LMFDB - Embedded newform 1900.3.e.g.1101.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 $ x^14 + 89x^12 + 3026x^10 + 49092x^8 + 390297x^6 + 1440495x^4 + 1994425x^2 + 151875
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.6
Root			$-1.84332i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.g.1101.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84332i q^{3} -10.5375 q^{7} +5.60216 q^{9} +O(q^{10})$	$q-1.84332i q^{3} -10.5375 q^{7} +5.60216 q^{9} +12.7272 q^{11} -23.0282i q^{13} +4.77411 q^{17} +(1.66986 + 18.9265i) q^{19} +19.4239i q^{21} +35.7018 q^{23} -26.9165i q^{27} +13.7704i q^{29} +30.5166i q^{31} -23.4603i q^{33} +54.8524i q^{37} -42.4484 q^{39} -18.6001i q^{41} -29.0112 q^{43} -11.3148 q^{47} +62.0382 q^{49} -8.80023i q^{51} -10.7760i q^{53} +(34.8876 - 3.07808i) q^{57} -54.5728i q^{59} +17.7698 q^{61} -59.0326 q^{63} -60.2136i q^{67} -65.8100i q^{69} -39.9041i q^{71} +24.6243 q^{73} -134.112 q^{77} +6.95992i q^{79} +0.803679 q^{81} +130.188 q^{83} +25.3833 q^{87} -160.576i q^{89} +242.659i q^{91} +56.2519 q^{93} -71.2997i q^{97} +71.2998 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) $ 14 * q - 4 * q^7 - 52 * q^9	$ 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) $ 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.84332i		−	0.614441i	−0.951638	−	0.307220i	$-0.900601\pi$
$3$		−	1.84332i		−	0.614441i	0.951638	−	0.307220i	$-0.0993989\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−10.5375			−1.50535			−0.752676	−	0.658391i	$-0.771237\pi$
$7$	−10.5375			−1.50535			−0.752676	+	0.658391i	$0.771237\pi$
$8$		0			0
$9$	5.60216			0.622462
$10$		0			0
$11$	12.7272			1.15702			0.578509	−	0.815676i	$-0.303635\pi$
$11$	12.7272			1.15702			0.578509	+	0.815676i	$0.303635\pi$
$12$		0			0
$13$		−	23.0282i		−	1.77140i	−0.464258	−	0.885700i	$-0.653679\pi$
$13$		−	23.0282i		−	1.77140i	0.464258	−	0.885700i	$-0.346321\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.77411			0.280830			0.140415	−	0.990093i	$-0.455156\pi$
$17$	4.77411			0.280830			0.140415	+	0.990093i	$0.455156\pi$
$18$		0			0
$19$	1.66986	+	18.9265i	0.0878871	+	0.996130i
$19$	1.66986	+	18.9265i	0.0878871	+	0.996130i
$20$		0			0
$21$			19.4239i			0.924950i
$22$		0			0
$23$	35.7018			1.55225			0.776127	−	0.630576i	$-0.217181\pi$
$23$	35.7018			1.55225			0.776127	+	0.630576i	$0.217181\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	26.9165i		−	0.996907i
$28$		0			0
$29$			13.7704i			0.474842i	0.971407	+	0.237421i	$0.0763020\pi$
$29$			13.7704i			0.474842i	−0.971407	+	0.237421i	$0.923698\pi$
$30$		0			0
$31$			30.5166i			0.984405i	0.870481	+	0.492203i	$0.163808\pi$
$31$			30.5166i			0.984405i	−0.870481	+	0.492203i	$0.836192\pi$
$32$		0			0
$33$		−	23.4603i		−	0.710919i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			54.8524i			1.48250i	0.671230	+	0.741249i	$0.265766\pi$
$37$			54.8524i			1.48250i	−0.671230	+	0.741249i	$0.734234\pi$
$38$		0			0
$39$	−42.4484			−1.08842
$40$		0			0
$41$		−	18.6001i		−	0.453660i	−0.973934	−	0.226830i	$-0.927164\pi$
$41$		−	18.6001i		−	0.453660i	0.973934	−	0.226830i	$-0.0728362\pi$
$42$		0			0
$43$	−29.0112			−0.674679			−0.337340	−	0.941383i	$-0.609527\pi$
$43$	−29.0112			−0.674679			−0.337340	+	0.941383i	$0.609527\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−11.3148			−0.240740			−0.120370	−	0.992729i	$-0.538408\pi$
$47$	−11.3148			−0.240740			−0.120370	+	0.992729i	$0.538408\pi$
$48$		0			0
$49$	62.0382			1.26609
$50$		0			0
$51$		−	8.80023i		−	0.172553i
$52$		0			0
$53$		−	10.7760i		−	0.203321i	−0.994819	−	0.101660i	$-0.967585\pi$
$53$		−	10.7760i		−	0.203321i	0.994819	−	0.101660i	$-0.0324155\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	34.8876	−	3.07808i	0.612063	−	0.0540014i
$58$		0			0
$59$		−	54.5728i		−	0.924963i	−0.886629	−	0.462482i	$-0.846959\pi$
$59$		−	54.5728i		−	0.924963i	0.886629	−	0.462482i	$-0.153041\pi$
$60$		0			0
$61$	17.7698			0.291308			0.145654	−	0.989336i	$-0.453471\pi$
$61$	17.7698			0.291308			0.145654	+	0.989336i	$0.453471\pi$
$62$		0			0
$63$	−59.0326			−0.937025
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	60.2136i		−	0.898710i	−0.893353	−	0.449355i	$-0.851654\pi$
$67$		−	60.2136i		−	0.898710i	0.893353	−	0.449355i	$-0.148346\pi$
$68$		0			0
$69$		−	65.8100i		−	0.953768i
$70$		0			0
$71$		−	39.9041i		−	0.562030i	−0.959703	−	0.281015i	$-0.909329\pi$
$71$		−	39.9041i		−	0.562030i	0.959703	−	0.281015i	$-0.0906710\pi$
$72$		0			0
$73$	24.6243			0.337319			0.168660	−	0.985674i	$-0.446056\pi$
$73$	24.6243			0.337319			0.168660	+	0.985674i	$0.446056\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−134.112			−1.74172
$78$		0			0
$79$			6.95992i			0.0881003i	0.999029	+	0.0440502i	$0.0140261\pi$
$79$			6.95992i			0.0881003i	−0.999029	+	0.0440502i	$0.985974\pi$
$80$		0			0
$81$	0.803679			0.00992196
$82$		0			0
$83$	130.188			1.56853			0.784263	−	0.620429i	$-0.213041\pi$
$83$	130.188			1.56853			0.784263	+	0.620429i	$0.213041\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	25.3833			0.291762
$88$		0			0
$89$		−	160.576i		−	1.80423i	−0.431498	−	0.902114i	$-0.642015\pi$
$89$		−	160.576i		−	1.80423i	0.431498	−	0.902114i	$-0.357985\pi$
$90$		0			0
$91$			242.659i			2.66658i
$92$		0			0
$93$	56.2519			0.604859
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	71.2997i		−	0.735049i	−0.930014	−	0.367524i	$-0.880205\pi$
$97$		−	71.2997i		−	0.735049i	0.930014	−	0.367524i	$-0.119795\pi$
$98$		0			0
$99$	71.2998			0.720200
$100$		0			0
$101$	107.366			1.06303			0.531514	−	0.847049i	$-0.321623\pi$
$101$	107.366			1.06303			0.531514	+	0.847049i	$0.321623\pi$
$102$		0			0
$103$			16.6106i			0.161268i	0.996744	+	0.0806341i	$0.0256945\pi$
$103$			16.6106i			0.161268i	−0.996744	+	0.0806341i	$0.974305\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	133.725i		−	1.24977i	−0.780718	−	0.624884i	$-0.785146\pi$
$107$		−	133.725i		−	1.24977i	0.780718	−	0.624884i	$-0.214854\pi$
$108$		0			0
$109$		−	176.340i		−	1.61780i	−0.587947	−	0.808899i	$-0.700064\pi$
$109$		−	176.340i		−	1.61780i	0.587947	−	0.808899i	$-0.299936\pi$
$110$		0			0
$111$	101.111			0.910908
$112$		0			0
$113$		−	224.945i		−	1.99066i	−0.0965339	−	0.995330i	$-0.530776\pi$
$113$		−	224.945i		−	1.99066i	0.0965339	−	0.995330i	$-0.469224\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	129.008i		−	1.10263i
$118$		0			0
$119$	−50.3070			−0.422748
$120$		0			0
$121$	40.9815			0.338690
$122$		0			0
$123$	−34.2859			−0.278747
$124$		0			0
$125$		0			0
$126$		0			0
$127$			125.867i			0.991076i	0.868586	+	0.495538i	$0.165029\pi$
$127$			125.867i			0.991076i	−0.868586	+	0.495538i	$0.834971\pi$
$128$		0			0
$129$			53.4770i			0.414551i
$130$		0			0
$131$	−169.986			−1.29760			−0.648800	−	0.760959i	$-0.724729\pi$
$131$	−169.986			−1.29760			−0.648800	+	0.760959i	$0.724729\pi$
$132$		0			0
$133$	−17.5960	−	199.437i	−0.132301	−	1.49953i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	169.636			1.23822			0.619109	−	0.785305i	$-0.287494\pi$
$137$	169.636			1.23822			0.619109	+	0.785305i	$0.287494\pi$
$138$		0			0
$139$	250.882			1.80491			0.902455	−	0.430784i	$-0.141763\pi$
$139$	250.882			1.80491			0.902455	+	0.430784i	$0.141763\pi$
$140$		0			0
$141$			20.8568i			0.147920i
$142$		0			0
$143$		−	293.084i		−	2.04954i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	114.356i		−	0.777935i
$148$		0			0
$149$	−156.146			−1.04796			−0.523979	−	0.851731i	$-0.675553\pi$
$149$	−156.146			−1.04796			−0.523979	+	0.851731i	$0.675553\pi$
$150$		0			0
$151$			86.4696i			0.572646i	0.958133	+	0.286323i	$0.0924331\pi$
$151$			86.4696i			0.572646i	−0.958133	+	0.286323i	$0.907567\pi$
$152$		0			0
$153$	26.7453			0.174806
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.9932			0.382122			0.191061	−	0.981578i	$-0.438807\pi$
$157$	59.9932			0.382122			0.191061	+	0.981578i	$0.438807\pi$
$158$		0			0
$159$	−19.8636			−0.124929
$160$		0			0
$161$	−376.207			−2.33669
$162$		0			0
$163$	−206.566			−1.26727			−0.633637	−	0.773630i	$-0.718439\pi$
$163$	−206.566			−1.26727			−0.633637	+	0.773630i	$0.718439\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			175.728i			1.05227i	0.850403	+	0.526133i	$0.176358\pi$
$167$			175.728i			1.05227i	−0.850403	+	0.526133i	$0.823642\pi$
$168$		0			0
$169$	−361.298			−2.13786
$170$		0			0
$171$	9.35480	+	106.029i	0.0547064	+	0.620054i
$172$		0			0
$173$		−	36.0672i		−	0.208481i	−0.994552	−	0.104241i	$-0.966759\pi$
$173$		−	36.0672i		−	0.208481i	0.994552	−	0.104241i	$-0.0332412\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−100.595			−0.568335
$178$		0			0
$179$		−	219.282i		−	1.22504i	−0.790456	−	0.612518i	$-0.790157\pi$
$179$		−	219.282i		−	1.22504i	0.790456	−	0.612518i	$-0.209843\pi$
$180$		0			0
$181$			40.5089i			0.223806i	0.993719	+	0.111903i	$0.0356946\pi$
$181$			40.5089i			0.223806i	−0.993719	+	0.111903i	$0.964305\pi$
$182$		0			0
$183$		−	32.7555i		−	0.178992i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	60.7611			0.324925
$188$		0			0
$189$			283.632i			1.50070i
$190$		0			0
$191$	−83.0565			−0.434851			−0.217425	−	0.976077i	$-0.569766\pi$
$191$	−83.0565			−0.434851			−0.217425	+	0.976077i	$0.569766\pi$
$192$		0			0
$193$		−	187.911i		−	0.973630i	−0.873505	−	0.486815i	$-0.838159\pi$
$193$		−	187.911i		−	0.973630i	0.873505	−	0.486815i	$-0.161841\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−170.319			−0.864562			−0.432281	−	0.901739i	$-0.642291\pi$
$197$	−170.319			−0.864562			−0.432281	+	0.901739i	$0.642291\pi$
$198$		0			0
$199$	121.271			0.609400			0.304700	−	0.952448i	$-0.401444\pi$
$199$	121.271			0.609400			0.304700	+	0.952448i	$0.401444\pi$
$200$		0			0
$201$	−110.993			−0.552204
$202$		0			0
$203$		−	145.105i		−	0.714804i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	200.008			0.966220
$208$		0			0
$209$	21.2526	+	240.881i	0.101687	+	1.15254i
$210$		0			0
$211$			182.151i			0.863276i	0.902047	+	0.431638i	$0.142064\pi$
$211$			182.151i			0.863276i	−0.902047	+	0.431638i	$0.857936\pi$
$212$		0			0
$213$	−73.5561			−0.345334
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	321.567i		−	1.48188i
$218$		0			0
$219$		−	45.3906i		−	0.207263i
$220$		0			0
$221$		−	109.939i		−	0.497462i
$222$		0			0
$223$			215.134i			0.964727i	0.875971	+	0.482363i	$0.160222\pi$
$223$			215.134i			0.964727i	−0.875971	+	0.482363i	$0.839778\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			265.837i			1.17109i	0.810640	+	0.585545i	$0.199119\pi$
$227$			265.837i			1.17109i	−0.810640	+	0.585545i	$0.800881\pi$
$228$		0			0
$229$	115.365			0.503777			0.251889	−	0.967756i	$-0.418948\pi$
$229$	115.365			0.503777			0.251889	+	0.967756i	$0.418948\pi$
$230$		0			0
$231$			247.212i			1.07018i
$232$		0			0
$233$	−189.784			−0.814524			−0.407262	−	0.913311i	$-0.633516\pi$
$233$	−189.784			−0.814524			−0.407262	+	0.913311i	$0.633516\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	12.8294			0.0541324
$238$		0			0
$239$	99.3829			0.415828			0.207914	−	0.978147i	$-0.433333\pi$
$239$	99.3829			0.415828			0.207914	+	0.978147i	$0.433333\pi$
$240$		0			0
$241$			130.340i			0.540829i	0.962744	+	0.270414i	$0.0871607\pi$
$241$			130.340i			0.540829i	−0.962744	+	0.270414i	$0.912839\pi$
$242$		0			0
$243$		−	243.730i		−	1.00300i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	435.843	−	38.4538i	1.76455	−	0.155683i
$248$		0			0
$249$		−	239.978i		−	0.963766i
$250$		0			0
$251$	501.184			1.99675			0.998374	−	0.0569953i	$-0.0181520\pi$
$251$	501.184			1.99675			0.998374	+	0.0569953i	$0.0181520\pi$
$252$		0			0
$253$	454.384			1.79599
$254$		0			0
$255$		0			0
$256$		0			0
$257$			162.791i			0.633428i	0.948521	+	0.316714i	$0.102580\pi$
$257$			162.791i			0.633428i	−0.948521	+	0.316714i	$0.897420\pi$
$258$		0			0
$259$		−	578.006i		−	2.23168i
$260$		0			0
$261$			77.1440i			0.295571i
$262$		0			0
$263$	−50.4816			−0.191945			−0.0959727	−	0.995384i	$-0.530596\pi$
$263$	−50.4816			−0.191945			−0.0959727	+	0.995384i	$0.530596\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−295.994			−1.10859
$268$		0			0
$269$			28.7859i			0.107011i	0.998568	+	0.0535053i	$0.0170394\pi$
$269$			28.7859i			0.107011i	−0.998568	+	0.0535053i	$0.982961\pi$
$270$		0			0
$271$	−400.882			−1.47927			−0.739634	−	0.673009i	$-0.765001\pi$
$271$	−400.882			−1.47927			−0.739634	+	0.673009i	$0.765001\pi$
$272$		0			0
$273$	447.299			1.63846
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−213.844			−0.772001			−0.386000	−	0.922499i	$-0.626144\pi$
$277$	−213.844			−0.772001			−0.386000	+	0.922499i	$0.626144\pi$
$278$		0			0
$279$			170.959i			0.612755i
$280$		0			0
$281$		−	229.607i		−	0.817108i	−0.912734	−	0.408554i	$-0.866033\pi$
$281$		−	229.607i		−	0.817108i	0.912734	−	0.408554i	$-0.133967\pi$
$282$		0			0
$283$	−201.356			−0.711506			−0.355753	−	0.934580i	$-0.615776\pi$
$283$	−201.356			−0.711506			−0.355753	+	0.934580i	$0.615776\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			195.998i			0.682919i
$288$		0			0
$289$	−266.208			−0.921134
$290$		0			0
$291$	−131.428			−0.451644
$292$		0			0
$293$		−	285.899i		−	0.975765i	−0.872909	−	0.487883i	$-0.837769\pi$
$293$		−	285.899i		−	0.975765i	0.872909	−	0.487883i	$-0.162231\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	342.572i		−	1.15344i
$298$		0			0
$299$		−	822.149i		−	2.74966i
$300$		0			0
$301$	305.705			1.01563
$302$		0			0
$303$		−	197.910i		−	0.653168i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			38.7426i			0.126197i	0.998007	+	0.0630986i	$0.0200983\pi$
$307$			38.7426i			0.126197i	−0.998007	+	0.0630986i	$0.979902\pi$
$308$		0			0
$309$	30.6187			0.0990898
$310$		0			0
$311$	607.655			1.95387			0.976937	−	0.213526i	$-0.0684949\pi$
$311$	607.655			1.95387			0.976937	+	0.213526i	$0.0684949\pi$
$312$		0			0
$313$	−293.775			−0.938577			−0.469289	−	0.883045i	$-0.655490\pi$
$313$	−293.775			−0.938577			−0.469289	+	0.883045i	$0.655490\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	558.422i		−	1.76158i	−0.473503	−	0.880792i	$-0.657011\pi$
$317$		−	558.422i		−	1.76158i	0.473503	−	0.880792i	$-0.342989\pi$
$318$		0			0
$319$			175.259i			0.549400i
$320$		0			0
$321$	−246.498			−0.767908
$322$		0			0
$323$	7.97208	+	90.3571i	0.0246814	+	0.279743i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−325.052			−0.994041
$328$		0			0
$329$	119.229			0.362398
$330$		0			0
$331$		−	366.890i		−	1.10843i	−0.832374	−	0.554214i	$-0.813019\pi$
$331$		−	366.890i		−	1.10843i	0.832374	−	0.554214i	$-0.186981\pi$
$332$		0			0
$333$			307.292i			0.922800i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	466.486i		−	1.38423i	−0.721787	−	0.692115i	$-0.756679\pi$
$337$		−	466.486i		−	1.38423i	0.721787	−	0.692115i	$-0.243321\pi$
$338$		0			0
$339$	−414.645			−1.22314
$340$		0			0
$341$			388.390i			1.13897i
$342$		0			0
$343$	−137.389			−0.400552
$344$		0			0
$345$		0			0
$346$		0			0
$347$	490.615			1.41388			0.706938	−	0.707275i	$-0.250076\pi$
$347$	490.615			1.41388			0.706938	+	0.707275i	$0.250076\pi$
$348$		0			0
$349$	503.588			1.44295			0.721473	−	0.692443i	$-0.243465\pi$
$349$	503.588			1.44295			0.721473	+	0.692443i	$0.243465\pi$
$350$		0			0
$351$	−619.838			−1.76592
$352$		0			0
$353$	−353.912			−1.00258			−0.501292	−	0.865278i	$-0.667142\pi$
$353$	−353.912			−1.00258			−0.501292	+	0.865278i	$0.667142\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			92.7321i			0.259754i
$358$		0			0
$359$	226.398			0.630634			0.315317	−	0.948986i	$-0.397889\pi$
$359$	226.398			0.630634			0.315317	+	0.948986i	$0.397889\pi$
$360$		0			0
$361$	−355.423	+	63.2090i	−0.984552	+	0.175094i
$362$		0			0
$363$		−	75.5422i		−	0.208105i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−427.998			−1.16621			−0.583104	−	0.812398i	$-0.698162\pi$
$367$	−427.998			−1.16621			−0.583104	+	0.812398i	$0.698162\pi$
$368$		0			0
$369$		−	104.201i		−	0.282387i
$370$		0			0
$371$			113.552i			0.306069i
$372$		0			0
$373$			275.474i			0.738537i	0.929323	+	0.369268i	$0.120392\pi$
$373$			275.474i			0.738537i	−0.929323	+	0.369268i	$0.879608\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	317.108			0.841134
$378$		0			0
$379$			250.621i			0.661268i	0.943759	+	0.330634i	$0.107263\pi$
$379$			250.621i			0.661268i	−0.943759	+	0.330634i	$0.892737\pi$
$380$		0			0
$381$	232.013			0.608957
$382$		0			0
$383$			148.912i			0.388805i	0.980922	+	0.194403i	$0.0622768\pi$
$383$			148.912i			0.388805i	−0.980922	+	0.194403i	$0.937723\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−162.526			−0.419963
$388$		0			0
$389$	57.2486			0.147169			0.0735843	−	0.997289i	$-0.476556\pi$
$389$	57.2486			0.147169			0.0735843	+	0.997289i	$0.476556\pi$
$390$		0			0
$391$	170.445			0.435920
$392$		0			0
$393$			313.338i			0.797298i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	597.128			1.50410			0.752050	−	0.659106i	$-0.229065\pi$
$397$	597.128			1.50410			0.752050	+	0.659106i	$0.229065\pi$
$398$		0			0
$399$	−367.627	+	32.4352i	−0.921371	+	0.0812912i
$400$		0			0
$401$		−	422.141i		−	1.05272i	−0.850262	−	0.526360i	$-0.823556\pi$
$401$		−	422.141i		−	1.05272i	0.850262	−	0.526360i	$-0.176444\pi$
$402$		0			0
$403$	702.741			1.74378
$404$		0			0
$405$		0			0
$406$		0			0
$407$			698.118i			1.71528i
$408$		0			0
$409$			244.086i			0.596786i	0.954443	+	0.298393i	$0.0964506\pi$
$409$			244.086i			0.596786i	−0.954443	+	0.298393i	$0.903549\pi$
$410$		0			0
$411$		−	312.693i		−	0.760811i
$412$		0			0
$413$			575.059i			1.39240i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	462.457i		−	1.10901i
$418$		0			0
$419$	499.432			1.19196			0.595980	−	0.802999i	$-0.296764\pi$
$419$	499.432			1.19196			0.595980	+	0.802999i	$0.296764\pi$
$420$		0			0
$421$		−	521.240i		−	1.23810i	−0.785351	−	0.619050i	$-0.787518\pi$
$421$		−	521.240i		−	1.23810i	0.785351	−	0.619050i	$-0.212482\pi$
$422$		0			0
$423$	−63.3872			−0.149851
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−187.249			−0.438522
$428$		0			0
$429$	−540.249			−1.25932
$430$		0			0
$431$			252.375i			0.585557i	0.956180	+	0.292778i	$0.0945798\pi$
$431$			252.375i			0.585557i	−0.956180	+	0.292778i	$0.905420\pi$
$432$		0			0
$433$		−	516.782i		−	1.19349i	−0.802430	−	0.596746i	$-0.796460\pi$
$433$		−	516.782i		−	1.19349i	0.802430	−	0.596746i	$-0.203540\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	59.6169	+	675.710i	0.136423	+	1.54625i
$438$		0			0
$439$			706.635i			1.60965i	0.593514	+	0.804823i	$0.297740\pi$
$439$			706.635i			1.60965i	−0.593514	+	0.804823i	$0.702260\pi$
$440$		0			0
$441$	347.548			0.788091
$442$		0			0
$443$	405.631			0.915645			0.457823	−	0.889044i	$-0.348630\pi$
$443$	405.631			0.915645			0.457823	+	0.889044i	$0.348630\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			287.827i			0.643908i
$448$		0			0
$449$			585.708i			1.30447i	0.758015	+	0.652237i	$0.226169\pi$
$449$			585.708i			1.30447i	−0.758015	+	0.652237i	$0.773831\pi$
$450$		0			0
$451$		−	236.727i		−	0.524893i
$452$		0			0
$453$	159.391			0.351857
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−445.652			−0.975169			−0.487585	−	0.873076i	$-0.662122\pi$
$457$	−445.652			−0.975169			−0.487585	+	0.873076i	$0.662122\pi$
$458$		0			0
$459$		−	128.502i		−	0.279962i
$460$		0			0
$461$	334.203			0.724953			0.362476	−	0.931993i	$-0.381931\pi$
$461$	334.203			0.724953			0.362476	+	0.931993i	$0.381931\pi$
$462$		0			0
$463$	180.792			0.390479			0.195240	−	0.980756i	$-0.437452\pi$
$463$	180.792			0.390479			0.195240	+	0.980756i	$0.437452\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−629.520			−1.34801			−0.674004	−	0.738727i	$-0.735427\pi$
$467$	−629.520			−1.34801			−0.674004	+	0.738727i	$0.735427\pi$
$468$		0			0
$469$			634.498i			1.35287i
$470$		0			0
$471$		−	110.587i		−	0.234792i
$472$		0			0
$473$	−369.231			−0.780616
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	60.3689i		−	0.126559i
$478$		0			0
$479$	304.093			0.634849			0.317425	−	0.948284i	$-0.397182\pi$
$479$	304.093			0.634849			0.317425	+	0.948284i	$0.397182\pi$
$480$		0			0
$481$	1263.15			2.62610
$482$		0			0
$483$			693.471i			1.43576i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			235.191i			0.482938i	0.970409	+	0.241469i	$0.0776291\pi$
$487$			235.191i			0.482938i	−0.970409	+	0.241469i	$0.922371\pi$
$488$		0			0
$489$			380.767i			0.778665i
$490$		0			0
$491$	289.350			0.589308			0.294654	−	0.955604i	$-0.404796\pi$
$491$	289.350			0.589308			0.294654	+	0.955604i	$0.404796\pi$
$492$		0			0
$493$			65.7415i			0.133350i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			420.488i			0.846053i
$498$		0			0
$499$	−754.978			−1.51298			−0.756491	−	0.654004i	$-0.773088\pi$
$499$	−754.978			−1.51298			−0.756491	+	0.654004i	$0.773088\pi$
$500$		0			0
$501$	323.924			0.646555
$502$		0			0
$503$	−347.663			−0.691180			−0.345590	−	0.938386i	$-0.612321\pi$
$503$	−347.663			−0.691180			−0.345590	+	0.938386i	$0.612321\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			665.989i			1.31359i
$508$		0			0
$509$		−	117.878i		−	0.231587i	−0.993273	−	0.115794i	$-0.963059\pi$
$509$		−	117.878i		−	0.231587i	0.993273	−	0.115794i	$-0.0369411\pi$
$510$		0			0
$511$	−259.478			−0.507784
$512$		0			0
$513$	509.434	−	44.9467i	0.993050	−	0.0876153i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−144.005			−0.278540
$518$		0			0
$519$	−66.4836			−0.128099
$520$		0			0
$521$			844.507i			1.62093i	0.585784	+	0.810467i	$0.300787\pi$
$521$			844.507i			1.62093i	−0.585784	+	0.810467i	$0.699213\pi$
$522$		0			0
$523$			482.142i			0.921879i	0.887432	+	0.460939i	$0.152487\pi$
$523$			482.142i			0.921879i	−0.887432	+	0.460939i	$0.847513\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			145.689i			0.276451i
$528$		0			0
$529$	745.622			1.40949
$530$		0			0
$531$		−	305.726i		−	0.575755i
$532$		0			0
$533$	−428.326			−0.803614
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−404.207			−0.752713
$538$		0			0
$539$	789.572			1.46488
$540$		0			0
$541$	−973.284			−1.79905			−0.899524	−	0.436872i	$-0.856086\pi$
$541$	−973.284			−1.79905			−0.899524	+	0.436872i	$0.856086\pi$
$542$		0			0
$543$	74.6709			0.137516
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	425.904i		−	0.778618i	−0.921107	−	0.389309i	$-0.872714\pi$
$547$		−	425.904i		−	0.778618i	0.921107	−	0.389309i	$-0.127286\pi$
$548$		0			0
$549$	99.5493			0.181328
$550$		0			0
$551$	−260.625	+	22.9946i	−0.473004	+	0.0417325i
$552$		0			0
$553$		−	73.3400i		−	0.132622i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−401.955			−0.721643			−0.360822	−	0.932635i	$-0.617504\pi$
$557$	−401.955			−0.721643			−0.360822	+	0.932635i	$0.617504\pi$
$558$		0			0
$559$			668.076i			1.19513i
$560$		0			0
$561$		−	112.002i		−	0.199647i
$562$		0			0
$563$			655.640i			1.16455i	0.812993	+	0.582274i	$0.197837\pi$
$563$			655.640i			1.16455i	−0.812993	+	0.582274i	$0.802163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−8.46874			−0.0149361
$568$		0			0
$569$			485.171i			0.852674i	0.904564	+	0.426337i	$0.140196\pi$
$569$			485.171i			0.852674i	−0.904564	+	0.426337i	$0.859804\pi$
$570$		0			0
$571$	1001.50			1.75395			0.876974	−	0.480537i	$-0.159558\pi$
$571$	1001.50			1.75395			0.876974	+	0.480537i	$0.159558\pi$
$572$		0			0
$573$			153.100i			0.267190i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	33.1105			0.0573838			0.0286919	−	0.999588i	$-0.490866\pi$
$577$	33.1105			0.0573838			0.0286919	+	0.999588i	$0.490866\pi$
$578$		0			0
$579$	−346.380			−0.598238
$580$		0			0
$581$	−1371.85			−2.36118
$582$		0			0
$583$		−	137.148i		−	0.235246i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	635.756			1.08306			0.541530	−	0.840681i	$-0.317845\pi$
$587$	635.756			1.08306			0.541530	+	0.840681i	$0.317845\pi$
$588$		0			0
$589$	−577.571	+	50.9582i	−0.980596	+	0.0865165i
$590$		0			0
$591$			313.952i			0.531222i
$592$		0			0
$593$	59.1677			0.0997770			0.0498885	−	0.998755i	$-0.484113\pi$
$593$	59.1677			0.0997770			0.0498885	+	0.998755i	$0.484113\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	223.541i		−	0.374440i
$598$		0			0
$599$		−	941.332i		−	1.57151i	−0.618540	−	0.785753i	$-0.712276\pi$
$599$		−	941.332i		−	1.57151i	0.618540	−	0.785753i	$-0.287724\pi$
$600$		0			0
$601$		−	333.283i		−	0.554547i	−0.960791	−	0.277274i	$-0.910569\pi$
$601$		−	333.283i		−	0.554547i	0.960791	−	0.277274i	$-0.0894308\pi$
$602$		0			0
$603$		−	337.326i		−	0.559413i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			747.914i			1.23215i	0.787688	+	0.616074i	$0.211278\pi$
$607$			747.914i			1.23215i	−0.787688	+	0.616074i	$0.788722\pi$
$608$		0			0
$609$	−267.476			−0.439205
$610$		0			0
$611$			260.559i			0.426446i
$612$		0			0
$613$	−226.642			−0.369726			−0.184863	−	0.982764i	$-0.559184\pi$
$613$	−226.642			−0.369726			−0.184863	+	0.982764i	$0.559184\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	796.334			1.29065			0.645327	−	0.763906i	$-0.276721\pi$
$617$	796.334			1.29065			0.645327	+	0.763906i	$0.276721\pi$
$618$		0			0
$619$	−31.3780			−0.0506914			−0.0253457	−	0.999679i	$-0.508069\pi$
$619$	−31.3780			−0.0506914			−0.0253457	+	0.999679i	$0.508069\pi$
$620$		0			0
$621$		−	960.969i		−	1.54745i
$622$		0			0
$623$			1692.07i			2.71600i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	444.021	−	39.1754i	0.708168	−	0.0624806i
$628$		0			0
$629$			261.872i			0.416330i
$630$		0			0
$631$	−493.470			−0.782045			−0.391022	−	0.920381i	$-0.627878\pi$
$631$	−493.470			−0.782045			−0.391022	+	0.920381i	$0.627878\pi$
$632$		0			0
$633$	335.763			0.530432
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	1428.63i		−	2.24274i
$638$		0			0
$639$		−	223.549i		−	0.349842i
$640$		0			0
$641$		−	946.351i		−	1.47637i	−0.674600	−	0.738183i	$-0.735684\pi$
$641$		−	946.351i		−	1.47637i	0.674600	−	0.738183i	$-0.264316\pi$
$642$		0			0
$643$	−816.065			−1.26915			−0.634576	−	0.772860i	$-0.718825\pi$
$643$	−816.065			−1.26915			−0.634576	+	0.772860i	$0.718825\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	573.382			0.886217			0.443108	−	0.896468i	$-0.353876\pi$
$647$	573.382			0.886217			0.443108	+	0.896468i	$0.353876\pi$
$648$		0			0
$649$		−	694.559i		−	1.07020i
$650$		0			0
$651$	−592.752			−0.910525
$652$		0			0
$653$	443.988			0.679920			0.339960	−	0.940440i	$-0.389586\pi$
$653$	443.988			0.679920			0.339960	+	0.940440i	$0.389586\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	137.949			0.209969
$658$		0			0
$659$		−	571.755i		−	0.867609i	−0.901007	−	0.433805i	$-0.857171\pi$
$659$		−	571.755i		−	0.867609i	0.901007	−	0.433805i	$-0.142829\pi$
$660$		0			0
$661$		−	1244.42i		−	1.88264i	−0.337519	−	0.941319i	$-0.609588\pi$
$661$		−	1244.42i		−	1.88264i	0.337519	−	0.941319i	$-0.390412\pi$
$662$		0			0
$663$	−202.653			−0.305661
$664$		0			0
$665$		0			0
$666$		0			0
$667$			491.629i			0.737075i
$668$		0			0
$669$	396.562			0.592768
$670$		0			0
$671$	226.160			0.337049
$672$		0			0
$673$		−	236.320i		−	0.351144i	−0.984467	−	0.175572i	$-0.943823\pi$
$673$		−	236.320i		−	0.351144i	0.984467	−	0.175572i	$-0.0561775\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1066.35i			1.57512i	0.616239	+	0.787559i	$0.288655\pi$
$677$			1066.35i			1.57512i	−0.616239	+	0.787559i	$0.711345\pi$
$678$		0			0
$679$			751.318i			1.10651i
$680$		0			0
$681$	490.024			0.719566
$682$		0			0
$683$			1176.02i			1.72184i	0.508739	+	0.860921i	$0.330112\pi$
$683$			1176.02i			1.72184i	−0.508739	+	0.860921i	$0.669888\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	212.655i		−	0.309541i
$688$		0			0
$689$	−248.152			−0.360162
$690$		0			0
$691$	18.1811			0.0263113			0.0131557	−	0.999913i	$-0.495812\pi$
$691$	18.1811			0.0263113			0.0131557	+	0.999913i	$0.495812\pi$
$692$		0			0
$693$	−751.319			−1.08415
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	88.7988i		−	0.127401i
$698$		0			0
$699$			349.833i			0.500477i
$700$		0			0
$701$	−438.306			−0.625258			−0.312629	−	0.949875i	$-0.601210\pi$
$701$	−438.306			−0.625258			−0.312629	+	0.949875i	$0.601210\pi$
$702$		0			0
$703$	−1038.16	+	91.5957i	−1.47676	+	0.130293i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1131.36			−1.60023
$708$		0			0
$709$	−629.057			−0.887245			−0.443622	−	0.896214i	$-0.646307\pi$
$709$	−629.057			−0.887245			−0.443622	+	0.896214i	$0.646307\pi$
$710$		0			0
$711$			38.9906i			0.0548391i
$712$		0			0
$713$			1089.50i			1.52805i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	183.195i		−	0.255502i
$718$		0			0
$719$	−1112.31			−1.54702			−0.773509	−	0.633786i	$-0.781500\pi$
$719$	−1112.31			−1.54702			−0.773509	+	0.633786i	$0.781500\pi$
$720$		0			0
$721$		−	175.034i		−	0.242765i
$722$		0			0
$723$	240.258			0.332307
$724$		0			0
$725$		0			0
$726$		0			0
$727$	684.037			0.940904			0.470452	−	0.882426i	$-0.344091\pi$
$727$	684.037			0.940904			0.470452	+	0.882426i	$0.344091\pi$
$728$		0			0
$729$	−442.040			−0.606364
$730$		0			0
$731$	−138.503			−0.189470
$732$		0			0
$733$	171.268			0.233653			0.116826	−	0.993152i	$-0.462728\pi$
$733$	171.268			0.233653			0.116826	+	0.993152i	$0.462728\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	766.350i		−	1.03982i
$738$		0			0
$739$	317.424			0.429532			0.214766	−	0.976665i	$-0.431101\pi$
$739$	317.424			0.429532			0.214766	+	0.976665i	$0.431101\pi$
$740$		0			0
$741$	−70.8827	−	803.399i	−0.0956582	−	1.08421i
$742$		0			0
$743$			140.619i			0.189258i	0.995513	+	0.0946289i	$0.0301664\pi$
$743$			140.619i			0.189258i	−0.995513	+	0.0946289i	$0.969834\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	729.332			0.976348
$748$		0			0
$749$			1409.12i			1.88134i
$750$		0			0
$751$		−	825.612i		−	1.09935i	−0.835378	−	0.549675i	$-0.814751\pi$
$751$		−	825.612i		−	1.09935i	0.835378	−	0.549675i	$-0.185249\pi$
$752$		0			0
$753$		−	923.844i		−	1.22688i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	693.756			0.916454			0.458227	−	0.888835i	$-0.348485\pi$
$757$	693.756			0.916454			0.458227	+	0.888835i	$0.348485\pi$
$758$		0			0
$759$		−	837.577i		−	1.10353i
$760$		0			0
$761$	−148.006			−0.194489			−0.0972447	−	0.995261i	$-0.531003\pi$
$761$	−148.006			−0.194489			−0.0972447	+	0.995261i	$0.531003\pi$
$762$		0			0
$763$			1858.18i			2.43536i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1256.71			−1.63848
$768$		0			0
$769$	1090.86			1.41854			0.709270	−	0.704937i	$-0.249025\pi$
$769$	1090.86			1.41854			0.709270	+	0.704937i	$0.249025\pi$
$770$		0			0
$771$	300.076			0.389204
$772$		0			0
$773$			498.925i			0.645440i	0.946494	+	0.322720i	$0.104597\pi$
$773$			498.925i			0.645440i	−0.946494	+	0.322720i	$0.895403\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1065.45			−1.37124
$778$		0			0
$779$	352.034	−	31.0594i	0.451905	−	0.0398709i
$780$		0			0
$781$		−	507.867i		−	0.650278i
$782$		0			0
$783$	370.651			0.473373
$784$		0			0
$785$		0			0
$786$		0			0
$787$			703.635i			0.894072i	0.894516	+	0.447036i	$0.147520\pi$
$787$			703.635i			0.894072i	−0.894516	+	0.447036i	$0.852480\pi$
$788$		0			0
$789$			93.0539i			0.117939i
$790$		0			0
$791$			2370.35i			2.99664i
$792$		0			0
$793$		−	409.207i		−	0.516023i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	28.1611i		−	0.0353339i	−0.999844	−	0.0176669i	$-0.994376\pi$
$797$		−	28.1611i		−	0.0353339i	0.999844	−	0.0176669i	$-0.00562385\pi$
$798$		0			0
$799$	−54.0180			−0.0676070
$800$		0			0
$801$		−	899.574i		−	1.12306i
$802$		0			0
$803$	313.399			0.390285
$804$		0			0
$805$		0			0
$806$		0			0
$807$	53.0616			0.0657517
$808$		0			0
$809$	666.095			0.823356			0.411678	−	0.911329i	$-0.364943\pi$
$809$	666.095			0.823356			0.411678	+	0.911329i	$0.364943\pi$
$810$		0			0
$811$		−	803.237i		−	0.990428i	−0.868771	−	0.495214i	$-0.835090\pi$
$811$		−	803.237i		−	0.990428i	0.868771	−	0.495214i	$-0.164910\pi$
$812$		0			0
$813$			738.954i			0.908923i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−48.4445	−	549.080i	−0.0592956	−	0.672069i
$818$		0			0
$819$			1359.41i			1.65985i
$820$		0			0
$821$	433.616			0.528157			0.264078	−	0.964501i	$-0.414932\pi$
$821$	433.616			0.528157			0.264078	+	0.964501i	$0.414932\pi$
$822$		0			0
$823$	69.2993			0.0842033			0.0421016	−	0.999113i	$-0.486595\pi$
$823$	69.2993			0.0842033			0.0421016	+	0.999113i	$0.486595\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			282.525i			0.341626i	0.985303	+	0.170813i	$0.0546394\pi$
$827$			282.525i			0.341626i	−0.985303	+	0.170813i	$0.945361\pi$
$828$		0			0
$829$			1136.04i			1.37037i	0.728369	+	0.685185i	$0.240278\pi$
$829$			1136.04i			1.37037i	−0.728369	+	0.685185i	$0.759722\pi$
$830$		0			0
$831$			394.184i			0.474349i
$832$		0			0
$833$	296.177			0.355555
$834$		0			0
$835$		0			0
$836$		0			0
$837$	821.399			0.981361
$838$		0			0
$839$		−	474.238i		−	0.565242i	−0.959232	−	0.282621i	$-0.908796\pi$
$839$		−	474.238i		−	0.565242i	0.959232	−	0.282621i	$-0.0912039\pi$
$840$		0			0
$841$	651.376			0.774526
$842$		0			0
$843$	−423.240			−0.502065
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−431.842			−0.509848
$848$		0			0
$849$			371.164i			0.437178i
$850$		0			0
$851$			1958.33i			2.30121i
$852$		0			0
$853$	−843.815			−0.989232			−0.494616	−	0.869112i	$-0.664691\pi$
$853$	−843.815			−0.989232			−0.494616	+	0.869112i	$0.664691\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			435.275i			0.507906i	0.967217	+	0.253953i	$0.0817308\pi$
$857$			435.275i			0.507906i	−0.967217	+	0.253953i	$0.918269\pi$
$858$		0			0
$859$	715.550			0.833003			0.416502	−	0.909135i	$-0.363256\pi$
$859$	715.550			0.833003			0.416502	+	0.909135i	$0.363256\pi$
$860$		0			0
$861$	361.287			0.419613
$862$		0			0
$863$		−	1083.60i		−	1.25562i	−0.778366	−	0.627811i	$-0.783951\pi$
$863$		−	1083.60i		−	1.25562i	0.778366	−	0.627811i	$-0.216049\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			490.707i			0.565983i
$868$		0			0
$869$			88.5803i			0.101934i
$870$		0			0
$871$	−1386.61			−1.59197
$872$		0			0
$873$		−	399.433i		−	0.457540i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	737.245i		−	0.840644i	−0.907375	−	0.420322i	$-0.861917\pi$
$877$		−	737.245i		−	0.840644i	0.907375	−	0.420322i	$-0.138083\pi$
$878$		0			0
$879$	−527.005			−0.599550
$880$		0			0
$881$	−64.9086			−0.0736761			−0.0368380	−	0.999321i	$-0.511729\pi$
$881$	−64.9086			−0.0736761			−0.0368380	+	0.999321i	$0.511729\pi$
$882$		0			0
$883$	1325.97			1.50166			0.750830	−	0.660495i	$-0.229654\pi$
$883$	1325.97			1.50166			0.750830	+	0.660495i	$0.229654\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			842.262i			0.949563i	0.880104	+	0.474781i	$0.157473\pi$
$887$			842.262i			0.949563i	−0.880104	+	0.474781i	$0.842527\pi$
$888$		0			0
$889$		−	1326.32i		−	1.49192i
$890$		0			0
$891$	10.2286			0.0114799
$892$		0			0
$893$	−18.8940	−	214.149i	−0.0211579	−	0.239808i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1515.49			−1.68951
$898$		0			0
$899$	−420.225			−0.467436
$900$		0			0
$901$		−	51.4458i		−	0.0570986i
$902$		0			0
$903$		−	563.512i		−	0.624045i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1212.41i			1.33673i	0.743834	+	0.668364i	$0.233005\pi$
$907$			1212.41i			1.33673i	−0.743834	+	0.668364i	$0.766995\pi$
$908$		0			0
$909$	601.481			0.661695
$910$		0			0
$911$		−	1204.26i		−	1.32191i	−0.750427	−	0.660954i	$-0.770152\pi$
$911$		−	1204.26i		−	1.32191i	0.750427	−	0.660954i	$-0.229848\pi$
$912$		0			0
$913$	1656.92			1.81481
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1791.22			1.95334
$918$		0			0
$919$	−117.855			−0.128243			−0.0641214	−	0.997942i	$-0.520424\pi$
$919$	−117.855			−0.128243			−0.0641214	+	0.997942i	$0.520424\pi$
$920$		0			0
$921$	71.4150			0.0775408
$922$		0			0
$923$	−918.920			−0.995579
$924$		0			0
$925$		0			0
$926$		0			0
$927$			93.0554i			0.100383i
$928$		0			0
$929$	−1171.59			−1.26113			−0.630566	−	0.776136i	$-0.717177\pi$
$929$	−1171.59			−1.26113			−0.630566	+	0.776136i	$0.717177\pi$
$930$		0			0
$931$	103.595	+	1174.16i	0.111273	+	1.26119i
$932$		0			0
$933$		−	1120.10i		−	1.20054i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1657.08			1.76849			0.884247	−	0.467020i	$-0.154672\pi$
$937$	1657.08			1.76849			0.884247	+	0.467020i	$0.154672\pi$
$938$		0			0
$939$			541.522i			0.576700i
$940$		0			0
$941$		−	897.326i		−	0.953588i	−0.879015	−	0.476794i	$-0.841799\pi$
$941$		−	897.326i		−	0.953588i	0.879015	−	0.476794i	$-0.158201\pi$
$942$		0			0
$943$		−	664.057i		−	0.704196i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1458.19			−1.53980			−0.769901	−	0.638163i	$-0.779695\pi$
$947$	−1458.19			−1.53980			−0.769901	+	0.638163i	$0.779695\pi$
$948$		0			0
$949$		−	567.054i		−	0.597528i
$950$		0			0
$951$	−1029.35			−1.08239
$952$		0			0
$953$		−	1049.86i		−	1.10164i	−0.834624	−	0.550820i	$-0.814315\pi$
$953$		−	1049.86i		−	1.10164i	0.834624	−	0.550820i	$-0.185685\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	323.058			0.337574
$958$		0			0
$959$	−1787.53			−1.86395
$960$		0			0
$961$	29.7397			0.0309466
$962$		0			0
$963$		−	749.150i		−	0.777933i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	236.232			0.244293			0.122147	−	0.992512i	$-0.461022\pi$
$967$	236.232			0.244293			0.122147	+	0.992512i	$0.461022\pi$
$968$		0			0
$969$	166.557	−	14.6951i	0.171886	−	0.0151652i
$970$		0			0
$971$			1281.82i			1.32010i	0.751220	+	0.660052i	$0.229466\pi$
$971$			1281.82i			1.32010i	−0.751220	+	0.660052i	$0.770534\pi$
$972$		0			0
$973$	−2643.67			−2.71702
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1475.99i			1.51074i	0.655300	+	0.755369i	$0.272542\pi$
$977$			1475.99i			1.51074i	−0.655300	+	0.755369i	$0.727458\pi$
$978$		0			0
$979$		−	2043.69i		−	2.08752i
$980$		0			0
$981$		−	987.885i		−	1.00702i
$982$		0			0
$983$			574.909i			0.584851i	0.956288	+	0.292426i	$0.0944624\pi$
$983$			574.909i			0.584851i	−0.956288	+	0.292426i	$0.905538\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	219.777i		−	0.222672i
$988$		0			0
$989$	−1035.75			−1.04727
$990$		0			0
$991$			47.2482i			0.0476773i	0.999716	+	0.0238387i	$0.00758880\pi$
$991$			47.2482i			0.0476773i	−0.999716	+	0.0238387i	$0.992411\pi$
$992$		0			0
$993$	−676.296			−0.681064
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−671.475			−0.673495			−0.336748	−	0.941595i	$-0.609327\pi$
$997$	−671.475			−0.673495			−0.336748	+	0.941595i	$0.609327\pi$
$998$		0			0
$999$	1476.44			1.47791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.g.1101.6	✓	14
5.2	odd	4		1900.3.g.d.949.12		28
5.3	odd	4		1900.3.g.d.949.17		28
5.4	even	2		1900.3.e.h.1101.9	yes	14
19.18	odd	2	inner	1900.3.e.g.1101.9	yes	14
95.18	even	4		1900.3.g.d.949.11		28
95.37	even	4		1900.3.g.d.949.18		28
95.94	odd	2		1900.3.e.h.1101.6	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.3.e.g.1101.6	✓	14	1.1	even	1	trivial
1900.3.e.g.1101.9	yes	14	19.18	odd	2	inner
1900.3.e.h.1101.6	yes	14	95.94	odd	2
1900.3.e.h.1101.9	yes	14	5.4	even	2
1900.3.g.d.949.11		28	95.18	even	4
1900.3.g.d.949.12		28	5.2	odd	4
1900.3.g.d.949.17		28	5.3	odd	4
1900.3.g.d.949.18		28	95.37	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.84332i q^{3} -10.5375 q^{7} +5.60216 q^{9} +O(q^{10})\)	\(q-1.84332i q^{3} -10.5375 q^{7} +5.60216 q^{9} +12.7272 q^{11} -23.0282i q^{13} +4.77411 q^{17} +(1.66986 + 18.9265i) q^{19} +19.4239i q^{21} +35.7018 q^{23} -26.9165i q^{27} +13.7704i q^{29} +30.5166i q^{31} -23.4603i q^{33} +54.8524i q^{37} -42.4484 q^{39} -18.6001i q^{41} -29.0112 q^{43} -11.3148 q^{47} +62.0382 q^{49} -8.80023i q^{51} -10.7760i q^{53} +(34.8876 - 3.07808i) q^{57} -54.5728i q^{59} +17.7698 q^{61} -59.0326 q^{63} -60.2136i q^{67} -65.8100i q^{69} -39.9041i q^{71} +24.6243 q^{73} -134.112 q^{77} +6.95992i q^{79} +0.803679 q^{81} +130.188 q^{83} +25.3833 q^{87} -160.576i q^{89} +242.659i q^{91} +56.2519 q^{93} -71.2997i q^{97} +71.2998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) \) 14 * q - 4 * q^7 - 52 * q^9	\( 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) \) 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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