645$	0	0
$646$	−17.2474	−0.678592
$647$	40.2474	1.58229	0.791145	−	0.611628i	$-0.209485\pi$
$647$	40.2474	1.58229	0.791145	+	0.611628i	$0.209485\pi$
$648$	5.49490	0.215860
$649$	−6.89898	−0.270809
$650$	0	0
$651$	−15.7980	−0.619171
$652$	−10.1010	−0.395586
$653$	10.0454	0.393107	0.196554	−	0.980493i	$-0.437025\pi$
$653$	10.0454	0.393107	0.196554	+	0.980493i	$0.437025\pi$
$654$	−20.2929	−0.793513
$655$	0	0
$656$	1.00000	0.0390434
$657$	11.5051	0.448857
$658$	−24.0000	−0.935617
$659$	6.55051	0.255172	0.127586	−	0.991828i	$-0.459277\pi$
$659$	6.55051	0.255172	0.127586	+	0.991828i	$0.459277\pi$
$660$	0	0
$661$	25.7980	1.00342	0.501712	−	0.865035i	$-0.332704\pi$
$661$	25.7980	1.00342	0.501712	+	0.865035i	$0.332704\pi$
$662$	−19.8990	−0.773396
$663$	−2.24745	−0.0872837
$664$	3.44949	0.133866
$665$	0	0
$666$	0.898979	0.0348347
$667$	1.79796	0.0696172
$668$	10.4495	0.404303
$669$	−31.5959	−1.22157
$670$	0	0
$671$	−22.2474	−0.858853
$672$	3.55051	0.136964
$673$	−5.79796	−0.223495	−0.111747	−	0.993737i	$-0.535645\pi$
$673$	−5.79796	−0.223495	−0.111747	+	0.993737i	$0.535645\pi$
$674$	15.6969	0.604623
$675$	0	0
$676$	−12.7980	−0.492229
$677$	43.5959	1.67553	0.837764	−	0.546033i	$-0.183863\pi$
$677$	43.5959	1.67553	0.837764	+	0.546033i	$0.183863\pi$
$678$	−6.59592	−0.253315
$679$	34.2929	1.31604
$680$	0	0
$681$	24.2020	0.927424
$682$	−15.3485	−0.587723
$683$	7.00000	0.267848	0.133924	−	0.990992i	$-0.457242\pi$
$683$	7.00000	0.267848	0.133924	+	0.990992i	$0.457242\pi$
$684$	4.49490	0.171867
$685$	0	0
$686$	−19.5959	−0.748176
$687$	8.40408	0.320636
$688$	1.10102	0.0419760
$689$	−2.69694	−0.102745
$690$	0	0
$691$	29.6515	1.12800	0.563999	−	0.825776i	$-0.309262\pi$
$691$	29.6515	1.12800	0.563999	+	0.825776i	$0.309262\pi$
$692$	6.69694	0.254579
$693$	7.59592	0.288545
$694$	−18.7980	−0.713561
$695$	0	0
$696$	1.30306	0.0493924
$697$	−3.44949	−0.130659
$698$	12.4495	0.471220
$699$	41.3031	1.56223
$700$	0	0
$701$	−34.2474	−1.29351	−0.646754	−	0.762699i	$-0.723874\pi$
$701$	−34.2474	−1.29351	−0.646754	+	0.762699i	$0.723874\pi$
$702$	2.54031	0.0958776
$703$	−5.00000	−0.188579
$704$	3.44949	0.130008
$705$	0	0
$706$	14.8990	0.560730
$707$	−2.69694	−0.101429
$708$	−2.89898	−0.108950
$709$	16.8990	0.634654	0.317327	−	0.948316i	$-0.397215\pi$
$709$	16.8990	0.634654	0.317327	+	0.948316i	$0.397215\pi$
$710$	0	0
$711$	−7.01021	−0.262903
$712$	−14.3485	−0.537732
$713$	−8.89898	−0.333269
$714$	−12.2474	−0.458349
$715$	0	0
$716$	−0.797959	−0.0298211
$717$	1.30306	0.0486637
$718$	31.3939	1.17161
$719$	−0.651531	−0.0242980	−0.0121490	−	0.999926i	$-0.503867\pi$
$719$	−0.651531	−0.0242980	−0.0121490	+	0.999926i	$0.503867\pi$
$720$	0	0
$721$	15.7980	0.588347
$722$	−6.00000	−0.223297
$723$	−3.69694	−0.137491
$724$	6.89898	0.256399
$725$	0	0
$726$	−1.30306	−0.0483611
$727$	−36.0000	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$728$	1.10102	0.0408065
$729$	29.5153	1.09316
$730$	0	0
$731$	−3.79796	−0.140473
$732$	−9.34847	−0.345529
$733$	15.5959	0.576048	0.288024	−	0.957623i	$-0.407002\pi$
$733$	15.5959	0.576048	0.288024	+	0.957623i	$0.407002\pi$
$734$	−14.2474	−0.525883
$735$	0	0
$736$	2.00000	0.0737210
$737$	−15.6969	−0.578204
$738$	0.898979	0.0330919
$739$	−31.5959	−1.16227	−0.581137	−	0.813806i	$-0.697392\pi$
$739$	−31.5959	−1.16227	−0.581137	+	0.813806i	$0.697392\pi$
$740$	0	0
$741$	−3.25765	−0.119673
$742$	−14.6969	−0.539542
$743$	6.40408	0.234943	0.117471	−	0.993076i	$-0.462521\pi$
$743$	6.40408	0.234943	0.117471	+	0.993076i	$0.462521\pi$
$744$	−6.44949	−0.236450
$745$	0	0
$746$	−20.0454	−0.733915
$747$	3.10102	0.113460
$748$	−11.8990	−0.435070
$749$	−20.9444	−0.765291
$750$	0	0
$751$	−30.6969	−1.12015	−0.560074	−	0.828443i	$-0.689227\pi$
$751$	−30.6969	−1.12015	−0.560074	+	0.828443i	$0.689227\pi$
$752$	−9.79796	−0.357295
$753$	−44.6413	−1.62682
$754$	0.404082	0.0147158
$755$	0	0
$756$	13.8434	0.503478
$757$	13.7980	0.501495	0.250748	−	0.968052i	$-0.419324\pi$
$757$	13.7980	0.501495	0.250748	+	0.968052i	$0.419324\pi$
$758$	21.0454	0.764404
$759$	−10.0000	−0.362977
$760$	0	0
$761$	−14.3031	−0.518486	−0.259243	−	0.965812i	$-0.583473\pi$
$761$	−14.3031	−0.518486	−0.259243	+	0.965812i	$0.583473\pi$
$762$	−23.5505	−0.853145
$763$	−34.2929	−1.24148
$764$	−7.79796	−0.282120
$765$	0	0
$766$	−5.34847	−0.193248
$767$	−0.898979	−0.0324603
$768$	1.44949	0.0523040
$769$	53.2474	1.92015	0.960076	−	0.279739i	$-0.0902480\pi$
$769$	53.2474	1.92015	0.960076	+	0.279739i	$0.0902480\pi$
$770$	0	0
$771$	10.0000	0.360141
$772$	2.55051	0.0917949
$773$	−22.0454	−0.792918	−0.396459	−	0.918052i	$-0.629761\pi$
$773$	−22.0454	−0.792918	−0.396459	+	0.918052i	$0.629761\pi$
$774$	0.989795	0.0355774
$775$	0	0
$776$	14.0000	0.502571
$777$	−3.55051	−0.127374
$778$	−13.1464	−0.471322
$779$	−5.00000	−0.179144
$780$	0	0
$781$	−26.0454	−0.931978
$782$	−6.89898	−0.246707
$783$	5.08061	0.181566
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−12.6061	−0.449646
$787$	−30.6969	−1.09423	−0.547114	−	0.837058i	$-0.684274\pi$
$787$	−30.6969	−1.09423	−0.547114	+	0.837058i	$0.684274\pi$
$788$	17.3485	0.618014
$789$	−0.292856	−0.0104260
$790$	0	0
$791$	−11.1464	−0.396321
$792$	3.10102	0.110190
$793$	−2.89898	−0.102946
$794$	−15.3485	−0.544697
$795$	0	0
$796$	−15.5959	−0.552783
$797$	−4.24745	−0.150452	−0.0752262	−	0.997166i	$-0.523968\pi$
$797$	−4.24745	−0.150452	−0.0752262	+	0.997166i	$0.523968\pi$
$798$	−17.7526	−0.628434
$799$	33.7980	1.19569
$800$	0	0
$801$	−12.8990	−0.455763
$802$	−15.9444	−0.563016
$803$	−44.1464	−1.55789
$804$	−6.59592	−0.232620
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	5.79796	0.204098
$808$	−1.10102	−0.0387338
$809$	−39.3939	−1.38501	−0.692507	−	0.721411i	$-0.743494\pi$
$809$	−39.3939	−1.38501	−0.692507	+	0.721411i	$0.743494\pi$
$810$	0	0
$811$	5.79796	0.203594	0.101797	−	0.994805i	$-0.467541\pi$
$811$	5.79796	0.203594	0.101797	+	0.994805i	$0.467541\pi$
$812$	2.20204	0.0772765
$813$	43.5505	1.52738
$814$	−3.44949	−0.120905
$815$	0	0
$816$	−5.00000	−0.175035
$817$	−5.50510	−0.192599
$818$	34.1464	1.19390
$819$	0.989795	0.0345862
$820$	0	0
$821$	4.04541	0.141186	0.0705929	−	0.997505i	$-0.477511\pi$
$821$	4.04541	0.141186	0.0705929	+	0.997505i	$0.477511\pi$
$822$	14.0556	0.490246
$823$	−33.8434	−1.17971	−0.589853	−	0.807511i	$-0.700814\pi$
$823$	−33.8434	−1.17971	−0.589853	+	0.807511i	$0.700814\pi$
$824$	6.44949	0.224679
$825$	0	0
$826$	−4.89898	−0.170457
$827$	−27.0000	−0.938882	−0.469441	−	0.882964i	$-0.655545\pi$
$827$	−27.0000	−0.938882	−0.469441	+	0.882964i	$0.655545\pi$
$828$	1.79796	0.0624834
$829$	21.3485	0.741463	0.370731	−	0.928740i	$-0.379107\pi$
$829$	21.3485	0.741463	0.370731	+	0.928740i	$0.379107\pi$
$830$	0	0
$831$	−8.40408	−0.291534
$832$	0.449490	0.0155833
$833$	3.44949	0.119518
$834$	18.1918	0.629932
$835$	0	0
$836$	−17.2474	−0.596515
$837$	−25.1464	−0.869188
$838$	21.4495	0.740960
$839$	−1.34847	−0.0465543	−0.0232772	−	0.999729i	$-0.507410\pi$
$839$	−1.34847	−0.0465543	−0.0232772	+	0.999729i	$0.507410\pi$
$840$	0	0
$841$	−28.1918	−0.972132
$842$	−32.8990	−1.13377
$843$	12.8990	0.444264
$844$	−4.55051	−0.156635
$845$	0	0
$846$	−8.80816	−0.302831
$847$	−2.20204	−0.0756630
$848$	−6.00000	−0.206041
$849$	16.9546	0.581880
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	−10.9444	−0.374949
$853$	19.3485	0.662479	0.331239	−	0.943547i	$-0.392533\pi$
$853$	19.3485	0.662479	0.331239	+	0.943547i	$0.392533\pi$
$854$	−15.7980	−0.540595
$855$	0	0
$856$	−8.55051	−0.292250
$857$	32.1464	1.09810	0.549051	−	0.835789i	$-0.314989\pi$
$857$	32.1464	1.09810	0.549051	+	0.835789i	$0.314989\pi$
$858$	−2.24745	−0.0767266
$859$	−9.49490	−0.323962	−0.161981	−	0.986794i	$-0.551788\pi$
$859$	−9.49490	−0.323962	−0.161981	+	0.986794i	$0.551788\pi$
$860$	0	0
$861$	−3.55051	−0.121001
$862$	20.6969	0.704941
$863$	−18.2474	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$863$	−18.2474	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$864$	5.65153	0.192269
$865$	0	0
$866$	19.0000	0.645646
$867$	−7.39388	−0.251109
$868$	−10.8990	−0.369935
$869$	26.8990	0.912485
$870$	0	0
$871$	−2.04541	−0.0693060
$872$	−14.0000	−0.474100
$873$	12.5857	0.425962
$874$	−10.0000	−0.338255
$875$	0	0
$876$	−18.5505	−0.626764
$877$	−4.65153	−0.157071	−0.0785355	−	0.996911i	$-0.525024\pi$
$877$	−4.65153	−0.157071	−0.0785355	+	0.996911i	$0.525024\pi$
$878$	19.5959	0.661330
$879$	40.6515	1.37114
$880$	0	0
$881$	34.2929	1.15536	0.577678	−	0.816265i	$-0.303959\pi$
$881$	34.2929	1.15536	0.577678	+	0.816265i	$0.303959\pi$
$882$	−0.898979	−0.0302702
$883$	−26.1010	−0.878369	−0.439185	−	0.898397i	$-0.644733\pi$
$883$	−26.1010	−0.878369	−0.439185	+	0.898397i	$0.644733\pi$
$884$	−1.55051	−0.0521493
$885$	0	0
$886$	−17.4495	−0.586227
$887$	−50.6969	−1.70224	−0.851118	−	0.524974i	$-0.824075\pi$
$887$	−50.6969	−1.70224	−0.851118	+	0.524974i	$0.824075\pi$
$888$	−1.44949	−0.0486417
$889$	−39.7980	−1.33478
$890$	0	0
$891$	−18.9546	−0.635003
$892$	−21.7980	−0.729850
$893$	48.9898	1.63938
$894$	6.74235	0.225498
$895$	0	0
$896$	2.44949	0.0818317
$897$	−1.30306	−0.0435080
$898$	−33.2474	−1.10948
$899$	−4.00000	−0.133407
$900$	0	0
$901$	20.6969	0.689515
$902$	−3.44949	−0.114855
$903$	−3.90918	−0.130090
$904$	−4.55051	−0.151348
$905$	0	0
$906$	20.2929	0.674185
$907$	34.8990	1.15880	0.579401	−	0.815043i	$-0.303287\pi$
$907$	34.8990	1.15880	0.579401	+	0.815043i	$0.303287\pi$
$908$	16.6969	0.554107
$909$	−0.989795	−0.0328294
$910$	0	0
$911$	−16.2474	−0.538302	−0.269151	−	0.963098i	$-0.586743\pi$
$911$	−16.2474	−0.538302	−0.269151	+	0.963098i	$0.586743\pi$
$912$	−7.24745	−0.239987
$913$	−11.8990	−0.393799
$914$	15.2474	0.504341
$915$	0	0
$916$	5.79796	0.191570
$917$	−21.3031	−0.703489
$918$	−19.4949	−0.643427
$919$	−51.7980	−1.70866	−0.854329	−	0.519733i	$-0.826031\pi$
$919$	−51.7980	−1.70866	−0.854329	+	0.519733i	$0.826031\pi$
$920$	0	0
$921$	−11.0908	−0.365455
$922$	−9.30306	−0.306380
$923$	−3.39388	−0.111711
$924$	−12.2474	−0.402911
$925$	0	0
$926$	9.55051	0.313849
$927$	5.79796	0.190430
$928$	0.898979	0.0295104
$929$	−5.79796	−0.190225	−0.0951124	−	0.995467i	$-0.530321\pi$
$929$	−5.79796	−0.190225	−0.0951124	+	0.995467i	$0.530321\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	28.4949	0.933381
$933$	−12.2474	−0.400963
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0.404082	0.0132078
$937$	−14.5959	−0.476828	−0.238414	−	0.971164i	$-0.576627\pi$
$937$	−14.5959	−0.476828	−0.238414	+	0.971164i	$0.576627\pi$
$938$	−11.1464	−0.363944
$939$	−12.8990	−0.420942
$940$	0	0
$941$	−41.3939	−1.34940	−0.674701	−	0.738091i	$-0.735727\pi$
$941$	−41.3939	−1.34940	−0.674701	+	0.738091i	$0.735727\pi$
$942$	23.8434	0.776859
$943$	−2.00000	−0.0651290
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	−3.79796	−0.123482
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	11.3031	0.367106
$949$	−5.75255	−0.186736
$950$	0	0
$951$	13.8434	0.448902
$952$	−8.44949	−0.273850
$953$	−25.4949	−0.825861	−0.412930	−	0.910763i	$-0.635495\pi$
$953$	−25.4949	−0.825861	−0.412930	+	0.910763i	$0.635495\pi$
$954$	−5.39388	−0.174633
$955$	0	0
$956$	0.898979	0.0290751
$957$	−4.49490	−0.145299
$958$	−6.24745	−0.201846
$959$	23.7526	0.767010
$960$	0	0
$961$	−11.2020	−0.361356
$962$	−0.449490	−0.0144921
$963$	−7.68673	−0.247702
$964$	−2.55051	−0.0821464
$965$	0	0
$966$	−7.10102	−0.228472
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	−0.898979	−0.0288943
$969$	25.0000	0.803116
$970$	0	0
$971$	−53.5403	−1.71819	−0.859095	−	0.511816i	$-0.828973\pi$
$971$	−53.5403	−1.71819	−0.859095	+	0.511816i	$0.828973\pi$
$972$	8.98979	0.288348
$973$	30.7423	0.985554
$974$	−30.7423	−0.985048
$975$	0	0
$976$	−6.44949	−0.206443
$977$	23.7423	0.759585	0.379792	−	0.925072i	$-0.375995\pi$
$977$	23.7423	0.759585	0.379792	+	0.925072i	$0.375995\pi$
$978$	14.6413	0.468178
$979$	49.4949	1.58186
$980$	0	0
$981$	−12.5857	−0.401831
$982$	41.7980	1.33383
$983$	−18.4949	−0.589896	−0.294948	−	0.955513i	$-0.595302\pi$
$983$	−18.4949	−0.589896	−0.294948	+	0.955513i	$0.595302\pi$
$984$	−1.44949	−0.0462080
$985$	0	0
$986$	−3.10102	−0.0987566
$987$	34.7878	1.10731
$988$	−2.24745	−0.0715009
$989$	−2.20204	−0.0700208
$990$	0	0
$991$	−16.4495	−0.522535	−0.261268	−	0.965266i	$-0.584141\pi$
$991$	−16.4495	−0.522535	−0.261268	+	0.965266i	$0.584141\pi$
$992$	−4.44949	−0.141271
$993$	28.8434	0.915317
$994$	−18.4949	−0.586623
$995$	0	0
$996$	−5.00000	−0.158431
$997$	−39.5505	−1.25258	−0.626289	−	0.779591i	$-0.715427\pi$
$997$	−39.5505	−1.25258	−0.626289	+	0.779591i	$0.715427\pi$
$998$	−22.0000	−0.696398
$999$	−5.65153	−0.178807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.r.1.2	✓	2
5.2	odd	4		1850.2.b.k.149.1		4
5.3	odd	4		1850.2.b.k.149.4		4
5.4	even	2		1850.2.a.w.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.r.1.2	✓	2	1.1	even	1	trivial
1850.2.a.w.1.1	yes	2	5.4	even	2
1850.2.b.k.149.1		4	5.2	odd	4
1850.2.b.k.149.4		4	5.3	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.44949 q^{3} +1.00000 q^{4} -1.44949 q^{6} -2.44949 q^{7} -1.00000 q^{8} -0.898979 q^{9} +3.44949 q^{11} +1.44949 q^{12} +0.449490 q^{13} +2.44949 q^{14} +1.00000 q^{16} -3.44949 q^{17} +0.898979 q^{18} -5.00000 q^{19} -3.55051 q^{21} -3.44949 q^{22} -2.00000 q^{23} -1.44949 q^{24} -0.449490 q^{26} -5.65153 q^{27} -2.44949 q^{28} -0.898979 q^{29} +4.44949 q^{31} -1.00000 q^{32} +5.00000 q^{33} +3.44949 q^{34} -0.898979 q^{36} +1.00000 q^{37} +5.00000 q^{38} +0.651531 q^{39} +1.00000 q^{41} +3.55051 q^{42} +1.10102 q^{43} +3.44949 q^{44} +2.00000 q^{46} -9.79796 q^{47} +1.44949 q^{48} -1.00000 q^{49} -5.00000 q^{51} +0.449490 q^{52} -6.00000 q^{53} +5.65153 q^{54} +2.44949 q^{56} -7.24745 q^{57} +0.898979 q^{58} -2.00000 q^{59} -6.44949 q^{61} -4.44949 q^{62} +2.20204 q^{63} +1.00000 q^{64} -5.00000 q^{66} -4.55051 q^{67} -3.44949 q^{68} -2.89898 q^{69} -7.55051 q^{71} +0.898979 q^{72} -12.7980 q^{73} -1.00000 q^{74} -5.00000 q^{76} -8.44949 q^{77} -0.651531 q^{78} +7.79796 q^{79} -5.49490 q^{81} -1.00000 q^{82} -3.44949 q^{83} -3.55051 q^{84} -1.10102 q^{86} -1.30306 q^{87} -3.44949 q^{88} +14.3485 q^{89} -1.10102 q^{91} -2.00000 q^{92} +6.44949 q^{93} +9.79796 q^{94} -1.44949 q^{96} -14.0000 q^{97} +1.00000 q^{98} -3.10102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 8 q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{16} - 2 q^{17} - 8 q^{18} - 10 q^{19} - 12 q^{21} - 2 q^{22} - 4 q^{23} + 2 q^{24} + 4 q^{26} - 26 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^8 + 8 * q^9 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^16 - 2 * q^17 - 8 * q^18 - 10 * q^19 - 12 * q^21 - 2 * q^22 - 4 * q^23 + 2 * q^24 + 4 * q^26 - 26 * q^27 + 8 * q^29 + 4 * q^31 - 2 * q^32 + 10 * q^33 + 2 * q^34 + 8 * q^36 + 2 * q^37 + 10 * q^38 + 16 * q^39 + 2 * q^41 + 12 * q^42 + 12 * q^43 + 2 * q^44 + 4 * q^46 - 2 * q^48 - 2 * q^49 - 10 * q^51 - 4 * q^52 - 12 * q^53 + 26 * q^54 + 10 * q^57 - 8 * q^58 - 4 * q^59 - 8 * q^61 - 4 * q^62 + 24 * q^63 + 2 * q^64 - 10 * q^66 - 14 * q^67 - 2 * q^68 + 4 * q^69 - 20 * q^71 - 8 * q^72 - 6 * q^73 - 2 * q^74 - 10 * q^76 - 12 * q^77 - 16 * q^78 - 4 * q^79 + 38 * q^81 - 2 * q^82 - 2 * q^83 - 12 * q^84 - 12 * q^86 - 32 * q^87 - 2 * q^88 + 14 * q^89 - 12 * q^91 - 4 * q^92 + 8 * q^93 + 2 * q^96 - 28 * q^97 + 2 * q^98 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more