LMFDB - Embedded newform 7500.2.a.m.1.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 $ x^12 - 6x^11 - 11x^10 + 94x^9 + 27x^8 - 460x^7 + 55x^6 + 812x^5 - 127x^4 - 512x^3 + 40x^2 + 72*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.879783$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.957526 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.957526 q^{7} +1.00000 q^{9} -5.41590 q^{11} +2.02271 q^{13} +0.642866 q^{17} -5.04105 q^{19} -0.957526 q^{21} -3.51960 q^{23} -1.00000 q^{27} +10.1408 q^{29} +3.69178 q^{31} +5.41590 q^{33} +11.3153 q^{37} -2.02271 q^{39} +3.52277 q^{41} -0.766348 q^{43} -4.93421 q^{47} -6.08314 q^{49} -0.642866 q^{51} -5.94634 q^{53} +5.04105 q^{57} +4.71878 q^{59} +4.34485 q^{61} +0.957526 q^{63} -9.51778 q^{67} +3.51960 q^{69} -11.9450 q^{71} +5.43304 q^{73} -5.18586 q^{77} +11.8494 q^{79} +1.00000 q^{81} +1.39408 q^{83} -10.1408 q^{87} -1.70812 q^{89} +1.93680 q^{91} -3.69178 q^{93} -14.5878 q^{97} -5.41590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * 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
$7$	0.957526	0.361911	0.180955	−	0.983491i	$-0.442081\pi$
$7$	0.957526	0.361911	0.180955	+	0.983491i	$0.442081\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.41590	−1.63295	−0.816477	−	0.577378i	$-0.804076\pi$
$11$	−5.41590	−1.63295	−0.816477	+	0.577378i	$0.804076\pi$
$12$	0	0
$13$	2.02271	0.560999	0.280499	−	0.959854i	$-0.409500\pi$
$13$	2.02271	0.560999	0.280499	+	0.959854i	$0.409500\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.642866	0.155918	0.0779590	−	0.996957i	$-0.475160\pi$
$17$	0.642866	0.155918	0.0779590	+	0.996957i	$0.475160\pi$
$18$	0	0
$19$	−5.04105	−1.15650	−0.578248	−	0.815861i	$-0.696263\pi$
$19$	−5.04105	−1.15650	−0.578248	+	0.815861i	$0.696263\pi$
$20$	0	0
$21$	−0.957526	−0.208949
$22$	0	0
$23$	−3.51960	−0.733888	−0.366944	−	0.930243i	$-0.619596\pi$
$23$	−3.51960	−0.733888	−0.366944	+	0.930243i	$0.619596\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.1408	1.88309	0.941546	−	0.336884i	$-0.109373\pi$
$29$	10.1408	1.88309	0.941546	+	0.336884i	$0.109373\pi$
$30$	0	0
$31$	3.69178	0.663063	0.331531	−	0.943444i	$-0.392435\pi$
$31$	3.69178	0.663063	0.331531	+	0.943444i	$0.392435\pi$
$32$	0	0
$33$	5.41590	0.942787
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.3153	1.86023	0.930114	−	0.367272i	$-0.119708\pi$
$37$	11.3153	1.86023	0.930114	+	0.367272i	$0.119708\pi$
$38$	0	0
$39$	−2.02271	−0.323893
$40$	0	0
$41$	3.52277	0.550164	0.275082	−	0.961421i	$-0.411295\pi$
$41$	3.52277	0.550164	0.275082	+	0.961421i	$0.411295\pi$
$42$	0	0
$43$	−0.766348	−0.116867	−0.0584335	−	0.998291i	$-0.518611\pi$
$43$	−0.766348	−0.116867	−0.0584335	+	0.998291i	$0.518611\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.93421	−0.719729	−0.359865	−	0.933005i	$-0.617177\pi$
$47$	−4.93421	−0.719729	−0.359865	+	0.933005i	$0.617177\pi$
$48$	0	0
$49$	−6.08314	−0.869021
$50$	0	0
$51$	−0.642866	−0.0900193
$52$	0	0
$53$	−5.94634	−0.816793	−0.408396	−	0.912805i	$-0.633912\pi$
$53$	−5.94634	−0.816793	−0.408396	+	0.912805i	$0.633912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.04105	0.667703
$58$	0	0
$59$	4.71878	0.614333	0.307166	−	0.951656i	$-0.400619\pi$
$59$	4.71878	0.614333	0.307166	+	0.951656i	$0.400619\pi$
$60$	0	0
$61$	4.34485	0.556301	0.278150	−	0.960538i	$-0.410279\pi$
$61$	4.34485	0.556301	0.278150	+	0.960538i	$0.410279\pi$
$62$	0	0
$63$	0.957526	0.120637
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.51778	−1.16278	−0.581391	−	0.813624i	$-0.697491\pi$
$67$	−9.51778	−1.16278	−0.581391	+	0.813624i	$0.697491\pi$
$68$	0	0
$69$	3.51960	0.423710
$70$	0	0
$71$	−11.9450	−1.41761	−0.708803	−	0.705406i	$-0.750765\pi$
$71$	−11.9450	−1.41761	−0.708803	+	0.705406i	$0.750765\pi$
$72$	0	0
$73$	5.43304	0.635889	0.317944	−	0.948109i	$-0.397007\pi$
$73$	5.43304	0.635889	0.317944	+	0.948109i	$0.397007\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.18586	−0.590984
$78$	0	0
$79$	11.8494	1.33316	0.666581	−	0.745432i	$-0.267757\pi$
$79$	11.8494	1.33316	0.666581	+	0.745432i	$0.267757\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.39408	0.153020	0.0765101	−	0.997069i	$-0.475622\pi$
$83$	1.39408	0.153020	0.0765101	+	0.997069i	$0.475622\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−10.1408	−1.08720
$88$	0	0
$89$	−1.70812	−0.181061	−0.0905303	−	0.995894i	$-0.528856\pi$
$89$	−1.70812	−0.181061	−0.0905303	+	0.995894i	$0.528856\pi$
$90$	0	0
$91$	1.93680	0.203032
$92$	0	0
$93$	−3.69178	−0.382819
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.5878	−1.48117	−0.740585	−	0.671963i	$-0.765451\pi$
$97$	−14.5878	−1.48117	−0.740585	+	0.671963i	$0.765451\pi$
$98$	0	0
$99$	−5.41590	−0.544318
$100$	0	0
$101$	−10.2832	−1.02322	−0.511610	−	0.859218i	$-0.670951\pi$
$101$	−10.2832	−1.02322	−0.511610	+	0.859218i	$0.670951\pi$
$102$	0	0
$103$	−13.9848	−1.37797	−0.688983	−	0.724777i	$-0.741943\pi$
$103$	−13.9848	−1.37797	−0.688983	+	0.724777i	$0.741943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.06727	−0.876566	−0.438283	−	0.898837i	$-0.644413\pi$
$107$	−9.06727	−0.876566	−0.438283	+	0.898837i	$0.644413\pi$
$108$	0	0
$109$	2.37535	0.227518	0.113759	−	0.993508i	$-0.463711\pi$
$109$	2.37535	0.227518	0.113759	+	0.993508i	$0.463711\pi$
$110$	0	0
$111$	−11.3153	−1.07400
$112$	0	0
$113$	13.4374	1.26408	0.632041	−	0.774935i	$-0.282218\pi$
$113$	13.4374	1.26408	0.632041	+	0.774935i	$0.282218\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.02271	0.187000
$118$	0	0
$119$	0.615561	0.0564284
$120$	0	0
$121$	18.3319	1.66654
$122$	0	0
$123$	−3.52277	−0.317637
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.2629	−1.26563	−0.632815	−	0.774303i	$-0.718101\pi$
$127$	−14.2629	−1.26563	−0.632815	+	0.774303i	$0.718101\pi$
$128$	0	0
$129$	0.766348	0.0674732
$130$	0	0
$131$	−0.128666	−0.0112416	−0.00562082	−	0.999984i	$-0.501789\pi$
$131$	−0.128666	−0.0112416	−0.00562082	+	0.999984i	$0.501789\pi$
$132$	0	0
$133$	−4.82694	−0.418548
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.7022	−1.17066	−0.585331	−	0.810795i	$-0.699035\pi$
$137$	−13.7022	−1.17066	−0.585331	+	0.810795i	$0.699035\pi$
$138$	0	0
$139$	23.4718	1.99085	0.995425	−	0.0955488i	$-0.0304606\pi$
$139$	23.4718	1.99085	0.995425	+	0.0955488i	$0.0304606\pi$
$140$	0	0
$141$	4.93421	0.415536
$142$	0	0
$143$	−10.9548	−0.916086
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.08314	0.501729
$148$	0	0
$149$	−10.6938	−0.876071	−0.438035	−	0.898958i	$-0.644326\pi$
$149$	−10.6938	−0.876071	−0.438035	+	0.898958i	$0.644326\pi$
$150$	0	0
$151$	7.37520	0.600185	0.300092	−	0.953910i	$-0.402982\pi$
$151$	7.37520	0.600185	0.300092	+	0.953910i	$0.402982\pi$
$152$	0	0
$153$	0.642866	0.0519727
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.0329	−1.04014	−0.520070	−	0.854124i	$-0.674094\pi$
$157$	−13.0329	−1.04014	−0.520070	+	0.854124i	$0.674094\pi$
$158$	0	0
$159$	5.94634	0.471575
$160$	0	0
$161$	−3.37011	−0.265602
$162$	0	0
$163$	−7.39602	−0.579301	−0.289650	−	0.957133i	$-0.593539\pi$
$163$	−7.39602	−0.579301	−0.289650	+	0.957133i	$0.593539\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.1265	−1.55744	−0.778718	−	0.627374i	$-0.784130\pi$
$167$	−20.1265	−1.55744	−0.778718	+	0.627374i	$0.784130\pi$
$168$	0	0
$169$	−8.90864	−0.685280
$170$	0	0
$171$	−5.04105	−0.385499
$172$	0	0
$173$	2.22112	0.168869	0.0844344	−	0.996429i	$-0.473092\pi$
$173$	2.22112	0.168869	0.0844344	+	0.996429i	$0.473092\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.71878	−0.354685
$178$	0	0
$179$	−0.0385919	−0.00288449	−0.00144225	−	0.999999i	$-0.500459\pi$
$179$	−0.0385919	−0.00288449	−0.00144225	+	0.999999i	$0.500459\pi$
$180$	0	0
$181$	0.146945	0.0109223	0.00546116	−	0.999985i	$-0.498262\pi$
$181$	0.146945	0.0109223	0.00546116	+	0.999985i	$0.498262\pi$
$182$	0	0
$183$	−4.34485	−0.321180
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.48170	−0.254607
$188$	0	0
$189$	−0.957526	−0.0696498
$190$	0	0
$191$	−0.459682	−0.0332614	−0.0166307	−	0.999862i	$-0.505294\pi$
$191$	−0.459682	−0.0332614	−0.0166307	+	0.999862i	$0.505294\pi$
$192$	0	0
$193$	19.0231	1.36932	0.684658	−	0.728864i	$-0.259952\pi$
$193$	19.0231	1.36932	0.684658	+	0.728864i	$0.259952\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.1151	−0.863164	−0.431582	−	0.902074i	$-0.642045\pi$
$197$	−12.1151	−0.863164	−0.431582	+	0.902074i	$0.642045\pi$
$198$	0	0
$199$	16.4872	1.16875	0.584375	−	0.811484i	$-0.301340\pi$
$199$	16.4872	1.16875	0.584375	+	0.811484i	$0.301340\pi$
$200$	0	0
$201$	9.51778	0.671333
$202$	0	0
$203$	9.71005	0.681512
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.51960	−0.244629
$208$	0	0
$209$	27.3018	1.88850
$210$	0	0
$211$	18.1862	1.25199	0.625996	−	0.779827i	$-0.284693\pi$
$211$	18.1862	1.25199	0.625996	+	0.779827i	$0.284693\pi$
$212$	0	0
$213$	11.9450	0.818455
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.53497	0.239970
$218$	0	0
$219$	−5.43304	−0.367131
$220$	0	0
$221$	1.30033	0.0874698
$222$	0	0
$223$	23.5103	1.57436	0.787182	−	0.616721i	$-0.211539\pi$
$223$	23.5103	1.57436	0.787182	+	0.616721i	$0.211539\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.6803	1.30623	0.653114	−	0.757260i	$-0.273462\pi$
$227$	19.6803	1.30623	0.653114	+	0.757260i	$0.273462\pi$
$228$	0	0
$229$	5.30698	0.350695	0.175347	−	0.984507i	$-0.443895\pi$
$229$	5.30698	0.350695	0.175347	+	0.984507i	$0.443895\pi$
$230$	0	0
$231$	5.18586	0.341205
$232$	0	0
$233$	23.2693	1.52442	0.762212	−	0.647327i	$-0.224113\pi$
$233$	23.2693	1.52442	0.762212	+	0.647327i	$0.224113\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.8494	−0.769702
$238$	0	0
$239$	−15.0750	−0.975118	−0.487559	−	0.873090i	$-0.662113\pi$
$239$	−15.0750	−0.975118	−0.487559	+	0.873090i	$0.662113\pi$
$240$	0	0
$241$	−18.9133	−1.21831	−0.609157	−	0.793049i	$-0.708492\pi$
$241$	−18.9133	−1.21831	−0.609157	+	0.793049i	$0.708492\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.1966	−0.648793
$248$	0	0
$249$	−1.39408	−0.0883463
$250$	0	0
$251$	−4.56761	−0.288305	−0.144153	−	0.989555i	$-0.546046\pi$
$251$	−4.56761	−0.288305	−0.144153	+	0.989555i	$0.546046\pi$
$252$	0	0
$253$	19.0618	1.19841
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.2556	−1.26351	−0.631754	−	0.775169i	$-0.717665\pi$
$257$	−20.2556	−1.26351	−0.631754	+	0.775169i	$0.717665\pi$
$258$	0	0
$259$	10.8347	0.673237
$260$	0	0
$261$	10.1408	0.627698
$262$	0	0
$263$	−30.7789	−1.89791	−0.948953	−	0.315418i	$-0.897855\pi$
$263$	−30.7789	−1.89791	−0.948953	+	0.315418i	$0.897855\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.70812	0.104535
$268$	0	0
$269$	−26.6739	−1.62634	−0.813168	−	0.582030i	$-0.802259\pi$
$269$	−26.6739	−1.62634	−0.813168	+	0.582030i	$0.802259\pi$
$270$	0	0
$271$	−5.52945	−0.335890	−0.167945	−	0.985796i	$-0.553713\pi$
$271$	−5.52945	−0.335890	−0.167945	+	0.985796i	$0.553713\pi$
$272$	0	0
$273$	−1.93680	−0.117220
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−30.9233	−1.85800	−0.929001	−	0.370077i	$-0.879331\pi$
$277$	−30.9233	−1.85800	−0.929001	+	0.370077i	$0.879331\pi$
$278$	0	0
$279$	3.69178	0.221021
$280$	0	0
$281$	−16.8489	−1.00512	−0.502560	−	0.864542i	$-0.667608\pi$
$281$	−16.8489	−1.00512	−0.502560	+	0.864542i	$0.667608\pi$
$282$	0	0
$283$	−11.2627	−0.669501	−0.334750	−	0.942307i	$-0.608652\pi$
$283$	−11.2627	−0.669501	−0.334750	+	0.942307i	$0.608652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.37314	0.199110
$288$	0	0
$289$	−16.5867	−0.975690
$290$	0	0
$291$	14.5878	0.855154
$292$	0	0
$293$	−11.1995	−0.654284	−0.327142	−	0.944975i	$-0.606086\pi$
$293$	−11.1995	−0.654284	−0.327142	+	0.944975i	$0.606086\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.41590	0.314262
$298$	0	0
$299$	−7.11914	−0.411710
$300$	0	0
$301$	−0.733798	−0.0422954
$302$	0	0
$303$	10.2832	0.590756
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.1289	1.37711	0.688554	−	0.725185i	$-0.258246\pi$
$307$	24.1289	1.37711	0.688554	+	0.725185i	$0.258246\pi$
$308$	0	0
$309$	13.9848	0.795569
$310$	0	0
$311$	2.07312	0.117556	0.0587778	−	0.998271i	$-0.481280\pi$
$311$	2.07312	0.117556	0.0587778	+	0.998271i	$0.481280\pi$
$312$	0	0
$313$	−8.51966	−0.481560	−0.240780	−	0.970580i	$-0.577403\pi$
$313$	−8.51966	−0.481560	−0.240780	+	0.970580i	$0.577403\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.6070	0.764243	0.382122	−	0.924112i	$-0.375194\pi$
$317$	13.6070	0.764243	0.382122	+	0.924112i	$0.375194\pi$
$318$	0	0
$319$	−54.9213	−3.07500
$320$	0	0
$321$	9.06727	0.506086
$322$	0	0
$323$	−3.24072	−0.180318
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.37535	−0.131357
$328$	0	0
$329$	−4.72464	−0.260478
$330$	0	0
$331$	−22.6320	−1.24397	−0.621985	−	0.783029i	$-0.713673\pi$
$331$	−22.6320	−1.24397	−0.621985	+	0.783029i	$0.713673\pi$
$332$	0	0
$333$	11.3153	0.620076
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−35.9900	−1.96050	−0.980250	−	0.197762i	$-0.936633\pi$
$337$	−35.9900	−1.96050	−0.980250	+	0.197762i	$0.936633\pi$
$338$	0	0
$339$	−13.4374	−0.729818
$340$	0	0
$341$	−19.9943	−1.08275
$342$	0	0
$343$	−12.5275	−0.676419
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.96344	0.481183	0.240591	−	0.970627i	$-0.422659\pi$
$347$	8.96344	0.481183	0.240591	+	0.970627i	$0.422659\pi$
$348$	0	0
$349$	−11.8276	−0.633114	−0.316557	−	0.948573i	$-0.602527\pi$
$349$	−11.8276	−0.633114	−0.316557	+	0.948573i	$0.602527\pi$
$350$	0	0
$351$	−2.02271	−0.107964
$352$	0	0
$353$	−23.0886	−1.22888	−0.614442	−	0.788962i	$-0.710619\pi$
$353$	−23.0886	−1.22888	−0.614442	+	0.788962i	$0.710619\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.615561	−0.0325790
$358$	0	0
$359$	21.1188	1.11461	0.557304	−	0.830309i	$-0.311836\pi$
$359$	21.1188	1.11461	0.557304	+	0.830309i	$0.311836\pi$
$360$	0	0
$361$	6.41217	0.337482
$362$	0	0
$363$	−18.3319	−0.962177
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.39044	−0.437977	−0.218989	−	0.975727i	$-0.570276\pi$
$367$	−8.39044	−0.437977	−0.218989	+	0.975727i	$0.570276\pi$
$368$	0	0
$369$	3.52277	0.183388
$370$	0	0
$371$	−5.69378	−0.295606
$372$	0	0
$373$	−6.32863	−0.327684	−0.163842	−	0.986487i	$-0.552389\pi$
$373$	−6.32863	−0.327684	−0.163842	+	0.986487i	$0.552389\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.5118	1.05641
$378$	0	0
$379$	15.0788	0.774546	0.387273	−	0.921965i	$-0.373417\pi$
$379$	15.0788	0.774546	0.387273	+	0.921965i	$0.373417\pi$
$380$	0	0
$381$	14.2629	0.730712
$382$	0	0
$383$	−32.5782	−1.66467	−0.832333	−	0.554276i	$-0.812995\pi$
$383$	−32.5782	−1.66467	−0.832333	+	0.554276i	$0.812995\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.766348	−0.0389557
$388$	0	0
$389$	−5.47168	−0.277425	−0.138713	−	0.990333i	$-0.544296\pi$
$389$	−5.47168	−0.277425	−0.138713	+	0.990333i	$0.544296\pi$
$390$	0	0
$391$	−2.26263	−0.114426
$392$	0	0
$393$	0.128666	0.00649036
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.2686	0.716120	0.358060	−	0.933699i	$-0.383438\pi$
$397$	14.2686	0.716120	0.358060	+	0.933699i	$0.383438\pi$
$398$	0	0
$399$	4.82694	0.241649
$400$	0	0
$401$	17.8291	0.890342	0.445171	−	0.895446i	$-0.353143\pi$
$401$	17.8291	0.890342	0.445171	+	0.895446i	$0.353143\pi$
$402$	0	0
$403$	7.46740	0.371977
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−61.2826	−3.03767
$408$	0	0
$409$	24.0428	1.18884	0.594420	−	0.804154i	$-0.297382\pi$
$409$	24.0428	1.18884	0.594420	+	0.804154i	$0.297382\pi$
$410$	0	0
$411$	13.7022	0.675882
$412$	0	0
$413$	4.51835	0.222334
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−23.4718	−1.14942
$418$	0	0
$419$	−28.0809	−1.37184	−0.685922	−	0.727675i	$-0.740601\pi$
$419$	−28.0809	−1.37184	−0.685922	+	0.727675i	$0.740601\pi$
$420$	0	0
$421$	20.1952	0.984254	0.492127	−	0.870523i	$-0.336220\pi$
$421$	20.1952	0.984254	0.492127	+	0.870523i	$0.336220\pi$
$422$	0	0
$423$	−4.93421	−0.239910
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.16030	0.201331
$428$	0	0
$429$	10.9548	0.528902
$430$	0	0
$431$	−4.15210	−0.200000	−0.0999999	−	0.994987i	$-0.531884\pi$
$431$	−4.15210	−0.200000	−0.0999999	+	0.994987i	$0.531884\pi$
$432$	0	0
$433$	−30.3201	−1.45709	−0.728545	−	0.684998i	$-0.759803\pi$
$433$	−30.3201	−1.45709	−0.728545	+	0.684998i	$0.759803\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.7425	0.848738
$438$	0	0
$439$	−3.18451	−0.151988	−0.0759941	−	0.997108i	$-0.524213\pi$
$439$	−3.18451	−0.151988	−0.0759941	+	0.997108i	$0.524213\pi$
$440$	0	0
$441$	−6.08314	−0.289674
$442$	0	0
$443$	30.4607	1.44723	0.723615	−	0.690204i	$-0.242479\pi$
$443$	30.4607	1.44723	0.723615	+	0.690204i	$0.242479\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6938	0.505800
$448$	0	0
$449$	1.35787	0.0640820	0.0320410	−	0.999487i	$-0.489799\pi$
$449$	1.35787	0.0640820	0.0320410	+	0.999487i	$0.489799\pi$
$450$	0	0
$451$	−19.0789	−0.898392
$452$	0	0
$453$	−7.37520	−0.346517
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.89208	0.462732	0.231366	−	0.972867i	$-0.425680\pi$
$457$	9.89208	0.462732	0.231366	+	0.972867i	$0.425680\pi$
$458$	0	0
$459$	−0.642866	−0.0300064
$460$	0	0
$461$	−21.0835	−0.981954	−0.490977	−	0.871172i	$-0.663360\pi$
$461$	−21.0835	−0.981954	−0.490977	+	0.871172i	$0.663360\pi$
$462$	0	0
$463$	−0.514142	−0.0238942	−0.0119471	−	0.999929i	$-0.503803\pi$
$463$	−0.514142	−0.0238942	−0.0119471	+	0.999929i	$0.503803\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.5740	−0.859502	−0.429751	−	0.902947i	$-0.641399\pi$
$467$	−18.5740	−0.859502	−0.429751	+	0.902947i	$0.641399\pi$
$468$	0	0
$469$	−9.11353	−0.420824
$470$	0	0
$471$	13.0329	0.600525
$472$	0	0
$473$	4.15046	0.190838
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.94634	−0.272264
$478$	0	0
$479$	22.7247	1.03832	0.519159	−	0.854677i	$-0.326245\pi$
$479$	22.7247	1.03832	0.519159	+	0.854677i	$0.326245\pi$
$480$	0	0
$481$	22.8876	1.04359
$482$	0	0
$483$	3.37011	0.153345
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.5852	−1.20469	−0.602346	−	0.798235i	$-0.705767\pi$
$487$	−26.5852	−1.20469	−0.602346	+	0.798235i	$0.705767\pi$
$488$	0	0
$489$	7.39602	0.334459
$490$	0	0
$491$	5.39133	0.243307	0.121654	−	0.992573i	$-0.461180\pi$
$491$	5.39133	0.243307	0.121654	+	0.992573i	$0.461180\pi$
$492$	0	0
$493$	6.51915	0.293608
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.4376	−0.513047
$498$	0	0
$499$	−14.5574	−0.651677	−0.325839	−	0.945425i	$-0.605647\pi$
$499$	−14.5574	−0.651677	−0.325839	+	0.945425i	$0.605647\pi$
$500$	0	0
$501$	20.1265	0.899187
$502$	0	0
$503$	33.9160	1.51224	0.756121	−	0.654432i	$-0.227092\pi$
$503$	33.9160	1.51224	0.756121	+	0.654432i	$0.227092\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.90864	0.395647
$508$	0	0
$509$	−16.7914	−0.744265	−0.372132	−	0.928180i	$-0.621373\pi$
$509$	−16.7914	−0.744265	−0.372132	+	0.928180i	$0.621373\pi$
$510$	0	0
$511$	5.20228	0.230135
$512$	0	0
$513$	5.04105	0.222568
$514$	0	0
$515$	0	0
$516$	0	0
$517$	26.7232	1.17528
$518$	0	0
$519$	−2.22112	−0.0974965
$520$	0	0
$521$	−17.0913	−0.748782	−0.374391	−	0.927271i	$-0.622148\pi$
$521$	−17.0913	−0.748782	−0.374391	+	0.927271i	$0.622148\pi$
$522$	0	0
$523$	−20.6505	−0.902983	−0.451491	−	0.892275i	$-0.649108\pi$
$523$	−20.6505	−0.902983	−0.451491	+	0.892275i	$0.649108\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.37332	0.103383
$528$	0	0
$529$	−10.6124	−0.461409
$530$	0	0
$531$	4.71878	0.204778
$532$	0	0
$533$	7.12554	0.308641
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.0385919	0.00166536
$538$	0	0
$539$	32.9457	1.41907
$540$	0	0
$541$	−12.1666	−0.523082	−0.261541	−	0.965192i	$-0.584231\pi$
$541$	−12.1666	−0.523082	−0.261541	+	0.965192i	$0.584231\pi$
$542$	0	0
$543$	−0.146945	−0.00630600
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11.1670	−0.477465	−0.238733	−	0.971085i	$-0.576732\pi$
$547$	−11.1670	−0.477465	−0.238733	+	0.971085i	$0.576732\pi$
$548$	0	0
$549$	4.34485	0.185434
$550$	0	0
$551$	−51.1201	−2.17779
$552$	0	0
$553$	11.3461	0.482486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.9282	−0.547787	−0.273894	−	0.961760i	$-0.588312\pi$
$557$	−12.9282	−0.547787	−0.273894	+	0.961760i	$0.588312\pi$
$558$	0	0
$559$	−1.55010	−0.0655622
$560$	0	0
$561$	3.48170	0.146997
$562$	0	0
$563$	31.7249	1.33705	0.668523	−	0.743692i	$-0.266927\pi$
$563$	31.7249	1.33705	0.668523	+	0.743692i	$0.266927\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.957526	0.0402123
$568$	0	0
$569$	28.8066	1.20764	0.603818	−	0.797122i	$-0.293646\pi$
$569$	28.8066	1.20764	0.603818	+	0.797122i	$0.293646\pi$
$570$	0	0
$571$	−18.1701	−0.760396	−0.380198	−	0.924905i	$-0.624144\pi$
$571$	−18.1701	−0.760396	−0.380198	+	0.924905i	$0.624144\pi$
$572$	0	0
$573$	0.459682	0.0192035
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−40.7956	−1.69834	−0.849172	−	0.528116i	$-0.822899\pi$
$577$	−40.7956	−1.69834	−0.849172	+	0.528116i	$0.822899\pi$
$578$	0	0
$579$	−19.0231	−0.790575
$580$	0	0
$581$	1.33487	0.0553797
$582$	0	0
$583$	32.2048	1.33379
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.20488	0.379926	0.189963	−	0.981791i	$-0.439163\pi$
$587$	9.20488	0.379926	0.189963	+	0.981791i	$0.439163\pi$
$588$	0	0
$589$	−18.6104	−0.766829
$590$	0	0
$591$	12.1151	0.498348
$592$	0	0
$593$	12.3856	0.508614	0.254307	−	0.967124i	$-0.418153\pi$
$593$	12.3856	0.508614	0.254307	+	0.967124i	$0.418153\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.4872	−0.674778
$598$	0	0
$599$	−18.1732	−0.742536	−0.371268	−	0.928526i	$-0.621077\pi$
$599$	−18.1732	−0.742536	−0.371268	+	0.928526i	$0.621077\pi$
$600$	0	0
$601$	29.8155	1.21620	0.608099	−	0.793861i	$-0.291932\pi$
$601$	29.8155	1.21620	0.608099	+	0.793861i	$0.291932\pi$
$602$	0	0
$603$	−9.51778	−0.387594
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.16747	0.372096	0.186048	−	0.982541i	$-0.440432\pi$
$607$	9.16747	0.372096	0.186048	+	0.982541i	$0.440432\pi$
$608$	0	0
$609$	−9.71005	−0.393471
$610$	0	0
$611$	−9.98049	−0.403767
$612$	0	0
$613$	14.0540	0.567638	0.283819	−	0.958878i	$-0.408399\pi$
$613$	14.0540	0.567638	0.283819	+	0.958878i	$0.408399\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.2343	−1.45874	−0.729368	−	0.684121i	$-0.760186\pi$
$617$	−36.2343	−1.45874	−0.729368	+	0.684121i	$0.760186\pi$
$618$	0	0
$619$	25.6751	1.03197	0.515986	−	0.856597i	$-0.327426\pi$
$619$	25.6751	1.03197	0.515986	+	0.856597i	$0.327426\pi$
$620$	0	0
$621$	3.51960	0.141237
$622$	0	0
$623$	−1.63557	−0.0655278
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−27.3018	−1.09033
$628$	0	0
$629$	7.27424	0.290043
$630$	0	0
$631$	−24.6723	−0.982190	−0.491095	−	0.871106i	$-0.663403\pi$
$631$	−24.6723	−0.982190	−0.491095	+	0.871106i	$0.663403\pi$
$632$	0	0
$633$	−18.1862	−0.722838
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.3044	−0.487520
$638$	0	0
$639$	−11.9450	−0.472535
$640$	0	0
$641$	−29.9378	−1.18247	−0.591236	−	0.806499i	$-0.701360\pi$
$641$	−29.9378	−1.18247	−0.591236	+	0.806499i	$0.701360\pi$
$642$	0	0
$643$	−10.1343	−0.399658	−0.199829	−	0.979831i	$-0.564039\pi$
$643$	−10.1343	−0.399658	−0.199829	+	0.979831i	$0.564039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.2593	−1.77932	−0.889662	−	0.456619i	$-0.849060\pi$
$647$	−45.2593	−1.77932	−0.889662	+	0.456619i	$0.849060\pi$
$648$	0	0
$649$	−25.5564	−1.00318
$650$	0	0
$651$	−3.53497	−0.138547
$652$	0	0
$653$	32.9848	1.29080	0.645398	−	0.763846i	$-0.276691\pi$
$653$	32.9848	1.29080	0.645398	+	0.763846i	$0.276691\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.43304	0.211963
$658$	0	0
$659$	2.55453	0.0995102	0.0497551	−	0.998761i	$-0.484156\pi$
$659$	2.55453	0.0995102	0.0497551	+	0.998761i	$0.484156\pi$
$660$	0	0
$661$	−45.4124	−1.76634	−0.883170	−	0.469054i	$-0.844595\pi$
$661$	−45.4124	−1.76634	−0.883170	+	0.469054i	$0.844595\pi$
$662$	0	0
$663$	−1.30033	−0.0505007
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35.6915	−1.38198
$668$	0	0
$669$	−23.5103	−0.908959
$670$	0	0
$671$	−23.5312	−0.908413
$672$	0	0
$673$	−11.3427	−0.437228	−0.218614	−	0.975811i	$-0.570153\pi$
$673$	−11.3427	−0.437228	−0.218614	+	0.975811i	$0.570153\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.4684	0.594499	0.297249	−	0.954800i	$-0.403931\pi$
$677$	15.4684	0.594499	0.297249	+	0.954800i	$0.403931\pi$
$678$	0	0
$679$	−13.9682	−0.536051
$680$	0	0
$681$	−19.6803	−0.754151
$682$	0	0
$683$	−6.79302	−0.259928	−0.129964	−	0.991519i	$-0.541486\pi$
$683$	−6.79302	−0.259928	−0.129964	+	0.991519i	$0.541486\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.30698	−0.202474
$688$	0	0
$689$	−12.0277	−0.458220
$690$	0	0
$691$	42.3195	1.60991	0.804954	−	0.593337i	$-0.202190\pi$
$691$	42.3195	1.60991	0.804954	+	0.593337i	$0.202190\pi$
$692$	0	0
$693$	−5.18586	−0.196995
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.26467	0.0857804
$698$	0	0
$699$	−23.2693	−0.880127
$700$	0	0
$701$	32.2924	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$701$	32.2924	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$702$	0	0
$703$	−57.0411	−2.15135
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.84647	−0.370314
$708$	0	0
$709$	−2.15883	−0.0810767	−0.0405383	−	0.999178i	$-0.512907\pi$
$709$	−2.15883	−0.0810767	−0.0405383	+	0.999178i	$0.512907\pi$
$710$	0	0
$711$	11.8494	0.444388
$712$	0	0
$713$	−12.9936	−0.486614
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.0750	0.562985
$718$	0	0
$719$	1.59152	0.0593537	0.0296768	−	0.999560i	$-0.490552\pi$
$719$	1.59152	0.0593537	0.0296768	+	0.999560i	$0.490552\pi$
$720$	0	0
$721$	−13.3908	−0.498701
$722$	0	0
$723$	18.9133	0.703394
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.33982	0.0496910	0.0248455	−	0.999691i	$-0.492091\pi$
$727$	1.33982	0.0496910	0.0248455	+	0.999691i	$0.492091\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.492659	−0.0182217
$732$	0	0
$733$	43.8507	1.61966	0.809831	−	0.586664i	$-0.199559\pi$
$733$	43.8507	1.61966	0.809831	+	0.586664i	$0.199559\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	51.5473	1.89877
$738$	0	0
$739$	18.3674	0.675656	0.337828	−	0.941208i	$-0.390308\pi$
$739$	18.3674	0.675656	0.337828	+	0.941208i	$0.390308\pi$
$740$	0	0
$741$	10.1966	0.374581
$742$	0	0
$743$	−21.5051	−0.788947	−0.394474	−	0.918907i	$-0.629073\pi$
$743$	−21.5051	−0.788947	−0.394474	+	0.918907i	$0.629073\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.39408	0.0510068
$748$	0	0
$749$	−8.68215	−0.317239
$750$	0	0
$751$	−7.02810	−0.256459	−0.128230	−	0.991745i	$-0.540929\pi$
$751$	−7.02810	−0.256459	−0.128230	+	0.991745i	$0.540929\pi$
$752$	0	0
$753$	4.56761	0.166453
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.39361	0.196034	0.0980171	−	0.995185i	$-0.468750\pi$
$757$	5.39361	0.196034	0.0980171	+	0.995185i	$0.468750\pi$
$758$	0	0
$759$	−19.0618	−0.691900
$760$	0	0
$761$	−20.1500	−0.730438	−0.365219	−	0.930922i	$-0.619006\pi$
$761$	−20.1500	−0.730438	−0.365219	+	0.930922i	$0.619006\pi$
$762$	0	0
$763$	2.27446	0.0823411
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.54472	0.344640
$768$	0	0
$769$	30.9167	1.11488	0.557442	−	0.830216i	$-0.311783\pi$
$769$	30.9167	1.11488	0.557442	+	0.830216i	$0.311783\pi$
$770$	0	0
$771$	20.2556	0.729486
$772$	0	0
$773$	−31.9913	−1.15065	−0.575323	−	0.817927i	$-0.695124\pi$
$773$	−31.9913	−1.15065	−0.575323	+	0.817927i	$0.695124\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10.8347	−0.388693
$778$	0	0
$779$	−17.7584	−0.636262
$780$	0	0
$781$	64.6927	2.31489
$782$	0	0
$783$	−10.1408	−0.362401
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.4220	0.870550	0.435275	−	0.900298i	$-0.356651\pi$
$787$	24.4220	0.870550	0.435275	+	0.900298i	$0.356651\pi$
$788$	0	0
$789$	30.7789	1.09576
$790$	0	0
$791$	12.8666	0.457485
$792$	0	0
$793$	8.78837	0.312084
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−49.6380	−1.75827	−0.879133	−	0.476576i	$-0.841878\pi$
$797$	−49.6380	−1.75827	−0.879133	+	0.476576i	$0.841878\pi$
$798$	0	0
$799$	−3.17204	−0.112219
$800$	0	0
$801$	−1.70812	−0.0603535
$802$	0	0
$803$	−29.4248	−1.03838
$804$	0	0
$805$	0	0
$806$	0	0
$807$	26.6739	0.938965
$808$	0	0
$809$	−18.8392	−0.662350	−0.331175	−	0.943569i	$-0.607445\pi$
$809$	−18.8392	−0.662350	−0.331175	+	0.943569i	$0.607445\pi$
$810$	0	0
$811$	18.7021	0.656721	0.328360	−	0.944553i	$-0.393504\pi$
$811$	18.7021	0.656721	0.328360	+	0.944553i	$0.393504\pi$
$812$	0	0
$813$	5.52945	0.193926
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.86320	0.135156
$818$	0	0
$819$	1.93680	0.0676772
$820$	0	0
$821$	36.7240	1.28168	0.640838	−	0.767676i	$-0.278587\pi$
$821$	36.7240	1.28168	0.640838	+	0.767676i	$0.278587\pi$
$822$	0	0
$823$	−40.5157	−1.41229	−0.706145	−	0.708068i	$-0.749567\pi$
$823$	−40.5157	−1.41229	−0.706145	+	0.708068i	$0.749567\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.1733	−0.388534	−0.194267	−	0.980949i	$-0.562233\pi$
$827$	−11.1733	−0.388534	−0.194267	+	0.980949i	$0.562233\pi$
$828$	0	0
$829$	36.8737	1.28068	0.640338	−	0.768093i	$-0.278794\pi$
$829$	36.8737	1.28068	0.640338	+	0.768093i	$0.278794\pi$
$830$	0	0
$831$	30.9233	1.07272
$832$	0	0
$833$	−3.91065	−0.135496
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3.69178	−0.127606
$838$	0	0
$839$	−11.4389	−0.394913	−0.197457	−	0.980312i	$-0.563268\pi$
$839$	−11.4389	−0.394913	−0.197457	+	0.980312i	$0.563268\pi$
$840$	0	0
$841$	73.8351	2.54604
$842$	0	0
$843$	16.8489	0.580306
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.5533	0.603139
$848$	0	0
$849$	11.2627	0.386537
$850$	0	0
$851$	−39.8254	−1.36520
$852$	0	0
$853$	24.6432	0.843768	0.421884	−	0.906650i	$-0.361369\pi$
$853$	24.6432	0.843768	0.421884	+	0.906650i	$0.361369\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.2342	−0.383752	−0.191876	−	0.981419i	$-0.561457\pi$
$857$	−11.2342	−0.383752	−0.191876	+	0.981419i	$0.561457\pi$
$858$	0	0
$859$	−15.5193	−0.529511	−0.264756	−	0.964316i	$-0.585291\pi$
$859$	−15.5193	−0.529511	−0.264756	+	0.964316i	$0.585291\pi$
$860$	0	0
$861$	−3.37314	−0.114956
$862$	0	0
$863$	19.5189	0.664432	0.332216	−	0.943203i	$-0.392204\pi$
$863$	19.5189	0.664432	0.332216	+	0.943203i	$0.392204\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.5867	0.563315
$868$	0	0
$869$	−64.1752	−2.17699
$870$	0	0
$871$	−19.2517	−0.652320
$872$	0	0
$873$	−14.5878	−0.493723
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.7361	0.463836	0.231918	−	0.972735i	$-0.425500\pi$
$877$	13.7361	0.463836	0.231918	+	0.972735i	$0.425500\pi$
$878$	0	0
$879$	11.1995	0.377751
$880$	0	0
$881$	−52.4110	−1.76577	−0.882886	−	0.469588i	$-0.844402\pi$
$881$	−52.4110	−1.76577	−0.882886	+	0.469588i	$0.844402\pi$
$882$	0	0
$883$	16.7017	0.562056	0.281028	−	0.959700i	$-0.409325\pi$
$883$	16.7017	0.562056	0.281028	+	0.959700i	$0.409325\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.9008	0.869664	0.434832	−	0.900512i	$-0.356808\pi$
$887$	25.9008	0.869664	0.434832	+	0.900512i	$0.356808\pi$
$888$	0	0
$889$	−13.6571	−0.458045
$890$	0	0
$891$	−5.41590	−0.181439
$892$	0	0
$893$	24.8736	0.832364
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.11914	0.237701
$898$	0	0
$899$	37.4374	1.24861
$900$	0	0
$901$	−3.82270	−0.127353
$902$	0	0
$903$	0.733798	0.0244193
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.74821	−0.157662	−0.0788308	−	0.996888i	$-0.525119\pi$
$907$	−4.74821	−0.157662	−0.0788308	+	0.996888i	$0.525119\pi$
$908$	0	0
$909$	−10.2832	−0.341073
$910$	0	0
$911$	−12.2828	−0.406947	−0.203473	−	0.979080i	$-0.565223\pi$
$911$	−12.2828	−0.406947	−0.203473	+	0.979080i	$0.565223\pi$
$912$	0	0
$913$	−7.55020	−0.249875
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.123201	−0.00406847
$918$	0	0
$919$	20.6802	0.682177	0.341089	−	0.940031i	$-0.389204\pi$
$919$	20.6802	0.682177	0.341089	+	0.940031i	$0.389204\pi$
$920$	0	0
$921$	−24.1289	−0.795073
$922$	0	0
$923$	−24.1612	−0.795276
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−13.9848	−0.459322
$928$	0	0
$929$	24.0728	0.789803	0.394902	−	0.918723i	$-0.370779\pi$
$929$	24.0728	0.789803	0.394902	+	0.918723i	$0.370779\pi$
$930$	0	0
$931$	30.6654	1.00502
$932$	0	0
$933$	−2.07312	−0.0678708
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−44.7993	−1.46353	−0.731765	−	0.681557i	$-0.761303\pi$
$937$	−44.7993	−1.46353	−0.731765	+	0.681557i	$0.761303\pi$
$938$	0	0
$939$	8.51966	0.278029
$940$	0	0
$941$	−47.0851	−1.53493	−0.767465	−	0.641091i	$-0.778482\pi$
$941$	−47.0851	−1.53493	−0.767465	+	0.641091i	$0.778482\pi$
$942$	0	0
$943$	−12.3987	−0.403759
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.9753	0.584120	0.292060	−	0.956400i	$-0.405659\pi$
$947$	17.9753	0.584120	0.292060	+	0.956400i	$0.405659\pi$
$948$	0	0
$949$	10.9895	0.356733
$950$	0	0
$951$	−13.6070	−0.441236
$952$	0	0
$953$	−35.9516	−1.16459	−0.582293	−	0.812979i	$-0.697844\pi$
$953$	−35.9516	−1.16459	−0.582293	+	0.812979i	$0.697844\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	54.9213	1.77535
$958$	0	0
$959$	−13.1203	−0.423675
$960$	0	0
$961$	−17.3708	−0.560348
$962$	0	0
$963$	−9.06727	−0.292189
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9.59917	0.308689	0.154344	−	0.988017i	$-0.450674\pi$
$967$	9.59917	0.308689	0.154344	+	0.988017i	$0.450674\pi$
$968$	0	0
$969$	3.24072	0.104107
$970$	0	0
$971$	−27.8888	−0.894995	−0.447497	−	0.894285i	$-0.647685\pi$
$971$	−27.8888	−0.894995	−0.447497	+	0.894285i	$0.647685\pi$
$972$	0	0
$973$	22.4748	0.720510
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.7436	1.81539	0.907694	−	0.419632i	$-0.137841\pi$
$977$	56.7436	1.81539	0.907694	+	0.419632i	$0.137841\pi$
$978$	0	0
$979$	9.25101	0.295664
$980$	0	0
$981$	2.37535	0.0758392
$982$	0	0
$983$	−11.2511	−0.358855	−0.179427	−	0.983771i	$-0.557425\pi$
$983$	−11.2511	−0.358855	−0.179427	+	0.983771i	$0.557425\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.72464	0.150387
$988$	0	0
$989$	2.69724	0.0857672
$990$	0	0
$991$	34.4601	1.09466	0.547331	−	0.836916i	$-0.315644\pi$
$991$	34.4601	1.09466	0.547331	+	0.836916i	$0.315644\pi$
$992$	0	0
$993$	22.6320	0.718206
$994$	0	0
$995$	0	0
$996$	0	0
$997$	51.1902	1.62121	0.810606	−	0.585592i	$-0.199138\pi$
$997$	51.1902	1.62121	0.810606	+	0.585592i	$0.199138\pi$
$998$	0	0
$999$	−11.3153	−0.358001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.m.1.8		12
5.2	odd	4		7500.2.d.g.1249.20		24
5.3	odd	4		7500.2.d.g.1249.5		24
5.4	even	2		7500.2.a.n.1.5		12
25.2	odd	20		1500.2.o.c.649.5		24
25.9	even	10		1500.2.m.c.901.3		24
25.11	even	5		1500.2.m.d.601.4		24
25.12	odd	20		300.2.o.a.169.2	✓	24
25.13	odd	20		1500.2.o.c.349.5		24
25.14	even	10		1500.2.m.c.601.3		24
25.16	even	5		1500.2.m.d.901.4		24
25.23	odd	20		300.2.o.a.229.2	yes	24
75.23	even	20		900.2.w.c.829.4		24
75.62	even	20		900.2.w.c.469.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.2	✓	24	25.12	odd	20
300.2.o.a.229.2	yes	24	25.23	odd	20
900.2.w.c.469.4		24	75.62	even	20
900.2.w.c.829.4		24	75.23	even	20
1500.2.m.c.601.3		24	25.14	even	10
1500.2.m.c.901.3		24	25.9	even	10
1500.2.m.d.601.4		24	25.11	even	5
1500.2.m.d.901.4		24	25.16	even	5
1500.2.o.c.349.5		24	25.13	odd	20
1500.2.o.c.649.5		24	25.2	odd	20
7500.2.a.m.1.8		12	1.1	even	1	trivial
7500.2.a.n.1.5		12	5.4	even	2
7500.2.d.g.1249.5		24	5.3	odd	4
7500.2.d.g.1249.20		24	5.2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.957526 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.957526 q^{7} +1.00000 q^{9} -5.41590 q^{11} +2.02271 q^{13} +0.642866 q^{17} -5.04105 q^{19} -0.957526 q^{21} -3.51960 q^{23} -1.00000 q^{27} +10.1408 q^{29} +3.69178 q^{31} +5.41590 q^{33} +11.3153 q^{37} -2.02271 q^{39} +3.52277 q^{41} -0.766348 q^{43} -4.93421 q^{47} -6.08314 q^{49} -0.642866 q^{51} -5.94634 q^{53} +5.04105 q^{57} +4.71878 q^{59} +4.34485 q^{61} +0.957526 q^{63} -9.51778 q^{67} +3.51960 q^{69} -11.9450 q^{71} +5.43304 q^{73} -5.18586 q^{77} +11.8494 q^{79} +1.00000 q^{81} +1.39408 q^{83} -10.1408 q^{87} -1.70812 q^{89} +1.93680 q^{91} -3.69178 q^{93} -14.5878 q^{97} -5.41590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 8 * q^7 + 12 * q^9 + 2 * q^11 - 8 * q^17 + 10 * q^19 + 8 * q^21 - 18 * q^23 - 12 * q^27 + 8 * q^29 - 2 * q^31 - 2 * q^33 - 4 * q^37 + 10 * q^41 - 28 * q^43 - 22 * q^47 + 28 * q^49 + 8 * q^51 - 16 * q^53 - 10 * q^57 - 2 * q^59 + 34 * q^61 - 8 * q^63 - 32 * q^67 + 18 * q^69 - 24 * q^73 - 18 * q^77 + 6 * q^79 + 12 * q^81 - 28 * q^83 - 8 * q^87 + 10 * q^89 + 20 * q^91 + 2 * q^93 - 16 * q^97 + 2 * q^99

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