LMFDB - Embedded newform 7500.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.70636$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.32440 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.32440 q^{7} +1.00000 q^{9} +0.584296 q^{11} -0.966262 q^{13} +2.32440 q^{17} -5.37899 q^{19} -4.32440 q^{21} +1.35813 q^{23} +1.00000 q^{27} -0.706362 q^{29} +8.48817 q^{31} +0.584296 q^{33} +6.11909 q^{37} -0.966262 q^{39} +11.0615 q^{41} +7.03076 q^{43} -9.06932 q^{47} +11.7004 q^{49} +2.32440 q^{51} -8.90571 q^{53} -5.37899 q^{57} -7.29250 q^{59} -4.81370 q^{61} -4.32440 q^{63} +0.376006 q^{67} +1.35813 q^{69} +10.5605 q^{71} -0.213375 q^{73} -2.52673 q^{77} -7.02594 q^{79} +1.00000 q^{81} -16.1113 q^{83} -0.706362 q^{87} -14.0197 q^{89} +4.17850 q^{91} +8.48817 q^{93} -15.7520 q^{97} +0.584296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - q^{11} - 5 q^{13} - 4 q^{17} - 5 q^{19} - 4 q^{21} - 9 q^{23} + 4 q^{27} + 6 q^{29} + 11 q^{31} - q^{33} - 2 q^{37} - 5 q^{39} + 6 q^{43} - 16 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} - 5 q^{57} + q^{59} - 22 q^{61} - 4 q^{63} - 36 q^{67} - 9 q^{69} + 20 q^{71} - 12 q^{73} + 11 q^{77} - 3 q^{79} + 4 q^{81} - 14 q^{83} + 6 q^{87} - 15 q^{89} - 10 q^{91} + 11 q^{93} + 12 q^{97} - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9 - q^11 - 5 * q^13 - 4 * q^17 - 5 * q^19 - 4 * q^21 - 9 * q^23 + 4 * q^27 + 6 * q^29 + 11 * q^31 - q^33 - 2 * q^37 - 5 * q^39 + 6 * q^43 - 16 * q^47 - 4 * q^49 - 4 * q^51 + 2 * q^53 - 5 * q^57 + q^59 - 22 * q^61 - 4 * q^63 - 36 * q^67 - 9 * q^69 + 20 * q^71 - 12 * q^73 + 11 * q^77 - 3 * q^79 + 4 * q^81 - 14 * q^83 + 6 * q^87 - 15 * q^89 - 10 * q^91 + 11 * q^93 + 12 * q^97 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.32440	−1.63447	−0.817234	−	0.576306i	$-0.804494\pi$
$7$	−4.32440	−1.63447	−0.817234	+	0.576306i	$0.804494\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.584296	0.176172	0.0880859	−	0.996113i	$-0.471925\pi$
$11$	0.584296	0.176172	0.0880859	+	0.996113i	$0.471925\pi$
$12$	0	0
$13$	−0.966262	−0.267993	−0.133996	−	0.990982i	$-0.542781\pi$
$13$	−0.966262	−0.267993	−0.133996	+	0.990982i	$0.542781\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.32440	0.563749	0.281874	−	0.959451i	$-0.409044\pi$
$17$	2.32440	0.563749	0.281874	+	0.959451i	$0.409044\pi$
$18$	0	0
$19$	−5.37899	−1.23402	−0.617012	−	0.786954i	$-0.711657\pi$
$19$	−5.37899	−1.23402	−0.617012	+	0.786954i	$0.711657\pi$
$20$	0	0
$21$	−4.32440	−0.943661
$22$	0	0
$23$	1.35813	0.283191	0.141595	−	0.989925i	$-0.454777\pi$
$23$	1.35813	0.283191	0.141595	+	0.989925i	$0.454777\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.706362	−0.131168	−0.0655841	−	0.997847i	$-0.520891\pi$
$29$	−0.706362	−0.131168	−0.0655841	+	0.997847i	$0.520891\pi$
$30$	0	0
$31$	8.48817	1.52452	0.762260	−	0.647271i	$-0.224090\pi$
$31$	8.48817	1.52452	0.762260	+	0.647271i	$0.224090\pi$
$32$	0	0
$33$	0.584296	0.101713
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.11909	1.00597	0.502986	−	0.864295i	$-0.332235\pi$
$37$	6.11909	1.00597	0.502986	+	0.864295i	$0.332235\pi$
$38$	0	0
$39$	−0.966262	−0.154726
$40$	0	0
$41$	11.0615	1.72752	0.863759	−	0.503905i	$-0.168104\pi$
$41$	11.0615	1.72752	0.863759	+	0.503905i	$0.168104\pi$
$42$	0	0
$43$	7.03076	1.07218	0.536090	−	0.844161i	$-0.319901\pi$
$43$	7.03076	1.07218	0.536090	+	0.844161i	$0.319901\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.06932	−1.32290	−0.661448	−	0.749991i	$-0.730058\pi$
$47$	−9.06932	−1.32290	−0.661448	+	0.749991i	$0.730058\pi$
$48$	0	0
$49$	11.7004	1.67149
$50$	0	0
$51$	2.32440	0.325481
$52$	0	0
$53$	−8.90571	−1.22329	−0.611647	−	0.791131i	$-0.709493\pi$
$53$	−8.90571	−1.22329	−0.611647	+	0.791131i	$0.709493\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.37899	−0.712464
$58$	0	0
$59$	−7.29250	−0.949403	−0.474701	−	0.880147i	$-0.657444\pi$
$59$	−7.29250	−0.949403	−0.474701	+	0.880147i	$0.657444\pi$
$60$	0	0
$61$	−4.81370	−0.616331	−0.308166	−	0.951333i	$-0.599715\pi$
$61$	−4.81370	−0.616331	−0.308166	+	0.951333i	$0.599715\pi$
$62$	0	0
$63$	−4.32440	−0.544823
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.376006	0.0459364	0.0229682	−	0.999736i	$-0.492688\pi$
$67$	0.376006	0.0459364	0.0229682	+	0.999736i	$0.492688\pi$
$68$	0	0
$69$	1.35813	0.163500
$70$	0	0
$71$	10.5605	1.25330	0.626648	−	0.779302i	$-0.284426\pi$
$71$	10.5605	1.25330	0.626648	+	0.779302i	$0.284426\pi$
$72$	0	0
$73$	−0.213375	−0.0249736	−0.0124868	−	0.999922i	$-0.503975\pi$
$73$	−0.213375	−0.0249736	−0.0124868	+	0.999922i	$0.503975\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.52673	−0.287947
$78$	0	0
$79$	−7.02594	−0.790480	−0.395240	−	0.918578i	$-0.629339\pi$
$79$	−7.02594	−0.790480	−0.395240	+	0.918578i	$0.629339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.1113	−1.76844	−0.884222	−	0.467068i	$-0.845310\pi$
$83$	−16.1113	−1.76844	−0.884222	+	0.467068i	$0.845310\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.706362	−0.0757300
$88$	0	0
$89$	−14.0197	−1.48609	−0.743043	−	0.669243i	$-0.766618\pi$
$89$	−14.0197	−1.48609	−0.743043	+	0.669243i	$0.766618\pi$
$90$	0	0
$91$	4.17850	0.438026
$92$	0	0
$93$	8.48817	0.880182
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.7520	−1.59937	−0.799687	−	0.600417i	$-0.795001\pi$
$97$	−15.7520	−1.59937	−0.799687	+	0.600417i	$0.795001\pi$
$98$	0	0
$99$	0.584296	0.0587239
$100$	0	0
$101$	7.14178	0.710634	0.355317	−	0.934746i	$-0.384373\pi$
$101$	7.14178	0.710634	0.355317	+	0.934746i	$0.384373\pi$
$102$	0	0
$103$	−1.32553	−0.130609	−0.0653044	−	0.997865i	$-0.520802\pi$
$103$	−1.32553	−0.130609	−0.0653044	+	0.997865i	$0.520802\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.6796	−1.70915	−0.854573	−	0.519331i	$-0.826181\pi$
$107$	−17.6796	−1.70915	−0.854573	+	0.519331i	$0.826181\pi$
$108$	0	0
$109$	3.53978	0.339049	0.169524	−	0.985526i	$-0.445777\pi$
$109$	3.53978	0.339049	0.169524	+	0.985526i	$0.445777\pi$
$110$	0	0
$111$	6.11909	0.580798
$112$	0	0
$113$	−17.4882	−1.64515	−0.822574	−	0.568658i	$-0.807463\pi$
$113$	−17.4882	−1.64515	−0.822574	+	0.568658i	$0.807463\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.966262	−0.0893309
$118$	0	0
$119$	−10.0516	−0.921430
$120$	0	0
$121$	−10.6586	−0.968964
$122$	0	0
$123$	11.0615	0.997383
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.5060	−1.55341	−0.776705	−	0.629865i	$-0.783110\pi$
$127$	−17.5060	−1.55341	−0.776705	+	0.629865i	$0.783110\pi$
$128$	0	0
$129$	7.03076	0.619024
$130$	0	0
$131$	−18.8660	−1.64833	−0.824166	−	0.566349i	$-0.808355\pi$
$131$	−18.8660	−1.64833	−0.824166	+	0.566349i	$0.808355\pi$
$132$	0	0
$133$	23.2609	2.01697
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.2508	−1.13209	−0.566046	−	0.824374i	$-0.691528\pi$
$137$	−13.2508	−1.13209	−0.566046	+	0.824374i	$0.691528\pi$
$138$	0	0
$139$	−16.9026	−1.43366	−0.716829	−	0.697249i	$-0.754407\pi$
$139$	−16.9026	−1.43366	−0.716829	+	0.697249i	$0.754407\pi$
$140$	0	0
$141$	−9.06932	−0.763774
$142$	0	0
$143$	−0.564582	−0.0472128
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.7004	0.965033
$148$	0	0
$149$	12.1625	0.996388	0.498194	−	0.867066i	$-0.333997\pi$
$149$	12.1625	0.996388	0.498194	+	0.867066i	$0.333997\pi$
$150$	0	0
$151$	−9.84446	−0.801131	−0.400565	−	0.916268i	$-0.631186\pi$
$151$	−9.84446	−0.801131	−0.400565	+	0.916268i	$0.631186\pi$
$152$	0	0
$153$	2.32440	0.187916
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.4804	1.47489	0.737447	−	0.675405i	$-0.236031\pi$
$157$	18.4804	1.47489	0.737447	+	0.675405i	$0.236031\pi$
$158$	0	0
$159$	−8.90571	−0.706269
$160$	0	0
$161$	−5.87311	−0.462866
$162$	0	0
$163$	14.5368	1.13861	0.569305	−	0.822127i	$-0.307212\pi$
$163$	14.5368	1.13861	0.569305	+	0.822127i	$0.307212\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.49299	0.347678	0.173839	−	0.984774i	$-0.444383\pi$
$167$	4.49299	0.347678	0.173839	+	0.984774i	$0.444383\pi$
$168$	0	0
$169$	−12.0663	−0.928180
$170$	0	0
$171$	−5.37899	−0.411341
$172$	0	0
$173$	−1.99816	−0.151917	−0.0759586	−	0.997111i	$-0.524202\pi$
$173$	−1.99816	−0.151917	−0.0759586	+	0.997111i	$0.524202\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.29250	−0.548138
$178$	0	0
$179$	13.9793	1.04486	0.522431	−	0.852681i	$-0.325025\pi$
$179$	13.9793	1.04486	0.522431	+	0.852681i	$0.325025\pi$
$180$	0	0
$181$	13.5379	1.00627	0.503133	−	0.864209i	$-0.332180\pi$
$181$	13.5379	1.00627	0.503133	+	0.864209i	$0.332180\pi$
$182$	0	0
$183$	−4.81370	−0.355839
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.35813	0.0993166
$188$	0	0
$189$	−4.32440	−0.314554
$190$	0	0
$191$	2.54547	0.184184	0.0920920	−	0.995751i	$-0.470645\pi$
$191$	2.54547	0.184184	0.0920920	+	0.995751i	$0.470645\pi$
$192$	0	0
$193$	−5.60541	−0.403486	−0.201743	−	0.979438i	$-0.564661\pi$
$193$	−5.60541	−0.403486	−0.201743	+	0.979438i	$0.564661\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.85621	0.559732	0.279866	−	0.960039i	$-0.409710\pi$
$197$	7.85621	0.559732	0.279866	+	0.960039i	$0.409710\pi$
$198$	0	0
$199$	16.9970	1.20489	0.602443	−	0.798162i	$-0.294194\pi$
$199$	16.9970	1.20489	0.602443	+	0.798162i	$0.294194\pi$
$200$	0	0
$201$	0.376006	0.0265214
$202$	0	0
$203$	3.05459	0.214390
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.35813	0.0943969
$208$	0	0
$209$	−3.14292	−0.217400
$210$	0	0
$211$	−12.9553	−0.891881	−0.445940	−	0.895063i	$-0.647131\pi$
$211$	−12.9553	−0.891881	−0.445940	+	0.895063i	$0.647131\pi$
$212$	0	0
$213$	10.5605	0.723591
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−36.7062	−2.49178
$218$	0	0
$219$	−0.213375	−0.0144185
$220$	0	0
$221$	−2.24597	−0.151081
$222$	0	0
$223$	−18.9801	−1.27100	−0.635501	−	0.772100i	$-0.719207\pi$
$223$	−18.9801	−1.27100	−0.635501	+	0.772100i	$0.719207\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.385109	0.0255606	0.0127803	−	0.999918i	$-0.495932\pi$
$227$	0.385109	0.0255606	0.0127803	+	0.999918i	$0.495932\pi$
$228$	0	0
$229$	10.2685	0.678562	0.339281	−	0.940685i	$-0.389816\pi$
$229$	10.2685	0.678562	0.339281	+	0.940685i	$0.389816\pi$
$230$	0	0
$231$	−2.52673	−0.166246
$232$	0	0
$233$	21.6380	1.41755	0.708777	−	0.705433i	$-0.249247\pi$
$233$	21.6380	1.41755	0.708777	+	0.705433i	$0.249247\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7.02594	−0.456384
$238$	0	0
$239$	−7.71601	−0.499107	−0.249553	−	0.968361i	$-0.580284\pi$
$239$	−7.71601	−0.499107	−0.249553	+	0.968361i	$0.580284\pi$
$240$	0	0
$241$	−25.3264	−1.63142	−0.815708	−	0.578463i	$-0.803653\pi$
$241$	−25.3264	−1.63142	−0.815708	+	0.578463i	$0.803653\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.19751	0.330710
$248$	0	0
$249$	−16.1113	−1.02101
$250$	0	0
$251$	−2.39913	−0.151432	−0.0757160	−	0.997129i	$-0.524124\pi$
$251$	−2.39913	−0.151432	−0.0757160	+	0.997129i	$0.524124\pi$
$252$	0	0
$253$	0.793552	0.0498902
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.3086	0.892548	0.446274	−	0.894896i	$-0.352751\pi$
$257$	14.3086	0.892548	0.446274	+	0.894896i	$0.352751\pi$
$258$	0	0
$259$	−26.4614	−1.64423
$260$	0	0
$261$	−0.706362	−0.0437227
$262$	0	0
$263$	−13.0805	−0.806580	−0.403290	−	0.915072i	$-0.632133\pi$
$263$	−13.0805	−0.806580	−0.403290	+	0.915072i	$0.632133\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.0197	−0.857993
$268$	0	0
$269$	30.6749	1.87028	0.935141	−	0.354277i	$-0.115273\pi$
$269$	30.6749	1.87028	0.935141	+	0.354277i	$0.115273\pi$
$270$	0	0
$271$	14.2362	0.864789	0.432395	−	0.901684i	$-0.357669\pi$
$271$	14.2362	0.864789	0.432395	+	0.901684i	$0.357669\pi$
$272$	0	0
$273$	4.17850	0.252894
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−20.1307	−1.20953	−0.604767	−	0.796402i	$-0.706734\pi$
$277$	−20.1307	−1.20953	−0.604767	+	0.796402i	$0.706734\pi$
$278$	0	0
$279$	8.48817	0.508173
$280$	0	0
$281$	−9.81651	−0.585604	−0.292802	−	0.956173i	$-0.594588\pi$
$281$	−9.81651	−0.585604	−0.292802	+	0.956173i	$0.594588\pi$
$282$	0	0
$283$	−6.54547	−0.389088	−0.194544	−	0.980894i	$-0.562323\pi$
$283$	−6.54547	−0.389088	−0.194544	+	0.980894i	$0.562323\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−47.8344	−2.82357
$288$	0	0
$289$	−11.5972	−0.682187
$290$	0	0
$291$	−15.7520	−0.923399
$292$	0	0
$293$	−0.869428	−0.0507925	−0.0253963	−	0.999677i	$-0.508085\pi$
$293$	−0.869428	−0.0507925	−0.0253963	+	0.999677i	$0.508085\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.584296	0.0339043
$298$	0	0
$299$	−1.31231	−0.0758930
$300$	0	0
$301$	−30.4038	−1.75244
$302$	0	0
$303$	7.14178	0.410285
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.7261	−1.23997	−0.619986	−	0.784613i	$-0.712862\pi$
$307$	−21.7261	−1.23997	−0.619986	+	0.784613i	$0.712862\pi$
$308$	0	0
$309$	−1.32553	−0.0754070
$310$	0	0
$311$	18.9831	1.07643	0.538216	−	0.842807i	$-0.319098\pi$
$311$	18.9831	1.07643	0.538216	+	0.842807i	$0.319098\pi$
$312$	0	0
$313$	6.08210	0.343781	0.171890	−	0.985116i	$-0.445012\pi$
$313$	6.08210	0.343781	0.171890	+	0.985116i	$0.445012\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.97817	−0.111105	−0.0555526	−	0.998456i	$-0.517692\pi$
$317$	−1.97817	−0.111105	−0.0555526	+	0.998456i	$0.517692\pi$
$318$	0	0
$319$	−0.412724	−0.0231081
$320$	0	0
$321$	−17.6796	−0.986776
$322$	0	0
$323$	−12.5029	−0.695680
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.53978	0.195750
$328$	0	0
$329$	39.2193	2.16223
$330$	0	0
$331$	−13.1307	−0.721730	−0.360865	−	0.932618i	$-0.617519\pi$
$331$	−13.1307	−0.721730	−0.360865	+	0.932618i	$0.617519\pi$
$332$	0	0
$333$	6.11909	0.335324
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.4960	−0.626225	−0.313113	−	0.949716i	$-0.601372\pi$
$337$	−11.4960	−0.626225	−0.313113	+	0.949716i	$0.601372\pi$
$338$	0	0
$339$	−17.4882	−0.949827
$340$	0	0
$341$	4.95960	0.268577
$342$	0	0
$343$	−20.3264	−1.09752
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.7775	−1.59854	−0.799271	−	0.600971i	$-0.794781\pi$
$347$	−29.7775	−1.59854	−0.799271	+	0.600971i	$0.794781\pi$
$348$	0	0
$349$	10.0870	0.539946	0.269973	−	0.962868i	$-0.412985\pi$
$349$	10.0870	0.539946	0.269973	+	0.962868i	$0.412985\pi$
$350$	0	0
$351$	−0.966262	−0.0515752
$352$	0	0
$353$	14.2620	0.759090	0.379545	−	0.925173i	$-0.376081\pi$
$353$	14.2620	0.759090	0.379545	+	0.925173i	$0.376081\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−10.0516	−0.531988
$358$	0	0
$359$	−6.60909	−0.348815	−0.174407	−	0.984674i	$-0.555801\pi$
$359$	−6.60909	−0.348815	−0.174407	+	0.984674i	$0.555801\pi$
$360$	0	0
$361$	9.93349	0.522815
$362$	0	0
$363$	−10.6586	−0.559431
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−9.54259	−0.498119	−0.249060	−	0.968488i	$-0.580122\pi$
$367$	−9.54259	−0.498119	−0.249060	+	0.968488i	$0.580122\pi$
$368$	0	0
$369$	11.0615	0.575840
$370$	0	0
$371$	38.5118	1.99943
$372$	0	0
$373$	1.30338	0.0674866	0.0337433	−	0.999431i	$-0.489257\pi$
$373$	1.30338	0.0674866	0.0337433	+	0.999431i	$0.489257\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.682531	0.0351521
$378$	0	0
$379$	24.2855	1.24746	0.623731	−	0.781639i	$-0.285616\pi$
$379$	24.2855	1.24746	0.623731	+	0.781639i	$0.285616\pi$
$380$	0	0
$381$	−17.5060	−0.896861
$382$	0	0
$383$	−12.1319	−0.619910	−0.309955	−	0.950751i	$-0.600314\pi$
$383$	−12.1319	−0.619910	−0.309955	+	0.950751i	$0.600314\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.03076	0.357394
$388$	0	0
$389$	−29.2598	−1.48353	−0.741766	−	0.670659i	$-0.766012\pi$
$389$	−29.2598	−1.48353	−0.741766	+	0.670659i	$0.766012\pi$
$390$	0	0
$391$	3.15684	0.159648
$392$	0	0
$393$	−18.8660	−0.951664
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.29380	−0.315877	−0.157938	−	0.987449i	$-0.550485\pi$
$397$	−6.29380	−0.315877	−0.157938	+	0.987449i	$0.550485\pi$
$398$	0	0
$399$	23.2609	1.16450
$400$	0	0
$401$	30.3064	1.51343	0.756714	−	0.653747i	$-0.226804\pi$
$401$	30.3064	1.51343	0.756714	+	0.653747i	$0.226804\pi$
$402$	0	0
$403$	−8.20179	−0.408560
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.57536	0.177224
$408$	0	0
$409$	18.4761	0.913583	0.456792	−	0.889574i	$-0.348998\pi$
$409$	18.4761	0.913583	0.456792	+	0.889574i	$0.348998\pi$
$410$	0	0
$411$	−13.2508	−0.653614
$412$	0	0
$413$	31.5357	1.55177
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−16.9026	−0.827722
$418$	0	0
$419$	−22.4453	−1.09653	−0.548263	−	0.836306i	$-0.684711\pi$
$419$	−22.4453	−1.09653	−0.548263	+	0.836306i	$0.684711\pi$
$420$	0	0
$421$	−6.25053	−0.304632	−0.152316	−	0.988332i	$-0.548673\pi$
$421$	−6.25053	−0.304632	−0.152316	+	0.988332i	$0.548673\pi$
$422$	0	0
$423$	−9.06932	−0.440965
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.8163	1.00737
$428$	0	0
$429$	−0.564582	−0.0272583
$430$	0	0
$431$	34.5914	1.66621	0.833104	−	0.553116i	$-0.186561\pi$
$431$	34.5914	1.66621	0.833104	+	0.553116i	$0.186561\pi$
$432$	0	0
$433$	3.93139	0.188930	0.0944652	−	0.995528i	$-0.469886\pi$
$433$	3.93139	0.188930	0.0944652	+	0.995528i	$0.469886\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.30539	−0.349464
$438$	0	0
$439$	4.98906	0.238115	0.119057	−	0.992887i	$-0.462013\pi$
$439$	4.98906	0.238115	0.119057	+	0.992887i	$0.462013\pi$
$440$	0	0
$441$	11.7004	0.557162
$442$	0	0
$443$	−9.92754	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$443$	−9.92754	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.1625	0.575265
$448$	0	0
$449$	9.87365	0.465966	0.232983	−	0.972481i	$-0.425151\pi$
$449$	9.87365	0.465966	0.232983	+	0.972481i	$0.425151\pi$
$450$	0	0
$451$	6.46320	0.304340
$452$	0	0
$453$	−9.84446	−0.462533
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.6424	−0.591387	−0.295693	−	0.955283i	$-0.595551\pi$
$457$	−12.6424	−0.591387	−0.295693	+	0.955283i	$0.595551\pi$
$458$	0	0
$459$	2.32440	0.108494
$460$	0	0
$461$	39.7747	1.85249	0.926246	−	0.376919i	$-0.123016\pi$
$461$	39.7747	1.85249	0.926246	+	0.376919i	$0.123016\pi$
$462$	0	0
$463$	−4.41868	−0.205354	−0.102677	−	0.994715i	$-0.532741\pi$
$463$	−4.41868	−0.205354	−0.102677	+	0.994715i	$0.532741\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.2746	0.521728	0.260864	−	0.965376i	$-0.415993\pi$
$467$	11.2746	0.521728	0.260864	+	0.965376i	$0.415993\pi$
$468$	0	0
$469$	−1.62600	−0.0750816
$470$	0	0
$471$	18.4804	0.851530
$472$	0	0
$473$	4.10804	0.188888
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−8.90571	−0.407765
$478$	0	0
$479$	−5.55895	−0.253995	−0.126997	−	0.991903i	$-0.540534\pi$
$479$	−5.55895	−0.253995	−0.126997	+	0.991903i	$0.540534\pi$
$480$	0	0
$481$	−5.91264	−0.269593
$482$	0	0
$483$	−5.87311	−0.267236
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.6341	0.753761	0.376881	−	0.926262i	$-0.376997\pi$
$487$	16.6341	0.753761	0.376881	+	0.926262i	$0.376997\pi$
$488$	0	0
$489$	14.5368	0.657377
$490$	0	0
$491$	1.05925	0.0478032	0.0239016	−	0.999714i	$-0.492391\pi$
$491$	1.05925	0.0478032	0.0239016	+	0.999714i	$0.492391\pi$
$492$	0	0
$493$	−1.64187	−0.0739459
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−45.6676	−2.04847
$498$	0	0
$499$	24.9700	1.11781	0.558906	−	0.829231i	$-0.311221\pi$
$499$	24.9700	1.11781	0.558906	+	0.829231i	$0.311221\pi$
$500$	0	0
$501$	4.49299	0.200732
$502$	0	0
$503$	−12.5464	−0.559415	−0.279708	−	0.960085i	$-0.590238\pi$
$503$	−12.5464	−0.559415	−0.279708	+	0.960085i	$0.590238\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0663	−0.535885
$508$	0	0
$509$	−15.4275	−0.683810	−0.341905	−	0.939735i	$-0.611072\pi$
$509$	−15.4275	−0.683810	−0.341905	+	0.939735i	$0.611072\pi$
$510$	0	0
$511$	0.922717	0.0408186
$512$	0	0
$513$	−5.37899	−0.237488
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.29916	−0.233057
$518$	0	0
$519$	−1.99816	−0.0877094
$520$	0	0
$521$	−19.3403	−0.847312	−0.423656	−	0.905823i	$-0.639253\pi$
$521$	−19.3403	−0.847312	−0.423656	+	0.905823i	$0.639253\pi$
$522$	0	0
$523$	−6.95039	−0.303920	−0.151960	−	0.988387i	$-0.548558\pi$
$523$	−6.95039	−0.303920	−0.151960	+	0.988387i	$0.548558\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.7299	0.859446
$528$	0	0
$529$	−21.1555	−0.919803
$530$	0	0
$531$	−7.29250	−0.316468
$532$	0	0
$533$	−10.6883	−0.462962
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.9793	0.603252
$538$	0	0
$539$	6.83649	0.294469
$540$	0	0
$541$	5.66780	0.243678	0.121839	−	0.992550i	$-0.461121\pi$
$541$	5.66780	0.243678	0.121839	+	0.992550i	$0.461121\pi$
$542$	0	0
$543$	13.5379	0.580968
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.3997	−0.530172	−0.265086	−	0.964225i	$-0.585400\pi$
$547$	−12.3997	−0.530172	−0.265086	+	0.964225i	$0.585400\pi$
$548$	0	0
$549$	−4.81370	−0.205444
$550$	0	0
$551$	3.79951	0.161865
$552$	0	0
$553$	30.3829	1.29201
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.8970	−1.05492	−0.527461	−	0.849579i	$-0.676856\pi$
$557$	−24.8970	−1.05492	−0.527461	+	0.849579i	$0.676856\pi$
$558$	0	0
$559$	−6.79355	−0.287337
$560$	0	0
$561$	1.35813	0.0573405
$562$	0	0
$563$	25.0285	1.05482	0.527412	−	0.849609i	$-0.323162\pi$
$563$	25.0285	1.05482	0.527412	+	0.849609i	$0.323162\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.32440	−0.181608
$568$	0	0
$569$	2.09245	0.0877199	0.0438600	−	0.999038i	$-0.486034\pi$
$569$	2.09245	0.0877199	0.0438600	+	0.999038i	$0.486034\pi$
$570$	0	0
$571$	−11.5320	−0.482598	−0.241299	−	0.970451i	$-0.577573\pi$
$571$	−11.5320	−0.482598	−0.241299	+	0.970451i	$0.577573\pi$
$572$	0	0
$573$	2.54547	0.106339
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.84516	0.243337	0.121669	−	0.992571i	$-0.461175\pi$
$577$	5.84516	0.243337	0.121669	+	0.992571i	$0.461175\pi$
$578$	0	0
$579$	−5.60541	−0.232953
$580$	0	0
$581$	69.6716	2.89046
$582$	0	0
$583$	−5.20357	−0.215510
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.20276	0.379838	0.189919	−	0.981800i	$-0.439177\pi$
$587$	9.20276	0.379838	0.189919	+	0.981800i	$0.439177\pi$
$588$	0	0
$589$	−45.6577	−1.88129
$590$	0	0
$591$	7.85621	0.323161
$592$	0	0
$593$	14.9033	0.612004	0.306002	−	0.952031i	$-0.401009\pi$
$593$	14.9033	0.612004	0.306002	+	0.952031i	$0.401009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.9970	0.695642
$598$	0	0
$599$	−7.52244	−0.307359	−0.153679	−	0.988121i	$-0.549112\pi$
$599$	−7.52244	−0.307359	−0.153679	+	0.988121i	$0.549112\pi$
$600$	0	0
$601$	18.6618	0.761232	0.380616	−	0.924733i	$-0.375712\pi$
$601$	18.6618	0.761232	0.380616	+	0.924733i	$0.375712\pi$
$602$	0	0
$603$	0.376006	0.0153121
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.810016	−0.0328776	−0.0164388	−	0.999865i	$-0.505233\pi$
$607$	−0.810016	−0.0328776	−0.0164388	+	0.999865i	$0.505233\pi$
$608$	0	0
$609$	3.05459	0.123778
$610$	0	0
$611$	8.76333	0.354527
$612$	0	0
$613$	−4.99106	−0.201587	−0.100794	−	0.994907i	$-0.532138\pi$
$613$	−4.99106	−0.201587	−0.100794	+	0.994907i	$0.532138\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.6100	1.71541	0.857706	−	0.514140i	$-0.171889\pi$
$617$	42.6100	1.71541	0.857706	+	0.514140i	$0.171889\pi$
$618$	0	0
$619$	36.9631	1.48567	0.742836	−	0.669473i	$-0.233480\pi$
$619$	36.9631	1.48567	0.742836	+	0.669473i	$0.233480\pi$
$620$	0	0
$621$	1.35813	0.0545001
$622$	0	0
$623$	60.6268	2.42896
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.14292	−0.125516
$628$	0	0
$629$	14.2232	0.567115
$630$	0	0
$631$	−33.2368	−1.32313	−0.661567	−	0.749886i	$-0.730108\pi$
$631$	−33.2368	−1.32313	−0.661567	+	0.749886i	$0.730108\pi$
$632$	0	0
$633$	−12.9553	−0.514928
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−11.3056	−0.447946
$638$	0	0
$639$	10.5605	0.417766
$640$	0	0
$641$	−27.2363	−1.07577	−0.537885	−	0.843018i	$-0.680777\pi$
$641$	−27.2363	−1.07577	−0.537885	+	0.843018i	$0.680777\pi$
$642$	0	0
$643$	12.8557	0.506978	0.253489	−	0.967338i	$-0.418422\pi$
$643$	12.8557	0.506978	0.253489	+	0.967338i	$0.418422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.8408	−1.25179	−0.625895	−	0.779907i	$-0.715266\pi$
$647$	−31.8408	−1.25179	−0.625895	+	0.779907i	$0.715266\pi$
$648$	0	0
$649$	−4.26098	−0.167258
$650$	0	0
$651$	−36.7062	−1.43863
$652$	0	0
$653$	14.5019	0.567504	0.283752	−	0.958898i	$-0.408421\pi$
$653$	14.5019	0.567504	0.283752	+	0.958898i	$0.408421\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.213375	−0.00832454
$658$	0	0
$659$	33.2386	1.29479	0.647396	−	0.762154i	$-0.275858\pi$
$659$	33.2386	1.29479	0.647396	+	0.762154i	$0.275858\pi$
$660$	0	0
$661$	−31.8531	−1.23894	−0.619472	−	0.785019i	$-0.712653\pi$
$661$	−31.8531	−1.23894	−0.619472	+	0.785019i	$0.712653\pi$
$662$	0	0
$663$	−2.24597	−0.0872264
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.959335	−0.0371456
$668$	0	0
$669$	−18.9801	−0.733814
$670$	0	0
$671$	−2.81262	−0.108580
$672$	0	0
$673$	26.1914	1.00960	0.504802	−	0.863235i	$-0.331566\pi$
$673$	26.1914	1.00960	0.504802	+	0.863235i	$0.331566\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.8442	0.570509	0.285254	−	0.958452i	$-0.407922\pi$
$677$	14.8442	0.570509	0.285254	+	0.958452i	$0.407922\pi$
$678$	0	0
$679$	68.1179	2.61413
$680$	0	0
$681$	0.385109	0.0147574
$682$	0	0
$683$	5.62626	0.215283	0.107641	−	0.994190i	$-0.465670\pi$
$683$	5.62626	0.215283	0.107641	+	0.994190i	$0.465670\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.2685	0.391768
$688$	0	0
$689$	8.60525	0.327834
$690$	0	0
$691$	34.4090	1.30898	0.654491	−	0.756070i	$-0.272883\pi$
$691$	34.4090	1.30898	0.654491	+	0.756070i	$0.272883\pi$
$692$	0	0
$693$	−2.52673	−0.0959824
$694$	0	0
$695$	0	0
$696$	0	0
$697$	25.7113	0.973887
$698$	0	0
$699$	21.6380	0.818425
$700$	0	0
$701$	8.86294	0.334749	0.167374	−	0.985893i	$-0.446471\pi$
$701$	8.86294	0.334749	0.167374	+	0.985893i	$0.446471\pi$
$702$	0	0
$703$	−32.9145	−1.24139
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.8839	−1.16151
$708$	0	0
$709$	1.53673	0.0577132	0.0288566	−	0.999584i	$-0.490813\pi$
$709$	1.53673	0.0577132	0.0288566	+	0.999584i	$0.490813\pi$
$710$	0	0
$711$	−7.02594	−0.263493
$712$	0	0
$713$	11.5281	0.431730
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.71601	−0.288160
$718$	0	0
$719$	−19.1353	−0.713625	−0.356813	−	0.934176i	$-0.616137\pi$
$719$	−19.1353	−0.713625	−0.356813	+	0.934176i	$0.616137\pi$
$720$	0	0
$721$	5.73214	0.213476
$722$	0	0
$723$	−25.3264	−0.941899
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−46.8094	−1.73606	−0.868032	−	0.496507i	$-0.834616\pi$
$727$	−46.8094	−1.73606	−0.868032	+	0.496507i	$0.834616\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.3423	0.604441
$732$	0	0
$733$	−51.0044	−1.88389	−0.941945	−	0.335768i	$-0.891004\pi$
$733$	−51.0044	−1.88389	−0.941945	+	0.335768i	$0.891004\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.219699	0.00809270
$738$	0	0
$739$	13.2056	0.485775	0.242887	−	0.970054i	$-0.421905\pi$
$739$	13.2056	0.485775	0.242887	+	0.970054i	$0.421905\pi$
$740$	0	0
$741$	5.19751	0.190935
$742$	0	0
$743$	−10.6063	−0.389107	−0.194553	−	0.980892i	$-0.562326\pi$
$743$	−10.6063	−0.389107	−0.194553	+	0.980892i	$0.562326\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−16.1113	−0.589481
$748$	0	0
$749$	76.4534	2.79355
$750$	0	0
$751$	28.2581	1.03115	0.515577	−	0.856843i	$-0.327578\pi$
$751$	28.2581	1.03115	0.515577	+	0.856843i	$0.327578\pi$
$752$	0	0
$753$	−2.39913	−0.0874293
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.7474	−1.00849	−0.504247	−	0.863559i	$-0.668230\pi$
$757$	−27.7474	−1.00849	−0.504247	+	0.863559i	$0.668230\pi$
$758$	0	0
$759$	0.793552	0.0288041
$760$	0	0
$761$	−16.1152	−0.584177	−0.292088	−	0.956391i	$-0.594350\pi$
$761$	−16.1152	−0.584177	−0.292088	+	0.956391i	$0.594350\pi$
$762$	0	0
$763$	−15.3074	−0.554165
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.04646	0.254433
$768$	0	0
$769$	−34.7435	−1.25288	−0.626441	−	0.779468i	$-0.715489\pi$
$769$	−34.7435	−1.25288	−0.626441	+	0.779468i	$0.715489\pi$
$770$	0	0
$771$	14.3086	0.515313
$772$	0	0
$773$	−23.4684	−0.844101	−0.422051	−	0.906572i	$-0.638690\pi$
$773$	−23.4684	−0.844101	−0.422051	+	0.906572i	$0.638690\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−26.4614	−0.949296
$778$	0	0
$779$	−59.4997	−2.13180
$780$	0	0
$781$	6.17043	0.220795
$782$	0	0
$783$	−0.706362	−0.0252433
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.6217	0.913316	0.456658	−	0.889642i	$-0.349046\pi$
$787$	25.6217	0.913316	0.456658	+	0.889642i	$0.349046\pi$
$788$	0	0
$789$	−13.0805	−0.465679
$790$	0	0
$791$	75.6258	2.68894
$792$	0	0
$793$	4.65129	0.165172
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.92304	0.0681177	0.0340588	−	0.999420i	$-0.489157\pi$
$797$	1.92304	0.0681177	0.0340588	+	0.999420i	$0.489157\pi$
$798$	0	0
$799$	−21.0807	−0.745781
$800$	0	0
$801$	−14.0197	−0.495362
$802$	0	0
$803$	−0.124674	−0.00439965
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.6749	1.07981
$808$	0	0
$809$	−45.4087	−1.59648	−0.798242	−	0.602336i	$-0.794237\pi$
$809$	−45.4087	−1.59648	−0.798242	+	0.602336i	$0.794237\pi$
$810$	0	0
$811$	−23.9207	−0.839970	−0.419985	−	0.907531i	$-0.637965\pi$
$811$	−23.9207	−0.839970	−0.419985	+	0.907531i	$0.637965\pi$
$812$	0	0
$813$	14.2362	0.499286
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−37.8184	−1.32310
$818$	0	0
$819$	4.17850	0.146009
$820$	0	0
$821$	−31.5862	−1.10237	−0.551184	−	0.834384i	$-0.685824\pi$
$821$	−31.5862	−1.10237	−0.551184	+	0.834384i	$0.685824\pi$
$822$	0	0
$823$	−8.53252	−0.297425	−0.148713	−	0.988880i	$-0.547513\pi$
$823$	−8.53252	−0.297425	−0.148713	+	0.988880i	$0.547513\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.7933	−0.618733	−0.309366	−	0.950943i	$-0.600117\pi$
$827$	−17.7933	−0.618733	−0.309366	+	0.950943i	$0.600117\pi$
$828$	0	0
$829$	−46.0216	−1.59840	−0.799199	−	0.601067i	$-0.794743\pi$
$829$	−46.0216	−1.59840	−0.799199	+	0.601067i	$0.794743\pi$
$830$	0	0
$831$	−20.1307	−0.698325
$832$	0	0
$833$	27.1964	0.942298
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.48817	0.293394
$838$	0	0
$839$	1.01208	0.0349410	0.0174705	−	0.999847i	$-0.494439\pi$
$839$	1.01208	0.0349410	0.0174705	+	0.999847i	$0.494439\pi$
$840$	0	0
$841$	−28.5011	−0.982795
$842$	0	0
$843$	−9.81651	−0.338099
$844$	0	0
$845$	0	0
$846$	0	0
$847$	46.0920	1.58374
$848$	0	0
$849$	−6.54547	−0.224640
$850$	0	0
$851$	8.31054	0.284882
$852$	0	0
$853$	9.81334	0.336002	0.168001	−	0.985787i	$-0.446269\pi$
$853$	9.81334	0.336002	0.168001	+	0.985787i	$0.446269\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	57.3424	1.95878	0.979390	−	0.201979i	$-0.0647372\pi$
$857$	57.3424	1.95878	0.979390	+	0.201979i	$0.0647372\pi$
$858$	0	0
$859$	10.7349	0.366271	0.183135	−	0.983088i	$-0.441375\pi$
$859$	10.7349	0.366271	0.183135	+	0.983088i	$0.441375\pi$
$860$	0	0
$861$	−47.8344	−1.63019
$862$	0	0
$863$	−9.63250	−0.327894	−0.163947	−	0.986469i	$-0.552423\pi$
$863$	−9.63250	−0.327894	−0.163947	+	0.986469i	$0.552423\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.5972	−0.393861
$868$	0	0
$869$	−4.10522	−0.139260
$870$	0	0
$871$	−0.363320	−0.0123106
$872$	0	0
$873$	−15.7520	−0.533125
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.703817	0.0237662	0.0118831	−	0.999929i	$-0.496217\pi$
$877$	0.703817	0.0237662	0.0118831	+	0.999929i	$0.496217\pi$
$878$	0	0
$879$	−0.869428	−0.0293251
$880$	0	0
$881$	9.92656	0.334434	0.167217	−	0.985920i	$-0.446522\pi$
$881$	9.92656	0.334434	0.167217	+	0.985920i	$0.446522\pi$
$882$	0	0
$883$	−16.8403	−0.566723	−0.283361	−	0.959013i	$-0.591450\pi$
$883$	−16.8403	−0.566723	−0.283361	+	0.959013i	$0.591450\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.3918	−1.55768	−0.778842	−	0.627221i	$-0.784192\pi$
$887$	−46.3918	−1.55768	−0.778842	+	0.627221i	$0.784192\pi$
$888$	0	0
$889$	75.7030	2.53900
$890$	0	0
$891$	0.584296	0.0195746
$892$	0	0
$893$	48.7837	1.63249
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.31231	−0.0438169
$898$	0	0
$899$	−5.99572	−0.199968
$900$	0	0
$901$	−20.7004	−0.689630
$902$	0	0
$903$	−30.4038	−1.01177
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.5101	−0.946662	−0.473331	−	0.880885i	$-0.656949\pi$
$907$	−28.5101	−0.946662	−0.473331	+	0.880885i	$0.656949\pi$
$908$	0	0
$909$	7.14178	0.236878
$910$	0	0
$911$	−6.50073	−0.215379	−0.107689	−	0.994185i	$-0.534345\pi$
$911$	−6.50073	−0.215379	−0.107689	+	0.994185i	$0.534345\pi$
$912$	0	0
$913$	−9.41375	−0.311550
$914$	0	0
$915$	0	0
$916$	0	0
$917$	81.5841	2.69414
$918$	0	0
$919$	16.8266	0.555058	0.277529	−	0.960717i	$-0.410485\pi$
$919$	16.8266	0.555058	0.277529	+	0.960717i	$0.410485\pi$
$920$	0	0
$921$	−21.7261	−0.715899
$922$	0	0
$923$	−10.2042	−0.335874
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.32553	−0.0435363
$928$	0	0
$929$	−31.6407	−1.03810	−0.519049	−	0.854744i	$-0.673714\pi$
$929$	−31.6407	−1.03810	−0.519049	+	0.854744i	$0.673714\pi$
$930$	0	0
$931$	−62.9363	−2.06265
$932$	0	0
$933$	18.9831	0.621479
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.35105	0.0768053	0.0384026	−	0.999262i	$-0.487773\pi$
$937$	2.35105	0.0768053	0.0384026	+	0.999262i	$0.487773\pi$
$938$	0	0
$939$	6.08210	0.198482
$940$	0	0
$941$	4.29802	0.140111	0.0700557	−	0.997543i	$-0.477682\pi$
$941$	4.29802	0.140111	0.0700557	+	0.997543i	$0.477682\pi$
$942$	0	0
$943$	15.0230	0.489217
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.8418	−0.904737	−0.452369	−	0.891831i	$-0.649421\pi$
$947$	−27.8418	−0.904737	−0.452369	+	0.891831i	$0.649421\pi$
$948$	0	0
$949$	0.206176	0.00669275
$950$	0	0
$951$	−1.97817	−0.0641467
$952$	0	0
$953$	44.6209	1.44541	0.722707	−	0.691154i	$-0.242898\pi$
$953$	44.6209	1.44541	0.722707	+	0.691154i	$0.242898\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.412724	−0.0133415
$958$	0	0
$959$	57.3017	1.85037
$960$	0	0
$961$	41.0490	1.32416
$962$	0	0
$963$	−17.6796	−0.569716
$964$	0	0
$965$	0	0
$966$	0	0
$967$	55.1855	1.77465	0.887323	−	0.461148i	$-0.152562\pi$
$967$	55.1855	1.77465	0.887323	+	0.461148i	$0.152562\pi$
$968$	0	0
$969$	−12.5029	−0.401651
$970$	0	0
$971$	−6.19165	−0.198699	−0.0993497	−	0.995053i	$-0.531676\pi$
$971$	−6.19165	−0.198699	−0.0993497	+	0.995053i	$0.531676\pi$
$972$	0	0
$973$	73.0934	2.34327
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.1646	−0.773095	−0.386548	−	0.922269i	$-0.626333\pi$
$977$	−24.1646	−0.773095	−0.386548	+	0.922269i	$0.626333\pi$
$978$	0	0
$979$	−8.19166	−0.261806
$980$	0	0
$981$	3.53978	0.113016
$982$	0	0
$983$	9.52304	0.303738	0.151869	−	0.988401i	$-0.451471\pi$
$983$	9.52304	0.303738	0.151869	+	0.988401i	$0.451471\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	39.2193	1.24836
$988$	0	0
$989$	9.54872	0.303631
$990$	0	0
$991$	−10.4117	−0.330740	−0.165370	−	0.986232i	$-0.552882\pi$
$991$	−10.4117	−0.330740	−0.165370	+	0.986232i	$0.552882\pi$
$992$	0	0
$993$	−13.1307	−0.416691
$994$	0	0
$995$	0	0
$996$	0	0
$997$	57.7687	1.82955	0.914777	−	0.403959i	$-0.132366\pi$
$997$	57.7687	1.82955	0.914777	+	0.403959i	$0.132366\pi$
$998$	0	0
$999$	6.11909	0.193599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.f.1.1		4
5.2	odd	4		7500.2.d.c.1249.1		8
5.3	odd	4		7500.2.d.c.1249.8		8
5.4	even	2		7500.2.a.e.1.4		4
25.2	odd	20		1500.2.o.b.649.1		16
25.9	even	10		300.2.m.b.181.1	yes	8
25.11	even	5		1500.2.m.a.601.1		8
25.12	odd	20		1500.2.o.b.349.3		16
25.13	odd	20		1500.2.o.b.349.2		16
25.14	even	10		300.2.m.b.121.1	✓	8
25.16	even	5		1500.2.m.a.901.1		8
25.23	odd	20		1500.2.o.b.649.4		16
75.14	odd	10		900.2.n.b.721.2		8
75.59	odd	10		900.2.n.b.181.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.121.1	✓	8	25.14	even	10
300.2.m.b.181.1	yes	8	25.9	even	10
900.2.n.b.181.2		8	75.59	odd	10
900.2.n.b.721.2		8	75.14	odd	10
1500.2.m.a.601.1		8	25.11	even	5
1500.2.m.a.901.1		8	25.16	even	5
1500.2.o.b.349.2		16	25.13	odd	20
1500.2.o.b.349.3		16	25.12	odd	20
1500.2.o.b.649.1		16	25.2	odd	20
1500.2.o.b.649.4		16	25.23	odd	20
7500.2.a.e.1.4		4	5.4	even	2
7500.2.a.f.1.1		4	1.1	even	1	trivial
7500.2.d.c.1249.1		8	5.2	odd	4
7500.2.d.c.1249.8		8	5.3	odd	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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.32440 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.32440 q^{7} +1.00000 q^{9} +0.584296 q^{11} -0.966262 q^{13} +2.32440 q^{17} -5.37899 q^{19} -4.32440 q^{21} +1.35813 q^{23} +1.00000 q^{27} -0.706362 q^{29} +8.48817 q^{31} +0.584296 q^{33} +6.11909 q^{37} -0.966262 q^{39} +11.0615 q^{41} +7.03076 q^{43} -9.06932 q^{47} +11.7004 q^{49} +2.32440 q^{51} -8.90571 q^{53} -5.37899 q^{57} -7.29250 q^{59} -4.81370 q^{61} -4.32440 q^{63} +0.376006 q^{67} +1.35813 q^{69} +10.5605 q^{71} -0.213375 q^{73} -2.52673 q^{77} -7.02594 q^{79} +1.00000 q^{81} -16.1113 q^{83} -0.706362 q^{87} -14.0197 q^{89} +4.17850 q^{91} +8.48817 q^{93} -15.7520 q^{97} +0.584296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - q^{11} - 5 q^{13} - 4 q^{17} - 5 q^{19} - 4 q^{21} - 9 q^{23} + 4 q^{27} + 6 q^{29} + 11 q^{31} - q^{33} - 2 q^{37} - 5 q^{39} + 6 q^{43} - 16 q^{47} - 4 q^{49} - 4 q^{51} + 2 q^{53} - 5 q^{57} + q^{59} - 22 q^{61} - 4 q^{63} - 36 q^{67} - 9 q^{69} + 20 q^{71} - 12 q^{73} + 11 q^{77} - 3 q^{79} + 4 q^{81} - 14 q^{83} + 6 q^{87} - 15 q^{89} - 10 q^{91} + 11 q^{93} + 12 q^{97} - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^7 + 4 * q^9 - q^11 - 5 * q^13 - 4 * q^17 - 5 * q^19 - 4 * q^21 - 9 * q^23 + 4 * q^27 + 6 * q^29 + 11 * q^31 - q^33 - 2 * q^37 - 5 * q^39 + 6 * q^43 - 16 * q^47 - 4 * q^49 - 4 * q^51 + 2 * q^53 - 5 * q^57 + q^59 - 22 * q^61 - 4 * q^63 - 36 * q^67 - 9 * q^69 + 20 * q^71 - 12 * q^73 + 11 * q^77 - 3 * q^79 + 4 * q^81 - 14 * q^83 + 6 * q^87 - 15 * q^89 - 10 * q^91 + 11 * q^93 + 12 * q^97 - q^99

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