LMFDB - Embedded newform 7600.2.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.a (trivial)

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:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49086 q^{3} +2.49086 q^{7} +3.20440 q^{9} +O(q^{10})$	$q-2.49086 q^{3} +2.49086 q^{7} +3.20440 q^{9} +3.49086 q^{11} +0.713538 q^{13} -3.91794 q^{17} +1.00000 q^{19} -6.20440 q^{21} -2.20440 q^{23} -0.509136 q^{27} +0.636712 q^{29} +2.14061 q^{31} -8.69527 q^{33} -2.06379 q^{37} -1.77733 q^{39} -4.35025 q^{41} -4.55465 q^{43} +0.268189 q^{47} -0.795598 q^{49} +9.75905 q^{51} -7.98173 q^{53} -2.49086 q^{57} +2.57292 q^{59} -10.8411 q^{61} +7.98173 q^{63} -1.08206 q^{67} +5.49086 q^{69} -6.83588 q^{71} -15.0403 q^{73} +8.69527 q^{77} +7.63671 q^{79} -8.34502 q^{81} +0.923174 q^{83} -1.58596 q^{87} +14.1679 q^{89} +1.77733 q^{91} -5.33198 q^{93} -6.14061 q^{97} +11.1861 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{9}+O(q^{10}) $ 3 * q + q^9	$ 3 q + q^{9} + 3 q^{11} + q^{13} - 2 q^{17} + 3 q^{19} - 10 q^{21} + 2 q^{23} - 9 q^{27} - q^{29} + 3 q^{31} - 10 q^{33} - q^{37} + q^{39} - 9 q^{41} - q^{43} - 13 q^{47} - 11 q^{49} + 8 q^{51} - 9 q^{53} + 10 q^{59} - 21 q^{61} + 9 q^{63} - 13 q^{67} + 9 q^{69} - q^{71} - 17 q^{73} + 10 q^{77} + 20 q^{79} - 13 q^{81} + q^{83} - 14 q^{87} + 4 q^{89} - q^{91} + 3 q^{93} - 15 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q + q^9 + 3 * q^11 + q^13 - 2 * q^17 + 3 * q^19 - 10 * q^21 + 2 * q^23 - 9 * q^27 - q^29 + 3 * q^31 - 10 * q^33 - q^37 + q^39 - 9 * q^41 - q^43 - 13 * q^47 - 11 * q^49 + 8 * q^51 - 9 * q^53 + 10 * q^59 - 21 * q^61 + 9 * q^63 - 13 * q^67 + 9 * q^69 - q^71 - 17 * q^73 + 10 * q^77 + 20 * q^79 - 13 * q^81 + q^83 - 14 * q^87 + 4 * q^89 - q^91 + 3 * q^93 - 15 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.49086	−1.43810	−0.719050	−	0.694958i	$-0.755423\pi$
$3$	−2.49086	−1.43810	−0.719050	+	0.694958i	$0.755423\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.49086	0.941458	0.470729	−	0.882278i	$-0.343991\pi$
$7$	2.49086	0.941458	0.470729	+	0.882278i	$0.343991\pi$
$8$	0	0
$9$	3.20440	1.06813
$10$	0	0
$11$	3.49086	1.05253	0.526267	−	0.850319i	$-0.323591\pi$
$11$	3.49086	1.05253	0.526267	+	0.850319i	$0.323591\pi$
$12$	0	0
$13$	0.713538	0.197900	0.0989499	−	0.995092i	$-0.468452\pi$
$13$	0.713538	0.197900	0.0989499	+	0.995092i	$0.468452\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.91794	−0.950240	−0.475120	−	0.879921i	$-0.657595\pi$
$17$	−3.91794	−0.950240	−0.475120	+	0.879921i	$0.657595\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.20440	−1.35391
$22$	0	0
$23$	−2.20440	−0.459649	−0.229825	−	0.973232i	$-0.573815\pi$
$23$	−2.20440	−0.459649	−0.229825	+	0.973232i	$0.573815\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−0.509136	−0.0979833
$28$	0	0
$29$	0.636712	0.118234	0.0591172	−	0.998251i	$-0.481171\pi$
$29$	0.636712	0.118234	0.0591172	+	0.998251i	$0.481171\pi$
$30$	0	0
$31$	2.14061	0.384466	0.192233	−	0.981349i	$-0.438427\pi$
$31$	2.14061	0.384466	0.192233	+	0.981349i	$0.438427\pi$
$32$	0	0
$33$	−8.69527	−1.51365
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.06379	−0.339285	−0.169642	−	0.985506i	$-0.554261\pi$
$37$	−2.06379	−0.339285	−0.169642	+	0.985506i	$0.554261\pi$
$38$	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
$43$	−4.55465	−0.694578	−0.347289	−	0.937758i	$-0.612898\pi$
$43$	−4.55465	−0.694578	−0.347289	+	0.937758i	$0.612898\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.268189	0.0391194	0.0195597	−	0.999809i	$-0.493774\pi$
$47$	0.268189	0.0391194	0.0195597	+	0.999809i	$0.493774\pi$
$48$	0	0
$49$	−0.795598	−0.113657
$50$	0	0
$51$	9.75905	1.36654
$52$	0	0
$53$	−7.98173	−1.09637	−0.548187	−	0.836356i	$-0.684682\pi$
$53$	−7.98173	−1.09637	−0.548187	+	0.836356i	$0.684682\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.49086	−0.329923
$58$	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
$63$	7.98173	1.00560
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.08206	−0.132195	−0.0660974	−	0.997813i	$-0.521055\pi$
$67$	−1.08206	−0.132195	−0.0660974	+	0.997813i	$0.521055\pi$
$68$	0	0
$69$	5.49086	0.661022
$70$	0	0
$71$	−6.83588	−0.811270	−0.405635	−	0.914035i	$-0.632950\pi$
$71$	−6.83588	−0.811270	−0.405635	+	0.914035i	$0.632950\pi$
$72$	0	0
$73$	−15.0403	−1.76033	−0.880166	−	0.474666i	$-0.842569\pi$
$73$	−15.0403	−1.76033	−0.880166	+	0.474666i	$0.842569\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.69527	0.990917
$78$	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
$83$	0.923174	0.101332	0.0506658	−	0.998716i	$-0.483866\pi$
$83$	0.923174	0.101332	0.0506658	+	0.998716i	$0.483866\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.58596	−0.170033
$88$	0	0
$89$	14.1679	1.50179	0.750895	−	0.660422i	$-0.229622\pi$
$89$	14.1679	1.50179	0.750895	+	0.660422i	$0.229622\pi$
$90$	0	0
$91$	1.77733	0.186314
$92$	0	0
$93$	−5.33198	−0.552900
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.14061	−0.623485	−0.311742	−	0.950167i	$-0.600913\pi$
$97$	−6.14061	−0.623485	−0.311742	+	0.950167i	$0.600913\pi$
$98$	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
$103$	−5.47259	−0.539230	−0.269615	−	0.962968i	$-0.586897\pi$
$103$	−5.47259	−0.539230	−0.269615	+	0.962968i	$0.586897\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.240947	−0.0232932	−0.0116466	−	0.999932i	$-0.503707\pi$
$107$	−0.240947	−0.0232932	−0.0116466	+	0.999932i	$0.503707\pi$
$108$	0	0
$109$	10.1861	0.975654	0.487827	−	0.872940i	$-0.337790\pi$
$109$	10.1861	0.975654	0.487827	+	0.872940i	$0.337790\pi$
$110$	0	0
$111$	5.14061	0.487925
$112$	0	0
$113$	−11.2682	−1.06002	−0.530011	−	0.847991i	$-0.677812\pi$
$113$	−11.2682	−1.06002	−0.530011	+	0.847991i	$0.677812\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.28646	0.211383
$118$	0	0
$119$	−9.75905	−0.894611
$120$	0	0
$121$	1.18613	0.107830
$122$	0	0
$123$	10.8359	0.977038
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.46736	0.662621	0.331310	−	0.943522i	$-0.392509\pi$
$127$	7.46736	0.662621	0.331310	+	0.943522i	$0.392509\pi$
$128$	0	0
$129$	11.3450	0.998873
$130$	0	0
$131$	7.77733	0.679508	0.339754	−	0.940514i	$-0.389656\pi$
$131$	7.77733	0.679508	0.339754	+	0.940514i	$0.389656\pi$
$132$	0	0
$133$	2.49086	0.215985
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.04028	0.772363	0.386182	−	0.922423i	$-0.373794\pi$
$137$	9.04028	0.772363	0.386182	+	0.922423i	$0.373794\pi$
$138$	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
$143$	2.49086	0.208296
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.98173	0.163450
$148$	0	0
$149$	−12.7591	−1.04526	−0.522631	−	0.852559i	$-0.675050\pi$
$149$	−12.7591	−1.04526	−0.522631	+	0.852559i	$0.675050\pi$
$150$	0	0
$151$	6.28646	0.511585	0.255793	−	0.966732i	$-0.417664\pi$
$151$	6.28646	0.511585	0.255793	+	0.966732i	$0.417664\pi$
$152$	0	0
$153$	−12.5547	−1.01498
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.4308	1.71036	0.855182	−	0.518327i	$-0.173445\pi$
$157$	21.4308	1.71036	0.855182	+	0.518327i	$0.173445\pi$
$158$	0	0
$159$	19.8814	1.57670
$160$	0	0
$161$	−5.49086	−0.432741
$162$	0	0
$163$	−18.1041	−1.41802	−0.709010	−	0.705198i	$-0.750858\pi$
$163$	−18.1041	−1.41802	−0.709010	+	0.705198i	$0.750858\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.3502	−0.955691	−0.477846	−	0.878444i	$-0.658582\pi$
$167$	−12.3502	−0.955691	−0.477846	+	0.878444i	$0.658582\pi$
$168$	0	0
$169$	−12.4909	−0.960836
$170$	0	0
$171$	3.20440	0.245047
$172$	0	0
$173$	7.60017	0.577830	0.288915	−	0.957355i	$-0.406706\pi$
$173$	7.60017	0.577830	0.288915	+	0.957355i	$0.406706\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.40880	−0.481715
$178$	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
$183$	27.0037	1.99617
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−13.6770	−1.00016
$188$	0	0
$189$	−1.26819	−0.0922472
$190$	0	0
$191$	−14.3085	−1.03532	−0.517662	−	0.855585i	$-0.673198\pi$
$191$	−14.3085	−1.03532	−0.517662	+	0.855585i	$0.673198\pi$
$192$	0	0
$193$	13.4491	0.968086	0.484043	−	0.875044i	$-0.339168\pi$
$193$	13.4491	0.968086	0.484043	+	0.875044i	$0.339168\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.79036	−0.198805	−0.0994026	−	0.995047i	$-0.531693\pi$
$197$	−2.79036	−0.198805	−0.0994026	+	0.995047i	$0.531693\pi$
$198$	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
$203$	1.58596	0.111313
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.06379	−0.490967
$208$	0	0
$209$	3.49086	0.241468
$210$	0	0
$211$	24.2264	1.66781	0.833907	−	0.551904i	$-0.186099\pi$
$211$	24.2264	1.66781	0.833907	+	0.551904i	$0.186099\pi$
$212$	0	0
$213$	17.0272	1.16669
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.33198	0.361958
$218$	0	0
$219$	37.4633	2.53153
$220$	0	0
$221$	−2.79560	−0.188052
$222$	0	0
$223$	−16.9269	−1.13351	−0.566755	−	0.823886i	$-0.691801\pi$
$223$	−16.9269	−1.13351	−0.566755	+	0.823886i	$0.691801\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.6315	−0.705636	−0.352818	−	0.935692i	$-0.614777\pi$
$227$	−10.6315	−0.705636	−0.352818	+	0.935692i	$0.614777\pi$
$228$	0	0
$229$	8.41404	0.556015	0.278008	−	0.960579i	$-0.410326\pi$
$229$	8.41404	0.556015	0.278008	+	0.960579i	$0.410326\pi$
$230$	0	0
$231$	−21.6587	−1.42504
$232$	0	0
$233$	−9.65498	−0.632519	−0.316260	−	0.948673i	$-0.602427\pi$
$233$	−9.65498	−0.632519	−0.316260	+	0.948673i	$0.602427\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−19.0220	−1.23561
$238$	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
$243$	22.3137	1.43142
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.713538	0.0454013
$248$	0	0
$249$	−2.29950	−0.145725
$250$	0	0
$251$	24.9034	1.57189	0.785944	−	0.618297i	$-0.212177\pi$
$251$	24.9034	1.57189	0.785944	+	0.618297i	$0.212177\pi$
$252$	0	0
$253$	−7.69527	−0.483797
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.20964	−0.387346	−0.193673	−	0.981066i	$-0.562040\pi$
$257$	−6.20964	−0.387346	−0.193673	+	0.981066i	$0.562040\pi$
$258$	0	0
$259$	−5.14061	−0.319422
$260$	0	0
$261$	2.04028	0.126290
$262$	0	0
$263$	−26.2316	−1.61751	−0.808756	−	0.588144i	$-0.799859\pi$
$263$	−26.2316	−1.61751	−0.808756	+	0.588144i	$0.799859\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−35.2902	−2.15972
$268$	0	0
$269$	−3.10407	−0.189258	−0.0946292	−	0.995513i	$-0.530167\pi$
$269$	−3.10407	−0.189258	−0.0946292	+	0.995513i	$0.530167\pi$
$270$	0	0
$271$	−17.0090	−1.03322	−0.516611	−	0.856220i	$-0.672807\pi$
$271$	−17.0090	−1.03322	−0.516611	+	0.856220i	$0.672807\pi$
$272$	0	0
$273$	−4.42708	−0.267939
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.9269	0.896871	0.448436	−	0.893815i	$-0.351981\pi$
$277$	14.9269	0.896871	0.448436	+	0.893815i	$0.351981\pi$
$278$	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
$283$	−8.22267	−0.488787	−0.244394	−	0.969676i	$-0.578589\pi$
$283$	−8.22267	−0.488787	−0.244394	+	0.969676i	$0.578589\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.8359	−0.639622
$288$	0	0
$289$	−1.64975	−0.0970441
$290$	0	0
$291$	15.2954	0.896634
$292$	0	0
$293$	20.1313	1.17608	0.588042	−	0.808830i	$-0.299899\pi$
$293$	20.1313	1.17608	0.588042	+	0.808830i	$0.299899\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.77733	−0.103131
$298$	0	0
$299$	−1.57292	−0.0909646
$300$	0	0
$301$	−11.3450	−0.653916
$302$	0	0
$303$	10.8684	0.624371
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.7225	−0.897331	−0.448665	−	0.893700i	$-0.648101\pi$
$307$	−15.7225	−0.897331	−0.448665	+	0.893700i	$0.648101\pi$
$308$	0	0
$309$	13.6315	0.775468
$310$	0	0
$311$	−24.1951	−1.37198	−0.685989	−	0.727612i	$-0.740630\pi$
$311$	−24.1951	−1.37198	−0.685989	+	0.727612i	$0.740630\pi$
$312$	0	0
$313$	17.8631	1.00968	0.504842	−	0.863212i	$-0.331551\pi$
$313$	17.8631	1.00968	0.504842	+	0.863212i	$0.331551\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.77733	−0.324487	−0.162243	−	0.986751i	$-0.551873\pi$
$317$	−5.77733	−0.324487	−0.162243	+	0.986751i	$0.551873\pi$
$318$	0	0
$319$	2.22267	0.124446
$320$	0	0
$321$	0.600166	0.0334980
$322$	0	0
$323$	−3.91794	−0.218000
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−25.3723	−1.40309
$328$	0	0
$329$	0.668023	0.0368293
$330$	0	0
$331$	1.11454	0.0612605	0.0306303	−	0.999531i	$-0.490249\pi$
$331$	1.11454	0.0612605	0.0306303	+	0.999531i	$0.490249\pi$
$332$	0	0
$333$	−6.61320	−0.362401
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.54161	0.0839770	0.0419885	−	0.999118i	$-0.486631\pi$
$337$	1.54161	0.0839770	0.0419885	+	0.999118i	$0.486631\pi$
$338$	0	0
$339$	28.0675	1.52442
$340$	0	0
$341$	7.47259	0.404663
$342$	0	0
$343$	−19.4178	−1.04846
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.3007	−1.57294	−0.786471	−	0.617627i	$-0.788094\pi$
$347$	−29.3007	−1.57294	−0.786471	+	0.617627i	$0.788094\pi$
$348$	0	0
$349$	−8.09626	−0.433383	−0.216692	−	0.976240i	$-0.569527\pi$
$349$	−8.09626	−0.433383	−0.216692	+	0.976240i	$0.569527\pi$
$350$	0	0
$351$	−0.363288	−0.0193909
$352$	0	0
$353$	−5.17716	−0.275552	−0.137776	−	0.990463i	$-0.543995\pi$
$353$	−5.17716	−0.275552	−0.137776	+	0.990463i	$0.543995\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.3085	1.28654
$358$	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
$363$	−2.95449	−0.155070
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.3398	−0.957329	−0.478664	−	0.877998i	$-0.658879\pi$
$367$	−18.3398	−0.957329	−0.478664	+	0.877998i	$0.658879\pi$
$368$	0	0
$369$	−13.9399	−0.725685
$370$	0	0
$371$	−19.8814	−1.03219
$372$	0	0
$373$	−32.8956	−1.70327	−0.851635	−	0.524136i	$-0.824388\pi$
$373$	−32.8956	−1.70327	−0.851635	+	0.524136i	$0.824388\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.454318	0.0233986
$378$	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
$383$	6.14061	0.313771	0.156885	−	0.987617i	$-0.449855\pi$
$383$	6.14061	0.313771	0.156885	+	0.987617i	$0.449855\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.5949	−0.741902
$388$	0	0
$389$	36.1899	1.83490	0.917449	−	0.397852i	$-0.130244\pi$
$389$	36.1899	1.83490	0.917449	+	0.397852i	$0.130244\pi$
$390$	0	0
$391$	8.63671	0.436777
$392$	0	0
$393$	−19.3723	−0.977201
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−32.3775	−1.62498	−0.812490	−	0.582975i	$-0.801888\pi$
$397$	−32.3775	−1.62498	−0.812490	+	0.582975i	$0.801888\pi$
$398$	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
$403$	1.52741	0.0760857
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.20440	−0.357109
$408$	0	0
$409$	−26.2954	−1.30023	−0.650113	−	0.759838i	$-0.725278\pi$
$409$	−26.2954	−1.30023	−0.650113	+	0.759838i	$0.725278\pi$
$410$	0	0
$411$	−22.5181	−1.11074
$412$	0	0
$413$	6.40880	0.315357
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−51.4905	−2.52150
$418$	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
$423$	0.859386	0.0417848
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−27.0037	−1.30680
$428$	0	0
$429$	−6.20440	−0.299551
$430$	0	0
$431$	10.4621	0.503943	0.251971	−	0.967735i	$-0.418921\pi$
$431$	10.4621	0.503943	0.251971	+	0.967735i	$0.418921\pi$
$432$	0	0
$433$	4.21487	0.202554	0.101277	−	0.994858i	$-0.467707\pi$
$433$	4.21487	0.202554	0.101277	+	0.994858i	$0.467707\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.20440	−0.105451
$438$	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
$443$	13.7303	0.652347	0.326173	−	0.945310i	$-0.394241\pi$
$443$	13.7303	0.652347	0.326173	+	0.945310i	$0.394241\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	31.7811	1.50319
$448$	0	0
$449$	−18.6039	−0.877972	−0.438986	−	0.898494i	$-0.644662\pi$
$449$	−18.6039	−0.877972	−0.438986	+	0.898494i	$0.644662\pi$
$450$	0	0
$451$	−15.1861	−0.715087
$452$	0	0
$453$	−15.6587	−0.735711
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.750083	−0.0350874	−0.0175437	−	0.999846i	$-0.505585\pi$
$457$	−0.750083	−0.0350874	−0.0175437	+	0.999846i	$0.505585\pi$
$458$	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
$463$	−13.4323	−0.624252	−0.312126	−	0.950041i	$-0.601041\pi$
$463$	−13.4323	−0.624252	−0.312126	+	0.950041i	$0.601041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.1757	1.11872	0.559358	−	0.828926i	$-0.311048\pi$
$467$	24.1757	1.11872	0.559358	+	0.828926i	$0.311048\pi$
$468$	0	0
$469$	−2.69527	−0.124456
$470$	0	0
$471$	−53.3812	−2.45968
$472$	0	0
$473$	−15.8997	−0.731067
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.5767	−1.17107
$478$	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
$483$	13.6770	0.622325
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.4125	0.879666	0.439833	−	0.898080i	$-0.355038\pi$
$487$	19.4125	0.879666	0.439833	+	0.898080i	$0.355038\pi$
$488$	0	0
$489$	45.0948	2.03926
$490$	0	0
$491$	15.7408	0.710371	0.355186	−	0.934796i	$-0.384418\pi$
$491$	15.7408	0.710371	0.355186	+	0.934796i	$0.384418\pi$
$492$	0	0
$493$	−2.49460	−0.112351
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.0272	−0.763776
$498$	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
$503$	−40.8161	−1.81990	−0.909950	−	0.414718i	$-0.863880\pi$
$503$	−40.8161	−1.81990	−0.909950	+	0.414718i	$0.863880\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.1130	1.38178
$508$	0	0
$509$	1.56919	0.0695531	0.0347765	−	0.999395i	$-0.488928\pi$
$509$	1.56919	0.0695531	0.0347765	+	0.999395i	$0.488928\pi$
$510$	0	0
$511$	−37.4633	−1.65728
$512$	0	0
$513$	−0.509136	−0.0224789
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.936212	0.0411746
$518$	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
$523$	33.3630	1.45886	0.729430	−	0.684055i	$-0.239785\pi$
$523$	33.3630	1.45886	0.729430	+	0.684055i	$0.239785\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.38680	−0.365335
$528$	0	0
$529$	−18.1406	−0.788722
$530$	0	0
$531$	8.24468	0.357789
$532$	0	0
$533$	−3.10407	−0.134452
$534$	0	0
$535$	0	0
$536$	0	0
$537$	18.1365	0.782650
$538$	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
$543$	−16.9362	−0.726802
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.5416	1.13484	0.567419	−	0.823429i	$-0.307942\pi$
$547$	26.5416	1.13484	0.567419	+	0.823429i	$0.307942\pi$
$548$	0	0
$549$	−34.7393	−1.48264
$550$	0	0
$551$	0.636712	0.0271248
$552$	0	0
$553$	19.0220	0.808898
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.89967	−0.122863	−0.0614314	−	0.998111i	$-0.519567\pi$
$557$	−2.89967	−0.122863	−0.0614314	+	0.998111i	$0.519567\pi$
$558$	0	0
$559$	−3.24992	−0.137457
$560$	0	0
$561$	34.0675	1.43833
$562$	0	0
$563$	36.0817	1.52066	0.760332	−	0.649535i	$-0.225036\pi$
$563$	36.0817	1.52066	0.760332	+	0.649535i	$0.225036\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−20.7863	−0.872942
$568$	0	0
$569$	−39.3540	−1.64980	−0.824902	−	0.565275i	$-0.808770\pi$
$569$	−39.3540	−1.64980	−0.824902	+	0.565275i	$0.808770\pi$
$570$	0	0
$571$	−17.4596	−0.730660	−0.365330	−	0.930878i	$-0.619044\pi$
$571$	−17.4596	−0.730660	−0.365330	+	0.930878i	$0.619044\pi$
$572$	0	0
$573$	35.6404	1.48890
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−21.1548	−0.880687	−0.440343	−	0.897829i	$-0.645143\pi$
$577$	−21.1548	−0.880687	−0.440343	+	0.897829i	$0.645143\pi$
$578$	0	0
$579$	−33.4998	−1.39221
$580$	0	0
$581$	2.29950	0.0953994
$582$	0	0
$583$	−27.8631	−1.15397
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.76429	0.361741	0.180870	−	0.983507i	$-0.442109\pi$
$587$	8.76429	0.361741	0.180870	+	0.983507i	$0.442109\pi$
$588$	0	0
$589$	2.14061	0.0882025
$590$	0	0
$591$	6.95042	0.285902
$592$	0	0
$593$	25.0127	1.02715	0.513574	−	0.858045i	$-0.328321\pi$
$593$	25.0127	1.02715	0.513574	+	0.858045i	$0.328321\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.2134	−0.909133
$598$	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
$603$	−3.46736	−0.141202
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.9672	1.58163	0.790815	−	0.612056i	$-0.209657\pi$
$607$	38.9672	1.58163	0.790815	+	0.612056i	$0.209657\pi$
$608$	0	0
$609$	−3.95042	−0.160079
$610$	0	0
$611$	0.191363	0.00774173
$612$	0	0
$613$	35.7680	1.44466	0.722328	−	0.691550i	$-0.243072\pi$
$613$	35.7680	1.44466	0.722328	+	0.691550i	$0.243072\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.2447	−0.815020	−0.407510	−	0.913201i	$-0.633603\pi$
$617$	−20.2447	−0.815020	−0.407510	+	0.913201i	$0.633603\pi$
$618$	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
$623$	35.2902	1.41387
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.69527	−0.347255
$628$	0	0
$629$	8.08580	0.322402
$630$	0	0
$631$	46.3212	1.84402	0.922008	−	0.387170i	$-0.126547\pi$
$631$	46.3212	1.84402	0.922008	+	0.387170i	$0.126547\pi$
$632$	0	0
$633$	−60.3447	−2.39849
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.567690	−0.0224927
$638$	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
$643$	−24.8542	−0.980152	−0.490076	−	0.871680i	$-0.663031\pi$
$643$	−24.8542	−0.980152	−0.490076	+	0.871680i	$0.663031\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.2589	1.74000	0.869998	−	0.493055i	$-0.164120\pi$
$647$	44.2589	1.74000	0.869998	+	0.493055i	$0.164120\pi$
$648$	0	0
$649$	8.98173	0.352564
$650$	0	0
$651$	−13.2812	−0.520532
$652$	0	0
$653$	0.113372	0.00443657	0.00221829	−	0.999998i	$-0.499294\pi$
$653$	0.113372	0.00443657	0.00221829	+	0.999998i	$0.499294\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−48.1951	−1.88027
$658$	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
$663$	6.96345	0.270438
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.40357	−0.0543464
$668$	0	0
$669$	42.1626	1.63010
$670$	0	0
$671$	−37.8448	−1.46098
$672$	0	0
$673$	29.2301	1.12674	0.563370	−	0.826205i	$-0.309505\pi$
$673$	29.2301	1.12674	0.563370	+	0.826205i	$0.309505\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.97276	0.152685	0.0763427	−	0.997082i	$-0.475676\pi$
$677$	3.97276	0.152685	0.0763427	+	0.997082i	$0.475676\pi$
$678$	0	0
$679$	−15.2954	−0.586985
$680$	0	0
$681$	26.4816	1.01478
$682$	0	0
$683$	11.8083	0.451832	0.225916	−	0.974147i	$-0.427462\pi$
$683$	11.8083	0.451832	0.225916	+	0.974147i	$0.427462\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−20.9582	−0.799606
$688$	0	0
$689$	−5.69527	−0.216972
$690$	0	0
$691$	−3.14585	−0.119674	−0.0598369	−	0.998208i	$-0.519058\pi$
$691$	−3.14585	−0.119674	−0.0598369	+	0.998208i	$0.519058\pi$
$692$	0	0
$693$	27.8631	1.05843
$694$	0	0
$695$	0	0
$696$	0	0
$697$	17.0440	0.645588
$698$	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
$703$	−2.06379	−0.0778372
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.8684	−0.408747
$708$	0	0
$709$	42.1951	1.58467	0.792335	−	0.610086i	$-0.208865\pi$
$709$	42.1951	1.58467	0.792335	+	0.610086i	$0.208865\pi$
$710$	0	0
$711$	24.4711	0.917738
$712$	0	0
$713$	−4.71877	−0.176719
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.985796	0.0368152
$718$	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
$723$	−5.12758	−0.190697
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.3189	0.642324	0.321162	−	0.947024i	$-0.395927\pi$
$727$	17.3189	0.642324	0.321162	+	0.947024i	$0.395927\pi$
$728$	0	0
$729$	−30.5453	−1.13131
$730$	0	0
$731$	17.8448	0.660016
$732$	0	0
$733$	22.4271	0.828363	0.414181	−	0.910194i	$-0.364068\pi$
$733$	22.4271	0.828363	0.414181	+	0.910194i	$0.364068\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.77733	−0.139140
$738$	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
$743$	−28.8423	−1.05812	−0.529060	−	0.848584i	$-0.677455\pi$
$743$	−28.8423	−1.05812	−0.529060	+	0.848584i	$0.677455\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.95822	0.108236
$748$	0	0
$749$	−0.600166	−0.0219296
$750$	0	0
$751$	7.79560	0.284465	0.142233	−	0.989833i	$-0.454572\pi$
$751$	7.79560	0.284465	0.142233	+	0.989833i	$0.454572\pi$
$752$	0	0
$753$	−62.0310	−2.26053
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.2227	−1.06212	−0.531058	−	0.847335i	$-0.678205\pi$
$757$	−29.2227	−1.06212	−0.531058	+	0.847335i	$0.678205\pi$
$758$	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
$763$	25.3723	0.918537
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.83588	0.0662897
$768$	0	0
$769$	−3.33605	−0.120301	−0.0601504	−	0.998189i	$-0.519158\pi$
$769$	−3.33605	−0.120301	−0.0601504	+	0.998189i	$0.519158\pi$
$770$	0	0
$771$	15.4674	0.557043
$772$	0	0
$773$	29.8866	1.07495	0.537474	−	0.843281i	$-0.319379\pi$
$773$	29.8866	1.07495	0.537474	+	0.843281i	$0.319379\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.8046	0.459361
$778$	0	0
$779$	−4.35025	−0.155864
$780$	0	0
$781$	−23.8631	−0.853890
$782$	0	0
$783$	−0.324173	−0.0115850
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.38796	−0.120768	−0.0603839	−	0.998175i	$-0.519232\pi$
$787$	−3.38796	−0.120768	−0.0603839	+	0.998175i	$0.519232\pi$
$788$	0	0
$789$	65.3394	2.32615
$790$	0	0
$791$	−28.0675	−0.997966
$792$	0	0
$793$	−7.73555	−0.274697
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.5296	−1.75443	−0.877215	−	0.480098i	$-0.840601\pi$
$797$	−49.5296	−1.75443	−0.877215	+	0.480098i	$0.840601\pi$
$798$	0	0
$799$	−1.05075	−0.0371728
$800$	0	0
$801$	45.3995	1.60411
$802$	0	0
$803$	−52.5036	−1.85281
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.73181	0.272173
$808$	0	0
$809$	−11.6042	−0.407983	−0.203992	−	0.978973i	$-0.565392\pi$
$809$	−11.6042	−0.407983	−0.203992	+	0.978973i	$0.565392\pi$
$810$	0	0
$811$	−29.1301	−1.02290	−0.511449	−	0.859314i	$-0.670891\pi$
$811$	−29.1301	−1.02290	−0.511449	+	0.859314i	$0.670891\pi$
$812$	0	0
$813$	42.3670	1.48588
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.55465	−0.159347
$818$	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
$823$	−3.98430	−0.138884	−0.0694419	−	0.997586i	$-0.522122\pi$
$823$	−3.98430	−0.138884	−0.0694419	+	0.997586i	$0.522122\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.6770	−0.892877	−0.446438	−	0.894814i	$-0.647308\pi$
$827$	−25.6770	−0.892877	−0.446438	+	0.894814i	$0.647308\pi$
$828$	0	0
$829$	−13.1533	−0.456834	−0.228417	−	0.973563i	$-0.573355\pi$
$829$	−13.1533	−0.456834	−0.228417	+	0.973563i	$0.573355\pi$
$830$	0	0
$831$	−37.1809	−1.28979
$832$	0	0
$833$	3.11711	0.108001
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.08986	−0.0376712
$838$	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
$843$	66.3902	2.28660
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.95449	0.101517
$848$	0	0
$849$	20.4816	0.702925
$850$	0	0
$851$	4.54942	0.155952
$852$	0	0
$853$	29.1186	0.997002	0.498501	−	0.866889i	$-0.333884\pi$
$853$	29.1186	0.997002	0.498501	+	0.866889i	$0.333884\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.8486	1.29288	0.646441	−	0.762964i	$-0.276256\pi$
$857$	37.8486	1.29288	0.646441	+	0.762964i	$0.276256\pi$
$858$	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
$863$	−7.16412	−0.243870	−0.121935	−	0.992538i	$-0.538910\pi$
$863$	−7.16412	−0.243870	−0.121935	+	0.992538i	$0.538910\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.10930	0.139559
$868$	0	0
$869$	26.6587	0.904335
$870$	0	0
$871$	−0.772091	−0.0261613
$872$	0	0
$873$	−19.6770	−0.665965
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.3801	−0.451813	−0.225906	−	0.974149i	$-0.572534\pi$
$877$	−13.3801	−0.451813	−0.225906	+	0.974149i	$0.572534\pi$
$878$	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
$883$	−1.20183	−0.0404449	−0.0202224	−	0.999796i	$-0.506437\pi$
$883$	−1.20183	−0.0404449	−0.0202224	+	0.999796i	$0.506437\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.3189	0.782973	0.391487	−	0.920184i	$-0.371961\pi$
$887$	23.3189	0.782973	0.391487	+	0.920184i	$0.371961\pi$
$888$	0	0
$889$	18.6002	0.623830
$890$	0	0
$891$	−29.1313	−0.975936
$892$	0	0
$893$	0.268189	0.00897461
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.91794	0.130816
$898$	0	0
$899$	1.36295	0.0454571
$900$	0	0
$901$	31.2719	1.04182
$902$	0	0
$903$	28.2589	0.940397
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31.3137	1.03975	0.519877	−	0.854241i	$-0.325978\pi$
$907$	31.3137	1.03975	0.519877	+	0.854241i	$0.325978\pi$
$908$	0	0
$909$	−13.9817	−0.463745
$910$	0	0
$911$	−20.6665	−0.684712	−0.342356	−	0.939570i	$-0.611225\pi$
$911$	−20.6665	−0.684712	−0.342356	+	0.939570i	$0.611225\pi$
$912$	0	0
$913$	3.22267	0.106655
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.3723	0.639728
$918$	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
$923$	−4.87766	−0.160550
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.5364	−0.575970
$928$	0	0
$929$	−40.6964	−1.33521	−0.667603	−	0.744517i	$-0.732680\pi$
$929$	−40.6964	−1.33521	−0.667603	+	0.744517i	$0.732680\pi$
$930$	0	0
$931$	−0.795598	−0.0260747
$932$	0	0
$933$	60.2667	1.97304
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5.06379	−0.165427	−0.0827134	−	0.996573i	$-0.526359\pi$
$937$	−5.06379	−0.165427	−0.0827134	+	0.996573i	$0.526359\pi$
$938$	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
$943$	9.58970	0.312284
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.0843	−0.717643	−0.358822	−	0.933406i	$-0.616821\pi$
$947$	−22.0843	−0.717643	−0.358822	+	0.933406i	$0.616821\pi$
$948$	0	0
$949$	−10.7318	−0.348369
$950$	0	0
$951$	14.3905	0.466645
$952$	0	0
$953$	−7.92051	−0.256570	−0.128285	−	0.991737i	$-0.540947\pi$
$953$	−7.92051	−0.256570	−0.128285	+	0.991737i	$0.540947\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.53638	−0.178966
$958$	0	0
$959$	22.5181	0.727148
$960$	0	0
$961$	−26.4178	−0.852186
$962$	0	0
$963$	−0.772091	−0.0248803
$964$	0	0
$965$	0	0
$966$	0	0
$967$	49.1041	1.57908	0.789540	−	0.613699i	$-0.210319\pi$
$967$	49.1041	1.57908	0.789540	+	0.613699i	$0.210319\pi$
$968$	0	0
$969$	9.75905	0.313506
$970$	0	0
$971$	−23.2264	−0.745371	−0.372685	−	0.927958i	$-0.621563\pi$
$971$	−23.2264	−0.745371	−0.372685	+	0.927958i	$0.621563\pi$
$972$	0	0
$973$	51.4905	1.65071
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.2290	0.999104	0.499552	−	0.866284i	$-0.333498\pi$
$977$	31.2290	0.999104	0.499552	+	0.866284i	$0.333498\pi$
$978$	0	0
$979$	49.4581	1.58069
$980$	0	0
$981$	32.6404	1.04213
$982$	0	0
$983$	−57.8251	−1.84433	−0.922167	−	0.386793i	$-0.873583\pi$
$983$	−57.8251	−1.84433	−0.922167	+	0.386793i	$0.873583\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.66395	−0.0529642
$988$	0	0
$989$	10.0403	0.319262
$990$	0	0
$991$	11.4125	0.362531	0.181266	−	0.983434i	$-0.441981\pi$
$991$	11.4125	0.362531	0.181266	+	0.983434i	$0.441981\pi$
$992$	0	0
$993$	−2.77616	−0.0880988
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−15.3137	−0.484990	−0.242495	−	0.970153i	$-0.577966\pi$
$997$	−15.3137	−0.484990	−0.242495	+	0.970153i	$0.577966\pi$
$998$	0	0
$999$	1.05075	0.0332442

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.u.1.1	✓	3	20.19	odd	2
3800.2.a.v.1.3	yes	3	4.3	odd	2
3800.2.d.m.3649.1		6	20.7	even	4
3800.2.d.m.3649.6		6	20.3	even	4
7600.2.a.bt.1.3		3	5.4	even	2
7600.2.a.bu.1.1		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49086 q^{3} +2.49086 q^{7} +3.20440 q^{9} +O(q^{10})\)	\(q-2.49086 q^{3} +2.49086 q^{7} +3.20440 q^{9} +3.49086 q^{11} +0.713538 q^{13} -3.91794 q^{17} +1.00000 q^{19} -6.20440 q^{21} -2.20440 q^{23} -0.509136 q^{27} +0.636712 q^{29} +2.14061 q^{31} -8.69527 q^{33} -2.06379 q^{37} -1.77733 q^{39} -4.35025 q^{41} -4.55465 q^{43} +0.268189 q^{47} -0.795598 q^{49} +9.75905 q^{51} -7.98173 q^{53} -2.49086 q^{57} +2.57292 q^{59} -10.8411 q^{61} +7.98173 q^{63} -1.08206 q^{67} +5.49086 q^{69} -6.83588 q^{71} -15.0403 q^{73} +8.69527 q^{77} +7.63671 q^{79} -8.34502 q^{81} +0.923174 q^{83} -1.58596 q^{87} +14.1679 q^{89} +1.77733 q^{91} -5.33198 q^{93} -6.14061 q^{97} +11.1861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{9}+O(q^{10}) \) 3 * q + q^9	\( 3 q + q^{9} + 3 q^{11} + q^{13} - 2 q^{17} + 3 q^{19} - 10 q^{21} + 2 q^{23} - 9 q^{27} - q^{29} + 3 q^{31} - 10 q^{33} - q^{37} + q^{39} - 9 q^{41} - q^{43} - 13 q^{47} - 11 q^{49} + 8 q^{51} - 9 q^{53} + 10 q^{59} - 21 q^{61} + 9 q^{63} - 13 q^{67} + 9 q^{69} - q^{71} - 17 q^{73} + 10 q^{77} + 20 q^{79} - 13 q^{81} + q^{83} - 14 q^{87} + 4 q^{89} - q^{91} + 3 q^{93} - 15 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q + q^9 + 3 * q^11 + q^13 - 2 * q^17 + 3 * q^19 - 10 * q^21 + 2 * q^23 - 9 * q^27 - q^29 + 3 * q^31 - 10 * q^33 - q^37 + q^39 - 9 * q^41 - q^43 - 13 * q^47 - 11 * q^49 + 8 * q^51 - 9 * q^53 + 10 * q^59 - 21 * q^61 + 9 * q^63 - 13 * q^67 + 9 * q^69 - q^71 - 17 * q^73 + 10 * q^77 + 20 * q^79 - 13 * q^81 + q^83 - 14 * q^87 + 4 * q^89 - q^91 + 3 * q^93 - 15 * q^97 + 10 * q^99

Introduction

L-functions

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