LMFDB - Embedded newform 3800.2.d.p.3649.6

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:	$12$
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} + 28x^{10} + 274x^{8} + 1078x^{6} + 1385x^{4} + 478x^{2} + 49 $ x^12 + 28x^10 + 274x^8 + 1078x^6 + 1385x^4 + 478*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.6
Root			$-0.471016i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.p.3649.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.471016i q^{3} +0.567324i q^{7} +2.77814 q^{9} +4.37804 q^{11} +0.165457i q^{13} -7.94693i q^{17} -1.00000 q^{19} +0.267219 q^{21} -3.87445i q^{23} -2.72160i q^{27} -3.53231 q^{29} +3.20380 q^{31} -2.06213i q^{33} +10.1779i q^{37} +0.0779330 q^{39} -5.97409 q^{41} -12.0904i q^{43} -5.46140i q^{47} +6.67814 q^{49} -3.74313 q^{51} +2.00333i q^{53} +0.471016i q^{57} -8.32164 q^{59} +11.8181 q^{61} +1.57611i q^{63} +8.79599i q^{67} -1.82493 q^{69} +0.720031 q^{71} -4.54017i q^{73} +2.48377i q^{77} -11.7123 q^{79} +7.05251 q^{81} +6.72351i q^{83} +1.66378i q^{87} -8.11941 q^{89} -0.0938678 q^{91} -1.50904i q^{93} -13.6321i q^{97} +12.1628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) $ 12 * q - 20 * q^9 + 6 * q^11 - 12 * q^19 + 22 * q^21 - 14 * q^29 + 10 * q^31 - 16 * q^39 + 22 * q^41 + 4 * q^49 + 26 * q^51 + 8 * q^59 + 26 * q^61 - 14 * q^69 + 58 * q^71 - 56 * q^79 + 76 * q^81 + 24 * q^89 - 20 * q^99

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
$3$	− 0.471016i	− 0.271941i	−0.990713	−	0.135971i	$-0.956585\pi$
$3$	− 0.471016i	− 0.271941i	0.990713	−	0.135971i	$-0.0434153\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.567324i	0.214428i	0.994236	+	0.107214i	$0.0341931\pi$
$7$	0.567324i	0.214428i	−0.994236	+	0.107214i	$0.965807\pi$
$8$	0	0
$9$	2.77814	0.926048
$10$	0	0
$11$	4.37804	1.32003	0.660014	−	0.751253i	$-0.270550\pi$
$11$	4.37804	1.32003	0.660014	+	0.751253i	$0.270550\pi$
$12$	0	0
$13$	0.165457i	0.0458895i	0.999737	+	0.0229448i	$0.00730419\pi$
$13$	0.165457i	0.0458895i	−0.999737	+	0.0229448i	$0.992696\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.94693i	− 1.92741i	−0.266963	−	0.963707i	$-0.586020\pi$
$17$	− 7.94693i	− 1.92741i	0.266963	−	0.963707i	$-0.413980\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.267219	0.0583119
$22$	0	0
$23$	− 3.87445i	− 0.807879i	−0.914786	−	0.403939i	$-0.867641\pi$
$23$	− 3.87445i	− 0.807879i	0.914786	−	0.403939i	$-0.132359\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.72160i	− 0.523772i
$28$	0	0
$29$	−3.53231	−0.655934	−0.327967	−	0.944689i	$-0.606364\pi$
$29$	−3.53231	−0.655934	−0.327967	+	0.944689i	$0.606364\pi$
$30$	0	0
$31$	3.20380	0.575419	0.287709	−	0.957718i	$-0.407106\pi$
$31$	3.20380	0.575419	0.287709	+	0.957718i	$0.407106\pi$
$32$	0	0
$33$	− 2.06213i	− 0.358970i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.1779i	1.67323i	0.547788	+	0.836617i	$0.315470\pi$
$37$	10.1779i	1.67323i	−0.547788	+	0.836617i	$0.684530\pi$
$38$	0	0
$39$	0.0779330	0.0124793
$40$	0	0
$41$	−5.97409	−0.932996	−0.466498	−	0.884522i	$-0.654485\pi$
$41$	−5.97409	−0.932996	−0.466498	+	0.884522i	$0.654485\pi$
$42$	0	0
$43$	− 12.0904i	− 1.84376i	−0.387472	−	0.921881i	$-0.626652\pi$
$43$	− 12.0904i	− 1.84376i	0.387472	−	0.921881i	$-0.373348\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 5.46140i	− 0.796627i	−0.917249	−	0.398314i	$-0.869596\pi$
$47$	− 5.46140i	− 0.796627i	0.917249	−	0.398314i	$-0.130404\pi$
$48$	0	0
$49$	6.67814	0.954020
$50$	0	0
$51$	−3.74313	−0.524143
$52$	0	0
$53$	2.00333i	0.275179i	0.990489	+	0.137589i	$0.0439354\pi$
$53$	2.00333i	0.275179i	−0.990489	+	0.137589i	$0.956065\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.471016i	0.0623876i
$58$	0	0
$59$	−8.32164	−1.08339	−0.541693	−	0.840577i	$-0.682216\pi$
$59$	−8.32164	−1.08339	−0.541693	+	0.840577i	$0.682216\pi$
$60$	0	0
$61$	11.8181	1.51315	0.756573	−	0.653909i	$-0.226872\pi$
$61$	11.8181	1.51315	0.756573	+	0.653909i	$0.226872\pi$
$62$	0	0
$63$	1.57611i	0.198571i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.79599i	1.07460i	0.843391	+	0.537300i	$0.180556\pi$
$67$	8.79599i	1.07460i	−0.843391	+	0.537300i	$0.819444\pi$
$68$	0	0
$69$	−1.82493	−0.219696
$70$	0	0
$71$	0.720031	0.0854520	0.0427260	−	0.999087i	$-0.486396\pi$
$71$	0.720031	0.0854520	0.0427260	+	0.999087i	$0.486396\pi$
$72$	0	0
$73$	− 4.54017i	− 0.531386i	−0.964058	−	0.265693i	$-0.914399\pi$
$73$	− 4.54017i	− 0.531386i	0.964058	−	0.265693i	$-0.0856008\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.48377i	0.283051i
$78$	0	0
$79$	−11.7123	−1.31774	−0.658870	−	0.752257i	$-0.728965\pi$
$79$	−11.7123	−1.31774	−0.658870	+	0.752257i	$0.728965\pi$
$80$	0	0
$81$	7.05251	0.783613
$82$	0	0
$83$	6.72351i	0.738001i	0.929429	+	0.369000i	$0.120300\pi$
$83$	6.72351i	0.738001i	−0.929429	+	0.369000i	$0.879700\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.66378i	0.178376i
$88$	0	0
$89$	−8.11941	−0.860656	−0.430328	−	0.902673i	$-0.641602\pi$
$89$	−8.11941	−0.860656	−0.430328	+	0.902673i	$0.641602\pi$
$90$	0	0
$91$	−0.0938678	−0.00984002
$92$	0	0
$93$	− 1.50904i	− 0.156480i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.6321i	− 1.38413i	−0.721835	−	0.692065i	$-0.756701\pi$
$97$	− 13.6321i	− 1.38413i	0.721835	−	0.692065i	$-0.243299\pi$
$98$	0	0
$99$	12.1628	1.22241
$100$	0	0
$101$	19.2686	1.91730	0.958649	−	0.284590i	$-0.0918575\pi$
$101$	19.2686	1.91730	0.958649	+	0.284590i	$0.0918575\pi$
$102$	0	0
$103$	19.2069i	1.89251i	0.323417	+	0.946257i	$0.395168\pi$
$103$	19.2069i	1.89251i	−0.323417	+	0.946257i	$0.604832\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0091i	− 1.16096i	−0.814275	−	0.580480i	$-0.802865\pi$
$107$	− 12.0091i	− 1.16096i	0.814275	−	0.580480i	$-0.197135\pi$
$108$	0	0
$109$	15.0032	1.43704	0.718521	−	0.695505i	$-0.244820\pi$
$109$	15.0032	1.43704	0.718521	+	0.695505i	$0.244820\pi$
$110$	0	0
$111$	4.79395	0.455022
$112$	0	0
$113$	− 8.13049i	− 0.764852i	−0.923986	−	0.382426i	$-0.875089\pi$
$113$	− 8.13049i	− 0.764852i	0.923986	−	0.382426i	$-0.124911\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.459664i	0.0424959i
$118$	0	0
$119$	4.50848	0.413292
$120$	0	0
$121$	8.16722	0.742474
$122$	0	0
$123$	2.81389i	0.253720i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.37051i	− 0.742762i	−0.928480	−	0.371381i	$-0.878884\pi$
$127$	− 8.37051i	− 0.742762i	0.928480	−	0.371381i	$-0.121116\pi$
$128$	0	0
$129$	−5.69476	−0.501395
$130$	0	0
$131$	−5.03206	−0.439653	−0.219826	−	0.975539i	$-0.570549\pi$
$131$	−5.03206	−0.439653	−0.219826	+	0.975539i	$0.570549\pi$
$132$	0	0
$133$	− 0.567324i	− 0.0491932i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.6498i	1.76423i	0.471033	+	0.882115i	$0.343881\pi$
$137$	20.6498i	1.76423i	−0.471033	+	0.882115i	$0.656119\pi$
$138$	0	0
$139$	−11.4726	−0.973092	−0.486546	−	0.873655i	$-0.661743\pi$
$139$	−11.4726	−0.973092	−0.486546	+	0.873655i	$0.661743\pi$
$140$	0	0
$141$	−2.57241	−0.216636
$142$	0	0
$143$	0.724378i	0.0605755i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.14551i	− 0.259438i
$148$	0	0
$149$	−13.0611	−1.07001	−0.535003	−	0.844850i	$-0.679689\pi$
$149$	−13.0611	−1.07001	−0.535003	+	0.844850i	$0.679689\pi$
$150$	0	0
$151$	17.8177	1.44998	0.724992	−	0.688758i	$-0.241844\pi$
$151$	17.8177	1.44998	0.724992	+	0.688758i	$0.241844\pi$
$152$	0	0
$153$	− 22.0777i	− 1.78488i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.3959i	− 0.909490i	−0.890622	−	0.454745i	$-0.849730\pi$
$157$	− 11.3959i	− 0.909490i	0.890622	−	0.454745i	$-0.150270\pi$
$158$	0	0
$159$	0.943601	0.0748324
$160$	0	0
$161$	2.19807	0.173232
$162$	0	0
$163$	− 16.5741i	− 1.29819i	−0.760709	−	0.649093i	$-0.775149\pi$
$163$	− 16.5741i	− 1.29819i	0.760709	−	0.649093i	$-0.224851\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 19.3378i	− 1.49640i	−0.663473	−	0.748200i	$-0.730918\pi$
$167$	− 19.3378i	− 1.49640i	0.663473	−	0.748200i	$-0.269082\pi$
$168$	0	0
$169$	12.9726	0.997894
$170$	0	0
$171$	−2.77814	−0.212450
$172$	0	0
$173$	− 9.31290i	− 0.708046i	−0.935237	−	0.354023i	$-0.884813\pi$
$173$	− 9.31290i	− 0.708046i	0.935237	−	0.354023i	$-0.115187\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.91963i	0.294617i
$178$	0	0
$179$	−24.1703	−1.80658	−0.903288	−	0.429035i	$-0.858854\pi$
$179$	−24.1703	−1.80658	−0.903288	+	0.429035i	$0.858854\pi$
$180$	0	0
$181$	18.6981	1.38982	0.694910	−	0.719097i	$-0.255444\pi$
$181$	18.6981	1.38982	0.694910	+	0.719097i	$0.255444\pi$
$182$	0	0
$183$	− 5.56649i	− 0.411487i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 34.7920i	− 2.54424i
$188$	0	0
$189$	1.54403	0.112312
$190$	0	0
$191$	6.27452	0.454008	0.227004	−	0.973894i	$-0.427107\pi$
$191$	6.27452	0.454008	0.227004	+	0.973894i	$0.427107\pi$
$192$	0	0
$193$	− 8.49096i	− 0.611193i	−0.952161	−	0.305596i	$-0.901144\pi$
$193$	− 8.49096i	− 0.611193i	0.952161	−	0.305596i	$-0.0988558\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.97358i	0.425600i	0.977096	+	0.212800i	$0.0682583\pi$
$197$	5.97358i	0.425600i	−0.977096	+	0.212800i	$0.931742\pi$
$198$	0	0
$199$	−19.8763	−1.40899	−0.704497	−	0.709707i	$-0.748827\pi$
$199$	−19.8763	−1.40899	−0.704497	+	0.709707i	$0.748827\pi$
$200$	0	0
$201$	4.14305	0.292228
$202$	0	0
$203$	− 2.00397i	− 0.140651i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 10.7638i	− 0.748135i
$208$	0	0
$209$	−4.37804	−0.302835
$210$	0	0
$211$	−2.84905	−0.196137	−0.0980685	−	0.995180i	$-0.531266\pi$
$211$	−2.84905	−0.196137	−0.0980685	+	0.995180i	$0.531266\pi$
$212$	0	0
$213$	− 0.339146i	− 0.0232379i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.81759i	0.123386i
$218$	0	0
$219$	−2.13849	−0.144506
$220$	0	0
$221$	1.31488	0.0884481
$222$	0	0
$223$	0.883914i	0.0591912i	0.999562	+	0.0295956i	$0.00942196\pi$
$223$	0.883914i	0.0591912i	−0.999562	+	0.0295956i	$0.990578\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.02836i	0.0682549i	0.999417	+	0.0341275i	$0.0108652\pi$
$227$	1.02836i	0.0682549i	−0.999417	+	0.0341275i	$0.989135\pi$
$228$	0	0
$229$	24.5878	1.62481	0.812404	−	0.583095i	$-0.198158\pi$
$229$	24.5878	1.62481	0.812404	+	0.583095i	$0.198158\pi$
$230$	0	0
$231$	1.16989	0.0769734
$232$	0	0
$233$	5.23585i	0.343012i	0.985183	+	0.171506i	$0.0548633\pi$
$233$	5.23585i	0.343012i	−0.985183	+	0.171506i	$0.945137\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.51669i	0.358348i
$238$	0	0
$239$	12.8124	0.828767	0.414384	−	0.910102i	$-0.363997\pi$
$239$	12.8124	0.828767	0.414384	+	0.910102i	$0.363997\pi$
$240$	0	0
$241$	−11.9996	−0.772962	−0.386481	−	0.922297i	$-0.626309\pi$
$241$	−11.9996	−0.772962	−0.386481	+	0.922297i	$0.626309\pi$
$242$	0	0
$243$	− 11.4866i	− 0.736869i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.165457i	− 0.0105278i
$248$	0	0
$249$	3.16688	0.200693
$250$	0	0
$251$	23.7189	1.49713	0.748563	−	0.663064i	$-0.230744\pi$
$251$	23.7189	1.49713	0.748563	+	0.663064i	$0.230744\pi$
$252$	0	0
$253$	− 16.9625i	− 1.06642i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.11678i	− 0.132041i	−0.997818	−	0.0660206i	$-0.978970\pi$
$257$	− 2.11678i	− 0.132041i	0.997818	−	0.0660206i	$-0.0210303\pi$
$258$	0	0
$259$	−5.77416	−0.358789
$260$	0	0
$261$	−9.81327	−0.607426
$262$	0	0
$263$	1.95674i	0.120658i	0.998179	+	0.0603288i	$0.0192149\pi$
$263$	1.95674i	0.120658i	−0.998179	+	0.0603288i	$0.980785\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.82437i	0.234048i
$268$	0	0
$269$	22.8278	1.39184	0.695919	−	0.718120i	$-0.254997\pi$
$269$	22.8278	1.39184	0.695919	+	0.718120i	$0.254997\pi$
$270$	0	0
$271$	−15.8203	−0.961014	−0.480507	−	0.876991i	$-0.659547\pi$
$271$	−15.8203	−0.961014	−0.480507	+	0.876991i	$0.659547\pi$
$272$	0	0
$273$	0.0442133i	0.00267591i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.47924i	0.569553i	0.958594	+	0.284776i	$0.0919193\pi$
$277$	9.47924i	0.569553i	−0.958594	+	0.284776i	$0.908081\pi$
$278$	0	0
$279$	8.90061	0.532866
$280$	0	0
$281$	13.3160	0.794367	0.397183	−	0.917739i	$-0.369988\pi$
$281$	13.3160	0.794367	0.397183	+	0.917739i	$0.369988\pi$
$282$	0	0
$283$	9.20676i	0.547285i	0.961831	+	0.273643i	$0.0882286\pi$
$283$	9.20676i	0.547285i	−0.961831	+	0.273643i	$0.911771\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.38924i	− 0.200061i
$288$	0	0
$289$	−46.1537	−2.71492
$290$	0	0
$291$	−6.42094	−0.376402
$292$	0	0
$293$	− 10.3435i	− 0.604273i	−0.953265	−	0.302137i	$-0.902300\pi$
$293$	− 10.3435i	− 0.604273i	0.953265	−	0.302137i	$-0.0976999\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 11.9153i	− 0.691394i
$298$	0	0
$299$	0.641056	0.0370732
$300$	0	0
$301$	6.85915	0.395355
$302$	0	0
$303$	− 9.07583i	− 0.521393i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0642i	1.25927i	0.776891	+	0.629636i	$0.216796\pi$
$307$	22.0642i	1.25927i	−0.776891	+	0.629636i	$0.783204\pi$
$308$	0	0
$309$	9.04677	0.514653
$310$	0	0
$311$	−31.4064	−1.78089	−0.890447	−	0.455087i	$-0.849608\pi$
$311$	−31.4064	−1.78089	−0.890447	+	0.455087i	$0.849608\pi$
$312$	0	0
$313$	15.1428i	0.855924i	0.903797	+	0.427962i	$0.140768\pi$
$313$	15.1428i	0.855924i	−0.903797	+	0.427962i	$0.859232\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 18.5791i	− 1.04350i	−0.853097	−	0.521752i	$-0.825279\pi$
$317$	− 18.5791i	− 1.04350i	0.853097	−	0.521752i	$-0.174721\pi$
$318$	0	0
$319$	−15.4646	−0.865852
$320$	0	0
$321$	−5.65646	−0.315713
$322$	0	0
$323$	7.94693i	0.442179i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7.06673i	− 0.390791i
$328$	0	0
$329$	3.09839	0.170820
$330$	0	0
$331$	22.8001	1.25320	0.626602	−	0.779339i	$-0.284445\pi$
$331$	22.8001	1.25320	0.626602	+	0.779339i	$0.284445\pi$
$332$	0	0
$333$	28.2756i	1.54950i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.9306i	1.57595i	0.615706	+	0.787976i	$0.288871\pi$
$337$	28.9306i	1.57595i	−0.615706	+	0.787976i	$0.711129\pi$
$338$	0	0
$339$	−3.82959	−0.207995
$340$	0	0
$341$	14.0263	0.759569
$342$	0	0
$343$	7.75994i	0.418997i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.2803i	− 1.08870i	−0.838857	−	0.544352i	$-0.816776\pi$
$347$	− 20.2803i	− 1.08870i	0.838857	−	0.544352i	$-0.183224\pi$
$348$	0	0
$349$	−20.4355	−1.09389	−0.546943	−	0.837170i	$-0.684208\pi$
$349$	−20.4355	−1.09389	−0.546943	+	0.837170i	$0.684208\pi$
$350$	0	0
$351$	0.450308	0.0240357
$352$	0	0
$353$	− 7.50588i	− 0.399498i	−0.979847	−	0.199749i	$-0.935987\pi$
$353$	− 7.50588i	− 0.399498i	0.979847	−	0.199749i	$-0.0640126\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.12357i	− 0.112391i
$358$	0	0
$359$	18.4383	0.973135	0.486567	−	0.873643i	$-0.338249\pi$
$359$	18.4383	0.973135	0.486567	+	0.873643i	$0.338249\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 3.84689i	− 0.201909i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 12.0105i	− 0.626943i	−0.949598	−	0.313471i	$-0.898508\pi$
$367$	− 12.0105i	− 0.626943i	0.949598	−	0.313471i	$-0.101492\pi$
$368$	0	0
$369$	−16.5969	−0.863999
$370$	0	0
$371$	−1.13654	−0.0590061
$372$	0	0
$373$	15.0719i	0.780395i	0.920731	+	0.390197i	$0.127593\pi$
$373$	15.0719i	0.780395i	−0.920731	+	0.390197i	$0.872407\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.584446i	− 0.0301005i
$378$	0	0
$379$	8.50216	0.436727	0.218363	−	0.975868i	$-0.429928\pi$
$379$	8.50216	0.436727	0.218363	+	0.975868i	$0.429928\pi$
$380$	0	0
$381$	−3.94264	−0.201988
$382$	0	0
$383$	34.9232i	1.78449i	0.451547	+	0.892247i	$0.350872\pi$
$383$	34.9232i	1.78449i	−0.451547	+	0.892247i	$0.649128\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 33.5888i	− 1.70741i
$388$	0	0
$389$	10.7455	0.544820	0.272410	−	0.962181i	$-0.412179\pi$
$389$	10.7455	0.544820	0.272410	+	0.962181i	$0.412179\pi$
$390$	0	0
$391$	−30.7900	−1.55712
$392$	0	0
$393$	2.37018i	0.119560i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.45462i	0.0730051i	0.999334	+	0.0365025i	$0.0116217\pi$
$397$	1.45462i	0.0730051i	−0.999334	+	0.0365025i	$0.988378\pi$
$398$	0	0
$399$	−0.267219	−0.0133777
$400$	0	0
$401$	11.5379	0.576176	0.288088	−	0.957604i	$-0.406980\pi$
$401$	11.5379	0.576176	0.288088	+	0.957604i	$0.406980\pi$
$402$	0	0
$403$	0.530091i	0.0264057i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.5592i	2.20872i
$408$	0	0
$409$	20.5596	1.01661	0.508304	−	0.861178i	$-0.330273\pi$
$409$	20.5596	1.01661	0.508304	+	0.861178i	$0.330273\pi$
$410$	0	0
$411$	9.72639	0.479767
$412$	0	0
$413$	− 4.72107i	− 0.232308i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.40377i	0.264624i
$418$	0	0
$419$	11.0760	0.541096	0.270548	−	0.962706i	$-0.412795\pi$
$419$	11.0760	0.541096	0.270548	+	0.962706i	$0.412795\pi$
$420$	0	0
$421$	18.9178	0.921999	0.461000	−	0.887400i	$-0.347491\pi$
$421$	18.9178	0.921999	0.461000	+	0.887400i	$0.347491\pi$
$422$	0	0
$423$	− 15.1726i	− 0.737715i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.70467i	0.324461i
$428$	0	0
$429$	0.341194	0.0164730
$430$	0	0
$431$	5.23923	0.252365	0.126182	−	0.992007i	$-0.459728\pi$
$431$	5.23923	0.252365	0.126182	+	0.992007i	$0.459728\pi$
$432$	0	0
$433$	8.26963i	0.397413i	0.980059	+	0.198707i	$0.0636741\pi$
$433$	8.26963i	0.397413i	−0.980059	+	0.198707i	$0.936326\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.87445i	0.185340i
$438$	0	0
$439$	−26.2786	−1.25421	−0.627104	−	0.778936i	$-0.715760\pi$
$439$	−26.2786	−1.25421	−0.627104	+	0.778936i	$0.715760\pi$
$440$	0	0
$441$	18.5528	0.883469
$442$	0	0
$443$	9.03372i	0.429205i	0.976701	+	0.214602i	$0.0688456\pi$
$443$	9.03372i	0.429205i	−0.976701	+	0.214602i	$0.931154\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.15198i	0.290979i
$448$	0	0
$449$	10.5553	0.498135	0.249067	−	0.968486i	$-0.419876\pi$
$449$	10.5553	0.498135	0.249067	+	0.968486i	$0.419876\pi$
$450$	0	0
$451$	−26.1548	−1.23158
$452$	0	0
$453$	− 8.39242i	− 0.394310i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.7694i	0.644105i	0.946722	+	0.322053i	$0.104373\pi$
$457$	13.7694i	0.644105i	−0.946722	+	0.322053i	$0.895627\pi$
$458$	0	0
$459$	−21.6284	−1.00953
$460$	0	0
$461$	−0.0771895	−0.00359507	−0.00179754	−	0.999998i	$-0.500572\pi$
$461$	−0.0771895	−0.00359507	−0.00179754	+	0.999998i	$0.500572\pi$
$462$	0	0
$463$	− 7.73598i	− 0.359521i	−0.983710	−	0.179761i	$-0.942468\pi$
$463$	− 7.73598i	− 0.359521i	0.983710	−	0.179761i	$-0.0575323\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.11290i	− 0.421695i	−0.977519	−	0.210847i	$-0.932378\pi$
$467$	− 9.11290i	− 0.421695i	0.977519	−	0.210847i	$-0.0676223\pi$
$468$	0	0
$469$	−4.99017	−0.230425
$470$	0	0
$471$	−5.36764	−0.247328
$472$	0	0
$473$	− 52.9321i	− 2.43382i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.56554i	0.254828i
$478$	0	0
$479$	−14.8868	−0.680197	−0.340099	−	0.940390i	$-0.610460\pi$
$479$	−14.8868	−0.680197	−0.340099	+	0.940390i	$0.610460\pi$
$480$	0	0
$481$	−1.68400	−0.0767840
$482$	0	0
$483$	− 1.03533i	− 0.0471090i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	37.1917i	1.68532i	0.538449	+	0.842658i	$0.319011\pi$
$487$	37.1917i	1.68532i	−0.538449	+	0.842658i	$0.680989\pi$
$488$	0	0
$489$	−7.80668	−0.353030
$490$	0	0
$491$	−6.94872	−0.313591	−0.156796	−	0.987631i	$-0.550116\pi$
$491$	−6.94872	−0.313591	−0.156796	+	0.987631i	$0.550116\pi$
$492$	0	0
$493$	28.0711i	1.26426i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.408491i	0.0183233i
$498$	0	0
$499$	−12.2261	−0.547314	−0.273657	−	0.961827i	$-0.588233\pi$
$499$	−12.2261	−0.547314	−0.273657	+	0.961827i	$0.588233\pi$
$500$	0	0
$501$	−9.10840	−0.406933
$502$	0	0
$503$	− 28.0625i	− 1.25125i	−0.780126	−	0.625623i	$-0.784845\pi$
$503$	− 28.0625i	− 1.25125i	0.780126	−	0.625623i	$-0.215155\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.11032i	− 0.271369i
$508$	0	0
$509$	−3.23221	−0.143265	−0.0716326	−	0.997431i	$-0.522821\pi$
$509$	−3.23221	−0.143265	−0.0716326	+	0.997431i	$0.522821\pi$
$510$	0	0
$511$	2.57575	0.113944
$512$	0	0
$513$	2.72160i	0.120162i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 23.9102i	− 1.05157i
$518$	0	0
$519$	−4.38653	−0.192547
$520$	0	0
$521$	−27.8354	−1.21949	−0.609745	−	0.792598i	$-0.708728\pi$
$521$	−27.8354	−1.21949	−0.609745	+	0.792598i	$0.708728\pi$
$522$	0	0
$523$	− 10.1695i	− 0.444682i	−0.974969	−	0.222341i	$-0.928630\pi$
$523$	− 10.1695i	− 0.444682i	0.974969	−	0.222341i	$-0.0713699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 25.4604i	− 1.10907i
$528$	0	0
$529$	7.98863	0.347332
$530$	0	0
$531$	−23.1187	−1.00327
$532$	0	0
$533$	− 0.988456i	− 0.0428148i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.3846i	0.491283i
$538$	0	0
$539$	29.2372	1.25933
$540$	0	0
$541$	27.4801	1.18146	0.590731	−	0.806869i	$-0.298840\pi$
$541$	27.4801	1.18146	0.590731	+	0.806869i	$0.298840\pi$
$542$	0	0
$543$	− 8.80711i	− 0.377949i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.84680i	− 0.249991i	−0.992157	−	0.124995i	$-0.960108\pi$
$547$	− 5.84680i	− 0.249991i	0.992157	−	0.124995i	$-0.0398916\pi$
$548$	0	0
$549$	32.8322	1.40125
$550$	0	0
$551$	3.53231	0.150482
$552$	0	0
$553$	− 6.64468i	− 0.282561i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.2691i	0.519860i	0.965628	+	0.259930i	$0.0836994\pi$
$557$	12.2691i	0.519860i	−0.965628	+	0.259930i	$0.916301\pi$
$558$	0	0
$559$	2.00044	0.0846095
$560$	0	0
$561$	−16.3876	−0.691884
$562$	0	0
$563$	17.3234i	0.730096i	0.930989	+	0.365048i	$0.118947\pi$
$563$	17.3234i	0.730096i	−0.930989	+	0.365048i	$0.881053\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.00106i	0.168029i
$568$	0	0
$569$	10.3824	0.435252	0.217626	−	0.976032i	$-0.430169\pi$
$569$	10.3824	0.435252	0.217626	+	0.976032i	$0.430169\pi$
$570$	0	0
$571$	10.3852	0.434607	0.217304	−	0.976104i	$-0.430274\pi$
$571$	10.3852	0.434607	0.217304	+	0.976104i	$0.430274\pi$
$572$	0	0
$573$	− 2.95540i	− 0.123464i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.6524i	0.693248i	0.938004	+	0.346624i	$0.112672\pi$
$577$	16.6524i	0.693248i	−0.938004	+	0.346624i	$0.887328\pi$
$578$	0	0
$579$	−3.99938	−0.166209
$580$	0	0
$581$	−3.81441	−0.158248
$582$	0	0
$583$	8.77065i	0.363243i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 13.4077i	− 0.553393i	−0.960957	−	0.276697i	$-0.910760\pi$
$587$	− 13.4077i	− 0.553393i	0.960957	−	0.276697i	$-0.0892397\pi$
$588$	0	0
$589$	−3.20380	−0.132010
$590$	0	0
$591$	2.81365	0.115738
$592$	0	0
$593$	− 20.8050i	− 0.854361i	−0.904166	−	0.427180i	$-0.859507\pi$
$593$	− 20.8050i	− 0.854361i	0.904166	−	0.427180i	$-0.140493\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.36207i	0.383164i
$598$	0	0
$599$	−9.75788	−0.398696	−0.199348	−	0.979929i	$-0.563882\pi$
$599$	−9.75788	−0.398696	−0.199348	+	0.979929i	$0.563882\pi$
$600$	0	0
$601$	3.27717	0.133679	0.0668393	−	0.997764i	$-0.478709\pi$
$601$	3.27717	0.133679	0.0668393	+	0.997764i	$0.478709\pi$
$602$	0	0
$603$	24.4365i	0.995132i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 31.1984i	− 1.26630i	−0.774028	−	0.633151i	$-0.781761\pi$
$607$	− 31.1984i	− 1.26630i	0.774028	−	0.633151i	$-0.218239\pi$
$608$	0	0
$609$	−0.943901	−0.0382488
$610$	0	0
$611$	0.903628	0.0365569
$612$	0	0
$613$	30.3455i	1.22564i	0.790222	+	0.612821i	$0.209965\pi$
$613$	30.3455i	1.22564i	−0.790222	+	0.612821i	$0.790035\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.7447i	1.64032i	0.572135	+	0.820159i	$0.306115\pi$
$617$	40.7447i	1.64032i	−0.572135	+	0.820159i	$0.693885\pi$
$618$	0	0
$619$	13.7115	0.551113	0.275557	−	0.961285i	$-0.411138\pi$
$619$	13.7115	0.551113	0.275557	+	0.961285i	$0.411138\pi$
$620$	0	0
$621$	−10.5447	−0.423144
$622$	0	0
$623$	− 4.60634i	− 0.184549i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.06213i	0.0823534i
$628$	0	0
$629$	80.8830	3.22501
$630$	0	0
$631$	−32.3748	−1.28882	−0.644411	−	0.764680i	$-0.722897\pi$
$631$	−32.3748	−1.28882	−0.644411	+	0.764680i	$0.722897\pi$
$632$	0	0
$633$	1.34195i	0.0533378i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.10495i	0.0437796i
$638$	0	0
$639$	2.00035	0.0791326
$640$	0	0
$641$	0.701397	0.0277035	0.0138518	−	0.999904i	$-0.495591\pi$
$641$	0.701397	0.0277035	0.0138518	+	0.999904i	$0.495591\pi$
$642$	0	0
$643$	12.5736i	0.495856i	0.968778	+	0.247928i	$0.0797497\pi$
$643$	12.5736i	0.495856i	−0.968778	+	0.247928i	$0.920250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 36.4494i	− 1.43297i	−0.697600	−	0.716487i	$-0.745749\pi$
$647$	− 36.4494i	− 1.43297i	0.697600	−	0.716487i	$-0.254251\pi$
$648$	0	0
$649$	−36.4325	−1.43010
$650$	0	0
$651$	0.856115	0.0335538
$652$	0	0
$653$	44.2535i	1.73177i	0.500240	+	0.865887i	$0.333245\pi$
$653$	44.2535i	1.73177i	−0.500240	+	0.865887i	$0.666755\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 12.6132i	− 0.492089i
$658$	0	0
$659$	18.4958	0.720493	0.360247	−	0.932857i	$-0.382693\pi$
$659$	18.4958	0.720493	0.360247	+	0.932857i	$0.382693\pi$
$660$	0	0
$661$	15.3505	0.597067	0.298533	−	0.954399i	$-0.403503\pi$
$661$	15.3505	0.597067	0.298533	+	0.954399i	$0.403503\pi$
$662$	0	0
$663$	− 0.619328i	− 0.0240527i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.6858i	0.529915i
$668$	0	0
$669$	0.416338	0.0160965
$670$	0	0
$671$	51.7399	1.99740
$672$	0	0
$673$	34.1219i	1.31530i	0.753322	+	0.657651i	$0.228450\pi$
$673$	34.1219i	1.31530i	−0.753322	+	0.657651i	$0.771550\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.7808i	0.529639i	0.964298	+	0.264819i	$0.0853124\pi$
$677$	13.7808i	0.529639i	−0.964298	+	0.264819i	$0.914688\pi$
$678$	0	0
$679$	7.73382	0.296797
$680$	0	0
$681$	0.484376	0.0185613
$682$	0	0
$683$	35.8693i	1.37250i	0.727366	+	0.686250i	$0.240744\pi$
$683$	35.8693i	1.37250i	−0.727366	+	0.686250i	$0.759256\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 11.5813i	− 0.441853i
$688$	0	0
$689$	−0.331465	−0.0126278
$690$	0	0
$691$	17.5888	0.669111	0.334555	−	0.942376i	$-0.391414\pi$
$691$	17.5888	0.669111	0.334555	+	0.942376i	$0.391414\pi$
$692$	0	0
$693$	6.90026i	0.262119i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	47.4757i	1.79827i
$698$	0	0
$699$	2.46617	0.0932792
$700$	0	0
$701$	−40.3585	−1.52432	−0.762159	−	0.647389i	$-0.775861\pi$
$701$	−40.3585	−1.52432	−0.762159	+	0.647389i	$0.775861\pi$
$702$	0	0
$703$	− 10.1779i	− 0.383866i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.9315i	0.411123i
$708$	0	0
$709$	−20.5258	−0.770862	−0.385431	−	0.922737i	$-0.625947\pi$
$709$	−20.5258	−0.770862	−0.385431	+	0.922737i	$0.625947\pi$
$710$	0	0
$711$	−32.5385	−1.22029
$712$	0	0
$713$	− 12.4130i	− 0.464869i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.03486i	− 0.225376i
$718$	0	0
$719$	29.8673	1.11386	0.556930	−	0.830559i	$-0.311979\pi$
$719$	29.8673	1.11386	0.556930	+	0.830559i	$0.311979\pi$
$720$	0	0
$721$	−10.8965	−0.405808
$722$	0	0
$723$	5.65200i	0.210200i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.4630i	0.981458i	0.871312	+	0.490729i	$0.163269\pi$
$727$	26.4630i	0.981458i	−0.871312	+	0.490729i	$0.836731\pi$
$728$	0	0
$729$	15.7471	0.583228
$730$	0	0
$731$	−96.0813	−3.55369
$732$	0	0
$733$	30.5488i	1.12834i	0.825657	+	0.564172i	$0.190804\pi$
$733$	30.5488i	1.12834i	−0.825657	+	0.564172i	$0.809196\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.5092i	1.41850i
$738$	0	0
$739$	34.4001	1.26543	0.632714	−	0.774386i	$-0.281941\pi$
$739$	34.4001	1.26543	0.632714	+	0.774386i	$0.281941\pi$
$740$	0	0
$741$	−0.0779330	−0.00286294
$742$	0	0
$743$	− 21.9370i	− 0.804790i	−0.915466	−	0.402395i	$-0.868178\pi$
$743$	− 21.9370i	− 0.804790i	0.915466	−	0.402395i	$-0.131822\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.6789i	0.683424i
$748$	0	0
$749$	6.81303	0.248943
$750$	0	0
$751$	−45.4750	−1.65941	−0.829703	−	0.558206i	$-0.811490\pi$
$751$	−45.4750	−1.65941	−0.829703	+	0.558206i	$0.811490\pi$
$752$	0	0
$753$	− 11.1720i	− 0.407130i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.91122i	0.323884i	0.986800	+	0.161942i	$0.0517757\pi$
$757$	8.91122i	0.323884i	−0.986800	+	0.161942i	$0.948224\pi$
$758$	0	0
$759$	−7.98961	−0.290005
$760$	0	0
$761$	−48.1068	−1.74387	−0.871934	−	0.489623i	$-0.837134\pi$
$761$	−48.1068	−1.74387	−0.871934	+	0.489623i	$0.837134\pi$
$762$	0	0
$763$	8.51165i	0.308142i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.37687i	− 0.0497161i
$768$	0	0
$769$	11.2615	0.406099	0.203049	−	0.979169i	$-0.434915\pi$
$769$	11.2615	0.406099	0.203049	+	0.979169i	$0.434915\pi$
$770$	0	0
$771$	−0.997039	−0.0359075
$772$	0	0
$773$	− 20.4897i	− 0.736964i	−0.929635	−	0.368482i	$-0.879878\pi$
$773$	− 20.4897i	− 0.736964i	0.929635	−	0.368482i	$-0.120122\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.71972i	0.0975695i
$778$	0	0
$779$	5.97409	0.214044
$780$	0	0
$781$	3.15232	0.112799
$782$	0	0
$783$	9.61354i	0.343560i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.45356i	0.194398i	0.995265	+	0.0971992i	$0.0309884\pi$
$787$	5.45356i	0.194398i	−0.995265	+	0.0971992i	$0.969012\pi$
$788$	0	0
$789$	0.921655	0.0328118
$790$	0	0
$791$	4.61262	0.164006
$792$	0	0
$793$	1.95538i	0.0694376i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 4.96228i	− 0.175773i	−0.996131	−	0.0878864i	$-0.971989\pi$
$797$	− 4.96228i	− 0.175773i	0.996131	−	0.0878864i	$-0.0280113\pi$
$798$	0	0
$799$	−43.4014	−1.53543
$800$	0	0
$801$	−22.5569	−0.797008
$802$	0	0
$803$	− 19.8770i	− 0.701445i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10.7523i	− 0.378498i
$808$	0	0
$809$	18.4521	0.648741	0.324370	−	0.945930i	$-0.394848\pi$
$809$	18.4521	0.648741	0.324370	+	0.945930i	$0.394848\pi$
$810$	0	0
$811$	31.3590	1.10116	0.550582	−	0.834781i	$-0.314406\pi$
$811$	31.3590	1.10116	0.550582	+	0.834781i	$0.314406\pi$
$812$	0	0
$813$	7.45161i	0.261339i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0904i	0.422988i
$818$	0	0
$819$	−0.260778	−0.00911233
$820$	0	0
$821$	−52.9504	−1.84798	−0.923991	−	0.382413i	$-0.875093\pi$
$821$	−52.9504	−1.84798	−0.923991	+	0.382413i	$0.875093\pi$
$822$	0	0
$823$	46.0815i	1.60630i	0.595776	+	0.803150i	$0.296845\pi$
$823$	46.0815i	1.60630i	−0.595776	+	0.803150i	$0.703155\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.9170i	− 0.379622i	−0.981821	−	0.189811i	$-0.939212\pi$
$827$	− 10.9170i	− 0.379622i	0.981821	−	0.189811i	$-0.0607875\pi$
$828$	0	0
$829$	−48.6141	−1.68844	−0.844219	−	0.535998i	$-0.819936\pi$
$829$	−48.6141	−1.68844	−0.844219	+	0.535998i	$0.819936\pi$
$830$	0	0
$831$	4.46488	0.154885
$832$	0	0
$833$	− 53.0707i	− 1.83879i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.71945i	− 0.301388i
$838$	0	0
$839$	35.2171	1.21583	0.607915	−	0.794002i	$-0.292006\pi$
$839$	35.2171	1.21583	0.607915	+	0.794002i	$0.292006\pi$
$840$	0	0
$841$	−16.5228	−0.569750
$842$	0	0
$843$	− 6.27206i	− 0.216021i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.63346i	0.159208i
$848$	0	0
$849$	4.33653	0.148829
$850$	0	0
$851$	39.4337	1.35177
$852$	0	0
$853$	− 7.77006i	− 0.266042i	−0.991113	−	0.133021i	$-0.957532\pi$
$853$	− 7.77006i	− 0.266042i	0.991113	−	0.133021i	$-0.0424677\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 20.8792i	− 0.713219i	−0.934253	−	0.356610i	$-0.883933\pi$
$857$	− 20.8792i	− 0.713219i	0.934253	−	0.356610i	$-0.116067\pi$
$858$	0	0
$859$	−5.48065	−0.186997	−0.0934987	−	0.995619i	$-0.529805\pi$
$859$	−5.48065	−0.186997	−0.0934987	+	0.995619i	$0.529805\pi$
$860$	0	0
$861$	−1.59639	−0.0544048
$862$	0	0
$863$	23.9562i	0.815479i	0.913098	+	0.407740i	$0.133683\pi$
$863$	23.9562i	0.815479i	−0.913098	+	0.407740i	$0.866317\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.7391i	0.738300i
$868$	0	0
$869$	−51.2770	−1.73945
$870$	0	0
$871$	−1.45536	−0.0493129
$872$	0	0
$873$	− 37.8719i	− 1.28177i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0925i	0.610941i	0.952201	+	0.305471i	$0.0988138\pi$
$877$	18.0925i	0.610941i	−0.952201	+	0.305471i	$0.901186\pi$
$878$	0	0
$879$	−4.87195	−0.164327
$880$	0	0
$881$	6.10173	0.205572	0.102786	−	0.994703i	$-0.467224\pi$
$881$	6.10173	0.205572	0.102786	+	0.994703i	$0.467224\pi$
$882$	0	0
$883$	11.8538i	0.398914i	0.979907	+	0.199457i	$0.0639178\pi$
$883$	11.8538i	0.398914i	−0.979907	+	0.199457i	$0.936082\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46.7513i	1.56976i	0.619651	+	0.784878i	$0.287274\pi$
$887$	46.7513i	1.56976i	−0.619651	+	0.784878i	$0.712726\pi$
$888$	0	0
$889$	4.74879	0.159269
$890$	0	0
$891$	30.8762	1.03439
$892$	0	0
$893$	5.46140i	0.182759i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.301948i	− 0.0100817i
$898$	0	0
$899$	−11.3168	−0.377437
$900$	0	0
$901$	15.9203	0.530383
$902$	0	0
$903$	− 3.23077i	− 0.107513i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.8753i	1.22442i	0.790693	+	0.612212i	$0.209720\pi$
$907$	36.8753i	1.22442i	−0.790693	+	0.612212i	$0.790280\pi$
$908$	0	0
$909$	53.5310	1.77551
$910$	0	0
$911$	−24.9642	−0.827099	−0.413550	−	0.910482i	$-0.635711\pi$
$911$	−24.9642	−0.827099	−0.413550	+	0.910482i	$0.635711\pi$
$912$	0	0
$913$	29.4358i	0.974182i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.85481i	− 0.0942740i
$918$	0	0
$919$	8.45204	0.278807	0.139404	−	0.990236i	$-0.455481\pi$
$919$	8.45204	0.278807	0.139404	+	0.990236i	$0.455481\pi$
$920$	0	0
$921$	10.3926	0.342448
$922$	0	0
$923$	0.119134i	0.00392135i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	53.3596i	1.75256i
$928$	0	0
$929$	−34.3786	−1.12793	−0.563963	−	0.825800i	$-0.690724\pi$
$929$	−34.3786	−1.12793	−0.563963	+	0.825800i	$0.690724\pi$
$930$	0	0
$931$	−6.67814	−0.218867
$932$	0	0
$933$	14.7929i	0.484299i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.23838i	− 0.171130i	−0.996333	−	0.0855652i	$-0.972730\pi$
$937$	− 5.23838i	− 0.171130i	0.996333	−	0.0855652i	$-0.0272696\pi$
$938$	0	0
$939$	7.13252	0.232761
$940$	0	0
$941$	14.3720	0.468514	0.234257	−	0.972175i	$-0.424734\pi$
$941$	14.3720	0.468514	0.234257	+	0.972175i	$0.424734\pi$
$942$	0	0
$943$	23.1463i	0.753748i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.8865i	0.906190i	0.891462	+	0.453095i	$0.149680\pi$
$947$	27.8865i	0.906190i	−0.891462	+	0.453095i	$0.850320\pi$
$948$	0	0
$949$	0.751203	0.0243851
$950$	0	0
$951$	−8.75104	−0.283772
$952$	0	0
$953$	18.8741i	0.611392i	0.952129	+	0.305696i	$0.0988891\pi$
$953$	18.8741i	0.611392i	−0.952129	+	0.305696i	$0.901111\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.28408i	0.235461i
$958$	0	0
$959$	−11.7151	−0.378301
$960$	0	0
$961$	−20.7357	−0.668893
$962$	0	0
$963$	− 33.3629i	− 1.07510i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.87426i	− 0.0924301i	−0.998932	−	0.0462150i	$-0.985284\pi$
$967$	− 2.87426i	− 0.0924301i	0.998932	−	0.0462150i	$-0.0147159\pi$
$968$	0	0
$969$	3.74313	0.120247
$970$	0	0
$971$	−50.4156	−1.61791	−0.808956	−	0.587869i	$-0.799967\pi$
$971$	−50.4156	−1.61791	−0.808956	+	0.587869i	$0.799967\pi$
$972$	0	0
$973$	− 6.50867i	− 0.208658i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 48.5411i	− 1.55297i	−0.630137	−	0.776484i	$-0.717001\pi$
$977$	− 48.5411i	− 1.55297i	0.630137	−	0.776484i	$-0.282999\pi$
$978$	0	0
$979$	−35.5471	−1.13609
$980$	0	0
$981$	41.6809	1.33077
$982$	0	0
$983$	− 5.00431i	− 0.159613i	−0.996810	−	0.0798063i	$-0.974570\pi$
$983$	− 5.00431i	− 0.159613i	0.996810	−	0.0798063i	$-0.0254302\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.45939i	− 0.0464529i
$988$	0	0
$989$	−46.8435	−1.48954
$990$	0	0
$991$	−43.9761	−1.39695	−0.698474	−	0.715635i	$-0.746137\pi$
$991$	−43.9761	−1.39695	−0.698474	+	0.715635i	$0.746137\pi$
$992$	0	0
$993$	− 10.7392i	− 0.340798i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 11.4138i	− 0.361478i	−0.983531	−	0.180739i	$-0.942151\pi$
$997$	− 11.4138i	− 0.361478i	0.983531	−	0.180739i	$-0.0578490\pi$
$998$	0	0
$999$	27.7001	0.876393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.p.3649.6		12
5.2	odd	4		3800.2.a.bd.1.3	yes	6
5.3	odd	4		3800.2.a.bb.1.4	✓	6
5.4	even	2	inner	3800.2.d.p.3649.7		12
20.3	even	4		7600.2.a.cm.1.3		6
20.7	even	4		7600.2.a.ci.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.4	✓	6	5.3	odd	4
3800.2.a.bd.1.3	yes	6	5.2	odd	4
3800.2.d.p.3649.6		12	1.1	even	1	trivial
3800.2.d.p.3649.7		12	5.4	even	2	inner
7600.2.a.ci.1.4		6	20.7	even	4
7600.2.a.cm.1.3		6	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)

\(f(q)\)	\(=\)	\(q-0.471016i q^{3} +0.567324i q^{7} +2.77814 q^{9} +4.37804 q^{11} +0.165457i q^{13} -7.94693i q^{17} -1.00000 q^{19} +0.267219 q^{21} -3.87445i q^{23} -2.72160i q^{27} -3.53231 q^{29} +3.20380 q^{31} -2.06213i q^{33} +10.1779i q^{37} +0.0779330 q^{39} -5.97409 q^{41} -12.0904i q^{43} -5.46140i q^{47} +6.67814 q^{49} -3.74313 q^{51} +2.00333i q^{53} +0.471016i q^{57} -8.32164 q^{59} +11.8181 q^{61} +1.57611i q^{63} +8.79599i q^{67} -1.82493 q^{69} +0.720031 q^{71} -4.54017i q^{73} +2.48377i q^{77} -11.7123 q^{79} +7.05251 q^{81} +6.72351i q^{83} +1.66378i q^{87} -8.11941 q^{89} -0.0938678 q^{91} -1.50904i q^{93} -13.6321i q^{97} +12.1628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) \) 12 * q - 20 * q^9 + 6 * q^11 - 12 * q^19 + 22 * q^21 - 14 * q^29 + 10 * q^31 - 16 * q^39 + 22 * q^41 + 4 * q^49 + 26 * q^51 + 8 * q^59 + 26 * q^61 - 14 * q^69 + 58 * q^71 - 56 * q^79 + 76 * q^81 + 24 * q^89 - 20 * q^99

Introduction

L-functions

Modular forms

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Database

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