LMFDB - Embedded newform 3800.2.d.p.3649.9

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			3649.9
Root			$1.22174i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.p.3649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22174i q^{3} -3.08602i q^{7} +1.50735 q^{9} -3.83920 q^{11} +5.85633i q^{13} -6.54564i q^{17} -1.00000 q^{19} +3.77031 q^{21} +4.37163i q^{23} +5.50681i q^{27} +4.41841 q^{29} +0.451431 q^{31} -4.69051i q^{33} -2.49067i q^{37} -7.15491 q^{39} -1.03924 q^{41} +3.19209i q^{43} +3.70641i q^{47} -2.52349 q^{49} +7.99707 q^{51} +5.19667i q^{53} -1.22174i q^{57} +7.18818 q^{59} +5.02265 q^{61} -4.65171i q^{63} +13.1631i q^{67} -5.34099 q^{69} +12.2993 q^{71} -2.49886i q^{73} +11.8478i q^{77} -11.0313 q^{79} -2.20584 q^{81} +0.245834i q^{83} +5.39814i q^{87} +13.4321 q^{89} +18.0727 q^{91} +0.551531i q^{93} -11.0992i q^{97} -5.78702 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.22174i	0.705372i	0.935742	+	0.352686i	$0.114732\pi$
$3$	1.22174i	0.705372i	−0.935742	+	0.352686i	$0.885268\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.08602i	− 1.16640i	−0.812327	−	0.583202i	$-0.801800\pi$
$7$	− 3.08602i	− 1.16640i	0.812327	−	0.583202i	$-0.198200\pi$
$8$	0	0
$9$	1.50735	0.502450
$10$	0	0
$11$	−3.83920	−1.15756	−0.578781	−	0.815483i	$-0.696472\pi$
$11$	−3.83920	−1.15756	−0.578781	+	0.815483i	$0.696472\pi$
$12$	0	0
$13$	5.85633i	1.62425i	0.583482	+	0.812126i	$0.301690\pi$
$13$	5.85633i	1.62425i	−0.583482	+	0.812126i	$0.698310\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.54564i	− 1.58755i	−0.608211	−	0.793775i	$-0.708113\pi$
$17$	− 6.54564i	− 1.58755i	0.608211	−	0.793775i	$-0.291887\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.77031	0.822749
$22$	0	0
$23$	4.37163i	0.911547i	0.890096	+	0.455774i	$0.150637\pi$
$23$	4.37163i	0.911547i	−0.890096	+	0.455774i	$0.849363\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.50681i	1.05979i
$28$	0	0
$29$	4.41841	0.820477	0.410239	−	0.911978i	$-0.365445\pi$
$29$	4.41841	0.820477	0.410239	+	0.911978i	$0.365445\pi$
$30$	0	0
$31$	0.451431	0.0810793	0.0405397	−	0.999178i	$-0.487092\pi$
$31$	0.451431	0.0810793	0.0405397	+	0.999178i	$0.487092\pi$
$32$	0	0
$33$	− 4.69051i	− 0.816512i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.49067i	− 0.409463i	−0.978818	−	0.204732i	$-0.934368\pi$
$37$	− 2.49067i	− 0.409463i	0.978818	−	0.204732i	$-0.0656322\pi$
$38$	0	0
$39$	−7.15491	−1.14570
$40$	0	0
$41$	−1.03924	−0.162302	−0.0811508	−	0.996702i	$-0.525860\pi$
$41$	−1.03924	−0.162302	−0.0811508	+	0.996702i	$0.525860\pi$
$42$	0	0
$43$	3.19209i	0.486789i	0.969927	+	0.243394i	$0.0782609\pi$
$43$	3.19209i	0.486789i	−0.969927	+	0.243394i	$0.921739\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.70641i	0.540635i	0.962771	+	0.270317i	$0.0871287\pi$
$47$	3.70641i	0.540635i	−0.962771	+	0.270317i	$0.912871\pi$
$48$	0	0
$49$	−2.52349	−0.360499
$50$	0	0
$51$	7.99707	1.11981
$52$	0	0
$53$	5.19667i	0.713817i	0.934139	+	0.356908i	$0.116169\pi$
$53$	5.19667i	0.713817i	−0.934139	+	0.356908i	$0.883831\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.22174i	− 0.161823i
$58$	0	0
$59$	7.18818	0.935821	0.467910	−	0.883776i	$-0.345007\pi$
$59$	7.18818	0.935821	0.467910	+	0.883776i	$0.345007\pi$
$60$	0	0
$61$	5.02265	0.643085	0.321542	−	0.946895i	$-0.395799\pi$
$61$	5.02265	0.643085	0.321542	+	0.946895i	$0.395799\pi$
$62$	0	0
$63$	− 4.65171i	− 0.586060i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.1631i	1.60813i	0.594542	+	0.804064i	$0.297333\pi$
$67$	13.1631i	1.60813i	−0.594542	+	0.804064i	$0.702667\pi$
$68$	0	0
$69$	−5.34099	−0.642980
$70$	0	0
$71$	12.2993	1.45965	0.729827	−	0.683632i	$-0.239601\pi$
$71$	12.2993	1.45965	0.729827	+	0.683632i	$0.239601\pi$
$72$	0	0
$73$	− 2.49886i	− 0.292470i	−0.989250	−	0.146235i	$-0.953285\pi$
$73$	− 2.49886i	− 0.292470i	0.989250	−	0.146235i	$-0.0467155\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.8478i	1.35019i
$78$	0	0
$79$	−11.0313	−1.24112	−0.620558	−	0.784160i	$-0.713094\pi$
$79$	−11.0313	−1.24112	−0.620558	+	0.784160i	$0.713094\pi$
$80$	0	0
$81$	−2.20584	−0.245093
$82$	0	0
$83$	0.245834i	0.0269838i	0.999909	+	0.0134919i	$0.00429473\pi$
$83$	0.245834i	0.0269838i	−0.999909	+	0.0134919i	$0.995705\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.39814i	0.578742i
$88$	0	0
$89$	13.4321	1.42380	0.711898	−	0.702283i	$-0.247836\pi$
$89$	13.4321	1.42380	0.711898	+	0.702283i	$0.247836\pi$
$90$	0	0
$91$	18.0727	1.89454
$92$	0	0
$93$	0.551531i	0.0571911i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 11.0992i	− 1.12696i	−0.826131	−	0.563478i	$-0.809463\pi$
$97$	− 11.0992i	− 1.12696i	0.826131	−	0.563478i	$-0.190537\pi$
$98$	0	0
$99$	−5.78702	−0.581618
$100$	0	0
$101$	16.0460	1.59664	0.798318	−	0.602236i	$-0.205724\pi$
$101$	16.0460	1.59664	0.798318	+	0.602236i	$0.205724\pi$
$102$	0	0
$103$	15.4569i	1.52301i	0.648157	+	0.761507i	$0.275540\pi$
$103$	15.4569i	1.52301i	−0.648157	+	0.761507i	$0.724460\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 9.23615i	− 0.892892i	−0.894811	−	0.446446i	$-0.852690\pi$
$107$	− 9.23615i	− 0.892892i	0.894811	−	0.446446i	$-0.147310\pi$
$108$	0	0
$109$	9.87147	0.945515	0.472758	−	0.881192i	$-0.343259\pi$
$109$	9.87147	0.945515	0.472758	+	0.881192i	$0.343259\pi$
$110$	0	0
$111$	3.04295	0.288824
$112$	0	0
$113$	18.4191i	1.73272i	0.499421	+	0.866360i	$0.333546\pi$
$113$	18.4191i	1.73272i	−0.499421	+	0.866360i	$0.666454\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.82754i	0.816106i
$118$	0	0
$119$	−20.1999	−1.85173
$120$	0	0
$121$	3.73946	0.339951
$122$	0	0
$123$	− 1.26968i	− 0.114483i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 14.1753i	− 1.25786i	−0.777464	−	0.628928i	$-0.783494\pi$
$127$	− 14.1753i	− 1.25786i	0.777464	−	0.628928i	$-0.216506\pi$
$128$	0	0
$129$	−3.89990	−0.343367
$130$	0	0
$131$	1.40424	0.122689	0.0613446	−	0.998117i	$-0.480461\pi$
$131$	1.40424	0.122689	0.0613446	+	0.998117i	$0.480461\pi$
$132$	0	0
$133$	3.08602i	0.267592i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.4212i	1.14665i	0.819326	+	0.573327i	$0.194348\pi$
$137$	13.4212i	1.14665i	−0.819326	+	0.573327i	$0.805652\pi$
$138$	0	0
$139$	−3.42929	−0.290868	−0.145434	−	0.989368i	$-0.546458\pi$
$139$	−3.42929	−0.290868	−0.145434	+	0.989368i	$0.546458\pi$
$140$	0	0
$141$	−4.52827	−0.381349
$142$	0	0
$143$	− 22.4836i	− 1.88017i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.08305i	− 0.254286i
$148$	0	0
$149$	20.3518	1.66729	0.833643	−	0.552304i	$-0.186251\pi$
$149$	20.3518	1.66729	0.833643	+	0.552304i	$0.186251\pi$
$150$	0	0
$151$	−2.34058	−0.190474	−0.0952369	−	0.995455i	$-0.530361\pi$
$151$	−2.34058	−0.190474	−0.0952369	+	0.995455i	$0.530361\pi$
$152$	0	0
$153$	− 9.86658i	− 0.797665i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.5097i	− 1.39742i	−0.715404	−	0.698711i	$-0.753757\pi$
$157$	− 17.5097i	− 1.39742i	0.715404	−	0.698711i	$-0.246243\pi$
$158$	0	0
$159$	−6.34897	−0.503506
$160$	0	0
$161$	13.4909	1.06323
$162$	0	0
$163$	− 6.65575i	− 0.521319i	−0.965431	−	0.260659i	$-0.916060\pi$
$163$	− 6.65575i	− 0.521319i	0.965431	−	0.260659i	$-0.0839399\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.32539i	0.644238i	0.946699	+	0.322119i	$0.104395\pi$
$167$	8.32539i	0.644238i	−0.946699	+	0.322119i	$0.895605\pi$
$168$	0	0
$169$	−21.2965	−1.63820
$170$	0	0
$171$	−1.50735	−0.115270
$172$	0	0
$173$	20.0259i	1.52254i	0.648433	+	0.761272i	$0.275425\pi$
$173$	20.0259i	1.52254i	−0.648433	+	0.761272i	$0.724575\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.78208i	0.660102i
$178$	0	0
$179$	12.1689	0.909544	0.454772	−	0.890608i	$-0.349721\pi$
$179$	12.1689	0.909544	0.454772	+	0.890608i	$0.349721\pi$
$180$	0	0
$181$	3.76508	0.279856	0.139928	−	0.990162i	$-0.455313\pi$
$181$	3.76508	0.279856	0.139928	+	0.990162i	$0.455313\pi$
$182$	0	0
$183$	6.13638i	0.453614i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	25.1300i	1.83769i
$188$	0	0
$189$	16.9941	1.23614
$190$	0	0
$191$	−13.0616	−0.945106	−0.472553	−	0.881302i	$-0.656667\pi$
$191$	−13.0616	−0.945106	−0.472553	+	0.881302i	$0.656667\pi$
$192$	0	0
$193$	9.44847i	0.680116i	0.940405	+	0.340058i	$0.110447\pi$
$193$	9.44847i	0.680116i	−0.940405	+	0.340058i	$0.889553\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.3270i	1.02075i	0.859951	+	0.510377i	$0.170494\pi$
$197$	14.3270i	1.02075i	−0.859951	+	0.510377i	$0.829506\pi$
$198$	0	0
$199$	16.1608	1.14561	0.572805	−	0.819692i	$-0.305855\pi$
$199$	16.1608	1.14561	0.572805	+	0.819692i	$0.305855\pi$
$200$	0	0
$201$	−16.0819	−1.13433
$202$	0	0
$203$	− 13.6353i	− 0.957008i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.58958i	0.458007i
$208$	0	0
$209$	3.83920	0.265563
$210$	0	0
$211$	4.61746	0.317879	0.158940	−	0.987288i	$-0.449193\pi$
$211$	4.61746	0.317879	0.158940	+	0.987288i	$0.449193\pi$
$212$	0	0
$213$	15.0265i	1.02960i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.39312i	− 0.0945713i
$218$	0	0
$219$	3.05296	0.206300
$220$	0	0
$221$	38.3334	2.57858
$222$	0	0
$223$	− 20.2785i	− 1.35795i	−0.734161	−	0.678975i	$-0.762424\pi$
$223$	− 20.2785i	− 1.35795i	0.734161	−	0.678975i	$-0.237576\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.5708i	0.767982i	0.923337	+	0.383991i	$0.125451\pi$
$227$	11.5708i	0.767982i	−0.923337	+	0.383991i	$0.874549\pi$
$228$	0	0
$229$	−14.0792	−0.930378	−0.465189	−	0.885211i	$-0.654014\pi$
$229$	−14.0792	−0.930378	−0.465189	+	0.885211i	$0.654014\pi$
$230$	0	0
$231$	−14.4750	−0.952383
$232$	0	0
$233$	3.95281i	0.258957i	0.991582	+	0.129479i	$0.0413304\pi$
$233$	3.95281i	0.258957i	−0.991582	+	0.129479i	$0.958670\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 13.4774i	− 0.875449i
$238$	0	0
$239$	−18.0661	−1.16860	−0.584300	−	0.811538i	$-0.698631\pi$
$239$	−18.0661	−1.16860	−0.584300	+	0.811538i	$0.698631\pi$
$240$	0	0
$241$	23.0999	1.48800	0.743998	−	0.668182i	$-0.232927\pi$
$241$	23.0999	1.48800	0.743998	+	0.668182i	$0.232927\pi$
$242$	0	0
$243$	13.8255i	0.886904i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.85633i	− 0.372629i
$248$	0	0
$249$	−0.300345	−0.0190336
$250$	0	0
$251$	−13.0987	−0.826784	−0.413392	−	0.910553i	$-0.635656\pi$
$251$	−13.0987	−0.826784	−0.413392	+	0.910553i	$0.635656\pi$
$252$	0	0
$253$	− 16.7836i	− 1.05517i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.5191i	0.843296i	0.906760	+	0.421648i	$0.138548\pi$
$257$	13.5191i	0.843296i	−0.906760	+	0.421648i	$0.861452\pi$
$258$	0	0
$259$	−7.68624	−0.477600
$260$	0	0
$261$	6.66009	0.412249
$262$	0	0
$263$	− 0.357938i	− 0.0220714i	−0.999939	−	0.0110357i	$-0.996487\pi$
$263$	− 0.357938i	− 0.0220714i	0.999939	−	0.0110357i	$-0.00351284\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	16.4105i	1.00431i
$268$	0	0
$269$	−15.9556	−0.972829	−0.486415	−	0.873728i	$-0.661696\pi$
$269$	−15.9556	−0.972829	−0.486415	+	0.873728i	$0.661696\pi$
$270$	0	0
$271$	0.795153	0.0483021	0.0241511	−	0.999708i	$-0.492312\pi$
$271$	0.795153	0.0483021	0.0241511	+	0.999708i	$0.492312\pi$
$272$	0	0
$273$	22.0802i	1.33635i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.9640i	0.778934i	0.921040	+	0.389467i	$0.127341\pi$
$277$	12.9640i	0.778934i	−0.921040	+	0.389467i	$0.872659\pi$
$278$	0	0
$279$	0.680465	0.0407383
$280$	0	0
$281$	−26.2769	−1.56755	−0.783775	−	0.621045i	$-0.786709\pi$
$281$	−26.2769	−1.56755	−0.783775	+	0.621045i	$0.786709\pi$
$282$	0	0
$283$	27.4599i	1.63232i	0.577826	+	0.816160i	$0.303901\pi$
$283$	27.4599i	1.63232i	−0.577826	+	0.816160i	$0.696099\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.20710i	0.189309i
$288$	0	0
$289$	−25.8454	−1.52032
$290$	0	0
$291$	13.5604	0.794923
$292$	0	0
$293$	− 19.7572i	− 1.15423i	−0.816663	−	0.577114i	$-0.804179\pi$
$293$	− 19.7572i	− 1.15423i	0.816663	−	0.577114i	$-0.195821\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 21.1418i	− 1.22677i
$298$	0	0
$299$	−25.6017	−1.48058
$300$	0	0
$301$	9.85083	0.567792
$302$	0	0
$303$	19.6040i	1.12622i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.9367i	1.65151i	0.564032	+	0.825753i	$0.309250\pi$
$307$	28.9367i	1.65151i	−0.564032	+	0.825753i	$0.690750\pi$
$308$	0	0
$309$	−18.8843	−1.07429
$310$	0	0
$311$	−28.0895	−1.59281	−0.796404	−	0.604764i	$-0.793267\pi$
$311$	−28.0895	−1.59281	−0.796404	+	0.604764i	$0.793267\pi$
$312$	0	0
$313$	29.7936i	1.68403i	0.539453	+	0.842016i	$0.318631\pi$
$313$	29.7936i	1.68403i	−0.539453	+	0.842016i	$0.681369\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.13494i	0.344572i	0.985047	+	0.172286i	$0.0551154\pi$
$317$	6.13494i	0.344572i	−0.985047	+	0.172286i	$0.944885\pi$
$318$	0	0
$319$	−16.9631	−0.949754
$320$	0	0
$321$	11.2842	0.629821
$322$	0	0
$323$	6.54564i	0.364209i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.0604i	0.666940i
$328$	0	0
$329$	11.4380	0.630599
$330$	0	0
$331$	−17.1019	−0.940003	−0.470002	−	0.882666i	$-0.655747\pi$
$331$	−17.1019	−0.940003	−0.470002	+	0.882666i	$0.655747\pi$
$332$	0	0
$333$	− 3.75431i	− 0.205735i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 17.0521i	− 0.928885i	−0.885603	−	0.464442i	$-0.846255\pi$
$337$	− 17.0521i	− 0.928885i	0.885603	−	0.464442i	$-0.153745\pi$
$338$	0	0
$339$	−22.5033	−1.22221
$340$	0	0
$341$	−1.73313	−0.0938544
$342$	0	0
$343$	− 13.8146i	− 0.745917i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1351i	1.40300i	0.712667	+	0.701502i	$0.247487\pi$
$347$	26.1351i	1.40300i	−0.712667	+	0.701502i	$0.752513\pi$
$348$	0	0
$349$	0.577782	0.0309279	0.0154640	−	0.999880i	$-0.495077\pi$
$349$	0.577782	0.0309279	0.0154640	+	0.999880i	$0.495077\pi$
$350$	0	0
$351$	−32.2497	−1.72136
$352$	0	0
$353$	30.0746i	1.60071i	0.599527	+	0.800355i	$0.295355\pi$
$353$	30.0746i	1.60071i	−0.599527	+	0.800355i	$0.704645\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.6791i	− 1.30616i
$358$	0	0
$359$	25.6793	1.35530	0.677651	−	0.735383i	$-0.262998\pi$
$359$	25.6793	1.35530	0.677651	+	0.735383i	$0.262998\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.56865i	0.239792i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7.57520i	− 0.395422i	−0.980260	−	0.197711i	$-0.936649\pi$
$367$	− 7.57520i	− 0.395422i	0.980260	−	0.197711i	$-0.0633508\pi$
$368$	0	0
$369$	−1.56650	−0.0815485
$370$	0	0
$371$	16.0370	0.832599
$372$	0	0
$373$	− 26.2562i	− 1.35949i	−0.733447	−	0.679747i	$-0.762090\pi$
$373$	− 26.2562i	− 1.35949i	0.733447	−	0.679747i	$-0.237910\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.8756i	1.33266i
$378$	0	0
$379$	15.6277	0.802742	0.401371	−	0.915915i	$-0.368534\pi$
$379$	15.6277	0.802742	0.401371	+	0.915915i	$0.368534\pi$
$380$	0	0
$381$	17.3185	0.887256
$382$	0	0
$383$	− 10.7189i	− 0.547711i	−0.961771	−	0.273856i	$-0.911701\pi$
$383$	− 10.7189i	− 0.547711i	0.961771	−	0.273856i	$-0.0882990\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.81159i	0.244587i
$388$	0	0
$389$	−7.83988	−0.397498	−0.198749	−	0.980050i	$-0.563688\pi$
$389$	−7.83988	−0.397498	−0.198749	+	0.980050i	$0.563688\pi$
$390$	0	0
$391$	28.6151	1.44713
$392$	0	0
$393$	1.71562i	0.0865416i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 37.9045i	− 1.90237i	−0.308614	−	0.951187i	$-0.599865\pi$
$397$	− 37.9045i	− 1.90237i	0.308614	−	0.951187i	$-0.400135\pi$
$398$	0	0
$399$	−3.77031	−0.188752
$400$	0	0
$401$	18.3901	0.918355	0.459178	−	0.888344i	$-0.348144\pi$
$401$	18.3901	0.918355	0.459178	+	0.888344i	$0.348144\pi$
$402$	0	0
$403$	2.64372i	0.131693i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.56218i	0.473979i
$408$	0	0
$409$	23.2744	1.15085	0.575423	−	0.817856i	$-0.304837\pi$
$409$	23.2744	1.15085	0.575423	+	0.817856i	$0.304837\pi$
$410$	0	0
$411$	−16.3973	−0.808818
$412$	0	0
$413$	− 22.1828i	− 1.09155i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.18970i	− 0.205170i
$418$	0	0
$419$	−27.5055	−1.34373	−0.671866	−	0.740673i	$-0.734507\pi$
$419$	−27.5055	−1.34373	−0.671866	+	0.740673i	$0.734507\pi$
$420$	0	0
$421$	−9.70130	−0.472812	−0.236406	−	0.971654i	$-0.575970\pi$
$421$	−9.70130	−0.472812	−0.236406	+	0.971654i	$0.575970\pi$
$422$	0	0
$423$	5.58686i	0.271642i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 15.5000i	− 0.750097i
$428$	0	0
$429$	27.4691	1.32622
$430$	0	0
$431$	15.6303	0.752886	0.376443	−	0.926440i	$-0.377147\pi$
$431$	15.6303	0.752886	0.376443	+	0.926440i	$0.377147\pi$
$432$	0	0
$433$	− 17.3839i	− 0.835416i	−0.908581	−	0.417708i	$-0.862834\pi$
$433$	− 17.3839i	− 0.835416i	0.908581	−	0.417708i	$-0.137166\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.37163i	− 0.209123i
$438$	0	0
$439$	25.3238	1.20864	0.604320	−	0.796742i	$-0.293445\pi$
$439$	25.3238	1.20864	0.604320	+	0.796742i	$0.293445\pi$
$440$	0	0
$441$	−3.80379	−0.181133
$442$	0	0
$443$	− 6.49504i	− 0.308589i	−0.988025	−	0.154294i	$-0.950690\pi$
$443$	− 6.49504i	− 0.308589i	0.988025	−	0.154294i	$-0.0493104\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	24.8646i	1.17606i
$448$	0	0
$449$	3.09007	0.145829	0.0729147	−	0.997338i	$-0.476770\pi$
$449$	3.09007	0.145829	0.0729147	+	0.997338i	$0.476770\pi$
$450$	0	0
$451$	3.98984	0.187874
$452$	0	0
$453$	− 2.85958i	− 0.134355i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 38.7215i	− 1.81131i	−0.424012	−	0.905657i	$-0.639378\pi$
$457$	− 38.7215i	− 1.81131i	0.424012	−	0.905657i	$-0.360622\pi$
$458$	0	0
$459$	36.0456	1.68246
$460$	0	0
$461$	−21.9817	−1.02379	−0.511895	−	0.859048i	$-0.671056\pi$
$461$	−21.9817	−1.02379	−0.511895	+	0.859048i	$0.671056\pi$
$462$	0	0
$463$	− 42.4385i	− 1.97228i	−0.165901	−	0.986142i	$-0.553053\pi$
$463$	− 42.4385i	− 1.97228i	0.165901	−	0.986142i	$-0.446947\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14.4303i	− 0.667754i	−0.942617	−	0.333877i	$-0.891643\pi$
$467$	− 14.4303i	− 0.667754i	0.942617	−	0.333877i	$-0.108357\pi$
$468$	0	0
$469$	40.6215	1.87573
$470$	0	0
$471$	21.3922	0.985703
$472$	0	0
$473$	− 12.2551i	− 0.563488i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.83320i	0.358658i
$478$	0	0
$479$	13.0612	0.596781	0.298390	−	0.954444i	$-0.403550\pi$
$479$	13.0612	0.596781	0.298390	+	0.954444i	$0.403550\pi$
$480$	0	0
$481$	14.5862	0.665072
$482$	0	0
$483$	16.4824i	0.749975i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.65696i	− 0.392284i	−0.980575	−	0.196142i	$-0.937159\pi$
$487$	− 8.65696i	− 0.392284i	0.980575	−	0.196142i	$-0.0628414\pi$
$488$	0	0
$489$	8.13160	0.367723
$490$	0	0
$491$	−8.37358	−0.377894	−0.188947	−	0.981987i	$-0.560507\pi$
$491$	−8.37358	−0.377894	−0.188947	+	0.981987i	$0.560507\pi$
$492$	0	0
$493$	− 28.9213i	− 1.30255i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 37.9557i	− 1.70255i
$498$	0	0
$499$	−35.0254	−1.56795	−0.783977	−	0.620790i	$-0.786812\pi$
$499$	−35.0254	−1.56795	−0.783977	+	0.620790i	$0.786812\pi$
$500$	0	0
$501$	−10.1715	−0.454427
$502$	0	0
$503$	14.7037i	0.655604i	0.944746	+	0.327802i	$0.106308\pi$
$503$	14.7037i	0.655604i	−0.944746	+	0.327802i	$0.893692\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 26.0188i	− 1.15554i
$508$	0	0
$509$	3.73411	0.165512	0.0827558	−	0.996570i	$-0.473628\pi$
$509$	3.73411	0.165512	0.0827558	+	0.996570i	$0.473628\pi$
$510$	0	0
$511$	−7.71153	−0.341138
$512$	0	0
$513$	− 5.50681i	− 0.243132i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 14.2296i	− 0.625819i
$518$	0	0
$519$	−24.4665	−1.07396
$520$	0	0
$521$	−3.38053	−0.148104	−0.0740518	−	0.997254i	$-0.523593\pi$
$521$	−3.38053	−0.148104	−0.0740518	+	0.997254i	$0.523593\pi$
$522$	0	0
$523$	− 13.7951i	− 0.603217i	−0.953432	−	0.301608i	$-0.902476\pi$
$523$	− 13.7951i	− 0.603217i	0.953432	−	0.301608i	$-0.0975235\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.95490i	− 0.128718i
$528$	0	0
$529$	3.88888	0.169082
$530$	0	0
$531$	10.8351	0.470203
$532$	0	0
$533$	− 6.08611i	− 0.263619i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.8672i	0.641567i
$538$	0	0
$539$	9.68820	0.417300
$540$	0	0
$541$	19.4420	0.835876	0.417938	−	0.908476i	$-0.362753\pi$
$541$	19.4420	0.835876	0.417938	+	0.908476i	$0.362753\pi$
$542$	0	0
$543$	4.59995i	0.197403i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.2715i	0.524689i	0.964974	+	0.262345i	$0.0844958\pi$
$547$	12.2715i	0.524689i	−0.964974	+	0.262345i	$0.915504\pi$
$548$	0	0
$549$	7.57090	0.323118
$550$	0	0
$551$	−4.41841	−0.188230
$552$	0	0
$553$	34.0427i	1.44764i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.01766i	− 0.254977i	−0.991840	−	0.127488i	$-0.959308\pi$
$557$	− 6.01766i	− 0.254977i	0.991840	−	0.127488i	$-0.0406915\pi$
$558$	0	0
$559$	−18.6939	−0.790668
$560$	0	0
$561$	−30.7024	−1.29625
$562$	0	0
$563$	− 30.1874i	− 1.27225i	−0.771587	−	0.636124i	$-0.780537\pi$
$563$	− 30.1874i	− 1.27225i	0.771587	−	0.636124i	$-0.219463\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.80725i	0.285878i
$568$	0	0
$569$	12.3451	0.517535	0.258768	−	0.965940i	$-0.416684\pi$
$569$	12.3451	0.517535	0.258768	+	0.965940i	$0.416684\pi$
$570$	0	0
$571$	3.13368	0.131140	0.0655702	−	0.997848i	$-0.479113\pi$
$571$	3.13368	0.131140	0.0655702	+	0.997848i	$0.479113\pi$
$572$	0	0
$573$	− 15.9579i	− 0.666651i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	27.2272i	1.13348i	0.823895	+	0.566742i	$0.191796\pi$
$577$	27.2272i	1.13348i	−0.823895	+	0.566742i	$0.808204\pi$
$578$	0	0
$579$	−11.5436	−0.479735
$580$	0	0
$581$	0.758647	0.0314740
$582$	0	0
$583$	− 19.9510i	− 0.826288i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.7445i	1.18641i	0.805050	+	0.593206i	$0.202138\pi$
$587$	28.7445i	1.18641i	−0.805050	+	0.593206i	$0.797862\pi$
$588$	0	0
$589$	−0.451431	−0.0186009
$590$	0	0
$591$	−17.5038	−0.720012
$592$	0	0
$593$	27.5257i	1.13034i	0.824973	+	0.565172i	$0.191190\pi$
$593$	27.5257i	1.13034i	−0.824973	+	0.565172i	$0.808810\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.7443i	0.808081i
$598$	0	0
$599$	28.2278	1.15335	0.576677	−	0.816972i	$-0.304349\pi$
$599$	28.2278	1.15335	0.576677	+	0.816972i	$0.304349\pi$
$600$	0	0
$601$	8.32545	0.339602	0.169801	−	0.985478i	$-0.445687\pi$
$601$	8.32545	0.339602	0.169801	+	0.985478i	$0.445687\pi$
$602$	0	0
$603$	19.8414i	0.808005i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11.7364i	− 0.476364i	−0.971220	−	0.238182i	$-0.923448\pi$
$607$	− 11.7364i	− 0.476364i	0.971220	−	0.238182i	$-0.0765515\pi$
$608$	0	0
$609$	16.6588	0.675047
$610$	0	0
$611$	−21.7059	−0.878128
$612$	0	0
$613$	− 31.5934i	− 1.27604i	−0.770018	−	0.638022i	$-0.779753\pi$
$613$	− 31.5934i	− 1.27604i	0.770018	−	0.638022i	$-0.220247\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 29.3247i	− 1.18057i	−0.807196	−	0.590284i	$-0.799016\pi$
$617$	− 29.3247i	− 1.18057i	0.807196	−	0.590284i	$-0.200984\pi$
$618$	0	0
$619$	37.8615	1.52178	0.760890	−	0.648881i	$-0.224763\pi$
$619$	37.8615	1.52178	0.760890	+	0.648881i	$0.224763\pi$
$620$	0	0
$621$	−24.0737	−0.966045
$622$	0	0
$623$	− 41.4516i	− 1.66072i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.69051i	0.187321i
$628$	0	0
$629$	−16.3030	−0.650044
$630$	0	0
$631$	−24.0801	−0.958615	−0.479308	−	0.877647i	$-0.659112\pi$
$631$	−24.0801	−0.958615	−0.479308	+	0.877647i	$0.659112\pi$
$632$	0	0
$633$	5.64134i	0.224223i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 14.7784i	− 0.585542i
$638$	0	0
$639$	18.5393	0.733404
$640$	0	0
$641$	−48.2114	−1.90424	−0.952118	−	0.305732i	$-0.901099\pi$
$641$	−48.2114	−1.90424	−0.952118	+	0.305732i	$0.901099\pi$
$642$	0	0
$643$	18.6087i	0.733856i	0.930249	+	0.366928i	$0.119590\pi$
$643$	18.6087i	0.733856i	−0.930249	+	0.366928i	$0.880410\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16.7599i	− 0.658900i	−0.944173	−	0.329450i	$-0.893137\pi$
$647$	− 16.7599i	− 0.658900i	0.944173	−	0.329450i	$-0.106863\pi$
$648$	0	0
$649$	−27.5968	−1.08327
$650$	0	0
$651$	1.70203	0.0667079
$652$	0	0
$653$	− 2.12145i	− 0.0830186i	−0.999138	−	0.0415093i	$-0.986783\pi$
$653$	− 2.12145i	− 0.0830186i	0.999138	−	0.0415093i	$-0.0132166\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.76666i	− 0.146951i
$658$	0	0
$659$	−0.175317	−0.00682937	−0.00341468	−	0.999994i	$-0.501087\pi$
$659$	−0.175317	−0.00682937	−0.00341468	+	0.999994i	$0.501087\pi$
$660$	0	0
$661$	16.9306	0.658526	0.329263	−	0.944238i	$-0.393200\pi$
$661$	16.9306	0.658526	0.329263	+	0.944238i	$0.393200\pi$
$662$	0	0
$663$	46.8334i	1.81886i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.3156i	0.747904i
$668$	0	0
$669$	24.7751	0.957860
$670$	0	0
$671$	−19.2830	−0.744411
$672$	0	0
$673$	− 21.8162i	− 0.840952i	−0.907304	−	0.420476i	$-0.861863\pi$
$673$	− 21.8162i	− 0.840952i	0.907304	−	0.420476i	$-0.138137\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.0305i	0.500803i	0.968142	+	0.250402i	$0.0805627\pi$
$677$	13.0305i	0.500803i	−0.968142	+	0.250402i	$0.919437\pi$
$678$	0	0
$679$	−34.2524	−1.31449
$680$	0	0
$681$	−14.1365	−0.541713
$682$	0	0
$683$	8.41103i	0.321839i	0.986968	+	0.160920i	$0.0514460\pi$
$683$	8.41103i	0.321839i	−0.986968	+	0.160920i	$0.948554\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 17.2011i	− 0.656263i
$688$	0	0
$689$	−30.4334	−1.15942
$690$	0	0
$691$	−8.74129	−0.332534	−0.166267	−	0.986081i	$-0.553171\pi$
$691$	−8.74129	−0.332534	−0.166267	+	0.986081i	$0.553171\pi$
$692$	0	0
$693$	17.8588i	0.678402i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.80247i	0.257662i
$698$	0	0
$699$	−4.82931	−0.182661
$700$	0	0
$701$	−45.5898	−1.72190	−0.860952	−	0.508686i	$-0.830131\pi$
$701$	−45.5898	−1.72190	−0.860952	+	0.508686i	$0.830131\pi$
$702$	0	0
$703$	2.49067i	0.0939373i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 49.5182i	− 1.86232i
$708$	0	0
$709$	34.2985	1.28811	0.644053	−	0.764981i	$-0.277252\pi$
$709$	34.2985	1.28811	0.644053	+	0.764981i	$0.277252\pi$
$710$	0	0
$711$	−16.6280	−0.623600
$712$	0	0
$713$	1.97349i	0.0739077i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 22.0721i	− 0.824297i
$718$	0	0
$719$	15.0753	0.562215	0.281107	−	0.959676i	$-0.409298\pi$
$719$	15.0753	0.562215	0.281107	+	0.959676i	$0.409298\pi$
$720$	0	0
$721$	47.7003	1.77645
$722$	0	0
$723$	28.2221i	1.04959i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 15.0189i	− 0.557021i	−0.960433	−	0.278510i	$-0.910159\pi$
$727$	− 15.0189i	− 0.557021i	0.960433	−	0.278510i	$-0.0898407\pi$
$728$	0	0
$729$	−23.5087	−0.870691
$730$	0	0
$731$	20.8942	0.772802
$732$	0	0
$733$	− 32.0731i	− 1.18465i	−0.805700	−	0.592324i	$-0.798211\pi$
$733$	− 32.0731i	− 1.18465i	0.805700	−	0.592324i	$-0.201789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 50.5358i	− 1.86151i
$738$	0	0
$739$	32.0663	1.17958	0.589788	−	0.807558i	$-0.299211\pi$
$739$	32.0663	1.17958	0.589788	+	0.807558i	$0.299211\pi$
$740$	0	0
$741$	7.15491	0.262842
$742$	0	0
$743$	− 14.3624i	− 0.526904i	−0.964673	−	0.263452i	$-0.915139\pi$
$743$	− 14.3624i	− 0.526904i	0.964673	−	0.263452i	$-0.0848611\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.370558i	0.0135580i
$748$	0	0
$749$	−28.5029	−1.04147
$750$	0	0
$751$	21.2055	0.773800	0.386900	−	0.922122i	$-0.373546\pi$
$751$	21.2055	0.773800	0.386900	+	0.922122i	$0.373546\pi$
$752$	0	0
$753$	− 16.0032i	− 0.583190i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.8045i	1.51941i	0.650267	+	0.759705i	$0.274657\pi$
$757$	41.8045i	1.51941i	−0.650267	+	0.759705i	$0.725343\pi$
$758$	0	0
$759$	20.5051	0.744289
$760$	0	0
$761$	−1.64223	−0.0595307	−0.0297654	−	0.999557i	$-0.509476\pi$
$761$	−1.64223	−0.0595307	−0.0297654	+	0.999557i	$0.509476\pi$
$762$	0	0
$763$	− 30.4635i	− 1.10285i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.0963i	1.52001i
$768$	0	0
$769$	−30.3955	−1.09609	−0.548044	−	0.836449i	$-0.684627\pi$
$769$	−30.3955	−1.09609	−0.548044	+	0.836449i	$0.684627\pi$
$770$	0	0
$771$	−16.5168	−0.594837
$772$	0	0
$773$	47.6753i	1.71476i	0.514684	+	0.857380i	$0.327909\pi$
$773$	47.6753i	1.71476i	−0.514684	+	0.857380i	$0.672091\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 9.39059i	− 0.336886i
$778$	0	0
$779$	1.03924	0.0372346
$780$	0	0
$781$	−47.2194	−1.68964
$782$	0	0
$783$	24.3313i	0.869531i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 34.3173i	− 1.22328i	−0.791137	−	0.611639i	$-0.790510\pi$
$787$	− 34.3173i	− 1.22328i	0.791137	−	0.611639i	$-0.209490\pi$
$788$	0	0
$789$	0.437307	0.0155685
$790$	0	0
$791$	56.8415	2.02105
$792$	0	0
$793$	29.4143i	1.04453i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 39.4643i	− 1.39790i	−0.715172	−	0.698949i	$-0.753651\pi$
$797$	− 39.4643i	− 1.39790i	0.715172	−	0.698949i	$-0.246349\pi$
$798$	0	0
$799$	24.2608	0.858285
$800$	0	0
$801$	20.2468	0.715387
$802$	0	0
$803$	9.59363i	0.338552i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 19.4936i	− 0.686207i
$808$	0	0
$809$	36.3760	1.27891	0.639456	−	0.768827i	$-0.279159\pi$
$809$	36.3760	1.27891	0.639456	+	0.768827i	$0.279159\pi$
$810$	0	0
$811$	−45.6700	−1.60369	−0.801845	−	0.597532i	$-0.796148\pi$
$811$	−45.6700	−1.60369	−0.801845	+	0.597532i	$0.796148\pi$
$812$	0	0
$813$	0.971471i	0.0340710i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.19209i	− 0.111677i
$818$	0	0
$819$	27.2419	0.951910
$820$	0	0
$821$	18.4543	0.644061	0.322030	−	0.946729i	$-0.395635\pi$
$821$	18.4543	0.644061	0.322030	+	0.946729i	$0.395635\pi$
$822$	0	0
$823$	− 13.0227i	− 0.453942i	−0.973902	−	0.226971i	$-0.927118\pi$
$823$	− 13.0227i	− 0.453942i	0.973902	−	0.226971i	$-0.0728822\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 38.0455i	− 1.32297i	−0.749958	−	0.661485i	$-0.769926\pi$
$827$	− 38.0455i	− 1.32297i	0.749958	−	0.661485i	$-0.230074\pi$
$828$	0	0
$829$	−35.9506	−1.24862	−0.624308	−	0.781178i	$-0.714619\pi$
$829$	−35.9506	−1.24862	−0.624308	+	0.781178i	$0.714619\pi$
$830$	0	0
$831$	−15.8387	−0.549438
$832$	0	0
$833$	16.5179i	0.572311i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.48594i	0.0859268i
$838$	0	0
$839$	3.66477	0.126522	0.0632610	−	0.997997i	$-0.479850\pi$
$839$	3.66477	0.126522	0.0632610	+	0.997997i	$0.479850\pi$
$840$	0	0
$841$	−9.47769	−0.326817
$842$	0	0
$843$	− 32.1036i	− 1.10571i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 11.5400i	− 0.396521i
$848$	0	0
$849$	−33.5488	−1.15139
$850$	0	0
$851$	10.8883	0.373245
$852$	0	0
$853$	− 31.5802i	− 1.08129i	−0.841252	−	0.540643i	$-0.818181\pi$
$853$	− 31.5802i	− 1.08129i	0.841252	−	0.540643i	$-0.181819\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.40752i	0.0822392i	0.999154	+	0.0411196i	$0.0130925\pi$
$857$	2.40752i	0.0822392i	−0.999154	+	0.0411196i	$0.986908\pi$
$858$	0	0
$859$	−41.4550	−1.41443	−0.707214	−	0.707000i	$-0.750048\pi$
$859$	−41.4550	−1.41443	−0.707214	+	0.707000i	$0.750048\pi$
$860$	0	0
$861$	−3.91825	−0.133534
$862$	0	0
$863$	− 6.99173i	− 0.238001i	−0.992894	−	0.119001i	$-0.962031\pi$
$863$	− 6.99173i	− 0.238001i	0.992894	−	0.119001i	$-0.0379690\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 31.5764i	− 1.07239i
$868$	0	0
$869$	42.3513	1.43667
$870$	0	0
$871$	−77.0874	−2.61201
$872$	0	0
$873$	− 16.7304i	− 0.566239i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.8619i	1.34604i	0.739624	+	0.673020i	$0.235003\pi$
$877$	39.8619i	1.34604i	−0.739624	+	0.673020i	$0.764997\pi$
$878$	0	0
$879$	24.1382	0.814161
$880$	0	0
$881$	24.7409	0.833541	0.416771	−	0.909012i	$-0.363162\pi$
$881$	24.7409	0.833541	0.416771	+	0.909012i	$0.363162\pi$
$882$	0	0
$883$	− 31.5648i	− 1.06224i	−0.847297	−	0.531120i	$-0.821771\pi$
$883$	− 31.5648i	− 1.06224i	0.847297	−	0.531120i	$-0.178229\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.7866i	0.395755i	0.980227	+	0.197877i	$0.0634048\pi$
$887$	11.7866i	0.395755i	−0.980227	+	0.197877i	$0.936595\pi$
$888$	0	0
$889$	−43.7452	−1.46717
$890$	0	0
$891$	8.46866	0.283711
$892$	0	0
$893$	− 3.70641i	− 0.124030i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 31.2786i	− 1.04436i
$898$	0	0
$899$	1.99460	0.0665238
$900$	0	0
$901$	34.0155	1.13322
$902$	0	0
$903$	12.0352i	0.400505i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.08545i	0.135655i	0.997697	+	0.0678276i	$0.0216068\pi$
$907$	4.08545i	0.135655i	−0.997697	+	0.0678276i	$0.978393\pi$
$908$	0	0
$909$	24.1869	0.802230
$910$	0	0
$911$	4.88428	0.161823	0.0809116	−	0.996721i	$-0.474217\pi$
$911$	4.88428	0.161823	0.0809116	+	0.996721i	$0.474217\pi$
$912$	0	0
$913$	− 0.943806i	− 0.0312354i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.33351i	− 0.143105i
$918$	0	0
$919$	53.8155	1.77521	0.887605	−	0.460606i	$-0.152368\pi$
$919$	53.8155	1.77521	0.887605	+	0.460606i	$0.152368\pi$
$920$	0	0
$921$	−35.3532	−1.16493
$922$	0	0
$923$	72.0285i	2.37085i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	23.2990i	0.765239i
$928$	0	0
$929$	43.3501	1.42227	0.711135	−	0.703055i	$-0.248181\pi$
$929$	43.3501	1.42227	0.711135	+	0.703055i	$0.248181\pi$
$930$	0	0
$931$	2.52349	0.0827042
$932$	0	0
$933$	− 34.3181i	− 1.12352i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 41.5106i	− 1.35609i	−0.735019	−	0.678046i	$-0.762827\pi$
$937$	− 41.5106i	− 1.35609i	0.735019	−	0.678046i	$-0.237173\pi$
$938$	0	0
$939$	−36.4000	−1.18787
$940$	0	0
$941$	10.2485	0.334091	0.167045	−	0.985949i	$-0.446577\pi$
$941$	10.2485	0.334091	0.167045	+	0.985949i	$0.446577\pi$
$942$	0	0
$943$	− 4.54316i	− 0.147946i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18.3985i	− 0.597871i	−0.954273	−	0.298936i	$-0.903368\pi$
$947$	− 18.3985i	− 0.597871i	0.954273	−	0.298936i	$-0.0966316\pi$
$948$	0	0
$949$	14.6341	0.475044
$950$	0	0
$951$	−7.49530	−0.243052
$952$	0	0
$953$	26.4170i	0.855732i	0.903842	+	0.427866i	$0.140734\pi$
$953$	26.4170i	0.855732i	−0.903842	+	0.427866i	$0.859266\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 20.7246i	− 0.669930i
$958$	0	0
$959$	41.4182	1.33746
$960$	0	0
$961$	−30.7962	−0.993426
$962$	0	0
$963$	− 13.9221i	− 0.448634i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.2597i	1.80919i	0.426271	+	0.904596i	$0.359827\pi$
$967$	56.2597i	1.80919i	−0.426271	+	0.904596i	$0.640173\pi$
$968$	0	0
$969$	−7.99707	−0.256903
$970$	0	0
$971$	−38.5451	−1.23697	−0.618485	−	0.785796i	$-0.712253\pi$
$971$	−38.5451	−1.23697	−0.618485	+	0.785796i	$0.712253\pi$
$972$	0	0
$973$	10.5828i	0.339270i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28.9699i	− 0.926831i	−0.886141	−	0.463415i	$-0.846624\pi$
$977$	− 28.9699i	− 0.926831i	0.886141	−	0.463415i	$-0.153376\pi$
$978$	0	0
$979$	−51.5684	−1.64813
$980$	0	0
$981$	14.8798	0.475075
$982$	0	0
$983$	1.75595i	0.0560061i	0.999608	+	0.0280030i	$0.00891481\pi$
$983$	1.75595i	0.0560061i	−0.999608	+	0.0280030i	$0.991085\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.9743i	0.444807i
$988$	0	0
$989$	−13.9546	−0.443731
$990$	0	0
$991$	1.13587	0.0360822	0.0180411	−	0.999837i	$-0.494257\pi$
$991$	1.13587	0.0360822	0.0180411	+	0.999837i	$0.494257\pi$
$992$	0	0
$993$	− 20.8940i	− 0.663052i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.2745i	0.515420i	0.966222	+	0.257710i	$0.0829679\pi$
$997$	16.2745i	0.515420i	−0.966222	+	0.257710i	$0.917032\pi$
$998$	0	0
$999$	13.7156	0.433944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.p.3649.9		12
5.2	odd	4		3800.2.a.bb.1.5	✓	6
5.3	odd	4		3800.2.a.bd.1.2	yes	6
5.4	even	2	inner	3800.2.d.p.3649.4		12
20.3	even	4		7600.2.a.ci.1.5		6
20.7	even	4		7600.2.a.cm.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.5	✓	6	5.2	odd	4
3800.2.a.bd.1.2	yes	6	5.3	odd	4
3800.2.d.p.3649.4		12	5.4	even	2	inner
3800.2.d.p.3649.9		12	1.1	even	1	trivial
7600.2.a.ci.1.5		6	20.3	even	4
7600.2.a.cm.1.2		6	20.7	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)

\(f(q)\)	\(=\)	\(q+1.22174i q^{3} -3.08602i q^{7} +1.50735 q^{9} -3.83920 q^{11} +5.85633i q^{13} -6.54564i q^{17} -1.00000 q^{19} +3.77031 q^{21} +4.37163i q^{23} +5.50681i q^{27} +4.41841 q^{29} +0.451431 q^{31} -4.69051i q^{33} -2.49067i q^{37} -7.15491 q^{39} -1.03924 q^{41} +3.19209i q^{43} +3.70641i q^{47} -2.52349 q^{49} +7.99707 q^{51} +5.19667i q^{53} -1.22174i q^{57} +7.18818 q^{59} +5.02265 q^{61} -4.65171i q^{63} +13.1631i q^{67} -5.34099 q^{69} +12.2993 q^{71} -2.49886i q^{73} +11.8478i q^{77} -11.0313 q^{79} -2.20584 q^{81} +0.245834i q^{83} +5.39814i q^{87} +13.4321 q^{89} +18.0727 q^{91} +0.551531i q^{93} -11.0992i q^{97} -5.78702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9} + 6 q^{11} - 12 q^{19} + 22 q^{21} - 14 q^{29} + 10 q^{31} - 16 q^{39} + 22 q^{41} + 4 q^{49} + 26 q^{51} + 8 q^{59} + 26 q^{61} - 14 q^{69} + 58 q^{71} - 56 q^{79} + 76 q^{81} + 24 q^{89}+ \cdots - 20 q^{99}+O(q^{100}) \) 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

Varieties

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Database

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