LMFDB - Embedded newform 7500.2.a.n.1.10

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 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			1.10
Root			$1.62661$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.80992 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.80992 q^{7} +1.00000 q^{9} +0.190733 q^{11} +1.67943 q^{13} -4.60324 q^{17} -2.64126 q^{19} +3.80992 q^{21} -6.35567 q^{23} +1.00000 q^{27} +2.52008 q^{29} +3.74937 q^{31} +0.190733 q^{33} +11.8626 q^{37} +1.67943 q^{39} -7.18953 q^{41} +9.22619 q^{43} +4.54848 q^{47} +7.51545 q^{49} -4.60324 q^{51} -9.43006 q^{53} -2.64126 q^{57} +7.14057 q^{59} +9.53012 q^{61} +3.80992 q^{63} -6.05168 q^{67} -6.35567 q^{69} +13.2983 q^{71} +5.21152 q^{73} +0.726676 q^{77} -3.11168 q^{79} +1.00000 q^{81} -4.67837 q^{83} +2.52008 q^{87} +13.9423 q^{89} +6.39850 q^{91} +3.74937 q^{93} -6.69507 q^{97} +0.190733 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	3.80992	1.44001	0.720006	−	0.693968i	$-0.244139\pi$
$7$	3.80992	1.44001	0.720006	+	0.693968i	$0.244139\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.190733	0.0575081	0.0287541	−	0.999587i	$-0.490846\pi$
$11$	0.190733	0.0575081	0.0287541	+	0.999587i	$0.490846\pi$
$12$	0	0
$13$	1.67943	0.465791	0.232896	−	0.972502i	$-0.425180\pi$
$13$	1.67943	0.465791	0.232896	+	0.972502i	$0.425180\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.60324	−1.11645	−0.558225	−	0.829690i	$-0.688517\pi$
$17$	−4.60324	−1.11645	−0.558225	+	0.829690i	$0.688517\pi$
$18$	0	0
$19$	−2.64126	−0.605946	−0.302973	−	0.952999i	$-0.597979\pi$
$19$	−2.64126	−0.605946	−0.302973	+	0.952999i	$0.597979\pi$
$20$	0	0
$21$	3.80992	0.831392
$22$	0	0
$23$	−6.35567	−1.32525	−0.662625	−	0.748951i	$-0.730558\pi$
$23$	−6.35567	−1.32525	−0.662625	+	0.748951i	$0.730558\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.52008	0.467966	0.233983	−	0.972241i	$-0.424824\pi$
$29$	2.52008	0.467966	0.233983	+	0.972241i	$0.424824\pi$
$30$	0	0
$31$	3.74937	0.673407	0.336704	−	0.941611i	$-0.390688\pi$
$31$	3.74937	0.673407	0.336704	+	0.941611i	$0.390688\pi$
$32$	0	0
$33$	0.190733	0.0332023
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.8626	1.95019	0.975097	−	0.221778i	$-0.0711861\pi$
$37$	11.8626	1.95019	0.975097	+	0.221778i	$0.0711861\pi$
$38$	0	0
$39$	1.67943	0.268925
$40$	0	0
$41$	−7.18953	−1.12282	−0.561408	−	0.827539i	$-0.689740\pi$
$41$	−7.18953	−1.12282	−0.561408	+	0.827539i	$0.689740\pi$
$42$	0	0
$43$	9.22619	1.40698	0.703491	−	0.710705i	$-0.251624\pi$
$43$	9.22619	1.40698	0.703491	+	0.710705i	$0.251624\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.54848	0.663464	0.331732	−	0.943374i	$-0.392367\pi$
$47$	4.54848	0.663464	0.331732	+	0.943374i	$0.392367\pi$
$48$	0	0
$49$	7.51545	1.07364
$50$	0	0
$51$	−4.60324	−0.644583
$52$	0	0
$53$	−9.43006	−1.29532	−0.647659	−	0.761930i	$-0.724252\pi$
$53$	−9.43006	−1.29532	−0.647659	+	0.761930i	$0.724252\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.64126	−0.349843
$58$	0	0
$59$	7.14057	0.929623	0.464812	−	0.885410i	$-0.346122\pi$
$59$	7.14057	0.929623	0.464812	+	0.885410i	$0.346122\pi$
$60$	0	0
$61$	9.53012	1.22021	0.610103	−	0.792322i	$-0.291128\pi$
$61$	9.53012	1.22021	0.610103	+	0.792322i	$0.291128\pi$
$62$	0	0
$63$	3.80992	0.480004
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.05168	−0.739330	−0.369665	−	0.929165i	$-0.620528\pi$
$67$	−6.05168	−0.739330	−0.369665	+	0.929165i	$0.620528\pi$
$68$	0	0
$69$	−6.35567	−0.765133
$70$	0	0
$71$	13.2983	1.57822	0.789110	−	0.614252i	$-0.210542\pi$
$71$	13.2983	1.57822	0.789110	+	0.614252i	$0.210542\pi$
$72$	0	0
$73$	5.21152	0.609963	0.304981	−	0.952358i	$-0.401350\pi$
$73$	5.21152	0.609963	0.304981	+	0.952358i	$0.401350\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.726676	0.0828124
$78$	0	0
$79$	−3.11168	−0.350092	−0.175046	−	0.984560i	$-0.556007\pi$
$79$	−3.11168	−0.350092	−0.175046	+	0.984560i	$0.556007\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.67837	−0.513518	−0.256759	−	0.966475i	$-0.582655\pi$
$83$	−4.67837	−0.513518	−0.256759	+	0.966475i	$0.582655\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.52008	0.270181
$88$	0	0
$89$	13.9423	1.47788	0.738939	−	0.673772i	$-0.235327\pi$
$89$	13.9423	1.47788	0.738939	+	0.673772i	$0.235327\pi$
$90$	0	0
$91$	6.39850	0.670745
$92$	0	0
$93$	3.74937	0.388792
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.69507	−0.679782	−0.339891	−	0.940465i	$-0.610390\pi$
$97$	−6.69507	−0.679782	−0.339891	+	0.940465i	$0.610390\pi$
$98$	0	0
$99$	0.190733	0.0191694
$100$	0	0
$101$	10.5147	1.04625	0.523127	−	0.852255i	$-0.324765\pi$
$101$	10.5147	1.04625	0.523127	+	0.852255i	$0.324765\pi$
$102$	0	0
$103$	12.9648	1.27746	0.638732	−	0.769429i	$-0.279459\pi$
$103$	12.9648	1.27746	0.638732	+	0.769429i	$0.279459\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.64606	0.932520	0.466260	−	0.884648i	$-0.345601\pi$
$107$	9.64606	0.932520	0.466260	+	0.884648i	$0.345601\pi$
$108$	0	0
$109$	20.4454	1.95831	0.979157	−	0.203103i	$-0.0651025\pi$
$109$	20.4454	1.95831	0.979157	+	0.203103i	$0.0651025\pi$
$110$	0	0
$111$	11.8626	1.12595
$112$	0	0
$113$	4.04417	0.380444	0.190222	−	0.981741i	$-0.439079\pi$
$113$	4.04417	0.380444	0.190222	+	0.981741i	$0.439079\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.67943	0.155264
$118$	0	0
$119$	−17.5380	−1.60770
$120$	0	0
$121$	−10.9636	−0.996693
$122$	0	0
$123$	−7.18953	−0.648258
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.7537	1.48665	0.743327	−	0.668928i	$-0.233247\pi$
$127$	16.7537	1.48665	0.743327	+	0.668928i	$0.233247\pi$
$128$	0	0
$129$	9.22619	0.812321
$130$	0	0
$131$	−3.30750	−0.288978	−0.144489	−	0.989506i	$-0.546154\pi$
$131$	−3.30750	−0.288978	−0.144489	+	0.989506i	$0.546154\pi$
$132$	0	0
$133$	−10.0630	−0.872570
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.09916	0.179343	0.0896717	−	0.995971i	$-0.471418\pi$
$137$	2.09916	0.179343	0.0896717	+	0.995971i	$0.471418\pi$
$138$	0	0
$139$	−13.2849	−1.12681	−0.563404	−	0.826181i	$-0.690509\pi$
$139$	−13.2849	−1.12681	−0.563404	+	0.826181i	$0.690509\pi$
$140$	0	0
$141$	4.54848	0.383051
$142$	0	0
$143$	0.320323	0.0267868
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.51545	0.619864
$148$	0	0
$149$	−2.79913	−0.229313	−0.114657	−	0.993405i	$-0.536577\pi$
$149$	−2.79913	−0.229313	−0.114657	+	0.993405i	$0.536577\pi$
$150$	0	0
$151$	−6.71330	−0.546320	−0.273160	−	0.961969i	$-0.588069\pi$
$151$	−6.71330	−0.546320	−0.273160	+	0.961969i	$0.588069\pi$
$152$	0	0
$153$	−4.60324	−0.372150
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.9495	1.11329	0.556646	−	0.830750i	$-0.312088\pi$
$157$	13.9495	1.11329	0.556646	+	0.830750i	$0.312088\pi$
$158$	0	0
$159$	−9.43006	−0.747853
$160$	0	0
$161$	−24.2146	−1.90838
$162$	0	0
$163$	−8.87503	−0.695146	−0.347573	−	0.937653i	$-0.612994\pi$
$163$	−8.87503	−0.695146	−0.347573	+	0.937653i	$0.612994\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.5944	1.90318	0.951588	−	0.307378i	$-0.0994515\pi$
$167$	24.5944	1.90318	0.951588	+	0.307378i	$0.0994515\pi$
$168$	0	0
$169$	−10.1795	−0.783039
$170$	0	0
$171$	−2.64126	−0.201982
$172$	0	0
$173$	−16.9350	−1.28755	−0.643774	−	0.765216i	$-0.722632\pi$
$173$	−16.9350	−1.28755	−0.643774	+	0.765216i	$0.722632\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.14057	0.536718
$178$	0	0
$179$	14.7497	1.10244	0.551222	−	0.834358i	$-0.314162\pi$
$179$	14.7497	1.10244	0.551222	+	0.834358i	$0.314162\pi$
$180$	0	0
$181$	−10.9888	−0.816791	−0.408396	−	0.912805i	$-0.633912\pi$
$181$	−10.9888	−0.816791	−0.408396	+	0.912805i	$0.633912\pi$
$182$	0	0
$183$	9.53012	0.704486
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.877989	−0.0642049
$188$	0	0
$189$	3.80992	0.277131
$190$	0	0
$191$	−24.0355	−1.73915	−0.869575	−	0.493801i	$-0.835607\pi$
$191$	−24.0355	−1.73915	−0.869575	+	0.493801i	$0.835607\pi$
$192$	0	0
$193$	−7.50843	−0.540469	−0.270234	−	0.962795i	$-0.587101\pi$
$193$	−7.50843	−0.540469	−0.270234	+	0.962795i	$0.587101\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.5522	0.894306	0.447153	−	0.894457i	$-0.352438\pi$
$197$	12.5522	0.894306	0.447153	+	0.894457i	$0.352438\pi$
$198$	0	0
$199$	−19.7618	−1.40088	−0.700440	−	0.713711i	$-0.747013\pi$
$199$	−19.7618	−1.40088	−0.700440	+	0.713711i	$0.747013\pi$
$200$	0	0
$201$	−6.05168	−0.426852
$202$	0	0
$203$	9.60128	0.673877
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.35567	−0.441750
$208$	0	0
$209$	−0.503774	−0.0348468
$210$	0	0
$211$	−11.2479	−0.774336	−0.387168	−	0.922009i	$-0.626547\pi$
$211$	−11.2479	−0.774336	−0.387168	+	0.922009i	$0.626547\pi$
$212$	0	0
$213$	13.2983	0.911186
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.2848	0.969715
$218$	0	0
$219$	5.21152	0.352162
$220$	0	0
$221$	−7.73084	−0.520033
$222$	0	0
$223$	7.27196	0.486967	0.243483	−	0.969905i	$-0.421710\pi$
$223$	7.27196	0.486967	0.243483	+	0.969905i	$0.421710\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.46929	−0.628499	−0.314249	−	0.949340i	$-0.601753\pi$
$227$	−9.46929	−0.628499	−0.314249	+	0.949340i	$0.601753\pi$
$228$	0	0
$229$	21.2301	1.40292	0.701461	−	0.712708i	$-0.252531\pi$
$229$	21.2301	1.40292	0.701461	+	0.712708i	$0.252531\pi$
$230$	0	0
$231$	0.726676	0.0478118
$232$	0	0
$233$	−7.08932	−0.464437	−0.232218	−	0.972664i	$-0.574598\pi$
$233$	−7.08932	−0.464437	−0.232218	+	0.972664i	$0.574598\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.11168	−0.202126
$238$	0	0
$239$	−20.0117	−1.29445	−0.647224	−	0.762300i	$-0.724070\pi$
$239$	−20.0117	−1.29445	−0.647224	+	0.762300i	$0.724070\pi$
$240$	0	0
$241$	20.8240	1.34139	0.670696	−	0.741732i	$-0.265995\pi$
$241$	20.8240	1.34139	0.670696	+	0.741732i	$0.265995\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.43582	−0.282244
$248$	0	0
$249$	−4.67837	−0.296480
$250$	0	0
$251$	−29.7741	−1.87932	−0.939662	−	0.342104i	$-0.888861\pi$
$251$	−29.7741	−1.87932	−0.939662	+	0.342104i	$0.888861\pi$
$252$	0	0
$253$	−1.21224	−0.0762126
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2853	1.32774	0.663869	−	0.747849i	$-0.268913\pi$
$257$	21.2853	1.32774	0.663869	+	0.747849i	$0.268913\pi$
$258$	0	0
$259$	45.1954	2.80830
$260$	0	0
$261$	2.52008	0.155989
$262$	0	0
$263$	−21.8653	−1.34827	−0.674135	−	0.738608i	$-0.735483\pi$
$263$	−21.8653	−1.34827	−0.674135	+	0.738608i	$0.735483\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.9423	0.853253
$268$	0	0
$269$	−4.56132	−0.278109	−0.139054	−	0.990285i	$-0.544406\pi$
$269$	−4.56132	−0.278109	−0.139054	+	0.990285i	$0.544406\pi$
$270$	0	0
$271$	−12.9193	−0.784791	−0.392395	−	0.919797i	$-0.628354\pi$
$271$	−12.9193	−0.784791	−0.392395	+	0.919797i	$0.628354\pi$
$272$	0	0
$273$	6.39850	0.387255
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8.83521	−0.530856	−0.265428	−	0.964131i	$-0.585513\pi$
$277$	−8.83521	−0.530856	−0.265428	+	0.964131i	$0.585513\pi$
$278$	0	0
$279$	3.74937	0.224469
$280$	0	0
$281$	0.0305495	0.00182243	0.000911215	−	1.00000i	$-0.499710\pi$
$281$	0.0305495	0.00182243	0.000911215		1.00000i	$0.499710\pi$
$282$	0	0
$283$	4.80719	0.285758	0.142879	−	0.989740i	$-0.454364\pi$
$283$	4.80719	0.285758	0.142879	+	0.989740i	$0.454364\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.3915	−1.61687
$288$	0	0
$289$	4.18984	0.246461
$290$	0	0
$291$	−6.69507	−0.392472
$292$	0	0
$293$	−14.9705	−0.874587	−0.437294	−	0.899319i	$-0.644063\pi$
$293$	−14.9705	−0.874587	−0.437294	+	0.899319i	$0.644063\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.190733	0.0110674
$298$	0	0
$299$	−10.6739	−0.617290
$300$	0	0
$301$	35.1510	2.02607
$302$	0	0
$303$	10.5147	0.604055
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.47622	0.255472	0.127736	−	0.991808i	$-0.459229\pi$
$307$	4.47622	0.255472	0.127736	+	0.991808i	$0.459229\pi$
$308$	0	0
$309$	12.9648	0.737544
$310$	0	0
$311$	0.296114	0.0167911	0.00839554	−	0.999965i	$-0.497328\pi$
$311$	0.296114	0.0167911	0.00839554	+	0.999965i	$0.497328\pi$
$312$	0	0
$313$	−21.1569	−1.19586	−0.597930	−	0.801548i	$-0.704010\pi$
$313$	−21.1569	−1.19586	−0.597930	+	0.801548i	$0.704010\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.9166	1.45562	0.727812	−	0.685777i	$-0.240537\pi$
$317$	25.9166	1.45562	0.727812	+	0.685777i	$0.240537\pi$
$318$	0	0
$319$	0.480661	0.0269119
$320$	0	0
$321$	9.64606	0.538391
$322$	0	0
$323$	12.1583	0.676509
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.4454	1.13063
$328$	0	0
$329$	17.3293	0.955397
$330$	0	0
$331$	−3.07039	−0.168764	−0.0843819	−	0.996433i	$-0.526892\pi$
$331$	−3.07039	−0.168764	−0.0843819	+	0.996433i	$0.526892\pi$
$332$	0	0
$333$	11.8626	0.650065
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.2521	1.53899	0.769495	−	0.638652i	$-0.220508\pi$
$337$	28.2521	1.53899	0.769495	+	0.638652i	$0.220508\pi$
$338$	0	0
$339$	4.04417	0.219649
$340$	0	0
$341$	0.715129	0.0387264
$342$	0	0
$343$	1.96383	0.106037
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.6011	1.64275	0.821376	−	0.570387i	$-0.193207\pi$
$347$	30.6011	1.64275	0.821376	+	0.570387i	$0.193207\pi$
$348$	0	0
$349$	−16.1178	−0.862764	−0.431382	−	0.902169i	$-0.641974\pi$
$349$	−16.1178	−0.862764	−0.431382	+	0.902169i	$0.641974\pi$
$350$	0	0
$351$	1.67943	0.0896416
$352$	0	0
$353$	−0.431010	−0.0229404	−0.0114702	−	0.999934i	$-0.503651\pi$
$353$	−0.431010	−0.0229404	−0.0114702	+	0.999934i	$0.503651\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−17.5380	−0.928207
$358$	0	0
$359$	18.4738	0.975008	0.487504	−	0.873121i	$-0.337907\pi$
$359$	18.4738	0.975008	0.487504	+	0.873121i	$0.337907\pi$
$360$	0	0
$361$	−12.0238	−0.632829
$362$	0	0
$363$	−10.9636	−0.575441
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.2757	0.536386	0.268193	−	0.963365i	$-0.413573\pi$
$367$	10.2757	0.536386	0.268193	+	0.963365i	$0.413573\pi$
$368$	0	0
$369$	−7.18953	−0.374272
$370$	0	0
$371$	−35.9277	−1.86528
$372$	0	0
$373$	14.6433	0.758201	0.379100	−	0.925356i	$-0.376233\pi$
$373$	14.6433	0.758201	0.379100	+	0.925356i	$0.376233\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.23230	0.217975
$378$	0	0
$379$	−10.7051	−0.549884	−0.274942	−	0.961461i	$-0.588659\pi$
$379$	−10.7051	−0.549884	−0.274942	+	0.961461i	$0.588659\pi$
$380$	0	0
$381$	16.7537	0.858320
$382$	0	0
$383$	26.3174	1.34476	0.672378	−	0.740208i	$-0.265273\pi$
$383$	26.3174	1.34476	0.672378	+	0.740208i	$0.265273\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.22619	0.468994
$388$	0	0
$389$	0.291780	0.0147938	0.00739692	−	0.999973i	$-0.497645\pi$
$389$	0.291780	0.0147938	0.00739692	+	0.999973i	$0.497645\pi$
$390$	0	0
$391$	29.2567	1.47958
$392$	0	0
$393$	−3.30750	−0.166841
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.01611	0.101185	0.0505927	−	0.998719i	$-0.483889\pi$
$397$	2.01611	0.101185	0.0505927	+	0.998719i	$0.483889\pi$
$398$	0	0
$399$	−10.0630	−0.503778
$400$	0	0
$401$	9.88760	0.493763	0.246882	−	0.969046i	$-0.420594\pi$
$401$	9.88760	0.493763	0.246882	+	0.969046i	$0.420594\pi$
$402$	0	0
$403$	6.29683	0.313667
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.26258	0.112152
$408$	0	0
$409$	−37.6809	−1.86320	−0.931601	−	0.363483i	$-0.881587\pi$
$409$	−37.6809	−1.86320	−0.931601	+	0.363483i	$0.881587\pi$
$410$	0	0
$411$	2.09916	0.103544
$412$	0	0
$413$	27.2050	1.33867
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.2849	−0.650563
$418$	0	0
$419$	−15.6537	−0.764734	−0.382367	−	0.924011i	$-0.624891\pi$
$419$	−15.6537	−0.764734	−0.382367	+	0.924011i	$0.624891\pi$
$420$	0	0
$421$	14.5277	0.708036	0.354018	−	0.935239i	$-0.384815\pi$
$421$	14.5277	0.708036	0.354018	+	0.935239i	$0.384815\pi$
$422$	0	0
$423$	4.54848	0.221155
$424$	0	0
$425$	0	0
$426$	0	0
$427$	36.3089	1.75711
$428$	0	0
$429$	0.320323	0.0154653
$430$	0	0
$431$	−12.6264	−0.608192	−0.304096	−	0.952641i	$-0.598354\pi$
$431$	−12.6264	−0.608192	−0.304096	+	0.952641i	$0.598354\pi$
$432$	0	0
$433$	−3.49014	−0.167725	−0.0838626	−	0.996477i	$-0.526726\pi$
$433$	−3.49014	−0.167725	−0.0838626	+	0.996477i	$0.526726\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.7870	0.803030
$438$	0	0
$439$	−19.5630	−0.933689	−0.466844	−	0.884340i	$-0.654609\pi$
$439$	−19.5630	−0.933689	−0.466844	+	0.884340i	$0.654609\pi$
$440$	0	0
$441$	7.51545	0.357879
$442$	0	0
$443$	−2.77485	−0.131837	−0.0659185	−	0.997825i	$-0.520998\pi$
$443$	−2.77485	−0.131837	−0.0659185	+	0.997825i	$0.520998\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.79913	−0.132394
$448$	0	0
$449$	−38.4261	−1.81344	−0.906721	−	0.421732i	$-0.861422\pi$
$449$	−38.4261	−1.81344	−0.906721	+	0.421732i	$0.861422\pi$
$450$	0	0
$451$	−1.37128	−0.0645710
$452$	0	0
$453$	−6.71330	−0.315418
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.0296	−1.68539	−0.842696	−	0.538389i	$-0.819033\pi$
$457$	−36.0296	−1.68539	−0.842696	+	0.538389i	$0.819033\pi$
$458$	0	0
$459$	−4.60324	−0.214861
$460$	0	0
$461$	0.857592	0.0399421	0.0199710	−	0.999801i	$-0.493643\pi$
$461$	0.857592	0.0399421	0.0199710	+	0.999801i	$0.493643\pi$
$462$	0	0
$463$	10.4068	0.483643	0.241822	−	0.970321i	$-0.422255\pi$
$463$	10.4068	0.483643	0.241822	+	0.970321i	$0.422255\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.5705	−0.951889	−0.475944	−	0.879475i	$-0.657894\pi$
$467$	−20.5705	−0.951889	−0.475944	+	0.879475i	$0.657894\pi$
$468$	0	0
$469$	−23.0564	−1.06464
$470$	0	0
$471$	13.9495	0.642759
$472$	0	0
$473$	1.75974	0.0809128
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.43006	−0.431773
$478$	0	0
$479$	31.9380	1.45929	0.729643	−	0.683828i	$-0.239686\pi$
$479$	31.9380	1.45929	0.729643	+	0.683828i	$0.239686\pi$
$480$	0	0
$481$	19.9224	0.908383
$482$	0	0
$483$	−24.2146	−1.10180
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15.6815	0.710599	0.355299	−	0.934753i	$-0.384379\pi$
$487$	15.6815	0.710599	0.355299	+	0.934753i	$0.384379\pi$
$488$	0	0
$489$	−8.87503	−0.401343
$490$	0	0
$491$	16.4831	0.743870	0.371935	−	0.928259i	$-0.378694\pi$
$491$	16.4831	0.743870	0.371935	+	0.928259i	$0.378694\pi$
$492$	0	0
$493$	−11.6005	−0.522461
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.6655	2.27266
$498$	0	0
$499$	12.4339	0.556618	0.278309	−	0.960492i	$-0.410226\pi$
$499$	12.4339	0.556618	0.278309	+	0.960492i	$0.410226\pi$
$500$	0	0
$501$	24.5944	1.09880
$502$	0	0
$503$	4.33327	0.193211	0.0966055	−	0.995323i	$-0.469201\pi$
$503$	4.33327	0.193211	0.0966055	+	0.995323i	$0.469201\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10.1795	−0.452088
$508$	0	0
$509$	14.0120	0.621072	0.310536	−	0.950562i	$-0.399492\pi$
$509$	14.0120	0.621072	0.310536	+	0.950562i	$0.399492\pi$
$510$	0	0
$511$	19.8555	0.878354
$512$	0	0
$513$	−2.64126	−0.116614
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.867545	0.0381546
$518$	0	0
$519$	−16.9350	−0.743366
$520$	0	0
$521$	1.92829	0.0844801	0.0422400	−	0.999107i	$-0.486551\pi$
$521$	1.92829	0.0844801	0.0422400	+	0.999107i	$0.486551\pi$
$522$	0	0
$523$	−1.95894	−0.0856583	−0.0428291	−	0.999082i	$-0.513637\pi$
$523$	−1.95894	−0.0856583	−0.0428291	+	0.999082i	$0.513637\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.2593	−0.751826
$528$	0	0
$529$	17.3946	0.756287
$530$	0	0
$531$	7.14057	0.309874
$532$	0	0
$533$	−12.0743	−0.522998
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.7497	0.636497
$538$	0	0
$539$	1.43344	0.0617428
$540$	0	0
$541$	−34.6132	−1.48814	−0.744069	−	0.668103i	$-0.767107\pi$
$541$	−34.6132	−1.48814	−0.744069	+	0.668103i	$0.767107\pi$
$542$	0	0
$543$	−10.9888	−0.471575
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.9494	0.681947	0.340973	−	0.940073i	$-0.389243\pi$
$547$	15.9494	0.681947	0.340973	+	0.940073i	$0.389243\pi$
$548$	0	0
$549$	9.53012	0.406735
$550$	0	0
$551$	−6.65617	−0.283562
$552$	0	0
$553$	−11.8552	−0.504136
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.0343	1.82342	0.911711	−	0.410831i	$-0.134761\pi$
$557$	43.0343	1.82342	0.911711	+	0.410831i	$0.134761\pi$
$558$	0	0
$559$	15.4948	0.655359
$560$	0	0
$561$	−0.877989	−0.0370687
$562$	0	0
$563$	18.8560	0.794686	0.397343	−	0.917670i	$-0.369932\pi$
$563$	18.8560	0.794686	0.397343	+	0.917670i	$0.369932\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.80992	0.160001
$568$	0	0
$569$	−24.3652	−1.02144	−0.510721	−	0.859747i	$-0.670621\pi$
$569$	−24.3652	−1.02144	−0.510721	+	0.859747i	$0.670621\pi$
$570$	0	0
$571$	15.3391	0.641923	0.320962	−	0.947092i	$-0.395994\pi$
$571$	15.3391	0.641923	0.320962	+	0.947092i	$0.395994\pi$
$572$	0	0
$573$	−24.0355	−1.00410
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.3401	0.596985	0.298493	−	0.954412i	$-0.403516\pi$
$577$	14.3401	0.596985	0.298493	+	0.954412i	$0.403516\pi$
$578$	0	0
$579$	−7.50843	−0.312040
$580$	0	0
$581$	−17.8242	−0.739472
$582$	0	0
$583$	−1.79862	−0.0744913
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.7026	−0.441744	−0.220872	−	0.975303i	$-0.570890\pi$
$587$	−10.7026	−0.441744	−0.220872	+	0.975303i	$0.570890\pi$
$588$	0	0
$589$	−9.90306	−0.408048
$590$	0	0
$591$	12.5522	0.516328
$592$	0	0
$593$	23.5756	0.968135	0.484067	−	0.875031i	$-0.339159\pi$
$593$	23.5756	0.968135	0.484067	+	0.875031i	$0.339159\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.7618	−0.808798
$598$	0	0
$599$	13.8055	0.564078	0.282039	−	0.959403i	$-0.408989\pi$
$599$	13.8055	0.564078	0.282039	+	0.959403i	$0.408989\pi$
$600$	0	0
$601$	9.61536	0.392219	0.196109	−	0.980582i	$-0.437169\pi$
$601$	9.61536	0.392219	0.196109	+	0.980582i	$0.437169\pi$
$602$	0	0
$603$	−6.05168	−0.246443
$604$	0	0
$605$	0	0
$606$	0	0
$607$	23.7884	0.965539	0.482770	−	0.875747i	$-0.339631\pi$
$607$	23.7884	0.965539	0.482770	+	0.875747i	$0.339631\pi$
$608$	0	0
$609$	9.60128	0.389063
$610$	0	0
$611$	7.63888	0.309036
$612$	0	0
$613$	−16.0716	−0.649127	−0.324563	−	0.945864i	$-0.605217\pi$
$613$	−16.0716	−0.649127	−0.324563	+	0.945864i	$0.605217\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.34595	−0.376254	−0.188127	−	0.982145i	$-0.560242\pi$
$617$	−9.34595	−0.376254	−0.188127	+	0.982145i	$0.560242\pi$
$618$	0	0
$619$	−18.4768	−0.742645	−0.371323	−	0.928504i	$-0.621096\pi$
$619$	−18.4768	−0.742645	−0.371323	+	0.928504i	$0.621096\pi$
$620$	0	0
$621$	−6.35567	−0.255044
$622$	0	0
$623$	53.1189	2.12816
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.503774	−0.0201188
$628$	0	0
$629$	−54.6063	−2.17729
$630$	0	0
$631$	38.5690	1.53541	0.767704	−	0.640804i	$-0.221399\pi$
$631$	38.5690	1.53541	0.767704	+	0.640804i	$0.221399\pi$
$632$	0	0
$633$	−11.2479	−0.447063
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.6217	0.500090
$638$	0	0
$639$	13.2983	0.526073
$640$	0	0
$641$	−12.8461	−0.507389	−0.253695	−	0.967284i	$-0.581646\pi$
$641$	−12.8461	−0.507389	−0.253695	+	0.967284i	$0.581646\pi$
$642$	0	0
$643$	−20.1030	−0.792784	−0.396392	−	0.918081i	$-0.629738\pi$
$643$	−20.1030	−0.792784	−0.396392	+	0.918081i	$0.629738\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.1895	−1.54070	−0.770350	−	0.637622i	$-0.779918\pi$
$647$	−39.1895	−1.54070	−0.770350	+	0.637622i	$0.779918\pi$
$648$	0	0
$649$	1.36194	0.0534609
$650$	0	0
$651$	14.2848	0.559865
$652$	0	0
$653$	−25.9894	−1.01704	−0.508522	−	0.861049i	$-0.669808\pi$
$653$	−25.9894	−1.01704	−0.508522	+	0.861049i	$0.669808\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.21152	0.203321
$658$	0	0
$659$	−6.42356	−0.250226	−0.125113	−	0.992142i	$-0.539929\pi$
$659$	−6.42356	−0.250226	−0.125113	+	0.992142i	$0.539929\pi$
$660$	0	0
$661$	21.5875	0.839657	0.419829	−	0.907603i	$-0.362090\pi$
$661$	21.5875	0.839657	0.419829	+	0.907603i	$0.362090\pi$
$662$	0	0
$663$	−7.73084	−0.300241
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.0168	−0.620172
$668$	0	0
$669$	7.27196	0.281150
$670$	0	0
$671$	1.81771	0.0701718
$672$	0	0
$673$	−5.94040	−0.228986	−0.114493	−	0.993424i	$-0.536524\pi$
$673$	−5.94040	−0.228986	−0.114493	+	0.993424i	$0.536524\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.1784	−1.42888	−0.714441	−	0.699696i	$-0.753319\pi$
$677$	−37.1784	−1.42888	−0.714441	+	0.699696i	$0.753319\pi$
$678$	0	0
$679$	−25.5077	−0.978894
$680$	0	0
$681$	−9.46929	−0.362864
$682$	0	0
$683$	45.1024	1.72580	0.862899	−	0.505377i	$-0.168647\pi$
$683$	45.1024	1.72580	0.862899	+	0.505377i	$0.168647\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.2301	0.809977
$688$	0	0
$689$	−15.8372	−0.603348
$690$	0	0
$691$	−28.5713	−1.08691	−0.543453	−	0.839440i	$-0.682883\pi$
$691$	−28.5713	−1.08691	−0.543453	+	0.839440i	$0.682883\pi$
$692$	0	0
$693$	0.726676	0.0276041
$694$	0	0
$695$	0	0
$696$	0	0
$697$	33.0952	1.25357
$698$	0	0
$699$	−7.08932	−0.268143
$700$	0	0
$701$	−44.2636	−1.67181	−0.835907	−	0.548871i	$-0.815058\pi$
$701$	−44.2636	−1.67181	−0.835907	+	0.548871i	$0.815058\pi$
$702$	0	0
$703$	−31.3321	−1.18171
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.0602	1.50662
$708$	0	0
$709$	17.0124	0.638913	0.319457	−	0.947601i	$-0.396500\pi$
$709$	17.0124	0.638913	0.319457	+	0.947601i	$0.396500\pi$
$710$	0	0
$711$	−3.11168	−0.116697
$712$	0	0
$713$	−23.8298	−0.892433
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−20.0117	−0.747350
$718$	0	0
$719$	8.29496	0.309350	0.154675	−	0.987965i	$-0.450567\pi$
$719$	8.29496	0.309350	0.154675	+	0.987965i	$0.450567\pi$
$720$	0	0
$721$	49.3950	1.83956
$722$	0	0
$723$	20.8240	0.774454
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.46319	0.276794	0.138397	−	0.990377i	$-0.455805\pi$
$727$	7.46319	0.276794	0.138397	+	0.990377i	$0.455805\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−42.4704	−1.57082
$732$	0	0
$733$	12.0201	0.443973	0.221987	−	0.975050i	$-0.428746\pi$
$733$	12.0201	0.443973	0.221987	+	0.975050i	$0.428746\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.15425	−0.0425175
$738$	0	0
$739$	25.3335	0.931908	0.465954	−	0.884809i	$-0.345711\pi$
$739$	25.3335	0.931908	0.465954	+	0.884809i	$0.345711\pi$
$740$	0	0
$741$	−4.43582	−0.162954
$742$	0	0
$743$	21.0959	0.773935	0.386968	−	0.922093i	$-0.373523\pi$
$743$	21.0959	0.773935	0.386968	+	0.922093i	$0.373523\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.67837	−0.171173
$748$	0	0
$749$	36.7507	1.34284
$750$	0	0
$751$	−34.3897	−1.25490	−0.627449	−	0.778658i	$-0.715901\pi$
$751$	−34.3897	−1.25490	−0.627449	+	0.778658i	$0.715901\pi$
$752$	0	0
$753$	−29.7741	−1.08503
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.9762	−1.23488	−0.617442	−	0.786616i	$-0.711831\pi$
$757$	−33.9762	−1.23488	−0.617442	+	0.786616i	$0.711831\pi$
$758$	0	0
$759$	−1.21224	−0.0440014
$760$	0	0
$761$	−13.3872	−0.485284	−0.242642	−	0.970116i	$-0.578014\pi$
$761$	−13.3872	−0.485284	−0.242642	+	0.970116i	$0.578014\pi$
$762$	0	0
$763$	77.8953	2.82000
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.9921	0.433010
$768$	0	0
$769$	−5.94485	−0.214377	−0.107188	−	0.994239i	$-0.534185\pi$
$769$	−5.94485	−0.214377	−0.107188	+	0.994239i	$0.534185\pi$
$770$	0	0
$771$	21.2853	0.766570
$772$	0	0
$773$	−34.0966	−1.22637	−0.613184	−	0.789940i	$-0.710112\pi$
$773$	−34.0966	−1.22637	−0.613184	+	0.789940i	$0.710112\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	45.1954	1.62138
$778$	0	0
$779$	18.9894	0.680366
$780$	0	0
$781$	2.53643	0.0907604
$782$	0	0
$783$	2.52008	0.0900602
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.97301	0.0703301	0.0351651	−	0.999382i	$-0.488804\pi$
$787$	1.97301	0.0703301	0.0351651	+	0.999382i	$0.488804\pi$
$788$	0	0
$789$	−21.8653	−0.778424
$790$	0	0
$791$	15.4079	0.547843
$792$	0	0
$793$	16.0052	0.568361
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.5512	−1.04676	−0.523379	−	0.852100i	$-0.675329\pi$
$797$	−29.5512	−1.04676	−0.523379	+	0.852100i	$0.675329\pi$
$798$	0	0
$799$	−20.9378	−0.740725
$800$	0	0
$801$	13.9423	0.492626
$802$	0	0
$803$	0.994009	0.0350778
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.56132	−0.160566
$808$	0	0
$809$	7.94507	0.279334	0.139667	−	0.990199i	$-0.455397\pi$
$809$	7.94507	0.279334	0.139667	+	0.990199i	$0.455397\pi$
$810$	0	0
$811$	18.5021	0.649698	0.324849	−	0.945766i	$-0.394687\pi$
$811$	18.5021	0.649698	0.324849	+	0.945766i	$0.394687\pi$
$812$	0	0
$813$	−12.9193	−0.453099
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.3687	−0.852555
$818$	0	0
$819$	6.39850	0.223582
$820$	0	0
$821$	−15.4242	−0.538307	−0.269154	−	0.963097i	$-0.586744\pi$
$821$	−15.4242	−0.538307	−0.269154	+	0.963097i	$0.586744\pi$
$822$	0	0
$823$	−41.5847	−1.44955	−0.724775	−	0.688986i	$-0.758056\pi$
$823$	−41.5847	−1.44955	−0.724775	+	0.688986i	$0.758056\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.8857	−1.21309	−0.606547	−	0.795048i	$-0.707446\pi$
$827$	−34.8857	−1.21309	−0.606547	+	0.795048i	$0.707446\pi$
$828$	0	0
$829$	46.6095	1.61882	0.809408	−	0.587247i	$-0.199788\pi$
$829$	46.6095	1.61882	0.809408	+	0.587247i	$0.199788\pi$
$830$	0	0
$831$	−8.83521	−0.306490
$832$	0	0
$833$	−34.5955	−1.19866
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.74937	0.129597
$838$	0	0
$839$	42.3953	1.46365	0.731825	−	0.681493i	$-0.238669\pi$
$839$	42.3953	1.46365	0.731825	+	0.681493i	$0.238669\pi$
$840$	0	0
$841$	−22.6492	−0.781007
$842$	0	0
$843$	0.0305495	0.00105218
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−41.7705	−1.43525
$848$	0	0
$849$	4.80719	0.164982
$850$	0	0
$851$	−75.3946	−2.58449
$852$	0	0
$853$	−3.74484	−0.128221	−0.0641104	−	0.997943i	$-0.520421\pi$
$853$	−3.74484	−0.128221	−0.0641104	+	0.997943i	$0.520421\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.9648	1.36517	0.682586	−	0.730806i	$-0.260855\pi$
$857$	39.9648	1.36517	0.682586	+	0.730806i	$0.260855\pi$
$858$	0	0
$859$	−56.1078	−1.91437	−0.957186	−	0.289472i	$-0.906520\pi$
$859$	−56.1078	−1.91437	−0.957186	+	0.289472i	$0.906520\pi$
$860$	0	0
$861$	−27.3915	−0.933500
$862$	0	0
$863$	18.5927	0.632904	0.316452	−	0.948609i	$-0.397508\pi$
$863$	18.5927	0.632904	0.316452	+	0.948609i	$0.397508\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4.18984	0.142294
$868$	0	0
$869$	−0.593500	−0.0201331
$870$	0	0
$871$	−10.1634	−0.344373
$872$	0	0
$873$	−6.69507	−0.226594
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.8792	0.367365	0.183682	−	0.982986i	$-0.441198\pi$
$877$	10.8792	0.367365	0.183682	+	0.982986i	$0.441198\pi$
$878$	0	0
$879$	−14.9705	−0.504943
$880$	0	0
$881$	33.9938	1.14528	0.572641	−	0.819806i	$-0.305919\pi$
$881$	33.9938	1.14528	0.572641	+	0.819806i	$0.305919\pi$
$882$	0	0
$883$	15.2549	0.513367	0.256683	−	0.966496i	$-0.417370\pi$
$883$	15.2549	0.513367	0.256683	+	0.966496i	$0.417370\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.5046	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$887$	−28.5046	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$888$	0	0
$889$	63.8303	2.14080
$890$	0	0
$891$	0.190733	0.00638979
$892$	0	0
$893$	−12.0137	−0.402024
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.6739	−0.356392
$898$	0	0
$899$	9.44871	0.315132
$900$	0	0
$901$	43.4089	1.44616
$902$	0	0
$903$	35.1510	1.16975
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.1919	−0.736868	−0.368434	−	0.929654i	$-0.620106\pi$
$907$	−22.1919	−0.736868	−0.368434	+	0.929654i	$0.620106\pi$
$908$	0	0
$909$	10.5147	0.348751
$910$	0	0
$911$	40.6047	1.34529	0.672647	−	0.739963i	$-0.265157\pi$
$911$	40.6047	1.34529	0.672647	+	0.739963i	$0.265157\pi$
$912$	0	0
$913$	−0.892319	−0.0295314
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.6013	−0.416132
$918$	0	0
$919$	12.5514	0.414031	0.207016	−	0.978338i	$-0.433625\pi$
$919$	12.5514	0.414031	0.207016	+	0.978338i	$0.433625\pi$
$920$	0	0
$921$	4.47622	0.147497
$922$	0	0
$923$	22.3336	0.735121
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.9648	0.425821
$928$	0	0
$929$	−56.4129	−1.85085	−0.925424	−	0.378932i	$-0.876291\pi$
$929$	−56.4129	−1.85085	−0.925424	+	0.378932i	$0.876291\pi$
$930$	0	0
$931$	−19.8502	−0.650565
$932$	0	0
$933$	0.296114	0.00969434
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.1247	−1.37616	−0.688078	−	0.725637i	$-0.741545\pi$
$937$	−42.1247	−1.37616	−0.688078	+	0.725637i	$0.741545\pi$
$938$	0	0
$939$	−21.1569	−0.690430
$940$	0	0
$941$	−30.7834	−1.00351	−0.501755	−	0.865010i	$-0.667312\pi$
$941$	−30.7834	−1.00351	−0.501755	+	0.865010i	$0.667312\pi$
$942$	0	0
$943$	45.6943	1.48801
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.741722	0.0241027	0.0120514	−	0.999927i	$-0.496164\pi$
$947$	0.741722	0.0241027	0.0120514	+	0.999927i	$0.496164\pi$
$948$	0	0
$949$	8.75241	0.284115
$950$	0	0
$951$	25.9166	0.840405
$952$	0	0
$953$	39.3063	1.27326	0.636628	−	0.771171i	$-0.280329\pi$
$953$	39.3063	1.27326	0.636628	+	0.771171i	$0.280329\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.480661	0.0155376
$958$	0	0
$959$	7.99763	0.258257
$960$	0	0
$961$	−16.9422	−0.546522
$962$	0	0
$963$	9.64606	0.310840
$964$	0	0
$965$	0	0
$966$	0	0
$967$	48.7793	1.56864	0.784319	−	0.620358i	$-0.213013\pi$
$967$	48.7793	1.56864	0.784319	+	0.620358i	$0.213013\pi$
$968$	0	0
$969$	12.1583	0.390582
$970$	0	0
$971$	−61.2669	−1.96615	−0.983074	−	0.183210i	$-0.941351\pi$
$971$	−61.2669	−1.96615	−0.983074	+	0.183210i	$0.941351\pi$
$972$	0	0
$973$	−50.6143	−1.62262
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.1646	0.357186	0.178593	−	0.983923i	$-0.442845\pi$
$977$	11.1646	0.357186	0.178593	+	0.983923i	$0.442845\pi$
$978$	0	0
$979$	2.65925	0.0849900
$980$	0	0
$981$	20.4454	0.652772
$982$	0	0
$983$	−31.0074	−0.988983	−0.494492	−	0.869182i	$-0.664646\pi$
$983$	−31.0074	−0.988983	−0.494492	+	0.869182i	$0.664646\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17.3293	0.551599
$988$	0	0
$989$	−58.6387	−1.86460
$990$	0	0
$991$	29.5575	0.938926	0.469463	−	0.882952i	$-0.344447\pi$
$991$	29.5575	0.938926	0.469463	+	0.882952i	$0.344447\pi$
$992$	0	0
$993$	−3.07039	−0.0974358
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54.0506	−1.71180	−0.855899	−	0.517142i	$-0.826996\pi$
$997$	−54.0506	−1.71180	−0.855899	+	0.517142i	$0.826996\pi$
$998$	0	0
$999$	11.8626	0.375315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.n.1.10		12
5.2	odd	4		7500.2.d.g.1249.10		24
5.3	odd	4		7500.2.d.g.1249.15		24
5.4	even	2		7500.2.a.m.1.3		12
25.2	odd	20		300.2.o.a.229.3	yes	24
25.9	even	10		1500.2.m.d.901.2		24
25.11	even	5		1500.2.m.c.601.5		24
25.12	odd	20		1500.2.o.c.349.4		24
25.13	odd	20		300.2.o.a.169.3	✓	24
25.14	even	10		1500.2.m.d.601.2		24
25.16	even	5		1500.2.m.c.901.5		24
25.23	odd	20		1500.2.o.c.649.4		24
75.2	even	20		900.2.w.c.829.1		24
75.38	even	20		900.2.w.c.469.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.3	✓	24	25.13	odd	20
300.2.o.a.229.3	yes	24	25.2	odd	20
900.2.w.c.469.1		24	75.38	even	20
900.2.w.c.829.1		24	75.2	even	20
1500.2.m.c.601.5		24	25.11	even	5
1500.2.m.c.901.5		24	25.16	even	5
1500.2.m.d.601.2		24	25.14	even	10
1500.2.m.d.901.2		24	25.9	even	10
1500.2.o.c.349.4		24	25.12	odd	20
1500.2.o.c.649.4		24	25.23	odd	20
7500.2.a.m.1.3		12	5.4	even	2
7500.2.a.n.1.10		12	1.1	even	1	trivial
7500.2.d.g.1249.10		24	5.2	odd	4
7500.2.d.g.1249.15		24	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.80992 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.80992 q^{7} +1.00000 q^{9} +0.190733 q^{11} +1.67943 q^{13} -4.60324 q^{17} -2.64126 q^{19} +3.80992 q^{21} -6.35567 q^{23} +1.00000 q^{27} +2.52008 q^{29} +3.74937 q^{31} +0.190733 q^{33} +11.8626 q^{37} +1.67943 q^{39} -7.18953 q^{41} +9.22619 q^{43} +4.54848 q^{47} +7.51545 q^{49} -4.60324 q^{51} -9.43006 q^{53} -2.64126 q^{57} +7.14057 q^{59} +9.53012 q^{61} +3.80992 q^{63} -6.05168 q^{67} -6.35567 q^{69} +13.2983 q^{71} +5.21152 q^{73} +0.726676 q^{77} -3.11168 q^{79} +1.00000 q^{81} -4.67837 q^{83} +2.52008 q^{87} +13.9423 q^{89} +6.39850 q^{91} +3.74937 q^{93} -6.69507 q^{97} +0.190733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 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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