LMFDB - Embedded newform 3364.2.a.q.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3364,2,Mod(1,3364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3364, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3364 = 2^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3364.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:	$26.8616752400$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3266578125.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1 $ x^8 - x^7 - 10x^6 + 6x^5 + 29x^4 - 9x^3 - 25*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.58001$ of defining polynomial
Character	$\chi$	$=$	3364.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.729419 q^{3} +0.767443 q^{5} -2.02171 q^{7} -2.46795 q^{9} +O(q^{10})$	$q+0.729419 q^{3} +0.767443 q^{5} -2.02171 q^{7} -2.46795 q^{9} +3.70678 q^{11} -1.24494 q^{13} +0.559788 q^{15} +3.58130 q^{17} -3.29207 q^{19} -1.47467 q^{21} -2.34621 q^{23} -4.41103 q^{25} -3.98843 q^{27} -5.93955 q^{31} +2.70380 q^{33} -1.55155 q^{35} -0.856538 q^{37} -0.908085 q^{39} -7.92138 q^{41} +8.30193 q^{43} -1.89401 q^{45} -2.40165 q^{47} -2.91270 q^{49} +2.61227 q^{51} +11.7655 q^{53} +2.84474 q^{55} -2.40130 q^{57} +1.61067 q^{59} -14.4889 q^{61} +4.98947 q^{63} -0.955422 q^{65} -4.41071 q^{67} -1.71137 q^{69} +15.2115 q^{71} -0.840227 q^{73} -3.21749 q^{75} -7.49402 q^{77} -9.84833 q^{79} +4.49461 q^{81} -12.8266 q^{83} +2.74844 q^{85} +2.46866 q^{89} +2.51691 q^{91} -4.33242 q^{93} -2.52648 q^{95} -16.1133 q^{97} -9.14813 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9	$ 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} + 17 q^{15} - 15 q^{17} - 11 q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 16 q^{27} + 9 q^{31} - 16 q^{33} + 9 q^{35} + 2 q^{37} - 9 q^{39} - 38 q^{41} + 3 q^{43} + 3 q^{45} + 7 q^{47} + 12 q^{49} + 3 q^{51} - 5 q^{53} - 34 q^{55} + 34 q^{57} - 7 q^{59} - 39 q^{61} - 6 q^{63} - 21 q^{65} - 26 q^{67} - 34 q^{69} + 52 q^{71} + 34 q^{73} - 10 q^{77} - 31 q^{79} + 24 q^{81} - 17 q^{83} - 18 q^{85} - 24 q^{89} + 54 q^{91} + 6 q^{93} - 9 q^{95} - 22 q^{97} + 11 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 + 17 * q^15 - 15 * q^17 - 11 * q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 16 * q^27 + 9 * q^31 - 16 * q^33 + 9 * q^35 + 2 * q^37 - 9 * q^39 - 38 * q^41 + 3 * q^43 + 3 * q^45 + 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 - 34 * q^55 + 34 * q^57 - 7 * q^59 - 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 - 34 * q^69 + 52 * q^71 + 34 * q^73 - 10 * q^77 - 31 * q^79 + 24 * q^81 - 17 * q^83 - 18 * q^85 - 24 * q^89 + 54 * q^91 + 6 * q^93 - 9 * q^95 - 22 * q^97 + 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.729419	0.421130	0.210565	−	0.977580i	$-0.432470\pi$
$3$	0.729419	0.421130	0.210565	+	0.977580i	$0.432470\pi$
$4$	0	0
$5$	0.767443	0.343211	0.171606	−	0.985166i	$-0.445105\pi$
$5$	0.767443	0.343211	0.171606	+	0.985166i	$0.445105\pi$
$6$	0	0
$7$	−2.02171	−0.764134	−0.382067	−	0.924135i	$-0.624788\pi$
$7$	−2.02171	−0.764134	−0.382067	+	0.924135i	$0.624788\pi$
$8$	0	0
$9$	−2.46795	−0.822649
$10$	0	0
$11$	3.70678	1.11764	0.558818	−	0.829290i	$-0.311255\pi$
$11$	3.70678	1.11764	0.558818	+	0.829290i	$0.311255\pi$
$12$	0	0
$13$	−1.24494	−0.345285	−0.172642	−	0.984985i	$-0.555231\pi$
$13$	−1.24494	−0.345285	−0.172642	+	0.984985i	$0.555231\pi$
$14$	0	0
$15$	0.559788	0.144537
$16$	0	0
$17$	3.58130	0.868593	0.434296	−	0.900770i	$-0.356997\pi$
$17$	3.58130	0.868593	0.434296	+	0.900770i	$0.356997\pi$
$18$	0	0
$19$	−3.29207	−0.755254	−0.377627	−	0.925958i	$-0.623260\pi$
$19$	−3.29207	−0.755254	−0.377627	+	0.925958i	$0.623260\pi$
$20$	0	0
$21$	−1.47467	−0.321800
$22$	0	0
$23$	−2.34621	−0.489219	−0.244609	−	0.969622i	$-0.578660\pi$
$23$	−2.34621	−0.489219	−0.244609	+	0.969622i	$0.578660\pi$
$24$	0	0
$25$	−4.41103	−0.882206
$26$	0	0
$27$	−3.98843	−0.767573
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−5.93955	−1.06677	−0.533387	−	0.845871i	$-0.679081\pi$
$31$	−5.93955	−1.06677	−0.533387	+	0.845871i	$0.679081\pi$
$32$	0	0
$33$	2.70380	0.470670
$34$	0	0
$35$	−1.55155	−0.262259
$36$	0	0
$37$	−0.856538	−0.140814	−0.0704070	−	0.997518i	$-0.522430\pi$
$37$	−0.856538	−0.140814	−0.0704070	+	0.997518i	$0.522430\pi$
$38$	0	0
$39$	−0.908085	−0.145410
$40$	0	0
$41$	−7.92138	−1.23711	−0.618556	−	0.785741i	$-0.712282\pi$
$41$	−7.92138	−1.23711	−0.618556	+	0.785741i	$0.712282\pi$
$42$	0	0
$43$	8.30193	1.26603	0.633016	−	0.774139i	$-0.281817\pi$
$43$	8.30193	1.26603	0.633016	+	0.774139i	$0.281817\pi$
$44$	0	0
$45$	−1.89401	−0.282342
$46$	0	0
$47$	−2.40165	−0.350316	−0.175158	−	0.984540i	$-0.556044\pi$
$47$	−2.40165	−0.350316	−0.175158	+	0.984540i	$0.556044\pi$
$48$	0	0
$49$	−2.91270	−0.416100
$50$	0	0
$51$	2.61227	0.365791
$52$	0	0
$53$	11.7655	1.61611	0.808055	−	0.589107i	$-0.200520\pi$
$53$	11.7655	1.61611	0.808055	+	0.589107i	$0.200520\pi$
$54$	0	0
$55$	2.84474	0.383585
$56$	0	0
$57$	−2.40130	−0.318060
$58$	0	0
$59$	1.61067	0.209692	0.104846	−	0.994488i	$-0.466565\pi$
$59$	1.61067	0.209692	0.104846	+	0.994488i	$0.466565\pi$
$60$	0	0
$61$	−14.4889	−1.85511	−0.927556	−	0.373684i	$-0.878095\pi$
$61$	−14.4889	−1.85511	−0.927556	+	0.373684i	$0.878095\pi$
$62$	0	0
$63$	4.98947	0.628614
$64$	0	0
$65$	−0.955422	−0.118506
$66$	0	0
$67$	−4.41071	−0.538854	−0.269427	−	0.963021i	$-0.586834\pi$
$67$	−4.41071	−0.538854	−0.269427	+	0.963021i	$0.586834\pi$
$68$	0	0
$69$	−1.71137	−0.206025
$70$	0	0
$71$	15.2115	1.80527	0.902636	−	0.430404i	$-0.141629\pi$
$71$	15.2115	1.80527	0.902636	+	0.430404i	$0.141629\pi$
$72$	0	0
$73$	−0.840227	−0.0983412	−0.0491706	−	0.998790i	$-0.515658\pi$
$73$	−0.840227	−0.0983412	−0.0491706	+	0.998790i	$0.515658\pi$
$74$	0	0
$75$	−3.21749	−0.371524
$76$	0	0
$77$	−7.49402	−0.854023
$78$	0	0
$79$	−9.84833	−1.10802	−0.554012	−	0.832509i	$-0.686904\pi$
$79$	−9.84833	−1.10802	−0.554012	+	0.832509i	$0.686904\pi$
$80$	0	0
$81$	4.49461	0.499401
$82$	0	0
$83$	−12.8266	−1.40790	−0.703949	−	0.710250i	$-0.748582\pi$
$83$	−12.8266	−1.40790	−0.703949	+	0.710250i	$0.748582\pi$
$84$	0	0
$85$	2.74844	0.298111
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.46866	0.261677	0.130839	−	0.991404i	$-0.458233\pi$
$89$	2.46866	0.261677	0.130839	+	0.991404i	$0.458233\pi$
$90$	0	0
$91$	2.51691	0.263844
$92$	0	0
$93$	−4.33242	−0.449251
$94$	0	0
$95$	−2.52648	−0.259211
$96$	0	0
$97$	−16.1133	−1.63606	−0.818030	−	0.575175i	$-0.804934\pi$
$97$	−16.1133	−1.63606	−0.818030	+	0.575175i	$0.804934\pi$
$98$	0	0
$99$	−9.14813	−0.919422
$100$	0	0
$101$	12.7240	1.26608	0.633042	−	0.774117i	$-0.281806\pi$
$101$	12.7240	1.26608	0.633042	+	0.774117i	$0.281806\pi$
$102$	0	0
$103$	−17.3194	−1.70653	−0.853264	−	0.521480i	$-0.825380\pi$
$103$	−17.3194	−1.70653	−0.853264	+	0.521480i	$0.825380\pi$
$104$	0	0
$105$	−1.13173	−0.110445
$106$	0	0
$107$	−6.79254	−0.656660	−0.328330	−	0.944563i	$-0.606486\pi$
$107$	−6.79254	−0.656660	−0.328330	+	0.944563i	$0.606486\pi$
$108$	0	0
$109$	9.13608	0.875078	0.437539	−	0.899199i	$-0.355850\pi$
$109$	9.13608	0.875078	0.437539	+	0.899199i	$0.355850\pi$
$110$	0	0
$111$	−0.624776	−0.0593011
$112$	0	0
$113$	−12.0253	−1.13125	−0.565623	−	0.824664i	$-0.691364\pi$
$113$	−12.0253	−1.13125	−0.565623	+	0.824664i	$0.691364\pi$
$114$	0	0
$115$	−1.80058	−0.167905
$116$	0	0
$117$	3.07245	0.284048
$118$	0	0
$119$	−7.24034	−0.663721
$120$	0	0
$121$	2.74021	0.249110
$122$	0	0
$123$	−5.77801	−0.520986
$124$	0	0
$125$	−7.22243	−0.645994
$126$	0	0
$127$	8.30315	0.736786	0.368393	−	0.929670i	$-0.379908\pi$
$127$	8.30315	0.736786	0.368393	+	0.929670i	$0.379908\pi$
$128$	0	0
$129$	6.05559	0.533165
$130$	0	0
$131$	2.12640	0.185784	0.0928920	−	0.995676i	$-0.470389\pi$
$131$	2.12640	0.185784	0.0928920	+	0.995676i	$0.470389\pi$
$132$	0	0
$133$	6.65561	0.577115
$134$	0	0
$135$	−3.06089	−0.263440
$136$	0	0
$137$	−18.2911	−1.56272	−0.781359	−	0.624082i	$-0.785473\pi$
$137$	−18.2911	−1.56272	−0.781359	+	0.624082i	$0.785473\pi$
$138$	0	0
$139$	−18.6779	−1.58424	−0.792119	−	0.610367i	$-0.791022\pi$
$139$	−18.6779	−1.58424	−0.792119	+	0.610367i	$0.791022\pi$
$140$	0	0
$141$	−1.75181	−0.147529
$142$	0	0
$143$	−4.61472	−0.385903
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−2.12458	−0.175232
$148$	0	0
$149$	22.3871	1.83403	0.917013	−	0.398858i	$-0.130593\pi$
$149$	22.3871	1.83403	0.917013	+	0.398858i	$0.130593\pi$
$150$	0	0
$151$	0.319407	0.0259930	0.0129965	−	0.999916i	$-0.495863\pi$
$151$	0.319407	0.0259930	0.0129965	+	0.999916i	$0.495863\pi$
$152$	0	0
$153$	−8.83846	−0.714547
$154$	0	0
$155$	−4.55827	−0.366129
$156$	0	0
$157$	−17.1206	−1.36637	−0.683187	−	0.730243i	$-0.739407\pi$
$157$	−17.1206	−1.36637	−0.683187	+	0.730243i	$0.739407\pi$
$158$	0	0
$159$	8.58195	0.680593
$160$	0	0
$161$	4.74335	0.373828
$162$	0	0
$163$	15.9530	1.24954	0.624769	−	0.780810i	$-0.285193\pi$
$163$	15.9530	1.24954	0.624769	+	0.780810i	$0.285193\pi$
$164$	0	0
$165$	2.07501	0.161539
$166$	0	0
$167$	−6.06511	−0.469332	−0.234666	−	0.972076i	$-0.575400\pi$
$167$	−6.06511	−0.469332	−0.234666	+	0.972076i	$0.575400\pi$
$168$	0	0
$169$	−11.4501	−0.880778
$170$	0	0
$171$	8.12467	0.621309
$172$	0	0
$173$	0.821089	0.0624263	0.0312131	−	0.999513i	$-0.490063\pi$
$173$	0.821089	0.0624263	0.0312131	+	0.999513i	$0.490063\pi$
$174$	0	0
$175$	8.91782	0.674124
$176$	0	0
$177$	1.17486	0.0883076
$178$	0	0
$179$	20.0466	1.49835	0.749177	−	0.662370i	$-0.230449\pi$
$179$	20.0466	1.49835	0.749177	+	0.662370i	$0.230449\pi$
$180$	0	0
$181$	−6.89043	−0.512162	−0.256081	−	0.966655i	$-0.582431\pi$
$181$	−6.89043	−0.512162	−0.256081	+	0.966655i	$0.582431\pi$
$182$	0	0
$183$	−10.5685	−0.781244
$184$	0	0
$185$	−0.657344	−0.0483289
$186$	0	0
$187$	13.2751	0.970770
$188$	0	0
$189$	8.06343	0.586529
$190$	0	0
$191$	15.4002	1.11432	0.557159	−	0.830406i	$-0.311891\pi$
$191$	15.4002	1.11432	0.557159	+	0.830406i	$0.311891\pi$
$192$	0	0
$193$	−18.0067	−1.29615	−0.648077	−	0.761575i	$-0.724427\pi$
$193$	−18.0067	−1.29615	−0.648077	+	0.761575i	$0.724427\pi$
$194$	0	0
$195$	−0.696904	−0.0499063
$196$	0	0
$197$	−7.34986	−0.523656	−0.261828	−	0.965115i	$-0.584325\pi$
$197$	−7.34986	−0.523656	−0.261828	+	0.965115i	$0.584325\pi$
$198$	0	0
$199$	−10.5618	−0.748709	−0.374355	−	0.927286i	$-0.622136\pi$
$199$	−10.5618	−0.748709	−0.374355	+	0.927286i	$0.622136\pi$
$200$	0	0
$201$	−3.21726	−0.226928
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.07921	−0.424590
$206$	0	0
$207$	5.79032	0.402455
$208$	0	0
$209$	−12.2030	−0.844098
$210$	0	0
$211$	−7.13512	−0.491202	−0.245601	−	0.969371i	$-0.578985\pi$
$211$	−7.13512	−0.491202	−0.245601	+	0.969371i	$0.578985\pi$
$212$	0	0
$213$	11.0956	0.760255
$214$	0	0
$215$	6.37126	0.434516
$216$	0	0
$217$	12.0080	0.815159
$218$	0	0
$219$	−0.612878	−0.0414145
$220$	0	0
$221$	−4.45851	−0.299912
$222$	0	0
$223$	0.144735	0.00969217	0.00484608	−	0.999988i	$-0.498457\pi$
$223$	0.144735	0.00969217	0.00484608	+	0.999988i	$0.498457\pi$
$224$	0	0
$225$	10.8862	0.725746
$226$	0	0
$227$	14.0201	0.930545	0.465272	−	0.885168i	$-0.345956\pi$
$227$	14.0201	0.930545	0.465272	+	0.885168i	$0.345956\pi$
$228$	0	0
$229$	−4.15050	−0.274273	−0.137136	−	0.990552i	$-0.543790\pi$
$229$	−4.15050	−0.274273	−0.137136	+	0.990552i	$0.543790\pi$
$230$	0	0
$231$	−5.46629	−0.359655
$232$	0	0
$233$	21.6954	1.42131	0.710656	−	0.703540i	$-0.248398\pi$
$233$	21.6954	1.42131	0.710656	+	0.703540i	$0.248398\pi$
$234$	0	0
$235$	−1.84313	−0.120232
$236$	0	0
$237$	−7.18357	−0.466623
$238$	0	0
$239$	24.5869	1.59039	0.795197	−	0.606351i	$-0.207367\pi$
$239$	24.5869	1.59039	0.795197	+	0.606351i	$0.207367\pi$
$240$	0	0
$241$	2.48439	0.160034	0.0800168	−	0.996794i	$-0.474503\pi$
$241$	2.48439	0.160034	0.0800168	+	0.996794i	$0.474503\pi$
$242$	0	0
$243$	15.2437	0.977886
$244$	0	0
$245$	−2.23533	−0.142810
$246$	0	0
$247$	4.09844	0.260778
$248$	0	0
$249$	−9.35595	−0.592909
$250$	0	0
$251$	7.75412	0.489435	0.244718	−	0.969594i	$-0.421305\pi$
$251$	7.75412	0.489435	0.244718	+	0.969594i	$0.421305\pi$
$252$	0	0
$253$	−8.69688	−0.546768
$254$	0	0
$255$	2.00477	0.125543
$256$	0	0
$257$	−31.2147	−1.94712	−0.973559	−	0.228434i	$-0.926639\pi$
$257$	−31.2147	−1.94712	−0.973559	+	0.228434i	$0.926639\pi$
$258$	0	0
$259$	1.73167	0.107601
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.4590	−0.644929	−0.322464	−	0.946582i	$-0.604511\pi$
$263$	−10.4590	−0.644929	−0.322464	+	0.946582i	$0.604511\pi$
$264$	0	0
$265$	9.02932	0.554667
$266$	0	0
$267$	1.80069	0.110200
$268$	0	0
$269$	−0.685928	−0.0418218	−0.0209109	−	0.999781i	$-0.506657\pi$
$269$	−0.685928	−0.0418218	−0.0209109	+	0.999781i	$0.506657\pi$
$270$	0	0
$271$	−16.1931	−0.983660	−0.491830	−	0.870691i	$-0.663672\pi$
$271$	−16.1931	−0.983660	−0.491830	+	0.870691i	$0.663672\pi$
$272$	0	0
$273$	1.83588	0.111113
$274$	0	0
$275$	−16.3507	−0.985985
$276$	0	0
$277$	14.8322	0.891183	0.445591	−	0.895236i	$-0.352993\pi$
$277$	14.8322	0.891183	0.445591	+	0.895236i	$0.352993\pi$
$278$	0	0
$279$	14.6585	0.877581
$280$	0	0
$281$	−7.37493	−0.439951	−0.219976	−	0.975505i	$-0.570598\pi$
$281$	−7.37493	−0.439951	−0.219976	+	0.975505i	$0.570598\pi$
$282$	0	0
$283$	23.9057	1.42105	0.710524	−	0.703673i	$-0.248458\pi$
$283$	23.9057	1.42105	0.710524	+	0.703673i	$0.248458\pi$
$284$	0	0
$285$	−1.84286	−0.109162
$286$	0	0
$287$	16.0147	0.945319
$288$	0	0
$289$	−4.17430	−0.245547
$290$	0	0
$291$	−11.7534	−0.688995
$292$	0	0
$293$	−3.37749	−0.197315	−0.0986576	−	0.995121i	$-0.531455\pi$
$293$	−3.37749	−0.197315	−0.0986576	+	0.995121i	$0.531455\pi$
$294$	0	0
$295$	1.23610	0.0719685
$296$	0	0
$297$	−14.7842	−0.857867
$298$	0	0
$299$	2.92090	0.168920
$300$	0	0
$301$	−16.7841	−0.967418
$302$	0	0
$303$	9.28113	0.533187
$304$	0	0
$305$	−11.1194	−0.636695
$306$	0	0
$307$	−10.1287	−0.578075	−0.289038	−	0.957318i	$-0.593335\pi$
$307$	−10.1287	−0.578075	−0.289038	+	0.957318i	$0.593335\pi$
$308$	0	0
$309$	−12.6331	−0.718671
$310$	0	0
$311$	−14.3866	−0.815786	−0.407893	−	0.913030i	$-0.633736\pi$
$311$	−14.3866	−0.815786	−0.407893	+	0.913030i	$0.633736\pi$
$312$	0	0
$313$	29.7699	1.68269	0.841346	−	0.540497i	$-0.181764\pi$
$313$	29.7699	1.68269	0.841346	+	0.540497i	$0.181764\pi$
$314$	0	0
$315$	3.82913	0.215747
$316$	0	0
$317$	3.38943	0.190369	0.0951846	−	0.995460i	$-0.469656\pi$
$317$	3.38943	0.190369	0.0951846	+	0.995460i	$0.469656\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.95461	−0.276540
$322$	0	0
$323$	−11.7899	−0.656008
$324$	0	0
$325$	5.49148	0.304612
$326$	0	0
$327$	6.66404	0.368522
$328$	0	0
$329$	4.85543	0.267688
$330$	0	0
$331$	−14.6345	−0.804385	−0.402192	−	0.915555i	$-0.631752\pi$
$331$	−14.6345	−0.804385	−0.402192	+	0.915555i	$0.631752\pi$
$332$	0	0
$333$	2.11389	0.115841
$334$	0	0
$335$	−3.38497	−0.184941
$336$	0	0
$337$	−11.2957	−0.615315	−0.307657	−	0.951497i	$-0.599545\pi$
$337$	−11.2957	−0.615315	−0.307657	+	0.951497i	$0.599545\pi$
$338$	0	0
$339$	−8.77149	−0.476402
$340$	0	0
$341$	−22.0166	−1.19227
$342$	0	0
$343$	20.0406	1.08209
$344$	0	0
$345$	−1.31338	−0.0707100
$346$	0	0
$347$	1.57899	0.0847648	0.0423824	−	0.999101i	$-0.486505\pi$
$347$	1.57899	0.0847648	0.0423824	+	0.999101i	$0.486505\pi$
$348$	0	0
$349$	−4.99875	−0.267577	−0.133788	−	0.991010i	$-0.542714\pi$
$349$	−4.99875	−0.267577	−0.133788	+	0.991010i	$0.542714\pi$
$350$	0	0
$351$	4.96536	0.265031
$352$	0	0
$353$	−8.23964	−0.438552	−0.219276	−	0.975663i	$-0.570370\pi$
$353$	−8.23964	−0.438552	−0.219276	+	0.975663i	$0.570370\pi$
$354$	0	0
$355$	11.6740	0.619589
$356$	0	0
$357$	−5.28124	−0.279513
$358$	0	0
$359$	−6.20747	−0.327617	−0.163809	−	0.986492i	$-0.552378\pi$
$359$	−6.20747	−0.327617	−0.163809	+	0.986492i	$0.552378\pi$
$360$	0	0
$361$	−8.16225	−0.429592
$362$	0	0
$363$	1.99876	0.104908
$364$	0	0
$365$	−0.644827	−0.0337518
$366$	0	0
$367$	26.1215	1.36353	0.681765	−	0.731571i	$-0.261212\pi$
$367$	26.1215	1.36353	0.681765	+	0.731571i	$0.261212\pi$
$368$	0	0
$369$	19.5496	1.01771
$370$	0	0
$371$	−23.7863	−1.23492
$372$	0	0
$373$	14.2786	0.739316	0.369658	−	0.929168i	$-0.379475\pi$
$373$	14.2786	0.739316	0.369658	+	0.929168i	$0.379475\pi$
$374$	0	0
$375$	−5.26818	−0.272048
$376$	0	0
$377$	0	0
$378$	0	0
$379$	19.6128	1.00744	0.503720	−	0.863867i	$-0.331964\pi$
$379$	19.6128	1.00744	0.503720	+	0.863867i	$0.331964\pi$
$380$	0	0
$381$	6.05648	0.310283
$382$	0	0
$383$	−15.9953	−0.817322	−0.408661	−	0.912686i	$-0.634004\pi$
$383$	−15.9953	−0.817322	−0.408661	+	0.912686i	$0.634004\pi$
$384$	0	0
$385$	−5.75124	−0.293110
$386$	0	0
$387$	−20.4887	−1.04150
$388$	0	0
$389$	−4.89046	−0.247956	−0.123978	−	0.992285i	$-0.539565\pi$
$389$	−4.89046	−0.247956	−0.123978	+	0.992285i	$0.539565\pi$
$390$	0	0
$391$	−8.40248	−0.424932
$392$	0	0
$393$	1.55103	0.0782393
$394$	0	0
$395$	−7.55804	−0.380286
$396$	0	0
$397$	27.4917	1.37977	0.689884	−	0.723920i	$-0.257661\pi$
$397$	27.4917	1.37977	0.689884	+	0.723920i	$0.257661\pi$
$398$	0	0
$399$	4.85473	0.243041
$400$	0	0
$401$	−18.2792	−0.912821	−0.456410	−	0.889769i	$-0.650865\pi$
$401$	−18.2792	−0.912821	−0.456410	+	0.889769i	$0.650865\pi$
$402$	0	0
$403$	7.39440	0.368341
$404$	0	0
$405$	3.44935	0.171400
$406$	0	0
$407$	−3.17500	−0.157379
$408$	0	0
$409$	1.66374	0.0822665	0.0411332	−	0.999154i	$-0.486903\pi$
$409$	1.66374	0.0822665	0.0411332	+	0.999154i	$0.486903\pi$
$410$	0	0
$411$	−13.3419	−0.658108
$412$	0	0
$413$	−3.25631	−0.160233
$414$	0	0
$415$	−9.84366	−0.483206
$416$	0	0
$417$	−13.6240	−0.667171
$418$	0	0
$419$	8.97520	0.438467	0.219234	−	0.975672i	$-0.429644\pi$
$419$	8.97520	0.438467	0.219234	+	0.975672i	$0.429644\pi$
$420$	0	0
$421$	19.7189	0.961041	0.480521	−	0.876983i	$-0.340448\pi$
$421$	19.7189	0.961041	0.480521	+	0.876983i	$0.340448\pi$
$422$	0	0
$423$	5.92714	0.288187
$424$	0	0
$425$	−15.7972	−0.766278
$426$	0	0
$427$	29.2923	1.41755
$428$	0	0
$429$	−3.36607	−0.162515
$430$	0	0
$431$	30.2008	1.45472	0.727362	−	0.686254i	$-0.240746\pi$
$431$	30.2008	1.45472	0.727362	+	0.686254i	$0.240746\pi$
$432$	0	0
$433$	9.82867	0.472336	0.236168	−	0.971712i	$-0.424108\pi$
$433$	9.82867	0.472336	0.236168	+	0.971712i	$0.424108\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.72390	0.369484
$438$	0	0
$439$	−12.4301	−0.593257	−0.296628	−	0.954993i	$-0.595862\pi$
$439$	−12.4301	−0.593257	−0.296628	+	0.954993i	$0.595862\pi$
$440$	0	0
$441$	7.18838	0.342304
$442$	0	0
$443$	28.9783	1.37680	0.688399	−	0.725332i	$-0.258314\pi$
$443$	28.9783	1.37680	0.688399	+	0.725332i	$0.258314\pi$
$444$	0	0
$445$	1.89456	0.0898106
$446$	0	0
$447$	16.3296	0.772364
$448$	0	0
$449$	−5.23266	−0.246945	−0.123472	−	0.992348i	$-0.539403\pi$
$449$	−5.23266	−0.246945	−0.123472	+	0.992348i	$0.539403\pi$
$450$	0	0
$451$	−29.3628	−1.38264
$452$	0	0
$453$	0.232982	0.0109464
$454$	0	0
$455$	1.93158	0.0905541
$456$	0	0
$457$	−28.4744	−1.33198	−0.665989	−	0.745962i	$-0.731990\pi$
$457$	−28.4744	−1.33198	−0.665989	+	0.745962i	$0.731990\pi$
$458$	0	0
$459$	−14.2838	−0.666708
$460$	0	0
$461$	−34.7454	−1.61825	−0.809127	−	0.587634i	$-0.800059\pi$
$461$	−34.7454	−1.61825	−0.809127	+	0.587634i	$0.800059\pi$
$462$	0	0
$463$	20.2963	0.943247	0.471624	−	0.881800i	$-0.343668\pi$
$463$	20.2963	0.943247	0.471624	+	0.881800i	$0.343668\pi$
$464$	0	0
$465$	−3.32489	−0.154188
$466$	0	0
$467$	−3.44474	−0.159404	−0.0797018	−	0.996819i	$-0.525397\pi$
$467$	−3.44474	−0.159404	−0.0797018	+	0.996819i	$0.525397\pi$
$468$	0	0
$469$	8.91716	0.411756
$470$	0	0
$471$	−12.4881	−0.575422
$472$	0	0
$473$	30.7734	1.41496
$474$	0	0
$475$	14.5214	0.666289
$476$	0	0
$477$	−29.0365	−1.32949
$478$	0	0
$479$	−36.4072	−1.66349	−0.831745	−	0.555158i	$-0.812658\pi$
$479$	−36.4072	−1.66349	−0.831745	+	0.555158i	$0.812658\pi$
$480$	0	0
$481$	1.06634	0.0486209
$482$	0	0
$483$	3.45989	0.157431
$484$	0	0
$485$	−12.3661	−0.561514
$486$	0	0
$487$	−0.113357	−0.00513670	−0.00256835	−	0.999997i	$-0.500818\pi$
$487$	−0.113357	−0.00513670	−0.00256835	+	0.999997i	$0.500818\pi$
$488$	0	0
$489$	11.6365	0.526218
$490$	0	0
$491$	−20.9051	−0.943436	−0.471718	−	0.881749i	$-0.656366\pi$
$491$	−20.9051	−0.943436	−0.471718	+	0.881749i	$0.656366\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−7.02067	−0.315556
$496$	0	0
$497$	−30.7532	−1.37947
$498$	0	0
$499$	36.9570	1.65442	0.827211	−	0.561891i	$-0.189926\pi$
$499$	36.9570	1.65442	0.827211	+	0.561891i	$0.189926\pi$
$500$	0	0
$501$	−4.42401	−0.197650
$502$	0	0
$503$	34.8086	1.55204	0.776019	−	0.630709i	$-0.217236\pi$
$503$	34.8086	1.55204	0.776019	+	0.630709i	$0.217236\pi$
$504$	0	0
$505$	9.76494	0.434534
$506$	0	0
$507$	−8.35194	−0.370923
$508$	0	0
$509$	21.1709	0.938382	0.469191	−	0.883097i	$-0.344546\pi$
$509$	21.1709	0.938382	0.469191	+	0.883097i	$0.344546\pi$
$510$	0	0
$511$	1.69869	0.0751458
$512$	0	0
$513$	13.1302	0.579712
$514$	0	0
$515$	−13.2916	−0.585699
$516$	0	0
$517$	−8.90237	−0.391526
$518$	0	0
$519$	0.598919	0.0262896
$520$	0	0
$521$	0.988926	0.0433257	0.0216628	−	0.999765i	$-0.493104\pi$
$521$	0.988926	0.0433257	0.0216628	+	0.999765i	$0.493104\pi$
$522$	0	0
$523$	−24.5416	−1.07313	−0.536564	−	0.843859i	$-0.680278\pi$
$523$	−24.5416	−1.07313	−0.536564	+	0.843859i	$0.680278\pi$
$524$	0	0
$525$	6.50483	0.283894
$526$	0	0
$527$	−21.2713	−0.926593
$528$	0	0
$529$	−17.4953	−0.760665
$530$	0	0
$531$	−3.97506	−0.172503
$532$	0	0
$533$	9.86166	0.427156
$534$	0	0
$535$	−5.21289	−0.225373
$536$	0	0
$537$	14.6224	0.631002
$538$	0	0
$539$	−10.7967	−0.465048
$540$	0	0
$541$	19.6472	0.844698	0.422349	−	0.906433i	$-0.361206\pi$
$541$	19.6472	0.844698	0.422349	+	0.906433i	$0.361206\pi$
$542$	0	0
$543$	−5.02601	−0.215687
$544$	0	0
$545$	7.01142	0.300336
$546$	0	0
$547$	19.9629	0.853551	0.426775	−	0.904358i	$-0.359650\pi$
$547$	19.9629	0.853551	0.426775	+	0.904358i	$0.359650\pi$
$548$	0	0
$549$	35.7578	1.52611
$550$	0	0
$551$	0	0
$552$	0	0
$553$	19.9105	0.846679
$554$	0	0
$555$	−0.479480	−0.0203528
$556$	0	0
$557$	11.4997	0.487257	0.243629	−	0.969869i	$-0.421662\pi$
$557$	11.4997	0.487257	0.243629	+	0.969869i	$0.421662\pi$
$558$	0	0
$559$	−10.3354	−0.437142
$560$	0	0
$561$	9.68310	0.408821
$562$	0	0
$563$	44.7407	1.88560	0.942798	−	0.333365i	$-0.108184\pi$
$563$	44.7407	1.88560	0.942798	+	0.333365i	$0.108184\pi$
$564$	0	0
$565$	−9.22874	−0.388256
$566$	0	0
$567$	−9.08678	−0.381609
$568$	0	0
$569$	−25.6562	−1.07556	−0.537781	−	0.843084i	$-0.680737\pi$
$569$	−25.6562	−1.07556	−0.537781	+	0.843084i	$0.680737\pi$
$570$	0	0
$571$	−22.1209	−0.925730	−0.462865	−	0.886429i	$-0.653178\pi$
$571$	−22.1209	−0.925730	−0.462865	+	0.886429i	$0.653178\pi$
$572$	0	0
$573$	11.2332	0.469273
$574$	0	0
$575$	10.3492	0.431592
$576$	0	0
$577$	6.41863	0.267211	0.133606	−	0.991035i	$-0.457344\pi$
$577$	6.41863	0.267211	0.133606	+	0.991035i	$0.457344\pi$
$578$	0	0
$579$	−13.1345	−0.545850
$580$	0	0
$581$	25.9316	1.07582
$582$	0	0
$583$	43.6119	1.80622
$584$	0	0
$585$	2.35793	0.0974885
$586$	0	0
$587$	18.3752	0.758427	0.379213	−	0.925309i	$-0.376195\pi$
$587$	18.3752	0.758427	0.379213	+	0.925309i	$0.376195\pi$
$588$	0	0
$589$	19.5534	0.805686
$590$	0	0
$591$	−5.36113	−0.220528
$592$	0	0
$593$	−42.9405	−1.76336	−0.881678	−	0.471851i	$-0.843586\pi$
$593$	−42.9405	−1.76336	−0.881678	+	0.471851i	$0.843586\pi$
$594$	0	0
$595$	−5.55655	−0.227796
$596$	0	0
$597$	−7.70402	−0.315304
$598$	0	0
$599$	36.0434	1.47269	0.736346	−	0.676605i	$-0.236550\pi$
$599$	36.0434	1.47269	0.736346	+	0.676605i	$0.236550\pi$
$600$	0	0
$601$	36.9395	1.50679	0.753396	−	0.657567i	$-0.228414\pi$
$601$	36.9395	1.50679	0.753396	+	0.657567i	$0.228414\pi$
$602$	0	0
$603$	10.8854	0.443288
$604$	0	0
$605$	2.10295	0.0854972
$606$	0	0
$607$	−22.4913	−0.912893	−0.456446	−	0.889751i	$-0.650878\pi$
$607$	−22.4913	−0.912893	−0.456446	+	0.889751i	$0.650878\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.98991	0.120959
$612$	0	0
$613$	−4.90637	−0.198167	−0.0990833	−	0.995079i	$-0.531591\pi$
$613$	−4.90637	−0.198167	−0.0990833	+	0.995079i	$0.531591\pi$
$614$	0	0
$615$	−4.43429	−0.178808
$616$	0	0
$617$	−3.17098	−0.127659	−0.0638295	−	0.997961i	$-0.520331\pi$
$617$	−3.17098	−0.127659	−0.0638295	+	0.997961i	$0.520331\pi$
$618$	0	0
$619$	20.7571	0.834299	0.417149	−	0.908838i	$-0.363029\pi$
$619$	20.7571	0.834299	0.417149	+	0.908838i	$0.363029\pi$
$620$	0	0
$621$	9.35769	0.375511
$622$	0	0
$623$	−4.99091	−0.199957
$624$	0	0
$625$	16.5124	0.660494
$626$	0	0
$627$	−8.90110	−0.355476
$628$	0	0
$629$	−3.06752	−0.122310
$630$	0	0
$631$	−14.0960	−0.561151	−0.280576	−	0.959832i	$-0.590525\pi$
$631$	−14.0960	−0.561151	−0.280576	+	0.959832i	$0.590525\pi$
$632$	0	0
$633$	−5.20449	−0.206860
$634$	0	0
$635$	6.37220	0.252873
$636$	0	0
$637$	3.62614	0.143673
$638$	0	0
$639$	−37.5412	−1.48511
$640$	0	0
$641$	−26.1964	−1.03470	−0.517349	−	0.855775i	$-0.673081\pi$
$641$	−26.1964	−1.03470	−0.517349	+	0.855775i	$0.673081\pi$
$642$	0	0
$643$	22.3949	0.883170	0.441585	−	0.897219i	$-0.354416\pi$
$643$	22.3949	0.883170	0.441585	+	0.897219i	$0.354416\pi$
$644$	0	0
$645$	4.64732	0.182988
$646$	0	0
$647$	32.6252	1.28263	0.641314	−	0.767278i	$-0.278389\pi$
$647$	32.6252	1.28263	0.641314	+	0.767278i	$0.278389\pi$
$648$	0	0
$649$	5.97041	0.234359
$650$	0	0
$651$	8.75890	0.343288
$652$	0	0
$653$	−22.2236	−0.869678	−0.434839	−	0.900508i	$-0.643195\pi$
$653$	−22.2236	−0.869678	−0.434839	+	0.900508i	$0.643195\pi$
$654$	0	0
$655$	1.63189	0.0637631
$656$	0	0
$657$	2.07364	0.0809003
$658$	0	0
$659$	35.3819	1.37828	0.689142	−	0.724627i	$-0.257988\pi$
$659$	35.3819	1.37828	0.689142	+	0.724627i	$0.257988\pi$
$660$	0	0
$661$	41.8752	1.62876	0.814378	−	0.580334i	$-0.197078\pi$
$661$	41.8752	1.62876	0.814378	+	0.580334i	$0.197078\pi$
$662$	0	0
$663$	−3.25212	−0.126302
$664$	0	0
$665$	5.10780	0.198072
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0.105572	0.00408167
$670$	0	0
$671$	−53.7071	−2.07334
$672$	0	0
$673$	−32.3743	−1.24794	−0.623970	−	0.781449i	$-0.714481\pi$
$673$	−32.3743	−1.24794	−0.623970	+	0.781449i	$0.714481\pi$
$674$	0	0
$675$	17.5931	0.677158
$676$	0	0
$677$	38.9484	1.49691	0.748454	−	0.663186i	$-0.230796\pi$
$677$	38.9484	1.49691	0.748454	+	0.663186i	$0.230796\pi$
$678$	0	0
$679$	32.5764	1.25017
$680$	0	0
$681$	10.2265	0.391881
$682$	0	0
$683$	39.9899	1.53017	0.765086	−	0.643928i	$-0.222696\pi$
$683$	39.9899	1.53017	0.765086	+	0.643928i	$0.222696\pi$
$684$	0	0
$685$	−14.0374	−0.536342
$686$	0	0
$687$	−3.02745	−0.115505
$688$	0	0
$689$	−14.6473	−0.558018
$690$	0	0
$691$	−19.7880	−0.752770	−0.376385	−	0.926463i	$-0.622833\pi$
$691$	−19.7880	−0.752770	−0.376385	+	0.926463i	$0.622833\pi$
$692$	0	0
$693$	18.4949	0.702561
$694$	0	0
$695$	−14.3342	−0.543728
$696$	0	0
$697$	−28.3688	−1.07455
$698$	0	0
$699$	15.8250	0.598557
$700$	0	0
$701$	−29.4121	−1.11088	−0.555440	−	0.831557i	$-0.687450\pi$
$701$	−29.4121	−1.11088	−0.555440	+	0.831557i	$0.687450\pi$
$702$	0	0
$703$	2.81979	0.106350
$704$	0	0
$705$	−1.34441	−0.0506335
$706$	0	0
$707$	−25.7242	−0.967458
$708$	0	0
$709$	−40.5367	−1.52239	−0.761193	−	0.648525i	$-0.775386\pi$
$709$	−40.5367	−1.52239	−0.761193	+	0.648525i	$0.775386\pi$
$710$	0	0
$711$	24.3052	0.911515
$712$	0	0
$713$	13.9354	0.521886
$714$	0	0
$715$	−3.54154	−0.132446
$716$	0	0
$717$	17.9342	0.669763
$718$	0	0
$719$	5.08472	0.189628	0.0948141	−	0.995495i	$-0.469774\pi$
$719$	5.08472	0.189628	0.0948141	+	0.995495i	$0.469774\pi$
$720$	0	0
$721$	35.0147	1.30402
$722$	0	0
$723$	1.81216	0.0673951
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.33242	−0.197768	−0.0988842	−	0.995099i	$-0.531527\pi$
$727$	−5.33242	−0.197768	−0.0988842	+	0.995099i	$0.531527\pi$
$728$	0	0
$729$	−2.36474	−0.0875830
$730$	0	0
$731$	29.7317	1.09967
$732$	0	0
$733$	8.74448	0.322985	0.161492	−	0.986874i	$-0.448369\pi$
$733$	8.74448	0.322985	0.161492	+	0.986874i	$0.448369\pi$
$734$	0	0
$735$	−1.63049	−0.0601416
$736$	0	0
$737$	−16.3495	−0.602242
$738$	0	0
$739$	−25.1780	−0.926186	−0.463093	−	0.886310i	$-0.653260\pi$
$739$	−25.1780	−0.926186	−0.463093	+	0.886310i	$0.653260\pi$
$740$	0	0
$741$	2.98948	0.109821
$742$	0	0
$743$	50.9579	1.86946	0.934731	−	0.355355i	$-0.115640\pi$
$743$	50.9579	1.86946	0.934731	+	0.355355i	$0.115640\pi$
$744$	0	0
$745$	17.1808	0.629458
$746$	0	0
$747$	31.6553	1.15821
$748$	0	0
$749$	13.7325	0.501776
$750$	0	0
$751$	22.5333	0.822253	0.411126	−	0.911578i	$-0.365135\pi$
$751$	22.5333	0.822253	0.411126	+	0.911578i	$0.365135\pi$
$752$	0	0
$753$	5.65600	0.206116
$754$	0	0
$755$	0.245127	0.00892108
$756$	0	0
$757$	20.2369	0.735522	0.367761	−	0.929920i	$-0.380124\pi$
$757$	20.2369	0.735522	0.367761	+	0.929920i	$0.380124\pi$
$758$	0	0
$759$	−6.34367	−0.230261
$760$	0	0
$761$	41.6120	1.50844	0.754218	−	0.656625i	$-0.228016\pi$
$761$	41.6120	1.50844	0.754218	+	0.656625i	$0.228016\pi$
$762$	0	0
$763$	−18.4705	−0.668677
$764$	0	0
$765$	−6.78301	−0.245240
$766$	0	0
$767$	−2.00519	−0.0724034
$768$	0	0
$769$	3.47614	0.125353	0.0626764	−	0.998034i	$-0.480036\pi$
$769$	3.47614	0.125353	0.0626764	+	0.998034i	$0.480036\pi$
$770$	0	0
$771$	−22.7686	−0.819991
$772$	0	0
$773$	21.1005	0.758932	0.379466	−	0.925206i	$-0.376108\pi$
$773$	21.1005	0.758932	0.379466	+	0.925206i	$0.376108\pi$
$774$	0	0
$775$	26.1995	0.941115
$776$	0	0
$777$	1.26311	0.0453140
$778$	0	0
$779$	26.0778	0.934333
$780$	0	0
$781$	56.3857	2.01764
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.1391	−0.468955
$786$	0	0
$787$	14.2337	0.507378	0.253689	−	0.967286i	$-0.418356\pi$
$787$	14.2337	0.507378	0.253689	+	0.967286i	$0.418356\pi$
$788$	0	0
$789$	−7.62899	−0.271599
$790$	0	0
$791$	24.3117	0.864423
$792$	0	0
$793$	18.0378	0.640542
$794$	0	0
$795$	6.58616	0.233587
$796$	0	0
$797$	−50.5995	−1.79233	−0.896163	−	0.443724i	$-0.853657\pi$
$797$	−50.5995	−1.79233	−0.896163	+	0.443724i	$0.853657\pi$
$798$	0	0
$799$	−8.60101	−0.304282
$800$	0	0
$801$	−6.09252	−0.215269
$802$	0	0
$803$	−3.11454	−0.109910
$804$	0	0
$805$	3.64025	0.128302
$806$	0	0
$807$	−0.500329	−0.0176124
$808$	0	0
$809$	−53.0220	−1.86415	−0.932077	−	0.362261i	$-0.882005\pi$
$809$	−53.0220	−1.86415	−0.932077	+	0.362261i	$0.882005\pi$
$810$	0	0
$811$	36.6813	1.28805	0.644027	−	0.765003i	$-0.277262\pi$
$811$	36.6813	1.28805	0.644027	+	0.765003i	$0.277262\pi$
$812$	0	0
$813$	−11.8116	−0.414249
$814$	0	0
$815$	12.2430	0.428855
$816$	0	0
$817$	−27.3306	−0.956175
$818$	0	0
$819$	−6.21160	−0.217051
$820$	0	0
$821$	26.0472	0.909054	0.454527	−	0.890733i	$-0.349808\pi$
$821$	26.0472	0.909054	0.454527	+	0.890733i	$0.349808\pi$
$822$	0	0
$823$	−0.553811	−0.0193046	−0.00965231	−	0.999953i	$-0.503072\pi$
$823$	−0.553811	−0.0193046	−0.00965231	+	0.999953i	$0.503072\pi$
$824$	0	0
$825$	−11.9265	−0.415228
$826$	0	0
$827$	−46.5323	−1.61809	−0.809043	−	0.587750i	$-0.800014\pi$
$827$	−46.5323	−1.61809	−0.809043	+	0.587750i	$0.800014\pi$
$828$	0	0
$829$	−15.5265	−0.539258	−0.269629	−	0.962964i	$-0.586901\pi$
$829$	−15.5265	−0.539258	−0.269629	+	0.962964i	$0.586901\pi$
$830$	0	0
$831$	10.8189	0.375304
$832$	0	0
$833$	−10.4312	−0.361421
$834$	0	0
$835$	−4.65463	−0.161080
$836$	0	0
$837$	23.6895	0.818828
$838$	0	0
$839$	−44.4093	−1.53318	−0.766590	−	0.642137i	$-0.778048\pi$
$839$	−44.4093	−1.53318	−0.766590	+	0.642137i	$0.778048\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	−5.37942	−0.185277
$844$	0	0
$845$	−8.78732	−0.302293
$846$	0	0
$847$	−5.53989	−0.190353
$848$	0	0
$849$	17.4373	0.598446
$850$	0	0
$851$	2.00962	0.0688888
$852$	0	0
$853$	−3.03167	−0.103803	−0.0519013	−	0.998652i	$-0.516528\pi$
$853$	−3.03167	−0.103803	−0.0519013	+	0.998652i	$0.516528\pi$
$854$	0	0
$855$	6.23522	0.213240
$856$	0	0
$857$	−3.80022	−0.129813	−0.0649064	−	0.997891i	$-0.520675\pi$
$857$	−3.80022	−0.129813	−0.0649064	+	0.997891i	$0.520675\pi$
$858$	0	0
$859$	−11.1993	−0.382115	−0.191057	−	0.981579i	$-0.561192\pi$
$859$	−11.1993	−0.382115	−0.191057	+	0.981579i	$0.561192\pi$
$860$	0	0
$861$	11.6814	0.398103
$862$	0	0
$863$	−36.4426	−1.24052	−0.620260	−	0.784396i	$-0.712973\pi$
$863$	−36.4426	−1.24052	−0.620260	+	0.784396i	$0.712973\pi$
$864$	0	0
$865$	0.630139	0.0214254
$866$	0	0
$867$	−3.04481	−0.103407
$868$	0	0
$869$	−36.5056	−1.23837
$870$	0	0
$871$	5.49108	0.186058
$872$	0	0
$873$	39.7669	1.34590
$874$	0	0
$875$	14.6016	0.493626
$876$	0	0
$877$	−27.6875	−0.934940	−0.467470	−	0.884009i	$-0.654834\pi$
$877$	−27.6875	−0.934940	−0.467470	+	0.884009i	$0.654834\pi$
$878$	0	0
$879$	−2.46361	−0.0830954
$880$	0	0
$881$	−22.6489	−0.763062	−0.381531	−	0.924356i	$-0.624603\pi$
$881$	−22.6489	−0.763062	−0.381531	+	0.924356i	$0.624603\pi$
$882$	0	0
$883$	−28.0182	−0.942887	−0.471443	−	0.881896i	$-0.656267\pi$
$883$	−28.0182	−0.942887	−0.471443	+	0.881896i	$0.656267\pi$
$884$	0	0
$885$	0.901635	0.0303081
$886$	0	0
$887$	6.85335	0.230113	0.115057	−	0.993359i	$-0.463295\pi$
$887$	6.85335	0.230113	0.115057	+	0.993359i	$0.463295\pi$
$888$	0	0
$889$	−16.7866	−0.563003
$890$	0	0
$891$	16.6605	0.558148
$892$	0	0
$893$	7.90640	0.264578
$894$	0	0
$895$	15.3846	0.514252
$896$	0	0
$897$	2.13056	0.0711373
$898$	0	0
$899$	0	0
$900$	0	0
$901$	42.1356	1.40374
$902$	0	0
$903$	−12.2426	−0.407409
$904$	0	0
$905$	−5.28801	−0.175780
$906$	0	0
$907$	−2.49857	−0.0829636	−0.0414818	−	0.999139i	$-0.513208\pi$
$907$	−2.49857	−0.0829636	−0.0414818	+	0.999139i	$0.513208\pi$
$908$	0	0
$909$	−31.4021	−1.04154
$910$	0	0
$911$	39.7230	1.31608	0.658041	−	0.752982i	$-0.271385\pi$
$911$	39.7230	1.31608	0.658041	+	0.752982i	$0.271385\pi$
$912$	0	0
$913$	−47.5452	−1.57352
$914$	0	0
$915$	−8.11071	−0.268132
$916$	0	0
$917$	−4.29895	−0.141964
$918$	0	0
$919$	0.936263	0.0308845	0.0154422	−	0.999881i	$-0.495084\pi$
$919$	0.936263	0.0308845	0.0154422	+	0.999881i	$0.495084\pi$
$920$	0	0
$921$	−7.38807	−0.243445
$922$	0	0
$923$	−18.9374	−0.623333
$924$	0	0
$925$	3.77822	0.124227
$926$	0	0
$927$	42.7433	1.40387
$928$	0	0
$929$	12.6079	0.413653	0.206827	−	0.978378i	$-0.433686\pi$
$929$	12.6079	0.413653	0.206827	+	0.978378i	$0.433686\pi$
$930$	0	0
$931$	9.58882	0.314261
$932$	0	0
$933$	−10.4938	−0.343553
$934$	0	0
$935$	10.1879	0.333179
$936$	0	0
$937$	2.55653	0.0835183	0.0417592	−	0.999128i	$-0.486704\pi$
$937$	2.55653	0.0835183	0.0417592	+	0.999128i	$0.486704\pi$
$938$	0	0
$939$	21.7147	0.708633
$940$	0	0
$941$	35.4450	1.15547	0.577736	−	0.816223i	$-0.303936\pi$
$941$	35.4450	1.15547	0.577736	+	0.816223i	$0.303936\pi$
$942$	0	0
$943$	18.5852	0.605218
$944$	0	0
$945$	6.18823	0.201303
$946$	0	0
$947$	13.9515	0.453362	0.226681	−	0.973969i	$-0.427213\pi$
$947$	13.9515	0.453362	0.226681	+	0.973969i	$0.427213\pi$
$948$	0	0
$949$	1.04603	0.0339557
$950$	0	0
$951$	2.47232	0.0801703
$952$	0	0
$953$	−2.34460	−0.0759490	−0.0379745	−	0.999279i	$-0.512091\pi$
$953$	−2.34460	−0.0759490	−0.0379745	+	0.999279i	$0.512091\pi$
$954$	0	0
$955$	11.8188	0.382446
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.9793	1.19413
$960$	0	0
$961$	4.27827	0.138009
$962$	0	0
$963$	16.7636	0.540201
$964$	0	0
$965$	−13.8192	−0.444854
$966$	0	0
$967$	−36.0689	−1.15990	−0.579949	−	0.814653i	$-0.696928\pi$
$967$	−36.0689	−1.15990	−0.579949	+	0.814653i	$0.696928\pi$
$968$	0	0
$969$	−8.59978	−0.276265
$970$	0	0
$971$	−33.0640	−1.06108	−0.530538	−	0.847661i	$-0.678010\pi$
$971$	−33.0640	−1.06108	−0.530538	+	0.847661i	$0.678010\pi$
$972$	0	0
$973$	37.7612	1.21057
$974$	0	0
$975$	4.00559	0.128282
$976$	0	0
$977$	40.2148	1.28659	0.643293	−	0.765620i	$-0.277568\pi$
$977$	40.2148	1.28659	0.643293	+	0.765620i	$0.277568\pi$
$978$	0	0
$979$	9.15077	0.292460
$980$	0	0
$981$	−22.5474	−0.719882
$982$	0	0
$983$	5.32325	0.169785	0.0848927	−	0.996390i	$-0.472945\pi$
$983$	5.32325	0.169785	0.0848927	+	0.996390i	$0.472945\pi$
$984$	0	0
$985$	−5.64060	−0.179725
$986$	0	0
$987$	3.54164	0.112732
$988$	0	0
$989$	−19.4781	−0.619367
$990$	0	0
$991$	−27.9955	−0.889307	−0.444654	−	0.895703i	$-0.646673\pi$
$991$	−27.9955	−0.889307	−0.444654	+	0.895703i	$0.646673\pi$
$992$	0	0
$993$	−10.6747	−0.338751
$994$	0	0
$995$	−8.10562	−0.256965
$996$	0	0
$997$	−19.4084	−0.614671	−0.307336	−	0.951601i	$-0.599437\pi$
$997$	−19.4084	−0.614671	−0.307336	+	0.951601i	$0.599437\pi$
$998$	0	0
$999$	3.41624	0.108085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3364.2.a.q.1.7	✓	8
29.12	odd	4		3364.2.c.k.1681.12		16
29.17	odd	4		3364.2.c.k.1681.5		16
29.28	even	2		3364.2.a.r.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3364.2.a.q.1.7	✓	8	1.1	even	1	trivial
3364.2.a.r.1.2	yes	8	29.28	even	2
3364.2.c.k.1681.5		16	29.17	odd	4
3364.2.c.k.1681.12		16	29.12	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3364 = 2^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3364.a (trivial)

\(f(q)\)	\(=\)	\(q+0.729419 q^{3} +0.767443 q^{5} -2.02171 q^{7} -2.46795 q^{9} +O(q^{10})\)	\(q+0.729419 q^{3} +0.767443 q^{5} -2.02171 q^{7} -2.46795 q^{9} +3.70678 q^{11} -1.24494 q^{13} +0.559788 q^{15} +3.58130 q^{17} -3.29207 q^{19} -1.47467 q^{21} -2.34621 q^{23} -4.41103 q^{25} -3.98843 q^{27} -5.93955 q^{31} +2.70380 q^{33} -1.55155 q^{35} -0.856538 q^{37} -0.908085 q^{39} -7.92138 q^{41} +8.30193 q^{43} -1.89401 q^{45} -2.40165 q^{47} -2.91270 q^{49} +2.61227 q^{51} +11.7655 q^{53} +2.84474 q^{55} -2.40130 q^{57} +1.61067 q^{59} -14.4889 q^{61} +4.98947 q^{63} -0.955422 q^{65} -4.41071 q^{67} -1.71137 q^{69} +15.2115 q^{71} -0.840227 q^{73} -3.21749 q^{75} -7.49402 q^{77} -9.84833 q^{79} +4.49461 q^{81} -12.8266 q^{83} +2.74844 q^{85} +2.46866 q^{89} +2.51691 q^{91} -4.33242 q^{93} -2.52648 q^{95} -16.1133 q^{97} -9.14813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9	\( 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} + 17 q^{15} - 15 q^{17} - 11 q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 16 q^{27} + 9 q^{31} - 16 q^{33} + 9 q^{35} + 2 q^{37} - 9 q^{39} - 38 q^{41} + 3 q^{43} + 3 q^{45} + 7 q^{47} + 12 q^{49} + 3 q^{51} - 5 q^{53} - 34 q^{55} + 34 q^{57} - 7 q^{59} - 39 q^{61} - 6 q^{63} - 21 q^{65} - 26 q^{67} - 34 q^{69} + 52 q^{71} + 34 q^{73} - 10 q^{77} - 31 q^{79} + 24 q^{81} - 17 q^{83} - 18 q^{85} - 24 q^{89} + 54 q^{91} + 6 q^{93} - 9 q^{95} - 22 q^{97} + 11 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 + 17 * q^15 - 15 * q^17 - 11 * q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 16 * q^27 + 9 * q^31 - 16 * q^33 + 9 * q^35 + 2 * q^37 - 9 * q^39 - 38 * q^41 + 3 * q^43 + 3 * q^45 + 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 - 34 * q^55 + 34 * q^57 - 7 * q^59 - 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 - 34 * q^69 + 52 * q^71 + 34 * q^73 - 10 * q^77 - 31 * q^79 + 24 * q^81 - 17 * q^83 - 18 * q^85 - 24 * q^89 + 54 * q^91 + 6 * q^93 - 9 * q^95 - 22 * q^97 + 11 * q^99

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