LMFDB - Embedded newform 1900.3.e.e.1101.1

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			1101.1
Root			$1.46321 + 2.80718i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.e.1101.11

$q$-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	$-0.667563\pi$
$7$	−7.03410			−1.00487			−0.502436	+	0.864615i	$0.667563\pi$
$8$		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	$-0.337854\pi$
$17$	16.5801			0.975303			0.487651	+	0.873038i	$0.337854\pi$
$18$		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	$-0.222949\pi$
$23$	35.1705			1.52915			0.764576	+	0.644534i	$0.222949\pi$
$24$		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	$0.0364765\pi$
$29$			6.63193i			0.228687i	−0.993441	+	0.114344i	$0.963523\pi$
$30$		0			0
$31$		−	25.3890i		−	0.818998i	−0.912310	−	0.409499i	$-0.865703\pi$
$31$		−	25.3890i		−	0.818998i	0.912310	−	0.409499i	$-0.134297\pi$
$32$		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	$-0.952759\pi$
$41$		−	12.1251i		−	0.295734i	0.989007	−	0.147867i	$-0.0472407\pi$
$42$		0			0
$43$	−50.0927			−1.16495			−0.582474	−	0.812850i	$-0.697915\pi$
$43$	−50.0927			−1.16495			−0.582474	+	0.812850i	$0.697915\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	78.3252			1.66649			0.833247	−	0.552901i	$-0.186479\pi$
$47$	78.3252			1.66649			0.833247	+	0.552901i	$0.186479\pi$
$48$		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	$-0.974484\pi$
$59$		−	9.44897i		−	0.160152i	0.996789	−	0.0800760i	$-0.0255163\pi$
$60$		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	$-0.908331\pi$
$71$		−	40.3311i		−	0.568043i	0.958818	−	0.284022i	$-0.0916687\pi$
$72$		0			0
$73$	45.4211			0.622207			0.311104	−	0.950376i	$-0.399301\pi$
$73$	45.4211			0.622207			0.311104	+	0.950376i	$0.399301\pi$
$74$		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	$-0.631520\pi$
$79$		−	144.704i		−	1.83170i	0.401526	−	0.915848i	$-0.368480\pi$
$80$		0			0
$81$	223.507			2.75935
$82$		0			0
$83$	56.8706			0.685188			0.342594	−	0.939484i	$-0.388694\pi$
$83$	56.8706			0.685188			0.342594	+	0.939484i	$0.388694\pi$
$84$		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	$0.151991\pi$
$89$			81.8009i			0.919111i	−0.888149	+	0.459556i	$0.848009\pi$
$90$		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	$0.149852\pi$
$109$			98.8797i			0.907153i	−0.891217	+	0.453577i	$0.850148\pi$
$110$		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	$-0.519745\pi$
$137$	−16.9858			−0.123984			−0.0619920	+	0.998077i	$0.519745\pi$
$138$		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	$0.235205\pi$
$151$			203.393i			1.34698i	−0.739198	+	0.673488i	$0.764795\pi$
$152$		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	$-0.634615\pi$
$157$	−128.869			−0.820825			−0.410412	+	0.911900i	$0.634615\pi$
$158$		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	$-0.196158\pi$
$163$	266.033			1.63210			0.816052	+	0.577979i	$0.196158\pi$
$164$		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	$0.199230\pi$
$179$			209.726i			1.17165i	−0.810437	+	0.585826i	$0.800770\pi$
$180$		0			0
$181$		−	177.564i		−	0.981016i	−0.871437	−	0.490508i	$-0.836811\pi$
$181$		−	177.564i		−	0.981016i	0.871437	−	0.490508i	$-0.163189\pi$
$182$		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	$-0.227293\pi$
$197$	297.749			1.51141			0.755707	+	0.654909i	$0.227293\pi$
$198$		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	$0.259058\pi$
$211$			306.768i			1.45388i	−0.686702	+	0.726939i	$0.740942\pi$
$212$		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	$-0.395449\pi$
$233$	150.324			0.645168			0.322584	+	0.946541i	$0.395449\pi$
$234$		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	$-0.757632\pi$
$241$		−	332.556i		−	1.37990i	0.723857	−	0.689950i	$-0.242368\pi$
$242$		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	$-0.465552\pi$
$263$	56.8141			0.216023			0.108012	+	0.994150i	$0.465552\pi$
$264$		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	$-0.638112\pi$
$269$		−	488.147i		−	1.81467i	0.420404	−	0.907337i	$-0.361888\pi$
$270$		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	$-0.538057\pi$
$277$	−66.0782			−0.238549			−0.119275	+	0.992861i	$0.538057\pi$
$278$		0			0
$279$			571.785i			2.04941i
$280$		0			0
$281$		−	412.679i		−	1.46861i	−0.678821	−	0.734304i	$-0.737509\pi$
$281$		−	412.679i		−	1.46861i	0.678821	−	0.734304i	$-0.262491\pi$
$282$		0			0
$283$	402.243			1.42135			0.710676	−	0.703519i	$-0.248389\pi$
$283$	402.243			1.42135			0.710676	+	0.703519i	$0.248389\pi$
$284$		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	$-0.325793\pi$
$313$	325.754			1.04075			0.520374	+	0.853939i	$0.325793\pi$
$314$		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	$0.0351196\pi$
$331$			72.8914i			0.220216i	−0.993920	+	0.110108i	$0.964880\pi$
$332$		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	$-0.377050\pi$
$347$	261.448			0.753452			0.376726	+	0.926325i	$0.377050\pi$
$348$		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	$-0.622125\pi$
$353$	−264.271			−0.748644			−0.374322	+	0.927299i	$0.622125\pi$
$354$		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	$-0.507475\pi$
$367$	−17.2357			−0.0469637			−0.0234818	+	0.999724i	$0.507475\pi$
$368$		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	$0.141228\pi$
$379$			325.385i			0.858534i	−0.903178	+	0.429267i	$0.858772\pi$
$380$		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	$-0.550390\pi$
$397$	−125.169			−0.315288			−0.157644	+	0.987496i	$0.550390\pi$
$398$		0			0
$399$	−648.853	+	376.843i	−1.62620	+	0.944468i
$400$		0			0
$401$			350.633i			0.874395i	0.899365	+	0.437198i	$0.144029\pi$
$401$			350.633i			0.874395i	−0.899365	+	0.437198i	$0.855971\pi$
$402$		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	$0.126822\pi$
$409$			317.356i			0.775932i	−0.921674	+	0.387966i	$0.873178\pi$
$410$		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	$0.0718822\pi$
$421$			188.532i			0.447820i	−0.974610	+	0.223910i	$0.928118\pi$
$422$		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	$-0.870296\pi$
$431$		−	341.606i		−	0.792590i	0.918123	−	0.396295i	$-0.129704\pi$
$432$		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	$-0.803070\pi$
$439$		−	509.200i		−	1.15991i	0.814649	−	0.579954i	$-0.196930\pi$
$440$		0			0
$441$	−10.7769			−0.0244374
$442$		0			0
$443$	772.768			1.74440			0.872198	−	0.489152i	$-0.162694\pi$
$443$	772.768			1.74440			0.872198	+	0.489152i	$0.162694\pi$
$444$		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	$0.267750\pi$
$449$			669.385i			1.49084i	−0.666598	+	0.745418i	$0.732250\pi$
$450$		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	$-0.234389\pi$
$457$	677.202			1.48184			0.740922	+	0.671591i	$0.234389\pi$
$458$		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	$-0.320578\pi$
$463$	494.756			1.06859			0.534293	+	0.845299i	$0.320578\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−79.7450			−0.170760			−0.0853801	−	0.996348i	$-0.527210\pi$
$467$	−79.7450			−0.170760			−0.0853801	+	0.996348i	$0.527210\pi$
$468$		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	$-0.571075\pi$
$503$	−222.767			−0.442877			−0.221438	+	0.975174i	$0.571075\pi$
$504$		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	$-0.902932\pi$
$509$		−	305.647i		−	0.600486i	0.953863	−	0.300243i	$-0.0970678\pi$
$510$		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	$-0.921564\pi$
$521$		−	254.171i		−	0.487853i	0.969794	−	0.243926i	$-0.0784355\pi$
$522$		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	$-0.597452\pi$
$557$	−335.753			−0.602787			−0.301394	+	0.953500i	$0.597452\pi$
$558$		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	$0.189073\pi$
$569$			636.906i			1.11934i	−0.828715	+	0.559671i	$0.810927\pi$
$570$		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	$-0.517338\pi$
$577$	−62.8264			−0.108885			−0.0544423	+	0.998517i	$0.517338\pi$
$578$		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	$-0.290354\pi$
$587$	718.521			1.22406			0.612028	+	0.790836i	$0.290354\pi$
$588$		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	$-0.478326\pi$
$593$	80.6940			0.136078			0.0680388	+	0.997683i	$0.478326\pi$
$594$		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	$0.0943811\pi$
$599$			350.033i			0.584363i	−0.956363	+	0.292181i	$0.905619\pi$
$600$		0			0
$601$			255.825i			0.425666i	0.977089	+	0.212833i	$0.0682691\pi$
$601$			255.825i			0.425666i	−0.977089	+	0.212833i	$0.931731\pi$
$602$		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	$-0.517038\pi$
$613$	−65.5926			−0.107003			−0.0535013	+	0.998568i	$0.517038\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−386.387			−0.626236			−0.313118	−	0.949714i	$-0.601373\pi$
$617$	−386.387			−0.626236			−0.313118	+	0.949714i	$0.601373\pi$
$618$		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	$0.114809\pi$
$641$			452.435i			0.705826i	−0.935656	+	0.352913i	$0.885191\pi$
$642$		0			0
$643$	235.639			0.366468			0.183234	−	0.983069i	$-0.441343\pi$
$643$	235.639			0.366468			0.183234	+	0.983069i	$0.441343\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−563.621			−0.871130			−0.435565	−	0.900157i	$-0.643451\pi$
$647$	−563.621			−0.871130			−0.435565	+	0.900157i	$0.643451\pi$
$648$		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	$-0.375273\pi$
$653$	498.748			0.763780			0.381890	+	0.924208i	$0.375273\pi$
$654$		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	$0.00880568\pi$
$659$			36.4563i			0.0553207i	−0.999617	+	0.0276603i	$0.991194\pi$
$660$		0			0
$661$		−	260.521i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$661$		−	260.521i		−	0.394132i	0.980390	−	0.197066i	$-0.0631414\pi$
$662$		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	$-0.541899\pi$
$727$	−190.836			−0.262499			−0.131249	+	0.991349i	$0.541899\pi$
$728$		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	$-0.334251\pi$
$733$	729.335			0.995001			0.497500	+	0.867464i	$0.334251\pi$
$734$		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	$0.00244261\pi$
$751$			11.5258i			0.0153472i	−0.999971	+	0.00767361i	$0.997557\pi$
$752$		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	$-0.610784\pi$
$757$	−516.354			−0.682106			−0.341053	+	0.940044i	$0.610784\pi$
$758$		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	$-0.683520\pi$
$811$		−	1359.80i		−	1.67670i	0.545131	−	0.838351i	$-0.316480\pi$
$812$		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	$-0.717843\pi$
$823$	−1040.58			−1.26438			−0.632188	+	0.774815i	$0.717843\pi$
$824$		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	$0.135211\pi$
$829$			683.294i			0.824239i	−0.911130	+	0.412119i	$0.864789\pi$
$830$		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	$0.255304\pi$
$839$			1206.13i			1.43758i	−0.695226	+	0.718791i	$0.744696\pi$
$840$		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	$-0.408425\pi$
$853$	484.057			0.567476			0.283738	+	0.958902i	$0.408425\pi$
$854$		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	$-0.263974\pi$
$883$	1192.75			1.35079			0.675394	+	0.737457i	$0.263974\pi$
$884$		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	$-0.714908\pi$
$911$		−	1422.27i		−	1.56122i	0.625017	−	0.780611i	$-0.285092\pi$
$912$		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	$-0.264727\pi$
$937$	1262.41			1.34729			0.673647	+	0.739053i	$0.264727\pi$
$938$		0			0
$939$		−	1828.90i		−	1.94771i
$940$		0			0
$941$		−	65.8851i		−	0.0700161i	−0.999387	−	0.0350080i	$-0.988854\pi$
$941$		−	65.8851i		−	0.0700161i	0.999387	−	0.0350080i	$-0.0111457\pi$
$942$		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	$-0.559975\pi$
$947$	−354.752			−0.374606			−0.187303	+	0.982302i	$0.559975\pi$
$948$		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	$-0.624428\pi$
$967$	−736.898			−0.762045			−0.381023	+	0.924566i	$0.624428\pi$
$968$		0			0
$969$	1529.42	−	888.260i	1.57835	−	0.916677i
$970$		0			0
$971$			1371.95i			1.41292i	0.707752	+	0.706461i	$0.249710\pi$
$971$			1371.95i			1.41292i	−0.707752	+	0.706461i	$0.750290\pi$
$972$		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	$0.00431537\pi$
$991$			26.8694i			0.0271134i	−0.999908	+	0.0135567i	$0.995685\pi$
$992$		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	$-0.273288\pi$
$997$	1303.14			1.30706			0.653529	+	0.756902i	$0.273288\pi$
$998$		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.1101.1		12
5.2	odd	4		380.3.g.c.189.1	✓	12
5.3	odd	4		380.3.g.c.189.12	yes	12
5.4	even	2	inner	1900.3.e.e.1101.12		12
15.2	even	4		3420.3.h.e.2089.8		12
15.8	even	4		3420.3.h.e.2089.5		12
19.18	odd	2	inner	1900.3.e.e.1101.11		12
95.18	even	4		380.3.g.c.189.2	yes	12
95.37	even	4		380.3.g.c.189.11	yes	12
95.94	odd	2	inner	1900.3.e.e.1101.2		12
285.113	odd	4		3420.3.h.e.2089.6		12
285.227	odd	4		3420.3.h.e.2089.7		12

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
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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