LMFDB - Embedded newform 8325.2.a.cr.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8325,2,Mod(1,8325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8325 = 3^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,5,0,11,0,0,-8,15,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.68489$ of defining polynomial
Character	$\chi$	$=$	8325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68489 q^{2} +5.20864 q^{4} +3.48285 q^{7} +8.61484 q^{8} +3.18991 q^{11} +1.81648 q^{13} +9.35107 q^{14} +12.7126 q^{16} +2.82645 q^{17} -5.72714 q^{19} +8.56456 q^{22} -1.43274 q^{23} +4.87704 q^{26} +18.1409 q^{28} +2.30634 q^{29} +6.00714 q^{31} +16.9023 q^{32} +7.58870 q^{34} +1.00000 q^{37} -15.3767 q^{38} -7.82452 q^{41} -11.6967 q^{43} +16.6151 q^{44} -3.84676 q^{46} +2.42814 q^{47} +5.13024 q^{49} +9.46137 q^{52} -9.32573 q^{53} +30.0042 q^{56} +6.19227 q^{58} -2.86539 q^{59} -4.67275 q^{61} +16.1285 q^{62} +19.9557 q^{64} -6.00752 q^{67} +14.7219 q^{68} -3.40566 q^{71} -1.89975 q^{73} +2.68489 q^{74} -29.8306 q^{76} +11.1100 q^{77} -1.60903 q^{79} -21.0080 q^{82} +7.37571 q^{83} -31.4043 q^{86} +27.4806 q^{88} +8.48746 q^{89} +6.32652 q^{91} -7.46264 q^{92} +6.51929 q^{94} -9.00003 q^{97} +13.7741 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{4} - 8 q^{7} + 15 q^{8} - 6 q^{13} + 4 q^{14} + 11 q^{16} + 18 q^{17} - 4 q^{19} - 6 q^{22} + 16 q^{23} + 6 q^{26} + 20 q^{28} + 2 q^{29} - 6 q^{31} + 35 q^{32} + 6 q^{34} + 9 q^{37}+ \cdots + 21 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^4 - 8 * q^7 + 15 * q^8 - 6 * q^13 + 4 * q^14 + 11 * q^16 + 18 * q^17 - 4 * q^19 - 6 * q^22 + 16 * q^23 + 6 * q^26 + 20 * q^28 + 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^34 + 9 * q^37 + 4 * q^38 - 2 * q^41 - 14 * q^43 - 4 * q^44 + 16 * q^46 + 34 * q^47 - 7 * q^49 - 16 * q^52 - 4 * q^53 + 24 * q^56 + 22 * q^58 - 10 * q^59 + 8 * q^61 + 26 * q^62 + 15 * q^64 - 16 * q^67 + 36 * q^68 - 8 * q^71 - 8 * q^73 + 5 * q^74 - 48 * q^76 + 14 * q^77 - 26 * q^79 + 4 * q^82 + 70 * q^83 + 16 * q^86 + 36 * q^88 + 8 * q^89 - 18 * q^91 + 16 * q^92 - 26 * q^94 + 2 * q^97 + 21 * 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 $

$2$	2.68489	1.89850	0.949252	−	0.314516i	$-0.101842\pi$
$2$	2.68489	1.89850	0.949252	+	0.314516i	$0.101842\pi$
$3$	0	0
$3$	0	0
$4$	5.20864	2.60432
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.48285	1.31639	0.658197	−	0.752846i	$-0.271320\pi$
$7$	3.48285	1.31639	0.658197	+	0.752846i	$0.271320\pi$
$8$	8.61484	3.04581
$9$	0	0
$10$	0	0
$11$	3.18991	0.961794	0.480897	−	0.876777i	$-0.340311\pi$
$11$	3.18991	0.961794	0.480897	+	0.876777i	$0.340311\pi$
$12$	0	0
$13$	1.81648	0.503800	0.251900	−	0.967753i	$-0.418945\pi$
$13$	1.81648	0.503800	0.251900	+	0.967753i	$0.418945\pi$
$14$	9.35107	2.49918
$15$	0	0
$16$	12.7126	3.17816
$17$	2.82645	0.685514	0.342757	−	0.939424i	$-0.388639\pi$
$17$	2.82645	0.685514	0.342757	+	0.939424i	$0.388639\pi$
$18$	0	0
$19$	−5.72714	−1.31389	−0.656947	−	0.753936i	$-0.728153\pi$
$19$	−5.72714	−1.31389	−0.656947	+	0.753936i	$0.728153\pi$
$20$	0	0
$21$	0	0
$22$	8.56456	1.82597
$23$	−1.43274	−0.298748	−0.149374	−	0.988781i	$-0.547726\pi$
$23$	−1.43274	−0.298748	−0.149374	+	0.988781i	$0.547726\pi$
$24$	0	0
$25$	0	0
$26$	4.87704	0.956467
$27$	0	0
$28$	18.1409	3.42831
$29$	2.30634	0.428276	0.214138	−	0.976803i	$-0.431306\pi$
$29$	2.30634	0.428276	0.214138	+	0.976803i	$0.431306\pi$
$30$	0	0
$31$	6.00714	1.07891	0.539457	−	0.842013i	$-0.318629\pi$
$31$	6.00714	1.07891	0.539457	+	0.842013i	$0.318629\pi$
$32$	16.9023	2.98794
$33$	0	0
$34$	7.58870	1.30145
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−15.3767	−2.49444
$39$	0	0
$40$	0	0
$41$	−7.82452	−1.22198	−0.610992	−	0.791636i	$-0.709229\pi$
$41$	−7.82452	−1.22198	−0.610992	+	0.791636i	$0.709229\pi$
$42$	0	0
$43$	−11.6967	−1.78373	−0.891863	−	0.452305i	$-0.850602\pi$
$43$	−11.6967	−1.78373	−0.891863	+	0.452305i	$0.850602\pi$
$44$	16.6151	2.50482
$45$	0	0
$46$	−3.84676	−0.567174
$47$	2.42814	0.354181	0.177090	−	0.984195i	$-0.443332\pi$
$47$	2.42814	0.354181	0.177090	+	0.984195i	$0.443332\pi$
$48$	0	0
$49$	5.13024	0.732892
$50$	0	0
$51$	0	0
$52$	9.46137	1.31206
$53$	−9.32573	−1.28099	−0.640493	−	0.767964i	$-0.721270\pi$
$53$	−9.32573	−1.28099	−0.640493	+	0.767964i	$0.721270\pi$
$54$	0	0
$55$	0	0
$56$	30.0042	4.00948
$57$	0	0
$58$	6.19227	0.813084
$59$	−2.86539	−0.373042	−0.186521	−	0.982451i	$-0.559721\pi$
$59$	−2.86539	−0.373042	−0.186521	+	0.982451i	$0.559721\pi$
$60$	0	0
$61$	−4.67275	−0.598285	−0.299142	−	0.954208i	$-0.596701\pi$
$61$	−4.67275	−0.598285	−0.299142	+	0.954208i	$0.596701\pi$
$62$	16.1285	2.04832
$63$	0	0
$64$	19.9557	2.49446
$65$	0	0
$66$	0	0
$67$	−6.00752	−0.733936	−0.366968	−	0.930234i	$-0.619604\pi$
$67$	−6.00752	−0.733936	−0.366968	+	0.930234i	$0.619604\pi$
$68$	14.7219	1.78530
$69$	0	0
$70$	0	0
$71$	−3.40566	−0.404177	−0.202089	−	0.979367i	$-0.564773\pi$
$71$	−3.40566	−0.404177	−0.202089	+	0.979367i	$0.564773\pi$
$72$	0	0
$73$	−1.89975	−0.222349	−0.111174	−	0.993801i	$-0.535461\pi$
$73$	−1.89975	−0.222349	−0.111174	+	0.993801i	$0.535461\pi$
$74$	2.68489	0.312112
$75$	0	0
$76$	−29.8306	−3.42180
$77$	11.1100	1.26610
$78$	0	0
$79$	−1.60903	−0.181031	−0.0905153	−	0.995895i	$-0.528851\pi$
$79$	−1.60903	−0.181031	−0.0905153	+	0.995895i	$0.528851\pi$
$80$	0	0
$81$	0	0
$82$	−21.0080	−2.31994
$83$	7.37571	0.809589	0.404795	−	0.914408i	$-0.367343\pi$
$83$	7.37571	0.809589	0.404795	+	0.914408i	$0.367343\pi$
$84$	0	0
$85$	0	0
$86$	−31.4043	−3.38641
$87$	0	0
$88$	27.4806	2.92944
$89$	8.48746	0.899669	0.449834	−	0.893112i	$-0.351483\pi$
$89$	8.48746	0.899669	0.449834	+	0.893112i	$0.351483\pi$
$90$	0	0
$91$	6.32652	0.663199
$92$	−7.46264	−0.778034
$93$	0	0
$94$	6.51929	0.672413
$95$	0	0
$96$	0	0
$97$	−9.00003	−0.913814	−0.456907	−	0.889514i	$-0.651043\pi$
$97$	−9.00003	−0.913814	−0.456907	+	0.889514i	$0.651043\pi$
$98$	13.7741	1.39140
$99$	0	0
$100$	0	0
$101$	12.9479	1.28836	0.644182	−	0.764872i	$-0.277198\pi$
$101$	12.9479	1.28836	0.644182	+	0.764872i	$0.277198\pi$
$102$	0	0
$103$	−7.46684	−0.735729	−0.367865	−	0.929879i	$-0.619911\pi$
$103$	−7.46684	−0.735729	−0.367865	+	0.929879i	$0.619911\pi$
$104$	15.6487	1.53448
$105$	0	0
$106$	−25.0386	−2.43196
$107$	11.2215	1.08482	0.542411	−	0.840113i	$-0.317511\pi$
$107$	11.2215	1.08482	0.542411	+	0.840113i	$0.317511\pi$
$108$	0	0
$109$	2.67262	0.255991	0.127996	−	0.991775i	$-0.459146\pi$
$109$	2.67262	0.255991	0.127996	+	0.991775i	$0.459146\pi$
$110$	0	0
$111$	0	0
$112$	44.2762	4.18371
$113$	−18.6735	−1.75666	−0.878329	−	0.478057i	$-0.841341\pi$
$113$	−18.6735	−1.75666	−0.878329	+	0.478057i	$0.841341\pi$
$114$	0	0
$115$	0	0
$116$	12.0129	1.11537
$117$	0	0
$118$	−7.69326	−0.708222
$119$	9.84409	0.902406
$120$	0	0
$121$	−0.824470	−0.0749518
$122$	−12.5458	−1.13585
$123$	0	0
$124$	31.2890	2.80984
$125$	0	0
$126$	0	0
$127$	−2.42257	−0.214968	−0.107484	−	0.994207i	$-0.534279\pi$
$127$	−2.42257	−0.214968	−0.107484	+	0.994207i	$0.534279\pi$
$128$	19.7741	1.74780
$129$	0	0
$130$	0	0
$131$	−11.2791	−0.985462	−0.492731	−	0.870182i	$-0.664001\pi$
$131$	−11.2791	−0.985462	−0.492731	+	0.870182i	$0.664001\pi$
$132$	0	0
$133$	−19.9467	−1.72960
$134$	−16.1295	−1.39338
$135$	0	0
$136$	24.3494	2.08794
$137$	−5.25383	−0.448865	−0.224433	−	0.974490i	$-0.572053\pi$
$137$	−5.25383	−0.448865	−0.224433	+	0.974490i	$0.572053\pi$
$138$	0	0
$139$	−3.02727	−0.256769	−0.128385	−	0.991724i	$-0.540979\pi$
$139$	−3.02727	−0.256769	−0.128385	+	0.991724i	$0.540979\pi$
$140$	0	0
$141$	0	0
$142$	−9.14382	−0.767332
$143$	5.79440	0.484552
$144$	0	0
$145$	0	0
$146$	−5.10061	−0.422130
$147$	0	0
$148$	5.20864	0.428147
$149$	−0.862201	−0.0706343	−0.0353171	−	0.999376i	$-0.511244\pi$
$149$	−0.862201	−0.0706343	−0.0353171	+	0.999376i	$0.511244\pi$
$150$	0	0
$151$	−1.86529	−0.151795	−0.0758976	−	0.997116i	$-0.524182\pi$
$151$	−1.86529	−0.151795	−0.0758976	+	0.997116i	$0.524182\pi$
$152$	−49.3384	−4.00187
$153$	0	0
$154$	29.8291	2.40370
$155$	0	0
$156$	0	0
$157$	19.7022	1.57241	0.786204	−	0.617968i	$-0.212044\pi$
$157$	19.7022	1.57241	0.786204	+	0.617968i	$0.212044\pi$
$158$	−4.32008	−0.343687
$159$	0	0
$160$	0	0
$161$	−4.99003	−0.393269
$162$	0	0
$163$	−21.8944	−1.71490	−0.857452	−	0.514564i	$-0.827954\pi$
$163$	−21.8944	−1.71490	−0.857452	+	0.514564i	$0.827954\pi$
$164$	−40.7551	−3.18244
$165$	0	0
$166$	19.8030	1.53701
$167$	17.7381	1.37262	0.686308	−	0.727311i	$-0.259230\pi$
$167$	17.7381	1.37262	0.686308	+	0.727311i	$0.259230\pi$
$168$	0	0
$169$	−9.70041	−0.746185
$170$	0	0
$171$	0	0
$172$	−60.9238	−4.64539
$173$	18.1222	1.37780	0.688901	−	0.724855i	$-0.258093\pi$
$173$	18.1222	1.37780	0.688901	+	0.724855i	$0.258093\pi$
$174$	0	0
$175$	0	0
$176$	40.5522	3.05673
$177$	0	0
$178$	22.7879	1.70802
$179$	18.6485	1.39385	0.696926	−	0.717143i	$-0.254551\pi$
$179$	18.6485	1.39385	0.696926	+	0.717143i	$0.254551\pi$
$180$	0	0
$181$	8.72883	0.648809	0.324404	−	0.945919i	$-0.394836\pi$
$181$	8.72883	0.648809	0.324404	+	0.945919i	$0.394836\pi$
$182$	16.9860	1.25909
$183$	0	0
$184$	−12.3429	−0.909928
$185$	0	0
$186$	0	0
$187$	9.01611	0.659324
$188$	12.6473	0.922399
$189$	0	0
$190$	0	0
$191$	24.5046	1.77309	0.886544	−	0.462644i	$-0.153099\pi$
$191$	24.5046	1.77309	0.886544	+	0.462644i	$0.153099\pi$
$192$	0	0
$193$	5.02850	0.361959	0.180980	−	0.983487i	$-0.442073\pi$
$193$	5.02850	0.361959	0.180980	+	0.983487i	$0.442073\pi$
$194$	−24.1641	−1.73488
$195$	0	0
$196$	26.7216	1.90868
$197$	5.67179	0.404098	0.202049	−	0.979375i	$-0.435240\pi$
$197$	5.67179	0.404098	0.202049	+	0.979375i	$0.435240\pi$
$198$	0	0
$199$	−9.87685	−0.700151	−0.350076	−	0.936721i	$-0.613844\pi$
$199$	−9.87685	−0.700151	−0.350076	+	0.936721i	$0.613844\pi$
$200$	0	0
$201$	0	0
$202$	34.7637	2.44597
$203$	8.03263	0.563780
$204$	0	0
$205$	0	0
$206$	−20.0476	−1.39679
$207$	0	0
$208$	23.0922	1.60116
$209$	−18.2690	−1.26370
$210$	0	0
$211$	9.45435	0.650864	0.325432	−	0.945565i	$-0.394490\pi$
$211$	9.45435	0.650864	0.325432	+	0.945565i	$0.394490\pi$
$212$	−48.5743	−3.33610
$213$	0	0
$214$	30.1285	2.05954
$215$	0	0
$216$	0	0
$217$	20.9220	1.42028
$218$	7.17570	0.486000
$219$	0	0
$220$	0	0
$221$	5.13418	0.345362
$222$	0	0
$223$	12.4583	0.834270	0.417135	−	0.908844i	$-0.363034\pi$
$223$	12.4583	0.834270	0.417135	+	0.908844i	$0.363034\pi$
$224$	58.8683	3.93331
$225$	0	0
$226$	−50.1364	−3.33502
$227$	21.2771	1.41221	0.706105	−	0.708107i	$-0.250451\pi$
$227$	21.2771	1.41221	0.706105	+	0.708107i	$0.250451\pi$
$228$	0	0
$229$	−15.8819	−1.04950	−0.524752	−	0.851255i	$-0.675842\pi$
$229$	−15.8819	−1.04950	−0.524752	+	0.851255i	$0.675842\pi$
$230$	0	0
$231$	0	0
$232$	19.8687	1.30445
$233$	−15.5581	−1.01924	−0.509621	−	0.860399i	$-0.670214\pi$
$233$	−15.5581	−1.01924	−0.509621	+	0.860399i	$0.670214\pi$
$234$	0	0
$235$	0	0
$236$	−14.9248	−0.971521
$237$	0	0
$238$	26.4303	1.71322
$239$	−3.01333	−0.194916	−0.0974581	−	0.995240i	$-0.531071\pi$
$239$	−3.01333	−0.194916	−0.0974581	+	0.995240i	$0.531071\pi$
$240$	0	0
$241$	−12.5699	−0.809698	−0.404849	−	0.914384i	$-0.632676\pi$
$241$	−12.5699	−0.809698	−0.404849	+	0.914384i	$0.632676\pi$
$242$	−2.21361	−0.142296
$243$	0	0
$244$	−24.3387	−1.55812
$245$	0	0
$246$	0	0
$247$	−10.4032	−0.661940
$248$	51.7506	3.28617
$249$	0	0
$250$	0	0
$251$	−5.84923	−0.369200	−0.184600	−	0.982814i	$-0.559099\pi$
$251$	−5.84923	−0.369200	−0.184600	+	0.982814i	$0.559099\pi$
$252$	0	0
$253$	−4.57032	−0.287334
$254$	−6.50432	−0.408117
$255$	0	0
$256$	13.1800	0.823752
$257$	11.7124	0.730597	0.365298	−	0.930891i	$-0.380967\pi$
$257$	11.7124	0.730597	0.365298	+	0.930891i	$0.380967\pi$
$258$	0	0
$259$	3.48285	0.216414
$260$	0	0
$261$	0	0
$262$	−30.2832	−1.87090
$263$	−0.696631	−0.0429561	−0.0214780	−	0.999769i	$-0.506837\pi$
$263$	−0.696631	−0.0429561	−0.0214780	+	0.999769i	$0.506837\pi$
$264$	0	0
$265$	0	0
$266$	−53.5548	−3.28366
$267$	0	0
$268$	−31.2910	−1.91140
$269$	−19.4644	−1.18677	−0.593383	−	0.804921i	$-0.702208\pi$
$269$	−19.4644	−1.18677	−0.593383	+	0.804921i	$0.702208\pi$
$270$	0	0
$271$	−20.7858	−1.26265	−0.631325	−	0.775518i	$-0.717489\pi$
$271$	−20.7858	−1.26265	−0.631325	+	0.775518i	$0.717489\pi$
$272$	35.9316	2.17867
$273$	0	0
$274$	−14.1060	−0.852173
$275$	0	0
$276$	0	0
$277$	−6.29904	−0.378472	−0.189236	−	0.981932i	$-0.560601\pi$
$277$	−6.29904	−0.378472	−0.189236	+	0.981932i	$0.560601\pi$
$278$	−8.12788	−0.487478
$279$	0	0
$280$	0	0
$281$	1.46824	0.0875878	0.0437939	−	0.999041i	$-0.486056\pi$
$281$	1.46824	0.0875878	0.0437939	+	0.999041i	$0.486056\pi$
$282$	0	0
$283$	0.325582	0.0193538	0.00967692	−	0.999953i	$-0.496920\pi$
$283$	0.325582	0.0193538	0.00967692	+	0.999953i	$0.496920\pi$
$284$	−17.7388	−1.05261
$285$	0	0
$286$	15.5573	0.919924
$287$	−27.2516	−1.60861
$288$	0	0
$289$	−9.01119	−0.530070
$290$	0	0
$291$	0	0
$292$	−9.89509	−0.579067
$293$	1.23431	0.0721090	0.0360545	−	0.999350i	$-0.488521\pi$
$293$	1.23431	0.0721090	0.0360545	+	0.999350i	$0.488521\pi$
$294$	0	0
$295$	0	0
$296$	8.61484	0.500728
$297$	0	0
$298$	−2.31492	−0.134100
$299$	−2.60255	−0.150509
$300$	0	0
$301$	−40.7378	−2.34809
$302$	−5.00810	−0.288184
$303$	0	0
$304$	−72.8070	−4.17577
$305$	0	0
$306$	0	0
$307$	−11.0805	−0.632397	−0.316198	−	0.948693i	$-0.602406\pi$
$307$	−11.0805	−0.632397	−0.316198	+	0.948693i	$0.602406\pi$
$308$	57.8679	3.29733
$309$	0	0
$310$	0	0
$311$	33.7268	1.91247	0.956236	−	0.292597i	$-0.0945196\pi$
$311$	33.7268	1.91247	0.956236	+	0.292597i	$0.0945196\pi$
$312$	0	0
$313$	31.0364	1.75428	0.877140	−	0.480235i	$-0.159448\pi$
$313$	31.0364	1.75428	0.877140	+	0.480235i	$0.159448\pi$
$314$	52.8983	2.98522
$315$	0	0
$316$	−8.38088	−0.471461
$317$	9.28000	0.521217	0.260608	−	0.965445i	$-0.416077\pi$
$317$	9.28000	0.521217	0.260608	+	0.965445i	$0.416077\pi$
$318$	0	0
$319$	7.35701	0.411914
$320$	0	0
$321$	0	0
$322$	−13.3977	−0.746624
$323$	−16.1874	−0.900694
$324$	0	0
$325$	0	0
$326$	−58.7842	−3.25575
$327$	0	0
$328$	−67.4070	−3.72193
$329$	8.45684	0.466241
$330$	0	0
$331$	−10.7993	−0.593584	−0.296792	−	0.954942i	$-0.595917\pi$
$331$	−10.7993	−0.593584	−0.296792	+	0.954942i	$0.595917\pi$
$332$	38.4174	2.10843
$333$	0	0
$334$	47.6249	2.60592
$335$	0	0
$336$	0	0
$337$	−3.06665	−0.167051	−0.0835256	−	0.996506i	$-0.526618\pi$
$337$	−3.06665	−0.167051	−0.0835256	+	0.996506i	$0.526618\pi$
$338$	−26.0445	−1.41664
$339$	0	0
$340$	0	0
$341$	19.1622	1.03769
$342$	0	0
$343$	−6.51209	−0.351620
$344$	−100.765	−5.43289
$345$	0	0
$346$	48.6560	2.61576
$347$	−26.0726	−1.39965	−0.699825	−	0.714315i	$-0.746739\pi$
$347$	−26.0726	−1.39965	−0.699825	+	0.714315i	$0.746739\pi$
$348$	0	0
$349$	12.2055	0.653346	0.326673	−	0.945137i	$-0.394072\pi$
$349$	12.2055	0.653346	0.326673	+	0.945137i	$0.394072\pi$
$350$	0	0
$351$	0	0
$352$	53.9170	2.87378
$353$	11.8648	0.631498	0.315749	−	0.948843i	$-0.397744\pi$
$353$	11.8648	0.631498	0.315749	+	0.948843i	$0.397744\pi$
$354$	0	0
$355$	0	0
$356$	44.2081	2.34302
$357$	0	0
$358$	50.0691	2.64623
$359$	−7.36628	−0.388777	−0.194389	−	0.980925i	$-0.562272\pi$
$359$	−7.36628	−0.388777	−0.194389	+	0.980925i	$0.562272\pi$
$360$	0	0
$361$	13.8001	0.726320
$362$	23.4359	1.23177
$363$	0	0
$364$	32.9525	1.72718
$365$	0	0
$366$	0	0
$367$	−20.4826	−1.06918	−0.534590	−	0.845111i	$-0.679534\pi$
$367$	−20.4826	−1.06918	−0.534590	+	0.845111i	$0.679534\pi$
$368$	−18.2139	−0.949467
$369$	0	0
$370$	0	0
$371$	−32.4801	−1.68628
$372$	0	0
$373$	−16.5915	−0.859075	−0.429538	−	0.903049i	$-0.641323\pi$
$373$	−16.5915	−0.859075	−0.429538	+	0.903049i	$0.641323\pi$
$374$	24.2073	1.25173
$375$	0	0
$376$	20.9180	1.07877
$377$	4.18941	0.215766
$378$	0	0
$379$	−15.9199	−0.817752	−0.408876	−	0.912590i	$-0.634079\pi$
$379$	−15.9199	−0.817752	−0.408876	+	0.912590i	$0.634079\pi$
$380$	0	0
$381$	0	0
$382$	65.7921	3.36622
$383$	30.4997	1.55846	0.779230	−	0.626738i	$-0.215610\pi$
$383$	30.4997	1.55846	0.779230	+	0.626738i	$0.215610\pi$
$384$	0	0
$385$	0	0
$386$	13.5010	0.687181
$387$	0	0
$388$	−46.8779	−2.37986
$389$	29.8769	1.51482	0.757409	−	0.652941i	$-0.226465\pi$
$389$	29.8769	1.51482	0.757409	+	0.652941i	$0.226465\pi$
$390$	0	0
$391$	−4.04957	−0.204796
$392$	44.1962	2.23225
$393$	0	0
$394$	15.2281	0.767182
$395$	0	0
$396$	0	0
$397$	−0.946163	−0.0474866	−0.0237433	−	0.999718i	$-0.507558\pi$
$397$	−0.946163	−0.0474866	−0.0237433	+	0.999718i	$0.507558\pi$
$398$	−26.5183	−1.32924
$399$	0	0
$400$	0	0
$401$	−37.3648	−1.86591	−0.932955	−	0.359994i	$-0.882779\pi$
$401$	−37.3648	−1.86591	−0.932955	+	0.359994i	$0.882779\pi$
$402$	0	0
$403$	10.9118	0.543557
$404$	67.4409	3.35531
$405$	0	0
$406$	21.5667	1.07034
$407$	3.18991	0.158118
$408$	0	0
$409$	22.9171	1.13318	0.566589	−	0.824000i	$-0.308263\pi$
$409$	22.9171	1.13318	0.566589	+	0.824000i	$0.308263\pi$
$410$	0	0
$411$	0	0
$412$	−38.8920	−1.91607
$413$	−9.97973	−0.491070
$414$	0	0
$415$	0	0
$416$	30.7027	1.50533
$417$	0	0
$418$	−49.0504	−2.39913
$419$	−18.8458	−0.920678	−0.460339	−	0.887743i	$-0.652272\pi$
$419$	−18.8458	−0.920678	−0.460339	+	0.887743i	$0.652272\pi$
$420$	0	0
$421$	14.7293	0.717861	0.358931	−	0.933364i	$-0.383141\pi$
$421$	14.7293	0.717861	0.358931	+	0.933364i	$0.383141\pi$
$422$	25.3839	1.23567
$423$	0	0
$424$	−80.3397	−3.90164
$425$	0	0
$426$	0	0
$427$	−16.2745	−0.787578
$428$	58.4487	2.82523
$429$	0	0
$430$	0	0
$431$	24.7947	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$431$	24.7947	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$432$	0	0
$433$	26.6307	1.27979	0.639894	−	0.768463i	$-0.278978\pi$
$433$	26.6307	1.27979	0.639894	+	0.768463i	$0.278978\pi$
$434$	56.1732	2.69640
$435$	0	0
$436$	13.9207	0.666682
$437$	8.20552	0.392523
$438$	0	0
$439$	2.93896	0.140269	0.0701344	−	0.997538i	$-0.477657\pi$
$439$	2.93896	0.140269	0.0701344	+	0.997538i	$0.477657\pi$
$440$	0	0
$441$	0	0
$442$	13.7847	0.655672
$443$	12.1586	0.577671	0.288836	−	0.957379i	$-0.406732\pi$
$443$	12.1586	0.577671	0.288836	+	0.957379i	$0.406732\pi$
$444$	0	0
$445$	0	0
$446$	33.4492	1.58387
$447$	0	0
$448$	69.5027	3.28369
$449$	12.0740	0.569809	0.284904	−	0.958556i	$-0.408038\pi$
$449$	12.0740	0.569809	0.284904	+	0.958556i	$0.408038\pi$
$450$	0	0
$451$	−24.9595	−1.17530
$452$	−97.2637	−4.57490
$453$	0	0
$454$	57.1266	2.68109
$455$	0	0
$456$	0	0
$457$	−14.3829	−0.672803	−0.336401	−	0.941719i	$-0.609210\pi$
$457$	−14.3829	−0.672803	−0.336401	+	0.941719i	$0.609210\pi$
$458$	−42.6411	−1.99249
$459$	0	0
$460$	0	0
$461$	16.2105	0.754997	0.377498	−	0.926010i	$-0.376784\pi$
$461$	16.2105	0.754997	0.377498	+	0.926010i	$0.376784\pi$
$462$	0	0
$463$	26.8128	1.24609	0.623047	−	0.782184i	$-0.285894\pi$
$463$	26.8128	1.24609	0.623047	+	0.782184i	$0.285894\pi$
$464$	29.3196	1.36113
$465$	0	0
$466$	−41.7717	−1.93504
$467$	8.58230	0.397141	0.198571	−	0.980087i	$-0.436370\pi$
$467$	8.58230	0.397141	0.198571	+	0.980087i	$0.436370\pi$
$468$	0	0
$469$	−20.9233	−0.966148
$470$	0	0
$471$	0	0
$472$	−24.6849	−1.13621
$473$	−37.3114	−1.71558
$474$	0	0
$475$	0	0
$476$	51.2743	2.35015
$477$	0	0
$478$	−8.09047	−0.370049
$479$	−2.01066	−0.0918693	−0.0459346	−	0.998944i	$-0.514627\pi$
$479$	−2.01066	−0.0918693	−0.0459346	+	0.998944i	$0.514627\pi$
$480$	0	0
$481$	1.81648	0.0828242
$482$	−33.7488	−1.53722
$483$	0	0
$484$	−4.29437	−0.195199
$485$	0	0
$486$	0	0
$487$	38.1485	1.72867	0.864336	−	0.502915i	$-0.167739\pi$
$487$	38.1485	1.72867	0.864336	+	0.502915i	$0.167739\pi$
$488$	−40.2550	−1.82226
$489$	0	0
$490$	0	0
$491$	−21.6136	−0.975409	−0.487705	−	0.873009i	$-0.662166\pi$
$491$	−21.6136	−0.975409	−0.487705	+	0.873009i	$0.662166\pi$
$492$	0	0
$493$	6.51874	0.293589
$494$	−27.9315	−1.25670
$495$	0	0
$496$	76.3666	3.42896
$497$	−11.8614	−0.532056
$498$	0	0
$499$	37.8811	1.69579	0.847896	−	0.530162i	$-0.177869\pi$
$499$	37.8811	1.69579	0.847896	+	0.530162i	$0.177869\pi$
$500$	0	0
$501$	0	0
$502$	−15.7045	−0.700928
$503$	−0.494731	−0.0220590	−0.0110295	−	0.999939i	$-0.503511\pi$
$503$	−0.494731	−0.0220590	−0.0110295	+	0.999939i	$0.503511\pi$
$504$	0	0
$505$	0	0
$506$	−12.2708	−0.545504
$507$	0	0
$508$	−12.6183	−0.559845
$509$	2.62220	0.116227	0.0581136	−	0.998310i	$-0.481491\pi$
$509$	2.62220	0.116227	0.0581136	+	0.998310i	$0.481491\pi$
$510$	0	0
$511$	−6.61653	−0.292698
$512$	−4.16132	−0.183906
$513$	0	0
$514$	31.4464	1.38704
$515$	0	0
$516$	0	0
$517$	7.74555	0.340649
$518$	9.35107	0.410862
$519$	0	0
$520$	0	0
$521$	−5.05777	−0.221585	−0.110792	−	0.993844i	$-0.535339\pi$
$521$	−5.05777	−0.221585	−0.110792	+	0.993844i	$0.535339\pi$
$522$	0	0
$523$	−18.1730	−0.794648	−0.397324	−	0.917678i	$-0.630061\pi$
$523$	−18.1730	−0.794648	−0.397324	+	0.917678i	$0.630061\pi$
$524$	−58.7489	−2.56646
$525$	0	0
$526$	−1.87038	−0.0815523
$527$	16.9789	0.739611
$528$	0	0
$529$	−20.9472	−0.910750
$530$	0	0
$531$	0	0
$532$	−103.895	−4.50444
$533$	−14.2131	−0.615636
$534$	0	0
$535$	0	0
$536$	−51.7538	−2.23543
$537$	0	0
$538$	−52.2598	−2.25308
$539$	16.3650	0.704891
$540$	0	0
$541$	6.96392	0.299402	0.149701	−	0.988731i	$-0.452169\pi$
$541$	6.96392	0.299402	0.149701	+	0.988731i	$0.452169\pi$
$542$	−55.8077	−2.39715
$543$	0	0
$544$	47.7736	2.04828
$545$	0	0
$546$	0	0
$547$	28.2295	1.20701	0.603504	−	0.797360i	$-0.293771\pi$
$547$	28.2295	1.20701	0.603504	+	0.797360i	$0.293771\pi$
$548$	−27.3653	−1.16899
$549$	0	0
$550$	0	0
$551$	−13.2087	−0.562710
$552$	0	0
$553$	−5.60402	−0.238307
$554$	−16.9122	−0.718531
$555$	0	0
$556$	−15.7679	−0.668710
$557$	−20.4748	−0.867545	−0.433773	−	0.901022i	$-0.642818\pi$
$557$	−20.4748	−0.867545	−0.433773	+	0.901022i	$0.642818\pi$
$558$	0	0
$559$	−21.2467	−0.898642
$560$	0	0
$561$	0	0
$562$	3.94206	0.166286
$563$	−4.03262	−0.169955	−0.0849774	−	0.996383i	$-0.527082\pi$
$563$	−4.03262	−0.169955	−0.0849774	+	0.996383i	$0.527082\pi$
$564$	0	0
$565$	0	0
$566$	0.874152	0.0367433
$567$	0	0
$568$	−29.3392	−1.23105
$569$	−24.6352	−1.03276	−0.516381	−	0.856359i	$-0.672721\pi$
$569$	−24.6352	−1.03276	−0.516381	+	0.856359i	$0.672721\pi$
$570$	0	0
$571$	31.7960	1.33062	0.665311	−	0.746566i	$-0.268299\pi$
$571$	31.7960	1.33062	0.665311	+	0.746566i	$0.268299\pi$
$572$	30.1809	1.26193
$573$	0	0
$574$	−73.1676	−3.05396
$575$	0	0
$576$	0	0
$577$	−39.0173	−1.62431	−0.812155	−	0.583442i	$-0.801706\pi$
$577$	−39.0173	−1.62431	−0.812155	+	0.583442i	$0.801706\pi$
$578$	−24.1941	−1.00634
$579$	0	0
$580$	0	0
$581$	25.6885	1.06574
$582$	0	0
$583$	−29.7482	−1.23205
$584$	−16.3660	−0.677231
$585$	0	0
$586$	3.31398	0.136899
$587$	28.0766	1.15884	0.579422	−	0.815028i	$-0.303278\pi$
$587$	28.0766	1.15884	0.579422	+	0.815028i	$0.303278\pi$
$588$	0	0
$589$	−34.4037	−1.41758
$590$	0	0
$591$	0	0
$592$	12.7126	0.522486
$593$	21.1330	0.867829	0.433915	−	0.900954i	$-0.357132\pi$
$593$	21.1330	0.867829	0.433915	+	0.900954i	$0.357132\pi$
$594$	0	0
$595$	0	0
$596$	−4.49090	−0.183954
$597$	0	0
$598$	−6.98755	−0.285742
$599$	−14.1739	−0.579129	−0.289564	−	0.957159i	$-0.593510\pi$
$599$	−14.1739	−0.579129	−0.289564	+	0.957159i	$0.593510\pi$
$600$	0	0
$601$	48.8255	1.99164	0.995818	−	0.0913627i	$-0.0291223\pi$
$601$	48.8255	1.99164	0.995818	+	0.0913627i	$0.0291223\pi$
$602$	−109.376	−4.45785
$603$	0	0
$604$	−9.71563	−0.395323
$605$	0	0
$606$	0	0
$607$	−8.91773	−0.361959	−0.180980	−	0.983487i	$-0.557927\pi$
$607$	−8.91773	−0.361959	−0.180980	+	0.983487i	$0.557927\pi$
$608$	−96.8020	−3.92584
$609$	0	0
$610$	0	0
$611$	4.41066	0.178436
$612$	0	0
$613$	43.5970	1.76087	0.880433	−	0.474171i	$-0.157252\pi$
$613$	43.5970	1.76087	0.880433	+	0.474171i	$0.157252\pi$
$614$	−29.7499	−1.20061
$615$	0	0
$616$	95.7107	3.85629
$617$	−25.4049	−1.02276	−0.511382	−	0.859353i	$-0.670866\pi$
$617$	−25.4049	−1.02276	−0.511382	+	0.859353i	$0.670866\pi$
$618$	0	0
$619$	−4.97246	−0.199860	−0.0999299	−	0.994994i	$-0.531862\pi$
$619$	−4.97246	−0.199860	−0.0999299	+	0.994994i	$0.531862\pi$
$620$	0	0
$621$	0	0
$622$	90.5528	3.63084
$623$	29.5605	1.18432
$624$	0	0
$625$	0	0
$626$	83.3293	3.33051
$627$	0	0
$628$	102.622	4.09505
$629$	2.82645	0.112698
$630$	0	0
$631$	−0.771437	−0.0307104	−0.0153552	−	0.999882i	$-0.504888\pi$
$631$	−0.771437	−0.0307104	−0.0153552	+	0.999882i	$0.504888\pi$
$632$	−13.8616	−0.551384
$633$	0	0
$634$	24.9158	0.989532
$635$	0	0
$636$	0	0
$637$	9.31897	0.369231
$638$	19.7528	0.782020
$639$	0	0
$640$	0	0
$641$	−41.6238	−1.64404	−0.822020	−	0.569459i	$-0.807153\pi$
$641$	−41.6238	−1.64404	−0.822020	+	0.569459i	$0.807153\pi$
$642$	0	0
$643$	7.31649	0.288534	0.144267	−	0.989539i	$-0.453918\pi$
$643$	7.31649	0.288534	0.144267	+	0.989539i	$0.453918\pi$
$644$	−25.9913	−1.02420
$645$	0	0
$646$	−43.4615	−1.70997
$647$	−43.4618	−1.70866	−0.854329	−	0.519733i	$-0.826032\pi$
$647$	−43.4618	−1.70866	−0.854329	+	0.519733i	$0.826032\pi$
$648$	0	0
$649$	−9.14034	−0.358790
$650$	0	0
$651$	0	0
$652$	−114.040	−4.46616
$653$	22.9698	0.898878	0.449439	−	0.893311i	$-0.351624\pi$
$653$	22.9698	0.898878	0.449439	+	0.893311i	$0.351624\pi$
$654$	0	0
$655$	0	0
$656$	−99.4703	−3.88366
$657$	0	0
$658$	22.7057	0.885160
$659$	−35.7491	−1.39259	−0.696294	−	0.717757i	$-0.745169\pi$
$659$	−35.7491	−1.39259	−0.696294	+	0.717757i	$0.745169\pi$
$660$	0	0
$661$	27.3305	1.06303	0.531516	−	0.847048i	$-0.321622\pi$
$661$	27.3305	1.06303	0.531516	+	0.847048i	$0.321622\pi$
$662$	−28.9950	−1.12692
$663$	0	0
$664$	63.5406	2.46585
$665$	0	0
$666$	0	0
$667$	−3.30439	−0.127947
$668$	92.3913	3.57473
$669$	0	0
$670$	0	0
$671$	−14.9057	−0.575427
$672$	0	0
$673$	−19.6281	−0.756609	−0.378305	−	0.925681i	$-0.623493\pi$
$673$	−19.6281	−0.756609	−0.378305	+	0.925681i	$0.623493\pi$
$674$	−8.23363	−0.317147
$675$	0	0
$676$	−50.5259	−1.94330
$677$	3.03052	0.116472	0.0582362	−	0.998303i	$-0.481452\pi$
$677$	3.03052	0.116472	0.0582362	+	0.998303i	$0.481452\pi$
$678$	0	0
$679$	−31.3457	−1.20294
$680$	0	0
$681$	0	0
$682$	51.4485	1.97007
$683$	44.7410	1.71197	0.855983	−	0.517004i	$-0.172953\pi$
$683$	44.7410	1.71197	0.855983	+	0.517004i	$0.172953\pi$
$684$	0	0
$685$	0	0
$686$	−17.4843	−0.667552
$687$	0	0
$688$	−148.696	−5.66897
$689$	−16.9400	−0.645361
$690$	0	0
$691$	−13.7681	−0.523764	−0.261882	−	0.965100i	$-0.584343\pi$
$691$	−13.7681	−0.523764	−0.261882	+	0.965100i	$0.584343\pi$
$692$	94.3917	3.58824
$693$	0	0
$694$	−70.0020	−2.65724
$695$	0	0
$696$	0	0
$697$	−22.1156	−0.837688
$698$	32.7705	1.24038
$699$	0	0
$700$	0	0
$701$	12.2193	0.461517	0.230758	−	0.973011i	$-0.425879\pi$
$701$	12.2193	0.461517	0.230758	+	0.973011i	$0.425879\pi$
$702$	0	0
$703$	−5.72714	−0.216003
$704$	63.6569	2.39916
$705$	0	0
$706$	31.8556	1.19890
$707$	45.0956	1.69599
$708$	0	0
$709$	−4.22619	−0.158718	−0.0793589	−	0.996846i	$-0.525287\pi$
$709$	−4.22619	−0.158718	−0.0793589	+	0.996846i	$0.525287\pi$
$710$	0	0
$711$	0	0
$712$	73.1181	2.74022
$713$	−8.60669	−0.322323
$714$	0	0
$715$	0	0
$716$	97.1331	3.63003
$717$	0	0
$718$	−19.7777	−0.738095
$719$	−19.2764	−0.718888	−0.359444	−	0.933167i	$-0.617034\pi$
$719$	−19.2764	−0.718888	−0.359444	+	0.933167i	$0.617034\pi$
$720$	0	0
$721$	−26.0059	−0.968509
$722$	37.0517	1.37892
$723$	0	0
$724$	45.4653	1.68970
$725$	0	0
$726$	0	0
$727$	−16.6423	−0.617227	−0.308614	−	0.951187i	$-0.599865\pi$
$727$	−16.6423	−0.617227	−0.308614	+	0.951187i	$0.599865\pi$
$728$	54.5019	2.01998
$729$	0	0
$730$	0	0
$731$	−33.0600	−1.22277
$732$	0	0
$733$	−1.77985	−0.0657404	−0.0328702	−	0.999460i	$-0.510465\pi$
$733$	−1.77985	−0.0657404	−0.0328702	+	0.999460i	$0.510465\pi$
$734$	−54.9934	−2.02984
$735$	0	0
$736$	−24.2167	−0.892640
$737$	−19.1635	−0.705895
$738$	0	0
$739$	32.6285	1.20026	0.600130	−	0.799903i	$-0.295116\pi$
$739$	32.6285	1.20026	0.600130	+	0.799903i	$0.295116\pi$
$740$	0	0
$741$	0	0
$742$	−87.2055	−3.20142
$743$	−21.1995	−0.777734	−0.388867	−	0.921294i	$-0.627134\pi$
$743$	−21.1995	−0.777734	−0.388867	+	0.921294i	$0.627134\pi$
$744$	0	0
$745$	0	0
$746$	−44.5464	−1.63096
$747$	0	0
$748$	46.9617	1.71709
$749$	39.0828	1.42805
$750$	0	0
$751$	−53.3277	−1.94595	−0.972977	−	0.230901i	$-0.925833\pi$
$751$	−53.3277	−1.94595	−0.972977	+	0.230901i	$0.925833\pi$
$752$	30.8680	1.12564
$753$	0	0
$754$	11.2481	0.409632
$755$	0	0
$756$	0	0
$757$	26.9137	0.978197	0.489098	−	0.872229i	$-0.337326\pi$
$757$	26.9137	0.978197	0.489098	+	0.872229i	$0.337326\pi$
$758$	−42.7433	−1.55251
$759$	0	0
$760$	0	0
$761$	19.7208	0.714879	0.357439	−	0.933936i	$-0.383650\pi$
$761$	19.7208	0.714879	0.357439	+	0.933936i	$0.383650\pi$
$762$	0	0
$763$	9.30835	0.336985
$764$	127.635	4.61769
$765$	0	0
$766$	81.8883	2.95874
$767$	−5.20492	−0.187939
$768$	0	0
$769$	−40.4230	−1.45769	−0.728846	−	0.684678i	$-0.759943\pi$
$769$	−40.4230	−1.45769	−0.728846	+	0.684678i	$0.759943\pi$
$770$	0	0
$771$	0	0
$772$	26.1916	0.942657
$773$	−14.1890	−0.510342	−0.255171	−	0.966896i	$-0.582132\pi$
$773$	−14.1890	−0.510342	−0.255171	+	0.966896i	$0.582132\pi$
$774$	0	0
$775$	0	0
$776$	−77.5338	−2.78330
$777$	0	0
$778$	80.2161	2.87589
$779$	44.8121	1.60556
$780$	0	0
$781$	−10.8637	−0.388735
$782$	−10.8727	−0.388806
$783$	0	0
$784$	65.2189	2.32925
$785$	0	0
$786$	0	0
$787$	−48.0720	−1.71358	−0.856790	−	0.515665i	$-0.827545\pi$
$787$	−48.0720	−1.71358	−0.856790	+	0.515665i	$0.827545\pi$
$788$	29.5423	1.05240
$789$	0	0
$790$	0	0
$791$	−65.0371	−2.31245
$792$	0	0
$793$	−8.48795	−0.301416
$794$	−2.54034	−0.0901535
$795$	0	0
$796$	−51.4449	−1.82342
$797$	35.1334	1.24449	0.622245	−	0.782822i	$-0.286221\pi$
$797$	35.1334	1.24449	0.622245	+	0.782822i	$0.286221\pi$
$798$	0	0
$799$	6.86301	0.242796
$800$	0	0
$801$	0	0
$802$	−100.320	−3.54244
$803$	−6.06002	−0.213854
$804$	0	0
$805$	0	0
$806$	29.2971	1.03195
$807$	0	0
$808$	111.544	3.92411
$809$	18.5994	0.653922	0.326961	−	0.945038i	$-0.393975\pi$
$809$	18.5994	0.653922	0.326961	+	0.945038i	$0.393975\pi$
$810$	0	0
$811$	−41.2856	−1.44973	−0.724867	−	0.688889i	$-0.758099\pi$
$811$	−41.2856	−1.44973	−0.724867	+	0.688889i	$0.758099\pi$
$812$	41.8391	1.46826
$813$	0	0
$814$	8.56456	0.300188
$815$	0	0
$816$	0	0
$817$	66.9884	2.34363
$818$	61.5300	2.15134
$819$	0	0
$820$	0	0
$821$	−5.64400	−0.196977	−0.0984885	−	0.995138i	$-0.531401\pi$
$821$	−5.64400	−0.196977	−0.0984885	+	0.995138i	$0.531401\pi$
$822$	0	0
$823$	52.1526	1.81792	0.908962	−	0.416879i	$-0.136876\pi$
$823$	52.1526	1.81792	0.908962	+	0.416879i	$0.136876\pi$
$824$	−64.3256	−2.24089
$825$	0	0
$826$	−26.7945	−0.932299
$827$	−10.2565	−0.356654	−0.178327	−	0.983971i	$-0.557069\pi$
$827$	−10.2565	−0.356654	−0.178327	+	0.983971i	$0.557069\pi$
$828$	0	0
$829$	−10.0803	−0.350105	−0.175052	−	0.984559i	$-0.556010\pi$
$829$	−10.0803	−0.350105	−0.175052	+	0.984559i	$0.556010\pi$
$830$	0	0
$831$	0	0
$832$	36.2491	1.25671
$833$	14.5004	0.502408
$834$	0	0
$835$	0	0
$836$	−95.1569	−3.29107
$837$	0	0
$838$	−50.5990	−1.74791
$839$	−42.0725	−1.45250	−0.726252	−	0.687428i	$-0.758740\pi$
$839$	−42.0725	−1.45250	−0.726252	+	0.687428i	$0.758740\pi$
$840$	0	0
$841$	−23.6808	−0.816579
$842$	39.5465	1.36286
$843$	0	0
$844$	49.2443	1.69506
$845$	0	0
$846$	0	0
$847$	−2.87151	−0.0986661
$848$	−118.555	−4.07118
$849$	0	0
$850$	0	0
$851$	−1.43274	−0.0491138
$852$	0	0
$853$	13.3261	0.456277	0.228138	−	0.973629i	$-0.426736\pi$
$853$	13.3261	0.456277	0.228138	+	0.973629i	$0.426736\pi$
$854$	−43.6952	−1.49522
$855$	0	0
$856$	96.6714	3.30416
$857$	−32.2245	−1.10077	−0.550383	−	0.834912i	$-0.685518\pi$
$857$	−32.2245	−1.10077	−0.550383	+	0.834912i	$0.685518\pi$
$858$	0	0
$859$	−28.0054	−0.955530	−0.477765	−	0.878488i	$-0.658553\pi$
$859$	−28.0054	−0.955530	−0.477765	+	0.878488i	$0.658553\pi$
$860$	0	0
$861$	0	0
$862$	66.5711	2.26742
$863$	−47.0680	−1.60221	−0.801107	−	0.598521i	$-0.795755\pi$
$863$	−47.0680	−1.60221	−0.801107	+	0.598521i	$0.795755\pi$
$864$	0	0
$865$	0	0
$866$	71.5005	2.42968
$867$	0	0
$868$	108.975	3.69885
$869$	−5.13268	−0.174114
$870$	0	0
$871$	−10.9125	−0.369757
$872$	23.0242	0.779699
$873$	0	0
$874$	22.0309	0.745207
$875$	0	0
$876$	0	0
$877$	−16.9389	−0.571986	−0.285993	−	0.958232i	$-0.592323\pi$
$877$	−16.9389	−0.571986	−0.285993	+	0.958232i	$0.592323\pi$
$878$	7.89078	0.266301
$879$	0	0
$880$	0	0
$881$	−34.0552	−1.14735	−0.573674	−	0.819084i	$-0.694482\pi$
$881$	−34.0552	−1.14735	−0.573674	+	0.819084i	$0.694482\pi$
$882$	0	0
$883$	7.85431	0.264319	0.132159	−	0.991228i	$-0.457809\pi$
$883$	7.85431	0.264319	0.132159	+	0.991228i	$0.457809\pi$
$884$	26.7421	0.899433
$885$	0	0
$886$	32.6444	1.09671
$887$	50.8348	1.70687	0.853433	−	0.521203i	$-0.174517\pi$
$887$	50.8348	1.70687	0.853433	+	0.521203i	$0.174517\pi$
$888$	0	0
$889$	−8.43743	−0.282982
$890$	0	0
$891$	0	0
$892$	64.8908	2.17271
$893$	−13.9063	−0.465356
$894$	0	0
$895$	0	0
$896$	68.8704	2.30080
$897$	0	0
$898$	32.4174	1.08178
$899$	13.8545	0.462074
$900$	0	0
$901$	−26.3587	−0.878135
$902$	−67.0136	−2.23131
$903$	0	0
$904$	−160.870	−5.35044
$905$	0	0
$906$	0	0
$907$	55.4966	1.84273	0.921366	−	0.388695i	$-0.127074\pi$
$907$	55.4966	1.84273	0.921366	+	0.388695i	$0.127074\pi$
$908$	110.825	3.67784
$909$	0	0
$910$	0	0
$911$	−35.8745	−1.18858	−0.594288	−	0.804252i	$-0.702566\pi$
$911$	−35.8745	−1.18858	−0.594288	+	0.804252i	$0.702566\pi$
$912$	0	0
$913$	23.5278	0.778658
$914$	−38.6165	−1.27732
$915$	0	0
$916$	−82.7230	−2.73324
$917$	−39.2835	−1.29726
$918$	0	0
$919$	−43.0966	−1.42162	−0.710812	−	0.703382i	$-0.751672\pi$
$919$	−43.0966	−1.42162	−0.710812	+	0.703382i	$0.751672\pi$
$920$	0	0
$921$	0	0
$922$	43.5233	1.43336
$923$	−6.18630	−0.203625
$924$	0	0
$925$	0	0
$926$	71.9893	2.36572
$927$	0	0
$928$	38.9825	1.27966
$929$	−41.5247	−1.36238	−0.681190	−	0.732107i	$-0.738537\pi$
$929$	−41.5247	−1.36238	−0.681190	+	0.732107i	$0.738537\pi$
$930$	0	0
$931$	−29.3816	−0.962942
$932$	−81.0363	−2.65443
$933$	0	0
$934$	23.0425	0.753975
$935$	0	0
$936$	0	0
$937$	18.2330	0.595647	0.297823	−	0.954621i	$-0.403739\pi$
$937$	18.2330	0.595647	0.297823	+	0.954621i	$0.403739\pi$
$938$	−56.1768	−1.83424
$939$	0	0
$940$	0	0
$941$	37.9004	1.23552	0.617759	−	0.786367i	$-0.288041\pi$
$941$	37.9004	1.23552	0.617759	+	0.786367i	$0.288041\pi$
$942$	0	0
$943$	11.2105	0.365065
$944$	−36.4267	−1.18559
$945$	0	0
$946$	−100.177	−3.25703
$947$	1.57146	0.0510655	0.0255328	−	0.999674i	$-0.491872\pi$
$947$	1.57146	0.0510655	0.0255328	+	0.999674i	$0.491872\pi$
$948$	0	0
$949$	−3.45085	−0.112019
$950$	0	0
$951$	0	0
$952$	84.8053	2.74856
$953$	−15.7102	−0.508904	−0.254452	−	0.967085i	$-0.581895\pi$
$953$	−15.7102	−0.508904	−0.254452	+	0.967085i	$0.581895\pi$
$954$	0	0
$955$	0	0
$956$	−15.6954	−0.507624
$957$	0	0
$958$	−5.39840	−0.174414
$959$	−18.2983	−0.590883
$960$	0	0
$961$	5.08577	0.164057
$962$	4.87704	0.157242
$963$	0	0
$964$	−65.4720	−2.10871
$965$	0	0
$966$	0	0
$967$	4.31176	0.138657	0.0693285	−	0.997594i	$-0.477914\pi$
$967$	4.31176	0.138657	0.0693285	+	0.997594i	$0.477914\pi$
$968$	−7.10268	−0.228289
$969$	0	0
$970$	0	0
$971$	−4.36056	−0.139937	−0.0699686	−	0.997549i	$-0.522290\pi$
$971$	−4.36056	−0.139937	−0.0699686	+	0.997549i	$0.522290\pi$
$972$	0	0
$973$	−10.5435	−0.338010
$974$	102.424	3.28189
$975$	0	0
$976$	−59.4030	−1.90144
$977$	−29.2433	−0.935576	−0.467788	−	0.883841i	$-0.654949\pi$
$977$	−29.2433	−0.935576	−0.467788	+	0.883841i	$0.654949\pi$
$978$	0	0
$979$	27.0742	0.865296
$980$	0	0
$981$	0	0
$982$	−58.0302	−1.85182
$983$	23.6085	0.752995	0.376498	−	0.926418i	$-0.377128\pi$
$983$	23.6085	0.752995	0.376498	+	0.926418i	$0.377128\pi$
$984$	0	0
$985$	0	0
$986$	17.5021	0.557381
$987$	0	0
$988$	−54.1866	−1.72390
$989$	16.7583	0.532884
$990$	0	0
$991$	−37.2039	−1.18182	−0.590910	−	0.806738i	$-0.701231\pi$
$991$	−37.2039	−1.18182	−0.590910	+	0.806738i	$0.701231\pi$
$992$	101.535	3.22373
$993$	0	0
$994$	−31.8465	−1.01011
$995$	0	0
$996$	0	0
$997$	57.1337	1.80944	0.904721	−	0.426004i	$-0.140079\pi$
$997$	57.1337	1.80944	0.904721	+	0.426004i	$0.140079\pi$
$998$	101.707	3.21947
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.cr.1.9		9
3.2	odd	2		925.2.a.l.1.1		9
5.2	odd	4		1665.2.c.e.334.18		18
5.3	odd	4		1665.2.c.e.334.1		18
5.4	even	2		8325.2.a.cq.1.1		9
15.2	even	4		185.2.b.a.149.1	✓	18
15.8	even	4		185.2.b.a.149.18	yes	18
15.14	odd	2		925.2.a.m.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.1	✓	18	15.2	even	4
185.2.b.a.149.18	yes	18	15.8	even	4
925.2.a.l.1.1		9	3.2	odd	2
925.2.a.m.1.9		9	15.14	odd	2
1665.2.c.e.334.1		18	5.3	odd	4
1665.2.c.e.334.18		18	5.2	odd	4
8325.2.a.cq.1.1		9	5.4	even	2
8325.2.a.cr.1.9		9	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68489 q^{2} +5.20864 q^{4} +3.48285 q^{7} +8.61484 q^{8} +3.18991 q^{11} +1.81648 q^{13} +9.35107 q^{14} +12.7126 q^{16} +2.82645 q^{17} -5.72714 q^{19} +8.56456 q^{22} -1.43274 q^{23} +4.87704 q^{26} +18.1409 q^{28} +2.30634 q^{29} +6.00714 q^{31} +16.9023 q^{32} +7.58870 q^{34} +1.00000 q^{37} -15.3767 q^{38} -7.82452 q^{41} -11.6967 q^{43} +16.6151 q^{44} -3.84676 q^{46} +2.42814 q^{47} +5.13024 q^{49} +9.46137 q^{52} -9.32573 q^{53} +30.0042 q^{56} +6.19227 q^{58} -2.86539 q^{59} -4.67275 q^{61} +16.1285 q^{62} +19.9557 q^{64} -6.00752 q^{67} +14.7219 q^{68} -3.40566 q^{71} -1.89975 q^{73} +2.68489 q^{74} -29.8306 q^{76} +11.1100 q^{77} -1.60903 q^{79} -21.0080 q^{82} +7.37571 q^{83} -31.4043 q^{86} +27.4806 q^{88} +8.48746 q^{89} +6.32652 q^{91} -7.46264 q^{92} +6.51929 q^{94} -9.00003 q^{97} +13.7741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} - 8 q^{7} + 15 q^{8} - 6 q^{13} + 4 q^{14} + 11 q^{16} + 18 q^{17} - 4 q^{19} - 6 q^{22} + 16 q^{23} + 6 q^{26} + 20 q^{28} + 2 q^{29} - 6 q^{31} + 35 q^{32} + 6 q^{34} + 9 q^{37}+ \cdots + 21 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^4 - 8 * q^7 + 15 * q^8 - 6 * q^13 + 4 * q^14 + 11 * q^16 + 18 * q^17 - 4 * q^19 - 6 * q^22 + 16 * q^23 + 6 * q^26 + 20 * q^28 + 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^34 + 9 * q^37 + 4 * q^38 - 2 * q^41 - 14 * q^43 - 4 * q^44 + 16 * q^46 + 34 * q^47 - 7 * q^49 - 16 * q^52 - 4 * q^53 + 24 * q^56 + 22 * q^58 - 10 * q^59 + 8 * q^61 + 26 * q^62 + 15 * q^64 - 16 * q^67 + 36 * q^68 - 8 * q^71 - 8 * q^73 + 5 * q^74 - 48 * q^76 + 14 * q^77 - 26 * q^79 + 4 * q^82 + 70 * q^83 + 16 * q^86 + 36 * q^88 + 8 * q^89 - 18 * q^91 + 16 * q^92 - 26 * q^94 + 2 * q^97 + 21 * q^98

Introduction

L-functions

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