LMFDB - Embedded newform 7497.2.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7497,2,Mod(1,7497)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7497, 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("7497.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7497 = 3^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7497.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:	$59.8638463954$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.17314349056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 40x^{4} - 32x^{2} + 4 $ x^8 - 12x^6 + 40x^4 - 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.03016$ of defining polynomial
Character	$\chi$	$=$	7497.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03016 q^{2} +2.12154 q^{4} -1.29601 q^{5} -0.246742 q^{8} +O(q^{10})$	$q-2.03016 q^{2} +2.12154 q^{4} -1.29601 q^{5} -0.246742 q^{8} +2.63111 q^{10} +0.899205 q^{11} +1.69747 q^{13} -3.74215 q^{16} +1.00000 q^{17} -7.59063 q^{19} -2.74954 q^{20} -1.82553 q^{22} -2.48530 q^{23} -3.32035 q^{25} -3.44614 q^{26} -0.813264 q^{29} -8.78144 q^{31} +8.09064 q^{32} -2.03016 q^{34} -8.15970 q^{37} +15.4102 q^{38} +0.319780 q^{40} +9.01696 q^{41} -3.64367 q^{43} +1.90770 q^{44} +5.04555 q^{46} +4.49040 q^{47} +6.74084 q^{50} +3.60126 q^{52} -7.53911 q^{53} -1.16538 q^{55} +1.65105 q^{58} -9.75911 q^{59} +3.44437 q^{61} +17.8277 q^{62} -8.94097 q^{64} -2.19995 q^{65} +13.0138 q^{67} +2.12154 q^{68} +6.41924 q^{71} +2.68053 q^{73} +16.5655 q^{74} -16.1038 q^{76} +2.62671 q^{79} +4.84987 q^{80} -18.3058 q^{82} +4.84987 q^{83} -1.29601 q^{85} +7.39722 q^{86} -0.221872 q^{88} +16.0551 q^{89} -5.27266 q^{92} -9.11622 q^{94} +9.83754 q^{95} -1.30493 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} + 12 q^{5}+O(q^{10}) $ 8 * q + 8 * q^4 + 12 * q^5	$ 8 q + 8 q^{4} + 12 q^{5} + 16 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{22} + 8 q^{25} - 4 q^{26} + 12 q^{37} + 60 q^{38} + 28 q^{41} - 24 q^{43} + 4 q^{46} + 20 q^{47} + 40 q^{58} + 12 q^{59} + 48 q^{62} + 48 q^{67} + 8 q^{68} + 60 q^{79} + 40 q^{80} + 40 q^{83} + 12 q^{85} - 8 q^{88} + 16 q^{89}+O(q^{100}) $ 8 * q + 8 * q^4 + 12 * q^5 + 16 * q^16 + 8 * q^17 - 8 * q^20 - 28 * q^22 + 8 * q^25 - 4 * q^26 + 12 * q^37 + 60 * q^38 + 28 * q^41 - 24 * q^43 + 4 * q^46 + 20 * q^47 + 40 * q^58 + 12 * q^59 + 48 * q^62 + 48 * q^67 + 8 * q^68 + 60 * q^79 + 40 * q^80 + 40 * q^83 + 12 * q^85 - 8 * q^88 + 16 * q^89

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 $

$2$	−2.03016	−1.43554	−0.717769	−	0.696281i	$-0.754837\pi$
$2$	−2.03016	−1.43554	−0.717769	+	0.696281i	$0.754837\pi$
$3$	0	0
$3$	0	0
$4$	2.12154	1.06077
$5$	−1.29601	−0.579594	−0.289797	−	0.957088i	$-0.593588\pi$
$5$	−1.29601	−0.579594	−0.289797	+	0.957088i	$0.593588\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−0.246742	−0.0872364
$9$	0	0
$10$	2.63111	0.832029
$11$	0.899205	0.271120	0.135560	−	0.990769i	$-0.456717\pi$
$11$	0.899205	0.271120	0.135560	+	0.990769i	$0.456717\pi$
$12$	0	0
$13$	1.69747	0.470795	0.235397	−	0.971899i	$-0.424361\pi$
$13$	1.69747	0.470795	0.235397	+	0.971899i	$0.424361\pi$
$14$	0	0
$15$	0	0
$16$	−3.74215	−0.935538
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−7.59063	−1.74141	−0.870704	−	0.491807i	$-0.836337\pi$
$19$	−7.59063	−1.74141	−0.870704	+	0.491807i	$0.836337\pi$
$20$	−2.74954	−0.614815
$21$	0	0
$22$	−1.82553	−0.389204
$23$	−2.48530	−0.518221	−0.259110	−	0.965848i	$-0.583429\pi$
$23$	−2.48530	−0.518221	−0.259110	+	0.965848i	$0.583429\pi$
$24$	0	0
$25$	−3.32035	−0.664071
$26$	−3.44614	−0.675844
$27$	0	0
$28$	0	0
$29$	−0.813264	−0.151019	−0.0755097	−	0.997145i	$-0.524058\pi$
$29$	−0.813264	−0.151019	−0.0755097	+	0.997145i	$0.524058\pi$
$30$	0	0
$31$	−8.78144	−1.57719	−0.788597	−	0.614910i	$-0.789192\pi$
$31$	−8.78144	−1.57719	−0.788597	+	0.614910i	$0.789192\pi$
$32$	8.09064	1.43024
$33$	0	0
$34$	−2.03016	−0.348169
$35$	0	0
$36$	0	0
$37$	−8.15970	−1.34145	−0.670723	−	0.741708i	$-0.734016\pi$
$37$	−8.15970	−1.34145	−0.670723	+	0.741708i	$0.734016\pi$
$38$	15.4102	2.49986
$39$	0	0
$40$	0.319780	0.0505617
$41$	9.01696	1.40821	0.704106	−	0.710095i	$-0.251348\pi$
$41$	9.01696	1.40821	0.704106	+	0.710095i	$0.251348\pi$
$42$	0	0
$43$	−3.64367	−0.555654	−0.277827	−	0.960631i	$-0.589614\pi$
$43$	−3.64367	−0.555654	−0.277827	+	0.960631i	$0.589614\pi$
$44$	1.90770	0.287596
$45$	0	0
$46$	5.04555	0.743926
$47$	4.49040	0.654992	0.327496	−	0.944853i	$-0.393795\pi$
$47$	4.49040	0.654992	0.327496	+	0.944853i	$0.393795\pi$
$48$	0	0
$49$	0	0
$50$	6.74084	0.953299
$51$	0	0
$52$	3.60126	0.499405
$53$	−7.53911	−1.03558	−0.517788	−	0.855509i	$-0.673245\pi$
$53$	−7.53911	−1.03558	−0.517788	+	0.855509i	$0.673245\pi$
$54$	0	0
$55$	−1.16538	−0.157140
$56$	0	0
$57$	0	0
$58$	1.65105	0.216794
$59$	−9.75911	−1.27053	−0.635264	−	0.772295i	$-0.719109\pi$
$59$	−9.75911	−1.27053	−0.635264	+	0.772295i	$0.719109\pi$
$60$	0	0
$61$	3.44437	0.441006	0.220503	−	0.975386i	$-0.429230\pi$
$61$	3.44437	0.441006	0.220503	+	0.975386i	$0.429230\pi$
$62$	17.8277	2.26412
$63$	0	0
$64$	−8.94097	−1.11762
$65$	−2.19995	−0.272870
$66$	0	0
$67$	13.0138	1.58989	0.794946	−	0.606681i	$-0.207499\pi$
$67$	13.0138	1.58989	0.794946	+	0.606681i	$0.207499\pi$
$68$	2.12154	0.257274
$69$	0	0
$70$	0	0
$71$	6.41924	0.761824	0.380912	−	0.924611i	$-0.375610\pi$
$71$	6.41924	0.761824	0.380912	+	0.924611i	$0.375610\pi$
$72$	0	0
$73$	2.68053	0.313732	0.156866	−	0.987620i	$-0.449861\pi$
$73$	2.68053	0.313732	0.156866	+	0.987620i	$0.449861\pi$
$74$	16.5655	1.92570
$75$	0	0
$76$	−16.1038	−1.84723
$77$	0	0
$78$	0	0
$79$	2.62671	0.295528	0.147764	−	0.989023i	$-0.452792\pi$
$79$	2.62671	0.295528	0.147764	+	0.989023i	$0.452792\pi$
$80$	4.84987	0.542232
$81$	0	0
$82$	−18.3058	−2.02154
$83$	4.84987	0.532342	0.266171	−	0.963926i	$-0.414241\pi$
$83$	4.84987	0.532342	0.266171	+	0.963926i	$0.414241\pi$
$84$	0	0
$85$	−1.29601	−0.140572
$86$	7.39722	0.797663
$87$	0	0
$88$	−0.221872	−0.0236516
$89$	16.0551	1.70184	0.850920	−	0.525296i	$-0.176045\pi$
$89$	16.0551	1.70184	0.850920	+	0.525296i	$0.176045\pi$
$90$	0	0
$91$	0	0
$92$	−5.27266	−0.549713
$93$	0	0
$94$	−9.11622	−0.940266
$95$	9.83754	1.00931
$96$	0	0
$97$	−1.30493	−0.132495	−0.0662476	−	0.997803i	$-0.521103\pi$
$97$	−1.30493	−0.132495	−0.0662476	+	0.997803i	$0.521103\pi$
$98$	0	0
$99$	0	0
$100$	−7.04426	−0.704426
$101$	6.12283	0.609244	0.304622	−	0.952473i	$-0.401470\pi$
$101$	6.12283	0.609244	0.304622	+	0.952473i	$0.401470\pi$
$102$	0	0
$103$	−3.85009	−0.379361	−0.189680	−	0.981846i	$-0.560745\pi$
$103$	−3.85009	−0.379361	−0.189680	+	0.981846i	$0.560745\pi$
$104$	−0.418838	−0.0410705
$105$	0	0
$106$	15.3056	1.48661
$107$	−7.31844	−0.707501	−0.353750	−	0.935340i	$-0.615094\pi$
$107$	−7.31844	−0.707501	−0.353750	+	0.935340i	$0.615094\pi$
$108$	0	0
$109$	12.4388	1.19142	0.595708	−	0.803201i	$-0.296871\pi$
$109$	12.4388	1.19142	0.595708	+	0.803201i	$0.296871\pi$
$110$	2.36590	0.225580
$111$	0	0
$112$	0	0
$113$	−20.7970	−1.95642	−0.978210	−	0.207616i	$-0.933429\pi$
$113$	−20.7970	−1.95642	−0.978210	+	0.207616i	$0.933429\pi$
$114$	0	0
$115$	3.22098	0.300358
$116$	−1.72537	−0.160197
$117$	0	0
$118$	19.8125	1.82389
$119$	0	0
$120$	0	0
$121$	−10.1914	−0.926494
$122$	−6.99261	−0.633082
$123$	0	0
$124$	−18.6302	−1.67304
$125$	10.7833	0.964485
$126$	0	0
$127$	−0.208209	−0.0184755	−0.00923777	−	0.999957i	$-0.502941\pi$
$127$	−0.208209	−0.0184755	−0.00923777	+	0.999957i	$0.502941\pi$
$128$	1.97029	0.174151
$129$	0	0
$130$	4.46624	0.391715
$131$	9.05293	0.790959	0.395479	−	0.918475i	$-0.370579\pi$
$131$	9.05293	0.790959	0.395479	+	0.918475i	$0.370579\pi$
$132$	0	0
$133$	0	0
$134$	−26.4201	−2.28235
$135$	0	0
$136$	−0.246742	−0.0211579
$137$	−15.9495	−1.36266	−0.681330	−	0.731976i	$-0.738598\pi$
$137$	−15.9495	−1.36266	−0.681330	+	0.731976i	$0.738598\pi$
$138$	0	0
$139$	4.27707	0.362776	0.181388	−	0.983412i	$-0.441941\pi$
$139$	4.27707	0.362776	0.181388	+	0.983412i	$0.441941\pi$
$140$	0	0
$141$	0	0
$142$	−13.0321	−1.09363
$143$	1.52638	0.127642
$144$	0	0
$145$	1.05400	0.0875299
$146$	−5.44189	−0.450374
$147$	0	0
$148$	−17.3111	−1.42297
$149$	−12.6726	−1.03818	−0.519090	−	0.854720i	$-0.673729\pi$
$149$	−12.6726	−1.03818	−0.519090	+	0.854720i	$0.673729\pi$
$150$	0	0
$151$	16.8047	1.36754	0.683772	−	0.729696i	$-0.260338\pi$
$151$	16.8047	1.36754	0.683772	+	0.729696i	$0.260338\pi$
$152$	1.87293	0.151914
$153$	0	0
$154$	0	0
$155$	11.3809	0.914132
$156$	0	0
$157$	−2.12654	−0.169717	−0.0848583	−	0.996393i	$-0.527044\pi$
$157$	−2.12654	−0.169717	−0.0848583	+	0.996393i	$0.527044\pi$
$158$	−5.33264	−0.424242
$159$	0	0
$160$	−10.4856	−0.828956
$161$	0	0
$162$	0	0
$163$	−13.0507	−1.02221	−0.511105	−	0.859519i	$-0.670764\pi$
$163$	−13.0507	−1.02221	−0.511105	+	0.859519i	$0.670764\pi$
$164$	19.1298	1.49379
$165$	0	0
$166$	−9.84600	−0.764198
$167$	0.212299	0.0164282	0.00821411	−	0.999966i	$-0.497385\pi$
$167$	0.212299	0.0164282	0.00821411	+	0.999966i	$0.497385\pi$
$168$	0	0
$169$	−10.1186	−0.778352
$170$	2.63111	0.201797
$171$	0	0
$172$	−7.73018	−0.589421
$173$	−4.55201	−0.346083	−0.173041	−	0.984915i	$-0.555359\pi$
$173$	−4.55201	−0.346083	−0.173041	+	0.984915i	$0.555359\pi$
$174$	0	0
$175$	0	0
$176$	−3.36496	−0.253643
$177$	0	0
$178$	−32.5944	−2.44306
$179$	18.9427	1.41584	0.707921	−	0.706292i	$-0.249633\pi$
$179$	18.9427	1.41584	0.707921	+	0.706292i	$0.249633\pi$
$180$	0	0
$181$	−21.7424	−1.61610	−0.808049	−	0.589115i	$-0.799477\pi$
$181$	−21.7424	−1.61610	−0.808049	+	0.589115i	$0.799477\pi$
$182$	0	0
$183$	0	0
$184$	0.613228	0.0452077
$185$	10.5751	0.777494
$186$	0	0
$187$	0.899205	0.0657564
$188$	9.52656	0.694796
$189$	0	0
$190$	−19.9717	−1.44890
$191$	15.6956	1.13570	0.567848	−	0.823134i	$-0.307776\pi$
$191$	15.6956	1.13570	0.567848	+	0.823134i	$0.307776\pi$
$192$	0	0
$193$	8.12372	0.584758	0.292379	−	0.956302i	$-0.405553\pi$
$193$	8.12372	0.584758	0.292379	+	0.956302i	$0.405553\pi$
$194$	2.64920	0.190202
$195$	0	0
$196$	0	0
$197$	−3.29856	−0.235013	−0.117506	−	0.993072i	$-0.537490\pi$
$197$	−3.29856	−0.235013	−0.117506	+	0.993072i	$0.537490\pi$
$198$	0	0
$199$	3.07062	0.217670	0.108835	−	0.994060i	$-0.465288\pi$
$199$	3.07062	0.217670	0.108835	+	0.994060i	$0.465288\pi$
$200$	0.819271	0.0579312
$201$	0	0
$202$	−12.4303	−0.874593
$203$	0	0
$204$	0	0
$205$	−11.6861	−0.816191
$206$	7.81629	0.544587
$207$	0	0
$208$	−6.35221	−0.440446
$209$	−6.82553	−0.472132
$210$	0	0
$211$	−12.5299	−0.862591	−0.431295	−	0.902211i	$-0.641943\pi$
$211$	−12.5299	−0.862591	−0.431295	+	0.902211i	$0.641943\pi$
$212$	−15.9945	−1.09851
$213$	0	0
$214$	14.8576	1.01564
$215$	4.72224	0.322054
$216$	0	0
$217$	0	0
$218$	−25.2526	−1.71032
$219$	0	0
$220$	−2.47240	−0.166689
$221$	1.69747	0.114185
$222$	0	0
$223$	18.5532	1.24242	0.621208	−	0.783646i	$-0.286642\pi$
$223$	18.5532	1.24242	0.621208	+	0.783646i	$0.286642\pi$
$224$	0	0
$225$	0	0
$226$	42.2213	2.80852
$227$	19.3659	1.28536	0.642680	−	0.766135i	$-0.277822\pi$
$227$	19.3659	1.28536	0.642680	+	0.766135i	$0.277822\pi$
$228$	0	0
$229$	−7.00665	−0.463013	−0.231506	−	0.972833i	$-0.574365\pi$
$229$	−7.00665	−0.463013	−0.231506	+	0.972833i	$0.574365\pi$
$230$	−6.53909	−0.431175
$231$	0	0
$232$	0.200666	0.0131744
$233$	17.8084	1.16667	0.583335	−	0.812232i	$-0.301747\pi$
$233$	17.8084	1.16667	0.583335	+	0.812232i	$0.301747\pi$
$234$	0	0
$235$	−5.81961	−0.379629
$236$	−20.7043	−1.34774
$237$	0	0
$238$	0	0
$239$	1.49362	0.0966145	0.0483072	−	0.998833i	$-0.484617\pi$
$239$	1.49362	0.0966145	0.0483072	+	0.998833i	$0.484617\pi$
$240$	0	0
$241$	−6.59112	−0.424571	−0.212286	−	0.977208i	$-0.568091\pi$
$241$	−6.59112	−0.424571	−0.212286	+	0.977208i	$0.568091\pi$
$242$	20.6902	1.33002
$243$	0	0
$244$	7.30736	0.467806
$245$	0	0
$246$	0	0
$247$	−12.8849	−0.819846
$248$	2.16675	0.137589
$249$	0	0
$250$	−21.8917	−1.38456
$251$	11.8585	0.748505	0.374252	−	0.927327i	$-0.377899\pi$
$251$	11.8585	0.748505	0.374252	+	0.927327i	$0.377899\pi$
$252$	0	0
$253$	−2.23479	−0.140500
$254$	0.422697	0.0265223
$255$	0	0
$256$	13.8819	0.867621
$257$	−9.46528	−0.590428	−0.295214	−	0.955431i	$-0.595391\pi$
$257$	−9.46528	−0.590428	−0.295214	+	0.955431i	$0.595391\pi$
$258$	0	0
$259$	0	0
$260$	−4.66727	−0.289452
$261$	0	0
$262$	−18.3789	−1.13545
$263$	−6.73902	−0.415546	−0.207773	−	0.978177i	$-0.566621\pi$
$263$	−6.73902	−0.415546	−0.207773	+	0.978177i	$0.566621\pi$
$264$	0	0
$265$	9.77077	0.600214
$266$	0	0
$267$	0	0
$268$	27.6093	1.68651
$269$	11.6263	0.708870	0.354435	−	0.935081i	$-0.384673\pi$
$269$	11.6263	0.708870	0.354435	+	0.935081i	$0.384673\pi$
$270$	0	0
$271$	9.35905	0.568522	0.284261	−	0.958747i	$-0.408252\pi$
$271$	9.35905	0.568522	0.284261	+	0.958747i	$0.408252\pi$
$272$	−3.74215	−0.226901
$273$	0	0
$274$	32.3801	1.95615
$275$	−2.98568	−0.180043
$276$	0	0
$277$	3.65363	0.219525	0.109763	−	0.993958i	$-0.464991\pi$
$277$	3.65363	0.219525	0.109763	+	0.993958i	$0.464991\pi$
$278$	−8.68312	−0.520779
$279$	0	0
$280$	0	0
$281$	−4.32795	−0.258184	−0.129092	−	0.991633i	$-0.541206\pi$
$281$	−4.32795	−0.258184	−0.129092	+	0.991633i	$0.541206\pi$
$282$	0	0
$283$	−0.272181	−0.0161795	−0.00808975	−	0.999967i	$-0.502575\pi$
$283$	−0.272181	−0.0161795	−0.00808975	+	0.999967i	$0.502575\pi$
$284$	13.6187	0.808119
$285$	0	0
$286$	−3.09879	−0.183235
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	−2.13978	−0.125652
$291$	0	0
$292$	5.68684	0.332797
$293$	6.57339	0.384022	0.192011	−	0.981393i	$-0.438499\pi$
$293$	6.57339	0.384022	0.192011	+	0.981393i	$0.438499\pi$
$294$	0	0
$295$	12.6479	0.736390
$296$	2.01334	0.117023
$297$	0	0
$298$	25.7274	1.49035
$299$	−4.21873	−0.243976
$300$	0	0
$301$	0	0
$302$	−34.1161	−1.96316
$303$	0	0
$304$	28.4053	1.62915
$305$	−4.46394	−0.255605
$306$	0	0
$307$	16.9092	0.965061	0.482531	−	0.875879i	$-0.339718\pi$
$307$	16.9092	0.965061	0.482531	+	0.875879i	$0.339718\pi$
$308$	0	0
$309$	0	0
$310$	−23.1049	−1.31227
$311$	−0.228305	−0.0129460	−0.00647300	−	0.999979i	$-0.502060\pi$
$311$	−0.228305	−0.0129460	−0.00647300	+	0.999979i	$0.502060\pi$
$312$	0	0
$313$	5.69139	0.321697	0.160848	−	0.986979i	$-0.448577\pi$
$313$	5.69139	0.321697	0.160848	+	0.986979i	$0.448577\pi$
$314$	4.31722	0.243635
$315$	0	0
$316$	5.57267	0.313487
$317$	15.9750	0.897244	0.448622	−	0.893722i	$-0.351915\pi$
$317$	15.9750	0.897244	0.448622	+	0.893722i	$0.351915\pi$
$318$	0	0
$319$	−0.731291	−0.0409444
$320$	11.5876	0.647766
$321$	0	0
$322$	0	0
$323$	−7.59063	−0.422354
$324$	0	0
$325$	−5.63622	−0.312641
$326$	26.4950	1.46742
$327$	0	0
$328$	−2.22486	−0.122847
$329$	0	0
$330$	0	0
$331$	−24.5025	−1.34678	−0.673391	−	0.739287i	$-0.735163\pi$
$331$	−24.5025	−1.34678	−0.673391	+	0.739287i	$0.735163\pi$
$332$	10.2892	0.564692
$333$	0	0
$334$	−0.431001	−0.0235833
$335$	−16.8661	−0.921491
$336$	0	0
$337$	32.3059	1.75982	0.879908	−	0.475144i	$-0.157604\pi$
$337$	32.3059	1.75982	0.879908	+	0.475144i	$0.157604\pi$
$338$	20.5423	1.11735
$339$	0	0
$340$	−2.74954	−0.149115
$341$	−7.89632	−0.427610
$342$	0	0
$343$	0	0
$344$	0.899046	0.0484733
$345$	0	0
$346$	9.24130	0.496815
$347$	13.0894	0.702676	0.351338	−	0.936249i	$-0.385727\pi$
$347$	13.0894	0.702676	0.351338	+	0.936249i	$0.385727\pi$
$348$	0	0
$349$	5.14639	0.275480	0.137740	−	0.990468i	$-0.456016\pi$
$349$	5.14639	0.275480	0.137740	+	0.990468i	$0.456016\pi$
$350$	0	0
$351$	0	0
$352$	7.27514	0.387766
$353$	−15.2434	−0.811325	−0.405663	−	0.914023i	$-0.632959\pi$
$353$	−15.2434	−0.811325	−0.405663	+	0.914023i	$0.632959\pi$
$354$	0	0
$355$	−8.31941	−0.441548
$356$	34.0616	1.80526
$357$	0	0
$358$	−38.4566	−2.03250
$359$	28.5952	1.50920	0.754600	−	0.656186i	$-0.227831\pi$
$359$	28.5952	1.50920	0.754600	+	0.656186i	$0.227831\pi$
$360$	0	0
$361$	38.6176	2.03251
$362$	44.1405	2.31997
$363$	0	0
$364$	0	0
$365$	−3.47399	−0.181837
$366$	0	0
$367$	−30.6481	−1.59982	−0.799909	−	0.600121i	$-0.795119\pi$
$367$	−30.6481	−1.59982	−0.799909	+	0.600121i	$0.795119\pi$
$368$	9.30037	0.484815
$369$	0	0
$370$	−21.4690	−1.11612
$371$	0	0
$372$	0	0
$373$	−7.20821	−0.373227	−0.186613	−	0.982433i	$-0.559751\pi$
$373$	−7.20821	−0.373227	−0.186613	+	0.982433i	$0.559751\pi$
$374$	−1.82553	−0.0943958
$375$	0	0
$376$	−1.10797	−0.0571392
$377$	−1.38050	−0.0710991
$378$	0	0
$379$	−20.3257	−1.04406	−0.522030	−	0.852927i	$-0.674825\pi$
$379$	−20.3257	−1.04406	−0.522030	+	0.852927i	$0.674825\pi$
$380$	20.8707	1.07064
$381$	0	0
$382$	−31.8646	−1.63033
$383$	−18.7674	−0.958969	−0.479484	−	0.877550i	$-0.659176\pi$
$383$	−18.7674	−0.958969	−0.479484	+	0.877550i	$0.659176\pi$
$384$	0	0
$385$	0	0
$386$	−16.4924	−0.839443
$387$	0	0
$388$	−2.76845	−0.140547
$389$	−27.3544	−1.38692	−0.693461	−	0.720494i	$-0.743915\pi$
$389$	−27.3544	−1.38692	−0.693461	+	0.720494i	$0.743915\pi$
$390$	0	0
$391$	−2.48530	−0.125687
$392$	0	0
$393$	0	0
$394$	6.69660	0.337370
$395$	−3.40425	−0.171286
$396$	0	0
$397$	35.2555	1.76942	0.884712	−	0.466138i	$-0.154355\pi$
$397$	35.2555	1.76942	0.884712	+	0.466138i	$0.154355\pi$
$398$	−6.23384	−0.312474
$399$	0	0
$400$	12.4253	0.621264
$401$	18.2693	0.912326	0.456163	−	0.889896i	$-0.349223\pi$
$401$	18.2693	0.912326	0.456163	+	0.889896i	$0.349223\pi$
$402$	0	0
$403$	−14.9063	−0.742535
$404$	12.9898	0.646267
$405$	0	0
$406$	0	0
$407$	−7.33724	−0.363694
$408$	0	0
$409$	−1.25352	−0.0619827	−0.0309914	−	0.999520i	$-0.509866\pi$
$409$	−1.25352	−0.0619827	−0.0309914	+	0.999520i	$0.509866\pi$
$410$	23.7246	1.17167
$411$	0	0
$412$	−8.16812	−0.402414
$413$	0	0
$414$	0	0
$415$	−6.28549	−0.308542
$416$	13.7337	0.673348
$417$	0	0
$418$	13.8569	0.677763
$419$	−27.2209	−1.32983	−0.664914	−	0.746920i	$-0.731532\pi$
$419$	−27.2209	−1.32983	−0.664914	+	0.746920i	$0.731532\pi$
$420$	0	0
$421$	−16.8065	−0.819099	−0.409550	−	0.912288i	$-0.634314\pi$
$421$	−16.8065	−0.819099	−0.409550	+	0.912288i	$0.634314\pi$
$422$	25.4376	1.23828
$423$	0	0
$424$	1.86022	0.0903400
$425$	−3.32035	−0.161061
$426$	0	0
$427$	0	0
$428$	−15.5264	−0.750495
$429$	0	0
$430$	−9.58688	−0.462320
$431$	−29.4400	−1.41807	−0.709036	−	0.705172i	$-0.750870\pi$
$431$	−29.4400	−1.41807	−0.709036	+	0.705172i	$0.750870\pi$
$432$	0	0
$433$	18.2507	0.877072	0.438536	−	0.898714i	$-0.355497\pi$
$433$	18.2507	0.877072	0.438536	+	0.898714i	$0.355497\pi$
$434$	0	0
$435$	0	0
$436$	26.3893	1.26382
$437$	18.8650	0.902434
$438$	0	0
$439$	10.8955	0.520011	0.260006	−	0.965607i	$-0.416276\pi$
$439$	10.8955	0.520011	0.260006	+	0.965607i	$0.416276\pi$
$440$	0.287548	0.0137083
$441$	0	0
$442$	−3.44614	−0.163916
$443$	22.7204	1.07948	0.539739	−	0.841833i	$-0.318523\pi$
$443$	22.7204	1.07948	0.539739	+	0.841833i	$0.318523\pi$
$444$	0	0
$445$	−20.8076	−0.986376
$446$	−37.6660	−1.78353
$447$	0	0
$448$	0	0
$449$	−10.5993	−0.500211	−0.250106	−	0.968219i	$-0.580465\pi$
$449$	−10.5993	−0.500211	−0.250106	+	0.968219i	$0.580465\pi$
$450$	0	0
$451$	8.10809	0.381795
$452$	−44.1217	−2.07531
$453$	0	0
$454$	−39.3158	−1.84518
$455$	0	0
$456$	0	0
$457$	−9.49079	−0.443961	−0.221980	−	0.975051i	$-0.571252\pi$
$457$	−9.49079	−0.443961	−0.221980	+	0.975051i	$0.571252\pi$
$458$	14.2246	0.664672
$459$	0	0
$460$	6.83342	0.318610
$461$	−0.940577	−0.0438071	−0.0219035	−	0.999760i	$-0.506973\pi$
$461$	−0.940577	−0.0438071	−0.0219035	+	0.999760i	$0.506973\pi$
$462$	0	0
$463$	14.2466	0.662093	0.331047	−	0.943614i	$-0.392598\pi$
$463$	14.2466	0.662093	0.331047	+	0.943614i	$0.392598\pi$
$464$	3.04336	0.141284
$465$	0	0
$466$	−36.1539	−1.67480
$467$	20.1097	0.930567	0.465284	−	0.885162i	$-0.345952\pi$
$467$	20.1097	0.930567	0.465284	+	0.885162i	$0.345952\pi$
$468$	0	0
$469$	0	0
$470$	11.8147	0.544973
$471$	0	0
$472$	2.40798	0.110836
$473$	−3.27640	−0.150649
$474$	0	0
$475$	25.2036	1.15642
$476$	0	0
$477$	0	0
$478$	−3.03229	−0.138694
$479$	42.3684	1.93586	0.967931	−	0.251215i	$-0.0808300\pi$
$479$	42.3684	1.93586	0.967931	+	0.251215i	$0.0808300\pi$
$480$	0	0
$481$	−13.8509	−0.631546
$482$	13.3810	0.609488
$483$	0	0
$484$	−21.6215	−0.982796
$485$	1.69120	0.0767934
$486$	0	0
$487$	32.0442	1.45206	0.726031	−	0.687662i	$-0.241363\pi$
$487$	32.0442	1.45206	0.726031	+	0.687662i	$0.241363\pi$
$488$	−0.849871	−0.0384718
$489$	0	0
$490$	0	0
$491$	25.0163	1.12897	0.564486	−	0.825443i	$-0.309075\pi$
$491$	25.0163	1.12897	0.564486	+	0.825443i	$0.309075\pi$
$492$	0	0
$493$	−0.813264	−0.0366276
$494$	26.1584	1.17692
$495$	0	0
$496$	32.8615	1.47552
$497$	0	0
$498$	0	0
$499$	18.2718	0.817960	0.408980	−	0.912543i	$-0.365885\pi$
$499$	18.2718	0.817960	0.408980	+	0.912543i	$0.365885\pi$
$500$	22.8771	1.02310
$501$	0	0
$502$	−24.0747	−1.07451
$503$	26.0511	1.16156	0.580780	−	0.814060i	$-0.302748\pi$
$503$	26.0511	1.16156	0.580780	+	0.814060i	$0.302748\pi$
$504$	0	0
$505$	−7.93525	−0.353114
$506$	4.53698	0.201693
$507$	0	0
$508$	−0.441723	−0.0195983
$509$	−31.2265	−1.38409	−0.692044	−	0.721855i	$-0.743290\pi$
$509$	−31.2265	−1.38409	−0.692044	+	0.721855i	$0.743290\pi$
$510$	0	0
$511$	0	0
$512$	−32.1231	−1.41965
$513$	0	0
$514$	19.2160	0.847582
$515$	4.98976	0.219875
$516$	0	0
$517$	4.03779	0.177582
$518$	0	0
$519$	0	0
$520$	0.542819	0.0238042
$521$	−40.5471	−1.77640	−0.888201	−	0.459454i	$-0.848045\pi$
$521$	−40.5471	−1.77640	−0.888201	+	0.459454i	$0.848045\pi$
$522$	0	0
$523$	−9.72888	−0.425414	−0.212707	−	0.977116i	$-0.568228\pi$
$523$	−9.72888	−0.425414	−0.212707	+	0.977116i	$0.568228\pi$
$524$	19.2061	0.839024
$525$	0	0
$526$	13.6813	0.596532
$527$	−8.78144	−0.382526
$528$	0	0
$529$	−16.8233	−0.731447
$530$	−19.8362	−0.861630
$531$	0	0
$532$	0	0
$533$	15.3061	0.662979
$534$	0	0
$535$	9.48479	0.410063
$536$	−3.21106	−0.138696
$537$	0	0
$538$	−23.6033	−1.01761
$539$	0	0
$540$	0	0
$541$	9.32331	0.400841	0.200420	−	0.979710i	$-0.435769\pi$
$541$	9.32331	0.400841	0.200420	+	0.979710i	$0.435769\pi$
$542$	−19.0003	−0.816134
$543$	0	0
$544$	8.09064	0.346883
$545$	−16.1208	−0.690538
$546$	0	0
$547$	25.7711	1.10189	0.550946	−	0.834541i	$-0.314267\pi$
$547$	25.7711	1.10189	0.550946	+	0.834541i	$0.314267\pi$
$548$	−33.8375	−1.44547
$549$	0	0
$550$	6.06140	0.258459
$551$	6.17318	0.262986
$552$	0	0
$553$	0	0
$554$	−7.41745	−0.315137
$555$	0	0
$556$	9.07396	0.384822
$557$	7.68283	0.325532	0.162766	−	0.986665i	$-0.447958\pi$
$557$	7.68283	0.325532	0.162766	+	0.986665i	$0.447958\pi$
$558$	0	0
$559$	−6.18503	−0.261599
$560$	0	0
$561$	0	0
$562$	8.78641	0.370632
$563$	12.7339	0.536669	0.268334	−	0.963326i	$-0.413527\pi$
$563$	12.7339	0.536669	0.268334	+	0.963326i	$0.413527\pi$
$564$	0	0
$565$	26.9532	1.13393
$566$	0.552571	0.0232263
$567$	0	0
$568$	−1.58390	−0.0664588
$569$	15.1722	0.636050	0.318025	−	0.948082i	$-0.396980\pi$
$569$	15.1722	0.636050	0.318025	+	0.948082i	$0.396980\pi$
$570$	0	0
$571$	−36.7696	−1.53876	−0.769380	−	0.638792i	$-0.779434\pi$
$571$	−36.7696	−1.53876	−0.769380	+	0.638792i	$0.779434\pi$
$572$	3.23827	0.135399
$573$	0	0
$574$	0	0
$575$	8.25208	0.344135
$576$	0	0
$577$	12.4329	0.517587	0.258794	−	0.965933i	$-0.416675\pi$
$577$	12.4329	0.517587	0.258794	+	0.965933i	$0.416675\pi$
$578$	−2.03016	−0.0844434
$579$	0	0
$580$	2.23610	0.0928490
$581$	0	0
$582$	0	0
$583$	−6.77921	−0.280766
$584$	−0.661399	−0.0273689
$585$	0	0
$586$	−13.3450	−0.551278
$587$	11.4461	0.472433	0.236216	−	0.971700i	$-0.424093\pi$
$587$	11.4461	0.472433	0.236216	+	0.971700i	$0.424093\pi$
$588$	0	0
$589$	66.6567	2.74654
$590$	−25.6773	−1.05712
$591$	0	0
$592$	30.5348	1.25497
$593$	40.1779	1.64991	0.824955	−	0.565198i	$-0.191200\pi$
$593$	40.1779	1.64991	0.824955	+	0.565198i	$0.191200\pi$
$594$	0	0
$595$	0	0
$596$	−26.8854	−1.10127
$597$	0	0
$598$	8.56469	0.350236
$599$	−10.7621	−0.439726	−0.219863	−	0.975531i	$-0.570561\pi$
$599$	−10.7621	−0.439726	−0.219863	+	0.975531i	$0.570561\pi$
$600$	0	0
$601$	12.5429	0.511634	0.255817	−	0.966725i	$-0.417655\pi$
$601$	12.5429	0.511634	0.255817	+	0.966725i	$0.417655\pi$
$602$	0	0
$603$	0	0
$604$	35.6517	1.45065
$605$	13.2082	0.536990
$606$	0	0
$607$	30.9466	1.25609	0.628043	−	0.778179i	$-0.283856\pi$
$607$	30.9466	1.25609	0.628043	+	0.778179i	$0.283856\pi$
$608$	−61.4130	−2.49063
$609$	0	0
$610$	9.06251	0.366930
$611$	7.62234	0.308367
$612$	0	0
$613$	−26.7550	−1.08063	−0.540313	−	0.841464i	$-0.681694\pi$
$613$	−26.7550	−1.08063	−0.540313	+	0.841464i	$0.681694\pi$
$614$	−34.3284	−1.38538
$615$	0	0
$616$	0	0
$617$	−46.6228	−1.87696	−0.938482	−	0.345328i	$-0.887768\pi$
$617$	−46.6228	−1.87696	−0.938482	+	0.345328i	$0.887768\pi$
$618$	0	0
$619$	−0.466775	−0.0187613	−0.00938064	−	0.999956i	$-0.502986\pi$
$619$	−0.466775	−0.0187613	−0.00938064	+	0.999956i	$0.502986\pi$
$620$	24.1449	0.969683
$621$	0	0
$622$	0.463495	0.0185845
$623$	0	0
$624$	0	0
$625$	2.62653	0.105061
$626$	−11.5544	−0.461808
$627$	0	0
$628$	−4.51154	−0.180030
$629$	−8.15970	−0.325349
$630$	0	0
$631$	−14.7761	−0.588226	−0.294113	−	0.955771i	$-0.595024\pi$
$631$	−14.7761	−0.588226	−0.294113	+	0.955771i	$0.595024\pi$
$632$	−0.648120	−0.0257808
$633$	0	0
$634$	−32.4317	−1.28803
$635$	0.269841	0.0107083
$636$	0	0
$637$	0	0
$638$	1.48464	0.0587773
$639$	0	0
$640$	−2.55352	−0.100937
$641$	−28.9547	−1.14364	−0.571821	−	0.820378i	$-0.693763\pi$
$641$	−28.9547	−1.14364	−0.571821	+	0.820378i	$0.693763\pi$
$642$	0	0
$643$	−35.2971	−1.39198	−0.695991	−	0.718050i	$-0.745035\pi$
$643$	−35.2971	−1.39198	−0.695991	+	0.718050i	$0.745035\pi$
$644$	0	0
$645$	0	0
$646$	15.4102	0.606305
$647$	26.4163	1.03853	0.519265	−	0.854613i	$-0.326206\pi$
$647$	26.4163	1.03853	0.519265	+	0.854613i	$0.326206\pi$
$648$	0	0
$649$	−8.77544	−0.344466
$650$	11.4424	0.448808
$651$	0	0
$652$	−27.6875	−1.08433
$653$	−20.4355	−0.799702	−0.399851	−	0.916580i	$-0.630938\pi$
$653$	−20.4355	−0.799702	−0.399851	+	0.916580i	$0.630938\pi$
$654$	0	0
$655$	−11.7327	−0.458435
$656$	−33.7428	−1.31744
$657$	0	0
$658$	0	0
$659$	−9.39750	−0.366075	−0.183037	−	0.983106i	$-0.558593\pi$
$659$	−9.39750	−0.366075	−0.183037	+	0.983106i	$0.558593\pi$
$660$	0	0
$661$	4.12525	0.160454	0.0802269	−	0.996777i	$-0.474436\pi$
$661$	4.12525	0.160454	0.0802269	+	0.996777i	$0.474436\pi$
$662$	49.7440	1.93336
$663$	0	0
$664$	−1.19667	−0.0464397
$665$	0	0
$666$	0	0
$667$	2.02121	0.0782614
$668$	0.450401	0.0174266
$669$	0	0
$670$	34.2407	1.32284
$671$	3.09719	0.119566
$672$	0	0
$673$	40.6984	1.56881	0.784404	−	0.620250i	$-0.212969\pi$
$673$	40.6984	1.56881	0.784404	+	0.620250i	$0.212969\pi$
$674$	−65.5861	−2.52628
$675$	0	0
$676$	−21.4670	−0.825652
$677$	23.5916	0.906700	0.453350	−	0.891333i	$-0.350229\pi$
$677$	23.5916	0.906700	0.453350	+	0.891333i	$0.350229\pi$
$678$	0	0
$679$	0	0
$680$	0.319780	0.0122630
$681$	0	0
$682$	16.0308	0.613850
$683$	−13.2252	−0.506049	−0.253024	−	0.967460i	$-0.581425\pi$
$683$	−13.2252	−0.506049	−0.253024	+	0.967460i	$0.581425\pi$
$684$	0	0
$685$	20.6708	0.789790
$686$	0	0
$687$	0	0
$688$	13.6352	0.519836
$689$	−12.7975	−0.487544
$690$	0	0
$691$	39.1181	1.48812	0.744062	−	0.668110i	$-0.232897\pi$
$691$	39.1181	1.48812	0.744062	+	0.668110i	$0.232897\pi$
$692$	−9.65726	−0.367114
$693$	0	0
$694$	−26.5736	−1.00872
$695$	−5.54313	−0.210263
$696$	0	0
$697$	9.01696	0.341542
$698$	−10.4480	−0.395462
$699$	0	0
$700$	0	0
$701$	46.3043	1.74889	0.874445	−	0.485125i	$-0.161226\pi$
$701$	46.3043	1.74889	0.874445	+	0.485125i	$0.161226\pi$
$702$	0	0
$703$	61.9372	2.33601
$704$	−8.03976	−0.303010
$705$	0	0
$706$	30.9465	1.16469
$707$	0	0
$708$	0	0
$709$	−38.5985	−1.44960	−0.724798	−	0.688962i	$-0.758067\pi$
$709$	−38.5985	−1.44960	−0.724798	+	0.688962i	$0.758067\pi$
$710$	16.8897	0.633859
$711$	0	0
$712$	−3.96147	−0.148462
$713$	21.8245	0.817335
$714$	0	0
$715$	−1.97820	−0.0739806
$716$	40.1876	1.50188
$717$	0	0
$718$	−58.0528	−2.16651
$719$	37.7065	1.40622	0.703108	−	0.711083i	$-0.251795\pi$
$719$	37.7065	1.40622	0.703108	+	0.711083i	$0.251795\pi$
$720$	0	0
$721$	0	0
$722$	−78.3998	−2.91774
$723$	0	0
$724$	−46.1273	−1.71431
$725$	2.70033	0.100288
$726$	0	0
$727$	32.8847	1.21963	0.609813	−	0.792546i	$-0.291245\pi$
$727$	32.8847	1.21963	0.609813	+	0.792546i	$0.291245\pi$
$728$	0	0
$729$	0	0
$730$	7.05275	0.261034
$731$	−3.64367	−0.134766
$732$	0	0
$733$	−17.7674	−0.656253	−0.328126	−	0.944634i	$-0.606417\pi$
$733$	−17.7674	−0.656253	−0.328126	+	0.944634i	$0.606417\pi$
$734$	62.2205	2.29660
$735$	0	0
$736$	−20.1077	−0.741178
$737$	11.7021	0.431052
$738$	0	0
$739$	−39.7705	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$739$	−39.7705	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$740$	22.4354	0.824742
$741$	0	0
$742$	0	0
$743$	−45.6666	−1.67534	−0.837672	−	0.546173i	$-0.816084\pi$
$743$	−45.6666	−1.67534	−0.837672	+	0.546173i	$0.816084\pi$
$744$	0	0
$745$	16.4238	0.601723
$746$	14.6338	0.535781
$747$	0	0
$748$	1.90770	0.0697523
$749$	0	0
$750$	0	0
$751$	−35.8133	−1.30685	−0.653424	−	0.756993i	$-0.726668\pi$
$751$	−35.8133	−1.30685	−0.653424	+	0.756993i	$0.726668\pi$
$752$	−16.8038	−0.612770
$753$	0	0
$754$	2.80262	0.102065
$755$	−21.7790	−0.792620
$756$	0	0
$757$	−14.5236	−0.527869	−0.263935	−	0.964541i	$-0.585020\pi$
$757$	−14.5236	−0.527869	−0.263935	+	0.964541i	$0.585020\pi$
$758$	41.2643	1.49879
$759$	0	0
$760$	−2.42733	−0.0880486
$761$	46.2796	1.67763	0.838817	−	0.544414i	$-0.183248\pi$
$761$	46.2796	1.67763	0.838817	+	0.544414i	$0.183248\pi$
$762$	0	0
$763$	0	0
$764$	33.2989	1.20471
$765$	0	0
$766$	38.1008	1.37664
$767$	−16.5658	−0.598158
$768$	0	0
$769$	−18.6526	−0.672630	−0.336315	−	0.941750i	$-0.609181\pi$
$769$	−18.6526	−0.672630	−0.336315	+	0.941750i	$0.609181\pi$
$770$	0	0
$771$	0	0
$772$	17.2348	0.620294
$773$	22.1689	0.797361	0.398681	−	0.917090i	$-0.369468\pi$
$773$	22.1689	0.797361	0.398681	+	0.917090i	$0.369468\pi$
$774$	0	0
$775$	29.1575	1.04737
$776$	0.321980	0.0115584
$777$	0	0
$778$	55.5337	1.99098
$779$	−68.4444	−2.45227
$780$	0	0
$781$	5.77221	0.206546
$782$	5.04555	0.180428
$783$	0	0
$784$	0	0
$785$	2.75602	0.0983667
$786$	0	0
$787$	−23.3764	−0.833278	−0.416639	−	0.909072i	$-0.636792\pi$
$787$	−23.3764	−0.833278	−0.416639	+	0.909072i	$0.636792\pi$
$788$	−6.99803	−0.249294
$789$	0	0
$790$	6.91116	0.245888
$791$	0	0
$792$	0	0
$793$	5.84673	0.207624
$794$	−71.5742	−2.54007
$795$	0	0
$796$	6.51444	0.230898
$797$	−9.66634	−0.342399	−0.171200	−	0.985236i	$-0.554764\pi$
$797$	−9.66634	−0.342399	−0.171200	+	0.985236i	$0.554764\pi$
$798$	0	0
$799$	4.49040	0.158859
$800$	−26.8638	−0.949779
$801$	0	0
$802$	−37.0896	−1.30968
$803$	2.41034	0.0850592
$804$	0	0
$805$	0	0
$806$	30.2621	1.06594
$807$	0	0
$808$	−1.51076	−0.0531483
$809$	53.8682	1.89391	0.946953	−	0.321372i	$-0.104144\pi$
$809$	53.8682	1.89391	0.946953	+	0.321372i	$0.104144\pi$
$810$	0	0
$811$	4.24730	0.149143	0.0745714	−	0.997216i	$-0.476241\pi$
$811$	4.24730	0.149143	0.0745714	+	0.997216i	$0.476241\pi$
$812$	0	0
$813$	0	0
$814$	14.8958	0.522096
$815$	16.9138	0.592466
$816$	0	0
$817$	27.6577	0.967621
$818$	2.54485	0.0889785
$819$	0	0
$820$	−24.7925	−0.865790
$821$	−17.2732	−0.602838	−0.301419	−	0.953492i	$-0.597460\pi$
$821$	−17.2732	−0.602838	−0.301419	+	0.953492i	$0.597460\pi$
$822$	0	0
$823$	54.4791	1.89902	0.949511	−	0.313733i	$-0.101580\pi$
$823$	54.4791	1.89902	0.949511	+	0.313733i	$0.101580\pi$
$824$	0.949979	0.0330941
$825$	0	0
$826$	0	0
$827$	35.6045	1.23809	0.619046	−	0.785355i	$-0.287520\pi$
$827$	35.6045	1.23809	0.619046	+	0.785355i	$0.287520\pi$
$828$	0	0
$829$	52.3490	1.81816	0.909078	−	0.416626i	$-0.136787\pi$
$829$	52.3490	1.81816	0.909078	+	0.416626i	$0.136787\pi$
$830$	12.7605	0.442924
$831$	0	0
$832$	−15.1771	−0.526170
$833$	0	0
$834$	0	0
$835$	−0.275143	−0.00952170
$836$	−14.4806	−0.500823
$837$	0	0
$838$	55.2627	1.90902
$839$	21.5916	0.745426	0.372713	−	0.927947i	$-0.378428\pi$
$839$	21.5916	0.745426	0.372713	+	0.927947i	$0.378428\pi$
$840$	0	0
$841$	−28.3386	−0.977193
$842$	34.1199	1.17585
$843$	0	0
$844$	−26.5826	−0.915010
$845$	13.1138	0.451128
$846$	0	0
$847$	0	0
$848$	28.2125	0.968821
$849$	0	0
$850$	6.74084	0.231209
$851$	20.2793	0.695166
$852$	0	0
$853$	−17.6350	−0.603812	−0.301906	−	0.953338i	$-0.597623\pi$
$853$	−17.6350	−0.603812	−0.301906	+	0.953338i	$0.597623\pi$
$854$	0	0
$855$	0	0
$856$	1.80577	0.0617199
$857$	1.07818	0.0368298	0.0184149	−	0.999830i	$-0.494138\pi$
$857$	1.07818	0.0368298	0.0184149	+	0.999830i	$0.494138\pi$
$858$	0	0
$859$	7.66631	0.261571	0.130786	−	0.991411i	$-0.458250\pi$
$859$	7.66631	0.261571	0.130786	+	0.991411i	$0.458250\pi$
$860$	10.0184	0.341625
$861$	0	0
$862$	59.7677	2.03570
$863$	5.57191	0.189670	0.0948351	−	0.995493i	$-0.469768\pi$
$863$	5.57191	0.189670	0.0948351	+	0.995493i	$0.469768\pi$
$864$	0	0
$865$	5.89946	0.200588
$866$	−37.0518	−1.25907
$867$	0	0
$868$	0	0
$869$	2.36195	0.0801237
$870$	0	0
$871$	22.0906	0.748512
$872$	−3.06916	−0.103935
$873$	0	0
$874$	−38.2989	−1.29548
$875$	0	0
$876$	0	0
$877$	15.3182	0.517258	0.258629	−	0.965977i	$-0.416729\pi$
$877$	15.3182	0.517258	0.258629	+	0.965977i	$0.416729\pi$
$878$	−22.1195	−0.746496
$879$	0	0
$880$	4.36103	0.147010
$881$	−48.8420	−1.64553	−0.822765	−	0.568381i	$-0.807570\pi$
$881$	−48.8420	−1.64553	−0.822765	+	0.568381i	$0.807570\pi$
$882$	0	0
$883$	−2.46902	−0.0830890	−0.0415445	−	0.999137i	$-0.513228\pi$
$883$	−2.46902	−0.0830890	−0.0415445	+	0.999137i	$0.513228\pi$
$884$	3.60126	0.121123
$885$	0	0
$886$	−46.1259	−1.54963
$887$	54.9820	1.84612	0.923058	−	0.384661i	$-0.125682\pi$
$887$	54.9820	1.84612	0.923058	+	0.384661i	$0.125682\pi$
$888$	0	0
$889$	0	0
$890$	42.2427	1.41598
$891$	0	0
$892$	39.3614	1.31792
$893$	−34.0850	−1.14061
$894$	0	0
$895$	−24.5499	−0.820613
$896$	0	0
$897$	0	0
$898$	21.5182	0.718072
$899$	7.14163	0.238187
$900$	0	0
$901$	−7.53911	−0.251164
$902$	−16.4607	−0.548081
$903$	0	0
$904$	5.13150	0.170671
$905$	28.1784	0.936681
$906$	0	0
$907$	10.7779	0.357875	0.178937	−	0.983860i	$-0.442734\pi$
$907$	10.7779	0.357875	0.178937	+	0.983860i	$0.442734\pi$
$908$	41.0855	1.36347
$909$	0	0
$910$	0	0
$911$	43.6354	1.44571	0.722853	−	0.691002i	$-0.242830\pi$
$911$	43.6354	1.44571	0.722853	+	0.691002i	$0.242830\pi$
$912$	0	0
$913$	4.36103	0.144329
$914$	19.2678	0.637322
$915$	0	0
$916$	−14.8649	−0.491149
$917$	0	0
$918$	0	0
$919$	11.4463	0.377579	0.188790	−	0.982018i	$-0.439544\pi$
$919$	11.4463	0.377579	0.188790	+	0.982018i	$0.439544\pi$
$920$	−0.794750	−0.0262021
$921$	0	0
$922$	1.90952	0.0628867
$923$	10.8965	0.358663
$924$	0	0
$925$	27.0931	0.890816
$926$	−28.9227	−0.950460
$927$	0	0
$928$	−6.57983	−0.215993
$929$	22.5540	0.739971	0.369986	−	0.929037i	$-0.379363\pi$
$929$	22.5540	0.739971	0.369986	+	0.929037i	$0.379363\pi$
$930$	0	0
$931$	0	0
$932$	37.7813	1.23757
$933$	0	0
$934$	−40.8259	−1.33586
$935$	−1.16538	−0.0381120
$936$	0	0
$937$	36.4643	1.19124	0.595619	−	0.803267i	$-0.296907\pi$
$937$	36.4643	1.19124	0.595619	+	0.803267i	$0.296907\pi$
$938$	0	0
$939$	0	0
$940$	−12.3465	−0.402699
$941$	−24.2685	−0.791130	−0.395565	−	0.918438i	$-0.629451\pi$
$941$	−24.2685	−0.791130	−0.395565	+	0.918438i	$0.629451\pi$
$942$	0	0
$943$	−22.4098	−0.729765
$944$	36.5201	1.18863
$945$	0	0
$946$	6.65162	0.216263
$947$	−8.01869	−0.260573	−0.130286	−	0.991476i	$-0.541590\pi$
$947$	−8.01869	−0.260573	−0.130286	+	0.991476i	$0.541590\pi$
$948$	0	0
$949$	4.55013	0.147703
$950$	−51.1672	−1.66008
$951$	0	0
$952$	0	0
$953$	32.0907	1.03952	0.519760	−	0.854313i	$-0.326022\pi$
$953$	32.0907	1.03952	0.519760	+	0.854313i	$0.326022\pi$
$954$	0	0
$955$	−20.3417	−0.658242
$956$	3.16878	0.102486
$957$	0	0
$958$	−86.0146	−2.77900
$959$	0	0
$960$	0	0
$961$	46.1138	1.48754
$962$	28.1195	0.906608
$963$	0	0
$964$	−13.9833	−0.450372
$965$	−10.5284	−0.338922
$966$	0	0
$967$	44.7898	1.44034	0.720172	−	0.693796i	$-0.244063\pi$
$967$	44.7898	1.44034	0.720172	+	0.693796i	$0.244063\pi$
$968$	2.51465	0.0808240
$969$	0	0
$970$	−3.43340	−0.110240
$971$	−13.2205	−0.424267	−0.212134	−	0.977241i	$-0.568041\pi$
$971$	−13.2205	−0.424267	−0.212134	+	0.977241i	$0.568041\pi$
$972$	0	0
$973$	0	0
$974$	−65.0548	−2.08449
$975$	0	0
$976$	−12.8894	−0.412578
$977$	4.54724	0.145479	0.0727395	−	0.997351i	$-0.476826\pi$
$977$	4.54724	0.145479	0.0727395	+	0.997351i	$0.476826\pi$
$978$	0	0
$979$	14.4368	0.461403
$980$	0	0
$981$	0	0
$982$	−50.7871	−1.62068
$983$	−1.59073	−0.0507365	−0.0253683	−	0.999678i	$-0.508076\pi$
$983$	−1.59073	−0.0507365	−0.0253683	+	0.999678i	$0.508076\pi$
$984$	0	0
$985$	4.27498	0.136212
$986$	1.65105	0.0525803
$987$	0	0
$988$	−27.3358	−0.869668
$989$	9.05561	0.287952
$990$	0	0
$991$	−41.3044	−1.31208	−0.656039	−	0.754727i	$-0.727769\pi$
$991$	−41.3044	−1.31208	−0.656039	+	0.754727i	$0.727769\pi$
$992$	−71.0475	−2.25576
$993$	0	0
$994$	0	0
$995$	−3.97956	−0.126160
$996$	0	0
$997$	57.1037	1.80849	0.904245	−	0.427013i	$-0.140434\pi$
$997$	57.1037	1.80849	0.904245	+	0.427013i	$0.140434\pi$
$998$	−37.0947	−1.17421
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7497.2.a.ce.1.2	yes	8
3.2	odd	2		7497.2.a.cd.1.7	yes	8
7.6	odd	2		7497.2.a.cd.1.2	✓	8
21.20	even	2	inner	7497.2.a.ce.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7497.2.a.cd.1.2	✓	8	7.6	odd	2
7497.2.a.cd.1.7	yes	8	3.2	odd	2
7497.2.a.ce.1.2	yes	8	1.1	even	1	trivial
7497.2.a.ce.1.7	yes	8	21.20	even	2	inner

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:	\( N \)	\(=\)	\( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7497.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03016 q^{2} +2.12154 q^{4} -1.29601 q^{5} -0.246742 q^{8} +O(q^{10})\)	\(q-2.03016 q^{2} +2.12154 q^{4} -1.29601 q^{5} -0.246742 q^{8} +2.63111 q^{10} +0.899205 q^{11} +1.69747 q^{13} -3.74215 q^{16} +1.00000 q^{17} -7.59063 q^{19} -2.74954 q^{20} -1.82553 q^{22} -2.48530 q^{23} -3.32035 q^{25} -3.44614 q^{26} -0.813264 q^{29} -8.78144 q^{31} +8.09064 q^{32} -2.03016 q^{34} -8.15970 q^{37} +15.4102 q^{38} +0.319780 q^{40} +9.01696 q^{41} -3.64367 q^{43} +1.90770 q^{44} +5.04555 q^{46} +4.49040 q^{47} +6.74084 q^{50} +3.60126 q^{52} -7.53911 q^{53} -1.16538 q^{55} +1.65105 q^{58} -9.75911 q^{59} +3.44437 q^{61} +17.8277 q^{62} -8.94097 q^{64} -2.19995 q^{65} +13.0138 q^{67} +2.12154 q^{68} +6.41924 q^{71} +2.68053 q^{73} +16.5655 q^{74} -16.1038 q^{76} +2.62671 q^{79} +4.84987 q^{80} -18.3058 q^{82} +4.84987 q^{83} -1.29601 q^{85} +7.39722 q^{86} -0.221872 q^{88} +16.0551 q^{89} -5.27266 q^{92} -9.11622 q^{94} +9.83754 q^{95} -1.30493 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 12 q^{5}+O(q^{10}) \) 8 * q + 8 * q^4 + 12 * q^5	\( 8 q + 8 q^{4} + 12 q^{5} + 16 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{22} + 8 q^{25} - 4 q^{26} + 12 q^{37} + 60 q^{38} + 28 q^{41} - 24 q^{43} + 4 q^{46} + 20 q^{47} + 40 q^{58} + 12 q^{59} + 48 q^{62} + 48 q^{67} + 8 q^{68} + 60 q^{79} + 40 q^{80} + 40 q^{83} + 12 q^{85} - 8 q^{88} + 16 q^{89}+O(q^{100}) \) 8 * q + 8 * q^4 + 12 * q^5 + 16 * q^16 + 8 * q^17 - 8 * q^20 - 28 * q^22 + 8 * q^25 - 4 * q^26 + 12 * q^37 + 60 * q^38 + 28 * q^41 - 24 * q^43 + 4 * q^46 + 20 * q^47 + 40 * q^58 + 12 * q^59 + 48 * q^62 + 48 * q^67 + 8 * q^68 + 60 * q^79 + 40 * q^80 + 40 * q^83 + 12 * q^85 - 8 * q^88 + 16 * q^89

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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