LMFDB - Embedded newform 3800.2.d.m.3649.1

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:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.1
Root			$-0.713538i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.m.3649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49086i q^{3} -2.49086i q^{7} -3.20440 q^{9} -3.49086 q^{11} -0.713538i q^{13} -3.91794i q^{17} +1.00000 q^{19} -6.20440 q^{21} -2.20440i q^{23} +0.509136i q^{27} -0.636712 q^{29} -2.14061 q^{31} +8.69527i q^{33} -2.06379i q^{37} -1.77733 q^{39} -4.35025 q^{41} -4.55465i q^{43} -0.268189i q^{47} +0.795598 q^{49} -9.75905 q^{51} +7.98173i q^{53} -2.49086i q^{57} +2.57292 q^{59} -10.8411 q^{61} +7.98173i q^{63} +1.08206i q^{67} -5.49086 q^{69} +6.83588 q^{71} +15.0403i q^{73} +8.69527i q^{77} +7.63671 q^{79} -8.34502 q^{81} +0.923174i q^{83} +1.58596i q^{87} -14.1679 q^{89} -1.77733 q^{91} +5.33198i q^{93} -6.14061i q^{97} +11.1861 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9} - 6 q^{11} + 6 q^{19} - 20 q^{21} + 2 q^{29} - 6 q^{31} + 2 q^{39} - 18 q^{41} + 22 q^{49} - 16 q^{51} + 20 q^{59} - 42 q^{61} - 18 q^{69} + 2 q^{71} + 40 q^{79} - 26 q^{81} - 8 q^{89} + 2 q^{91}+ \cdots + 20 q^{99}+O(q^{100}) $ 6 * q - 2 * q^9 - 6 * q^11 + 6 * q^19 - 20 * q^21 + 2 * q^29 - 6 * q^31 + 2 * q^39 - 18 * q^41 + 22 * q^49 - 16 * q^51 + 20 * q^59 - 42 * q^61 - 18 * q^69 + 2 * q^71 + 40 * q^79 - 26 * q^81 - 8 * q^89 + 2 * q^91 + 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
$3$	− 2.49086i	− 1.43810i	−0.694958	−	0.719050i	$-0.744577\pi$
$3$	− 2.49086i	− 1.43810i	0.694958	−	0.719050i	$-0.255423\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.49086i	− 0.941458i	−0.882278	−	0.470729i	$-0.843991\pi$
$7$	− 2.49086i	− 0.941458i	0.882278	−	0.470729i	$-0.156009\pi$
$8$	0	0
$9$	−3.20440	−1.06813
$10$	0	0
$11$	−3.49086	−1.05253	−0.526267	−	0.850319i	$-0.676409\pi$
$11$	−3.49086	−1.05253	−0.526267	+	0.850319i	$0.676409\pi$
$12$	0	0
$13$	− 0.713538i	− 0.197900i	−0.995092	−	0.0989499i	$-0.968452\pi$
$13$	− 0.713538i	− 0.197900i	0.995092	−	0.0989499i	$-0.0315484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.91794i	− 0.950240i	−0.879921	−	0.475120i	$-0.842405\pi$
$17$	− 3.91794i	− 0.950240i	0.879921	−	0.475120i	$-0.157595\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.20440	−1.35391
$22$	0	0
$23$	− 2.20440i	− 0.459649i	−0.973232	−	0.229825i	$-0.926185\pi$
$23$	− 2.20440i	− 0.459649i	0.973232	−	0.229825i	$-0.0738153\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.509136i	0.0979833i
$28$	0	0
$29$	−0.636712	−0.118234	−0.0591172	−	0.998251i	$-0.518829\pi$
$29$	−0.636712	−0.118234	−0.0591172	+	0.998251i	$0.518829\pi$
$30$	0	0
$31$	−2.14061	−0.384466	−0.192233	−	0.981349i	$-0.561573\pi$
$31$	−2.14061	−0.384466	−0.192233	+	0.981349i	$0.561573\pi$
$32$	0	0
$33$	8.69527i	1.51365i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.06379i	− 0.339285i	−0.985506	−	0.169642i	$-0.945739\pi$
$37$	− 2.06379i	− 0.339285i	0.985506	−	0.169642i	$-0.0542612\pi$
$38$	0	0
$39$	−1.77733	−0.284600
$40$	0	0
$41$	−4.35025	−0.679395	−0.339697	−	0.940535i	$-0.610325\pi$
$41$	−4.35025	−0.679395	−0.339697	+	0.940535i	$0.610325\pi$
$42$	0	0
$43$	− 4.55465i	− 0.694578i	−0.937758	−	0.347289i	$-0.887102\pi$
$43$	− 4.55465i	− 0.694578i	0.937758	−	0.347289i	$-0.112898\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.268189i	− 0.0391194i	−0.999809	−	0.0195597i	$-0.993774\pi$
$47$	− 0.268189i	− 0.0391194i	0.999809	−	0.0195597i	$-0.00622645\pi$
$48$	0	0
$49$	0.795598	0.113657
$50$	0	0
$51$	−9.75905	−1.36654
$52$	0	0
$53$	7.98173i	1.09637i	0.836356	+	0.548187i	$0.184682\pi$
$53$	7.98173i	1.09637i	−0.836356	+	0.548187i	$0.815318\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.49086i	− 0.329923i
$58$	0	0
$59$	2.57292	0.334966	0.167483	−	0.985875i	$-0.446436\pi$
$59$	2.57292	0.334966	0.167483	+	0.985875i	$0.446436\pi$
$60$	0	0
$61$	−10.8411	−1.38806	−0.694031	−	0.719945i	$-0.744167\pi$
$61$	−10.8411	−1.38806	−0.694031	+	0.719945i	$0.744167\pi$
$62$	0	0
$63$	7.98173i	1.00560i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.08206i	0.132195i	0.997813	+	0.0660974i	$0.0210548\pi$
$67$	1.08206i	0.132195i	−0.997813	+	0.0660974i	$0.978945\pi$
$68$	0	0
$69$	−5.49086	−0.661022
$70$	0	0
$71$	6.83588	0.811270	0.405635	−	0.914035i	$-0.367050\pi$
$71$	6.83588	0.811270	0.405635	+	0.914035i	$0.367050\pi$
$72$	0	0
$73$	15.0403i	1.76033i	0.474666	+	0.880166i	$0.342569\pi$
$73$	15.0403i	1.76033i	−0.474666	+	0.880166i	$0.657431\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.69527i	0.990917i
$78$	0	0
$79$	7.63671	0.859197	0.429599	−	0.903020i	$-0.358655\pi$
$79$	7.63671	0.859197	0.429599	+	0.903020i	$0.358655\pi$
$80$	0	0
$81$	−8.34502	−0.927224
$82$	0	0
$83$	0.923174i	0.101332i	0.998716	+	0.0506658i	$0.0161343\pi$
$83$	0.923174i	0.101332i	−0.998716	+	0.0506658i	$0.983866\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.58596i	0.170033i
$88$	0	0
$89$	−14.1679	−1.50179	−0.750895	−	0.660422i	$-0.770378\pi$
$89$	−14.1679	−1.50179	−0.750895	+	0.660422i	$0.770378\pi$
$90$	0	0
$91$	−1.77733	−0.186314
$92$	0	0
$93$	5.33198i	0.552900i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.14061i	− 0.623485i	−0.950167	−	0.311742i	$-0.899087\pi$
$97$	− 6.14061i	− 0.623485i	0.950167	−	0.311742i	$-0.100913\pi$
$98$	0	0
$99$	11.1861	1.12425
$100$	0	0
$101$	−4.36329	−0.434163	−0.217082	−	0.976153i	$-0.569654\pi$
$101$	−4.36329	−0.434163	−0.217082	+	0.976153i	$0.569654\pi$
$102$	0	0
$103$	− 5.47259i	− 0.539230i	−0.962968	−	0.269615i	$-0.913103\pi$
$103$	− 5.47259i	− 0.539230i	0.962968	−	0.269615i	$-0.0868965\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.240947i	0.0232932i	0.999932	+	0.0116466i	$0.00370732\pi$
$107$	0.240947i	0.0232932i	−0.999932	+	0.0116466i	$0.996293\pi$
$108$	0	0
$109$	−10.1861	−0.975654	−0.487827	−	0.872940i	$-0.662210\pi$
$109$	−10.1861	−0.975654	−0.487827	+	0.872940i	$0.662210\pi$
$110$	0	0
$111$	−5.14061	−0.487925
$112$	0	0
$113$	11.2682i	1.06002i	0.847991	+	0.530011i	$0.177812\pi$
$113$	11.2682i	1.06002i	−0.847991	+	0.530011i	$0.822188\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.28646i	0.211383i
$118$	0	0
$119$	−9.75905	−0.894611
$120$	0	0
$121$	1.18613	0.107830
$122$	0	0
$123$	10.8359i	0.977038i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.46736i	− 0.662621i	−0.943522	−	0.331310i	$-0.892509\pi$
$127$	− 7.46736i	− 0.662621i	0.943522	−	0.331310i	$-0.107491\pi$
$128$	0	0
$129$	−11.3450	−0.998873
$130$	0	0
$131$	−7.77733	−0.679508	−0.339754	−	0.940514i	$-0.610344\pi$
$131$	−7.77733	−0.679508	−0.339754	+	0.940514i	$0.610344\pi$
$132$	0	0
$133$	− 2.49086i	− 0.215985i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.04028i	0.772363i	0.922423	+	0.386182i	$0.126206\pi$
$137$	9.04028i	0.772363i	−0.922423	+	0.386182i	$0.873794\pi$
$138$	0	0
$139$	20.6718	1.75336	0.876678	−	0.481078i	$-0.159755\pi$
$139$	20.6718	1.75336	0.876678	+	0.481078i	$0.159755\pi$
$140$	0	0
$141$	−0.668023	−0.0562577
$142$	0	0
$143$	2.49086i	0.208296i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.98173i	− 0.163450i
$148$	0	0
$149$	12.7591	1.04526	0.522631	−	0.852559i	$-0.324950\pi$
$149$	12.7591	1.04526	0.522631	+	0.852559i	$0.324950\pi$
$150$	0	0
$151$	−6.28646	−0.511585	−0.255793	−	0.966732i	$-0.582336\pi$
$151$	−6.28646	−0.511585	−0.255793	+	0.966732i	$0.582336\pi$
$152$	0	0
$153$	12.5547i	1.01498i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.4308i	1.71036i	0.518327	+	0.855182i	$0.326555\pi$
$157$	21.4308i	1.71036i	−0.518327	+	0.855182i	$0.673445\pi$
$158$	0	0
$159$	19.8814	1.57670
$160$	0	0
$161$	−5.49086	−0.432741
$162$	0	0
$163$	− 18.1041i	− 1.41802i	−0.705198	−	0.709010i	$-0.749142\pi$
$163$	− 18.1041i	− 1.41802i	0.705198	−	0.709010i	$-0.250858\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.3502i	0.955691i	0.878444	+	0.477846i	$0.158582\pi$
$167$	12.3502i	0.955691i	−0.878444	+	0.477846i	$0.841418\pi$
$168$	0	0
$169$	12.4909	0.960836
$170$	0	0
$171$	−3.20440	−0.245047
$172$	0	0
$173$	− 7.60017i	− 0.577830i	−0.957355	−	0.288915i	$-0.906706\pi$
$173$	− 7.60017i	− 0.577830i	0.957355	−	0.288915i	$-0.0932945\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 6.40880i	− 0.481715i
$178$	0	0
$179$	−7.28123	−0.544225	−0.272112	−	0.962266i	$-0.587722\pi$
$179$	−7.28123	−0.544225	−0.272112	+	0.962266i	$0.587722\pi$
$180$	0	0
$181$	6.79933	0.505390	0.252695	−	0.967546i	$-0.418683\pi$
$181$	6.79933	0.505390	0.252695	+	0.967546i	$0.418683\pi$
$182$	0	0
$183$	27.0037i	1.99617i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.6770i	1.00016i
$188$	0	0
$189$	1.26819	0.0922472
$190$	0	0
$191$	14.3085	1.03532	0.517662	−	0.855585i	$-0.326802\pi$
$191$	14.3085	1.03532	0.517662	+	0.855585i	$0.326802\pi$
$192$	0	0
$193$	− 13.4491i	− 0.968086i	−0.875044	−	0.484043i	$-0.839168\pi$
$193$	− 13.4491i	− 0.968086i	0.875044	−	0.484043i	$-0.160832\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.79036i	− 0.198805i	−0.995047	−	0.0994026i	$-0.968307\pi$
$197$	− 2.79036i	− 0.198805i	0.995047	−	0.0994026i	$-0.0316932\pi$
$198$	0	0
$199$	8.91794	0.632176	0.316088	−	0.948730i	$-0.397631\pi$
$199$	8.91794	0.632176	0.316088	+	0.948730i	$0.397631\pi$
$200$	0	0
$201$	2.69527	0.190109
$202$	0	0
$203$	1.58596i	0.111313i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.06379i	0.490967i
$208$	0	0
$209$	−3.49086	−0.241468
$210$	0	0
$211$	−24.2264	−1.66781	−0.833907	−	0.551904i	$-0.813901\pi$
$211$	−24.2264	−1.66781	−0.833907	+	0.551904i	$0.813901\pi$
$212$	0	0
$213$	− 17.0272i	− 1.16669i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.33198i	0.361958i
$218$	0	0
$219$	37.4633	2.53153
$220$	0	0
$221$	−2.79560	−0.188052
$222$	0	0
$223$	− 16.9269i	− 1.13351i	−0.823886	−	0.566755i	$-0.808199\pi$
$223$	− 16.9269i	− 1.13351i	0.823886	−	0.566755i	$-0.191801\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.6315i	0.705636i	0.935692	+	0.352818i	$0.114777\pi$
$227$	10.6315i	0.705636i	−0.935692	+	0.352818i	$0.885223\pi$
$228$	0	0
$229$	−8.41404	−0.556015	−0.278008	−	0.960579i	$-0.589674\pi$
$229$	−8.41404	−0.556015	−0.278008	+	0.960579i	$0.589674\pi$
$230$	0	0
$231$	21.6587	1.42504
$232$	0	0
$233$	9.65498i	0.632519i	0.948673	+	0.316260i	$0.102427\pi$
$233$	9.65498i	0.632519i	−0.948673	+	0.316260i	$0.897573\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 19.0220i	− 1.23561i
$238$	0	0
$239$	−0.395765	−0.0255999	−0.0127999	−	0.999918i	$-0.504074\pi$
$239$	−0.395765	−0.0255999	−0.0127999	+	0.999918i	$0.504074\pi$
$240$	0	0
$241$	2.05855	0.132603	0.0663015	−	0.997800i	$-0.478880\pi$
$241$	2.05855	0.132603	0.0663015	+	0.997800i	$0.478880\pi$
$242$	0	0
$243$	22.3137i	1.43142i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.713538i	− 0.0454013i
$248$	0	0
$249$	2.29950	0.145725
$250$	0	0
$251$	−24.9034	−1.57189	−0.785944	−	0.618297i	$-0.787823\pi$
$251$	−24.9034	−1.57189	−0.785944	+	0.618297i	$0.787823\pi$
$252$	0	0
$253$	7.69527i	0.483797i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.20964i	− 0.387346i	−0.981066	−	0.193673i	$-0.937960\pi$
$257$	− 6.20964i	− 0.387346i	0.981066	−	0.193673i	$-0.0620402\pi$
$258$	0	0
$259$	−5.14061	−0.319422
$260$	0	0
$261$	2.04028	0.126290
$262$	0	0
$263$	− 26.2316i	− 1.61751i	−0.588144	−	0.808756i	$-0.700141\pi$
$263$	− 26.2316i	− 1.61751i	0.588144	−	0.808756i	$-0.299859\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	35.2902i	2.15972i
$268$	0	0
$269$	3.10407	0.189258	0.0946292	−	0.995513i	$-0.469833\pi$
$269$	3.10407	0.189258	0.0946292	+	0.995513i	$0.469833\pi$
$270$	0	0
$271$	17.0090	1.03322	0.516611	−	0.856220i	$-0.327193\pi$
$271$	17.0090	1.03322	0.516611	+	0.856220i	$0.327193\pi$
$272$	0	0
$273$	4.42708i	0.267939i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.9269i	0.896871i	0.893815	+	0.448436i	$0.148019\pi$
$277$	14.9269i	0.896871i	−0.893815	+	0.448436i	$0.851981\pi$
$278$	0	0
$279$	6.85939	0.410661
$280$	0	0
$281$	−26.6535	−1.59001	−0.795007	−	0.606601i	$-0.792533\pi$
$281$	−26.6535	−1.59001	−0.795007	+	0.606601i	$0.792533\pi$
$282$	0	0
$283$	− 8.22267i	− 0.488787i	−0.969676	−	0.244394i	$-0.921411\pi$
$283$	− 8.22267i	− 0.488787i	0.969676	−	0.244394i	$-0.0785889\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.8359i	0.639622i
$288$	0	0
$289$	1.64975	0.0970441
$290$	0	0
$291$	−15.2954	−0.896634
$292$	0	0
$293$	− 20.1313i	− 1.17608i	−0.808830	−	0.588042i	$-0.799899\pi$
$293$	− 20.1313i	− 1.17608i	0.808830	−	0.588042i	$-0.200101\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.77733i	− 0.103131i
$298$	0	0
$299$	−1.57292	−0.0909646
$300$	0	0
$301$	−11.3450	−0.653916
$302$	0	0
$303$	10.8684i	0.624371i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.7225i	0.897331i	0.893700	+	0.448665i	$0.148101\pi$
$307$	15.7225i	0.897331i	−0.893700	+	0.448665i	$0.851899\pi$
$308$	0	0
$309$	−13.6315	−0.775468
$310$	0	0
$311$	24.1951	1.37198	0.685989	−	0.727612i	$-0.259370\pi$
$311$	24.1951	1.37198	0.685989	+	0.727612i	$0.259370\pi$
$312$	0	0
$313$	− 17.8631i	− 1.00968i	−0.863212	−	0.504842i	$-0.831551\pi$
$313$	− 17.8631i	− 1.00968i	0.863212	−	0.504842i	$-0.168449\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.77733i	− 0.324487i	−0.986751	−	0.162243i	$-0.948127\pi$
$317$	− 5.77733i	− 0.324487i	0.986751	−	0.162243i	$-0.0518730\pi$
$318$	0	0
$319$	2.22267	0.124446
$320$	0	0
$321$	0.600166	0.0334980
$322$	0	0
$323$	− 3.91794i	− 0.218000i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	25.3723i	1.40309i
$328$	0	0
$329$	−0.668023	−0.0368293
$330$	0	0
$331$	−1.11454	−0.0612605	−0.0306303	−	0.999531i	$-0.509751\pi$
$331$	−1.11454	−0.0612605	−0.0306303	+	0.999531i	$0.509751\pi$
$332$	0	0
$333$	6.61320i	0.362401i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.54161i	0.0839770i	0.999118	+	0.0419885i	$0.0133693\pi$
$337$	1.54161i	0.0839770i	−0.999118	+	0.0419885i	$0.986631\pi$
$338$	0	0
$339$	28.0675	1.52442
$340$	0	0
$341$	7.47259	0.404663
$342$	0	0
$343$	− 19.4178i	− 1.04846i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.3007i	1.57294i	0.617627	+	0.786471i	$0.288094\pi$
$347$	29.3007i	1.57294i	−0.617627	+	0.786471i	$0.711906\pi$
$348$	0	0
$349$	8.09626	0.433383	0.216692	−	0.976240i	$-0.430473\pi$
$349$	8.09626	0.433383	0.216692	+	0.976240i	$0.430473\pi$
$350$	0	0
$351$	0.363288	0.0193909
$352$	0	0
$353$	5.17716i	0.275552i	0.990463	+	0.137776i	$0.0439955\pi$
$353$	5.17716i	0.275552i	−0.990463	+	0.137776i	$0.956005\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.3085i	1.28654i
$358$	0	0
$359$	−30.3122	−1.59982	−0.799908	−	0.600122i	$-0.795119\pi$
$359$	−30.3122	−1.59982	−0.799908	+	0.600122i	$0.795119\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 2.95449i	− 0.155070i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.3398i	0.957329i	0.877998	+	0.478664i	$0.158879\pi$
$367$	18.3398i	0.957329i	−0.877998	+	0.478664i	$0.841121\pi$
$368$	0	0
$369$	13.9399	0.725685
$370$	0	0
$371$	19.8814	1.03219
$372$	0	0
$373$	32.8956i	1.70327i	0.524136	+	0.851635i	$0.324388\pi$
$373$	32.8956i	1.70327i	−0.524136	+	0.851635i	$0.675612\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.454318i	0.0233986i
$378$	0	0
$379$	15.2865	0.785213	0.392606	−	0.919707i	$-0.371573\pi$
$379$	15.2865	0.785213	0.392606	+	0.919707i	$0.371573\pi$
$380$	0	0
$381$	−18.6002	−0.952915
$382$	0	0
$383$	6.14061i	0.313771i	0.987617	+	0.156885i	$0.0501453\pi$
$383$	6.14061i	0.313771i	−0.987617	+	0.156885i	$0.949855\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.5949i	0.741902i
$388$	0	0
$389$	−36.1899	−1.83490	−0.917449	−	0.397852i	$-0.869756\pi$
$389$	−36.1899	−1.83490	−0.917449	+	0.397852i	$0.869756\pi$
$390$	0	0
$391$	−8.63671	−0.436777
$392$	0	0
$393$	19.3723i	0.977201i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 32.3775i	− 1.62498i	−0.582975	−	0.812490i	$-0.698112\pi$
$397$	− 32.3775i	− 1.62498i	0.582975	−	0.812490i	$-0.301888\pi$
$398$	0	0
$399$	−6.20440	−0.310609
$400$	0	0
$401$	−16.0090	−0.799450	−0.399725	−	0.916635i	$-0.630894\pi$
$401$	−16.0090	−0.799450	−0.399725	+	0.916635i	$0.630894\pi$
$402$	0	0
$403$	1.52741i	0.0760857i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.20440i	0.357109i
$408$	0	0
$409$	26.2954	1.30023	0.650113	−	0.759838i	$-0.274722\pi$
$409$	26.2954	1.30023	0.650113	+	0.759838i	$0.274722\pi$
$410$	0	0
$411$	22.5181	1.11074
$412$	0	0
$413$	− 6.40880i	− 0.315357i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 51.4905i	− 2.52150i
$418$	0	0
$419$	−22.9399	−1.12069	−0.560345	−	0.828259i	$-0.689331\pi$
$419$	−22.9399	−1.12069	−0.560345	+	0.828259i	$0.689331\pi$
$420$	0	0
$421$	−5.03131	−0.245211	−0.122606	−	0.992455i	$-0.539125\pi$
$421$	−5.03131	−0.245211	−0.122606	+	0.992455i	$0.539125\pi$
$422$	0	0
$423$	0.859386i	0.0417848i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.0037i	1.30680i
$428$	0	0
$429$	6.20440	0.299551
$430$	0	0
$431$	−10.4621	−0.503943	−0.251971	−	0.967735i	$-0.581079\pi$
$431$	−10.4621	−0.503943	−0.251971	+	0.967735i	$0.581079\pi$
$432$	0	0
$433$	− 4.21487i	− 0.202554i	−0.994858	−	0.101277i	$-0.967707\pi$
$433$	− 4.21487i	− 0.202554i	0.994858	−	0.101277i	$-0.0322928\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.20440i	− 0.105451i
$438$	0	0
$439$	−5.86312	−0.279832	−0.139916	−	0.990163i	$-0.544683\pi$
$439$	−5.86312	−0.279832	−0.139916	+	0.990163i	$0.544683\pi$
$440$	0	0
$441$	−2.54942	−0.121401
$442$	0	0
$443$	13.7303i	0.652347i	0.945310	+	0.326173i	$0.105759\pi$
$443$	13.7303i	0.652347i	−0.945310	+	0.326173i	$0.894241\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 31.7811i	− 1.50319i
$448$	0	0
$449$	18.6039	0.877972	0.438986	−	0.898494i	$-0.355338\pi$
$449$	18.6039	0.877972	0.438986	+	0.898494i	$0.355338\pi$
$450$	0	0
$451$	15.1861	0.715087
$452$	0	0
$453$	15.6587i	0.735711i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.750083i	− 0.0350874i	−0.999846	−	0.0175437i	$-0.994415\pi$
$457$	− 0.750083i	− 0.0350874i	0.999846	−	0.0175437i	$-0.00558462\pi$
$458$	0	0
$459$	1.99477	0.0931077
$460$	0	0
$461$	−16.9139	−0.787757	−0.393879	−	0.919162i	$-0.628867\pi$
$461$	−16.9139	−0.787757	−0.393879	+	0.919162i	$0.628867\pi$
$462$	0	0
$463$	− 13.4323i	− 0.624252i	−0.950041	−	0.312126i	$-0.898959\pi$
$463$	− 13.4323i	− 0.624252i	0.950041	−	0.312126i	$-0.101041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.1757i	− 1.11872i	−0.828926	−	0.559358i	$-0.811048\pi$
$467$	− 24.1757i	− 1.11872i	0.828926	−	0.559358i	$-0.188952\pi$
$468$	0	0
$469$	2.69527	0.124456
$470$	0	0
$471$	53.3812	2.45968
$472$	0	0
$473$	15.8997i	0.731067i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 25.5767i	− 1.17107i
$478$	0	0
$479$	−18.7915	−0.858607	−0.429303	−	0.903160i	$-0.641241\pi$
$479$	−18.7915	−0.858607	−0.429303	+	0.903160i	$0.641241\pi$
$480$	0	0
$481$	−1.47259	−0.0671444
$482$	0	0
$483$	13.6770i	0.622325i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 19.4125i	− 0.879666i	−0.898080	−	0.439833i	$-0.855038\pi$
$487$	− 19.4125i	− 0.879666i	0.898080	−	0.439833i	$-0.144962\pi$
$488$	0	0
$489$	−45.0948	−2.03926
$490$	0	0
$491$	−15.7408	−0.710371	−0.355186	−	0.934796i	$-0.615582\pi$
$491$	−15.7408	−0.710371	−0.355186	+	0.934796i	$0.615582\pi$
$492$	0	0
$493$	2.49460i	0.112351i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 17.0272i	− 0.763776i
$498$	0	0
$499$	−8.45432	−0.378467	−0.189234	−	0.981932i	$-0.560600\pi$
$499$	−8.45432	−0.378467	−0.189234	+	0.981932i	$0.560600\pi$
$500$	0	0
$501$	30.7628	1.37438
$502$	0	0
$503$	− 40.8161i	− 1.81990i	−0.414718	−	0.909950i	$-0.636120\pi$
$503$	− 40.8161i	− 1.81990i	0.414718	−	0.909950i	$-0.363880\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 31.1130i	− 1.38178i
$508$	0	0
$509$	−1.56919	−0.0695531	−0.0347765	−	0.999395i	$-0.511072\pi$
$509$	−1.56919	−0.0695531	−0.0347765	+	0.999395i	$0.511072\pi$
$510$	0	0
$511$	37.4633	1.65728
$512$	0	0
$513$	0.509136i	0.0224789i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.936212i	0.0411746i
$518$	0	0
$519$	−18.9310	−0.830978
$520$	0	0
$521$	38.2081	1.67393	0.836964	−	0.547257i	$-0.184328\pi$
$521$	38.2081	1.67393	0.836964	+	0.547257i	$0.184328\pi$
$522$	0	0
$523$	33.3630i	1.45886i	0.684055	+	0.729430i	$0.260215\pi$
$523$	33.3630i	1.45886i	−0.684055	+	0.729430i	$0.739785\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.38680i	0.365335i
$528$	0	0
$529$	18.1406	0.788722
$530$	0	0
$531$	−8.24468	−0.357789
$532$	0	0
$533$	3.10407i	0.134452i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.1365i	0.782650i
$538$	0	0
$539$	−2.77733	−0.119628
$540$	0	0
$541$	−45.2171	−1.94404	−0.972018	−	0.234908i	$-0.924521\pi$
$541$	−45.2171	−1.94404	−0.972018	+	0.234908i	$0.924521\pi$
$542$	0	0
$543$	− 16.9362i	− 0.726802i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 26.5416i	− 1.13484i	−0.823429	−	0.567419i	$-0.807942\pi$
$547$	− 26.5416i	− 1.13484i	0.823429	−	0.567419i	$-0.192058\pi$
$548$	0	0
$549$	34.7393	1.48264
$550$	0	0
$551$	−0.636712	−0.0271248
$552$	0	0
$553$	− 19.0220i	− 0.808898i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.89967i	− 0.122863i	−0.998111	−	0.0614314i	$-0.980433\pi$
$557$	− 2.89967i	− 0.122863i	0.998111	−	0.0614314i	$-0.0195665\pi$
$558$	0	0
$559$	−3.24992	−0.137457
$560$	0	0
$561$	34.0675	1.43833
$562$	0	0
$563$	36.0817i	1.52066i	0.649535	+	0.760332i	$0.274964\pi$
$563$	36.0817i	1.52066i	−0.649535	+	0.760332i	$0.725036\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	20.7863i	0.872942i
$568$	0	0
$569$	39.3540	1.64980	0.824902	−	0.565275i	$-0.191230\pi$
$569$	39.3540	1.64980	0.824902	+	0.565275i	$0.191230\pi$
$570$	0	0
$571$	17.4596	0.730660	0.365330	−	0.930878i	$-0.380956\pi$
$571$	17.4596	0.730660	0.365330	+	0.930878i	$0.380956\pi$
$572$	0	0
$573$	− 35.6404i	− 1.48890i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 21.1548i	− 0.880687i	−0.897829	−	0.440343i	$-0.854857\pi$
$577$	− 21.1548i	− 0.880687i	0.897829	−	0.440343i	$-0.145143\pi$
$578$	0	0
$579$	−33.4998	−1.39221
$580$	0	0
$581$	2.29950	0.0953994
$582$	0	0
$583$	− 27.8631i	− 1.15397i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.76429i	− 0.361741i	−0.983507	−	0.180870i	$-0.942109\pi$
$587$	− 8.76429i	− 0.361741i	0.983507	−	0.180870i	$-0.0578915\pi$
$588$	0	0
$589$	−2.14061	−0.0882025
$590$	0	0
$591$	−6.95042	−0.285902
$592$	0	0
$593$	− 25.0127i	− 1.02715i	−0.858045	−	0.513574i	$-0.828321\pi$
$593$	− 25.0127i	− 1.02715i	0.858045	−	0.513574i	$-0.171679\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 22.2134i	− 0.909133i
$598$	0	0
$599$	−3.97276	−0.162322	−0.0811612	−	0.996701i	$-0.525863\pi$
$599$	−3.97276	−0.162322	−0.0811612	+	0.996701i	$0.525863\pi$
$600$	0	0
$601$	21.0768	0.859742	0.429871	−	0.902890i	$-0.358559\pi$
$601$	21.0768	0.859742	0.429871	+	0.902890i	$0.358559\pi$
$602$	0	0
$603$	− 3.46736i	− 0.141202i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 38.9672i	− 1.58163i	−0.612056	−	0.790815i	$-0.709657\pi$
$607$	− 38.9672i	− 1.58163i	0.612056	−	0.790815i	$-0.290343\pi$
$608$	0	0
$609$	3.95042	0.160079
$610$	0	0
$611$	−0.191363	−0.00774173
$612$	0	0
$613$	− 35.7680i	− 1.44466i	−0.691550	−	0.722328i	$-0.743072\pi$
$613$	− 35.7680i	− 1.44466i	0.691550	−	0.722328i	$-0.256928\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.2447i	− 0.815020i	−0.913201	−	0.407510i	$-0.866397\pi$
$617$	− 20.2447i	− 0.815020i	0.913201	−	0.407510i	$-0.133603\pi$
$618$	0	0
$619$	−2.87616	−0.115603	−0.0578013	−	0.998328i	$-0.518409\pi$
$619$	−2.87616	−0.115603	−0.0578013	+	0.998328i	$0.518409\pi$
$620$	0	0
$621$	1.12234	0.0450380
$622$	0	0
$623$	35.2902i	1.41387i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.69527i	0.347255i
$628$	0	0
$629$	−8.08580	−0.322402
$630$	0	0
$631$	−46.3212	−1.84402	−0.922008	−	0.387170i	$-0.873453\pi$
$631$	−46.3212	−1.84402	−0.922008	+	0.387170i	$0.873453\pi$
$632$	0	0
$633$	60.3447i	2.39849i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.567690i	− 0.0224927i
$638$	0	0
$639$	−21.9049	−0.866544
$640$	0	0
$641$	−19.5804	−0.773379	−0.386690	−	0.922210i	$-0.626381\pi$
$641$	−19.5804	−0.773379	−0.386690	+	0.922210i	$0.626381\pi$
$642$	0	0
$643$	− 24.8542i	− 0.980152i	−0.871680	−	0.490076i	$-0.836969\pi$
$643$	− 24.8542i	− 0.980152i	0.871680	−	0.490076i	$-0.163031\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 44.2589i	− 1.74000i	−0.493055	−	0.869998i	$-0.664120\pi$
$647$	− 44.2589i	− 1.74000i	0.493055	−	0.869998i	$-0.335880\pi$
$648$	0	0
$649$	−8.98173	−0.352564
$650$	0	0
$651$	13.2812	0.520532
$652$	0	0
$653$	− 0.113372i	− 0.00443657i	−0.999998	−	0.00221829i	$-0.999294\pi$
$653$	− 0.113372i	− 0.00443657i	0.999998	−	0.00221829i	$-0.000706103\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 48.1951i	− 1.88027i
$658$	0	0
$659$	−7.65092	−0.298037	−0.149019	−	0.988834i	$-0.547611\pi$
$659$	−7.65092	−0.298037	−0.149019	+	0.988834i	$0.547611\pi$
$660$	0	0
$661$	−35.3488	−1.37491	−0.687454	−	0.726228i	$-0.741271\pi$
$661$	−35.3488	−1.37491	−0.687454	+	0.726228i	$0.741271\pi$
$662$	0	0
$663$	6.96345i	0.270438i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.40357i	0.0543464i
$668$	0	0
$669$	−42.1626	−1.63010
$670$	0	0
$671$	37.8448	1.46098
$672$	0	0
$673$	− 29.2301i	− 1.12674i	−0.826205	−	0.563370i	$-0.809505\pi$
$673$	− 29.2301i	− 1.12674i	0.826205	−	0.563370i	$-0.190495\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.97276i	0.152685i	0.997082	+	0.0763427i	$0.0243243\pi$
$677$	3.97276i	0.152685i	−0.997082	+	0.0763427i	$0.975676\pi$
$678$	0	0
$679$	−15.2954	−0.586985
$680$	0	0
$681$	26.4816	1.01478
$682$	0	0
$683$	11.8083i	0.451832i	0.974147	+	0.225916i	$0.0725375\pi$
$683$	11.8083i	0.451832i	−0.974147	+	0.225916i	$0.927462\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.9582i	0.799606i
$688$	0	0
$689$	5.69527	0.216972
$690$	0	0
$691$	3.14585	0.119674	0.0598369	−	0.998208i	$-0.480942\pi$
$691$	3.14585	0.119674	0.0598369	+	0.998208i	$0.480942\pi$
$692$	0	0
$693$	− 27.8631i	− 1.05843i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	17.0440i	0.645588i
$698$	0	0
$699$	24.0492	0.909626
$700$	0	0
$701$	−52.2313	−1.97275	−0.986375	−	0.164514i	$-0.947394\pi$
$701$	−52.2313	−1.97275	−0.986375	+	0.164514i	$0.947394\pi$
$702$	0	0
$703$	− 2.06379i	− 0.0778372i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.8684i	0.408747i
$708$	0	0
$709$	−42.1951	−1.58467	−0.792335	−	0.610086i	$-0.791135\pi$
$709$	−42.1951	−1.58467	−0.792335	+	0.610086i	$0.791135\pi$
$710$	0	0
$711$	−24.4711	−0.917738
$712$	0	0
$713$	4.71877i	0.176719i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.985796i	0.0368152i
$718$	0	0
$719$	14.6770	0.547359	0.273680	−	0.961821i	$-0.411759\pi$
$719$	14.6770	0.547359	0.273680	+	0.961821i	$0.411759\pi$
$720$	0	0
$721$	−13.6315	−0.507663
$722$	0	0
$723$	− 5.12758i	− 0.190697i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17.3189i	− 0.642324i	−0.947024	−	0.321162i	$-0.895927\pi$
$727$	− 17.3189i	− 0.642324i	0.947024	−	0.321162i	$-0.104073\pi$
$728$	0	0
$729$	30.5453	1.13131
$730$	0	0
$731$	−17.8448	−0.660016
$732$	0	0
$733$	− 22.4271i	− 0.828363i	−0.910194	−	0.414181i	$-0.864068\pi$
$733$	− 22.4271i	− 0.828363i	0.910194	−	0.414181i	$-0.135932\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.77733i	− 0.139140i
$738$	0	0
$739$	−7.60390	−0.279714	−0.139857	−	0.990172i	$-0.544664\pi$
$739$	−7.60390	−0.279714	−0.139857	+	0.990172i	$0.544664\pi$
$740$	0	0
$741$	−1.77733	−0.0652917
$742$	0	0
$743$	− 28.8423i	− 1.05812i	−0.848584	−	0.529060i	$-0.822545\pi$
$743$	− 28.8423i	− 1.05812i	0.848584	−	0.529060i	$-0.177455\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.95822i	− 0.108236i
$748$	0	0
$749$	0.600166	0.0219296
$750$	0	0
$751$	−7.79560	−0.284465	−0.142233	−	0.989833i	$-0.545428\pi$
$751$	−7.79560	−0.284465	−0.142233	+	0.989833i	$0.545428\pi$
$752$	0	0
$753$	62.0310i	2.26053i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 29.2227i	− 1.06212i	−0.847335	−	0.531058i	$-0.821795\pi$
$757$	− 29.2227i	− 1.06212i	0.847335	−	0.531058i	$-0.178205\pi$
$758$	0	0
$759$	19.1679	0.695749
$760$	0	0
$761$	0.996265	0.0361146	0.0180573	−	0.999837i	$-0.494252\pi$
$761$	0.996265	0.0361146	0.0180573	+	0.999837i	$0.494252\pi$
$762$	0	0
$763$	25.3723i	0.918537i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.83588i	− 0.0662897i
$768$	0	0
$769$	3.33605	0.120301	0.0601504	−	0.998189i	$-0.480842\pi$
$769$	3.33605	0.120301	0.0601504	+	0.998189i	$0.480842\pi$
$770$	0	0
$771$	−15.4674	−0.557043
$772$	0	0
$773$	− 29.8866i	− 1.07495i	−0.843281	−	0.537474i	$-0.819379\pi$
$773$	− 29.8866i	− 1.07495i	0.843281	−	0.537474i	$-0.180621\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.8046i	0.459361i
$778$	0	0
$779$	−4.35025	−0.155864
$780$	0	0
$781$	−23.8631	−0.853890
$782$	0	0
$783$	− 0.324173i	− 0.0115850i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.38796i	0.120768i	0.998175	+	0.0603839i	$0.0192325\pi$
$787$	3.38796i	0.120768i	−0.998175	+	0.0603839i	$0.980768\pi$
$788$	0	0
$789$	−65.3394	−2.32615
$790$	0	0
$791$	28.0675	0.997966
$792$	0	0
$793$	7.73555i	0.274697i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 49.5296i	− 1.75443i	−0.480098	−	0.877215i	$-0.659399\pi$
$797$	− 49.5296i	− 1.75443i	0.480098	−	0.877215i	$-0.340601\pi$
$798$	0	0
$799$	−1.05075	−0.0371728
$800$	0	0
$801$	45.3995	1.60411
$802$	0	0
$803$	− 52.5036i	− 1.85281i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.73181i	− 0.272173i
$808$	0	0
$809$	11.6042	0.407983	0.203992	−	0.978973i	$-0.434608\pi$
$809$	11.6042	0.407983	0.203992	+	0.978973i	$0.434608\pi$
$810$	0	0
$811$	29.1301	1.02290	0.511449	−	0.859314i	$-0.329109\pi$
$811$	29.1301	1.02290	0.511449	+	0.859314i	$0.329109\pi$
$812$	0	0
$813$	− 42.3670i	− 1.48588i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.55465i	− 0.159347i
$818$	0	0
$819$	5.69527	0.199009
$820$	0	0
$821$	4.06902	0.142010	0.0710049	−	0.997476i	$-0.477379\pi$
$821$	4.06902	0.142010	0.0710049	+	0.997476i	$0.477379\pi$
$822$	0	0
$823$	− 3.98430i	− 0.138884i	−0.997586	−	0.0694419i	$-0.977878\pi$
$823$	− 3.98430i	− 0.138884i	0.997586	−	0.0694419i	$-0.0221219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.6770i	0.892877i	0.894814	+	0.446438i	$0.147308\pi$
$827$	25.6770i	0.892877i	−0.894814	+	0.446438i	$0.852692\pi$
$828$	0	0
$829$	13.1533	0.456834	0.228417	−	0.973563i	$-0.426645\pi$
$829$	13.1533	0.456834	0.228417	+	0.973563i	$0.426645\pi$
$830$	0	0
$831$	37.1809	1.28979
$832$	0	0
$833$	− 3.11711i	− 0.108001i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.08986i	− 0.0376712i
$838$	0	0
$839$	−16.9948	−0.586724	−0.293362	−	0.956001i	$-0.594774\pi$
$839$	−16.9948	−0.586724	−0.293362	+	0.956001i	$0.594774\pi$
$840$	0	0
$841$	−28.5946	−0.986021
$842$	0	0
$843$	66.3902i	2.28660i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.95449i	− 0.101517i
$848$	0	0
$849$	−20.4816	−0.702925
$850$	0	0
$851$	−4.54942	−0.155952
$852$	0	0
$853$	− 29.1186i	− 0.997002i	−0.866889	−	0.498501i	$-0.833884\pi$
$853$	− 29.1186i	− 0.997002i	0.866889	−	0.498501i	$-0.166116\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.8486i	1.29288i	0.762964	+	0.646441i	$0.223744\pi$
$857$	37.8486i	1.29288i	−0.762964	+	0.646441i	$0.776256\pi$
$858$	0	0
$859$	−47.9019	−1.63439	−0.817196	−	0.576360i	$-0.804473\pi$
$859$	−47.9019	−1.63439	−0.817196	+	0.576360i	$0.804473\pi$
$860$	0	0
$861$	26.9907	0.919840
$862$	0	0
$863$	− 7.16412i	− 0.243870i	−0.992538	−	0.121935i	$-0.961090\pi$
$863$	− 7.16412i	− 0.243870i	0.992538	−	0.121935i	$-0.0389099\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.10930i	− 0.139559i
$868$	0	0
$869$	−26.6587	−0.904335
$870$	0	0
$871$	0.772091	0.0261613
$872$	0	0
$873$	19.6770i	0.665965i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 13.3801i	− 0.451813i	−0.974149	−	0.225906i	$-0.927466\pi$
$877$	− 13.3801i	− 0.451813i	0.974149	−	0.225906i	$-0.0725343\pi$
$878$	0	0
$879$	−50.1443	−1.69133
$880$	0	0
$881$	30.8579	1.03963	0.519814	−	0.854279i	$-0.326001\pi$
$881$	30.8579	1.03963	0.519814	+	0.854279i	$0.326001\pi$
$882$	0	0
$883$	− 1.20183i	− 0.0404449i	−0.999796	−	0.0202224i	$-0.993563\pi$
$883$	− 1.20183i	− 0.0404449i	0.999796	−	0.0202224i	$-0.00643744\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 23.3189i	− 0.782973i	−0.920184	−	0.391487i	$-0.871961\pi$
$887$	− 23.3189i	− 0.782973i	0.920184	−	0.391487i	$-0.128039\pi$
$888$	0	0
$889$	−18.6002	−0.623830
$890$	0	0
$891$	29.1313	0.975936
$892$	0	0
$893$	− 0.268189i	− 0.00897461i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.91794i	0.130816i
$898$	0	0
$899$	1.36295	0.0454571
$900$	0	0
$901$	31.2719	1.04182
$902$	0	0
$903$	28.2589i	0.940397i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 31.3137i	− 1.03975i	−0.854241	−	0.519877i	$-0.825978\pi$
$907$	− 31.3137i	− 1.03975i	0.854241	−	0.519877i	$-0.174022\pi$
$908$	0	0
$909$	13.9817	0.463745
$910$	0	0
$911$	20.6665	0.684712	0.342356	−	0.939570i	$-0.388775\pi$
$911$	20.6665	0.684712	0.342356	+	0.939570i	$0.388775\pi$
$912$	0	0
$913$	− 3.22267i	− 0.106655i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.3723i	0.639728i
$918$	0	0
$919$	43.1846	1.42453	0.712265	−	0.701911i	$-0.247670\pi$
$919$	43.1846	1.42453	0.712265	+	0.701911i	$0.247670\pi$
$920$	0	0
$921$	39.1626	1.29045
$922$	0	0
$923$	− 4.87766i	− 0.160550i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.5364i	0.575970i
$928$	0	0
$929$	40.6964	1.33521	0.667603	−	0.744517i	$-0.267320\pi$
$929$	40.6964	1.33521	0.667603	+	0.744517i	$0.267320\pi$
$930$	0	0
$931$	0.795598	0.0260747
$932$	0	0
$933$	− 60.2667i	− 1.97304i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.06379i	− 0.165427i	−0.996573	−	0.0827134i	$-0.973641\pi$
$937$	− 5.06379i	− 0.165427i	0.996573	−	0.0827134i	$-0.0263586\pi$
$938$	0	0
$939$	−44.4946	−1.45203
$940$	0	0
$941$	14.2876	0.465763	0.232882	−	0.972505i	$-0.425185\pi$
$941$	14.2876	0.465763	0.232882	+	0.972505i	$0.425185\pi$
$942$	0	0
$943$	9.58970i	0.312284i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.0843i	0.717643i	0.933406	+	0.358822i	$0.116821\pi$
$947$	22.0843i	0.717643i	−0.933406	+	0.358822i	$0.883179\pi$
$948$	0	0
$949$	10.7318	0.348369
$950$	0	0
$951$	−14.3905	−0.466645
$952$	0	0
$953$	7.92051i	0.256570i	0.991737	+	0.128285i	$0.0409473\pi$
$953$	7.92051i	0.256570i	−0.991737	+	0.128285i	$0.959053\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.53638i	− 0.178966i
$958$	0	0
$959$	22.5181	0.727148
$960$	0	0
$961$	−26.4178	−0.852186
$962$	0	0
$963$	− 0.772091i	− 0.0248803i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 49.1041i	− 1.57908i	−0.613699	−	0.789540i	$-0.710319\pi$
$967$	− 49.1041i	− 1.57908i	0.613699	−	0.789540i	$-0.289681\pi$
$968$	0	0
$969$	−9.75905	−0.313506
$970$	0	0
$971$	23.2264	0.745371	0.372685	−	0.927958i	$-0.378437\pi$
$971$	23.2264	0.745371	0.372685	+	0.927958i	$0.378437\pi$
$972$	0	0
$973$	− 51.4905i	− 1.65071i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.2290i	0.999104i	0.866284	+	0.499552i	$0.166502\pi$
$977$	31.2290i	0.999104i	−0.866284	+	0.499552i	$0.833498\pi$
$978$	0	0
$979$	49.4581	1.58069
$980$	0	0
$981$	32.6404	1.04213
$982$	0	0
$983$	− 57.8251i	− 1.84433i	−0.386793	−	0.922167i	$-0.626417\pi$
$983$	− 57.8251i	− 1.84433i	0.386793	−	0.922167i	$-0.373583\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.66395i	0.0529642i
$988$	0	0
$989$	−10.0403	−0.319262
$990$	0	0
$991$	−11.4125	−0.362531	−0.181266	−	0.983434i	$-0.558019\pi$
$991$	−11.4125	−0.362531	−0.181266	+	0.983434i	$0.558019\pi$
$992$	0	0
$993$	2.77616i	0.0880988i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 15.3137i	− 0.484990i	−0.970153	−	0.242495i	$-0.922034\pi$
$997$	− 15.3137i	− 0.484990i	0.970153	−	0.242495i	$-0.0779658\pi$
$998$	0	0
$999$	1.05075	0.0332442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.m.3649.1		6
5.2	odd	4		3800.2.a.u.1.1	✓	3
5.3	odd	4		3800.2.a.v.1.3	yes	3
5.4	even	2	inner	3800.2.d.m.3649.6		6
20.3	even	4		7600.2.a.bu.1.1		3
20.7	even	4		7600.2.a.bt.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.1	✓	3	5.2	odd	4
3800.2.a.v.1.3	yes	3	5.3	odd	4
3800.2.d.m.3649.1		6	1.1	even	1	trivial
3800.2.d.m.3649.6		6	5.4	even	2	inner
7600.2.a.bt.1.3		3	20.7	even	4
7600.2.a.bu.1.1		3	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)

\(f(q)\)	\(=\)	\(q-2.49086i q^{3} -2.49086i q^{7} -3.20440 q^{9} -3.49086 q^{11} -0.713538i q^{13} -3.91794i q^{17} +1.00000 q^{19} -6.20440 q^{21} -2.20440i q^{23} +0.509136i q^{27} -0.636712 q^{29} -2.14061 q^{31} +8.69527i q^{33} -2.06379i q^{37} -1.77733 q^{39} -4.35025 q^{41} -4.55465i q^{43} -0.268189i q^{47} +0.795598 q^{49} -9.75905 q^{51} +7.98173i q^{53} -2.49086i q^{57} +2.57292 q^{59} -10.8411 q^{61} +7.98173i q^{63} +1.08206i q^{67} -5.49086 q^{69} +6.83588 q^{71} +15.0403i q^{73} +8.69527i q^{77} +7.63671 q^{79} -8.34502 q^{81} +0.923174i q^{83} +1.58596i q^{87} -14.1679 q^{89} -1.77733 q^{91} +5.33198i q^{93} -6.14061i q^{97} +11.1861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} - 6 q^{11} + 6 q^{19} - 20 q^{21} + 2 q^{29} - 6 q^{31} + 2 q^{39} - 18 q^{41} + 22 q^{49} - 16 q^{51} + 20 q^{59} - 42 q^{61} - 18 q^{69} + 2 q^{71} + 40 q^{79} - 26 q^{81} - 8 q^{89} + 2 q^{91}+ \cdots + 20 q^{99}+O(q^{100}) \) 6 * q - 2 * q^9 - 6 * q^11 + 6 * q^19 - 20 * q^21 + 2 * q^29 - 6 * q^31 + 2 * q^39 - 18 * q^41 + 22 * q^49 - 16 * q^51 + 20 * q^59 - 42 * q^61 - 18 * q^69 + 2 * q^71 + 40 * q^79 - 26 * q^81 - 8 * q^89 + 2 * q^91 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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