LMFDB - Embedded newform 1850.2.a.u.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} -3.30278 q^{6} +2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} -2.30278 q^{11} -3.30278 q^{12} -1.30278 q^{13} +2.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} +7.90833 q^{18} +2.00000 q^{19} -8.60555 q^{21} -2.30278 q^{22} -3.90833 q^{23} -3.30278 q^{24} -1.30278 q^{26} -16.2111 q^{27} +2.60555 q^{28} -3.90833 q^{29} -0.302776 q^{31} +1.00000 q^{32} +7.60555 q^{33} +6.00000 q^{34} +7.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +4.30278 q^{39} +9.90833 q^{41} -8.60555 q^{42} -0.605551 q^{43} -2.30278 q^{44} -3.90833 q^{46} -4.60555 q^{47} -3.30278 q^{48} -0.211103 q^{49} -19.8167 q^{51} -1.30278 q^{52} +6.00000 q^{53} -16.2111 q^{54} +2.60555 q^{56} -6.60555 q^{57} -3.90833 q^{58} +10.6056 q^{59} +7.51388 q^{61} -0.302776 q^{62} +20.6056 q^{63} +1.00000 q^{64} +7.60555 q^{66} +3.51388 q^{67} +6.00000 q^{68} +12.9083 q^{69} +6.00000 q^{71} +7.90833 q^{72} +12.3028 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +4.30278 q^{78} +9.11943 q^{79} +29.8167 q^{81} +9.90833 q^{82} -2.78890 q^{83} -8.60555 q^{84} -0.605551 q^{86} +12.9083 q^{87} -2.30278 q^{88} -9.21110 q^{89} -3.39445 q^{91} -3.90833 q^{92} +1.00000 q^{93} -4.60555 q^{94} -3.30278 q^{96} +16.4222 q^{97} -0.211103 q^{98} -18.2111 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24}+ \cdots - 22 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 - q^11 - 3 * q^12 + q^13 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 5 * q^18 + 4 * q^19 - 10 * q^21 - q^22 + 3 * q^23 - 3 * q^24 + q^26 - 18 * q^27 - 2 * q^28 + 3 * q^29 + 3 * q^31 + 2 * q^32 + 8 * q^33 + 12 * q^34 + 5 * q^36 - 2 * q^37 + 4 * q^38 + 5 * q^39 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 14 * q^49 - 18 * q^51 + q^52 + 12 * q^53 - 18 * q^54 - 2 * q^56 - 6 * q^57 + 3 * q^58 + 14 * q^59 - 3 * q^61 + 3 * q^62 + 34 * q^63 + 2 * q^64 + 8 * q^66 - 11 * q^67 + 12 * q^68 + 15 * q^69 + 12 * q^71 + 5 * q^72 + 21 * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^77 + 5 * q^78 - 7 * q^79 + 38 * q^81 + 9 * q^82 - 20 * q^83 - 10 * q^84 + 6 * q^86 + 15 * q^87 - q^88 - 4 * q^89 - 14 * q^91 + 3 * q^92 + 2 * q^93 - 2 * q^94 - 3 * q^96 + 4 * q^97 + 14 * q^98 - 22 * 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
$3$	−3.30278	−1.90686	−0.953429	−	0.301617i	$-0.902474\pi$
$3$	−3.30278	−1.90686	−0.953429	+	0.301617i	$0.902474\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.30278	−1.34835
$7$	2.60555	0.984806	0.492403	−	0.870367i	$-0.336119\pi$
$7$	2.60555	0.984806	0.492403	+	0.870367i	$0.336119\pi$
$8$	1.00000	0.353553
$9$	7.90833	2.63611
$10$	0	0
$11$	−2.30278	−0.694313	−0.347156	−	0.937807i	$-0.612853\pi$
$11$	−2.30278	−0.694313	−0.347156	+	0.937807i	$0.612853\pi$
$12$	−3.30278	−0.953429
$13$	−1.30278	−0.361325	−0.180662	−	0.983545i	$-0.557824\pi$
$13$	−1.30278	−0.361325	−0.180662	+	0.983545i	$0.557824\pi$
$14$	2.60555	0.696363
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	7.90833	1.86401
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−8.60555	−1.87789
$22$	−2.30278	−0.490953
$23$	−3.90833	−0.814942	−0.407471	−	0.913218i	$-0.633589\pi$
$23$	−3.90833	−0.814942	−0.407471	+	0.913218i	$0.633589\pi$
$24$	−3.30278	−0.674176
$25$	0	0
$26$	−1.30278	−0.255495
$27$	−16.2111	−3.11983
$28$	2.60555	0.492403
$29$	−3.90833	−0.725758	−0.362879	−	0.931836i	$-0.618206\pi$
$29$	−3.90833	−0.725758	−0.362879	+	0.931836i	$0.618206\pi$
$30$	0	0
$31$	−0.302776	−0.0543801	−0.0271901	−	0.999630i	$-0.508656\pi$
$31$	−0.302776	−0.0543801	−0.0271901	+	0.999630i	$0.508656\pi$
$32$	1.00000	0.176777
$33$	7.60555	1.32396
$34$	6.00000	1.02899
$35$	0	0
$36$	7.90833	1.31805
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	2.00000	0.324443
$39$	4.30278	0.688996
$40$	0	0
$41$	9.90833	1.54742	0.773710	−	0.633540i	$-0.218399\pi$
$41$	9.90833	1.54742	0.773710	+	0.633540i	$0.218399\pi$
$42$	−8.60555	−1.32787
$43$	−0.605551	−0.0923457	−0.0461729	−	0.998933i	$-0.514703\pi$
$43$	−0.605551	−0.0923457	−0.0461729	+	0.998933i	$0.514703\pi$
$44$	−2.30278	−0.347156
$45$	0	0
$46$	−3.90833	−0.576251
$47$	−4.60555	−0.671789	−0.335894	−	0.941900i	$-0.609039\pi$
$47$	−4.60555	−0.671789	−0.335894	+	0.941900i	$0.609039\pi$
$48$	−3.30278	−0.476715
$49$	−0.211103	−0.0301575
$50$	0	0
$51$	−19.8167	−2.77489
$52$	−1.30278	−0.180662
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−16.2111	−2.20605
$55$	0	0
$56$	2.60555	0.348181
$57$	−6.60555	−0.874927
$58$	−3.90833	−0.513188
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	7.51388	0.962054	0.481027	−	0.876706i	$-0.340264\pi$
$61$	7.51388	0.962054	0.481027	+	0.876706i	$0.340264\pi$
$62$	−0.302776	−0.0384525
$63$	20.6056	2.59606
$64$	1.00000	0.125000
$65$	0	0
$66$	7.60555	0.936179
$67$	3.51388	0.429289	0.214644	−	0.976692i	$-0.431141\pi$
$67$	3.51388	0.429289	0.214644	+	0.976692i	$0.431141\pi$
$68$	6.00000	0.727607
$69$	12.9083	1.55398
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	7.90833	0.932005
$73$	12.3028	1.43993	0.719965	−	0.694010i	$-0.244158\pi$
$73$	12.3028	1.43993	0.719965	+	0.694010i	$0.244158\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	−6.00000	−0.683763
$78$	4.30278	0.487193
$79$	9.11943	1.02602	0.513008	−	0.858384i	$-0.328531\pi$
$79$	9.11943	1.02602	0.513008	+	0.858384i	$0.328531\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	9.90833	1.09419
$83$	−2.78890	−0.306121	−0.153061	−	0.988217i	$-0.548913\pi$
$83$	−2.78890	−0.306121	−0.153061	+	0.988217i	$0.548913\pi$
$84$	−8.60555	−0.938943
$85$	0	0
$86$	−0.605551	−0.0652983
$87$	12.9083	1.38392
$88$	−2.30278	−0.245477
$89$	−9.21110	−0.976375	−0.488187	−	0.872739i	$-0.662342\pi$
$89$	−9.21110	−0.976375	−0.488187	+	0.872739i	$0.662342\pi$
$90$	0	0
$91$	−3.39445	−0.355835
$92$	−3.90833	−0.407471
$93$	1.00000	0.103695
$94$	−4.60555	−0.475026
$95$	0	0
$96$	−3.30278	−0.337088
$97$	16.4222	1.66742	0.833711	−	0.552201i	$-0.186212\pi$
$97$	16.4222	1.66742	0.833711	+	0.552201i	$0.186212\pi$
$98$	−0.211103	−0.0213246
$99$	−18.2111	−1.83028
$100$	0	0
$101$	−12.4222	−1.23606	−0.618028	−	0.786156i	$-0.712068\pi$
$101$	−12.4222	−1.23606	−0.618028	+	0.786156i	$0.712068\pi$
$102$	−19.8167	−1.96214
$103$	0.302776	0.0298334	0.0149167	−	0.999889i	$-0.495252\pi$
$103$	0.302776	0.0298334	0.0149167	+	0.999889i	$0.495252\pi$
$104$	−1.30278	−0.127748
$105$	0	0
$106$	6.00000	0.582772
$107$	−0.697224	−0.0674032	−0.0337016	−	0.999432i	$-0.510730\pi$
$107$	−0.697224	−0.0674032	−0.0337016	+	0.999432i	$0.510730\pi$
$108$	−16.2111	−1.55991
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	3.30278	0.313486
$112$	2.60555	0.246201
$113$	3.21110	0.302075	0.151038	−	0.988528i	$-0.451739\pi$
$113$	3.21110	0.302075	0.151038	+	0.988528i	$0.451739\pi$
$114$	−6.60555	−0.618667
$115$	0	0
$116$	−3.90833	−0.362879
$117$	−10.3028	−0.952492
$118$	10.6056	0.976320
$119$	15.6333	1.43310
$120$	0	0
$121$	−5.69722	−0.517929
$122$	7.51388	0.680275
$123$	−32.7250	−2.95071
$124$	−0.302776	−0.0271901
$125$	0	0
$126$	20.6056	1.83569
$127$	19.2111	1.70471	0.852355	−	0.522964i	$-0.175174\pi$
$127$	19.2111	1.70471	0.852355	+	0.522964i	$0.175174\pi$
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	0	0
$131$	10.6056	0.926611	0.463306	−	0.886199i	$-0.346663\pi$
$131$	10.6056	0.926611	0.463306	+	0.886199i	$0.346663\pi$
$132$	7.60555	0.661978
$133$	5.21110	0.451860
$134$	3.51388	0.303553
$135$	0	0
$136$	6.00000	0.514496
$137$	−0.908327	−0.0776036	−0.0388018	−	0.999247i	$-0.512354\pi$
$137$	−0.908327	−0.0776036	−0.0388018	+	0.999247i	$0.512354\pi$
$138$	12.9083	1.09883
$139$	−1.90833	−0.161862	−0.0809311	−	0.996720i	$-0.525789\pi$
$139$	−1.90833	−0.161862	−0.0809311	+	0.996720i	$0.525789\pi$
$140$	0	0
$141$	15.2111	1.28101
$142$	6.00000	0.503509
$143$	3.00000	0.250873
$144$	7.90833	0.659027
$145$	0	0
$146$	12.3028	1.01818
$147$	0.697224	0.0575061
$148$	−1.00000	−0.0821995
$149$	19.8167	1.62344	0.811722	−	0.584044i	$-0.198531\pi$
$149$	19.8167	1.62344	0.811722	+	0.584044i	$0.198531\pi$
$150$	0	0
$151$	−20.6056	−1.67686	−0.838428	−	0.545012i	$-0.816525\pi$
$151$	−20.6056	−1.67686	−0.838428	+	0.545012i	$0.816525\pi$
$152$	2.00000	0.162221
$153$	47.4500	3.83610
$154$	−6.00000	−0.483494
$155$	0	0
$156$	4.30278	0.344498
$157$	7.21110	0.575509	0.287754	−	0.957704i	$-0.407091\pi$
$157$	7.21110	0.575509	0.287754	+	0.957704i	$0.407091\pi$
$158$	9.11943	0.725503
$159$	−19.8167	−1.57156
$160$	0	0
$161$	−10.1833	−0.802560
$162$	29.8167	2.34262
$163$	−8.42221	−0.659678	−0.329839	−	0.944037i	$-0.606994\pi$
$163$	−8.42221	−0.659678	−0.329839	+	0.944037i	$0.606994\pi$
$164$	9.90833	0.773710
$165$	0	0
$166$	−2.78890	−0.216460
$167$	5.51388	0.426677	0.213338	−	0.976978i	$-0.431566\pi$
$167$	5.51388	0.426677	0.213338	+	0.976978i	$0.431566\pi$
$168$	−8.60555	−0.663933
$169$	−11.3028	−0.869444
$170$	0	0
$171$	15.8167	1.20953
$172$	−0.605551	−0.0461729
$173$	8.78890	0.668207	0.334104	−	0.942536i	$-0.391566\pi$
$173$	8.78890	0.668207	0.334104	+	0.942536i	$0.391566\pi$
$174$	12.9083	0.978578
$175$	0	0
$176$	−2.30278	−0.173578
$177$	−35.0278	−2.63285
$178$	−9.21110	−0.690401
$179$	−13.8167	−1.03271	−0.516353	−	0.856376i	$-0.672711\pi$
$179$	−13.8167	−1.03271	−0.516353	+	0.856376i	$0.672711\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	−3.39445	−0.251613
$183$	−24.8167	−1.83450
$184$	−3.90833	−0.288126
$185$	0	0
$186$	1.00000	0.0733236
$187$	−13.8167	−1.01037
$188$	−4.60555	−0.335894
$189$	−42.2389	−3.07242
$190$	0	0
$191$	−5.51388	−0.398970	−0.199485	−	0.979901i	$-0.563927\pi$
$191$	−5.51388	−0.398970	−0.199485	+	0.979901i	$0.563927\pi$
$192$	−3.30278	−0.238357
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	16.4222	1.17905
$195$	0	0
$196$	−0.211103	−0.0150788
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−18.2111	−1.29421
$199$	26.4222	1.87302	0.936510	−	0.350640i	$-0.114036\pi$
$199$	26.4222	1.87302	0.936510	+	0.350640i	$0.114036\pi$
$200$	0	0
$201$	−11.6056	−0.818592
$202$	−12.4222	−0.874023
$203$	−10.1833	−0.714731
$204$	−19.8167	−1.38744
$205$	0	0
$206$	0.302776	0.0210954
$207$	−30.9083	−2.14828
$208$	−1.30278	−0.0903312
$209$	−4.60555	−0.318573
$210$	0	0
$211$	10.3028	0.709272	0.354636	−	0.935004i	$-0.384605\pi$
$211$	10.3028	0.709272	0.354636	+	0.935004i	$0.384605\pi$
$212$	6.00000	0.412082
$213$	−19.8167	−1.35781
$214$	−0.697224	−0.0476613
$215$	0	0
$216$	−16.2111	−1.10303
$217$	−0.788897	−0.0535538
$218$	2.00000	0.135457
$219$	−40.6333	−2.74574
$220$	0	0
$221$	−7.81665	−0.525805
$222$	3.30278	0.221668
$223$	5.81665	0.389512	0.194756	−	0.980852i	$-0.437609\pi$
$223$	5.81665	0.389512	0.194756	+	0.980852i	$0.437609\pi$
$224$	2.60555	0.174091
$225$	0	0
$226$	3.21110	0.213599
$227$	−13.8167	−0.917044	−0.458522	−	0.888683i	$-0.651621\pi$
$227$	−13.8167	−0.917044	−0.458522	+	0.888683i	$0.651621\pi$
$228$	−6.60555	−0.437463
$229$	24.6056	1.62598	0.812990	−	0.582277i	$-0.197838\pi$
$229$	24.6056	1.62598	0.812990	+	0.582277i	$0.197838\pi$
$230$	0	0
$231$	19.8167	1.30384
$232$	−3.90833	−0.256594
$233$	−8.51388	−0.557763	−0.278881	−	0.960326i	$-0.589964\pi$
$233$	−8.51388	−0.557763	−0.278881	+	0.960326i	$0.589964\pi$
$234$	−10.3028	−0.673514
$235$	0	0
$236$	10.6056	0.690363
$237$	−30.1194	−1.95647
$238$	15.6333	1.01336
$239$	−17.5139	−1.13288	−0.566439	−	0.824103i	$-0.691679\pi$
$239$	−17.5139	−1.13288	−0.566439	+	0.824103i	$0.691679\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−5.69722	−0.366231
$243$	−49.8444	−3.19752
$244$	7.51388	0.481027
$245$	0	0
$246$	−32.7250	−2.08647
$247$	−2.60555	−0.165787
$248$	−0.302776	−0.0192263
$249$	9.21110	0.583730
$250$	0	0
$251$	−21.2111	−1.33883	−0.669416	−	0.742887i	$-0.733456\pi$
$251$	−21.2111	−1.33883	−0.669416	+	0.742887i	$0.733456\pi$
$252$	20.6056	1.29803
$253$	9.00000	0.565825
$254$	19.2111	1.20541
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.21110	−0.200303	−0.100152	−	0.994972i	$-0.531933\pi$
$257$	−3.21110	−0.200303	−0.100152	+	0.994972i	$0.531933\pi$
$258$	2.00000	0.124515
$259$	−2.60555	−0.161901
$260$	0	0
$261$	−30.9083	−1.91318
$262$	10.6056	0.655213
$263$	−13.8167	−0.851971	−0.425986	−	0.904730i	$-0.640073\pi$
$263$	−13.8167	−0.851971	−0.425986	+	0.904730i	$0.640073\pi$
$264$	7.60555	0.468089
$265$	0	0
$266$	5.21110	0.319513
$267$	30.4222	1.86181
$268$	3.51388	0.214644
$269$	−21.2111	−1.29326	−0.646632	−	0.762802i	$-0.723823\pi$
$269$	−21.2111	−1.29326	−0.646632	+	0.762802i	$0.723823\pi$
$270$	0	0
$271$	−22.4222	−1.36205	−0.681026	−	0.732259i	$-0.738466\pi$
$271$	−22.4222	−1.36205	−0.681026	+	0.732259i	$0.738466\pi$
$272$	6.00000	0.363803
$273$	11.2111	0.678527
$274$	−0.908327	−0.0548740
$275$	0	0
$276$	12.9083	0.776990
$277$	−0.119429	−0.00717582	−0.00358791	−	0.999994i	$-0.501142\pi$
$277$	−0.119429	−0.00717582	−0.00358791	+	0.999994i	$0.501142\pi$
$278$	−1.90833	−0.114454
$279$	−2.39445	−0.143352
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	15.2111	0.905808
$283$	−24.6056	−1.46265	−0.731324	−	0.682030i	$-0.761097\pi$
$283$	−24.6056	−1.46265	−0.731324	+	0.682030i	$0.761097\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	3.00000	0.177394
$287$	25.8167	1.52391
$288$	7.90833	0.466003
$289$	19.0000	1.11765
$290$	0	0
$291$	−54.2389	−3.17954
$292$	12.3028	0.719965
$293$	−11.0278	−0.644248	−0.322124	−	0.946697i	$-0.604397\pi$
$293$	−11.0278	−0.644248	−0.322124	+	0.946697i	$0.604397\pi$
$294$	0.697224	0.0406630
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	37.3305	2.16614
$298$	19.8167	1.14795
$299$	5.09167	0.294459
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	−20.6056	−1.18572
$303$	41.0278	2.35698
$304$	2.00000	0.114708
$305$	0	0
$306$	47.4500	2.71253
$307$	−17.9083	−1.02208	−0.511041	−	0.859556i	$-0.670740\pi$
$307$	−17.9083	−1.02208	−0.511041	+	0.859556i	$0.670740\pi$
$308$	−6.00000	−0.341882
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	15.9083	0.902078	0.451039	−	0.892504i	$-0.351053\pi$
$311$	15.9083	0.902078	0.451039	+	0.892504i	$0.351053\pi$
$312$	4.30278	0.243597
$313$	9.02776	0.510279	0.255139	−	0.966904i	$-0.417879\pi$
$313$	9.02776	0.510279	0.255139	+	0.966904i	$0.417879\pi$
$314$	7.21110	0.406946
$315$	0	0
$316$	9.11943	0.513008
$317$	−9.21110	−0.517347	−0.258674	−	0.965965i	$-0.583285\pi$
$317$	−9.21110	−0.517347	−0.258674	+	0.965965i	$0.583285\pi$
$318$	−19.8167	−1.11126
$319$	9.00000	0.503903
$320$	0	0
$321$	2.30278	0.128528
$322$	−10.1833	−0.567496
$323$	12.0000	0.667698
$324$	29.8167	1.65648
$325$	0	0
$326$	−8.42221	−0.466463
$327$	−6.60555	−0.365288
$328$	9.90833	0.547096
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−13.2111	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$331$	−13.2111	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$332$	−2.78890	−0.153061
$333$	−7.90833	−0.433374
$334$	5.51388	0.301706
$335$	0	0
$336$	−8.60555	−0.469471
$337$	−6.11943	−0.333347	−0.166673	−	0.986012i	$-0.553303\pi$
$337$	−6.11943	−0.333347	−0.166673	+	0.986012i	$0.553303\pi$
$338$	−11.3028	−0.614790
$339$	−10.6056	−0.576014
$340$	0	0
$341$	0.697224	0.0377568
$342$	15.8167	0.855267
$343$	−18.7889	−1.01451
$344$	−0.605551	−0.0326491
$345$	0	0
$346$	8.78890	0.472494
$347$	−10.1833	−0.546671	−0.273335	−	0.961919i	$-0.588127\pi$
$347$	−10.1833	−0.546671	−0.273335	+	0.961919i	$0.588127\pi$
$348$	12.9083	0.691959
$349$	28.2389	1.51159	0.755796	−	0.654807i	$-0.227250\pi$
$349$	28.2389	1.51159	0.755796	+	0.654807i	$0.227250\pi$
$350$	0	0
$351$	21.1194	1.12727
$352$	−2.30278	−0.122738
$353$	−10.1833	−0.542005	−0.271002	−	0.962579i	$-0.587355\pi$
$353$	−10.1833	−0.542005	−0.271002	+	0.962579i	$0.587355\pi$
$354$	−35.0278	−1.86170
$355$	0	0
$356$	−9.21110	−0.488187
$357$	−51.6333	−2.73272
$358$	−13.8167	−0.730233
$359$	3.21110	0.169476	0.0847378	−	0.996403i	$-0.472995\pi$
$359$	3.21110	0.169476	0.0847378	+	0.996403i	$0.472995\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	20.0000	1.05118
$363$	18.8167	0.987618
$364$	−3.39445	−0.177917
$365$	0	0
$366$	−24.8167	−1.29719
$367$	−3.81665	−0.199228	−0.0996139	−	0.995026i	$-0.531761\pi$
$367$	−3.81665	−0.199228	−0.0996139	+	0.995026i	$0.531761\pi$
$368$	−3.90833	−0.203736
$369$	78.3583	4.07917
$370$	0	0
$371$	15.6333	0.811641
$372$	1.00000	0.0518476
$373$	17.8167	0.922511	0.461256	−	0.887267i	$-0.347399\pi$
$373$	17.8167	0.922511	0.461256	+	0.887267i	$0.347399\pi$
$374$	−13.8167	−0.714442
$375$	0	0
$376$	−4.60555	−0.237513
$377$	5.09167	0.262235
$378$	−42.2389	−2.17253
$379$	24.3305	1.24978	0.624888	−	0.780715i	$-0.285145\pi$
$379$	24.3305	1.24978	0.624888	+	0.780715i	$0.285145\pi$
$380$	0	0
$381$	−63.4500	−3.25064
$382$	−5.51388	−0.282115
$383$	36.8444	1.88266	0.941331	−	0.337486i	$-0.109576\pi$
$383$	36.8444	1.88266	0.941331	+	0.337486i	$0.109576\pi$
$384$	−3.30278	−0.168544
$385$	0	0
$386$	4.00000	0.203595
$387$	−4.78890	−0.243433
$388$	16.4222	0.833711
$389$	−37.1194	−1.88203	−0.941015	−	0.338365i	$-0.890126\pi$
$389$	−37.1194	−1.88203	−0.941015	+	0.338365i	$0.890126\pi$
$390$	0	0
$391$	−23.4500	−1.18592
$392$	−0.211103	−0.0106623
$393$	−35.0278	−1.76692
$394$	6.00000	0.302276
$395$	0	0
$396$	−18.2111	−0.915142
$397$	−6.18335	−0.310333	−0.155167	−	0.987888i	$-0.549591\pi$
$397$	−6.18335	−0.310333	−0.155167	+	0.987888i	$0.549591\pi$
$398$	26.4222	1.32443
$399$	−17.2111	−0.861633
$400$	0	0
$401$	−7.81665	−0.390345	−0.195173	−	0.980769i	$-0.562527\pi$
$401$	−7.81665	−0.390345	−0.195173	+	0.980769i	$0.562527\pi$
$402$	−11.6056	−0.578832
$403$	0.394449	0.0196489
$404$	−12.4222	−0.618028
$405$	0	0
$406$	−10.1833	−0.505391
$407$	2.30278	0.114144
$408$	−19.8167	−0.981071
$409$	31.0278	1.53422	0.767112	−	0.641513i	$-0.221693\pi$
$409$	31.0278	1.53422	0.767112	+	0.641513i	$0.221693\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0.302776	0.0149167
$413$	27.6333	1.35975
$414$	−30.9083	−1.51906
$415$	0	0
$416$	−1.30278	−0.0638738
$417$	6.30278	0.308648
$418$	−4.60555	−0.225265
$419$	36.1472	1.76591	0.882953	−	0.469462i	$-0.155552\pi$
$419$	36.1472	1.76591	0.882953	+	0.469462i	$0.155552\pi$
$420$	0	0
$421$	−3.72498	−0.181544	−0.0907722	−	0.995872i	$-0.528934\pi$
$421$	−3.72498	−0.181544	−0.0907722	+	0.995872i	$0.528934\pi$
$422$	10.3028	0.501531
$423$	−36.4222	−1.77091
$424$	6.00000	0.291386
$425$	0	0
$426$	−19.8167	−0.960120
$427$	19.5778	0.947436
$428$	−0.697224	−0.0337016
$429$	−9.90833	−0.478379
$430$	0	0
$431$	9.21110	0.443683	0.221842	−	0.975083i	$-0.428793\pi$
$431$	9.21110	0.443683	0.221842	+	0.975083i	$0.428793\pi$
$432$	−16.2111	−0.779957
$433$	−34.9361	−1.67892	−0.839461	−	0.543421i	$-0.817129\pi$
$433$	−34.9361	−1.67892	−0.839461	+	0.543421i	$0.817129\pi$
$434$	−0.788897	−0.0378683
$435$	0	0
$436$	2.00000	0.0957826
$437$	−7.81665	−0.373921
$438$	−40.6333	−1.94153
$439$	30.3305	1.44760	0.723799	−	0.690011i	$-0.242394\pi$
$439$	30.3305	1.44760	0.723799	+	0.690011i	$0.242394\pi$
$440$	0	0
$441$	−1.66947	−0.0794985
$442$	−7.81665	−0.371800
$443$	−32.7250	−1.55481	−0.777405	−	0.629000i	$-0.783465\pi$
$443$	−32.7250	−1.55481	−0.777405	+	0.629000i	$0.783465\pi$
$444$	3.30278	0.156743
$445$	0	0
$446$	5.81665	0.275427
$447$	−65.4500	−3.09568
$448$	2.60555	0.123101
$449$	−15.2111	−0.717856	−0.358928	−	0.933365i	$-0.616858\pi$
$449$	−15.2111	−0.717856	−0.358928	+	0.933365i	$0.616858\pi$
$450$	0	0
$451$	−22.8167	−1.07439
$452$	3.21110	0.151038
$453$	68.0555	3.19753
$454$	−13.8167	−0.648448
$455$	0	0
$456$	−6.60555	−0.309333
$457$	2.60555	0.121883	0.0609413	−	0.998141i	$-0.480590\pi$
$457$	2.60555	0.121883	0.0609413	+	0.998141i	$0.480590\pi$
$458$	24.6056	1.14974
$459$	−97.2666	−4.54002
$460$	0	0
$461$	12.4222	0.578560	0.289280	−	0.957245i	$-0.406584\pi$
$461$	12.4222	0.578560	0.289280	+	0.957245i	$0.406584\pi$
$462$	19.8167	0.921954
$463$	−26.6972	−1.24073	−0.620363	−	0.784315i	$-0.713015\pi$
$463$	−26.6972	−1.24073	−0.620363	+	0.784315i	$0.713015\pi$
$464$	−3.90833	−0.181440
$465$	0	0
$466$	−8.51388	−0.394398
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−10.3028	−0.476246
$469$	9.15559	0.422766
$470$	0	0
$471$	−23.8167	−1.09741
$472$	10.6056	0.488160
$473$	1.39445	0.0641168
$474$	−30.1194	−1.38343
$475$	0	0
$476$	15.6333	0.716551
$477$	47.4500	2.17258
$478$	−17.5139	−0.801066
$479$	−13.1194	−0.599442	−0.299721	−	0.954027i	$-0.596894\pi$
$479$	−13.1194	−0.599442	−0.299721	+	0.954027i	$0.596894\pi$
$480$	0	0
$481$	1.30278	0.0594015
$482$	8.00000	0.364390
$483$	33.6333	1.53037
$484$	−5.69722	−0.258965
$485$	0	0
$486$	−49.8444	−2.26099
$487$	37.2111	1.68620	0.843098	−	0.537760i	$-0.180729\pi$
$487$	37.2111	1.68620	0.843098	+	0.537760i	$0.180729\pi$
$488$	7.51388	0.340137
$489$	27.8167	1.25791
$490$	0	0
$491$	17.7250	0.799917	0.399959	−	0.916533i	$-0.369025\pi$
$491$	17.7250	0.799917	0.399959	+	0.916533i	$0.369025\pi$
$492$	−32.7250	−1.47536
$493$	−23.4500	−1.05613
$494$	−2.60555	−0.117229
$495$	0	0
$496$	−0.302776	−0.0135950
$497$	15.6333	0.701250
$498$	9.21110	0.412759
$499$	−42.2389	−1.89087	−0.945436	−	0.325809i	$-0.894363\pi$
$499$	−42.2389	−1.89087	−0.945436	+	0.325809i	$0.894363\pi$
$500$	0	0
$501$	−18.2111	−0.813612
$502$	−21.2111	−0.946698
$503$	6.48612	0.289202	0.144601	−	0.989490i	$-0.453810\pi$
$503$	6.48612	0.289202	0.144601	+	0.989490i	$0.453810\pi$
$504$	20.6056	0.917844
$505$	0	0
$506$	9.00000	0.400099
$507$	37.3305	1.65791
$508$	19.2111	0.852355
$509$	−4.18335	−0.185424	−0.0927118	−	0.995693i	$-0.529554\pi$
$509$	−4.18335	−0.185424	−0.0927118	+	0.995693i	$0.529554\pi$
$510$	0	0
$511$	32.0555	1.41805
$512$	1.00000	0.0441942
$513$	−32.4222	−1.43148
$514$	−3.21110	−0.141636
$515$	0	0
$516$	2.00000	0.0880451
$517$	10.6056	0.466432
$518$	−2.60555	−0.114481
$519$	−29.0278	−1.27418
$520$	0	0
$521$	33.6333	1.47350	0.736751	−	0.676164i	$-0.236359\pi$
$521$	33.6333	1.47350	0.736751	+	0.676164i	$0.236359\pi$
$522$	−30.9083	−1.35282
$523$	18.2389	0.797530	0.398765	−	0.917053i	$-0.369439\pi$
$523$	18.2389	0.797530	0.398765	+	0.917053i	$0.369439\pi$
$524$	10.6056	0.463306
$525$	0	0
$526$	−13.8167	−0.602435
$527$	−1.81665	−0.0791347
$528$	7.60555	0.330989
$529$	−7.72498	−0.335869
$530$	0	0
$531$	83.8722	3.63974
$532$	5.21110	0.225930
$533$	−12.9083	−0.559122
$534$	30.4222	1.31650
$535$	0	0
$536$	3.51388	0.151776
$537$	45.6333	1.96922
$538$	−21.2111	−0.914476
$539$	0.486122	0.0209387
$540$	0	0
$541$	25.9361	1.11508	0.557540	−	0.830150i	$-0.311745\pi$
$541$	25.9361	1.11508	0.557540	+	0.830150i	$0.311745\pi$
$542$	−22.4222	−0.963116
$543$	−66.0555	−2.83471
$544$	6.00000	0.257248
$545$	0	0
$546$	11.2111	0.479791
$547$	20.6056	0.881030	0.440515	−	0.897745i	$-0.354796\pi$
$547$	20.6056	0.881030	0.440515	+	0.897745i	$0.354796\pi$
$548$	−0.908327	−0.0388018
$549$	59.4222	2.53608
$550$	0	0
$551$	−7.81665	−0.333001
$552$	12.9083	0.549415
$553$	23.7611	1.01043
$554$	−0.119429	−0.00507407
$555$	0	0
$556$	−1.90833	−0.0809311
$557$	−11.5139	−0.487859	−0.243929	−	0.969793i	$-0.578436\pi$
$557$	−11.5139	−0.487859	−0.243929	+	0.969793i	$0.578436\pi$
$558$	−2.39445	−0.101365
$559$	0.788897	0.0333668
$560$	0	0
$561$	45.6333	1.92664
$562$	−12.0000	−0.506189
$563$	28.0555	1.18240	0.591199	−	0.806525i	$-0.298655\pi$
$563$	28.0555	1.18240	0.591199	+	0.806525i	$0.298655\pi$
$564$	15.2111	0.640503
$565$	0	0
$566$	−24.6056	−1.03425
$567$	77.6888	3.26262
$568$	6.00000	0.251754
$569$	18.4222	0.772299	0.386150	−	0.922436i	$-0.373805\pi$
$569$	18.4222	0.772299	0.386150	+	0.922436i	$0.373805\pi$
$570$	0	0
$571$	−16.6972	−0.698757	−0.349379	−	0.936982i	$-0.613607\pi$
$571$	−16.6972	−0.698757	−0.349379	+	0.936982i	$0.613607\pi$
$572$	3.00000	0.125436
$573$	18.2111	0.760780
$574$	25.8167	1.07757
$575$	0	0
$576$	7.90833	0.329514
$577$	−22.2389	−0.925816	−0.462908	−	0.886406i	$-0.653194\pi$
$577$	−22.2389	−0.925816	−0.462908	+	0.886406i	$0.653194\pi$
$578$	19.0000	0.790296
$579$	−13.2111	−0.549035
$580$	0	0
$581$	−7.26662	−0.301470
$582$	−54.2389	−2.24827
$583$	−13.8167	−0.572227
$584$	12.3028	0.509092
$585$	0	0
$586$	−11.0278	−0.455552
$587$	45.6333	1.88349	0.941744	−	0.336330i	$-0.109186\pi$
$587$	45.6333	1.88349	0.941744	+	0.336330i	$0.109186\pi$
$588$	0.697224	0.0287530
$589$	−0.605551	−0.0249513
$590$	0	0
$591$	−19.8167	−0.815148
$592$	−1.00000	−0.0410997
$593$	−18.4861	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$593$	−18.4861	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$594$	37.3305	1.53169
$595$	0	0
$596$	19.8167	0.811722
$597$	−87.2666	−3.57158
$598$	5.09167	0.208214
$599$	−20.7889	−0.849411	−0.424706	−	0.905331i	$-0.639622\pi$
$599$	−20.7889	−0.849411	−0.424706	+	0.905331i	$0.639622\pi$
$600$	0	0
$601$	−24.3028	−0.991331	−0.495665	−	0.868514i	$-0.665076\pi$
$601$	−24.3028	−0.991331	−0.495665	+	0.868514i	$0.665076\pi$
$602$	−1.57779	−0.0643061
$603$	27.7889	1.13165
$604$	−20.6056	−0.838428
$605$	0	0
$606$	41.0278	1.66664
$607$	13.4861	0.547385	0.273692	−	0.961817i	$-0.411755\pi$
$607$	13.4861	0.547385	0.273692	+	0.961817i	$0.411755\pi$
$608$	2.00000	0.0811107
$609$	33.6333	1.36289
$610$	0	0
$611$	6.00000	0.242734
$612$	47.4500	1.91805
$613$	29.8167	1.20428	0.602142	−	0.798389i	$-0.294314\pi$
$613$	29.8167	1.20428	0.602142	+	0.798389i	$0.294314\pi$
$614$	−17.9083	−0.722721
$615$	0	0
$616$	−6.00000	−0.241747
$617$	42.5694	1.71378	0.856890	−	0.515500i	$-0.172394\pi$
$617$	42.5694	1.71378	0.856890	+	0.515500i	$0.172394\pi$
$618$	−1.00000	−0.0402259
$619$	−6.30278	−0.253330	−0.126665	−	0.991946i	$-0.540427\pi$
$619$	−6.30278	−0.253330	−0.126665	+	0.991946i	$0.540427\pi$
$620$	0	0
$621$	63.3583	2.54248
$622$	15.9083	0.637866
$623$	−24.0000	−0.961540
$624$	4.30278	0.172249
$625$	0	0
$626$	9.02776	0.360822
$627$	15.2111	0.607473
$628$	7.21110	0.287754
$629$	−6.00000	−0.239236
$630$	0	0
$631$	14.6972	0.585087	0.292544	−	0.956252i	$-0.405498\pi$
$631$	14.6972	0.585087	0.292544	+	0.956252i	$0.405498\pi$
$632$	9.11943	0.362751
$633$	−34.0278	−1.35248
$634$	−9.21110	−0.365820
$635$	0	0
$636$	−19.8167	−0.785781
$637$	0.275019	0.0108967
$638$	9.00000	0.356313
$639$	47.4500	1.87709
$640$	0	0
$641$	−20.5139	−0.810249	−0.405125	−	0.914261i	$-0.632772\pi$
$641$	−20.5139	−0.810249	−0.405125	+	0.914261i	$0.632772\pi$
$642$	2.30278	0.0908833
$643$	8.18335	0.322720	0.161360	−	0.986896i	$-0.448412\pi$
$643$	8.18335	0.322720	0.161360	+	0.986896i	$0.448412\pi$
$644$	−10.1833	−0.401280
$645$	0	0
$646$	12.0000	0.472134
$647$	−20.9361	−0.823082	−0.411541	−	0.911391i	$-0.635009\pi$
$647$	−20.9361	−0.823082	−0.411541	+	0.911391i	$0.635009\pi$
$648$	29.8167	1.17131
$649$	−24.4222	−0.958655
$650$	0	0
$651$	2.60555	0.102120
$652$	−8.42221	−0.329839
$653$	3.90833	0.152945	0.0764723	−	0.997072i	$-0.475634\pi$
$653$	3.90833	0.152945	0.0764723	+	0.997072i	$0.475634\pi$
$654$	−6.60555	−0.258297
$655$	0	0
$656$	9.90833	0.386855
$657$	97.2944	3.79581
$658$	−12.0000	−0.467809
$659$	−16.8806	−0.657574	−0.328787	−	0.944404i	$-0.606640\pi$
$659$	−16.8806	−0.657574	−0.328787	+	0.944404i	$0.606640\pi$
$660$	0	0
$661$	−30.5139	−1.18685	−0.593426	−	0.804888i	$-0.702225\pi$
$661$	−30.5139	−1.18685	−0.593426	+	0.804888i	$0.702225\pi$
$662$	−13.2111	−0.513464
$663$	25.8167	1.00264
$664$	−2.78890	−0.108230
$665$	0	0
$666$	−7.90833	−0.306441
$667$	15.2750	0.591451
$668$	5.51388	0.213338
$669$	−19.2111	−0.742744
$670$	0	0
$671$	−17.3028	−0.667966
$672$	−8.60555	−0.331966
$673$	−20.6972	−0.797819	−0.398910	−	0.916990i	$-0.630611\pi$
$673$	−20.6972	−0.797819	−0.398910	+	0.916990i	$0.630611\pi$
$674$	−6.11943	−0.235712
$675$	0	0
$676$	−11.3028	−0.434722
$677$	−14.2389	−0.547244	−0.273622	−	0.961837i	$-0.588222\pi$
$677$	−14.2389	−0.547244	−0.273622	+	0.961837i	$0.588222\pi$
$678$	−10.6056	−0.407304
$679$	42.7889	1.64209
$680$	0	0
$681$	45.6333	1.74867
$682$	0.697224	0.0266981
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	15.8167	0.604765
$685$	0	0
$686$	−18.7889	−0.717363
$687$	−81.2666	−3.10051
$688$	−0.605551	−0.0230864
$689$	−7.81665	−0.297791
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	8.78890	0.334104
$693$	−47.4500	−1.80247
$694$	−10.1833	−0.386555
$695$	0	0
$696$	12.9083	0.489289
$697$	59.4500	2.25183
$698$	28.2389	1.06886
$699$	28.1194	1.06357
$700$	0	0
$701$	40.1194	1.51529	0.757645	−	0.652667i	$-0.226350\pi$
$701$	40.1194	1.51529	0.757645	+	0.652667i	$0.226350\pi$
$702$	21.1194	0.797101
$703$	−2.00000	−0.0754314
$704$	−2.30278	−0.0867891
$705$	0	0
$706$	−10.1833	−0.383255
$707$	−32.3667	−1.21727
$708$	−35.0278	−1.31642
$709$	−41.3305	−1.55220	−0.776100	−	0.630609i	$-0.782805\pi$
$709$	−41.3305	−1.55220	−0.776100	+	0.630609i	$0.782805\pi$
$710$	0	0
$711$	72.1194	2.70469
$712$	−9.21110	−0.345201
$713$	1.18335	0.0443167
$714$	−51.6333	−1.93233
$715$	0	0
$716$	−13.8167	−0.516353
$717$	57.8444	2.16024
$718$	3.21110	0.119837
$719$	−51.6333	−1.92560	−0.962799	−	0.270220i	$-0.912904\pi$
$719$	−51.6333	−1.92560	−0.962799	+	0.270220i	$0.912904\pi$
$720$	0	0
$721$	0.788897	0.0293801
$722$	−15.0000	−0.558242
$723$	−26.4222	−0.982652
$724$	20.0000	0.743294
$725$	0	0
$726$	18.8167	0.698352
$727$	−19.0917	−0.708071	−0.354035	−	0.935232i	$-0.615191\pi$
$727$	−19.0917	−0.708071	−0.354035	+	0.935232i	$0.615191\pi$
$728$	−3.39445	−0.125807
$729$	75.1749	2.78426
$730$	0	0
$731$	−3.63331	−0.134383
$732$	−24.8167	−0.917250
$733$	13.6333	0.503558	0.251779	−	0.967785i	$-0.418984\pi$
$733$	13.6333	0.503558	0.251779	+	0.967785i	$0.418984\pi$
$734$	−3.81665	−0.140875
$735$	0	0
$736$	−3.90833	−0.144063
$737$	−8.09167	−0.298061
$738$	78.3583	2.88441
$739$	−2.66947	−0.0981980	−0.0490990	−	0.998794i	$-0.515635\pi$
$739$	−2.66947	−0.0981980	−0.0490990	+	0.998794i	$0.515635\pi$
$740$	0	0
$741$	8.60555	0.316133
$742$	15.6333	0.573917
$743$	29.4500	1.08041	0.540207	−	0.841532i	$-0.318346\pi$
$743$	29.4500	1.08041	0.540207	+	0.841532i	$0.318346\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	17.8167	0.652314
$747$	−22.0555	−0.806969
$748$	−13.8167	−0.505187
$749$	−1.81665	−0.0663791
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−4.60555	−0.167947
$753$	70.0555	2.55296
$754$	5.09167	0.185428
$755$	0	0
$756$	−42.2389	−1.53621
$757$	−5.69722	−0.207069	−0.103535	−	0.994626i	$-0.533015\pi$
$757$	−5.69722	−0.207069	−0.103535	+	0.994626i	$0.533015\pi$
$758$	24.3305	0.883725
$759$	−29.7250	−1.07895
$760$	0	0
$761$	16.8806	0.611920	0.305960	−	0.952044i	$-0.401023\pi$
$761$	16.8806	0.611920	0.305960	+	0.952044i	$0.401023\pi$
$762$	−63.4500	−2.29855
$763$	5.21110	0.188655
$764$	−5.51388	−0.199485
$765$	0	0
$766$	36.8444	1.33124
$767$	−13.8167	−0.498890
$768$	−3.30278	−0.119179
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	10.6056	0.381950
$772$	4.00000	0.143963
$773$	−22.0555	−0.793282	−0.396641	−	0.917974i	$-0.629824\pi$
$773$	−22.0555	−0.793282	−0.396641	+	0.917974i	$0.629824\pi$
$774$	−4.78890	−0.172133
$775$	0	0
$776$	16.4222	0.589523
$777$	8.60555	0.308722
$778$	−37.1194	−1.33080
$779$	19.8167	0.710005
$780$	0	0
$781$	−13.8167	−0.494399
$782$	−23.4500	−0.838569
$783$	63.3583	2.26424
$784$	−0.211103	−0.00753938
$785$	0	0
$786$	−35.0278	−1.24940
$787$	−10.7889	−0.384583	−0.192291	−	0.981338i	$-0.561592\pi$
$787$	−10.7889	−0.384583	−0.192291	+	0.981338i	$0.561592\pi$
$788$	6.00000	0.213741
$789$	45.6333	1.62459
$790$	0	0
$791$	8.36669	0.297485
$792$	−18.2111	−0.647103
$793$	−9.78890	−0.347614
$794$	−6.18335	−0.219439
$795$	0	0
$796$	26.4222	0.936510
$797$	−22.3305	−0.790988	−0.395494	−	0.918469i	$-0.629427\pi$
$797$	−22.3305	−0.790988	−0.395494	+	0.918469i	$0.629427\pi$
$798$	−17.2111	−0.609266
$799$	−27.6333	−0.977596
$800$	0	0
$801$	−72.8444	−2.57383
$802$	−7.81665	−0.276016
$803$	−28.3305	−0.999763
$804$	−11.6056	−0.409296
$805$	0	0
$806$	0.394449	0.0138939
$807$	70.0555	2.46607
$808$	−12.4222	−0.437012
$809$	−35.4500	−1.24635	−0.623177	−	0.782081i	$-0.714158\pi$
$809$	−35.4500	−1.24635	−0.623177	+	0.782081i	$0.714158\pi$
$810$	0	0
$811$	−7.14719	−0.250972	−0.125486	−	0.992095i	$-0.540049\pi$
$811$	−7.14719	−0.250972	−0.125486	+	0.992095i	$0.540049\pi$
$812$	−10.1833	−0.357365
$813$	74.0555	2.59724
$814$	2.30278	0.0807122
$815$	0	0
$816$	−19.8167	−0.693722
$817$	−1.21110	−0.0423711
$818$	31.0278	1.08486
$819$	−26.8444	−0.938020
$820$	0	0
$821$	−3.21110	−0.112068	−0.0560341	−	0.998429i	$-0.517846\pi$
$821$	−3.21110	−0.112068	−0.0560341	+	0.998429i	$0.517846\pi$
$822$	3.00000	0.104637
$823$	−44.8444	−1.56318	−0.781589	−	0.623794i	$-0.785590\pi$
$823$	−44.8444	−1.56318	−0.781589	+	0.623794i	$0.785590\pi$
$824$	0.302776	0.0105477
$825$	0	0
$826$	27.6333	0.961486
$827$	−34.6056	−1.20335	−0.601676	−	0.798740i	$-0.705500\pi$
$827$	−34.6056	−1.20335	−0.601676	+	0.798740i	$0.705500\pi$
$828$	−30.9083	−1.07414
$829$	−27.7250	−0.962928	−0.481464	−	0.876466i	$-0.659895\pi$
$829$	−27.7250	−0.962928	−0.481464	+	0.876466i	$0.659895\pi$
$830$	0	0
$831$	0.394449	0.0136833
$832$	−1.30278	−0.0451656
$833$	−1.26662	−0.0438856
$834$	6.30278	0.218247
$835$	0	0
$836$	−4.60555	−0.159286
$837$	4.90833	0.169657
$838$	36.1472	1.24868
$839$	−12.9722	−0.447852	−0.223926	−	0.974606i	$-0.571887\pi$
$839$	−12.9722	−0.447852	−0.223926	+	0.974606i	$0.571887\pi$
$840$	0	0
$841$	−13.7250	−0.473275
$842$	−3.72498	−0.128371
$843$	39.6333	1.36504
$844$	10.3028	0.354636
$845$	0	0
$846$	−36.4222	−1.25222
$847$	−14.8444	−0.510060
$848$	6.00000	0.206041
$849$	81.2666	2.78906
$850$	0	0
$851$	3.90833	0.133976
$852$	−19.8167	−0.678907
$853$	−42.5416	−1.45660	−0.728299	−	0.685260i	$-0.759689\pi$
$853$	−42.5416	−1.45660	−0.728299	+	0.685260i	$0.759689\pi$
$854$	19.5778	0.669938
$855$	0	0
$856$	−0.697224	−0.0238306
$857$	−42.8444	−1.46354	−0.731769	−	0.681553i	$-0.761305\pi$
$857$	−42.8444	−1.46354	−0.731769	+	0.681553i	$0.761305\pi$
$858$	−9.90833	−0.338265
$859$	48.0555	1.63963	0.819816	−	0.572626i	$-0.194075\pi$
$859$	48.0555	1.63963	0.819816	+	0.572626i	$0.194075\pi$
$860$	0	0
$861$	−85.2666	−2.90588
$862$	9.21110	0.313731
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	−16.2111	−0.551513
$865$	0	0
$866$	−34.9361	−1.18718
$867$	−62.7527	−2.13119
$868$	−0.788897	−0.0267769
$869$	−21.0000	−0.712376
$870$	0	0
$871$	−4.57779	−0.155113
$872$	2.00000	0.0677285
$873$	129.872	4.39551
$874$	−7.81665	−0.264402
$875$	0	0
$876$	−40.6333	−1.37287
$877$	7.21110	0.243502	0.121751	−	0.992561i	$-0.461149\pi$
$877$	7.21110	0.243502	0.121751	+	0.992561i	$0.461149\pi$
$878$	30.3305	1.02361
$879$	36.4222	1.22849
$880$	0	0
$881$	−28.5416	−0.961592	−0.480796	−	0.876832i	$-0.659652\pi$
$881$	−28.5416	−0.961592	−0.480796	+	0.876832i	$0.659652\pi$
$882$	−1.66947	−0.0562139
$883$	−26.4222	−0.889178	−0.444589	−	0.895735i	$-0.646650\pi$
$883$	−26.4222	−0.889178	−0.444589	+	0.895735i	$0.646650\pi$
$884$	−7.81665	−0.262903
$885$	0	0
$886$	−32.7250	−1.09942
$887$	0.422205	0.0141763	0.00708813	−	0.999975i	$-0.497744\pi$
$887$	0.422205	0.0141763	0.00708813	+	0.999975i	$0.497744\pi$
$888$	3.30278	0.110834
$889$	50.0555	1.67881
$890$	0	0
$891$	−68.6611	−2.30023
$892$	5.81665	0.194756
$893$	−9.21110	−0.308238
$894$	−65.4500	−2.18897
$895$	0	0
$896$	2.60555	0.0870454
$897$	−16.8167	−0.561492
$898$	−15.2111	−0.507601
$899$	1.18335	0.0394668
$900$	0	0
$901$	36.0000	1.19933
$902$	−22.8167	−0.759711
$903$	5.21110	0.173415
$904$	3.21110	0.106800
$905$	0	0
$906$	68.0555	2.26099
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	−13.8167	−0.458522
$909$	−98.2389	−3.25838
$910$	0	0
$911$	−17.5778	−0.582378	−0.291189	−	0.956665i	$-0.594051\pi$
$911$	−17.5778	−0.582378	−0.291189	+	0.956665i	$0.594051\pi$
$912$	−6.60555	−0.218732
$913$	6.42221	0.212544
$914$	2.60555	0.0861840
$915$	0	0
$916$	24.6056	0.812990
$917$	27.6333	0.912532
$918$	−97.2666	−3.21028
$919$	−9.57779	−0.315942	−0.157971	−	0.987444i	$-0.550495\pi$
$919$	−9.57779	−0.315942	−0.157971	+	0.987444i	$0.550495\pi$
$920$	0	0
$921$	59.1472	1.94897
$922$	12.4222	0.409104
$923$	−7.81665	−0.257288
$924$	19.8167	0.651920
$925$	0	0
$926$	−26.6972	−0.877325
$927$	2.39445	0.0786440
$928$	−3.90833	−0.128297
$929$	−18.4861	−0.606510	−0.303255	−	0.952909i	$-0.598073\pi$
$929$	−18.4861	−0.606510	−0.303255	+	0.952909i	$0.598073\pi$
$930$	0	0
$931$	−0.422205	−0.0138372
$932$	−8.51388	−0.278881
$933$	−52.5416	−1.72014
$934$	0	0
$935$	0	0
$936$	−10.3028	−0.336757
$937$	18.0917	0.591029	0.295515	−	0.955338i	$-0.404509\pi$
$937$	18.0917	0.591029	0.295515	+	0.955338i	$0.404509\pi$
$938$	9.15559	0.298941
$939$	−29.8167	−0.973030
$940$	0	0
$941$	13.8167	0.450410	0.225205	−	0.974311i	$-0.427695\pi$
$941$	13.8167	0.450410	0.225205	+	0.974311i	$0.427695\pi$
$942$	−23.8167	−0.775989
$943$	−38.7250	−1.26106
$944$	10.6056	0.345181
$945$	0	0
$946$	1.39445	0.0453374
$947$	3.63331	0.118067	0.0590333	−	0.998256i	$-0.481198\pi$
$947$	3.63331	0.118067	0.0590333	+	0.998256i	$0.481198\pi$
$948$	−30.1194	−0.978234
$949$	−16.0278	−0.520283
$950$	0	0
$951$	30.4222	0.986508
$952$	15.6333	0.506678
$953$	−49.7527	−1.61165	−0.805825	−	0.592154i	$-0.798278\pi$
$953$	−49.7527	−1.61165	−0.805825	+	0.592154i	$0.798278\pi$
$954$	47.4500	1.53625
$955$	0	0
$956$	−17.5139	−0.566439
$957$	−29.7250	−0.960872
$958$	−13.1194	−0.423870
$959$	−2.36669	−0.0764245
$960$	0	0
$961$	−30.9083	−0.997043
$962$	1.30278	0.0420032
$963$	−5.51388	−0.177682
$964$	8.00000	0.257663
$965$	0	0
$966$	33.6333	1.08213
$967$	6.72498	0.216261	0.108130	−	0.994137i	$-0.465514\pi$
$967$	6.72498	0.216261	0.108130	+	0.994137i	$0.465514\pi$
$968$	−5.69722	−0.183116
$969$	−39.6333	−1.27321
$970$	0	0
$971$	−22.5416	−0.723395	−0.361698	−	0.932295i	$-0.617803\pi$
$971$	−22.5416	−0.723395	−0.361698	+	0.932295i	$0.617803\pi$
$972$	−49.8444	−1.59876
$973$	−4.97224	−0.159403
$974$	37.2111	1.19232
$975$	0	0
$976$	7.51388	0.240513
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	27.8167	0.889479
$979$	21.2111	0.677910
$980$	0	0
$981$	15.8167	0.504987
$982$	17.7250	0.565627
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	−32.7250	−1.04323
$985$	0	0
$986$	−23.4500	−0.746799
$987$	39.6333	1.26154
$988$	−2.60555	−0.0828936
$989$	2.36669	0.0752564
$990$	0	0
$991$	50.6972	1.61045	0.805225	−	0.592969i	$-0.202044\pi$
$991$	50.6972	1.61045	0.805225	+	0.592969i	$0.202044\pi$
$992$	−0.302776	−0.00961314
$993$	43.6333	1.38466
$994$	15.6333	0.495858
$995$	0	0
$996$	9.21110	0.291865
$997$	52.4222	1.66023	0.830114	−	0.557594i	$-0.188275\pi$
$997$	52.4222	1.66023	0.830114	+	0.557594i	$0.188275\pi$
$998$	−42.2389	−1.33705
$999$	16.2111	0.512897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.u.1.1		2
5.2	odd	4		1850.2.b.i.149.4		4
5.3	odd	4		1850.2.b.i.149.1		4
5.4	even	2		74.2.a.a.1.2	✓	2
15.14	odd	2		666.2.a.j.1.2		2
20.19	odd	2		592.2.a.f.1.1		2
35.34	odd	2		3626.2.a.a.1.1		2
40.19	odd	2		2368.2.a.ba.1.2		2
40.29	even	2		2368.2.a.s.1.1		2
55.54	odd	2		8954.2.a.p.1.2		2
60.59	even	2		5328.2.a.bf.1.2		2
185.184	even	2		2738.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.a.a.1.2	✓	2	5.4	even	2
592.2.a.f.1.1		2	20.19	odd	2
666.2.a.j.1.2		2	15.14	odd	2
1850.2.a.u.1.1		2	1.1	even	1	trivial
1850.2.b.i.149.1		4	5.3	odd	4
1850.2.b.i.149.4		4	5.2	odd	4
2368.2.a.s.1.1		2	40.29	even	2
2368.2.a.ba.1.2		2	40.19	odd	2
2738.2.a.l.1.2		2	185.184	even	2
3626.2.a.a.1.1		2	35.34	odd	2
5328.2.a.bf.1.2		2	60.59	even	2
8954.2.a.p.1.2		2	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} -3.30278 q^{6} +2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} -2.30278 q^{11} -3.30278 q^{12} -1.30278 q^{13} +2.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} +7.90833 q^{18} +2.00000 q^{19} -8.60555 q^{21} -2.30278 q^{22} -3.90833 q^{23} -3.30278 q^{24} -1.30278 q^{26} -16.2111 q^{27} +2.60555 q^{28} -3.90833 q^{29} -0.302776 q^{31} +1.00000 q^{32} +7.60555 q^{33} +6.00000 q^{34} +7.90833 q^{36} -1.00000 q^{37} +2.00000 q^{38} +4.30278 q^{39} +9.90833 q^{41} -8.60555 q^{42} -0.605551 q^{43} -2.30278 q^{44} -3.90833 q^{46} -4.60555 q^{47} -3.30278 q^{48} -0.211103 q^{49} -19.8167 q^{51} -1.30278 q^{52} +6.00000 q^{53} -16.2111 q^{54} +2.60555 q^{56} -6.60555 q^{57} -3.90833 q^{58} +10.6056 q^{59} +7.51388 q^{61} -0.302776 q^{62} +20.6056 q^{63} +1.00000 q^{64} +7.60555 q^{66} +3.51388 q^{67} +6.00000 q^{68} +12.9083 q^{69} +6.00000 q^{71} +7.90833 q^{72} +12.3028 q^{73} -1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} +4.30278 q^{78} +9.11943 q^{79} +29.8167 q^{81} +9.90833 q^{82} -2.78890 q^{83} -8.60555 q^{84} -0.605551 q^{86} +12.9083 q^{87} -2.30278 q^{88} -9.21110 q^{89} -3.39445 q^{91} -3.90833 q^{92} +1.00000 q^{93} -4.60555 q^{94} -3.30278 q^{96} +16.4222 q^{97} -0.211103 q^{98} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{11} - 3 q^{12} + q^{13} - 2 q^{14} + 2 q^{16} + 12 q^{17} + 5 q^{18} + 4 q^{19} - 10 q^{21} - q^{22} + 3 q^{23} - 3 q^{24}+ \cdots - 22 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 2 * q^7 + 2 * q^8 + 5 * q^9 - q^11 - 3 * q^12 + q^13 - 2 * q^14 + 2 * q^16 + 12 * q^17 + 5 * q^18 + 4 * q^19 - 10 * q^21 - q^22 + 3 * q^23 - 3 * q^24 + q^26 - 18 * q^27 - 2 * q^28 + 3 * q^29 + 3 * q^31 + 2 * q^32 + 8 * q^33 + 12 * q^34 + 5 * q^36 - 2 * q^37 + 4 * q^38 + 5 * q^39 + 9 * q^41 - 10 * q^42 + 6 * q^43 - q^44 + 3 * q^46 - 2 * q^47 - 3 * q^48 + 14 * q^49 - 18 * q^51 + q^52 + 12 * q^53 - 18 * q^54 - 2 * q^56 - 6 * q^57 + 3 * q^58 + 14 * q^59 - 3 * q^61 + 3 * q^62 + 34 * q^63 + 2 * q^64 + 8 * q^66 - 11 * q^67 + 12 * q^68 + 15 * q^69 + 12 * q^71 + 5 * q^72 + 21 * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^77 + 5 * q^78 - 7 * q^79 + 38 * q^81 + 9 * q^82 - 20 * q^83 - 10 * q^84 + 6 * q^86 + 15 * q^87 - q^88 - 4 * q^89 - 14 * q^91 + 3 * q^92 + 2 * q^93 - 2 * q^94 - 3 * q^96 + 4 * q^97 + 14 * q^98 - 22 * q^99

Introduction

L-functions

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