LMFDB - Embedded newform 1900.3.e.e.1101.2

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 44x^{8} + 270x^{6} + 36676x^{4} + 71664x^{2} + 687241 $ x^12 - 20x^10 + 44x^8 + 270x^6 + 36676x^4 + 71664*x^2 + 687241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.2
Root			$-1.46321 + 2.80718i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.e.1101.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.61436i q^{3} +7.03410 q^{7} -22.5210 q^{9} +O(q^{10})$	$q-5.61436i q^{3} +7.03410 q^{7} -22.5210 q^{9} -1.02125 q^{11} +11.3480i q^{13} -16.5801 q^{17} +(9.54227 - 16.4300i) q^{19} -39.4919i q^{21} -35.1705 q^{23} +75.9119i q^{27} -6.63193i q^{29} +25.3890i q^{31} +5.73365i q^{33} +47.8401i q^{37} +63.7118 q^{39} +12.1251i q^{41} +50.0927 q^{43} -78.3252 q^{47} +0.478526 q^{49} +93.0869i q^{51} -45.0342i q^{53} +(-92.2439 - 53.5737i) q^{57} +9.44897i q^{59} -43.1487 q^{61} -158.415 q^{63} +112.523i q^{67} +197.460i q^{69} +40.3311i q^{71} -45.4211 q^{73} -7.18356 q^{77} +144.704i q^{79} +223.507 q^{81} -56.8706 q^{83} -37.2341 q^{87} -81.8009i q^{89} +79.8230i q^{91} +142.543 q^{93} -103.506i q^{97} +22.9995 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 76 q^{9}+O(q^{10}) $ 12 * q - 76 * q^9	$ 12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99}+O(q^{100}) $ 12 * q - 76 * q^9 - 24 * q^11 - 68 * q^19 + 264 * q^39 - 212 * q^49 - 600 * q^61 + 492 * q^81 - 136 * 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
$3$		−	5.61436i		−	1.87145i	−0.352727	−	0.935726i	$-0.614745\pi$
$3$		−	5.61436i		−	1.87145i	0.352727	−	0.935726i	$-0.385255\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.03410			1.00487			0.502436	−	0.864615i	$-0.332437\pi$
$7$	7.03410			1.00487			0.502436	+	0.864615i	$0.332437\pi$
$8$		0			0
$9$	−22.5210			−2.50234
$10$		0			0
$11$	−1.02125			−0.0928407			−0.0464204	−	0.998922i	$-0.514781\pi$
$11$	−1.02125			−0.0928407			−0.0464204	+	0.998922i	$0.514781\pi$
$12$		0			0
$13$			11.3480i			0.872924i	0.899723	+	0.436462i	$0.143769\pi$
$13$			11.3480i			0.872924i	−0.899723	+	0.436462i	$0.856231\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−16.5801			−0.975303			−0.487651	−	0.873038i	$-0.662146\pi$
$17$	−16.5801			−0.975303			−0.487651	+	0.873038i	$0.662146\pi$
$18$		0			0
$19$	9.54227	−	16.4300i	0.502225	−	0.864737i
$19$	9.54227	−	16.4300i	0.502225	−	0.864737i
$20$		0			0
$21$		−	39.4919i		−	1.88057i
$22$		0			0
$23$	−35.1705			−1.52915			−0.764576	−	0.644534i	$-0.777051\pi$
$23$	−35.1705			−1.52915			−0.764576	+	0.644534i	$0.777051\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			75.9119i			2.81155i
$28$		0			0
$29$		−	6.63193i		−	0.228687i	−0.993441	−	0.114344i	$-0.963523\pi$
$29$		−	6.63193i		−	0.228687i	0.993441	−	0.114344i	$-0.0364765\pi$
$30$		0			0
$31$			25.3890i			0.818998i	0.912310	+	0.409499i	$0.134297\pi$
$31$			25.3890i			0.818998i	−0.912310	+	0.409499i	$0.865703\pi$
$32$		0			0
$33$			5.73365i			0.173747i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			47.8401i			1.29297i	0.762925	+	0.646487i	$0.223763\pi$
$37$			47.8401i			1.29297i	−0.762925	+	0.646487i	$0.776237\pi$
$38$		0			0
$39$	63.7118			1.63364
$40$		0			0
$41$			12.1251i			0.295734i	0.989007	+	0.147867i	$0.0472407\pi$
$41$			12.1251i			0.295734i	−0.989007	+	0.147867i	$0.952759\pi$
$42$		0			0
$43$	50.0927			1.16495			0.582474	−	0.812850i	$-0.302085\pi$
$43$	50.0927			1.16495			0.582474	+	0.812850i	$0.302085\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−78.3252			−1.66649			−0.833247	−	0.552901i	$-0.813521\pi$
$47$	−78.3252			−1.66649			−0.833247	+	0.552901i	$0.813521\pi$
$48$		0			0
$49$	0.478526			0.00976584
$50$		0			0
$51$			93.0869i			1.82523i
$52$		0			0
$53$		−	45.0342i		−	0.849701i	−0.905263	−	0.424851i	$-0.860327\pi$
$53$		−	45.0342i		−	0.849701i	0.905263	−	0.424851i	$-0.139673\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−92.2439	−	53.5737i	−1.61831	−	0.939890i
$58$		0			0
$59$			9.44897i			0.160152i	0.996789	+	0.0800760i	$0.0255163\pi$
$59$			9.44897i			0.160152i	−0.996789	+	0.0800760i	$0.974484\pi$
$60$		0			0
$61$	−43.1487			−0.707356			−0.353678	−	0.935367i	$-0.615069\pi$
$61$	−43.1487			−0.707356			−0.353678	+	0.935367i	$0.615069\pi$
$62$		0			0
$63$	−158.415			−2.51452
$64$		0			0
$65$		0			0
$66$		0			0
$67$			112.523i			1.67945i	0.543011	+	0.839726i	$0.317284\pi$
$67$			112.523i			1.67945i	−0.543011	+	0.839726i	$0.682716\pi$
$68$		0			0
$69$			197.460i			2.86174i
$70$		0			0
$71$			40.3311i			0.568043i	0.958818	+	0.284022i	$0.0916687\pi$
$71$			40.3311i			0.568043i	−0.958818	+	0.284022i	$0.908331\pi$
$72$		0			0
$73$	−45.4211			−0.622207			−0.311104	−	0.950376i	$-0.600699\pi$
$73$	−45.4211			−0.622207			−0.311104	+	0.950376i	$0.600699\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.18356			−0.0932930
$78$		0			0
$79$			144.704i			1.83170i	0.401526	+	0.915848i	$0.368480\pi$
$79$			144.704i			1.83170i	−0.401526	+	0.915848i	$0.631520\pi$
$80$		0			0
$81$	223.507			2.75935
$82$		0			0
$83$	−56.8706			−0.685188			−0.342594	−	0.939484i	$-0.611306\pi$
$83$	−56.8706			−0.685188			−0.342594	+	0.939484i	$0.611306\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−37.2341			−0.427978
$88$		0			0
$89$		−	81.8009i		−	0.919111i	−0.888149	−	0.459556i	$-0.848009\pi$
$89$		−	81.8009i		−	0.919111i	0.888149	−	0.459556i	$-0.151991\pi$
$90$		0			0
$91$			79.8230i			0.877176i
$92$		0			0
$93$	142.543			1.53272
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	103.506i		−	1.06708i	−0.845776	−	0.533539i	$-0.820862\pi$
$97$		−	103.506i		−	1.06708i	0.845776	−	0.533539i	$-0.179138\pi$
$98$		0			0
$99$	22.9995			0.232319
$100$		0			0
$101$	−19.5967			−0.194027			−0.0970136	−	0.995283i	$-0.530929\pi$
$101$	−19.5967			−0.194027			−0.0970136	+	0.995283i	$0.530929\pi$
$102$		0			0
$103$			110.433i			1.07217i	0.844165	+	0.536083i	$0.180097\pi$
$103$			110.433i			1.07217i	−0.844165	+	0.536083i	$0.819903\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	38.8233i		−	0.362835i	−0.983406	−	0.181417i	$-0.941931\pi$
$107$		−	38.8233i		−	0.362835i	0.983406	−	0.181417i	$-0.0580685\pi$
$108$		0			0
$109$		−	98.8797i		−	0.907153i	−0.891217	−	0.453577i	$-0.850148\pi$
$109$		−	98.8797i		−	0.907153i	0.891217	−	0.453577i	$-0.149852\pi$
$110$		0			0
$111$	268.591			2.41974
$112$		0			0
$113$			74.5996i			0.660174i	0.943951	+	0.330087i	$0.107078\pi$
$113$			74.5996i			0.660174i	−0.943951	+	0.330087i	$0.892922\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	255.569i		−	2.18435i
$118$		0			0
$119$	−116.626			−0.980053
$120$		0			0
$121$	−119.957			−0.991381
$122$		0			0
$123$	68.0746			0.553452
$124$		0			0
$125$		0			0
$126$		0			0
$127$			193.575i			1.52421i	0.647453	+	0.762106i	$0.275834\pi$
$127$			193.575i			1.52421i	−0.647453	+	0.762106i	$0.724166\pi$
$128$		0			0
$129$		−	281.239i		−	2.18014i
$130$		0			0
$131$	−32.7455			−0.249965			−0.124983	−	0.992159i	$-0.539888\pi$
$131$	−32.7455			−0.249965			−0.124983	+	0.992159i	$0.539888\pi$
$132$		0			0
$133$	67.1213	−	115.570i	0.504671	−	0.868949i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	16.9858			0.123984			0.0619920	−	0.998077i	$-0.480255\pi$
$137$	16.9858			0.123984			0.0619920	+	0.998077i	$0.480255\pi$
$138$		0			0
$139$	114.212			0.821666			0.410833	−	0.911711i	$-0.365238\pi$
$139$	114.212			0.821666			0.410833	+	0.911711i	$0.365238\pi$
$140$		0			0
$141$			439.746i			3.11876i
$142$		0			0
$143$		−	11.5891i		−	0.0810429i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	2.68662i		−	0.0182763i
$148$		0			0
$149$	179.871			1.20719			0.603593	−	0.797293i	$-0.293735\pi$
$149$	179.871			1.20719			0.603593	+	0.797293i	$0.293735\pi$
$150$		0			0
$151$		−	203.393i		−	1.34698i	−0.739198	−	0.673488i	$-0.764795\pi$
$151$		−	203.393i		−	1.34698i	0.739198	−	0.673488i	$-0.235205\pi$
$152$		0			0
$153$	373.402			2.44053
$154$		0			0
$155$		0			0
$156$		0			0
$157$	128.869			0.820825			0.410412	−	0.911900i	$-0.365385\pi$
$157$	128.869			0.820825			0.410412	+	0.911900i	$0.365385\pi$
$158$		0			0
$159$	−252.838			−1.59018
$160$		0			0
$161$	−247.393			−1.53660
$162$		0			0
$163$	−266.033			−1.63210			−0.816052	−	0.577979i	$-0.803842\pi$
$163$	−266.033			−1.63210			−0.816052	+	0.577979i	$0.803842\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	296.182i		−	1.77354i	−0.462209	−	0.886771i	$-0.652943\pi$
$167$		−	296.182i		−	1.77354i	0.462209	−	0.886771i	$-0.347057\pi$
$168$		0			0
$169$	40.2226			0.238004
$170$		0			0
$171$	−214.902	+	370.021i	−1.25673	+	2.16386i
$172$		0			0
$173$			289.021i			1.67064i	0.549761	+	0.835322i	$0.314719\pi$
$173$			289.021i			1.67064i	−0.549761	+	0.835322i	$0.685281\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	53.0499			0.299717
$178$		0			0
$179$		−	209.726i		−	1.17165i	−0.810437	−	0.585826i	$-0.800770\pi$
$179$		−	209.726i		−	1.17165i	0.810437	−	0.585826i	$-0.199230\pi$
$180$		0			0
$181$			177.564i			0.981016i	0.871437	+	0.490508i	$0.163189\pi$
$181$			177.564i			0.981016i	−0.871437	+	0.490508i	$0.836811\pi$
$182$		0			0
$183$			242.252i			1.32378i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.9324			0.0905478
$188$		0			0
$189$			533.971i			2.82525i
$190$		0			0
$191$	318.581			1.66796			0.833982	−	0.551792i	$-0.186056\pi$
$191$	318.581			1.66796			0.833982	+	0.551792i	$0.186056\pi$
$192$		0			0
$193$		−	278.981i		−	1.44550i	−0.691112	−	0.722748i	$-0.742879\pi$
$193$		−	278.981i		−	1.44550i	0.691112	−	0.722748i	$-0.257121\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−297.749			−1.51141			−0.755707	−	0.654909i	$-0.772707\pi$
$197$	−297.749			−1.51141			−0.755707	+	0.654909i	$0.772707\pi$
$198$		0			0
$199$	−44.1173			−0.221695			−0.110848	−	0.993837i	$-0.535357\pi$
$199$	−44.1173			−0.221695			−0.110848	+	0.993837i	$0.535357\pi$
$200$		0			0
$201$	631.746			3.14301
$202$		0			0
$203$		−	46.6497i		−	0.229801i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	792.075			3.82645
$208$		0			0
$209$	−9.74502	+	16.7791i	−0.0466269	+	0.0802828i
$210$		0			0
$211$		−	306.768i		−	1.45388i	−0.686702	−	0.726939i	$-0.740942\pi$
$211$		−	306.768i		−	1.45388i	0.686702	−	0.726939i	$-0.259058\pi$
$212$		0			0
$213$	226.433			1.06307
$214$		0			0
$215$		0			0
$216$		0			0
$217$			178.588i			0.822988i
$218$		0			0
$219$			255.010i			1.16443i
$220$		0			0
$221$		−	188.152i		−	0.851365i
$222$		0			0
$223$		−	157.741i		−	0.707360i	−0.935366	−	0.353680i	$-0.884930\pi$
$223$		−	157.741i		−	0.707360i	0.935366	−	0.353680i	$-0.115070\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	27.8332i		−	0.122613i	−0.998119	−	0.0613066i	$-0.980473\pi$
$227$		−	27.8332i		−	0.122613i	0.998119	−	0.0613066i	$-0.0195267\pi$
$228$		0			0
$229$	164.314			0.717529			0.358765	−	0.933428i	$-0.383198\pi$
$229$	164.314			0.717529			0.358765	+	0.933428i	$0.383198\pi$
$230$		0			0
$231$			40.3311i			0.174593i
$232$		0			0
$233$	−150.324			−0.645168			−0.322584	−	0.946541i	$-0.604551\pi$
$233$	−150.324			−0.645168			−0.322584	+	0.946541i	$0.604551\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	812.420			3.42793
$238$		0			0
$239$	240.092			1.00457			0.502285	−	0.864702i	$-0.332493\pi$
$239$	240.092			1.00457			0.502285	+	0.864702i	$0.332493\pi$
$240$		0			0
$241$			332.556i			1.37990i	0.723857	+	0.689950i	$0.242368\pi$
$241$			332.556i			1.37990i	−0.723857	+	0.689950i	$0.757632\pi$
$242$		0			0
$243$		−	571.643i		−	2.35244i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	186.448	+	108.286i	0.754850	+	0.438404i
$248$		0			0
$249$			319.292i			1.28230i
$250$		0			0
$251$	275.443			1.09738			0.548692	−	0.836024i	$-0.315126\pi$
$251$	275.443			1.09738			0.548692	+	0.836024i	$0.315126\pi$
$252$		0			0
$253$	35.9178			0.141968
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	307.172i		−	1.19522i	−0.801787	−	0.597610i	$-0.796117\pi$
$257$		−	307.172i		−	1.19522i	0.801787	−	0.597610i	$-0.203883\pi$
$258$		0			0
$259$			336.512i			1.29927i
$260$		0			0
$261$			149.358i			0.572253i
$262$		0			0
$263$	−56.8141			−0.216023			−0.108012	−	0.994150i	$-0.534448\pi$
$263$	−56.8141			−0.216023			−0.108012	+	0.994150i	$0.534448\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−459.260			−1.72007
$268$		0			0
$269$			488.147i			1.81467i	0.420404	+	0.907337i	$0.361888\pi$
$269$			488.147i			1.81467i	−0.420404	+	0.907337i	$0.638112\pi$
$270$		0			0
$271$	123.352			0.455175			0.227587	−	0.973758i	$-0.426916\pi$
$271$	123.352			0.455175			0.227587	+	0.973758i	$0.426916\pi$
$272$		0			0
$273$	448.155			1.64159
$274$		0			0
$275$		0			0
$276$		0			0
$277$	66.0782			0.238549			0.119275	−	0.992861i	$-0.461943\pi$
$277$	66.0782			0.238549			0.119275	+	0.992861i	$0.461943\pi$
$278$		0			0
$279$		−	571.785i		−	2.04941i
$280$		0			0
$281$			412.679i			1.46861i	0.678821	+	0.734304i	$0.262491\pi$
$281$			412.679i			1.46861i	−0.678821	+	0.734304i	$0.737509\pi$
$282$		0			0
$283$	−402.243			−1.42135			−0.710676	−	0.703519i	$-0.751611\pi$
$283$	−402.243			−1.42135			−0.710676	+	0.703519i	$0.751611\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			85.2890i			0.297174i
$288$		0			0
$289$	−14.0988			−0.0487847
$290$		0			0
$291$	−581.123			−1.99698
$292$		0			0
$293$		−	9.43424i		−	0.0321988i	−0.999870	−	0.0160994i	$-0.994875\pi$
$293$		−	9.43424i		−	0.0321988i	0.999870	−	0.0160994i	$-0.00512482\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	77.5248i		−	0.261026i
$298$		0			0
$299$		−	399.115i		−	1.33483i
$300$		0			0
$301$	352.357			1.17062
$302$		0			0
$303$			110.023i			0.363113i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			266.792i			0.869031i	0.900664	+	0.434515i	$0.143080\pi$
$307$			266.792i			0.869031i	−0.900664	+	0.434515i	$0.856920\pi$
$308$		0			0
$309$	620.011			2.00651
$310$		0			0
$311$	−436.217			−1.40263			−0.701314	−	0.712853i	$-0.747403\pi$
$311$	−436.217			−1.40263			−0.701314	+	0.712853i	$0.747403\pi$
$312$		0			0
$313$	−325.754			−1.04075			−0.520374	−	0.853939i	$-0.674207\pi$
$313$	−325.754			−1.04075			−0.520374	+	0.853939i	$0.674207\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	91.1420i		−	0.287514i	−0.989613	−	0.143757i	$-0.954082\pi$
$317$		−	91.1420i		−	0.287514i	0.989613	−	0.143757i	$-0.0459184\pi$
$318$		0			0
$319$			6.77285i			0.0212315i
$320$		0			0
$321$	−217.968			−0.679028
$322$		0			0
$323$	−158.212	+	272.412i	−0.489821	+	0.843380i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−555.146			−1.69769
$328$		0			0
$329$	−550.947			−1.67461
$330$		0			0
$331$		−	72.8914i		−	0.220216i	−0.993920	−	0.110108i	$-0.964880\pi$
$331$		−	72.8914i		−	0.220216i	0.993920	−	0.110108i	$-0.0351196\pi$
$332$		0			0
$333$		−	1077.41i		−	3.23546i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			76.9804i			0.228429i	0.993456	+	0.114214i	$0.0364351\pi$
$337$			76.9804i			0.228429i	−0.993456	+	0.114214i	$0.963565\pi$
$338$		0			0
$339$	418.829			1.23548
$340$		0			0
$341$		−	25.9284i		−	0.0760364i
$342$		0			0
$343$	−341.305			−0.995058
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−261.448			−0.753452			−0.376726	−	0.926325i	$-0.622950\pi$
$347$	−261.448			−0.753452			−0.376726	+	0.926325i	$0.622950\pi$
$348$		0			0
$349$	−220.895			−0.632937			−0.316469	−	0.948603i	$-0.602497\pi$
$349$	−220.895			−0.632937			−0.316469	+	0.948603i	$0.602497\pi$
$350$		0			0
$351$	−861.449			−2.45427
$352$		0			0
$353$	264.271			0.748644			0.374322	−	0.927299i	$-0.377875\pi$
$353$	264.271			0.748644			0.374322	+	0.927299i	$0.377875\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			654.782i			1.83412i
$358$		0			0
$359$	−347.108			−0.966874			−0.483437	−	0.875379i	$-0.660612\pi$
$359$	−347.108			−0.966874			−0.483437	+	0.875379i	$0.660612\pi$
$360$		0			0
$361$	−178.890	−	313.559i	−0.495541	−	0.868585i
$362$		0			0
$363$			673.482i			1.85532i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	17.2357			0.0469637			0.0234818	−	0.999724i	$-0.492525\pi$
$367$	17.2357			0.0469637			0.0234818	+	0.999724i	$0.492525\pi$
$368$		0			0
$369$		−	273.069i		−	0.740025i
$370$		0			0
$371$		−	316.775i		−	0.853840i
$372$		0			0
$373$		−	116.587i		−	0.312565i	−0.987712	−	0.156283i	$-0.950049\pi$
$373$		−	116.587i		−	0.312565i	0.987712	−	0.156283i	$-0.0499510\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	75.2593			0.199627
$378$		0			0
$379$		−	325.385i		−	0.858534i	−0.903178	−	0.429267i	$-0.858772\pi$
$379$		−	325.385i		−	0.858534i	0.903178	−	0.429267i	$-0.141228\pi$
$380$		0			0
$381$	1086.80			2.85249
$382$		0			0
$383$			300.829i			0.785454i	0.919655	+	0.392727i	$0.128468\pi$
$383$			300.829i			0.785454i	−0.919655	+	0.392727i	$0.871532\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1128.14			−2.91509
$388$		0			0
$389$	48.0199			0.123444			0.0617222	−	0.998093i	$-0.480341\pi$
$389$	48.0199			0.123444			0.0617222	+	0.998093i	$0.480341\pi$
$390$		0			0
$391$	583.132			1.49139
$392$		0			0
$393$			183.845i			0.467799i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	125.169			0.315288			0.157644	−	0.987496i	$-0.449610\pi$
$397$	125.169			0.315288			0.157644	+	0.987496i	$0.449610\pi$
$398$		0			0
$399$	−648.853	−	376.843i	−1.62620	−	0.944468i
$400$		0			0
$401$		−	350.633i		−	0.874395i	−0.899365	−	0.437198i	$-0.855971\pi$
$401$		−	350.633i		−	0.874395i	0.899365	−	0.437198i	$-0.144029\pi$
$402$		0			0
$403$	−288.114			−0.714923
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	48.8566i		−	0.120041i
$408$		0			0
$409$		−	317.356i		−	0.775932i	−0.921674	−	0.387966i	$-0.873178\pi$
$409$		−	317.356i		−	0.775932i	0.921674	−	0.387966i	$-0.126822\pi$
$410$		0			0
$411$		−	95.3645i		−	0.232030i
$412$		0			0
$413$			66.4650i			0.160932i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	641.225i		−	1.53771i
$418$		0			0
$419$	−240.313			−0.573539			−0.286770	−	0.958000i	$-0.592581\pi$
$419$	−240.313			−0.573539			−0.286770	+	0.958000i	$0.592581\pi$
$420$		0			0
$421$		−	188.532i		−	0.447820i	−0.974610	−	0.223910i	$-0.928118\pi$
$421$		−	188.532i		−	0.447820i	0.974610	−	0.223910i	$-0.0718822\pi$
$422$		0			0
$423$	1763.96			4.17013
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−303.512			−0.710802
$428$		0			0
$429$	−65.0655			−0.151668
$430$		0			0
$431$			341.606i			0.792590i	0.918123	+	0.396295i	$0.129704\pi$
$431$			341.606i			0.792590i	−0.918123	+	0.396295i	$0.870296\pi$
$432$		0			0
$433$			487.775i			1.12650i	0.826286	+	0.563251i	$0.190450\pi$
$433$			487.775i			1.12650i	−0.826286	+	0.563251i	$0.809550\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−335.606	+	577.851i	−0.767978	+	1.32231i
$438$		0			0
$439$			509.200i			1.15991i	0.814649	+	0.579954i	$0.196930\pi$
$439$			509.200i			1.15991i	−0.814649	+	0.579954i	$0.803070\pi$
$440$		0			0
$441$	−10.7769			−0.0244374
$442$		0			0
$443$	−772.768			−1.74440			−0.872198	−	0.489152i	$-0.837306\pi$
$443$	−772.768			−1.74440			−0.872198	+	0.489152i	$0.837306\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	1009.86i		−	2.25919i
$448$		0			0
$449$		−	669.385i		−	1.49084i	−0.666598	−	0.745418i	$-0.732250\pi$
$449$		−	669.385i		−	1.49084i	0.666598	−	0.745418i	$-0.267750\pi$
$450$		0			0
$451$		−	12.3827i		−	0.0274561i
$452$		0			0
$453$	−1141.92			−2.52080
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−677.202			−1.48184			−0.740922	−	0.671591i	$-0.765611\pi$
$457$	−677.202			−1.48184			−0.740922	+	0.671591i	$0.765611\pi$
$458$		0			0
$459$		−	1258.63i		−	2.74211i
$460$		0			0
$461$	77.2391			0.167547			0.0837735	−	0.996485i	$-0.473303\pi$
$461$	77.2391			0.167547			0.0837735	+	0.996485i	$0.473303\pi$
$462$		0			0
$463$	−494.756			−1.06859			−0.534293	−	0.845299i	$-0.679422\pi$
$463$	−494.756			−1.06859			−0.534293	+	0.845299i	$0.679422\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	79.7450			0.170760			0.0853801	−	0.996348i	$-0.472790\pi$
$467$	79.7450			0.170760			0.0853801	+	0.996348i	$0.472790\pi$
$468$		0			0
$469$			791.499i			1.68763i
$470$		0			0
$471$		−	723.520i		−	1.53613i
$472$		0			0
$473$	−51.1571			−0.108155
$474$		0			0
$475$		0			0
$476$		0			0
$477$			1014.22i			2.12624i
$478$		0			0
$479$	−522.458			−1.09073			−0.545364	−	0.838200i	$-0.683608\pi$
$479$	−522.458			−1.09073			−0.545364	+	0.838200i	$0.683608\pi$
$480$		0			0
$481$	−542.890			−1.12867
$482$		0			0
$483$			1388.95i			2.87567i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			362.955i			0.745287i	0.927975	+	0.372644i	$0.121549\pi$
$487$			362.955i			0.745287i	−0.927975	+	0.372644i	$0.878451\pi$
$488$		0			0
$489$			1493.60i			3.05440i
$490$		0			0
$491$	858.606			1.74869			0.874344	−	0.485307i	$-0.161292\pi$
$491$	858.606			1.74869			0.874344	+	0.485307i	$0.161292\pi$
$492$		0			0
$493$			109.958i			0.223039i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			283.693i			0.570810i
$498$		0			0
$499$	−467.505			−0.936883			−0.468441	−	0.883495i	$-0.655184\pi$
$499$	−467.505			−0.936883			−0.468441	+	0.883495i	$0.655184\pi$
$500$		0			0
$501$	−1662.87			−3.31910
$502$		0			0
$503$	222.767			0.442877			0.221438	−	0.975174i	$-0.428925\pi$
$503$	222.767			0.442877			0.221438	+	0.975174i	$0.428925\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	225.824i		−	0.445413i
$508$		0			0
$509$			305.647i			0.600486i	0.953863	+	0.300243i	$0.0970678\pi$
$509$			305.647i			0.600486i	−0.953863	+	0.300243i	$0.902932\pi$
$510$		0			0
$511$	−319.497			−0.625238
$512$		0			0
$513$	1247.23	+	724.372i	2.43125	+	1.41203i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	79.9895			0.154718
$518$		0			0
$519$	1622.67			3.12653
$520$		0			0
$521$			254.171i			0.487853i	0.969794	+	0.243926i	$0.0784355\pi$
$521$			254.171i			0.487853i	−0.969794	+	0.243926i	$0.921564\pi$
$522$		0			0
$523$		−	498.490i		−	0.953136i	−0.879138	−	0.476568i	$-0.841881\pi$
$523$		−	498.490i		−	0.953136i	0.879138	−	0.476568i	$-0.158119\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	420.953i		−	0.798771i
$528$		0			0
$529$	707.963			1.33830
$530$		0			0
$531$		−	212.800i		−	0.400754i
$532$		0			0
$533$	−137.596			−0.258153
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1177.48			−2.19269
$538$		0			0
$539$	−0.488694			−0.000906668
$540$		0			0
$541$	−106.934			−0.197659			−0.0988295	−	0.995104i	$-0.531510\pi$
$541$	−106.934			−0.197659			−0.0988295	+	0.995104i	$0.531510\pi$
$542$		0			0
$543$	996.908			1.83593
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	243.162i		−	0.444538i	−0.974985	−	0.222269i	$-0.928654\pi$
$547$		−	243.162i		−	0.444538i	0.974985	−	0.222269i	$-0.0713463\pi$
$548$		0			0
$549$	971.754			1.77004
$550$		0			0
$551$	−108.963	−	63.2837i	−0.197754	−	0.114852i
$552$		0			0
$553$			1017.86i			1.84062i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	335.753			0.602787			0.301394	−	0.953500i	$-0.402548\pi$
$557$	335.753			0.602787			0.301394	+	0.953500i	$0.402548\pi$
$558$		0			0
$559$			568.453i			1.01691i
$560$		0			0
$561$		−	95.0648i		−	0.169456i
$562$		0			0
$563$			271.689i			0.482573i	0.970454	+	0.241286i	$0.0775693\pi$
$563$			271.689i			0.482573i	−0.970454	+	0.241286i	$0.922431\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1572.17			2.77279
$568$		0			0
$569$		−	636.906i		−	1.11934i	−0.828715	−	0.559671i	$-0.810927\pi$
$569$		−	636.906i		−	1.11934i	0.828715	−	0.559671i	$-0.189073\pi$
$570$		0			0
$571$	15.3268			0.0268421			0.0134210	−	0.999910i	$-0.495728\pi$
$571$	15.3268			0.0268421			0.0134210	+	0.999910i	$0.495728\pi$
$572$		0			0
$573$		−	1788.63i		−	3.12152i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	62.8264			0.108885			0.0544423	−	0.998517i	$-0.482662\pi$
$577$	62.8264			0.108885			0.0544423	+	0.998517i	$0.482662\pi$
$578$		0			0
$579$	−1566.30			−2.70518
$580$		0			0
$581$	−400.033			−0.688526
$582$		0			0
$583$			45.9910i			0.0788869i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−718.521			−1.22406			−0.612028	−	0.790836i	$-0.709646\pi$
$587$	−718.521			−1.22406			−0.612028	+	0.790836i	$0.709646\pi$
$588$		0			0
$589$	417.141	+	242.268i	0.708218	+	0.411321i
$590$		0			0
$591$			1671.67i			2.82854i
$592$		0			0
$593$	−80.6940			−0.136078			−0.0680388	−	0.997683i	$-0.521674\pi$
$593$	−80.6940			−0.136078			−0.0680388	+	0.997683i	$0.521674\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			247.690i			0.414892i
$598$		0			0
$599$		−	350.033i		−	0.584363i	−0.956363	−	0.292181i	$-0.905619\pi$
$599$		−	350.033i		−	0.584363i	0.956363	−	0.292181i	$-0.0943811\pi$
$600$		0			0
$601$		−	255.825i		−	0.425666i	−0.977089	−	0.212833i	$-0.931731\pi$
$601$		−	255.825i		−	0.425666i	0.977089	−	0.212833i	$-0.0682691\pi$
$602$		0			0
$603$		−	2534.14i		−	4.20255i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			760.876i			1.25350i	0.779219	+	0.626751i	$0.215616\pi$
$607$			760.876i			1.25350i	−0.779219	+	0.626751i	$0.784384\pi$
$608$		0			0
$609$	−261.908			−0.430062
$610$		0			0
$611$		−	888.835i		−	1.45472i
$612$		0			0
$613$	65.5926			0.107003			0.0535013	−	0.998568i	$-0.482962\pi$
$613$	65.5926			0.107003			0.0535013	+	0.998568i	$0.482962\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	386.387			0.626236			0.313118	−	0.949714i	$-0.398627\pi$
$617$	386.387			0.626236			0.313118	+	0.949714i	$0.398627\pi$
$618$		0			0
$619$	−107.682			−0.173961			−0.0869804	−	0.996210i	$-0.527722\pi$
$619$	−107.682			−0.173961			−0.0869804	+	0.996210i	$0.527722\pi$
$620$		0			0
$621$		−	2669.86i		−	4.29929i
$622$		0			0
$623$		−	575.396i		−	0.923589i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	94.2039	+	54.7121i	0.150246	+	0.0872601i
$628$		0			0
$629$		−	793.195i		−	1.26104i
$630$		0			0
$631$	−617.627			−0.978807			−0.489403	−	0.872058i	$-0.662785\pi$
$631$	−617.627			−0.978807			−0.489403	+	0.872058i	$0.662785\pi$
$632$		0			0
$633$	−1722.31			−2.72087
$634$		0			0
$635$		0			0
$636$		0			0
$637$			5.43032i			0.00852484i
$638$		0			0
$639$		−	908.297i		−	1.42143i
$640$		0			0
$641$		−	452.435i		−	0.705826i	−0.935656	−	0.352913i	$-0.885191\pi$
$641$		−	452.435i		−	0.705826i	0.935656	−	0.352913i	$-0.114809\pi$
$642$		0			0
$643$	−235.639			−0.366468			−0.183234	−	0.983069i	$-0.558657\pi$
$643$	−235.639			−0.366468			−0.183234	+	0.983069i	$0.558657\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	563.621			0.871130			0.435565	−	0.900157i	$-0.356549\pi$
$647$	563.621			0.871130			0.435565	+	0.900157i	$0.356549\pi$
$648$		0			0
$649$		−	9.64974i		−	0.0148686i
$650$		0			0
$651$	1002.66			1.54018
$652$		0			0
$653$	−498.748			−0.763780			−0.381890	−	0.924208i	$-0.624727\pi$
$653$	−498.748			−0.763780			−0.381890	+	0.924208i	$0.624727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1022.93			1.55697
$658$		0			0
$659$		−	36.4563i		−	0.0553207i	−0.999617	−	0.0276603i	$-0.991194\pi$
$659$		−	36.4563i		−	0.0553207i	0.999617	−	0.0276603i	$-0.00880568\pi$
$660$		0			0
$661$			260.521i			0.394132i	0.980390	+	0.197066i	$0.0631414\pi$
$661$			260.521i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$662$		0			0
$663$	−1056.35			−1.59329
$664$		0			0
$665$		0			0
$666$		0			0
$667$			233.248i			0.349698i
$668$		0			0
$669$	−885.617			−1.32379
$670$		0			0
$671$	44.0656			0.0656715
$672$		0			0
$673$			448.262i			0.666065i	0.942915	+	0.333032i	$0.108072\pi$
$673$			448.262i			0.666065i	−0.942915	+	0.333032i	$0.891928\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	106.193i		−	0.156858i	−0.996920	−	0.0784292i	$-0.975010\pi$
$677$		−	106.193i		−	0.156858i	0.996920	−	0.0784292i	$-0.0249905\pi$
$678$		0			0
$679$		−	728.075i		−	1.07227i
$680$		0			0
$681$	−156.266			−0.229465
$682$		0			0
$683$			604.017i			0.884359i	0.896927	+	0.442179i	$0.145795\pi$
$683$			604.017i			0.884359i	−0.896927	+	0.442179i	$0.854205\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	922.519i		−	1.34282i
$688$		0			0
$689$	511.048			0.741724
$690$		0			0
$691$	531.425			0.769067			0.384534	−	0.923111i	$-0.374362\pi$
$691$	531.425			0.769067			0.384534	+	0.923111i	$0.374362\pi$
$692$		0			0
$693$	161.781			0.233450
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	201.036i		−	0.288430i
$698$		0			0
$699$			843.973i			1.20740i
$700$		0			0
$701$	−868.092			−1.23836			−0.619181	−	0.785248i	$-0.712535\pi$
$701$	−868.092			−1.23836			−0.619181	+	0.785248i	$0.712535\pi$
$702$		0			0
$703$	786.013	+	456.503i	1.11808	+	0.649364i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−137.845			−0.194972
$708$		0			0
$709$	−285.032			−0.402020			−0.201010	−	0.979589i	$-0.564422\pi$
$709$	−285.032			−0.402020			−0.201010	+	0.979589i	$0.564422\pi$
$710$		0			0
$711$		−	3258.88i		−	4.58352i
$712$		0			0
$713$		−	892.942i		−	1.25237i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	1347.97i		−	1.88001i
$718$		0			0
$719$	−450.375			−0.626390			−0.313195	−	0.949689i	$-0.601399\pi$
$719$	−450.375			−0.626390			−0.313195	+	0.949689i	$0.601399\pi$
$720$		0			0
$721$			776.797i			1.07739i
$722$		0			0
$723$	1867.09			2.58242
$724$		0			0
$725$		0			0
$726$		0			0
$727$	190.836			0.262499			0.131249	−	0.991349i	$-0.458101\pi$
$727$	190.836			0.262499			0.131249	+	0.991349i	$0.458101\pi$
$728$		0			0
$729$	−1197.84			−1.64313
$730$		0			0
$731$	−830.545			−1.13618
$732$		0			0
$733$	−729.335			−0.995001			−0.497500	−	0.867464i	$-0.665749\pi$
$733$	−729.335			−0.995001			−0.497500	+	0.867464i	$0.665749\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	114.914i		−	0.155921i
$738$		0			0
$739$	−1321.24			−1.78787			−0.893936	−	0.448194i	$-0.852067\pi$
$739$	−1321.24			−1.78787			−0.893936	+	0.448194i	$0.852067\pi$
$740$		0			0
$741$	607.955	−	1046.79i	0.820452	−	1.41267i
$742$		0			0
$743$		−	76.4830i		−	0.102938i	−0.998675	−	0.0514690i	$-0.983610\pi$
$743$		−	76.4830i		−	0.102938i	0.998675	−	0.0514690i	$-0.0163903\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1280.78			1.71457
$748$		0			0
$749$		−	273.087i		−	0.364602i
$750$		0			0
$751$		−	11.5258i		−	0.0153472i	−0.999971	−	0.00767361i	$-0.997557\pi$
$751$		−	11.5258i		−	0.0153472i	0.999971	−	0.00767361i	$-0.00244261\pi$
$752$		0			0
$753$		−	1546.44i		−	2.05370i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	516.354			0.682106			0.341053	−	0.940044i	$-0.389216\pi$
$757$	516.354			0.682106			0.341053	+	0.940044i	$0.389216\pi$
$758$		0			0
$759$		−	201.655i		−	0.265686i
$760$		0			0
$761$	−437.830			−0.575335			−0.287667	−	0.957730i	$-0.592880\pi$
$761$	−437.830			−0.575335			−0.287667	+	0.957730i	$0.592880\pi$
$762$		0			0
$763$		−	695.529i		−	0.911572i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−107.227			−0.139800
$768$		0			0
$769$	514.298			0.668788			0.334394	−	0.942433i	$-0.391468\pi$
$769$	514.298			0.668788			0.334394	+	0.942433i	$0.391468\pi$
$770$		0			0
$771$	−1724.57			−2.23680
$772$		0			0
$773$			899.495i			1.16364i	0.813317	+	0.581821i	$0.197660\pi$
$773$			899.495i			1.16364i	−0.813317	+	0.581821i	$0.802340\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	1889.30			2.43153
$778$		0			0
$779$	199.215	+	115.701i	0.255732	+	0.148525i
$780$		0			0
$781$		−	41.1880i		−	0.0527375i
$782$		0			0
$783$	503.443			0.642966
$784$		0			0
$785$		0			0
$786$		0			0
$787$			84.0809i			0.106837i	0.998572	+	0.0534186i	$0.0170118\pi$
$787$			84.0809i			0.106837i	−0.998572	+	0.0534186i	$0.982988\pi$
$788$		0			0
$789$			318.975i			0.404277i
$790$		0			0
$791$			524.741i			0.663390i
$792$		0			0
$793$		−	489.652i		−	0.617468i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			826.262i			1.03672i	0.855164	+	0.518358i	$0.173456\pi$
$797$			826.262i			1.03672i	−0.855164	+	0.518358i	$0.826544\pi$
$798$		0			0
$799$	1298.64			1.62534
$800$		0			0
$801$			1842.24i			2.29993i
$802$		0			0
$803$	46.3862			0.0577662
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2740.63			3.39608
$808$		0			0
$809$	26.0920			0.0322522			0.0161261	−	0.999870i	$-0.494867\pi$
$809$	26.0920			0.0322522			0.0161261	+	0.999870i	$0.494867\pi$
$810$		0			0
$811$			1359.80i			1.67670i	0.545131	+	0.838351i	$0.316480\pi$
$811$			1359.80i			1.67670i	−0.545131	+	0.838351i	$0.683520\pi$
$812$		0			0
$813$		−	692.545i		−	0.851838i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	477.998	−	823.024i	0.585065	−	1.00737i
$818$		0			0
$819$		−	1797.70i		−	2.19499i
$820$		0			0
$821$	−164.334			−0.200163			−0.100081	−	0.994979i	$-0.531910\pi$
$821$	−164.334			−0.200163			−0.100081	+	0.994979i	$0.531910\pi$
$822$		0			0
$823$	1040.58			1.26438			0.632188	−	0.774815i	$-0.282157\pi$
$823$	1040.58			1.26438			0.632188	+	0.774815i	$0.282157\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1418.49i			1.71522i	0.514299	+	0.857611i	$0.328052\pi$
$827$			1418.49i			1.71522i	−0.514299	+	0.857611i	$0.671948\pi$
$828$		0			0
$829$		−	683.294i		−	0.824239i	−0.911130	−	0.412119i	$-0.864789\pi$
$829$		−	683.294i		−	0.824239i	0.911130	−	0.412119i	$-0.135211\pi$
$830$		0			0
$831$		−	370.987i		−	0.446434i
$832$		0			0
$833$	−7.93404			−0.00952465
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1927.32			−2.30266
$838$		0			0
$839$		−	1206.13i		−	1.43758i	−0.695226	−	0.718791i	$-0.744696\pi$
$839$		−	1206.13i		−	1.43758i	0.695226	−	0.718791i	$-0.255304\pi$
$840$		0			0
$841$	797.017			0.947702
$842$		0			0
$843$	2316.93			2.74843
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−843.790			−0.996210
$848$		0			0
$849$			2258.34i			2.65999i
$850$		0			0
$851$		−	1682.56i		−	1.97715i
$852$		0			0
$853$	−484.057			−0.567476			−0.283738	−	0.958902i	$-0.591575\pi$
$853$	−484.057			−0.567476			−0.283738	+	0.958902i	$0.591575\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	142.789i		−	0.166615i	−0.996524	−	0.0833076i	$-0.973452\pi$
$857$		−	142.789i		−	0.166615i	0.996524	−	0.0833076i	$-0.0265484\pi$
$858$		0			0
$859$	737.199			0.858206			0.429103	−	0.903256i	$-0.358830\pi$
$859$	737.199			0.858206			0.429103	+	0.903256i	$0.358830\pi$
$860$		0			0
$861$	478.843			0.556148
$862$		0			0
$863$			1101.44i			1.27629i	0.769916	+	0.638145i	$0.220298\pi$
$863$			1101.44i			1.27629i	−0.769916	+	0.638145i	$0.779702\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			79.1556i			0.0912982i
$868$		0			0
$869$		−	147.779i		−	0.170056i
$870$		0			0
$871$	−1276.91			−1.46603
$872$		0			0
$873$			2331.07i			2.67019i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1668.88i		−	1.90295i	−0.307733	−	0.951473i	$-0.599570\pi$
$877$		−	1668.88i		−	1.90295i	0.307733	−	0.951473i	$-0.400430\pi$
$878$		0			0
$879$	−52.9672			−0.0602585
$880$		0			0
$881$	1534.98			1.74231			0.871156	−	0.491007i	$-0.163371\pi$
$881$	1534.98			1.74231			0.871156	+	0.491007i	$0.163371\pi$
$882$		0			0
$883$	−1192.75			−1.35079			−0.675394	−	0.737457i	$-0.736026\pi$
$883$	−1192.75			−1.35079			−0.675394	+	0.737457i	$0.736026\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			90.7193i			0.102277i	0.998692	+	0.0511383i	$0.0162849\pi$
$887$			90.7193i			0.102277i	−0.998692	+	0.0511383i	$0.983715\pi$
$888$		0			0
$889$			1361.62i			1.53164i
$890$		0			0
$891$	−228.256			−0.256180
$892$		0			0
$893$	−747.400	+	1286.88i	−0.836954	+	1.44108i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2240.78			−2.49808
$898$		0			0
$899$	168.378			0.187295
$900$		0			0
$901$			746.673i			0.828716i
$902$		0			0
$903$		−	1978.26i		−	2.19076i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	756.388i		−	0.833944i	−0.908919	−	0.416972i	$-0.863091\pi$
$907$		−	756.388i		−	0.833944i	0.908919	−	0.416972i	$-0.136909\pi$
$908$		0			0
$909$	441.339			0.485521
$910$		0			0
$911$			1422.27i			1.56122i	0.625017	+	0.780611i	$0.285092\pi$
$911$			1422.27i			1.56122i	−0.625017	+	0.780611i	$0.714908\pi$
$912$		0			0
$913$	58.0790			0.0636134
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−230.335			−0.251183
$918$		0			0
$919$	802.847			0.873609			0.436804	−	0.899556i	$-0.356110\pi$
$919$	802.847			0.873609			0.436804	+	0.899556i	$0.356110\pi$
$920$		0			0
$921$	1497.87			1.62635
$922$		0			0
$923$	−457.677			−0.495859
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	2487.07i		−	2.68292i
$928$		0			0
$929$	1818.21			1.95717			0.978584	−	0.205849i	$-0.0659955\pi$
$929$	1818.21			1.95717			0.978584	+	0.205849i	$0.0659955\pi$
$930$		0			0
$931$	4.56623	−	7.86219i	0.00490465	−	0.00844489i
$932$		0			0
$933$			2449.08i			2.62495i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1262.41			−1.34729			−0.673647	−	0.739053i	$-0.735273\pi$
$937$	−1262.41			−1.34729			−0.673647	+	0.739053i	$0.735273\pi$
$938$		0			0
$939$			1828.90i			1.94771i
$940$		0			0
$941$			65.8851i			0.0700161i	0.999387	+	0.0350080i	$0.0111457\pi$
$941$			65.8851i			0.0700161i	−0.999387	+	0.0350080i	$0.988854\pi$
$942$		0			0
$943$		−	426.445i		−	0.452222i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	354.752			0.374606			0.187303	−	0.982302i	$-0.440025\pi$
$947$	354.752			0.374606			0.187303	+	0.982302i	$0.440025\pi$
$948$		0			0
$949$		−	515.439i		−	0.543140i
$950$		0			0
$951$	−511.704			−0.538069
$952$		0			0
$953$			198.204i			0.207979i	0.994578	+	0.103990i	$0.0331609\pi$
$953$			198.204i			0.207979i	−0.994578	+	0.103990i	$0.966839\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	38.0252			0.0397338
$958$		0			0
$959$	119.480			0.124588
$960$		0			0
$961$	316.401			0.329242
$962$		0			0
$963$			874.341i			0.907935i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	736.898			0.762045			0.381023	−	0.924566i	$-0.375572\pi$
$967$	736.898			0.762045			0.381023	+	0.924566i	$0.375572\pi$
$968$		0			0
$969$	1529.42	+	888.260i	1.57835	+	0.916677i
$970$		0			0
$971$		−	1371.95i		−	1.41292i	−0.707752	−	0.706461i	$-0.750290\pi$
$971$		−	1371.95i		−	1.41292i	0.707752	−	0.706461i	$-0.249710\pi$
$972$		0			0
$973$	803.375			0.825668
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1562.15i			1.59893i	0.600713	+	0.799464i	$0.294883\pi$
$977$			1562.15i			1.59893i	−0.600713	+	0.799464i	$0.705117\pi$
$978$		0			0
$979$			83.5390i			0.0853310i
$980$		0			0
$981$			2226.87i			2.27000i
$982$		0			0
$983$		−	1552.98i		−	1.57983i	−0.613215	−	0.789916i	$-0.710124\pi$
$983$		−	1552.98i		−	1.57983i	0.613215	−	0.789916i	$-0.289876\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			3093.21i			3.13396i
$988$		0			0
$989$	−1761.79			−1.78138
$990$		0			0
$991$		−	26.8694i		−	0.0271134i	−0.999908	−	0.0135567i	$-0.995685\pi$
$991$		−	26.8694i		−	0.0271134i	0.999908	−	0.0135567i	$-0.00431537\pi$
$992$		0			0
$993$	−409.239			−0.412123
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1303.14			−1.30706			−0.653529	−	0.756902i	$-0.726712\pi$
$997$	−1303.14			−1.30706			−0.653529	+	0.756902i	$0.726712\pi$
$998$		0			0
$999$	−3631.63			−3.63526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.e.1101.2		12
5.2	odd	4		380.3.g.c.189.2	yes	12
5.3	odd	4		380.3.g.c.189.11	yes	12
5.4	even	2	inner	1900.3.e.e.1101.11		12
15.2	even	4		3420.3.h.e.2089.6		12
15.8	even	4		3420.3.h.e.2089.7		12
19.18	odd	2	inner	1900.3.e.e.1101.12		12
95.18	even	4		380.3.g.c.189.1	✓	12
95.37	even	4		380.3.g.c.189.12	yes	12
95.94	odd	2	inner	1900.3.e.e.1101.1		12
285.113	odd	4		3420.3.h.e.2089.8		12
285.227	odd	4		3420.3.h.e.2089.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.c.189.1	✓	12	95.18	even	4
380.3.g.c.189.2	yes	12	5.2	odd	4
380.3.g.c.189.11	yes	12	5.3	odd	4
380.3.g.c.189.12	yes	12	95.37	even	4
1900.3.e.e.1101.1		12	95.94	odd	2	inner
1900.3.e.e.1101.2		12	1.1	even	1	trivial
1900.3.e.e.1101.11		12	5.4	even	2	inner
1900.3.e.e.1101.12		12	19.18	odd	2	inner
3420.3.h.e.2089.5		12	285.227	odd	4
3420.3.h.e.2089.6		12	15.2	even	4
3420.3.h.e.2089.7		12	15.8	even	4
3420.3.h.e.2089.8		12	285.113	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.61436i q^{3} +7.03410 q^{7} -22.5210 q^{9} +O(q^{10})\)	\(q-5.61436i q^{3} +7.03410 q^{7} -22.5210 q^{9} -1.02125 q^{11} +11.3480i q^{13} -16.5801 q^{17} +(9.54227 - 16.4300i) q^{19} -39.4919i q^{21} -35.1705 q^{23} +75.9119i q^{27} -6.63193i q^{29} +25.3890i q^{31} +5.73365i q^{33} +47.8401i q^{37} +63.7118 q^{39} +12.1251i q^{41} +50.0927 q^{43} -78.3252 q^{47} +0.478526 q^{49} +93.0869i q^{51} -45.0342i q^{53} +(-92.2439 - 53.5737i) q^{57} +9.44897i q^{59} -43.1487 q^{61} -158.415 q^{63} +112.523i q^{67} +197.460i q^{69} +40.3311i q^{71} -45.4211 q^{73} -7.18356 q^{77} +144.704i q^{79} +223.507 q^{81} -56.8706 q^{83} -37.2341 q^{87} -81.8009i q^{89} +79.8230i q^{91} +142.543 q^{93} -103.506i q^{97} +22.9995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 76 q^{9}+O(q^{10}) \) 12 * q - 76 * q^9	\( 12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99}+O(q^{100}) \) 12 * q - 76 * q^9 - 24 * q^11 - 68 * q^19 + 264 * q^39 - 212 * q^49 - 600 * q^61 + 492 * q^81 - 136 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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