LMFDB - Embedded newform 9196.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9196,2,Mod(1,9196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9196, 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("9196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.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:	$73.4304296988$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} - 3x^{7} + 84x^{6} + 16x^{5} - 174x^{4} - 16x^{3} + 122x^{2} - 9 $ x^10 - 16x^8 - 3x^7 + 84x^6 + 16x^5 - 174x^4 - 16x^3 + 122*x^2 - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.43917$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43917 q^{3} -1.46040 q^{5} +2.94667 q^{7} -0.928777 q^{9} +O(q^{10})$	$q-1.43917 q^{3} -1.46040 q^{5} +2.94667 q^{7} -0.928777 q^{9} +2.95605 q^{13} +2.10178 q^{15} +1.11490 q^{17} +1.00000 q^{19} -4.24077 q^{21} +6.23162 q^{23} -2.86722 q^{25} +5.65420 q^{27} -5.44747 q^{29} -4.95247 q^{31} -4.30332 q^{35} -2.84952 q^{37} -4.25427 q^{39} -12.2615 q^{41} -1.95512 q^{43} +1.35639 q^{45} -4.03225 q^{47} +1.68285 q^{49} -1.60453 q^{51} +0.0749606 q^{53} -1.43917 q^{57} +2.93575 q^{59} -0.0740859 q^{61} -2.73680 q^{63} -4.31703 q^{65} +3.18622 q^{67} -8.96839 q^{69} +11.5779 q^{71} -0.483015 q^{73} +4.12643 q^{75} +5.27944 q^{79} -5.35104 q^{81} -8.39444 q^{83} -1.62820 q^{85} +7.83986 q^{87} +7.78559 q^{89} +8.71050 q^{91} +7.12746 q^{93} -1.46040 q^{95} +4.75458 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{5} - 5 q^{7} + 2 q^{9}+O(q^{10}) $ 10 * q + 8 * q^5 - 5 * q^7 + 2 * q^9	$ 10 q + 8 q^{5} - 5 q^{7} + 2 q^{9} + 2 q^{13} - 10 q^{15} - 4 q^{17} + 10 q^{19} - 6 q^{21} - 8 q^{23} - 10 q^{25} - 9 q^{27} + 3 q^{29} - 12 q^{31} - 9 q^{35} - 23 q^{37} + 18 q^{39} + 5 q^{41} + 14 q^{43} - 27 q^{47} - 17 q^{49} + 12 q^{51} - 10 q^{53} - 9 q^{59} - 4 q^{61} - 14 q^{63} - 7 q^{65} - 17 q^{67} - 3 q^{69} - 34 q^{71} + 9 q^{73} - 26 q^{75} + 16 q^{79} - 22 q^{81} - 6 q^{83} - 24 q^{85} - q^{87} - 35 q^{89} - 35 q^{91} - 44 q^{93} + 8 q^{95} + 22 q^{97}+O(q^{100}) $ 10 * q + 8 * q^5 - 5 * q^7 + 2 * q^9 + 2 * q^13 - 10 * q^15 - 4 * q^17 + 10 * q^19 - 6 * q^21 - 8 * q^23 - 10 * q^25 - 9 * q^27 + 3 * q^29 - 12 * q^31 - 9 * q^35 - 23 * q^37 + 18 * q^39 + 5 * q^41 + 14 * q^43 - 27 * q^47 - 17 * q^49 + 12 * q^51 - 10 * q^53 - 9 * q^59 - 4 * q^61 - 14 * q^63 - 7 * q^65 - 17 * q^67 - 3 * q^69 - 34 * q^71 + 9 * q^73 - 26 * q^75 + 16 * q^79 - 22 * q^81 - 6 * q^83 - 24 * q^85 - q^87 - 35 * q^89 - 35 * q^91 - 44 * q^93 + 8 * q^95 + 22 * q^97

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
$3$	−1.43917	−0.830908	−0.415454	−	0.909614i	$-0.636377\pi$
$3$	−1.43917	−0.830908	−0.415454	+	0.909614i	$0.636377\pi$
$4$	0	0
$5$	−1.46040	−0.653112	−0.326556	−	0.945178i	$-0.605888\pi$
$5$	−1.46040	−0.653112	−0.326556	+	0.945178i	$0.605888\pi$
$6$	0	0
$7$	2.94667	1.11374	0.556868	−	0.830601i	$-0.312003\pi$
$7$	2.94667	1.11374	0.556868	+	0.830601i	$0.312003\pi$
$8$	0	0
$9$	−0.928777	−0.309592
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.95605	0.819861	0.409931	−	0.912117i	$-0.365553\pi$
$13$	2.95605	0.819861	0.409931	+	0.912117i	$0.365553\pi$
$14$	0	0
$15$	2.10178	0.542676
$16$	0	0
$17$	1.11490	0.270402	0.135201	−	0.990818i	$-0.456832\pi$
$17$	1.11490	0.270402	0.135201	+	0.990818i	$0.456832\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.24077	−0.925411
$22$	0	0
$23$	6.23162	1.29938	0.649691	−	0.760198i	$-0.274898\pi$
$23$	6.23162	1.29938	0.649691	+	0.760198i	$0.274898\pi$
$24$	0	0
$25$	−2.86722	−0.573444
$26$	0	0
$27$	5.65420	1.08815
$28$	0	0
$29$	−5.44747	−1.01157	−0.505785	−	0.862660i	$-0.668797\pi$
$29$	−5.44747	−1.01157	−0.505785	+	0.862660i	$0.668797\pi$
$30$	0	0
$31$	−4.95247	−0.889489	−0.444745	−	0.895657i	$-0.646706\pi$
$31$	−4.95247	−0.889489	−0.444745	+	0.895657i	$0.646706\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.30332	−0.727394
$36$	0	0
$37$	−2.84952	−0.468458	−0.234229	−	0.972181i	$-0.575257\pi$
$37$	−2.84952	−0.468458	−0.234229	+	0.972181i	$0.575257\pi$
$38$	0	0
$39$	−4.25427	−0.681229
$40$	0	0
$41$	−12.2615	−1.91493	−0.957464	−	0.288551i	$-0.906826\pi$
$41$	−12.2615	−1.91493	−0.957464	+	0.288551i	$0.906826\pi$
$42$	0	0
$43$	−1.95512	−0.298154	−0.149077	−	0.988826i	$-0.547630\pi$
$43$	−1.95512	−0.298154	−0.149077	+	0.988826i	$0.547630\pi$
$44$	0	0
$45$	1.35639	0.202199
$46$	0	0
$47$	−4.03225	−0.588164	−0.294082	−	0.955780i	$-0.595014\pi$
$47$	−4.03225	−0.588164	−0.294082	+	0.955780i	$0.595014\pi$
$48$	0	0
$49$	1.68285	0.240407
$50$	0	0
$51$	−1.60453	−0.224679
$52$	0	0
$53$	0.0749606	0.0102966	0.00514832	−	0.999987i	$-0.498361\pi$
$53$	0.0749606	0.0102966	0.00514832	+	0.999987i	$0.498361\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.43917	−0.190623
$58$	0	0
$59$	2.93575	0.382202	0.191101	−	0.981570i	$-0.438794\pi$
$59$	2.93575	0.382202	0.191101	+	0.981570i	$0.438794\pi$
$60$	0	0
$61$	−0.0740859	−0.00948573	−0.00474287	−	0.999989i	$-0.501510\pi$
$61$	−0.0740859	−0.00948573	−0.00474287	+	0.999989i	$0.501510\pi$
$62$	0	0
$63$	−2.73680	−0.344804
$64$	0	0
$65$	−4.31703	−0.535462
$66$	0	0
$67$	3.18622	0.389259	0.194630	−	0.980877i	$-0.437650\pi$
$67$	3.18622	0.389259	0.194630	+	0.980877i	$0.437650\pi$
$68$	0	0
$69$	−8.96839	−1.07967
$70$	0	0
$71$	11.5779	1.37405	0.687023	−	0.726636i	$-0.258917\pi$
$71$	11.5779	1.37405	0.687023	+	0.726636i	$0.258917\pi$
$72$	0	0
$73$	−0.483015	−0.0565326	−0.0282663	−	0.999600i	$-0.508999\pi$
$73$	−0.483015	−0.0565326	−0.0282663	+	0.999600i	$0.508999\pi$
$74$	0	0
$75$	4.12643	0.476479
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.27944	0.593983	0.296991	−	0.954880i	$-0.404017\pi$
$79$	5.27944	0.593983	0.296991	+	0.954880i	$0.404017\pi$
$80$	0	0
$81$	−5.35104	−0.594560
$82$	0	0
$83$	−8.39444	−0.921410	−0.460705	−	0.887553i	$-0.652403\pi$
$83$	−8.39444	−0.921410	−0.460705	+	0.887553i	$0.652403\pi$
$84$	0	0
$85$	−1.62820	−0.176603
$86$	0	0
$87$	7.83986	0.840521
$88$	0	0
$89$	7.78559	0.825271	0.412635	−	0.910896i	$-0.364608\pi$
$89$	7.78559	0.825271	0.412635	+	0.910896i	$0.364608\pi$
$90$	0	0
$91$	8.71050	0.913109
$92$	0	0
$93$	7.12746	0.739083
$94$	0	0
$95$	−1.46040	−0.149834
$96$	0	0
$97$	4.75458	0.482754	0.241377	−	0.970431i	$-0.422401\pi$
$97$	4.75458	0.482754	0.241377	+	0.970431i	$0.422401\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.87406	−0.186476	−0.0932380	−	0.995644i	$-0.529722\pi$
$101$	−1.87406	−0.186476	−0.0932380	+	0.995644i	$0.529722\pi$
$102$	0	0
$103$	−2.68834	−0.264890	−0.132445	−	0.991190i	$-0.542283\pi$
$103$	−2.68834	−0.264890	−0.132445	+	0.991190i	$0.542283\pi$
$104$	0	0
$105$	6.19323	0.604398
$106$	0	0
$107$	2.65438	0.256609	0.128304	−	0.991735i	$-0.459047\pi$
$107$	2.65438	0.256609	0.128304	+	0.991735i	$0.459047\pi$
$108$	0	0
$109$	8.05431	0.771463	0.385731	−	0.922611i	$-0.373949\pi$
$109$	8.05431	0.771463	0.385731	+	0.922611i	$0.373949\pi$
$110$	0	0
$111$	4.10095	0.389245
$112$	0	0
$113$	0.257951	0.0242660	0.0121330	−	0.999926i	$-0.496138\pi$
$113$	0.257951	0.0242660	0.0121330	+	0.999926i	$0.496138\pi$
$114$	0	0
$115$	−9.10068	−0.848643
$116$	0	0
$117$	−2.74551	−0.253823
$118$	0	0
$119$	3.28523	0.301157
$120$	0	0
$121$	0	0
$122$	0	0
$123$	17.6465	1.59113
$124$	0	0
$125$	11.4893	1.02764
$126$	0	0
$127$	−19.7585	−1.75329	−0.876643	−	0.481142i	$-0.840222\pi$
$127$	−19.7585	−1.75329	−0.876643	+	0.481142i	$0.840222\pi$
$128$	0	0
$129$	2.81377	0.247738
$130$	0	0
$131$	4.64754	0.406057	0.203029	−	0.979173i	$-0.434922\pi$
$131$	4.64754	0.406057	0.203029	+	0.979173i	$0.434922\pi$
$132$	0	0
$133$	2.94667	0.255508
$134$	0	0
$135$	−8.25741	−0.710684
$136$	0	0
$137$	−11.6941	−0.999096	−0.499548	−	0.866286i	$-0.666501\pi$
$137$	−11.6941	−0.999096	−0.499548	+	0.866286i	$0.666501\pi$
$138$	0	0
$139$	16.8580	1.42988	0.714938	−	0.699188i	$-0.246455\pi$
$139$	16.8580	1.42988	0.714938	+	0.699188i	$0.246455\pi$
$140$	0	0
$141$	5.80311	0.488710
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.95550	0.660668
$146$	0	0
$147$	−2.42191	−0.199756
$148$	0	0
$149$	−14.1985	−1.16318	−0.581592	−	0.813480i	$-0.697570\pi$
$149$	−14.1985	−1.16318	−0.581592	+	0.813480i	$0.697570\pi$
$150$	0	0
$151$	15.2687	1.24255	0.621273	−	0.783594i	$-0.286616\pi$
$151$	15.2687	1.24255	0.621273	+	0.783594i	$0.286616\pi$
$152$	0	0
$153$	−1.03549	−0.0837145
$154$	0	0
$155$	7.23260	0.580936
$156$	0	0
$157$	19.4455	1.55192	0.775959	−	0.630783i	$-0.217266\pi$
$157$	19.4455	1.55192	0.775959	+	0.630783i	$0.217266\pi$
$158$	0	0
$159$	−0.107881	−0.00855555
$160$	0	0
$161$	18.3625	1.44717
$162$	0	0
$163$	−19.4845	−1.52615	−0.763074	−	0.646312i	$-0.776311\pi$
$163$	−19.4845	−1.52615	−0.763074	+	0.646312i	$0.776311\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.64762	−0.591791	−0.295895	−	0.955220i	$-0.595618\pi$
$167$	−7.64762	−0.591791	−0.295895	+	0.955220i	$0.595618\pi$
$168$	0	0
$169$	−4.26176	−0.327827
$170$	0	0
$171$	−0.928777	−0.0710254
$172$	0	0
$173$	−1.71505	−0.130393	−0.0651963	−	0.997872i	$-0.520767\pi$
$173$	−1.71505	−0.130393	−0.0651963	+	0.997872i	$0.520767\pi$
$174$	0	0
$175$	−8.44875	−0.638665
$176$	0	0
$177$	−4.22505	−0.317574
$178$	0	0
$179$	−8.39932	−0.627795	−0.313897	−	0.949457i	$-0.601635\pi$
$179$	−8.39932	−0.627795	−0.313897	+	0.949457i	$0.601635\pi$
$180$	0	0
$181$	−20.0508	−1.49037	−0.745183	−	0.666860i	$-0.767638\pi$
$181$	−20.0508	−1.49037	−0.745183	+	0.666860i	$0.767638\pi$
$182$	0	0
$183$	0.106623	0.00788177
$184$	0	0
$185$	4.16145	0.305956
$186$	0	0
$187$	0	0
$188$	0	0
$189$	16.6610	1.21191
$190$	0	0
$191$	−13.4448	−0.972832	−0.486416	−	0.873727i	$-0.661696\pi$
$191$	−13.4448	−0.972832	−0.486416	+	0.873727i	$0.661696\pi$
$192$	0	0
$193$	−11.9698	−0.861608	−0.430804	−	0.902445i	$-0.641770\pi$
$193$	−11.9698	−0.861608	−0.430804	+	0.902445i	$0.641770\pi$
$194$	0	0
$195$	6.21296	0.444919
$196$	0	0
$197$	−18.0316	−1.28469	−0.642347	−	0.766414i	$-0.722039\pi$
$197$	−18.0316	−1.28469	−0.642347	+	0.766414i	$0.722039\pi$
$198$	0	0
$199$	21.3508	1.51352	0.756760	−	0.653693i	$-0.226781\pi$
$199$	21.3508	1.51352	0.756760	+	0.653693i	$0.226781\pi$
$200$	0	0
$201$	−4.58553	−0.323438
$202$	0	0
$203$	−16.0519	−1.12662
$204$	0	0
$205$	17.9068	1.25066
$206$	0	0
$207$	−5.78778	−0.402279
$208$	0	0
$209$	0	0
$210$	0	0
$211$	27.8923	1.92018	0.960092	−	0.279684i	$-0.0902297\pi$
$211$	27.8923	1.92018	0.960092	+	0.279684i	$0.0902297\pi$
$212$	0	0
$213$	−16.6626	−1.14171
$214$	0	0
$215$	2.85527	0.194728
$216$	0	0
$217$	−14.5933	−0.990656
$218$	0	0
$219$	0.695143	0.0469734
$220$	0	0
$221$	3.29570	0.221692
$222$	0	0
$223$	7.46194	0.499689	0.249844	−	0.968286i	$-0.419621\pi$
$223$	7.46194	0.499689	0.249844	+	0.968286i	$0.419621\pi$
$224$	0	0
$225$	2.66301	0.177534
$226$	0	0
$227$	−27.6861	−1.83759	−0.918796	−	0.394733i	$-0.870837\pi$
$227$	−27.6861	−1.83759	−0.918796	+	0.394733i	$0.870837\pi$
$228$	0	0
$229$	−21.0639	−1.39194	−0.695970	−	0.718071i	$-0.745025\pi$
$229$	−21.0639	−1.39194	−0.695970	+	0.718071i	$0.745025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.8659	1.62902	0.814509	−	0.580151i	$-0.197006\pi$
$233$	24.8659	1.62902	0.814509	+	0.580151i	$0.197006\pi$
$234$	0	0
$235$	5.88871	0.384137
$236$	0	0
$237$	−7.59803	−0.493545
$238$	0	0
$239$	−15.7089	−1.01613	−0.508063	−	0.861320i	$-0.669638\pi$
$239$	−15.7089	−1.01613	−0.508063	+	0.861320i	$0.669638\pi$
$240$	0	0
$241$	−14.3016	−0.921249	−0.460624	−	0.887595i	$-0.652374\pi$
$241$	−14.3016	−0.921249	−0.460624	+	0.887595i	$0.652374\pi$
$242$	0	0
$243$	−9.26150	−0.594126
$244$	0	0
$245$	−2.45764	−0.157013
$246$	0	0
$247$	2.95605	0.188089
$248$	0	0
$249$	12.0811	0.765607
$250$	0	0
$251$	0.351776	0.0222039	0.0111019	−	0.999938i	$-0.496466\pi$
$251$	0.351776	0.0222039	0.0111019	+	0.999938i	$0.496466\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.34326	0.146741
$256$	0	0
$257$	1.17252	0.0731401	0.0365700	−	0.999331i	$-0.488357\pi$
$257$	1.17252	0.0731401	0.0365700	+	0.999331i	$0.488357\pi$
$258$	0	0
$259$	−8.39658	−0.521738
$260$	0	0
$261$	5.05948	0.313174
$262$	0	0
$263$	−11.5816	−0.714154	−0.357077	−	0.934075i	$-0.616227\pi$
$263$	−11.5816	−0.714154	−0.357077	+	0.934075i	$0.616227\pi$
$264$	0	0
$265$	−0.109473	−0.00672486
$266$	0	0
$267$	−11.2048	−0.685724
$268$	0	0
$269$	0.822788	0.0501662	0.0250831	−	0.999685i	$-0.492015\pi$
$269$	0.822788	0.0501662	0.0250831	+	0.999685i	$0.492015\pi$
$270$	0	0
$271$	11.6980	0.710602	0.355301	−	0.934752i	$-0.384378\pi$
$271$	11.6980	0.710602	0.355301	+	0.934752i	$0.384378\pi$
$272$	0	0
$273$	−12.5359	−0.758709
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.53225	0.0920643	0.0460321	−	0.998940i	$-0.485342\pi$
$277$	1.53225	0.0920643	0.0460321	+	0.998940i	$0.485342\pi$
$278$	0	0
$279$	4.59974	0.275379
$280$	0	0
$281$	15.2564	0.910121	0.455060	−	0.890461i	$-0.349618\pi$
$281$	15.2564	0.910121	0.455060	+	0.890461i	$0.349618\pi$
$282$	0	0
$283$	8.96708	0.533038	0.266519	−	0.963830i	$-0.414127\pi$
$283$	8.96708	0.533038	0.266519	+	0.963830i	$0.414127\pi$
$284$	0	0
$285$	2.10178	0.124498
$286$	0	0
$287$	−36.1306	−2.13272
$288$	0	0
$289$	−15.7570	−0.926883
$290$	0	0
$291$	−6.84267	−0.401124
$292$	0	0
$293$	−14.7384	−0.861026	−0.430513	−	0.902584i	$-0.641667\pi$
$293$	−14.7384	−0.861026	−0.430513	+	0.902584i	$0.641667\pi$
$294$	0	0
$295$	−4.28738	−0.249621
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.4210	1.06531
$300$	0	0
$301$	−5.76110	−0.332064
$302$	0	0
$303$	2.69710	0.154944
$304$	0	0
$305$	0.108195	0.00619525
$306$	0	0
$307$	5.58452	0.318726	0.159363	−	0.987220i	$-0.449056\pi$
$307$	5.58452	0.318726	0.159363	+	0.987220i	$0.449056\pi$
$308$	0	0
$309$	3.86899	0.220099
$310$	0	0
$311$	−28.5036	−1.61629	−0.808145	−	0.588984i	$-0.799528\pi$
$311$	−28.5036	−1.61629	−0.808145	+	0.588984i	$0.799528\pi$
$312$	0	0
$313$	−16.5910	−0.937781	−0.468890	−	0.883256i	$-0.655346\pi$
$313$	−16.5910	−0.937781	−0.468890	+	0.883256i	$0.655346\pi$
$314$	0	0
$315$	3.99683	0.225196
$316$	0	0
$317$	22.1448	1.24378	0.621888	−	0.783106i	$-0.286366\pi$
$317$	22.1448	1.24378	0.621888	+	0.783106i	$0.286366\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.82011	−0.213218
$322$	0	0
$323$	1.11490	0.0620346
$324$	0	0
$325$	−8.47566	−0.470145
$326$	0	0
$327$	−11.5916	−0.641014
$328$	0	0
$329$	−11.8817	−0.655059
$330$	0	0
$331$	23.4297	1.28781	0.643907	−	0.765104i	$-0.277312\pi$
$331$	23.4297	1.28781	0.643907	+	0.765104i	$0.277312\pi$
$332$	0	0
$333$	2.64657	0.145031
$334$	0	0
$335$	−4.65317	−0.254230
$336$	0	0
$337$	−16.0969	−0.876854	−0.438427	−	0.898767i	$-0.644464\pi$
$337$	−16.0969	−0.876854	−0.438427	+	0.898767i	$0.644464\pi$
$338$	0	0
$339$	−0.371237	−0.0201628
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.6679	−0.845986
$344$	0	0
$345$	13.0975	0.705144
$346$	0	0
$347$	−3.83321	−0.205778	−0.102889	−	0.994693i	$-0.532809\pi$
$347$	−3.83321	−0.205778	−0.102889	+	0.994693i	$0.532809\pi$
$348$	0	0
$349$	−4.53771	−0.242898	−0.121449	−	0.992598i	$-0.538754\pi$
$349$	−4.53771	−0.242898	−0.121449	+	0.992598i	$0.538754\pi$
$350$	0	0
$351$	16.7141	0.892132
$352$	0	0
$353$	27.3402	1.45517	0.727586	−	0.686017i	$-0.240642\pi$
$353$	27.3402	1.45517	0.727586	+	0.686017i	$0.240642\pi$
$354$	0	0
$355$	−16.9084	−0.897406
$356$	0	0
$357$	−4.72802	−0.250233
$358$	0	0
$359$	24.9406	1.31631	0.658157	−	0.752881i	$-0.271336\pi$
$359$	24.9406	1.31631	0.658157	+	0.752881i	$0.271336\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.705397	0.0369221
$366$	0	0
$367$	−17.7477	−0.926422	−0.463211	−	0.886248i	$-0.653303\pi$
$367$	−17.7477	−0.926422	−0.463211	+	0.886248i	$0.653303\pi$
$368$	0	0
$369$	11.3882	0.592847
$370$	0	0
$371$	0.220884	0.0114677
$372$	0	0
$373$	21.7030	1.12374	0.561870	−	0.827225i	$-0.310082\pi$
$373$	21.7030	1.12374	0.561870	+	0.827225i	$0.310082\pi$
$374$	0	0
$375$	−16.5351	−0.853871
$376$	0	0
$377$	−16.1030	−0.829347
$378$	0	0
$379$	−1.58372	−0.0813504	−0.0406752	−	0.999172i	$-0.512951\pi$
$379$	−1.58372	−0.0813504	−0.0406752	+	0.999172i	$0.512951\pi$
$380$	0	0
$381$	28.4360	1.45682
$382$	0	0
$383$	−33.2985	−1.70147	−0.850736	−	0.525593i	$-0.823844\pi$
$383$	−33.2985	−1.70147	−0.850736	+	0.525593i	$0.823844\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.81588	0.0923061
$388$	0	0
$389$	29.9928	1.52070	0.760349	−	0.649515i	$-0.225028\pi$
$389$	29.9928	1.52070	0.760349	+	0.649515i	$0.225028\pi$
$390$	0	0
$391$	6.94762	0.351356
$392$	0	0
$393$	−6.68862	−0.337396
$394$	0	0
$395$	−7.71011	−0.387938
$396$	0	0
$397$	3.94977	0.198234	0.0991168	−	0.995076i	$-0.468398\pi$
$397$	3.94977	0.198234	0.0991168	+	0.995076i	$0.468398\pi$
$398$	0	0
$399$	−4.24077	−0.212304
$400$	0	0
$401$	−34.8660	−1.74112	−0.870561	−	0.492060i	$-0.836244\pi$
$401$	−34.8660	−1.74112	−0.870561	+	0.492060i	$0.836244\pi$
$402$	0	0
$403$	−14.6397	−0.729258
$404$	0	0
$405$	7.81468	0.388315
$406$	0	0
$407$	0	0
$408$	0	0
$409$	3.93351	0.194500	0.0972498	−	0.995260i	$-0.468995\pi$
$409$	3.93351	0.194500	0.0972498	+	0.995260i	$0.468995\pi$
$410$	0	0
$411$	16.8299	0.830157
$412$	0	0
$413$	8.65067	0.425672
$414$	0	0
$415$	12.2593	0.601784
$416$	0	0
$417$	−24.2616	−1.18810
$418$	0	0
$419$	−22.8214	−1.11490	−0.557449	−	0.830211i	$-0.688220\pi$
$419$	−22.8214	−1.11490	−0.557449	+	0.830211i	$0.688220\pi$
$420$	0	0
$421$	27.3763	1.33424	0.667119	−	0.744951i	$-0.267527\pi$
$421$	27.3763	1.33424	0.667119	+	0.744951i	$0.267527\pi$
$422$	0	0
$423$	3.74506	0.182091
$424$	0	0
$425$	−3.19666	−0.155061
$426$	0	0
$427$	−0.218307	−0.0105646
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.6547	−0.754062	−0.377031	−	0.926201i	$-0.623055\pi$
$431$	−15.6547	−0.754062	−0.377031	+	0.926201i	$0.623055\pi$
$432$	0	0
$433$	21.0192	1.01012	0.505060	−	0.863085i	$-0.331471\pi$
$433$	21.0192	1.01012	0.505060	+	0.863085i	$0.331471\pi$
$434$	0	0
$435$	−11.4494	−0.548955
$436$	0	0
$437$	6.23162	0.298099
$438$	0	0
$439$	−12.7001	−0.606144	−0.303072	−	0.952968i	$-0.598012\pi$
$439$	−12.7001	−0.606144	−0.303072	+	0.952968i	$0.598012\pi$
$440$	0	0
$441$	−1.56299	−0.0744281
$442$	0	0
$443$	−16.0928	−0.764593	−0.382296	−	0.924040i	$-0.624867\pi$
$443$	−16.0928	−0.764593	−0.382296	+	0.924040i	$0.624867\pi$
$444$	0	0
$445$	−11.3701	−0.538995
$446$	0	0
$447$	20.4341	0.966499
$448$	0	0
$449$	−0.908087	−0.0428553	−0.0214276	−	0.999770i	$-0.506821\pi$
$449$	−0.908087	−0.0428553	−0.0214276	+	0.999770i	$0.506821\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−21.9743	−1.03244
$454$	0	0
$455$	−12.7208	−0.596363
$456$	0	0
$457$	−21.1577	−0.989718	−0.494859	−	0.868973i	$-0.664780\pi$
$457$	−21.1577	−0.989718	−0.494859	+	0.868973i	$0.664780\pi$
$458$	0	0
$459$	6.30385	0.294238
$460$	0	0
$461$	−27.9700	−1.30269	−0.651347	−	0.758780i	$-0.725796\pi$
$461$	−27.9700	−1.30269	−0.651347	+	0.758780i	$0.725796\pi$
$462$	0	0
$463$	−29.9663	−1.39265	−0.696325	−	0.717726i	$-0.745183\pi$
$463$	−29.9663	−1.39265	−0.696325	+	0.717726i	$0.745183\pi$
$464$	0	0
$465$	−10.4090	−0.482704
$466$	0	0
$467$	−2.78850	−0.129036	−0.0645181	−	0.997917i	$-0.520551\pi$
$467$	−2.78850	−0.129036	−0.0645181	+	0.997917i	$0.520551\pi$
$468$	0	0
$469$	9.38874	0.433532
$470$	0	0
$471$	−27.9854	−1.28950
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.86722	−0.131557
$476$	0	0
$477$	−0.0696217	−0.00318776
$478$	0	0
$479$	15.7397	0.719167	0.359583	−	0.933113i	$-0.382919\pi$
$479$	15.7397	0.719167	0.359583	+	0.933113i	$0.382919\pi$
$480$	0	0
$481$	−8.42333	−0.384071
$482$	0	0
$483$	−26.4268	−1.20246
$484$	0	0
$485$	−6.94360	−0.315293
$486$	0	0
$487$	−28.9843	−1.31340	−0.656702	−	0.754151i	$-0.728049\pi$
$487$	−28.9843	−1.31340	−0.656702	+	0.754151i	$0.728049\pi$
$488$	0	0
$489$	28.0417	1.26809
$490$	0	0
$491$	5.29830	0.239109	0.119554	−	0.992828i	$-0.461853\pi$
$491$	5.29830	0.239109	0.119554	+	0.992828i	$0.461853\pi$
$492$	0	0
$493$	−6.07337	−0.273531
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.1163	1.53032
$498$	0	0
$499$	−27.4614	−1.22934	−0.614670	−	0.788784i	$-0.710711\pi$
$499$	−27.4614	−1.22934	−0.614670	+	0.788784i	$0.710711\pi$
$500$	0	0
$501$	11.0063	0.491723
$502$	0	0
$503$	−24.7566	−1.10384	−0.551920	−	0.833897i	$-0.686105\pi$
$503$	−24.7566	−1.10384	−0.551920	+	0.833897i	$0.686105\pi$
$504$	0	0
$505$	2.73688	0.121790
$506$	0	0
$507$	6.13341	0.272394
$508$	0	0
$509$	−31.0388	−1.37577	−0.687886	−	0.725819i	$-0.741461\pi$
$509$	−31.0388	−1.37577	−0.687886	+	0.725819i	$0.741461\pi$
$510$	0	0
$511$	−1.42328	−0.0629624
$512$	0	0
$513$	5.65420	0.249639
$514$	0	0
$515$	3.92606	0.173003
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.46825	0.108344
$520$	0	0
$521$	−5.95793	−0.261021	−0.130511	−	0.991447i	$-0.541662\pi$
$521$	−5.95793	−0.261021	−0.130511	+	0.991447i	$0.541662\pi$
$522$	0	0
$523$	7.62722	0.333515	0.166758	−	0.985998i	$-0.446670\pi$
$523$	7.62722	0.333515	0.166758	+	0.985998i	$0.446670\pi$
$524$	0	0
$525$	12.1592	0.530672
$526$	0	0
$527$	−5.52149	−0.240520
$528$	0	0
$529$	15.8331	0.688394
$530$	0	0
$531$	−2.72666	−0.118327
$532$	0	0
$533$	−36.2457	−1.56998
$534$	0	0
$535$	−3.87647	−0.167594
$536$	0	0
$537$	12.0881	0.521639
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.8559	−0.767687	−0.383844	−	0.923398i	$-0.625400\pi$
$541$	−17.8559	−0.767687	−0.383844	+	0.923398i	$0.625400\pi$
$542$	0	0
$543$	28.8566	1.23836
$544$	0	0
$545$	−11.7625	−0.503852
$546$	0	0
$547$	19.8352	0.848092	0.424046	−	0.905641i	$-0.360610\pi$
$547$	19.8352	0.848092	0.424046	+	0.905641i	$0.360610\pi$
$548$	0	0
$549$	0.0688093	0.00293671
$550$	0	0
$551$	−5.44747	−0.232070
$552$	0	0
$553$	15.5567	0.661540
$554$	0	0
$555$	−5.98905	−0.254221
$556$	0	0
$557$	−3.04144	−0.128870	−0.0644350	−	0.997922i	$-0.520525\pi$
$557$	−3.04144	−0.128870	−0.0644350	+	0.997922i	$0.520525\pi$
$558$	0	0
$559$	−5.77945	−0.244445
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.5682	−0.613975	−0.306988	−	0.951713i	$-0.599321\pi$
$563$	−14.5682	−0.613975	−0.306988	+	0.951713i	$0.599321\pi$
$564$	0	0
$565$	−0.376713	−0.0158484
$566$	0	0
$567$	−15.7677	−0.662183
$568$	0	0
$569$	−15.0474	−0.630819	−0.315409	−	0.948956i	$-0.602142\pi$
$569$	−15.0474	−0.630819	−0.315409	+	0.948956i	$0.602142\pi$
$570$	0	0
$571$	9.64735	0.403729	0.201864	−	0.979413i	$-0.435300\pi$
$571$	9.64735	0.403729	0.201864	+	0.979413i	$0.435300\pi$
$572$	0	0
$573$	19.3494	0.808334
$574$	0	0
$575$	−17.8674	−0.745123
$576$	0	0
$577$	2.58272	0.107520	0.0537600	−	0.998554i	$-0.482879\pi$
$577$	2.58272	0.107520	0.0537600	+	0.998554i	$0.482879\pi$
$578$	0	0
$579$	17.2267	0.715917
$580$	0	0
$581$	−24.7356	−1.02621
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.00956	0.165775
$586$	0	0
$587$	−22.2977	−0.920325	−0.460163	−	0.887835i	$-0.652209\pi$
$587$	−22.2977	−0.920325	−0.460163	+	0.887835i	$0.652209\pi$
$588$	0	0
$589$	−4.95247	−0.204063
$590$	0	0
$591$	25.9505	1.06746
$592$	0	0
$593$	−15.2005	−0.624208	−0.312104	−	0.950048i	$-0.601034\pi$
$593$	−15.2005	−0.624208	−0.312104	+	0.950048i	$0.601034\pi$
$594$	0	0
$595$	−4.79777	−0.196689
$596$	0	0
$597$	−30.7276	−1.25759
$598$	0	0
$599$	−22.0761	−0.902006	−0.451003	−	0.892523i	$-0.648934\pi$
$599$	−22.0761	−0.902006	−0.451003	+	0.892523i	$0.648934\pi$
$600$	0	0
$601$	−16.5255	−0.674088	−0.337044	−	0.941489i	$-0.609427\pi$
$601$	−16.5255	−0.674088	−0.337044	+	0.941489i	$0.609427\pi$
$602$	0	0
$603$	−2.95929	−0.120512
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.92037	0.321478	0.160739	−	0.986997i	$-0.448612\pi$
$607$	7.92037	0.321478	0.160739	+	0.986997i	$0.448612\pi$
$608$	0	0
$609$	23.1014	0.936118
$610$	0	0
$611$	−11.9195	−0.482213
$612$	0	0
$613$	18.8271	0.760419	0.380209	−	0.924900i	$-0.375852\pi$
$613$	18.8271	0.760419	0.380209	+	0.924900i	$0.375852\pi$
$614$	0	0
$615$	−25.7710	−1.03919
$616$	0	0
$617$	−13.2885	−0.534975	−0.267487	−	0.963561i	$-0.586193\pi$
$617$	−13.2885	−0.534975	−0.267487	+	0.963561i	$0.586193\pi$
$618$	0	0
$619$	−39.1318	−1.57284	−0.786420	−	0.617692i	$-0.788068\pi$
$619$	−39.1318	−1.57284	−0.786420	+	0.617692i	$0.788068\pi$
$620$	0	0
$621$	35.2348	1.41392
$622$	0	0
$623$	22.9415	0.919134
$624$	0	0
$625$	−2.44294	−0.0977174
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.17692	−0.126672
$630$	0	0
$631$	−22.1395	−0.881361	−0.440680	−	0.897664i	$-0.645263\pi$
$631$	−22.1395	−0.881361	−0.440680	+	0.897664i	$0.645263\pi$
$632$	0	0
$633$	−40.1419	−1.59550
$634$	0	0
$635$	28.8554	1.14509
$636$	0	0
$637$	4.97458	0.197100
$638$	0	0
$639$	−10.7533	−0.425394
$640$	0	0
$641$	−7.71948	−0.304901	−0.152451	−	0.988311i	$-0.548717\pi$
$641$	−7.71948	−0.304901	−0.152451	+	0.988311i	$0.548717\pi$
$642$	0	0
$643$	42.6046	1.68016	0.840081	−	0.542461i	$-0.182508\pi$
$643$	42.6046	1.68016	0.840081	+	0.542461i	$0.182508\pi$
$644$	0	0
$645$	−4.10923	−0.161801
$646$	0	0
$647$	−10.6127	−0.417228	−0.208614	−	0.977998i	$-0.566895\pi$
$647$	−10.6127	−0.417228	−0.208614	+	0.977998i	$0.566895\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	21.0023	0.823143
$652$	0	0
$653$	−48.3078	−1.89043	−0.945215	−	0.326448i	$-0.894148\pi$
$653$	−48.3078	−1.89043	−0.945215	+	0.326448i	$0.894148\pi$
$654$	0	0
$655$	−6.78728	−0.265201
$656$	0	0
$657$	0.448613	0.0175021
$658$	0	0
$659$	1.71267	0.0667161	0.0333581	−	0.999443i	$-0.489380\pi$
$659$	1.71267	0.0667161	0.0333581	+	0.999443i	$0.489380\pi$
$660$	0	0
$661$	48.7587	1.89649	0.948246	−	0.317536i	$-0.102855\pi$
$661$	48.7587	1.89649	0.948246	+	0.317536i	$0.102855\pi$
$662$	0	0
$663$	−4.74308	−0.184206
$664$	0	0
$665$	−4.30332	−0.166876
$666$	0	0
$667$	−33.9465	−1.31442
$668$	0	0
$669$	−10.7390	−0.415195
$670$	0	0
$671$	0	0
$672$	0	0
$673$	16.2999	0.628313	0.314157	−	0.949371i	$-0.398278\pi$
$673$	16.2999	0.628313	0.314157	+	0.949371i	$0.398278\pi$
$674$	0	0
$675$	−16.2118	−0.623994
$676$	0	0
$677$	−0.534093	−0.0205268	−0.0102634	−	0.999947i	$-0.503267\pi$
$677$	−0.534093	−0.0205268	−0.0102634	+	0.999947i	$0.503267\pi$
$678$	0	0
$679$	14.0102	0.537661
$680$	0	0
$681$	39.8451	1.52687
$682$	0	0
$683$	−22.9689	−0.878882	−0.439441	−	0.898271i	$-0.644823\pi$
$683$	−22.9689	−0.878882	−0.439441	+	0.898271i	$0.644823\pi$
$684$	0	0
$685$	17.0781	0.652522
$686$	0	0
$687$	30.3146	1.15657
$688$	0	0
$689$	0.221588	0.00844181
$690$	0	0
$691$	48.6707	1.85152	0.925760	−	0.378111i	$-0.123426\pi$
$691$	48.6707	1.85152	0.925760	+	0.378111i	$0.123426\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.6195	−0.933870
$696$	0	0
$697$	−13.6703	−0.517801
$698$	0	0
$699$	−35.7863	−1.35356
$700$	0	0
$701$	23.9477	0.904493	0.452247	−	0.891893i	$-0.350623\pi$
$701$	23.9477	0.904493	0.452247	+	0.891893i	$0.350623\pi$
$702$	0	0
$703$	−2.84952	−0.107472
$704$	0	0
$705$	−8.47488	−0.319182
$706$	0	0
$707$	−5.52223	−0.207685
$708$	0	0
$709$	−25.5406	−0.959199	−0.479600	−	0.877487i	$-0.659218\pi$
$709$	−25.5406	−0.959199	−0.479600	+	0.877487i	$0.659218\pi$
$710$	0	0
$711$	−4.90342	−0.183893
$712$	0	0
$713$	−30.8619	−1.15579
$714$	0	0
$715$	0	0
$716$	0	0
$717$	22.6079	0.844307
$718$	0	0
$719$	30.3372	1.13139	0.565693	−	0.824616i	$-0.308609\pi$
$719$	30.3372	1.13139	0.565693	+	0.824616i	$0.308609\pi$
$720$	0	0
$721$	−7.92164	−0.295017
$722$	0	0
$723$	20.5825	0.765473
$724$	0	0
$725$	15.6191	0.580079
$726$	0	0
$727$	−21.9102	−0.812605	−0.406302	−	0.913739i	$-0.633182\pi$
$727$	−21.9102	−0.812605	−0.406302	+	0.913739i	$0.633182\pi$
$728$	0	0
$729$	29.3820	1.08822
$730$	0	0
$731$	−2.17976	−0.0806215
$732$	0	0
$733$	−33.6541	−1.24304	−0.621522	−	0.783397i	$-0.713485\pi$
$733$	−33.6541	−1.24304	−0.621522	+	0.783397i	$0.713485\pi$
$734$	0	0
$735$	3.53697	0.130463
$736$	0	0
$737$	0	0
$738$	0	0
$739$	23.0890	0.849341	0.424671	−	0.905348i	$-0.360390\pi$
$739$	23.0890	0.849341	0.424671	+	0.905348i	$0.360390\pi$
$740$	0	0
$741$	−4.25427	−0.156285
$742$	0	0
$743$	42.0685	1.54334	0.771671	−	0.636021i	$-0.219421\pi$
$743$	42.0685	1.54334	0.771671	+	0.636021i	$0.219421\pi$
$744$	0	0
$745$	20.7355	0.759690
$746$	0	0
$747$	7.79657	0.285261
$748$	0	0
$749$	7.82157	0.285794
$750$	0	0
$751$	−32.2661	−1.17741	−0.588703	−	0.808349i	$-0.700361\pi$
$751$	−32.2661	−1.17741	−0.588703	+	0.808349i	$0.700361\pi$
$752$	0	0
$753$	−0.506267	−0.0184494
$754$	0	0
$755$	−22.2984	−0.811522
$756$	0	0
$757$	−39.0348	−1.41874	−0.709372	−	0.704834i	$-0.751021\pi$
$757$	−39.0348	−1.41874	−0.709372	+	0.704834i	$0.751021\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−53.7651	−1.94898	−0.974492	−	0.224424i	$-0.927950\pi$
$761$	−53.7651	−1.94898	−0.974492	+	0.224424i	$0.927950\pi$
$762$	0	0
$763$	23.7334	0.859206
$764$	0	0
$765$	1.51224	0.0546750
$766$	0	0
$767$	8.67822	0.313353
$768$	0	0
$769$	37.2325	1.34264	0.671319	−	0.741168i	$-0.265728\pi$
$769$	37.2325	1.34264	0.671319	+	0.741168i	$0.265728\pi$
$770$	0	0
$771$	−1.68747	−0.0607727
$772$	0	0
$773$	9.16819	0.329757	0.164878	−	0.986314i	$-0.447277\pi$
$773$	9.16819	0.329757	0.164878	+	0.986314i	$0.447277\pi$
$774$	0	0
$775$	14.1998	0.510072
$776$	0	0
$777$	12.0841	0.433516
$778$	0	0
$779$	−12.2615	−0.439315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−30.8010	−1.10074
$784$	0	0
$785$	−28.3982	−1.01358
$786$	0	0
$787$	2.26024	0.0805688	0.0402844	−	0.999188i	$-0.487174\pi$
$787$	2.26024	0.0805688	0.0402844	+	0.999188i	$0.487174\pi$
$788$	0	0
$789$	16.6680	0.593396
$790$	0	0
$791$	0.760097	0.0270259
$792$	0	0
$793$	−0.219002	−0.00777699
$794$	0	0
$795$	0.157550	0.00558774
$796$	0	0
$797$	29.5147	1.04546	0.522732	−	0.852497i	$-0.324913\pi$
$797$	29.5147	1.04546	0.522732	+	0.852497i	$0.324913\pi$
$798$	0	0
$799$	−4.49554	−0.159041
$800$	0	0
$801$	−7.23108	−0.255498
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−26.8167	−0.945163
$806$	0	0
$807$	−1.18413	−0.0416835
$808$	0	0
$809$	−45.6411	−1.60466	−0.802328	−	0.596883i	$-0.796406\pi$
$809$	−45.6411	−1.60466	−0.802328	+	0.596883i	$0.796406\pi$
$810$	0	0
$811$	−26.8997	−0.944577	−0.472288	−	0.881444i	$-0.656572\pi$
$811$	−26.8997	−0.944577	−0.472288	+	0.881444i	$0.656572\pi$
$812$	0	0
$813$	−16.8354	−0.590445
$814$	0	0
$815$	28.4553	0.996746
$816$	0	0
$817$	−1.95512	−0.0684012
$818$	0	0
$819$	−8.09011	−0.282691
$820$	0	0
$821$	−27.7319	−0.967850	−0.483925	−	0.875110i	$-0.660789\pi$
$821$	−27.7319	−0.967850	−0.483925	+	0.875110i	$0.660789\pi$
$822$	0	0
$823$	−0.507131	−0.0176775	−0.00883873	−	0.999961i	$-0.502813\pi$
$823$	−0.507131	−0.0176775	−0.00883873	+	0.999961i	$0.502813\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.86838	0.0649699	0.0324850	−	0.999472i	$-0.489658\pi$
$827$	1.86838	0.0649699	0.0324850	+	0.999472i	$0.489658\pi$
$828$	0	0
$829$	43.1546	1.49882	0.749410	−	0.662106i	$-0.230337\pi$
$829$	43.1546	1.49882	0.749410	+	0.662106i	$0.230337\pi$
$830$	0	0
$831$	−2.20518	−0.0764969
$832$	0	0
$833$	1.87620	0.0650066
$834$	0	0
$835$	11.1686	0.386506
$836$	0	0
$837$	−28.0022	−0.967898
$838$	0	0
$839$	−16.9808	−0.586243	−0.293121	−	0.956075i	$-0.594694\pi$
$839$	−16.9808	−0.586243	−0.293121	+	0.956075i	$0.594694\pi$
$840$	0	0
$841$	0.674908	0.0232727
$842$	0	0
$843$	−21.9566	−0.756226
$844$	0	0
$845$	6.22388	0.214108
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12.9052	−0.442905
$850$	0	0
$851$	−17.7571	−0.608706
$852$	0	0
$853$	20.4011	0.698519	0.349260	−	0.937026i	$-0.386433\pi$
$853$	20.4011	0.698519	0.349260	+	0.937026i	$0.386433\pi$
$854$	0	0
$855$	1.35639	0.0463875
$856$	0	0
$857$	−24.6779	−0.842981	−0.421490	−	0.906833i	$-0.638493\pi$
$857$	−24.6779	−0.842981	−0.421490	+	0.906833i	$0.638493\pi$
$858$	0	0
$859$	−3.52545	−0.120287	−0.0601433	−	0.998190i	$-0.519156\pi$
$859$	−3.52545	−0.120287	−0.0601433	+	0.998190i	$0.519156\pi$
$860$	0	0
$861$	51.9983	1.77210
$862$	0	0
$863$	−39.3399	−1.33915	−0.669573	−	0.742746i	$-0.733523\pi$
$863$	−39.3399	−1.33915	−0.669573	+	0.742746i	$0.733523\pi$
$864$	0	0
$865$	2.50466	0.0851610
$866$	0	0
$867$	22.6771	0.770154
$868$	0	0
$869$	0	0
$870$	0	0
$871$	9.41864	0.319139
$872$	0	0
$873$	−4.41594	−0.149457
$874$	0	0
$875$	33.8552	1.14451
$876$	0	0
$877$	−50.9812	−1.72151	−0.860756	−	0.509018i	$-0.830009\pi$
$877$	−50.9812	−1.72151	−0.860756	+	0.509018i	$0.830009\pi$
$878$	0	0
$879$	21.2111	0.715433
$880$	0	0
$881$	−11.9057	−0.401112	−0.200556	−	0.979682i	$-0.564275\pi$
$881$	−11.9057	−0.401112	−0.200556	+	0.979682i	$0.564275\pi$
$882$	0	0
$883$	51.5914	1.73619	0.868094	−	0.496400i	$-0.165345\pi$
$883$	51.5914	1.73619	0.868094	+	0.496400i	$0.165345\pi$
$884$	0	0
$885$	6.17028	0.207412
$886$	0	0
$887$	−12.0095	−0.403238	−0.201619	−	0.979464i	$-0.564620\pi$
$887$	−12.0095	−0.403238	−0.201619	+	0.979464i	$0.564620\pi$
$888$	0	0
$889$	−58.2218	−1.95270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.03225	−0.134934
$894$	0	0
$895$	12.2664	0.410020
$896$	0	0
$897$	−26.5110	−0.885177
$898$	0	0
$899$	26.9784	0.899780
$900$	0	0
$901$	0.0835735	0.00278424
$902$	0	0
$903$	8.29123	0.275915
$904$	0	0
$905$	29.2823	0.973377
$906$	0	0
$907$	32.4551	1.07765	0.538826	−	0.842417i	$-0.318868\pi$
$907$	32.4551	1.07765	0.538826	+	0.842417i	$0.318868\pi$
$908$	0	0
$909$	1.74058	0.0577315
$910$	0	0
$911$	2.82752	0.0936799	0.0468399	−	0.998902i	$-0.485085\pi$
$911$	2.82752	0.0936799	0.0468399	+	0.998902i	$0.485085\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−0.155712	−0.00514768
$916$	0	0
$917$	13.6947	0.452240
$918$	0	0
$919$	−34.3216	−1.13217	−0.566083	−	0.824348i	$-0.691542\pi$
$919$	−34.3216	−1.13217	−0.566083	+	0.824348i	$0.691542\pi$
$920$	0	0
$921$	−8.03710	−0.264832
$922$	0	0
$923$	34.2249	1.12653
$924$	0	0
$925$	8.17020	0.268635
$926$	0	0
$927$	2.49687	0.0820078
$928$	0	0
$929$	−43.1300	−1.41505	−0.707524	−	0.706689i	$-0.750188\pi$
$929$	−43.1300	−1.41505	−0.707524	+	0.706689i	$0.750188\pi$
$930$	0	0
$931$	1.68285	0.0551531
$932$	0	0
$933$	41.0216	1.34299
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55.2852	1.80609	0.903045	−	0.429547i	$-0.141327\pi$
$937$	55.2852	1.80609	0.903045	+	0.429547i	$0.141327\pi$
$938$	0	0
$939$	23.8774	0.779209
$940$	0	0
$941$	12.8579	0.419155	0.209578	−	0.977792i	$-0.432791\pi$
$941$	12.8579	0.419155	0.209578	+	0.977792i	$0.432791\pi$
$942$	0	0
$943$	−76.4092	−2.48822
$944$	0	0
$945$	−24.3318	−0.791515
$946$	0	0
$947$	−5.74390	−0.186652	−0.0933258	−	0.995636i	$-0.529750\pi$
$947$	−5.74390	−0.186652	−0.0933258	+	0.995636i	$0.529750\pi$
$948$	0	0
$949$	−1.42782	−0.0463489
$950$	0	0
$951$	−31.8702	−1.03346
$952$	0	0
$953$	−3.54118	−0.114710	−0.0573551	−	0.998354i	$-0.518267\pi$
$953$	−3.54118	−0.114710	−0.0573551	+	0.998354i	$0.518267\pi$
$954$	0	0
$955$	19.6348	0.635369
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.4587	−1.11273
$960$	0	0
$961$	−6.47308	−0.208809
$962$	0	0
$963$	−2.46533	−0.0794440
$964$	0	0
$965$	17.4808	0.562727
$966$	0	0
$967$	−48.5614	−1.56163	−0.780815	−	0.624762i	$-0.785196\pi$
$967$	−48.5614	−1.56163	−0.780815	+	0.624762i	$0.785196\pi$
$968$	0	0
$969$	−1.60453	−0.0515450
$970$	0	0
$971$	−17.7863	−0.570789	−0.285394	−	0.958410i	$-0.592125\pi$
$971$	−17.7863	−0.570789	−0.285394	+	0.958410i	$0.592125\pi$
$972$	0	0
$973$	49.6749	1.59250
$974$	0	0
$975$	12.1979	0.390647
$976$	0	0
$977$	21.2604	0.680181	0.340090	−	0.940393i	$-0.389542\pi$
$977$	21.2604	0.680181	0.340090	+	0.940393i	$0.389542\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−7.48066	−0.238839
$982$	0	0
$983$	−54.3152	−1.73238	−0.866192	−	0.499711i	$-0.833440\pi$
$983$	−54.3152	−1.73238	−0.866192	+	0.499711i	$0.833440\pi$
$984$	0	0
$985$	26.3333	0.839050
$986$	0	0
$987$	17.0998	0.544293
$988$	0	0
$989$	−12.1836	−0.387416
$990$	0	0
$991$	−15.7311	−0.499714	−0.249857	−	0.968283i	$-0.580384\pi$
$991$	−15.7311	−0.499714	−0.249857	+	0.968283i	$0.580384\pi$
$992$	0	0
$993$	−33.7194	−1.07005
$994$	0	0
$995$	−31.1808	−0.988498
$996$	0	0
$997$	27.8370	0.881605	0.440803	−	0.897604i	$-0.354694\pi$
$997$	27.8370	0.881605	0.440803	+	0.897604i	$0.354694\pi$
$998$	0	0
$999$	−16.1117	−0.509753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.s.1.3		10
11.5	even	5		836.2.j.b.685.2	yes	20
11.9	even	5		836.2.j.b.609.2	✓	20
11.10	odd	2		9196.2.a.t.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.b.609.2	✓	20	11.9	even	5
836.2.j.b.685.2	yes	20	11.5	even	5
9196.2.a.s.1.3		10	1.1	even	1	trivial
9196.2.a.t.1.3		10	11.10	odd	2

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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43917 q^{3} -1.46040 q^{5} +2.94667 q^{7} -0.928777 q^{9} +O(q^{10})\)	\(q-1.43917 q^{3} -1.46040 q^{5} +2.94667 q^{7} -0.928777 q^{9} +2.95605 q^{13} +2.10178 q^{15} +1.11490 q^{17} +1.00000 q^{19} -4.24077 q^{21} +6.23162 q^{23} -2.86722 q^{25} +5.65420 q^{27} -5.44747 q^{29} -4.95247 q^{31} -4.30332 q^{35} -2.84952 q^{37} -4.25427 q^{39} -12.2615 q^{41} -1.95512 q^{43} +1.35639 q^{45} -4.03225 q^{47} +1.68285 q^{49} -1.60453 q^{51} +0.0749606 q^{53} -1.43917 q^{57} +2.93575 q^{59} -0.0740859 q^{61} -2.73680 q^{63} -4.31703 q^{65} +3.18622 q^{67} -8.96839 q^{69} +11.5779 q^{71} -0.483015 q^{73} +4.12643 q^{75} +5.27944 q^{79} -5.35104 q^{81} -8.39444 q^{83} -1.62820 q^{85} +7.83986 q^{87} +7.78559 q^{89} +8.71050 q^{91} +7.12746 q^{93} -1.46040 q^{95} +4.75458 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{5} - 5 q^{7} + 2 q^{9}+O(q^{10}) \) 10 * q + 8 * q^5 - 5 * q^7 + 2 * q^9	\( 10 q + 8 q^{5} - 5 q^{7} + 2 q^{9} + 2 q^{13} - 10 q^{15} - 4 q^{17} + 10 q^{19} - 6 q^{21} - 8 q^{23} - 10 q^{25} - 9 q^{27} + 3 q^{29} - 12 q^{31} - 9 q^{35} - 23 q^{37} + 18 q^{39} + 5 q^{41} + 14 q^{43} - 27 q^{47} - 17 q^{49} + 12 q^{51} - 10 q^{53} - 9 q^{59} - 4 q^{61} - 14 q^{63} - 7 q^{65} - 17 q^{67} - 3 q^{69} - 34 q^{71} + 9 q^{73} - 26 q^{75} + 16 q^{79} - 22 q^{81} - 6 q^{83} - 24 q^{85} - q^{87} - 35 q^{89} - 35 q^{91} - 44 q^{93} + 8 q^{95} + 22 q^{97}+O(q^{100}) \) 10 * q + 8 * q^5 - 5 * q^7 + 2 * q^9 + 2 * q^13 - 10 * q^15 - 4 * q^17 + 10 * q^19 - 6 * q^21 - 8 * q^23 - 10 * q^25 - 9 * q^27 + 3 * q^29 - 12 * q^31 - 9 * q^35 - 23 * q^37 + 18 * q^39 + 5 * q^41 + 14 * q^43 - 27 * q^47 - 17 * q^49 + 12 * q^51 - 10 * q^53 - 9 * q^59 - 4 * q^61 - 14 * q^63 - 7 * q^65 - 17 * q^67 - 3 * q^69 - 34 * q^71 + 9 * q^73 - 26 * q^75 + 16 * q^79 - 22 * q^81 - 6 * q^83 - 24 * q^85 - q^87 - 35 * q^89 - 35 * q^91 - 44 * q^93 + 8 * q^95 + 22 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more