LMFDB - Embedded newform 3800.2.d.h.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (of order $2$, degree $1$, not 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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.h.3649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{3} +2.46410 q^{9} +O(q^{10})$	$q-0.732051i q^{3} +2.46410 q^{9} +2.00000 q^{11} +2.73205i q^{13} -0.535898i q^{17} -1.00000 q^{19} +5.46410i q^{23} -4.00000i q^{27} -3.46410 q^{29} +4.00000 q^{31} -1.46410i q^{33} +9.66025i q^{37} +2.00000 q^{39} +7.46410 q^{41} +10.9282i q^{43} -10.9282i q^{47} +7.00000 q^{49} -0.392305 q^{51} +5.66025i q^{53} +0.732051i q^{57} +5.46410 q^{59} -13.4641 q^{61} -6.19615i q^{67} +4.00000 q^{69} -2.92820 q^{71} +10.3923i q^{73} -12.3923 q^{79} +4.46410 q^{81} +1.46410i q^{83} +2.53590i q^{87} -3.46410 q^{89} -2.92820i q^{93} -1.66025i q^{97} +4.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 8 q^{11} - 4 q^{19} + 16 q^{31} + 8 q^{39} + 16 q^{41} + 28 q^{49} + 40 q^{51} + 8 q^{59} - 40 q^{61} + 16 q^{69} + 16 q^{71} - 8 q^{79} + 4 q^{81} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^11 - 4 * q^19 + 16 * q^31 + 8 * q^39 + 16 * q^41 + 28 * q^49 + 40 * q^51 + 8 * q^59 - 40 * q^61 + 16 * q^69 + 16 * q^71 - 8 * q^79 + 4 * q^81 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 0.732051i	− 0.422650i	−0.977416	−	0.211325i	$-0.932222\pi$
$3$	− 0.732051i	− 0.422650i	0.977416	−	0.211325i	$-0.0677778\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.73205i	0.757735i	0.925451	+	0.378867i	$0.123686\pi$
$13$	2.73205i	0.757735i	−0.925451	+	0.378867i	$0.876314\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.535898i	− 0.129974i	−0.997886	−	0.0649872i	$-0.979299\pi$
$17$	− 0.535898i	− 0.129974i	0.997886	−	0.0649872i	$-0.0207007\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.46410i	1.13934i	0.821872	+	0.569672i	$0.192930\pi$
$23$	5.46410i	1.13934i	−0.821872	+	0.569672i	$0.807070\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 1.46410i	− 0.254867i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.66025i	1.58814i	0.607829	+	0.794068i	$0.292041\pi$
$37$	9.66025i	1.58814i	−0.607829	+	0.794068i	$0.707959\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	7.46410	1.16570	0.582848	−	0.812581i	$-0.301938\pi$
$41$	7.46410	1.16570	0.582848	+	0.812581i	$0.301938\pi$
$42$	0	0
$43$	10.9282i	1.66654i	0.552870	+	0.833268i	$0.313533\pi$
$43$	10.9282i	1.66654i	−0.552870	+	0.833268i	$0.686467\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.9282i	− 1.59404i	−0.603951	−	0.797021i	$-0.706408\pi$
$47$	− 10.9282i	− 1.59404i	0.603951	−	0.797021i	$-0.293592\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	−0.392305	−0.0549337
$52$	0	0
$53$	5.66025i	0.777496i	0.921344	+	0.388748i	$0.127092\pi$
$53$	5.66025i	0.777496i	−0.921344	+	0.388748i	$0.872908\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.732051i	0.0969625i
$58$	0	0
$59$	5.46410	0.711365	0.355683	−	0.934607i	$-0.384248\pi$
$59$	5.46410	0.711365	0.355683	+	0.934607i	$0.384248\pi$
$60$	0	0
$61$	−13.4641	−1.72390	−0.861951	−	0.506992i	$-0.830757\pi$
$61$	−13.4641	−1.72390	−0.861951	+	0.506992i	$0.830757\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.19615i	− 0.756980i	−0.925605	−	0.378490i	$-0.876443\pi$
$67$	− 6.19615i	− 0.756980i	0.925605	−	0.378490i	$-0.123557\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−2.92820	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	0	0
$73$	10.3923i	1.21633i	0.793812	+	0.608164i	$0.208094\pi$
$73$	10.3923i	1.21633i	−0.793812	+	0.608164i	$0.791906\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.3923	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$79$	−12.3923	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	1.46410i	0.160706i	0.996766	+	0.0803530i	$0.0256048\pi$
$83$	1.46410i	0.160706i	−0.996766	+	0.0803530i	$0.974395\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.53590i	0.271877i
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 2.92820i	− 0.303641i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.66025i	− 0.168573i	−0.996442	−	0.0842866i	$-0.973139\pi$
$97$	− 1.66025i	− 0.168573i	0.996442	−	0.0842866i	$-0.0268611\pi$
$98$	0	0
$99$	4.92820	0.495303
$100$	0	0
$101$	−5.46410	−0.543698	−0.271849	−	0.962340i	$-0.587635\pi$
$101$	−5.46410	−0.543698	−0.271849	+	0.962340i	$0.587635\pi$
$102$	0	0
$103$	− 11.6603i	− 1.14892i	−0.818533	−	0.574459i	$-0.805212\pi$
$103$	− 11.6603i	− 1.14892i	0.818533	−	0.574459i	$-0.194788\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.12436i	0.882085i	0.897486	+	0.441042i	$0.145391\pi$
$107$	9.12436i	0.882085i	−0.897486	+	0.441042i	$0.854609\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	7.07180	0.671225
$112$	0	0
$113$	13.2679i	1.24814i	0.781367	+	0.624072i	$0.214523\pi$
$113$	13.2679i	1.24814i	−0.781367	+	0.624072i	$0.785477\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.73205i	0.622378i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 5.46410i	− 0.492681i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 0.339746i	− 0.0301476i	−0.999886	−	0.0150738i	$-0.995202\pi$
$127$	− 0.339746i	− 0.0301476i	0.999886	−	0.0150738i	$-0.00479832\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 7.46410i	− 0.637701i	−0.947805	−	0.318851i	$-0.896703\pi$
$137$	− 7.46410i	− 0.637701i	0.947805	−	0.318851i	$-0.103297\pi$
$138$	0	0
$139$	19.8564	1.68420	0.842099	−	0.539323i	$-0.181320\pi$
$139$	19.8564	1.68420	0.842099	+	0.539323i	$0.181320\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	5.46410i	0.456931i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5.12436i	− 0.422650i
$148$	0	0
$149$	19.3205	1.58280	0.791399	−	0.611300i	$-0.209353\pi$
$149$	19.3205	1.58280	0.791399	+	0.611300i	$0.209353\pi$
$150$	0	0
$151$	1.46410	0.119147	0.0595734	−	0.998224i	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	0	0
$153$	− 1.32051i	− 0.106757i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.535898i	0.0427693i	0.999771	+	0.0213847i	$0.00680747\pi$
$157$	0.535898i	0.0427693i	−0.999771	+	0.0213847i	$0.993193\pi$
$158$	0	0
$159$	4.14359	0.328608
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.26795i	0.252882i	0.991974	+	0.126441i	$0.0403554\pi$
$167$	3.26795i	0.252882i	−0.991974	+	0.126441i	$0.959645\pi$
$168$	0	0
$169$	5.53590	0.425838
$170$	0	0
$171$	−2.46410	−0.188435
$172$	0	0
$173$	7.12436i	0.541655i	0.962628	+	0.270827i	$0.0872973\pi$
$173$	7.12436i	0.541655i	−0.962628	+	0.270827i	$0.912703\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	2.53590	0.189542	0.0947710	−	0.995499i	$-0.469788\pi$
$179$	2.53590	0.189542	0.0947710	+	0.995499i	$0.469788\pi$
$180$	0	0
$181$	11.8564	0.881280	0.440640	−	0.897684i	$-0.354752\pi$
$181$	11.8564	0.881280	0.440640	+	0.897684i	$0.354752\pi$
$182$	0	0
$183$	9.85641i	0.728607i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.07180i	− 0.0783775i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.92820	0.211877	0.105939	−	0.994373i	$-0.466215\pi$
$191$	2.92820	0.211877	0.105939	+	0.994373i	$0.466215\pi$
$192$	0	0
$193$	3.80385i	0.273807i	0.990584	+	0.136903i	$0.0437150\pi$
$193$	3.80385i	0.273807i	−0.990584	+	0.136903i	$0.956285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.8564i	1.12972i	0.825186	+	0.564861i	$0.191070\pi$
$197$	15.8564i	1.12972i	−0.825186	+	0.564861i	$0.808930\pi$
$198$	0	0
$199$	5.07180	0.359530	0.179765	−	0.983710i	$-0.442466\pi$
$199$	5.07180	0.359530	0.179765	+	0.983710i	$0.442466\pi$
$200$	0	0
$201$	−4.53590	−0.319938
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.4641i	0.935820i
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	4.39230	0.302379	0.151189	−	0.988505i	$-0.451690\pi$
$211$	4.39230	0.302379	0.151189	+	0.988505i	$0.451690\pi$
$212$	0	0
$213$	2.14359i	0.146877i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.60770	0.514080
$220$	0	0
$221$	1.46410	0.0984861
$222$	0	0
$223$	− 12.0526i	− 0.807099i	−0.914958	−	0.403550i	$-0.867776\pi$
$223$	− 12.0526i	− 0.807099i	0.914958	−	0.403550i	$-0.132224\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.73205i	− 0.314077i	−0.987592	−	0.157039i	$-0.949805\pi$
$227$	− 4.73205i	− 0.314077i	0.987592	−	0.157039i	$-0.0501947\pi$
$228$	0	0
$229$	1.46410	0.0967506	0.0483753	−	0.998829i	$-0.484596\pi$
$229$	1.46410	0.0967506	0.0483753	+	0.998829i	$0.484596\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.07180i	0.589277i
$238$	0	0
$239$	5.07180	0.328067	0.164034	−	0.986455i	$-0.447549\pi$
$239$	5.07180	0.328067	0.164034	+	0.986455i	$0.447549\pi$
$240$	0	0
$241$	10.3923	0.669427	0.334714	−	0.942320i	$-0.391360\pi$
$241$	10.3923	0.669427	0.334714	+	0.942320i	$0.391360\pi$
$242$	0	0
$243$	− 15.2679i	− 0.979439i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.73205i	− 0.173836i
$248$	0	0
$249$	1.07180	0.0679224
$250$	0	0
$251$	25.8564	1.63204	0.816021	−	0.578022i	$-0.196175\pi$
$251$	25.8564	1.63204	0.816021	+	0.578022i	$0.196175\pi$
$252$	0	0
$253$	10.9282i	0.687050i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 21.2679i	− 1.32666i	−0.748328	−	0.663329i	$-0.769143\pi$
$257$	− 21.2679i	− 1.32666i	0.748328	−	0.663329i	$-0.230857\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.53590	−0.528359
$262$	0	0
$263$	− 15.3205i	− 0.944703i	−0.881410	−	0.472351i	$-0.843405\pi$
$263$	− 15.3205i	− 0.944703i	0.881410	−	0.472351i	$-0.156595\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.53590i	0.155194i
$268$	0	0
$269$	−4.92820	−0.300478	−0.150239	−	0.988650i	$-0.548004\pi$
$269$	−4.92820	−0.300478	−0.150239	+	0.988650i	$0.548004\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	27.4641i	1.65016i	0.565017	+	0.825079i	$0.308869\pi$
$277$	27.4641i	1.65016i	−0.565017	+	0.825079i	$0.691131\pi$
$278$	0	0
$279$	9.85641	0.590088
$280$	0	0
$281$	−1.60770	−0.0959071	−0.0479535	−	0.998850i	$-0.515270\pi$
$281$	−1.60770	−0.0959071	−0.0479535	+	0.998850i	$0.515270\pi$
$282$	0	0
$283$	− 2.53590i	− 0.150744i	−0.997156	−	0.0753718i	$-0.975986\pi$
$283$	− 2.53590i	− 0.150744i	0.997156	−	0.0753718i	$-0.0240143\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.7128	0.983107
$290$	0	0
$291$	−1.21539	−0.0712474
$292$	0	0
$293$	7.80385i	0.455906i	0.973672	+	0.227953i	$0.0732032\pi$
$293$	7.80385i	0.455906i	−0.973672	+	0.227953i	$0.926797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 8.00000i	− 0.464207i
$298$	0	0
$299$	−14.9282	−0.863320
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.00000i	0.229794i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 28.7321i	− 1.63982i	−0.572489	−	0.819912i	$-0.694022\pi$
$307$	− 28.7321i	− 1.63982i	0.572489	−	0.819912i	$-0.305978\pi$
$308$	0	0
$309$	−8.53590	−0.485590
$310$	0	0
$311$	−7.85641	−0.445496	−0.222748	−	0.974876i	$-0.571503\pi$
$311$	−7.85641	−0.445496	−0.222748	+	0.974876i	$0.571503\pi$
$312$	0	0
$313$	11.8564i	0.670164i	0.942189	+	0.335082i	$0.108764\pi$
$313$	11.8564i	0.670164i	−0.942189	+	0.335082i	$0.891236\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 24.5885i	− 1.38103i	−0.723320	−	0.690513i	$-0.757385\pi$
$317$	− 24.5885i	− 1.38103i	0.723320	−	0.690513i	$-0.242615\pi$
$318$	0	0
$319$	−6.92820	−0.387905
$320$	0	0
$321$	6.67949	0.372813
$322$	0	0
$323$	0.535898i	0.0298182i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 10.2487i	− 0.566755i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−34.2487	−1.88248	−0.941240	−	0.337739i	$-0.890338\pi$
$331$	−34.2487	−1.88248	−0.941240	+	0.337739i	$0.890338\pi$
$332$	0	0
$333$	23.8038i	1.30444i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.66025i	0.308334i	0.988045	+	0.154167i	$0.0492693\pi$
$337$	5.66025i	0.308334i	−0.988045	+	0.154167i	$0.950731\pi$
$338$	0	0
$339$	9.71281	0.527528
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.3923i	− 0.665254i	−0.943059	−	0.332627i	$-0.892065\pi$
$347$	− 12.3923i	− 0.665254i	0.943059	−	0.332627i	$-0.107935\pi$
$348$	0	0
$349$	−18.7846	−1.00552	−0.502759	−	0.864427i	$-0.667682\pi$
$349$	−18.7846	−1.00552	−0.502759	+	0.864427i	$0.667682\pi$
$350$	0	0
$351$	10.9282	0.583304
$352$	0	0
$353$	− 16.2487i	− 0.864832i	−0.901674	−	0.432416i	$-0.857661\pi$
$353$	− 16.2487i	− 0.864832i	0.901674	−	0.432416i	$-0.142339\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.92820	0.260101	0.130050	−	0.991507i	$-0.458486\pi$
$359$	4.92820	0.260101	0.130050	+	0.991507i	$0.458486\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.12436i	0.268959i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.92820i	0.152851i	0.997075	+	0.0764255i	$0.0243507\pi$
$367$	2.92820i	0.152851i	−0.997075	+	0.0764255i	$0.975649\pi$
$368$	0	0
$369$	18.3923	0.957465
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 0.196152i	− 0.0101564i	−0.999987	−	0.00507819i	$-0.998384\pi$
$373$	− 0.196152i	− 0.0101564i	0.999987	−	0.00507819i	$-0.00161645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.46410i	− 0.487426i
$378$	0	0
$379$	−34.6410	−1.77939	−0.889695	−	0.456556i	$-0.849083\pi$
$379$	−34.6410	−1.77939	−0.889695	+	0.456556i	$0.849083\pi$
$380$	0	0
$381$	−0.248711	−0.0127419
$382$	0	0
$383$	21.8038i	1.11412i	0.830471	+	0.557062i	$0.188072\pi$
$383$	21.8038i	1.11412i	−0.830471	+	0.557062i	$0.811928\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.9282i	1.36884i
$388$	0	0
$389$	−11.8564	−0.601144	−0.300572	−	0.953759i	$-0.597178\pi$
$389$	−11.8564	−0.601144	−0.300572	+	0.953759i	$0.597178\pi$
$390$	0	0
$391$	2.92820	0.148086
$392$	0	0
$393$	− 8.78461i	− 0.443125i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.5359i	0.629159i	0.949231	+	0.314579i	$0.101863\pi$
$397$	12.5359i	0.629159i	−0.949231	+	0.314579i	$0.898137\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	10.9282i	0.544373i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.3205i	0.957682i
$408$	0	0
$409$	−28.9282	−1.43041	−0.715204	−	0.698916i	$-0.753666\pi$
$409$	−28.9282	−1.43041	−0.715204	+	0.698916i	$0.753666\pi$
$410$	0	0
$411$	−5.46410	−0.269524
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 14.5359i	− 0.711826i
$418$	0	0
$419$	1.85641	0.0906914	0.0453457	−	0.998971i	$-0.485561\pi$
$419$	1.85641	0.0906914	0.0453457	+	0.998971i	$0.485561\pi$
$420$	0	0
$421$	−32.2487	−1.57171	−0.785853	−	0.618413i	$-0.787776\pi$
$421$	−32.2487	−1.57171	−0.785853	+	0.618413i	$0.787776\pi$
$422$	0	0
$423$	− 26.9282i	− 1.30929i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	7.60770	0.366450	0.183225	−	0.983071i	$-0.441346\pi$
$431$	7.60770	0.366450	0.183225	+	0.983071i	$0.441346\pi$
$432$	0	0
$433$	24.9808i	1.20050i	0.799813	+	0.600249i	$0.204932\pi$
$433$	24.9808i	1.20050i	−0.799813	+	0.600249i	$0.795068\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.46410i	− 0.261383i
$438$	0	0
$439$	22.2487	1.06187	0.530937	−	0.847412i	$-0.321840\pi$
$439$	22.2487	1.06187	0.530937	+	0.847412i	$0.321840\pi$
$440$	0	0
$441$	17.2487	0.821367
$442$	0	0
$443$	− 26.5359i	− 1.26076i	−0.776287	−	0.630379i	$-0.782899\pi$
$443$	− 26.5359i	− 1.26076i	0.776287	−	0.630379i	$-0.217101\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.1436i	− 0.668969i
$448$	0	0
$449$	−32.6410	−1.54042	−0.770212	−	0.637787i	$-0.779850\pi$
$449$	−32.6410	−1.54042	−0.770212	+	0.637787i	$0.779850\pi$
$450$	0	0
$451$	14.9282	0.702942
$452$	0	0
$453$	− 1.07180i	− 0.0503574i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 15.0718i	− 0.705029i	−0.935806	−	0.352514i	$-0.885327\pi$
$457$	− 15.0718i	− 0.705029i	0.935806	−	0.352514i	$-0.114673\pi$
$458$	0	0
$459$	−2.14359	−0.100054
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	− 38.2487i	− 1.77757i	−0.458326	−	0.888784i	$-0.651551\pi$
$463$	− 38.2487i	− 1.77757i	0.458326	−	0.888784i	$-0.348449\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.679492i	0.0314431i	0.999876	+	0.0157216i	$0.00500454\pi$
$467$	0.679492i	0.0314431i	−0.999876	+	0.0157216i	$0.994995\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.392305	0.0180765
$472$	0	0
$473$	21.8564i	1.00496i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.9474i	0.638609i
$478$	0	0
$479$	7.85641	0.358968	0.179484	−	0.983761i	$-0.442557\pi$
$479$	7.85641	0.358968	0.179484	+	0.983761i	$0.442557\pi$
$480$	0	0
$481$	−26.3923	−1.20339
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 39.2679i	− 1.77940i	−0.456545	−	0.889700i	$-0.650913\pi$
$487$	− 39.2679i	− 1.77940i	0.456545	−	0.889700i	$-0.349087\pi$
$488$	0	0
$489$	−2.92820	−0.132418
$490$	0	0
$491$	25.8564	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$491$	25.8564	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$492$	0	0
$493$	1.85641i	0.0836083i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.7846	0.482785	0.241393	−	0.970428i	$-0.422396\pi$
$499$	10.7846	0.482785	0.241393	+	0.970428i	$0.422396\pi$
$500$	0	0
$501$	2.39230	0.106880
$502$	0	0
$503$	− 11.3205i	− 0.504757i	−0.967629	−	0.252378i	$-0.918787\pi$
$503$	− 11.3205i	− 0.504757i	0.967629	−	0.252378i	$-0.0812127\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4.05256i	− 0.179980i
$508$	0	0
$509$	18.3923	0.815225	0.407612	−	0.913155i	$-0.366361\pi$
$509$	18.3923	0.815225	0.407612	+	0.913155i	$0.366361\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 21.8564i	− 0.961244i
$518$	0	0
$519$	5.21539	0.228930
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 15.2679i	− 0.667621i	−0.942640	−	0.333810i	$-0.891665\pi$
$523$	− 15.2679i	− 0.667621i	0.942640	−	0.333810i	$-0.108335\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.14359i	− 0.0933764i
$528$	0	0
$529$	−6.85641	−0.298105
$530$	0	0
$531$	13.4641	0.584292
$532$	0	0
$533$	20.3923i	0.883289i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.85641i	− 0.0801099i
$538$	0	0
$539$	14.0000	0.603023
$540$	0	0
$541$	−44.3923	−1.90857	−0.954287	−	0.298891i	$-0.903383\pi$
$541$	−44.3923	−1.90857	−0.954287	+	0.298891i	$0.903383\pi$
$542$	0	0
$543$	− 8.67949i	− 0.372473i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.5885i	0.623757i	0.950122	+	0.311879i	$0.100958\pi$
$547$	14.5885i	0.623757i	−0.950122	+	0.311879i	$0.899042\pi$
$548$	0	0
$549$	−33.1769	−1.41596
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	36.2487i	1.53591i	0.640505	+	0.767954i	$0.278725\pi$
$557$	36.2487i	1.53591i	−0.640505	+	0.767954i	$0.721275\pi$
$558$	0	0
$559$	−29.8564	−1.26279
$560$	0	0
$561$	−0.784610	−0.0331262
$562$	0	0
$563$	22.5885i	0.951990i	0.879448	+	0.475995i	$0.157912\pi$
$563$	22.5885i	0.951990i	−0.879448	+	0.475995i	$0.842088\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.7846	1.12287	0.561435	−	0.827521i	$-0.310250\pi$
$569$	26.7846	1.12287	0.561435	+	0.827521i	$0.310250\pi$
$570$	0	0
$571$	10.7846	0.451322	0.225661	−	0.974206i	$-0.427546\pi$
$571$	10.7846	0.451322	0.225661	+	0.974206i	$0.427546\pi$
$572$	0	0
$573$	− 2.14359i	− 0.0895499i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.14359i	− 0.339022i	−0.985528	−	0.169511i	$-0.945781\pi$
$577$	− 8.14359i	− 0.339022i	0.985528	−	0.169511i	$-0.0542189\pi$
$578$	0	0
$579$	2.78461	0.115724
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11.3205i	0.468848i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.1051i	1.65532i	0.561233	+	0.827658i	$0.310327\pi$
$587$	40.1051i	1.65532i	−0.561233	+	0.827658i	$0.689673\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	11.6077	0.477477
$592$	0	0
$593$	− 8.14359i	− 0.334417i	−0.985922	−	0.167209i	$-0.946525\pi$
$593$	− 8.14359i	− 0.334417i	0.985922	−	0.167209i	$-0.0534754\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3.71281i	− 0.151955i
$598$	0	0
$599$	−47.0333	−1.92173	−0.960865	−	0.277018i	$-0.910654\pi$
$599$	−47.0333	−1.92173	−0.960865	+	0.277018i	$0.910654\pi$
$600$	0	0
$601$	−43.4641	−1.77294	−0.886469	−	0.462788i	$-0.846849\pi$
$601$	−43.4641	−1.77294	−0.886469	+	0.462788i	$0.846849\pi$
$602$	0	0
$603$	− 15.2679i	− 0.621759i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.4449i	0.829831i	0.909860	+	0.414916i	$0.136189\pi$
$607$	20.4449i	0.829831i	−0.909860	+	0.414916i	$0.863811\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.8564	1.20786
$612$	0	0
$613$	− 4.24871i	− 0.171604i	−0.996312	−	0.0858019i	$-0.972655\pi$
$613$	− 4.24871i	− 0.171604i	0.996312	−	0.0858019i	$-0.0273452\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.46410i	0.139459i	0.997566	+	0.0697297i	$0.0222137\pi$
$617$	3.46410i	0.139459i	−0.997566	+	0.0697297i	$0.977786\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	0	0
$621$	21.8564	0.877067
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.46410i	0.0584706i
$628$	0	0
$629$	5.17691	0.206417
$630$	0	0
$631$	−48.6410	−1.93637	−0.968184	−	0.250239i	$-0.919491\pi$
$631$	−48.6410	−1.93637	−0.968184	+	0.250239i	$0.919491\pi$
$632$	0	0
$633$	− 3.21539i	− 0.127800i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	19.1244i	0.757735i
$638$	0	0
$639$	−7.21539	−0.285436
$640$	0	0
$641$	−45.0333	−1.77871	−0.889355	−	0.457218i	$-0.848846\pi$
$641$	−45.0333	−1.77871	−0.889355	+	0.457218i	$0.848846\pi$
$642$	0	0
$643$	50.2487i	1.98162i	0.135277	+	0.990808i	$0.456808\pi$
$643$	50.2487i	1.98162i	−0.135277	+	0.990808i	$0.543192\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.6795i	0.498482i	0.968441	+	0.249241i	$0.0801812\pi$
$647$	12.6795i	0.498482i	−0.968441	+	0.249241i	$0.919819\pi$
$648$	0	0
$649$	10.9282	0.428969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 13.6077i	− 0.532510i	−0.963903	−	0.266255i	$-0.914214\pi$
$653$	− 13.6077i	− 0.532510i	0.963903	−	0.266255i	$-0.0857864\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	25.6077i	0.999051i
$658$	0	0
$659$	−11.6077	−0.452172	−0.226086	−	0.974107i	$-0.572593\pi$
$659$	−11.6077	−0.452172	−0.226086	+	0.974107i	$0.572593\pi$
$660$	0	0
$661$	−23.4641	−0.912648	−0.456324	−	0.889814i	$-0.650834\pi$
$661$	−23.4641	−0.912648	−0.456324	+	0.889814i	$0.650834\pi$
$662$	0	0
$663$	− 1.07180i	− 0.0416251i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.9282i	− 0.732903i
$668$	0	0
$669$	−8.82309	−0.341120
$670$	0	0
$671$	−26.9282	−1.03955
$672$	0	0
$673$	− 44.1962i	− 1.70364i	−0.523837	−	0.851818i	$-0.675500\pi$
$673$	− 44.1962i	− 1.70364i	0.523837	−	0.851818i	$-0.324500\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.9090i	1.38009i	0.723765	+	0.690047i	$0.242410\pi$
$677$	35.9090i	1.38009i	−0.723765	+	0.690047i	$0.757590\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.46410	−0.132745
$682$	0	0
$683$	− 26.1962i	− 1.00237i	−0.865341	−	0.501184i	$-0.832898\pi$
$683$	− 26.1962i	− 1.00237i	0.865341	−	0.501184i	$-0.167102\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.07180i	− 0.0408916i
$688$	0	0
$689$	−15.4641	−0.589135
$690$	0	0
$691$	2.78461	0.105932	0.0529658	−	0.998596i	$-0.483133\pi$
$691$	2.78461	0.105932	0.0529658	+	0.998596i	$0.483133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	10.2487	0.387642
$700$	0	0
$701$	26.2487	0.991400	0.495700	−	0.868494i	$-0.334912\pi$
$701$	26.2487	0.991400	0.495700	+	0.868494i	$0.334912\pi$
$702$	0	0
$703$	− 9.66025i	− 0.364343i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22.7846	0.855694	0.427847	−	0.903851i	$-0.359272\pi$
$709$	22.7846	0.855694	0.427847	+	0.903851i	$0.359272\pi$
$710$	0	0
$711$	−30.5359	−1.14519
$712$	0	0
$713$	21.8564i	0.818529i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.71281i	− 0.138658i
$718$	0	0
$719$	−35.5692	−1.32651	−0.663254	−	0.748394i	$-0.730825\pi$
$719$	−35.5692	−1.32651	−0.663254	+	0.748394i	$0.730825\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 7.60770i	− 0.282933i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 34.9282i	− 1.29542i	−0.761889	−	0.647708i	$-0.775728\pi$
$727$	− 34.9282i	− 1.29542i	0.761889	−	0.647708i	$-0.224272\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	5.85641	0.216607
$732$	0	0
$733$	− 43.8564i	− 1.61987i	−0.586517	−	0.809937i	$-0.699501\pi$
$733$	− 43.8564i	− 1.61987i	0.586517	−	0.809937i	$-0.300499\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.3923i	− 0.456476i
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	− 43.3731i	− 1.59120i	−0.605820	−	0.795602i	$-0.707155\pi$
$743$	− 43.3731i	− 1.59120i	0.605820	−	0.795602i	$-0.292845\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.60770i	0.131999i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.5359	0.676385	0.338192	−	0.941077i	$-0.390185\pi$
$751$	18.5359	0.676385	0.338192	+	0.941077i	$0.390185\pi$
$752$	0	0
$753$	− 18.9282i	− 0.689782i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.7846i	1.55503i	0.628862	+	0.777517i	$0.283521\pi$
$757$	42.7846i	1.55503i	−0.628862	+	0.777517i	$0.716479\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	3.60770	0.130779	0.0653894	−	0.997860i	$-0.479171\pi$
$761$	3.60770	0.130779	0.0653894	+	0.997860i	$0.479171\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.9282i	0.539026i
$768$	0	0
$769$	−3.60770	−0.130097	−0.0650484	−	0.997882i	$-0.520720\pi$
$769$	−3.60770	−0.130097	−0.0650484	+	0.997882i	$0.520720\pi$
$770$	0	0
$771$	−15.5692	−0.560712
$772$	0	0
$773$	− 0.196152i	− 0.00705511i	−0.999994	−	0.00352756i	$-0.998877\pi$
$773$	− 0.196152i	− 0.00705511i	0.999994	−	0.00352756i	$-0.00112286\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.46410	−0.267429
$780$	0	0
$781$	−5.85641	−0.209559
$782$	0	0
$783$	13.8564i	0.495188i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 10.1962i	− 0.363454i	−0.983349	−	0.181727i	$-0.941831\pi$
$787$	− 10.1962i	− 0.363454i	0.983349	−	0.181727i	$-0.0581686\pi$
$788$	0	0
$789$	−11.2154	−0.399278
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 36.7846i	− 1.30626i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0526i	0.781142i	0.920573	+	0.390571i	$0.127722\pi$
$797$	22.0526i	0.781142i	−0.920573	+	0.390571i	$0.872278\pi$
$798$	0	0
$799$	−5.85641	−0.207185
$800$	0	0
$801$	−8.53590	−0.301601
$802$	0	0
$803$	20.7846i	0.733473i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.60770i	0.126997i
$808$	0	0
$809$	45.7128	1.60718	0.803588	−	0.595185i	$-0.202921\pi$
$809$	45.7128	1.60718	0.803588	+	0.595185i	$0.202921\pi$
$810$	0	0
$811$	−13.8564	−0.486564	−0.243282	−	0.969956i	$-0.578224\pi$
$811$	−13.8564	−0.486564	−0.243282	+	0.969956i	$0.578224\pi$
$812$	0	0
$813$	− 10.2487i	− 0.359438i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10.9282i	− 0.382329i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.7846	−0.655587	−0.327794	−	0.944749i	$-0.606305\pi$
$821$	−18.7846	−0.655587	−0.327794	+	0.944749i	$0.606305\pi$
$822$	0	0
$823$	− 13.0718i	− 0.455654i	−0.973702	−	0.227827i	$-0.926838\pi$
$823$	− 13.0718i	− 0.455654i	0.973702	−	0.227827i	$-0.0731621\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.9090i	1.04004i	0.854155	+	0.520018i	$0.174075\pi$
$827$	29.9090i	1.04004i	−0.854155	+	0.520018i	$0.825925\pi$
$828$	0	0
$829$	−21.7128	−0.754117	−0.377059	−	0.926189i	$-0.623064\pi$
$829$	−21.7128	−0.754117	−0.377059	+	0.926189i	$0.623064\pi$
$830$	0	0
$831$	20.1051	0.697439
$832$	0	0
$833$	− 3.75129i	− 0.129974i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	−44.3923	−1.53259	−0.766296	−	0.642487i	$-0.777903\pi$
$839$	−44.3923	−1.53259	−0.766296	+	0.642487i	$0.777903\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	1.17691i	0.0405351i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.85641	−0.0637117
$850$	0	0
$851$	−52.7846	−1.80943
$852$	0	0
$853$	− 26.7846i	− 0.917088i	−0.888672	−	0.458544i	$-0.848371\pi$
$853$	− 26.7846i	− 0.917088i	0.888672	−	0.458544i	$-0.151629\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.87564i	0.166549i	0.996527	+	0.0832744i	$0.0265378\pi$
$857$	4.87564i	0.166549i	−0.996527	+	0.0832744i	$0.973462\pi$
$858$	0	0
$859$	−57.8564	−1.97404	−0.987018	−	0.160612i	$-0.948653\pi$
$859$	−57.8564	−1.97404	−0.987018	+	0.160612i	$0.948653\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 28.7321i	− 0.978050i	−0.872270	−	0.489025i	$-0.837353\pi$
$863$	− 28.7321i	− 0.978050i	0.872270	−	0.489025i	$-0.162647\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 12.2346i	− 0.415510i
$868$	0	0
$869$	−24.7846	−0.840760
$870$	0	0
$871$	16.9282	0.573590
$872$	0	0
$873$	− 4.09103i	− 0.138461i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 46.4449i	− 1.56833i	−0.620551	−	0.784166i	$-0.713091\pi$
$877$	− 46.4449i	− 1.56833i	0.620551	−	0.784166i	$-0.286909\pi$
$878$	0	0
$879$	5.71281	0.192688
$880$	0	0
$881$	−8.39230	−0.282744	−0.141372	−	0.989957i	$-0.545151\pi$
$881$	−8.39230	−0.282744	−0.141372	+	0.989957i	$0.545151\pi$
$882$	0	0
$883$	− 30.6410i	− 1.03115i	−0.856844	−	0.515576i	$-0.827578\pi$
$883$	− 30.6410i	− 1.03115i	0.856844	−	0.515576i	$-0.172422\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.1244i	1.51513i	0.652761	+	0.757564i	$0.273611\pi$
$887$	45.1244i	1.51513i	−0.652761	+	0.757564i	$0.726389\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	8.92820	0.299106
$892$	0	0
$893$	10.9282i	0.365698i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.9282i	0.364882i
$898$	0	0
$899$	−13.8564	−0.462137
$900$	0	0
$901$	3.03332	0.101055
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 24.4449i	− 0.811678i	−0.913945	−	0.405839i	$-0.866979\pi$
$907$	− 24.4449i	− 0.811678i	0.913945	−	0.405839i	$-0.133021\pi$
$908$	0	0
$909$	−13.4641	−0.446576
$910$	0	0
$911$	42.5359	1.40928	0.704639	−	0.709566i	$-0.251109\pi$
$911$	42.5359	1.40928	0.704639	+	0.709566i	$0.251109\pi$
$912$	0	0
$913$	2.92820i	0.0969094i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10.9282	0.360488	0.180244	−	0.983622i	$-0.442311\pi$
$919$	10.9282	0.360488	0.180244	+	0.983622i	$0.442311\pi$
$920$	0	0
$921$	−21.0333	−0.693071
$922$	0	0
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 28.7321i	− 0.943684i
$928$	0	0
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	0	0
$933$	5.75129i	0.188289i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 50.3923i	− 1.64624i	−0.567864	−	0.823122i	$-0.692230\pi$
$937$	− 50.3923i	− 1.64624i	0.567864	−	0.823122i	$-0.307770\pi$
$938$	0	0
$939$	8.67949	0.283245
$940$	0	0
$941$	48.9282	1.59501	0.797507	−	0.603310i	$-0.206152\pi$
$941$	48.9282	1.59501	0.797507	+	0.603310i	$0.206152\pi$
$942$	0	0
$943$	40.7846i	1.32813i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.75129i	0.0569092i	0.999595	+	0.0284546i	$0.00905861\pi$
$947$	1.75129i	0.0569092i	−0.999595	+	0.0284546i	$0.990941\pi$
$948$	0	0
$949$	−28.3923	−0.921653
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	− 48.5885i	− 1.57393i	−0.616995	−	0.786967i	$-0.711650\pi$
$953$	− 48.5885i	− 1.57393i	0.616995	−	0.786967i	$-0.288350\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.07180i	0.163948i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	22.4833i	0.724515i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 5.17691i	− 0.166478i	−0.996530	−	0.0832392i	$-0.973473\pi$
$967$	− 5.17691i	− 0.166478i	0.996530	−	0.0832392i	$-0.0265265\pi$
$968$	0	0
$969$	0.392305	0.0126026
$970$	0	0
$971$	−6.92820	−0.222337	−0.111168	−	0.993802i	$-0.535459\pi$
$971$	−6.92820	−0.222337	−0.111168	+	0.993802i	$0.535459\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 50.4449i	− 1.61387i	−0.590637	−	0.806937i	$-0.701124\pi$
$977$	− 50.4449i	− 1.61387i	0.590637	−	0.806937i	$-0.298876\pi$
$978$	0	0
$979$	−6.92820	−0.221426
$980$	0	0
$981$	34.4974	1.10142
$982$	0	0
$983$	41.5167i	1.32418i	0.749426	+	0.662088i	$0.230329\pi$
$983$	41.5167i	1.32418i	−0.749426	+	0.662088i	$0.769671\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−59.7128	−1.89876
$990$	0	0
$991$	31.3205	0.994929	0.497464	−	0.867484i	$-0.334265\pi$
$991$	31.3205	0.994929	0.497464	+	0.867484i	$0.334265\pi$
$992$	0	0
$993$	25.0718i	0.795629i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	17.6077i	0.557641i	0.960343	+	0.278821i	$0.0899435\pi$
$997$	17.6077i	0.557641i	−0.960343	+	0.278821i	$0.910056\pi$
$998$	0	0
$999$	38.6410	1.22255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.h.3649.2		4
5.2	odd	4		760.2.a.g.1.1	✓	2
5.3	odd	4		3800.2.a.j.1.2		2
5.4	even	2	inner	3800.2.d.h.3649.3		4
15.2	even	4		6840.2.a.y.1.2		2
20.3	even	4		7600.2.a.bd.1.1		2
20.7	even	4		1520.2.a.k.1.2		2
40.27	even	4		6080.2.a.bk.1.1		2
40.37	odd	4		6080.2.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.g.1.1	✓	2	5.2	odd	4
1520.2.a.k.1.2		2	20.7	even	4
3800.2.a.j.1.2		2	5.3	odd	4
3800.2.d.h.3649.2		4	1.1	even	1	trivial
3800.2.d.h.3649.3		4	5.4	even	2	inner
6080.2.a.ba.1.2		2	40.37	odd	4
6080.2.a.bk.1.1		2	40.27	even	4
6840.2.a.y.1.2		2	15.2	even	4
7600.2.a.bd.1.1		2	20.3	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.732051i q^{3} +2.46410 q^{9} +O(q^{10})\)	\(q-0.732051i q^{3} +2.46410 q^{9} +2.00000 q^{11} +2.73205i q^{13} -0.535898i q^{17} -1.00000 q^{19} +5.46410i q^{23} -4.00000i q^{27} -3.46410 q^{29} +4.00000 q^{31} -1.46410i q^{33} +9.66025i q^{37} +2.00000 q^{39} +7.46410 q^{41} +10.9282i q^{43} -10.9282i q^{47} +7.00000 q^{49} -0.392305 q^{51} +5.66025i q^{53} +0.732051i q^{57} +5.46410 q^{59} -13.4641 q^{61} -6.19615i q^{67} +4.00000 q^{69} -2.92820 q^{71} +10.3923i q^{73} -12.3923 q^{79} +4.46410 q^{81} +1.46410i q^{83} +2.53590i q^{87} -3.46410 q^{89} -2.92820i q^{93} -1.66025i q^{97} +4.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 8 q^{11} - 4 q^{19} + 16 q^{31} + 8 q^{39} + 16 q^{41} + 28 q^{49} + 40 q^{51} + 8 q^{59} - 40 q^{61} + 16 q^{69} + 16 q^{71} - 8 q^{79} + 4 q^{81} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^11 - 4 * q^19 + 16 * q^31 + 8 * q^39 + 16 * q^41 + 28 * q^49 + 40 * q^51 + 8 * q^59 - 40 * q^61 + 16 * q^69 + 16 * q^71 - 8 * q^79 + 4 * q^81 - 8 * q^99

Introduction

L-functions

Modular forms

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