LMFDB - Embedded newform 5175.2.a.bz.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, 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("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.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:	$41.3225830460$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.98838128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 $ x^6 - x^5 - 10x^4 - x^3 + 16x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1035)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.493507$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.810417 q^{2} -1.34322 q^{4} +4.60617 q^{7} +2.70941 q^{8} +5.35375 q^{11} -5.51982 q^{13} -3.73292 q^{14} +0.490703 q^{16} -1.17660 q^{17} -1.73292 q^{19} -4.33877 q^{22} -1.00000 q^{23} +4.47336 q^{26} -6.18712 q^{28} -3.08635 q^{29} +4.34322 q^{31} -5.81648 q^{32} +0.953534 q^{34} -8.81926 q^{37} +1.40438 q^{38} -7.27347 q^{41} -8.52038 q^{43} -7.19129 q^{44} +0.810417 q^{46} -9.69252 q^{47} +14.2168 q^{49} +7.41436 q^{52} -10.2018 q^{53} +12.4800 q^{56} +2.50123 q^{58} -11.7468 q^{59} +1.33270 q^{61} -3.51982 q^{62} +3.73237 q^{64} +6.01391 q^{67} +1.58043 q^{68} -11.3991 q^{71} +1.51982 q^{73} +7.14728 q^{74} +2.32770 q^{76} +24.6603 q^{77} +1.26764 q^{79} +5.89454 q^{82} -1.15555 q^{83} +6.90506 q^{86} +14.5055 q^{88} -5.40223 q^{89} -25.4252 q^{91} +1.34322 q^{92} +7.85498 q^{94} -16.4440 q^{97} -11.5215 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{7} + 4 q^{11} - 12 q^{13} - 4 q^{14} + 14 q^{16} - 4 q^{17} + 8 q^{19} - 8 q^{22} - 6 q^{23} - 12 q^{26} - 24 q^{28} - 6 q^{29} + 8 q^{31} - 20 q^{32} - 12 q^{34} - 22 q^{37} - 36 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^7 + 4 * q^11 - 12 * q^13 - 4 * q^14 + 14 * q^16 - 4 * q^17 + 8 * q^19 - 8 * q^22 - 6 * q^23 - 12 * q^26 - 24 * q^28 - 6 * q^29 + 8 * q^31 - 20 * q^32 - 12 * q^34 - 22 * q^37 - 36 * q^38 - 18 * q^41 - 8 * q^43 + 4 * q^44 - 12 * q^47 + 18 * q^49 - 4 * q^53 + 4 * q^56 + 16 * q^58 - 6 * q^59 + 14 * q^64 - 10 * q^67 - 28 * q^68 - 22 * q^71 - 12 * q^73 - 20 * q^74 + 16 * q^76 + 4 * q^77 + 4 * q^79 + 12 * q^82 - 24 * q^83 + 20 * q^86 + 8 * q^88 - 8 * q^89 + 4 * q^91 - 10 * q^92 + 20 * q^94 - 12 * q^97 + 4 * q^98

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 $

$2$	−0.810417	−0.573051	−0.286526	−	0.958073i	$-0.592500\pi$
$2$	−0.810417	−0.573051	−0.286526	+	0.958073i	$0.592500\pi$
$3$	0	0
$3$	0	0
$4$	−1.34322	−0.671612
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.60617	1.74097	0.870484	−	0.492196i	$-0.163806\pi$
$7$	4.60617	1.74097	0.870484	+	0.492196i	$0.163806\pi$
$8$	2.70941	0.957919
$9$	0	0
$10$	0	0
$11$	5.35375	1.61422	0.807108	−	0.590404i	$-0.201031\pi$
$11$	5.35375	1.61422	0.807108	+	0.590404i	$0.201031\pi$
$12$	0	0
$13$	−5.51982	−1.53092	−0.765462	−	0.643482i	$-0.777489\pi$
$13$	−5.51982	−1.53092	−0.765462	+	0.643482i	$0.777489\pi$
$14$	−3.73292	−0.997664
$15$	0	0
$16$	0.490703	0.122676
$17$	−1.17660	−0.285367	−0.142683	−	0.989768i	$-0.545573\pi$
$17$	−1.17660	−0.285367	−0.142683	+	0.989768i	$0.545573\pi$
$18$	0	0
$19$	−1.73292	−0.397558	−0.198779	−	0.980044i	$-0.563698\pi$
$19$	−1.73292	−0.397558	−0.198779	+	0.980044i	$0.563698\pi$
$20$	0	0
$21$	0	0
$22$	−4.33877	−0.925028
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	4.47336	0.877297
$27$	0	0
$28$	−6.18712	−1.16926
$29$	−3.08635	−0.573120	−0.286560	−	0.958062i	$-0.592512\pi$
$29$	−3.08635	−0.573120	−0.286560	+	0.958062i	$0.592512\pi$
$30$	0	0
$31$	4.34322	0.780066	0.390033	−	0.920801i	$-0.372464\pi$
$31$	4.34322	0.780066	0.390033	+	0.920801i	$0.372464\pi$
$32$	−5.81648	−1.02822
$33$	0	0
$34$	0.953534	0.163530
$35$	0	0
$36$	0	0
$37$	−8.81926	−1.44988	−0.724939	−	0.688813i	$-0.758132\pi$
$37$	−8.81926	−1.44988	−0.724939	+	0.688813i	$0.758132\pi$
$38$	1.40438	0.227821
$39$	0	0
$40$	0	0
$41$	−7.27347	−1.13593	−0.567963	−	0.823054i	$-0.692268\pi$
$41$	−7.27347	−1.13593	−0.567963	+	0.823054i	$0.692268\pi$
$42$	0	0
$43$	−8.52038	−1.29935	−0.649673	−	0.760214i	$-0.725094\pi$
$43$	−8.52038	−1.29935	−0.649673	+	0.760214i	$0.725094\pi$
$44$	−7.19129	−1.08413
$45$	0	0
$46$	0.810417	0.119489
$47$	−9.69252	−1.41380	−0.706899	−	0.707314i	$-0.749907\pi$
$47$	−9.69252	−1.41380	−0.706899	+	0.707314i	$0.749907\pi$
$48$	0	0
$49$	14.2168	2.03097
$50$	0	0
$51$	0	0
$52$	7.41436	1.02819
$53$	−10.2018	−1.40133	−0.700663	−	0.713492i	$-0.747113\pi$
$53$	−10.2018	−1.40133	−0.700663	+	0.713492i	$0.747113\pi$
$54$	0	0
$55$	0	0
$56$	12.4800	1.66771
$57$	0	0
$58$	2.50123	0.328427
$59$	−11.7468	−1.52931	−0.764653	−	0.644442i	$-0.777090\pi$
$59$	−11.7468	−1.52931	−0.764653	+	0.644442i	$0.777090\pi$
$60$	0	0
$61$	1.33270	0.170635	0.0853174	−	0.996354i	$-0.472810\pi$
$61$	1.33270	0.170635	0.0853174	+	0.996354i	$0.472810\pi$
$62$	−3.51982	−0.447018
$63$	0	0
$64$	3.73237	0.466546
$65$	0	0
$66$	0	0
$67$	6.01391	0.734716	0.367358	−	0.930080i	$-0.380262\pi$
$67$	6.01391	0.734716	0.367358	+	0.930080i	$0.380262\pi$
$68$	1.58043	0.191656
$69$	0	0
$70$	0	0
$71$	−11.3991	−1.35283	−0.676415	−	0.736521i	$-0.736467\pi$
$71$	−11.3991	−1.35283	−0.676415	+	0.736521i	$0.736467\pi$
$72$	0	0
$73$	1.51982	0.177882	0.0889408	−	0.996037i	$-0.471652\pi$
$73$	1.51982	0.177882	0.0889408	+	0.996037i	$0.471652\pi$
$74$	7.14728	0.830854
$75$	0	0
$76$	2.32770	0.267005
$77$	24.6603	2.81030
$78$	0	0
$79$	1.26764	0.142621	0.0713103	−	0.997454i	$-0.477282\pi$
$79$	1.26764	0.142621	0.0713103	+	0.997454i	$0.477282\pi$
$80$	0	0
$81$	0	0
$82$	5.89454	0.650943
$83$	−1.15555	−0.126838	−0.0634189	−	0.997987i	$-0.520200\pi$
$83$	−1.15555	−0.126838	−0.0634189	+	0.997987i	$0.520200\pi$
$84$	0	0
$85$	0	0
$86$	6.90506	0.744591
$87$	0	0
$88$	14.5055	1.54629
$89$	−5.40223	−0.572635	−0.286317	−	0.958135i	$-0.592431\pi$
$89$	−5.40223	−0.572635	−0.286317	+	0.958135i	$0.592431\pi$
$90$	0	0
$91$	−25.4252	−2.66529
$92$	1.34322	0.140041
$93$	0	0
$94$	7.85498	0.810179
$95$	0	0
$96$	0	0
$97$	−16.4440	−1.66964	−0.834819	−	0.550524i	$-0.814428\pi$
$97$	−16.4440	−1.66964	−0.834819	+	0.550524i	$0.814428\pi$
$98$	−11.5215	−1.16385
$99$	0	0
$100$	0	0
$101$	−3.43403	−0.341699	−0.170849	−	0.985297i	$-0.554651\pi$
$101$	−3.43403	−0.341699	−0.170849	+	0.985297i	$0.554651\pi$
$102$	0	0
$103$	−9.42879	−0.929046	−0.464523	−	0.885561i	$-0.653774\pi$
$103$	−9.42879	−0.929046	−0.464523	+	0.885561i	$0.653774\pi$
$104$	−14.9554	−1.46650
$105$	0	0
$106$	8.26772	0.803032
$107$	14.9093	1.44134	0.720669	−	0.693279i	$-0.243835\pi$
$107$	14.9093	1.44134	0.720669	+	0.693279i	$0.243835\pi$
$108$	0	0
$109$	8.26216	0.791371	0.395686	−	0.918386i	$-0.370507\pi$
$109$	8.26216	0.791371	0.395686	+	0.918386i	$0.370507\pi$
$110$	0	0
$111$	0	0
$112$	2.26026	0.213575
$113$	−10.6280	−0.999798	−0.499899	−	0.866084i	$-0.666630\pi$
$113$	−10.6280	−0.999798	−0.499899	+	0.866084i	$0.666630\pi$
$114$	0	0
$115$	0	0
$116$	4.14566	0.384915
$117$	0	0
$118$	9.51982	0.876371
$119$	−5.41960	−0.496814
$120$	0	0
$121$	17.6626	1.60569
$122$	−1.08004	−0.0977825
$123$	0	0
$124$	−5.83393	−0.523902
$125$	0	0
$126$	0	0
$127$	2.23560	0.198377	0.0991887	−	0.995069i	$-0.468375\pi$
$127$	2.23560	0.198377	0.0991887	+	0.995069i	$0.468375\pi$
$128$	8.60819	0.760864
$129$	0	0
$130$	0	0
$131$	10.6831	0.933386	0.466693	−	0.884419i	$-0.345445\pi$
$131$	10.6831	0.933386	0.466693	+	0.884419i	$0.345445\pi$
$132$	0	0
$133$	−7.98211	−0.692136
$134$	−4.87377	−0.421030
$135$	0	0
$136$	−3.18788	−0.273358
$137$	−0.326633	−0.0279061	−0.0139531	−	0.999903i	$-0.504442\pi$
$137$	−0.326633	−0.0279061	−0.0139531	+	0.999903i	$0.504442\pi$
$138$	0	0
$139$	9.54894	0.809931	0.404965	−	0.914332i	$-0.367284\pi$
$139$	9.54894	0.809931	0.404965	+	0.914332i	$0.367284\pi$
$140$	0	0
$141$	0	0
$142$	9.23805	0.775240
$143$	−29.5517	−2.47124
$144$	0	0
$145$	0	0
$146$	−1.23169	−0.101935
$147$	0	0
$148$	11.8463	0.973756
$149$	22.5368	1.84628	0.923142	−	0.384460i	$-0.125612\pi$
$149$	22.5368	1.84628	0.923142	+	0.384460i	$0.125612\pi$
$150$	0	0
$151$	16.3305	1.32896	0.664478	−	0.747308i	$-0.268654\pi$
$151$	16.3305	1.32896	0.664478	+	0.747308i	$0.268654\pi$
$152$	−4.69517	−0.380829
$153$	0	0
$154$	−19.9851	−1.61045
$155$	0	0
$156$	0	0
$157$	13.2666	1.05879	0.529397	−	0.848374i	$-0.322418\pi$
$157$	13.2666	1.05879	0.529397	+	0.848374i	$0.322418\pi$
$158$	−1.02732	−0.0817289
$159$	0	0
$160$	0	0
$161$	−4.60617	−0.363017
$162$	0	0
$163$	15.7261	1.23176	0.615881	−	0.787839i	$-0.288800\pi$
$163$	15.7261	1.23176	0.615881	+	0.787839i	$0.288800\pi$
$164$	9.76990	0.762901
$165$	0	0
$166$	0.936475	0.0726846
$167$	−1.59282	−0.123256	−0.0616279	−	0.998099i	$-0.519629\pi$
$167$	−1.59282	−0.123256	−0.0616279	+	0.998099i	$0.519629\pi$
$168$	0	0
$169$	17.4684	1.34373
$170$	0	0
$171$	0	0
$172$	11.4448	0.872657
$173$	6.90506	0.524982	0.262491	−	0.964934i	$-0.415456\pi$
$173$	6.90506	0.524982	0.262491	+	0.964934i	$0.415456\pi$
$174$	0	0
$175$	0	0
$176$	2.62710	0.198025
$177$	0	0
$178$	4.37805	0.328149
$179$	−25.1515	−1.87991	−0.939957	−	0.341294i	$-0.889135\pi$
$179$	−25.1515	−1.87991	−0.939957	+	0.341294i	$0.889135\pi$
$180$	0	0
$181$	−12.9678	−0.963886	−0.481943	−	0.876203i	$-0.660069\pi$
$181$	−12.9678	−0.963886	−0.481943	+	0.876203i	$0.660069\pi$
$182$	20.6050	1.52735
$183$	0	0
$184$	−2.70941	−0.199740
$185$	0	0
$186$	0	0
$187$	−6.29920	−0.460643
$188$	13.0192	0.949525
$189$	0	0
$190$	0	0
$191$	3.90897	0.282843	0.141421	−	0.989949i	$-0.454833\pi$
$191$	3.90897	0.282843	0.141421	+	0.989949i	$0.454833\pi$
$192$	0	0
$193$	2.68645	0.193375	0.0966874	−	0.995315i	$-0.469175\pi$
$193$	2.68645	0.193375	0.0966874	+	0.995315i	$0.469175\pi$
$194$	13.3265	0.956788
$195$	0	0
$196$	−19.0964	−1.36403
$197$	−9.78139	−0.696895	−0.348448	−	0.937328i	$-0.613291\pi$
$197$	−9.78139	−0.696895	−0.348448	+	0.937328i	$0.613291\pi$
$198$	0	0
$199$	−19.0867	−1.35302	−0.676509	−	0.736434i	$-0.736508\pi$
$199$	−19.0867	−1.35302	−0.676509	+	0.736434i	$0.736508\pi$
$200$	0	0
$201$	0	0
$202$	2.78299	0.195811
$203$	−14.2162	−0.997784
$204$	0	0
$205$	0	0
$206$	7.64125	0.532391
$207$	0	0
$208$	−2.70859	−0.187807
$209$	−9.27760	−0.641745
$210$	0	0
$211$	−8.09524	−0.557299	−0.278650	−	0.960393i	$-0.589887\pi$
$211$	−8.09524	−0.557299	−0.278650	+	0.960393i	$0.589887\pi$
$212$	13.7033	0.941149
$213$	0	0
$214$	−12.0828	−0.825960
$215$	0	0
$216$	0	0
$217$	20.0056	1.35807
$218$	−6.69579	−0.453496
$219$	0	0
$220$	0	0
$221$	6.49460	0.436874
$222$	0	0
$223$	−14.2094	−0.951534	−0.475767	−	0.879571i	$-0.657829\pi$
$223$	−14.2094	−0.951534	−0.475767	+	0.879571i	$0.657829\pi$
$224$	−26.7917	−1.79010
$225$	0	0
$226$	8.61311	0.572936
$227$	−16.6256	−1.10348	−0.551740	−	0.834016i	$-0.686036\pi$
$227$	−16.6256	−1.10348	−0.551740	+	0.834016i	$0.686036\pi$
$228$	0	0
$229$	15.3640	1.01528	0.507640	−	0.861569i	$-0.330518\pi$
$229$	15.3640	1.01528	0.507640	+	0.861569i	$0.330518\pi$
$230$	0	0
$231$	0	0
$232$	−8.36217	−0.549003
$233$	−15.0878	−0.988433	−0.494217	−	0.869339i	$-0.664545\pi$
$233$	−15.0878	−0.988433	−0.494217	+	0.869339i	$0.664545\pi$
$234$	0	0
$235$	0	0
$236$	15.7786	1.02710
$237$	0	0
$238$	4.39214	0.284700
$239$	1.42847	0.0924000	0.0462000	−	0.998932i	$-0.485289\pi$
$239$	1.42847	0.0924000	0.0462000	+	0.998932i	$0.485289\pi$
$240$	0	0
$241$	10.5954	0.682511	0.341255	−	0.939971i	$-0.389148\pi$
$241$	10.5954	0.682511	0.341255	+	0.939971i	$0.389148\pi$
$242$	−14.3141	−0.920145
$243$	0	0
$244$	−1.79012	−0.114600
$245$	0	0
$246$	0	0
$247$	9.56539	0.608631
$248$	11.7676	0.747241
$249$	0	0
$250$	0	0
$251$	−15.3939	−0.971657	−0.485829	−	0.874054i	$-0.661482\pi$
$251$	−15.3939	−0.971657	−0.485829	+	0.874054i	$0.661482\pi$
$252$	0	0
$253$	−5.35375	−0.336587
$254$	−1.81177	−0.113680
$255$	0	0
$256$	−14.4410	−0.902560
$257$	−7.92004	−0.494038	−0.247019	−	0.969011i	$-0.579451\pi$
$257$	−7.92004	−0.494038	−0.247019	+	0.969011i	$0.579451\pi$
$258$	0	0
$259$	−40.6230	−2.52419
$260$	0	0
$261$	0	0
$262$	−8.65776	−0.534878
$263$	−17.7857	−1.09671	−0.548355	−	0.836246i	$-0.684746\pi$
$263$	−17.7857	−1.09671	−0.548355	+	0.836246i	$0.684746\pi$
$264$	0	0
$265$	0	0
$266$	6.46883	0.396629
$267$	0	0
$268$	−8.07803	−0.493444
$269$	0.155319	0.00946996	0.00473498	−	0.999989i	$-0.498493\pi$
$269$	0.155319	0.00946996	0.00473498	+	0.999989i	$0.498493\pi$
$270$	0	0
$271$	26.1967	1.59134	0.795668	−	0.605733i	$-0.207120\pi$
$271$	26.1967	1.59134	0.795668	+	0.605733i	$0.207120\pi$
$272$	−0.577359	−0.0350076
$273$	0	0
$274$	0.264709	0.0159916
$275$	0	0
$276$	0	0
$277$	17.6542	1.06074	0.530369	−	0.847767i	$-0.322053\pi$
$277$	17.6542	1.06074	0.530369	+	0.847767i	$0.322053\pi$
$278$	−7.73862	−0.464132
$279$	0	0
$280$	0	0
$281$	18.7534	1.11873	0.559366	−	0.828921i	$-0.311045\pi$
$281$	18.7534	1.11873	0.559366	+	0.828921i	$0.311045\pi$
$282$	0	0
$283$	−16.3826	−0.973847	−0.486923	−	0.873445i	$-0.661881\pi$
$283$	−16.3826	−0.973847	−0.486923	+	0.873445i	$0.661881\pi$
$284$	15.3116	0.908577
$285$	0	0
$286$	23.9492	1.41615
$287$	−33.5028	−1.97761
$288$	0	0
$289$	−15.6156	−0.918566
$290$	0	0
$291$	0	0
$292$	−2.04146	−0.119468
$293$	16.5506	0.966897	0.483448	−	0.875373i	$-0.339384\pi$
$293$	16.5506	0.966897	0.483448	+	0.875373i	$0.339384\pi$
$294$	0	0
$295$	0	0
$296$	−23.8950	−1.38887
$297$	0	0
$298$	−18.2642	−1.05801
$299$	5.51982	0.319220
$300$	0	0
$301$	−39.2463	−2.26212
$302$	−13.2345	−0.761560
$303$	0	0
$304$	−0.850347	−0.0487707
$305$	0	0
$306$	0	0
$307$	−17.3084	−0.987844	−0.493922	−	0.869506i	$-0.664437\pi$
$307$	−17.3084	−0.987844	−0.493922	+	0.869506i	$0.664437\pi$
$308$	−33.1243	−1.88743
$309$	0	0
$310$	0	0
$311$	−16.8210	−0.953834	−0.476917	−	0.878948i	$-0.658246\pi$
$311$	−16.8210	−0.953834	−0.476917	+	0.878948i	$0.658246\pi$
$312$	0	0
$313$	8.44566	0.477377	0.238688	−	0.971096i	$-0.423283\pi$
$313$	8.44566	0.477377	0.238688	+	0.971096i	$0.423283\pi$
$314$	−10.7515	−0.606743
$315$	0	0
$316$	−1.70273	−0.0957858
$317$	−32.7283	−1.83821	−0.919104	−	0.394016i	$-0.871085\pi$
$317$	−32.7283	−1.83821	−0.919104	+	0.394016i	$0.871085\pi$
$318$	0	0
$319$	−16.5235	−0.925140
$320$	0	0
$321$	0	0
$322$	3.73292	0.208027
$323$	2.03894	0.113450
$324$	0	0
$325$	0	0
$326$	−12.7447	−0.705863
$327$	0	0
$328$	−19.7068	−1.08812
$329$	−44.6454	−2.46138
$330$	0	0
$331$	−15.2008	−0.835512	−0.417756	−	0.908559i	$-0.637183\pi$
$331$	−15.2008	−0.835512	−0.417756	+	0.908559i	$0.637183\pi$
$332$	1.55216	0.0851859
$333$	0	0
$334$	1.29085	0.0706319
$335$	0	0
$336$	0	0
$337$	−11.9751	−0.652327	−0.326164	−	0.945313i	$-0.605756\pi$
$337$	−11.9751	−0.652327	−0.326164	+	0.945313i	$0.605756\pi$
$338$	−14.1567	−0.770023
$339$	0	0
$340$	0	0
$341$	23.2525	1.25920
$342$	0	0
$343$	33.2418	1.79489
$344$	−23.0852	−1.24467
$345$	0	0
$346$	−5.59597	−0.300841
$347$	15.6679	0.841095	0.420547	−	0.907271i	$-0.361838\pi$
$347$	15.6679	0.841095	0.420547	+	0.907271i	$0.361838\pi$
$348$	0	0
$349$	24.0603	1.28792	0.643958	−	0.765061i	$-0.277291\pi$
$349$	24.0603	1.28792	0.643958	+	0.765061i	$0.277291\pi$
$350$	0	0
$351$	0	0
$352$	−31.1400	−1.65977
$353$	22.5772	1.20166	0.600830	−	0.799376i	$-0.294837\pi$
$353$	22.5772	1.20166	0.600830	+	0.799376i	$0.294837\pi$
$354$	0	0
$355$	0	0
$356$	7.25641	0.384589
$357$	0	0
$358$	20.3832	1.07729
$359$	−24.3132	−1.28320	−0.641602	−	0.767038i	$-0.721730\pi$
$359$	−24.3132	−1.28320	−0.641602	+	0.767038i	$0.721730\pi$
$360$	0	0
$361$	−15.9970	−0.841947
$362$	10.5093	0.552356
$363$	0	0
$364$	34.1518	1.79004
$365$	0	0
$366$	0	0
$367$	−16.0482	−0.837708	−0.418854	−	0.908053i	$-0.637568\pi$
$367$	−16.0482	−0.837708	−0.418854	+	0.908053i	$0.637568\pi$
$368$	−0.490703	−0.0255797
$369$	0	0
$370$	0	0
$371$	−46.9913	−2.43967
$372$	0	0
$373$	−14.8470	−0.768747	−0.384373	−	0.923178i	$-0.625582\pi$
$373$	−14.8470	−0.768747	−0.384373	+	0.923178i	$0.625582\pi$
$374$	5.10498	0.263972
$375$	0	0
$376$	−26.2610	−1.35431
$377$	17.0361	0.877403
$378$	0	0
$379$	21.8303	1.12135	0.560673	−	0.828038i	$-0.310543\pi$
$379$	21.8303	1.12135	0.560673	+	0.828038i	$0.310543\pi$
$380$	0	0
$381$	0	0
$382$	−3.16789	−0.162083
$383$	11.4590	0.585528	0.292764	−	0.956185i	$-0.405425\pi$
$383$	11.4590	0.585528	0.292764	+	0.956185i	$0.405425\pi$
$384$	0	0
$385$	0	0
$386$	−2.17714	−0.110814
$387$	0	0
$388$	22.0880	1.12135
$389$	30.7060	1.55686	0.778428	−	0.627734i	$-0.216018\pi$
$389$	30.7060	1.55686	0.778428	+	0.627734i	$0.216018\pi$
$390$	0	0
$391$	1.17660	0.0595031
$392$	38.5191	1.94551
$393$	0	0
$394$	7.92700	0.399357
$395$	0	0
$396$	0	0
$397$	36.3445	1.82408	0.912039	−	0.410102i	$-0.134507\pi$
$397$	36.3445	1.82408	0.912039	+	0.410102i	$0.134507\pi$
$398$	15.4682	0.775348
$399$	0	0
$400$	0	0
$401$	−7.23228	−0.361163	−0.180581	−	0.983560i	$-0.557798\pi$
$401$	−7.23228	−0.361163	−0.180581	+	0.983560i	$0.557798\pi$
$402$	0	0
$403$	−23.9738	−1.19422
$404$	4.61267	0.229489
$405$	0	0
$406$	11.5211	0.571781
$407$	−47.2161	−2.34042
$408$	0	0
$409$	−13.2289	−0.654129	−0.327064	−	0.945002i	$-0.606059\pi$
$409$	−13.2289	−0.654129	−0.327064	+	0.945002i	$0.606059\pi$
$410$	0	0
$411$	0	0
$412$	12.6650	0.623959
$413$	−54.1079	−2.66247
$414$	0	0
$415$	0	0
$416$	32.1060	1.57412
$417$	0	0
$418$	7.51872	0.367753
$419$	−18.3115	−0.894574	−0.447287	−	0.894390i	$-0.647610\pi$
$419$	−18.3115	−0.894574	−0.447287	+	0.894390i	$0.647610\pi$
$420$	0	0
$421$	14.1996	0.692049	0.346024	−	0.938226i	$-0.387531\pi$
$421$	14.1996	0.692049	0.346024	+	0.938226i	$0.387531\pi$
$422$	6.56052	0.319361
$423$	0	0
$424$	−27.6408	−1.34236
$425$	0	0
$426$	0	0
$427$	6.13864	0.297070
$428$	−20.0266	−0.968020
$429$	0	0
$430$	0	0
$431$	−35.0475	−1.68818	−0.844090	−	0.536202i	$-0.819859\pi$
$431$	−35.0475	−1.68818	−0.844090	+	0.536202i	$0.819859\pi$
$432$	0	0
$433$	−3.14590	−0.151182	−0.0755911	−	0.997139i	$-0.524084\pi$
$433$	−3.14590	−0.151182	−0.0755911	+	0.997139i	$0.524084\pi$
$434$	−16.2129	−0.778244
$435$	0	0
$436$	−11.0979	−0.531495
$437$	1.73292	0.0828966
$438$	0	0
$439$	1.95513	0.0933135	0.0466567	−	0.998911i	$-0.485143\pi$
$439$	1.95513	0.0933135	0.0466567	+	0.998911i	$0.485143\pi$
$440$	0	0
$441$	0	0
$442$	−5.26334	−0.250351
$443$	27.4890	1.30604	0.653020	−	0.757340i	$-0.273502\pi$
$443$	27.4890	1.30604	0.653020	+	0.757340i	$0.273502\pi$
$444$	0	0
$445$	0	0
$446$	11.5156	0.545277
$447$	0	0
$448$	17.1919	0.812242
$449$	2.83238	0.133668	0.0668341	−	0.997764i	$-0.478710\pi$
$449$	2.83238	0.133668	0.0668341	+	0.997764i	$0.478710\pi$
$450$	0	0
$451$	−38.9403	−1.83363
$452$	14.2758	0.671477
$453$	0	0
$454$	13.4737	0.632350
$455$	0	0
$456$	0	0
$457$	2.97045	0.138952	0.0694760	−	0.997584i	$-0.477867\pi$
$457$	2.97045	0.138952	0.0694760	+	0.997584i	$0.477867\pi$
$458$	−12.4512	−0.581808
$459$	0	0
$460$	0	0
$461$	16.1855	0.753832	0.376916	−	0.926247i	$-0.376984\pi$
$461$	16.1855	0.753832	0.376916	+	0.926247i	$0.376984\pi$
$462$	0	0
$463$	2.40718	0.111871	0.0559356	−	0.998434i	$-0.482186\pi$
$463$	2.40718	0.111871	0.0559356	+	0.998434i	$0.482186\pi$
$464$	−1.51448	−0.0703080
$465$	0	0
$466$	12.2274	0.566423
$467$	−10.0639	−0.465700	−0.232850	−	0.972513i	$-0.574805\pi$
$467$	−10.0639	−0.465700	−0.232850	+	0.972513i	$0.574805\pi$
$468$	0	0
$469$	27.7011	1.27912
$470$	0	0
$471$	0	0
$472$	−31.8269	−1.46495
$473$	−45.6160	−2.09742
$474$	0	0
$475$	0	0
$476$	7.27975	0.333667
$477$	0	0
$478$	−1.15766	−0.0529499
$479$	−26.0812	−1.19168	−0.595841	−	0.803102i	$-0.703181\pi$
$479$	−26.0812	−1.19168	−0.595841	+	0.803102i	$0.703181\pi$
$480$	0	0
$481$	48.6808	2.21965
$482$	−8.58670	−0.391114
$483$	0	0
$484$	−23.7249	−1.07840
$485$	0	0
$486$	0	0
$487$	−23.7436	−1.07592	−0.537962	−	0.842969i	$-0.680806\pi$
$487$	−23.7436	−1.07592	−0.537962	+	0.842969i	$0.680806\pi$
$488$	3.61082	0.163454
$489$	0	0
$490$	0	0
$491$	−2.79506	−0.126139	−0.0630696	−	0.998009i	$-0.520089\pi$
$491$	−2.79506	−0.126139	−0.0630696	+	0.998009i	$0.520089\pi$
$492$	0	0
$493$	3.63139	0.163549
$494$	−7.75195	−0.348777
$495$	0	0
$496$	2.13123	0.0956952
$497$	−52.5064	−2.35523
$498$	0	0
$499$	−8.35214	−0.373893	−0.186947	−	0.982370i	$-0.559859\pi$
$499$	−8.35214	−0.373893	−0.186947	+	0.982370i	$0.559859\pi$
$500$	0	0
$501$	0	0
$502$	12.4755	0.556809
$503$	19.8111	0.883333	0.441667	−	0.897179i	$-0.354387\pi$
$503$	19.8111	0.883333	0.441667	+	0.897179i	$0.354387\pi$
$504$	0	0
$505$	0	0
$506$	4.33877	0.192882
$507$	0	0
$508$	−3.00291	−0.133233
$509$	−29.1471	−1.29192	−0.645961	−	0.763371i	$-0.723543\pi$
$509$	−29.1471	−1.29192	−0.645961	+	0.763371i	$0.723543\pi$
$510$	0	0
$511$	7.00056	0.309686
$512$	−5.51319	−0.243651
$513$	0	0
$514$	6.41853	0.283109
$515$	0	0
$516$	0	0
$517$	−51.8913	−2.28218
$518$	32.9216	1.44649
$519$	0	0
$520$	0	0
$521$	−3.73563	−0.163661	−0.0818306	−	0.996646i	$-0.526077\pi$
$521$	−3.73563	−0.163661	−0.0818306	+	0.996646i	$0.526077\pi$
$522$	0	0
$523$	28.4551	1.24425	0.622127	−	0.782916i	$-0.286269\pi$
$523$	28.4551	1.24425	0.622127	+	0.782916i	$0.286269\pi$
$524$	−14.3498	−0.626874
$525$	0	0
$526$	14.4138	0.628471
$527$	−5.11022	−0.222605
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	10.7218	0.464847
$533$	40.1483	1.73901
$534$	0	0
$535$	0	0
$536$	16.2941	0.703798
$537$	0	0
$538$	−0.125873	−0.00542677
$539$	76.1132	3.27843
$540$	0	0
$541$	37.3591	1.60619	0.803097	−	0.595849i	$-0.203184\pi$
$541$	37.3591	1.60619	0.803097	+	0.595849i	$0.203184\pi$
$542$	−21.2302	−0.911917
$543$	0	0
$544$	6.84366	0.293419
$545$	0	0
$546$	0	0
$547$	38.4683	1.64479	0.822394	−	0.568919i	$-0.192638\pi$
$547$	38.4683	1.64479	0.822394	+	0.568919i	$0.192638\pi$
$548$	0.438742	0.0187421
$549$	0	0
$550$	0	0
$551$	5.34838	0.227849
$552$	0	0
$553$	5.83896	0.248298
$554$	−14.3073	−0.607857
$555$	0	0
$556$	−12.8264	−0.543959
$557$	29.8981	1.26682	0.633412	−	0.773815i	$-0.281654\pi$
$557$	29.8981	1.26682	0.633412	+	0.773815i	$0.281654\pi$
$558$	0	0
$559$	47.0310	1.98920
$560$	0	0
$561$	0	0
$562$	−15.1980	−0.641090
$563$	36.7777	1.54999	0.774997	−	0.631965i	$-0.217751\pi$
$563$	36.7777	1.54999	0.774997	+	0.631965i	$0.217751\pi$
$564$	0	0
$565$	0	0
$566$	13.2768	0.558064
$567$	0	0
$568$	−30.8849	−1.29590
$569$	−19.0192	−0.797328	−0.398664	−	0.917097i	$-0.630526\pi$
$569$	−19.0192	−0.797328	−0.398664	+	0.917097i	$0.630526\pi$
$570$	0	0
$571$	41.8743	1.75238	0.876192	−	0.481962i	$-0.160076\pi$
$571$	41.8743	1.75238	0.876192	+	0.481962i	$0.160076\pi$
$572$	39.6946	1.65972
$573$	0	0
$574$	27.1513	1.13327
$575$	0	0
$576$	0	0
$577$	−30.6755	−1.27704	−0.638520	−	0.769605i	$-0.720453\pi$
$577$	−30.6755	−1.27704	−0.638520	+	0.769605i	$0.720453\pi$
$578$	12.6552	0.526385
$579$	0	0
$580$	0	0
$581$	−5.32265	−0.220821
$582$	0	0
$583$	−54.6180	−2.26204
$584$	4.11781	0.170396
$585$	0	0
$586$	−13.4129	−0.554081
$587$	1.52558	0.0629675	0.0314837	−	0.999504i	$-0.489977\pi$
$587$	1.52558	0.0629675	0.0314837	+	0.999504i	$0.489977\pi$
$588$	0	0
$589$	−7.52644	−0.310122
$590$	0	0
$591$	0	0
$592$	−4.32764	−0.177865
$593$	5.12072	0.210283	0.105141	−	0.994457i	$-0.466471\pi$
$593$	5.12072	0.210283	0.105141	+	0.994457i	$0.466471\pi$
$594$	0	0
$595$	0	0
$596$	−30.2719	−1.23999
$597$	0	0
$598$	−4.47336	−0.182929
$599$	18.5010	0.755931	0.377966	−	0.925820i	$-0.376624\pi$
$599$	18.5010	0.755931	0.377966	+	0.925820i	$0.376624\pi$
$600$	0	0
$601$	−30.5128	−1.24464	−0.622322	−	0.782761i	$-0.713811\pi$
$601$	−30.5128	−1.24464	−0.622322	+	0.782761i	$0.713811\pi$
$602$	31.8059	1.29631
$603$	0	0
$604$	−21.9355	−0.892544
$605$	0	0
$606$	0	0
$607$	20.6414	0.837808	0.418904	−	0.908031i	$-0.362414\pi$
$607$	20.6414	0.837808	0.418904	+	0.908031i	$0.362414\pi$
$608$	10.0795	0.408777
$609$	0	0
$610$	0	0
$611$	53.5010	2.16442
$612$	0	0
$613$	11.4793	0.463643	0.231822	−	0.972758i	$-0.425531\pi$
$613$	11.4793	0.463643	0.231822	+	0.972758i	$0.425531\pi$
$614$	14.0270	0.566085
$615$	0	0
$616$	66.8147	2.69204
$617$	19.8055	0.797340	0.398670	−	0.917095i	$-0.369472\pi$
$617$	19.8055	0.797340	0.398670	+	0.917095i	$0.369472\pi$
$618$	0	0
$619$	−24.4491	−0.982692	−0.491346	−	0.870965i	$-0.663495\pi$
$619$	−24.4491	−0.982692	−0.491346	+	0.870965i	$0.663495\pi$
$620$	0	0
$621$	0	0
$622$	13.6320	0.546595
$623$	−24.8836	−0.996939
$624$	0	0
$625$	0	0
$626$	−6.84450	−0.273561
$627$	0	0
$628$	−17.8201	−0.711099
$629$	10.3767	0.413747
$630$	0	0
$631$	46.5829	1.85444	0.927218	−	0.374522i	$-0.122193\pi$
$631$	46.5829	1.85444	0.927218	+	0.374522i	$0.122193\pi$
$632$	3.43455	0.136619
$633$	0	0
$634$	26.5236	1.05339
$635$	0	0
$636$	0	0
$637$	−78.4742	−3.10926
$638$	13.3909	0.530153
$639$	0	0
$640$	0	0
$641$	18.8170	0.743226	0.371613	−	0.928388i	$-0.378805\pi$
$641$	18.8170	0.743226	0.371613	+	0.928388i	$0.378805\pi$
$642$	0	0
$643$	17.1868	0.677783	0.338891	−	0.940826i	$-0.389948\pi$
$643$	17.1868	0.677783	0.338891	+	0.940826i	$0.389948\pi$
$644$	6.18712	0.243807
$645$	0	0
$646$	−1.65239	−0.0650126
$647$	16.6153	0.653214	0.326607	−	0.945160i	$-0.394095\pi$
$647$	16.6153	0.653214	0.326607	+	0.945160i	$0.394095\pi$
$648$	0	0
$649$	−62.8896	−2.46863
$650$	0	0
$651$	0	0
$652$	−21.1237	−0.827267
$653$	8.71155	0.340909	0.170455	−	0.985366i	$-0.445476\pi$
$653$	8.71155	0.340909	0.170455	+	0.985366i	$0.445476\pi$
$654$	0	0
$655$	0	0
$656$	−3.56911	−0.139350
$657$	0	0
$658$	36.1814	1.41050
$659$	24.1116	0.939256	0.469628	−	0.882865i	$-0.344388\pi$
$659$	24.1116	0.939256	0.469628	+	0.882865i	$0.344388\pi$
$660$	0	0
$661$	39.6144	1.54082	0.770411	−	0.637548i	$-0.220051\pi$
$661$	39.6144	1.54082	0.770411	+	0.637548i	$0.220051\pi$
$662$	12.3190	0.478791
$663$	0	0
$664$	−3.13085	−0.121500
$665$	0	0
$666$	0	0
$667$	3.08635	0.119504
$668$	2.13951	0.0827802
$669$	0	0
$670$	0	0
$671$	7.13494	0.275441
$672$	0	0
$673$	−20.0269	−0.771980	−0.385990	−	0.922503i	$-0.626140\pi$
$673$	−20.0269	−0.771980	−0.385990	+	0.922503i	$0.626140\pi$
$674$	9.70485	0.373817
$675$	0	0
$676$	−23.4640	−0.902463
$677$	51.3624	1.97402	0.987010	−	0.160662i	$-0.0513629\pi$
$677$	51.3624	1.97402	0.987010	+	0.160662i	$0.0513629\pi$
$678$	0	0
$679$	−75.7440	−2.90679
$680$	0	0
$681$	0	0
$682$	−18.8442	−0.721583
$683$	1.78656	0.0683609	0.0341804	−	0.999416i	$-0.489118\pi$
$683$	1.78656	0.0683609	0.0341804	+	0.999416i	$0.489118\pi$
$684$	0	0
$685$	0	0
$686$	−26.9397	−1.02856
$687$	0	0
$688$	−4.18097	−0.159398
$689$	56.3122	2.14532
$690$	0	0
$691$	−51.7197	−1.96751	−0.983755	−	0.179515i	$-0.942547\pi$
$691$	−51.7197	−1.96751	−0.983755	+	0.179515i	$0.942547\pi$
$692$	−9.27504	−0.352584
$693$	0	0
$694$	−12.6975	−0.481990
$695$	0	0
$696$	0	0
$697$	8.55794	0.324155
$698$	−19.4988	−0.738042
$699$	0	0
$700$	0	0
$701$	33.4929	1.26501	0.632504	−	0.774557i	$-0.282027\pi$
$701$	33.4929	1.26501	0.632504	+	0.774557i	$0.282027\pi$
$702$	0	0
$703$	15.2830	0.576411
$704$	19.9822	0.753106
$705$	0	0
$706$	−18.2969	−0.688613
$707$	−15.8177	−0.594887
$708$	0	0
$709$	−15.8720	−0.596085	−0.298043	−	0.954553i	$-0.596334\pi$
$709$	−15.8720	−0.596085	−0.298043	+	0.954553i	$0.596334\pi$
$710$	0	0
$711$	0	0
$712$	−14.6368	−0.548538
$713$	−4.34322	−0.162655
$714$	0	0
$715$	0	0
$716$	33.7842	1.26257
$717$	0	0
$718$	19.7038	0.735341
$719$	−15.3345	−0.571881	−0.285941	−	0.958247i	$-0.592306\pi$
$719$	−15.3345	−0.571881	−0.285941	+	0.958247i	$0.592306\pi$
$720$	0	0
$721$	−43.4306	−1.61744
$722$	12.9642	0.482479
$723$	0	0
$724$	17.4186	0.647358
$725$	0	0
$726$	0	0
$727$	−27.1804	−1.00806	−0.504032	−	0.863685i	$-0.668151\pi$
$727$	−27.1804	−1.00806	−0.504032	+	0.863685i	$0.668151\pi$
$728$	−68.8873	−2.55313
$729$	0	0
$730$	0	0
$731$	10.0250	0.370790
$732$	0	0
$733$	4.70308	0.173712	0.0868560	−	0.996221i	$-0.472318\pi$
$733$	4.70308	0.173712	0.0868560	+	0.996221i	$0.472318\pi$
$734$	13.0057	0.480050
$735$	0	0
$736$	5.81648	0.214398
$737$	32.1970	1.18599
$738$	0	0
$739$	1.19298	0.0438845	0.0219422	−	0.999759i	$-0.493015\pi$
$739$	1.19298	0.0438845	0.0219422	+	0.999759i	$0.493015\pi$
$740$	0	0
$741$	0	0
$742$	38.0825	1.39805
$743$	−10.0273	−0.367867	−0.183934	−	0.982939i	$-0.558883\pi$
$743$	−10.0273	−0.367867	−0.183934	+	0.982939i	$0.558883\pi$
$744$	0	0
$745$	0	0
$746$	12.0322	0.440531
$747$	0	0
$748$	8.46125	0.309374
$749$	68.6748	2.50932
$750$	0	0
$751$	−19.2248	−0.701524	−0.350762	−	0.936465i	$-0.614077\pi$
$751$	−19.2248	−0.701524	−0.350762	+	0.936465i	$0.614077\pi$
$752$	−4.75615	−0.173439
$753$	0	0
$754$	−13.8063	−0.502797
$755$	0	0
$756$	0	0
$757$	29.0674	1.05647	0.528236	−	0.849098i	$-0.322854\pi$
$757$	29.0674	1.05647	0.528236	+	0.849098i	$0.322854\pi$
$758$	−17.6916	−0.642588
$759$	0	0
$760$	0	0
$761$	−34.9604	−1.26731	−0.633657	−	0.773614i	$-0.718447\pi$
$761$	−34.9604	−1.26731	−0.633657	+	0.773614i	$0.718447\pi$
$762$	0	0
$763$	38.0569	1.37775
$764$	−5.25062	−0.189961
$765$	0	0
$766$	−9.28658	−0.335538
$767$	64.8404	2.34125
$768$	0	0
$769$	8.00084	0.288518	0.144259	−	0.989540i	$-0.453920\pi$
$769$	8.00084	0.288518	0.144259	+	0.989540i	$0.453920\pi$
$770$	0	0
$771$	0	0
$772$	−3.60851	−0.129873
$773$	25.3463	0.911644	0.455822	−	0.890071i	$-0.349345\pi$
$773$	25.3463	0.911644	0.455822	+	0.890071i	$0.349345\pi$
$774$	0	0
$775$	0	0
$776$	−44.5535	−1.59938
$777$	0	0
$778$	−24.8846	−0.892158
$779$	12.6043	0.451596
$780$	0	0
$781$	−61.0282	−2.18376
$782$	−0.953534	−0.0340983
$783$	0	0
$784$	6.97622	0.249151
$785$	0	0
$786$	0	0
$787$	5.10486	0.181969	0.0909843	−	0.995852i	$-0.470999\pi$
$787$	5.10486	0.181969	0.0909843	+	0.995852i	$0.470999\pi$
$788$	13.1386	0.468044
$789$	0	0
$790$	0	0
$791$	−48.9544	−1.74062
$792$	0	0
$793$	−7.35627	−0.261229
$794$	−29.4542	−1.04529
$795$	0	0
$796$	25.6377	0.908704
$797$	−44.0182	−1.55921	−0.779603	−	0.626274i	$-0.784579\pi$
$797$	−44.0182	−1.55921	−0.779603	+	0.626274i	$0.784579\pi$
$798$	0	0
$799$	11.4042	0.403451
$800$	0	0
$801$	0	0
$802$	5.86116	0.206965
$803$	8.13674	0.287139
$804$	0	0
$805$	0	0
$806$	19.4288	0.684350
$807$	0	0
$808$	−9.30418	−0.327320
$809$	6.15978	0.216566	0.108283	−	0.994120i	$-0.465465\pi$
$809$	6.15978	0.216566	0.108283	+	0.994120i	$0.465465\pi$
$810$	0	0
$811$	−2.23801	−0.0785873	−0.0392936	−	0.999228i	$-0.512511\pi$
$811$	−2.23801	−0.0785873	−0.0392936	+	0.999228i	$0.512511\pi$
$812$	19.0956	0.670124
$813$	0	0
$814$	38.2647	1.34118
$815$	0	0
$816$	0	0
$817$	14.7651	0.516565
$818$	10.7209	0.374849
$819$	0	0
$820$	0	0
$821$	−39.9914	−1.39571	−0.697855	−	0.716239i	$-0.745862\pi$
$821$	−39.9914	−1.39571	−0.697855	+	0.716239i	$0.745862\pi$
$822$	0	0
$823$	−23.3116	−0.812590	−0.406295	−	0.913742i	$-0.633179\pi$
$823$	−23.3116	−0.812590	−0.406295	+	0.913742i	$0.633179\pi$
$824$	−25.5464	−0.889951
$825$	0	0
$826$	43.8499	1.52573
$827$	−43.4988	−1.51260	−0.756301	−	0.654224i	$-0.772995\pi$
$827$	−43.4988	−1.51260	−0.756301	+	0.654224i	$0.772995\pi$
$828$	0	0
$829$	33.1090	1.14992	0.574961	−	0.818181i	$-0.305017\pi$
$829$	33.1090	1.14992	0.574961	+	0.818181i	$0.305017\pi$
$830$	0	0
$831$	0	0
$832$	−20.6020	−0.714246
$833$	−16.7274	−0.579571
$834$	0	0
$835$	0	0
$836$	12.4619	0.431004
$837$	0	0
$838$	14.8399	0.512637
$839$	52.5227	1.81329	0.906643	−	0.421899i	$-0.138636\pi$
$839$	52.5227	1.81329	0.906643	+	0.421899i	$0.138636\pi$
$840$	0	0
$841$	−19.4745	−0.671533
$842$	−11.5076	−0.396579
$843$	0	0
$844$	10.8737	0.374289
$845$	0	0
$846$	0	0
$847$	81.3571	2.79546
$848$	−5.00606	−0.171909
$849$	0	0
$850$	0	0
$851$	8.81926	0.302320
$852$	0	0
$853$	−22.9673	−0.786385	−0.393192	−	0.919456i	$-0.628629\pi$
$853$	−22.9673	−0.786385	−0.393192	+	0.919456i	$0.628629\pi$
$854$	−4.97486	−0.170236
$855$	0	0
$856$	40.3954	1.38069
$857$	−55.1228	−1.88296	−0.941480	−	0.337069i	$-0.890564\pi$
$857$	−55.1228	−1.88296	−0.941480	+	0.337069i	$0.890564\pi$
$858$	0	0
$859$	−49.9526	−1.70436	−0.852180	−	0.523249i	$-0.824720\pi$
$859$	−49.9526	−1.70436	−0.852180	+	0.523249i	$0.824720\pi$
$860$	0	0
$861$	0	0
$862$	28.4031	0.967413
$863$	12.0802	0.411214	0.205607	−	0.978635i	$-0.434083\pi$
$863$	12.0802	0.411214	0.205607	+	0.978635i	$0.434083\pi$
$864$	0	0
$865$	0	0
$866$	2.54949	0.0866351
$867$	0	0
$868$	−26.8721	−0.912097
$869$	6.78663	0.230221
$870$	0	0
$871$	−33.1957	−1.12479
$872$	22.3855	0.758070
$873$	0	0
$874$	−1.40438	−0.0475040
$875$	0	0
$876$	0	0
$877$	−45.5573	−1.53836	−0.769180	−	0.639033i	$-0.779335\pi$
$877$	−45.5573	−1.53836	−0.769180	+	0.639033i	$0.779335\pi$
$878$	−1.58447	−0.0534734
$879$	0	0
$880$	0	0
$881$	6.57232	0.221427	0.110714	−	0.993852i	$-0.464686\pi$
$881$	6.57232	0.221427	0.110714	+	0.993852i	$0.464686\pi$
$882$	0	0
$883$	−5.51200	−0.185494	−0.0927468	−	0.995690i	$-0.529565\pi$
$883$	−5.51200	−0.185494	−0.0927468	+	0.995690i	$0.529565\pi$
$884$	−8.72371	−0.293410
$885$	0	0
$886$	−22.2775	−0.748428
$887$	−31.2681	−1.04988	−0.524941	−	0.851139i	$-0.675913\pi$
$887$	−31.2681	−1.04988	−0.524941	+	0.851139i	$0.675913\pi$
$888$	0	0
$889$	10.2975	0.345369
$890$	0	0
$891$	0	0
$892$	19.0865	0.639062
$893$	16.7963	0.562067
$894$	0	0
$895$	0	0
$896$	39.6508	1.32464
$897$	0	0
$898$	−2.29541	−0.0765987
$899$	−13.4047	−0.447072
$900$	0	0
$901$	12.0034	0.399892
$902$	31.5579	1.05076
$903$	0	0
$904$	−28.7956	−0.957726
$905$	0	0
$906$	0	0
$907$	12.7989	0.424982	0.212491	−	0.977163i	$-0.431842\pi$
$907$	12.7989	0.424982	0.212491	+	0.977163i	$0.431842\pi$
$908$	22.3319	0.741110
$909$	0	0
$910$	0	0
$911$	31.6963	1.05015	0.525073	−	0.851057i	$-0.324038\pi$
$911$	31.6963	1.05015	0.525073	+	0.851057i	$0.324038\pi$
$912$	0	0
$913$	−6.18651	−0.204744
$914$	−2.40730	−0.0796265
$915$	0	0
$916$	−20.6373	−0.681875
$917$	49.2081	1.62500
$918$	0	0
$919$	7.63249	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$919$	7.63249	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$920$	0	0
$921$	0	0
$922$	−13.1170	−0.431984
$923$	62.9212	2.07108
$924$	0	0
$925$	0	0
$926$	−1.95082	−0.0641080
$927$	0	0
$928$	17.9517	0.589293
$929$	−17.2979	−0.567525	−0.283763	−	0.958895i	$-0.591583\pi$
$929$	−17.2979	−0.567525	−0.283763	+	0.958895i	$0.591583\pi$
$930$	0	0
$931$	−24.6365	−0.807429
$932$	20.2663	0.663844
$933$	0	0
$934$	8.15593	0.266870
$935$	0	0
$936$	0	0
$937$	−12.5967	−0.411517	−0.205759	−	0.978603i	$-0.565966\pi$
$937$	−12.5967	−0.411517	−0.205759	+	0.978603i	$0.565966\pi$
$938$	−22.4494	−0.732999
$939$	0	0
$940$	0	0
$941$	19.7213	0.642895	0.321448	−	0.946927i	$-0.395831\pi$
$941$	19.7213	0.642895	0.321448	+	0.946927i	$0.395831\pi$
$942$	0	0
$943$	7.27347	0.236857
$944$	−5.76420	−0.187609
$945$	0	0
$946$	36.9679	1.20193
$947$	−24.8824	−0.808569	−0.404285	−	0.914633i	$-0.632479\pi$
$947$	−24.8824	−0.808569	−0.404285	+	0.914633i	$0.632479\pi$
$948$	0	0
$949$	−8.38914	−0.272323
$950$	0	0
$951$	0	0
$952$	−14.6839	−0.475908
$953$	−24.9095	−0.806899	−0.403449	−	0.915002i	$-0.632189\pi$
$953$	−24.9095	−0.806899	−0.403449	+	0.915002i	$0.632189\pi$
$954$	0	0
$955$	0	0
$956$	−1.91876	−0.0620570
$957$	0	0
$958$	21.1367	0.682895
$959$	−1.50453	−0.0485837
$960$	0	0
$961$	−12.1364	−0.391497
$962$	−39.4517	−1.27197
$963$	0	0
$964$	−14.2320	−0.458383
$965$	0	0
$966$	0	0
$967$	29.4706	0.947709	0.473855	−	0.880603i	$-0.342862\pi$
$967$	29.4706	0.947709	0.473855	+	0.880603i	$0.342862\pi$
$968$	47.8552	1.53813
$969$	0	0
$970$	0	0
$971$	−36.7868	−1.18054	−0.590272	−	0.807204i	$-0.700980\pi$
$971$	−36.7868	−1.18054	−0.590272	+	0.807204i	$0.700980\pi$
$972$	0	0
$973$	43.9840	1.41006
$974$	19.2422	0.616560
$975$	0	0
$976$	0.653960	0.0209327
$977$	14.6435	0.468488	0.234244	−	0.972178i	$-0.424738\pi$
$977$	14.6435	0.468488	0.234244	+	0.972178i	$0.424738\pi$
$978$	0	0
$979$	−28.9222	−0.924357
$980$	0	0
$981$	0	0
$982$	2.26516	0.0722843
$983$	1.10622	0.0352831	0.0176415	−	0.999844i	$-0.494384\pi$
$983$	1.10622	0.0352831	0.0176415	+	0.999844i	$0.494384\pi$
$984$	0	0
$985$	0	0
$986$	−2.94294	−0.0937222
$987$	0	0
$988$	−12.8485	−0.408764
$989$	8.52038	0.270932
$990$	0	0
$991$	−42.7058	−1.35659	−0.678297	−	0.734788i	$-0.737282\pi$
$991$	−42.7058	−1.35659	−0.678297	+	0.734788i	$0.737282\pi$
$992$	−25.2623	−0.802079
$993$	0	0
$994$	42.5520	1.34967
$995$	0	0
$996$	0	0
$997$	−30.0771	−0.952553	−0.476276	−	0.879296i	$-0.658014\pi$
$997$	−30.0771	−0.952553	−0.476276	+	0.879296i	$0.658014\pi$
$998$	6.76871	0.214260
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bz.1.3		6
3.2	odd	2		5175.2.a.by.1.4		6
5.4	even	2		1035.2.a.p.1.4	✓	6
15.14	odd	2		1035.2.a.q.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.2.a.p.1.4	✓	6	5.4	even	2
1035.2.a.q.1.3	yes	6	15.14	odd	2
5175.2.a.by.1.4		6	3.2	odd	2
5175.2.a.bz.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q-0.810417 q^{2} -1.34322 q^{4} +4.60617 q^{7} +2.70941 q^{8} +5.35375 q^{11} -5.51982 q^{13} -3.73292 q^{14} +0.490703 q^{16} -1.17660 q^{17} -1.73292 q^{19} -4.33877 q^{22} -1.00000 q^{23} +4.47336 q^{26} -6.18712 q^{28} -3.08635 q^{29} +4.34322 q^{31} -5.81648 q^{32} +0.953534 q^{34} -8.81926 q^{37} +1.40438 q^{38} -7.27347 q^{41} -8.52038 q^{43} -7.19129 q^{44} +0.810417 q^{46} -9.69252 q^{47} +14.2168 q^{49} +7.41436 q^{52} -10.2018 q^{53} +12.4800 q^{56} +2.50123 q^{58} -11.7468 q^{59} +1.33270 q^{61} -3.51982 q^{62} +3.73237 q^{64} +6.01391 q^{67} +1.58043 q^{68} -11.3991 q^{71} +1.51982 q^{73} +7.14728 q^{74} +2.32770 q^{76} +24.6603 q^{77} +1.26764 q^{79} +5.89454 q^{82} -1.15555 q^{83} +6.90506 q^{86} +14.5055 q^{88} -5.40223 q^{89} -25.4252 q^{91} +1.34322 q^{92} +7.85498 q^{94} -16.4440 q^{97} -11.5215 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{7} + 4 q^{11} - 12 q^{13} - 4 q^{14} + 14 q^{16} - 4 q^{17} + 8 q^{19} - 8 q^{22} - 6 q^{23} - 12 q^{26} - 24 q^{28} - 6 q^{29} + 8 q^{31} - 20 q^{32} - 12 q^{34} - 22 q^{37} - 36 q^{38}+ \cdots + 4 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^7 + 4 * q^11 - 12 * q^13 - 4 * q^14 + 14 * q^16 - 4 * q^17 + 8 * q^19 - 8 * q^22 - 6 * q^23 - 12 * q^26 - 24 * q^28 - 6 * q^29 + 8 * q^31 - 20 * q^32 - 12 * q^34 - 22 * q^37 - 36 * q^38 - 18 * q^41 - 8 * q^43 + 4 * q^44 - 12 * q^47 + 18 * q^49 - 4 * q^53 + 4 * q^56 + 16 * q^58 - 6 * q^59 + 14 * q^64 - 10 * q^67 - 28 * q^68 - 22 * q^71 - 12 * q^73 - 20 * q^74 + 16 * q^76 + 4 * q^77 + 4 * q^79 + 12 * q^82 - 24 * q^83 + 20 * q^86 + 8 * q^88 - 8 * q^89 + 4 * q^91 - 10 * q^92 + 20 * q^94 - 12 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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