LMFDB - Embedded newform 693.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.561553 q^{2} -7.68466 q^{4} +3.31534 q^{5} +7.00000 q^{7} +8.80776 q^{8} -1.86174 q^{10} -11.0000 q^{11} -41.9157 q^{13} -3.93087 q^{14} +56.5312 q^{16} +68.8769 q^{17} -114.408 q^{19} -25.4773 q^{20} +6.17708 q^{22} +124.985 q^{23} -114.009 q^{25} +23.5379 q^{26} -53.7926 q^{28} +147.670 q^{29} -55.9697 q^{31} -102.207 q^{32} -38.6780 q^{34} +23.2074 q^{35} +162.948 q^{37} +64.2462 q^{38} +29.2007 q^{40} -258.617 q^{41} -106.739 q^{43} +84.5312 q^{44} -70.1856 q^{46} -110.779 q^{47} +49.0000 q^{49} +64.0218 q^{50} +322.108 q^{52} -10.4451 q^{53} -36.4688 q^{55} +61.6543 q^{56} -82.9242 q^{58} +182.283 q^{59} +189.879 q^{61} +31.4299 q^{62} -394.855 q^{64} -138.965 q^{65} +580.779 q^{67} -529.295 q^{68} -13.0322 q^{70} +1161.39 q^{71} -79.6998 q^{73} -91.5038 q^{74} +879.187 q^{76} -77.0000 q^{77} +1090.04 q^{79} +187.420 q^{80} +145.227 q^{82} +874.830 q^{83} +228.350 q^{85} +59.9394 q^{86} -96.8854 q^{88} +844.193 q^{89} -293.410 q^{91} -960.466 q^{92} +62.2084 q^{94} -379.302 q^{95} +925.097 q^{97} -27.5161 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - 3 q^{4} + 19 q^{5} + 14 q^{7} - 3 q^{8} + 54 q^{10} - 22 q^{11} + 11 q^{13} + 21 q^{14} - 23 q^{16} + 146 q^{17} - 101 q^{19} + 48 q^{20} - 33 q^{22} + 184 q^{23} + 7 q^{25} + 212 q^{26}+ \cdots + 147 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 - 3 * q^4 + 19 * q^5 + 14 * q^7 - 3 * q^8 + 54 * q^10 - 22 * q^11 + 11 * q^13 + 21 * q^14 - 23 * q^16 + 146 * q^17 - 101 * q^19 + 48 * q^20 - 33 * q^22 + 184 * q^23 + 7 * q^25 + 212 * q^26 - 21 * q^28 + 217 * q^29 + 20 * q^31 - 291 * q^32 + 236 * q^34 + 133 * q^35 - 173 * q^37 + 112 * q^38 - 156 * q^40 + 60 * q^41 - 164 * q^43 + 33 * q^44 + 140 * q^46 + 467 * q^47 + 98 * q^49 + 495 * q^50 + 570 * q^52 - 326 * q^53 - 209 * q^55 - 21 * q^56 + 164 * q^58 + 781 * q^59 - 148 * q^61 + 302 * q^62 - 431 * q^64 + 691 * q^65 + 473 * q^67 - 168 * q^68 + 378 * q^70 + 756 * q^71 - 213 * q^73 - 1288 * q^74 + 942 * q^76 - 154 * q^77 + 168 * q^79 - 1060 * q^80 + 1280 * q^82 + 2096 * q^83 + 1438 * q^85 - 144 * q^86 + 33 * q^88 - 736 * q^89 + 77 * q^91 - 684 * q^92 + 2120 * q^94 - 169 * q^95 + 638 * q^97 + 147 * q^98

Coefficient data

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

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

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

$2$	−0.561553	−0.198539	−0.0992695	−	0.995061i	$-0.531651\pi$
$2$	−0.561553	−0.198539	−0.0992695	+	0.995061i	$0.531651\pi$
$3$	0	0
$3$	0	0
$4$	−7.68466	−0.960582
$5$	3.31534	0.296533	0.148267	−	0.988947i	$-0.452631\pi$
$5$	3.31534	0.296533	0.148267	+	0.988947i	$0.452631\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.80776	0.389252
$9$	0	0
$10$	−1.86174	−0.0588734
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−41.9157	−0.894256	−0.447128	−	0.894470i	$-0.647553\pi$
$13$	−41.9157	−0.894256	−0.447128	+	0.894470i	$0.647553\pi$
$14$	−3.93087	−0.0750407
$15$	0	0
$16$	56.5312	0.883301
$17$	68.8769	0.982653	0.491326	−	0.870975i	$-0.336512\pi$
$17$	68.8769	0.982653	0.491326	+	0.870975i	$0.336512\pi$
$18$	0	0
$19$	−114.408	−1.38142	−0.690711	−	0.723131i	$-0.742702\pi$
$19$	−114.408	−1.38142	−0.690711	+	0.723131i	$0.742702\pi$
$20$	−25.4773	−0.284845
$21$	0	0
$22$	6.17708	0.0598617
$23$	124.985	1.13309	0.566547	−	0.824030i	$-0.308279\pi$
$23$	124.985	1.13309	0.566547	+	0.824030i	$0.308279\pi$
$24$	0	0
$25$	−114.009	−0.912068
$26$	23.5379	0.177545
$27$	0	0
$28$	−53.7926	−0.363066
$29$	147.670	0.945570	0.472785	−	0.881178i	$-0.343249\pi$
$29$	147.670	0.945570	0.472785	+	0.881178i	$0.343249\pi$
$30$	0	0
$31$	−55.9697	−0.324273	−0.162136	−	0.986768i	$-0.551838\pi$
$31$	−55.9697	−0.324273	−0.162136	+	0.986768i	$0.551838\pi$
$32$	−102.207	−0.564621
$33$	0	0
$34$	−38.6780	−0.195095
$35$	23.2074	0.112079
$36$	0	0
$37$	162.948	0.724013	0.362006	−	0.932176i	$-0.382092\pi$
$37$	162.948	0.724013	0.362006	+	0.932176i	$0.382092\pi$
$38$	64.2462	0.274266
$39$	0	0
$40$	29.2007	0.115426
$41$	−258.617	−0.985104	−0.492552	−	0.870283i	$-0.663936\pi$
$41$	−258.617	−0.985104	−0.492552	+	0.870283i	$0.663936\pi$
$42$	0	0
$43$	−106.739	−0.378546	−0.189273	−	0.981924i	$-0.560613\pi$
$43$	−106.739	−0.378546	−0.189273	+	0.981924i	$0.560613\pi$
$44$	84.5312	0.289626
$45$	0	0
$46$	−70.1856	−0.224963
$47$	−110.779	−0.343805	−0.171902	−	0.985114i	$-0.554991\pi$
$47$	−110.779	−0.343805	−0.171902	+	0.985114i	$0.554991\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	64.0218	0.181081
$51$	0	0
$52$	322.108	0.859006
$53$	−10.4451	−0.0270706	−0.0135353	−	0.999908i	$-0.504309\pi$
$53$	−10.4451	−0.0270706	−0.0135353	+	0.999908i	$0.504309\pi$
$54$	0	0
$55$	−36.4688	−0.0894081
$56$	61.6543	0.147123
$57$	0	0
$58$	−82.9242	−0.187732
$59$	182.283	0.402225	0.201112	−	0.979568i	$-0.435544\pi$
$59$	182.283	0.402225	0.201112	+	0.979568i	$0.435544\pi$
$60$	0	0
$61$	189.879	0.398549	0.199274	−	0.979944i	$-0.436141\pi$
$61$	189.879	0.398549	0.199274	+	0.979944i	$0.436141\pi$
$62$	31.4299	0.0643807
$63$	0	0
$64$	−394.855	−0.771201
$65$	−138.965	−0.265177
$66$	0	0
$67$	580.779	1.05901	0.529504	−	0.848308i	$-0.322378\pi$
$67$	580.779	1.05901	0.529504	+	0.848308i	$0.322378\pi$
$68$	−529.295	−0.943919
$69$	0	0
$70$	−13.0322	−0.0222520
$71$	1161.39	1.94129	0.970645	−	0.240515i	$-0.0773164\pi$
$71$	1161.39	1.94129	0.970645	+	0.240515i	$0.0773164\pi$
$72$	0	0
$73$	−79.6998	−0.127783	−0.0638915	−	0.997957i	$-0.520351\pi$
$73$	−79.6998	−0.127783	−0.0638915	+	0.997957i	$0.520351\pi$
$74$	−91.5038	−0.143745
$75$	0	0
$76$	879.187	1.32697
$77$	−77.0000	−0.113961
$78$	0	0
$79$	1090.04	1.55239	0.776195	−	0.630493i	$-0.217147\pi$
$79$	1090.04	1.55239	0.776195	+	0.630493i	$0.217147\pi$
$80$	187.420	0.261928
$81$	0	0
$82$	145.227	0.195581
$83$	874.830	1.15693	0.578464	−	0.815708i	$-0.303652\pi$
$83$	874.830	1.15693	0.578464	+	0.815708i	$0.303652\pi$
$84$	0	0
$85$	228.350	0.291389
$86$	59.9394	0.0751562
$87$	0	0
$88$	−96.8854	−0.117364
$89$	844.193	1.00544	0.502721	−	0.864449i	$-0.332332\pi$
$89$	844.193	1.00544	0.502721	+	0.864449i	$0.332332\pi$
$90$	0	0
$91$	−293.410	−0.337997
$92$	−960.466	−1.08843
$93$	0	0
$94$	62.2084	0.0682586
$95$	−379.302	−0.409638
$96$	0	0
$97$	925.097	0.968344	0.484172	−	0.874973i	$-0.339121\pi$
$97$	925.097	0.968344	0.484172	+	0.874973i	$0.339121\pi$
$98$	−27.5161	−0.0283627
$99$	0	0
$100$	876.116	0.876116
$101$	884.867	0.871758	0.435879	−	0.900005i	$-0.356438\pi$
$101$	884.867	0.871758	0.435879	+	0.900005i	$0.356438\pi$
$102$	0	0
$103$	1069.78	1.02339	0.511694	−	0.859168i	$-0.329018\pi$
$103$	1069.78	1.02339	0.511694	+	0.859168i	$0.329018\pi$
$104$	−369.184	−0.348091
$105$	0	0
$106$	5.86547	0.00537457
$107$	−178.196	−0.160999	−0.0804993	−	0.996755i	$-0.525651\pi$
$107$	−178.196	−0.160999	−0.0804993	+	0.996755i	$0.525651\pi$
$108$	0	0
$109$	1973.58	1.73426	0.867130	−	0.498082i	$-0.165962\pi$
$109$	1973.58	1.73426	0.867130	+	0.498082i	$0.165962\pi$
$110$	20.4791	0.0177510
$111$	0	0
$112$	395.719	0.333856
$113$	−1642.68	−1.36753	−0.683763	−	0.729704i	$-0.739658\pi$
$113$	−1642.68	−1.36753	−0.683763	+	0.729704i	$0.739658\pi$
$114$	0	0
$115$	414.367	0.336000
$116$	−1134.79	−0.908298
$117$	0	0
$118$	−102.362	−0.0798572
$119$	482.138	0.371408
$120$	0	0
$121$	121.000	0.0909091
$122$	−106.627	−0.0791275
$123$	0	0
$124$	430.108	0.311491
$125$	−792.395	−0.566992
$126$	0	0
$127$	−1796.94	−1.25554	−0.627768	−	0.778400i	$-0.716031\pi$
$127$	−1796.94	−1.25554	−0.627768	+	0.778400i	$0.716031\pi$
$128$	1039.39	0.717735
$129$	0	0
$130$	78.0361	0.0526479
$131$	−934.305	−0.623134	−0.311567	−	0.950224i	$-0.600854\pi$
$131$	−934.305	−0.623134	−0.311567	+	0.950224i	$0.600854\pi$
$132$	0	0
$133$	−800.857	−0.522129
$134$	−326.138	−0.210254
$135$	0	0
$136$	606.651	0.382499
$137$	1590.87	0.992097	0.496048	−	0.868295i	$-0.334784\pi$
$137$	1590.87	0.992097	0.496048	+	0.868295i	$0.334784\pi$
$138$	0	0
$139$	−1549.00	−0.945213	−0.472607	−	0.881274i	$-0.656687\pi$
$139$	−1549.00	−0.945213	−0.472607	+	0.881274i	$0.656687\pi$
$140$	−178.341	−0.107661
$141$	0	0
$142$	−652.182	−0.385422
$143$	461.073	0.269628
$144$	0	0
$145$	489.575	0.280393
$146$	44.7557	0.0253699
$147$	0	0
$148$	−1252.20	−0.695474
$149$	468.094	0.257367	0.128684	−	0.991686i	$-0.458925\pi$
$149$	468.094	0.257367	0.128684	+	0.991686i	$0.458925\pi$
$150$	0	0
$151$	1865.34	1.00529	0.502647	−	0.864492i	$-0.332360\pi$
$151$	1865.34	1.00529	0.502647	+	0.864492i	$0.332360\pi$
$152$	−1007.68	−0.537721
$153$	0	0
$154$	43.2396	0.0226256
$155$	−185.559	−0.0961576
$156$	0	0
$157$	1501.59	0.763310	0.381655	−	0.924305i	$-0.375354\pi$
$157$	1501.59	0.763310	0.381655	+	0.924305i	$0.375354\pi$
$158$	−612.114	−0.308210
$159$	0	0
$160$	−338.852	−0.167429
$161$	874.894	0.428269
$162$	0	0
$163$	524.401	0.251989	0.125995	−	0.992031i	$-0.459788\pi$
$163$	524.401	0.251989	0.125995	+	0.992031i	$0.459788\pi$
$164$	1987.39	0.946273
$165$	0	0
$166$	−491.263	−0.229695
$167$	3293.09	1.52591	0.762956	−	0.646450i	$-0.223747\pi$
$167$	3293.09	1.52591	0.762956	+	0.646450i	$0.223747\pi$
$168$	0	0
$169$	−440.073	−0.200306
$170$	−128.231	−0.0578521
$171$	0	0
$172$	820.250	0.363625
$173$	3036.96	1.33466	0.667330	−	0.744762i	$-0.267437\pi$
$173$	3036.96	1.33466	0.667330	+	0.744762i	$0.267437\pi$
$174$	0	0
$175$	−798.060	−0.344729
$176$	−621.844	−0.266325
$177$	0	0
$178$	−474.059	−0.199619
$179$	−690.152	−0.288181	−0.144090	−	0.989565i	$-0.546026\pi$
$179$	−690.152	−0.288181	−0.144090	+	0.989565i	$0.546026\pi$
$180$	0	0
$181$	−2533.94	−1.04059	−0.520294	−	0.853987i	$-0.674178\pi$
$181$	−2533.94	−1.04059	−0.520294	+	0.853987i	$0.674178\pi$
$182$	164.765	0.0671056
$183$	0	0
$184$	1100.84	0.441059
$185$	540.228	0.214694
$186$	0	0
$187$	−757.646	−0.296281
$188$	851.301	0.330253
$189$	0	0
$190$	212.998	0.0813290
$191$	−1845.13	−0.698998	−0.349499	−	0.936937i	$-0.613648\pi$
$191$	−1845.13	−0.698998	−0.349499	+	0.936937i	$0.613648\pi$
$192$	0	0
$193$	−2654.48	−0.990019	−0.495009	−	0.868888i	$-0.664835\pi$
$193$	−2654.48	−0.990019	−0.495009	+	0.868888i	$0.664835\pi$
$194$	−519.491	−0.192254
$195$	0	0
$196$	−376.548	−0.137226
$197$	−164.057	−0.0593328	−0.0296664	−	0.999560i	$-0.509445\pi$
$197$	−164.057	−0.0593328	−0.0296664	+	0.999560i	$0.509445\pi$
$198$	0	0
$199$	2888.41	1.02891	0.514457	−	0.857516i	$-0.327993\pi$
$199$	2888.41	1.02891	0.514457	+	0.857516i	$0.327993\pi$
$200$	−1004.16	−0.355024
$201$	0	0
$202$	−496.900	−0.173078
$203$	1033.69	0.357392
$204$	0	0
$205$	−857.405	−0.292116
$206$	−600.740	−0.203182
$207$	0	0
$208$	−2369.55	−0.789897
$209$	1258.49	0.416515
$210$	0	0
$211$	630.956	0.205862	0.102931	−	0.994689i	$-0.467178\pi$
$211$	630.956	0.205862	0.102931	+	0.994689i	$0.467178\pi$
$212$	80.2670	0.0260036
$213$	0	0
$214$	100.066	0.0319645
$215$	−353.875	−0.112252
$216$	0	0
$217$	−391.788	−0.122564
$218$	−1108.27	−0.344318
$219$	0	0
$220$	280.250	0.0858839
$221$	−2887.02	−0.878743
$222$	0	0
$223$	−6252.50	−1.87757	−0.938786	−	0.344501i	$-0.888048\pi$
$223$	−6252.50	−1.87757	−0.938786	+	0.344501i	$0.888048\pi$
$224$	−715.452	−0.213407
$225$	0	0
$226$	922.453	0.271507
$227$	3316.20	0.969621	0.484810	−	0.874619i	$-0.338889\pi$
$227$	3316.20	0.969621	0.484810	+	0.874619i	$0.338889\pi$
$228$	0	0
$229$	3077.20	0.887979	0.443989	−	0.896032i	$-0.353563\pi$
$229$	3077.20	0.887979	0.443989	+	0.896032i	$0.353563\pi$
$230$	−232.689	−0.0667090
$231$	0	0
$232$	1300.64	0.368065
$233$	−1358.08	−0.381849	−0.190924	−	0.981605i	$-0.561149\pi$
$233$	−1358.08	−0.381849	−0.190924	+	0.981605i	$0.561149\pi$
$234$	0	0
$235$	−367.271	−0.101950
$236$	−1400.78	−0.386370
$237$	0	0
$238$	−270.746	−0.0737389
$239$	4151.18	1.12350	0.561752	−	0.827306i	$-0.310128\pi$
$239$	4151.18	1.12350	0.561752	+	0.827306i	$0.310128\pi$
$240$	0	0
$241$	−3491.71	−0.933282	−0.466641	−	0.884447i	$-0.654536\pi$
$241$	−3491.71	−0.933282	−0.466641	+	0.884447i	$0.654536\pi$
$242$	−67.9479	−0.0180490
$243$	0	0
$244$	−1459.15	−0.382839
$245$	162.452	0.0423619
$246$	0	0
$247$	4795.50	1.23535
$248$	−492.968	−0.126224
$249$	0	0
$250$	444.972	0.112570
$251$	879.383	0.221140	0.110570	−	0.993868i	$-0.464732\pi$
$251$	879.383	0.221140	0.110570	+	0.993868i	$0.464732\pi$
$252$	0	0
$253$	−1374.83	−0.341640
$254$	1009.08	0.249273
$255$	0	0
$256$	2575.17	0.628703
$257$	−5691.10	−1.38133	−0.690663	−	0.723176i	$-0.742681\pi$
$257$	−5691.10	−1.38133	−0.690663	+	0.723176i	$0.742681\pi$
$258$	0	0
$259$	1140.64	0.273651
$260$	1067.90	0.254724
$261$	0	0
$262$	524.661	0.123716
$263$	−2024.55	−0.474673	−0.237337	−	0.971427i	$-0.576274\pi$
$263$	−2024.55	−0.474673	−0.237337	+	0.971427i	$0.576274\pi$
$264$	0	0
$265$	−34.6290	−0.00802734
$266$	449.723	0.103663
$267$	0	0
$268$	−4463.09	−1.01726
$269$	3431.96	0.777882	0.388941	−	0.921263i	$-0.372841\pi$
$269$	3431.96	0.777882	0.388941	+	0.921263i	$0.372841\pi$
$270$	0	0
$271$	−2974.80	−0.666812	−0.333406	−	0.942783i	$-0.608198\pi$
$271$	−2974.80	−0.666812	−0.333406	+	0.942783i	$0.608198\pi$
$272$	3893.70	0.867978
$273$	0	0
$274$	−893.358	−0.196970
$275$	1254.09	0.274999
$276$	0	0
$277$	7781.98	1.68799	0.843996	−	0.536350i	$-0.180197\pi$
$277$	7781.98	1.68799	0.843996	+	0.536350i	$0.180197\pi$
$278$	869.846	0.187662
$279$	0	0
$280$	204.405	0.0436270
$281$	2627.68	0.557845	0.278922	−	0.960314i	$-0.410023\pi$
$281$	2627.68	0.557845	0.278922	+	0.960314i	$0.410023\pi$
$282$	0	0
$283$	−2501.46	−0.525430	−0.262715	−	0.964873i	$-0.584618\pi$
$283$	−2501.46	−0.525430	−0.262715	+	0.964873i	$0.584618\pi$
$284$	−8924.89	−1.86477
$285$	0	0
$286$	−258.917	−0.0535317
$287$	−1810.32	−0.372334
$288$	0	0
$289$	−168.973	−0.0343931
$290$	−274.922	−0.0556689
$291$	0	0
$292$	612.466	0.122746
$293$	4646.04	0.926364	0.463182	−	0.886263i	$-0.346708\pi$
$293$	4646.04	0.926364	0.463182	+	0.886263i	$0.346708\pi$
$294$	0	0
$295$	604.331	0.119273
$296$	1435.21	0.281823
$297$	0	0
$298$	−262.859	−0.0510974
$299$	−5238.83	−1.01328
$300$	0	0
$301$	−747.170	−0.143077
$302$	−1047.49	−0.199590
$303$	0	0
$304$	−6467.63	−1.22021
$305$	629.513	0.118183
$306$	0	0
$307$	−4325.92	−0.804213	−0.402107	−	0.915593i	$-0.631722\pi$
$307$	−4325.92	−0.804213	−0.402107	+	0.915593i	$0.631722\pi$
$308$	591.719	0.109469
$309$	0	0
$310$	104.201	0.0190910
$311$	−1845.22	−0.336440	−0.168220	−	0.985749i	$-0.553802\pi$
$311$	−1845.22	−0.336440	−0.168220	+	0.985749i	$0.553802\pi$
$312$	0	0
$313$	−5287.15	−0.954784	−0.477392	−	0.878690i	$-0.658418\pi$
$313$	−5287.15	−0.954784	−0.477392	+	0.878690i	$0.658418\pi$
$314$	−843.220	−0.151547
$315$	0	0
$316$	−8376.57	−1.49120
$317$	−9353.74	−1.65728	−0.828641	−	0.559780i	$-0.810886\pi$
$317$	−9353.74	−1.65728	−0.828641	+	0.559780i	$0.810886\pi$
$318$	0	0
$319$	−1624.36	−0.285100
$320$	−1309.08	−0.228687
$321$	0	0
$322$	−491.299	−0.0850280
$323$	−7880.08	−1.35746
$324$	0	0
$325$	4778.75	0.815622
$326$	−294.479	−0.0500296
$327$	0	0
$328$	−2277.84	−0.383453
$329$	−775.455	−0.129946
$330$	0	0
$331$	11385.6	1.89066	0.945330	−	0.326115i	$-0.105740\pi$
$331$	11385.6	1.89066	0.945330	+	0.326115i	$0.105740\pi$
$332$	−6722.77	−1.11132
$333$	0	0
$334$	−1849.25	−0.302953
$335$	1925.48	0.314031
$336$	0	0
$337$	−10686.1	−1.72732	−0.863660	−	0.504075i	$-0.831833\pi$
$337$	−10686.1	−1.72732	−0.863660	+	0.504075i	$0.831833\pi$
$338$	247.124	0.0397686
$339$	0	0
$340$	−1754.80	−0.279903
$341$	615.667	0.0977719
$342$	0	0
$343$	343.000	0.0539949
$344$	−940.129	−0.147350
$345$	0	0
$346$	−1705.42	−0.264982
$347$	−6242.71	−0.965782	−0.482891	−	0.875680i	$-0.660413\pi$
$347$	−6242.71	−0.965782	−0.482891	+	0.875680i	$0.660413\pi$
$348$	0	0
$349$	4518.02	0.692964	0.346482	−	0.938057i	$-0.387376\pi$
$349$	4518.02	0.692964	0.346482	+	0.938057i	$0.387376\pi$
$350$	448.153	0.0684422
$351$	0	0
$352$	1124.28	0.170240
$353$	9270.29	1.39776	0.698878	−	0.715241i	$-0.253683\pi$
$353$	9270.29	1.39776	0.698878	+	0.715241i	$0.253683\pi$
$354$	0	0
$355$	3850.40	0.575657
$356$	−6487.34	−0.965809
$357$	0	0
$358$	387.557	0.0572151
$359$	−8219.53	−1.20838	−0.604192	−	0.796839i	$-0.706504\pi$
$359$	−8219.53	−1.20838	−0.604192	+	0.796839i	$0.706504\pi$
$360$	0	0
$361$	6230.22	0.908328
$362$	1422.94	0.206597
$363$	0	0
$364$	2254.76	0.324674
$365$	−264.232	−0.0378919
$366$	0	0
$367$	9001.57	1.28032	0.640161	−	0.768241i	$-0.278868\pi$
$367$	9001.57	1.28032	0.640161	+	0.768241i	$0.278868\pi$
$368$	7065.55	1.00086
$369$	0	0
$370$	−303.367	−0.0426251
$371$	−73.1156	−0.0102317
$372$	0	0
$373$	1966.15	0.272931	0.136466	−	0.990645i	$-0.456426\pi$
$373$	1966.15	0.272931	0.136466	+	0.990645i	$0.456426\pi$
$374$	425.458	0.0588233
$375$	0	0
$376$	−975.718	−0.133827
$377$	−6189.67	−0.845582
$378$	0	0
$379$	1022.30	0.138554	0.0692768	−	0.997597i	$-0.477931\pi$
$379$	1022.30	0.138554	0.0692768	+	0.997597i	$0.477931\pi$
$380$	2914.81	0.393491
$381$	0	0
$382$	1036.14	0.138778
$383$	−12825.9	−1.71116	−0.855579	−	0.517672i	$-0.826799\pi$
$383$	−12825.9	−1.71116	−0.855579	+	0.517672i	$0.826799\pi$
$384$	0	0
$385$	−255.281	−0.0337931
$386$	1490.63	0.196557
$387$	0	0
$388$	−7109.05	−0.930174
$389$	12229.1	1.59393	0.796964	−	0.604027i	$-0.206438\pi$
$389$	12229.1	1.59393	0.796964	+	0.604027i	$0.206438\pi$
$390$	0	0
$391$	8608.57	1.11344
$392$	431.580	0.0556074
$393$	0	0
$394$	92.1266	0.0117799
$395$	3613.85	0.460335
$396$	0	0
$397$	−3401.07	−0.429961	−0.214981	−	0.976618i	$-0.568969\pi$
$397$	−3401.07	−0.429961	−0.214981	+	0.976618i	$0.568969\pi$
$398$	−1622.00	−0.204280
$399$	0	0
$400$	−6445.04	−0.805630
$401$	6234.40	0.776386	0.388193	−	0.921578i	$-0.373099\pi$
$401$	6234.40	0.776386	0.388193	+	0.921578i	$0.373099\pi$
$402$	0	0
$403$	2346.01	0.289983
$404$	−6799.90	−0.837396
$405$	0	0
$406$	−580.470	−0.0709562
$407$	−1792.43	−0.218298
$408$	0	0
$409$	−11429.3	−1.38177	−0.690885	−	0.722964i	$-0.742779\pi$
$409$	−11429.3	−1.38177	−0.690885	+	0.722964i	$0.742779\pi$
$410$	481.478	0.0579964
$411$	0	0
$412$	−8220.93	−0.983048
$413$	1275.98	0.152027
$414$	0	0
$415$	2900.36	0.343068
$416$	4284.10	0.504916
$417$	0	0
$418$	−706.708	−0.0826943
$419$	−3827.73	−0.446293	−0.223146	−	0.974785i	$-0.571633\pi$
$419$	−3827.73	−0.446293	−0.223146	+	0.974785i	$0.571633\pi$
$420$	0	0
$421$	−11626.8	−1.34597	−0.672987	−	0.739654i	$-0.734989\pi$
$421$	−11626.8	−1.34597	−0.672987	+	0.739654i	$0.734989\pi$
$422$	−354.315	−0.0408716
$423$	0	0
$424$	−91.9979	−0.0105373
$425$	−7852.55	−0.896246
$426$	0	0
$427$	1329.15	0.150637
$428$	1369.38	0.154652
$429$	0	0
$430$	198.720	0.0222863
$431$	−7469.80	−0.834820	−0.417410	−	0.908718i	$-0.637062\pi$
$431$	−7469.80	−0.834820	−0.417410	+	0.908718i	$0.637062\pi$
$432$	0	0
$433$	1546.97	0.171692	0.0858459	−	0.996308i	$-0.472641\pi$
$433$	1546.97	0.171692	0.0858459	+	0.996308i	$0.472641\pi$
$434$	220.010	0.0243336
$435$	0	0
$436$	−15166.3	−1.66590
$437$	−14299.3	−1.56528
$438$	0	0
$439$	−8082.79	−0.878748	−0.439374	−	0.898304i	$-0.644800\pi$
$439$	−8082.79	−0.878748	−0.439374	+	0.898304i	$0.644800\pi$
$440$	−321.208	−0.0348023
$441$	0	0
$442$	1621.22	0.174465
$443$	13000.1	1.39426	0.697128	−	0.716947i	$-0.254461\pi$
$443$	13000.1	1.39426	0.697128	+	0.716947i	$0.254461\pi$
$444$	0	0
$445$	2798.79	0.298147
$446$	3511.11	0.372771
$447$	0	0
$448$	−2763.99	−0.291487
$449$	4572.68	0.480619	0.240310	−	0.970696i	$-0.422751\pi$
$449$	4572.68	0.480619	0.240310	+	0.970696i	$0.422751\pi$
$450$	0	0
$451$	2844.79	0.297020
$452$	12623.4	1.31362
$453$	0	0
$454$	−1862.22	−0.192507
$455$	−972.754	−0.100227
$456$	0	0
$457$	−10846.5	−1.11024	−0.555119	−	0.831771i	$-0.687327\pi$
$457$	−10846.5	−1.11024	−0.555119	+	0.831771i	$0.687327\pi$
$458$	−1728.01	−0.176298
$459$	0	0
$460$	−3184.27	−0.322755
$461$	1299.33	0.131270	0.0656352	−	0.997844i	$-0.479093\pi$
$461$	1299.33	0.131270	0.0656352	+	0.997844i	$0.479093\pi$
$462$	0	0
$463$	12116.9	1.21624	0.608120	−	0.793845i	$-0.291924\pi$
$463$	12116.9	1.21624	0.608120	+	0.793845i	$0.291924\pi$
$464$	8347.94	0.835223
$465$	0	0
$466$	762.633	0.0758118
$467$	13546.6	1.34232	0.671160	−	0.741313i	$-0.265796\pi$
$467$	13546.6	1.34232	0.671160	+	0.741313i	$0.265796\pi$
$468$	0	0
$469$	4065.46	0.400267
$470$	206.242	0.0202409
$471$	0	0
$472$	1605.51	0.156567
$473$	1174.12	0.114136
$474$	0	0
$475$	13043.5	1.25995
$476$	−3705.07	−0.356768
$477$	0	0
$478$	−2331.10	−0.223059
$479$	10663.2	1.01715	0.508573	−	0.861019i	$-0.330173\pi$
$479$	10663.2	1.01715	0.508573	+	0.861019i	$0.330173\pi$
$480$	0	0
$481$	−6830.08	−0.647453
$482$	1960.78	0.185293
$483$	0	0
$484$	−929.844	−0.0873257
$485$	3067.01	0.287146
$486$	0	0
$487$	8149.91	0.758332	0.379166	−	0.925329i	$-0.376211\pi$
$487$	8149.91	0.758332	0.379166	+	0.925329i	$0.376211\pi$
$488$	1672.41	0.155136
$489$	0	0
$490$	−91.2252	−0.00841048
$491$	15142.4	1.39178	0.695891	−	0.718147i	$-0.255010\pi$
$491$	15142.4	1.39178	0.695891	+	0.718147i	$0.255010\pi$
$492$	0	0
$493$	10171.0	0.929167
$494$	−2692.93	−0.245264
$495$	0	0
$496$	−3164.04	−0.286430
$497$	8129.73	0.733739
$498$	0	0
$499$	10430.8	0.935764	0.467882	−	0.883791i	$-0.345017\pi$
$499$	10430.8	0.935764	0.467882	+	0.883791i	$0.345017\pi$
$500$	6089.28	0.544642
$501$	0	0
$502$	−493.820	−0.0439049
$503$	−2173.55	−0.192671	−0.0963356	−	0.995349i	$-0.530712\pi$
$503$	−2173.55	−0.192671	−0.0963356	+	0.995349i	$0.530712\pi$
$504$	0	0
$505$	2933.64	0.258505
$506$	772.042	0.0678289
$507$	0	0
$508$	13808.9	1.20605
$509$	13729.6	1.19559	0.597793	−	0.801651i	$-0.296045\pi$
$509$	13729.6	1.19559	0.597793	+	0.801651i	$0.296045\pi$
$510$	0	0
$511$	−557.899	−0.0482974
$512$	−9761.22	−0.842557
$513$	0	0
$514$	3195.85	0.274247
$515$	3546.70	0.303468
$516$	0	0
$517$	1218.57	0.103661
$518$	−640.527	−0.0543304
$519$	0	0
$520$	−1223.97	−0.103220
$521$	−18257.2	−1.53525	−0.767623	−	0.640902i	$-0.778561\pi$
$521$	−18257.2	−1.53525	−0.767623	+	0.640902i	$0.778561\pi$
$522$	0	0
$523$	−19979.6	−1.67045	−0.835225	−	0.549908i	$-0.814663\pi$
$523$	−19979.6	−1.67045	−0.835225	+	0.549908i	$0.814663\pi$
$524$	7179.81	0.598572
$525$	0	0
$526$	1136.89	0.0942411
$527$	−3855.02	−0.318648
$528$	0	0
$529$	3454.21	0.283900
$530$	19.4460	0.00159374
$531$	0	0
$532$	6154.31	0.501548
$533$	10840.1	0.880935
$534$	0	0
$535$	−590.780	−0.0477414
$536$	5115.37	0.412221
$537$	0	0
$538$	−1927.23	−0.154440
$539$	−539.000	−0.0430730
$540$	0	0
$541$	7904.01	0.628133	0.314066	−	0.949401i	$-0.398309\pi$
$541$	7904.01	0.628133	0.314066	+	0.949401i	$0.398309\pi$
$542$	1670.51	0.132388
$543$	0	0
$544$	−7039.73	−0.554827
$545$	6543.08	0.514265
$546$	0	0
$547$	16806.6	1.31370	0.656852	−	0.754019i	$-0.271887\pi$
$547$	16806.6	1.31370	0.656852	+	0.754019i	$0.271887\pi$
$548$	−12225.3	−0.952991
$549$	0	0
$550$	−704.240	−0.0545980
$551$	−16894.6	−1.30623
$552$	0	0
$553$	7630.26	0.586748
$554$	−4369.99	−0.335132
$555$	0	0
$556$	11903.6	0.907955
$557$	8002.82	0.608780	0.304390	−	0.952548i	$-0.401547\pi$
$557$	8002.82	0.608780	0.304390	+	0.952548i	$0.401547\pi$
$558$	0	0
$559$	4474.03	0.338517
$560$	1311.94	0.0989995
$561$	0	0
$562$	−1475.58	−0.110754
$563$	12327.9	0.922840	0.461420	−	0.887182i	$-0.347340\pi$
$563$	12327.9	0.922840	0.461420	+	0.887182i	$0.347340\pi$
$564$	0	0
$565$	−5446.05	−0.405517
$566$	1404.70	0.104318
$567$	0	0
$568$	10229.2	0.755651
$569$	6786.77	0.500028	0.250014	−	0.968242i	$-0.419565\pi$
$569$	6786.77	0.500028	0.250014	+	0.968242i	$0.419565\pi$
$570$	0	0
$571$	16890.0	1.23787	0.618935	−	0.785442i	$-0.287564\pi$
$571$	16890.0	1.23787	0.618935	+	0.785442i	$0.287564\pi$
$572$	−3543.19	−0.259000
$573$	0	0
$574$	1016.59	0.0739228
$575$	−14249.3	−1.03346
$576$	0	0
$577$	1857.64	0.134029	0.0670144	−	0.997752i	$-0.478653\pi$
$577$	1857.64	0.134029	0.0670144	+	0.997752i	$0.478653\pi$
$578$	94.8875	0.00682837
$579$	0	0
$580$	−3762.22	−0.269341
$581$	6123.81	0.437278
$582$	0	0
$583$	114.896	0.00816210
$584$	−701.977	−0.0497398
$585$	0	0
$586$	−2609.00	−0.183919
$587$	−21977.7	−1.54534	−0.772672	−	0.634805i	$-0.781080\pi$
$587$	−21977.7	−1.54534	−0.772672	+	0.634805i	$0.781080\pi$
$588$	0	0
$589$	6403.39	0.447958
$590$	−339.364	−0.0236803
$591$	0	0
$592$	9211.65	0.639521
$593$	−17263.7	−1.19550	−0.597752	−	0.801681i	$-0.703939\pi$
$593$	−17263.7	−1.19550	−0.597752	+	0.801681i	$0.703939\pi$
$594$	0	0
$595$	1598.45	0.110135
$596$	−3597.14	−0.247223
$597$	0	0
$598$	2941.88	0.201175
$599$	23211.4	1.58329	0.791646	−	0.610981i	$-0.209225\pi$
$599$	23211.4	1.58329	0.791646	+	0.610981i	$0.209225\pi$
$600$	0	0
$601$	−20051.0	−1.36089	−0.680446	−	0.732799i	$-0.738214\pi$
$601$	−20051.0	−1.36089	−0.680446	+	0.732799i	$0.738214\pi$
$602$	419.576	0.0284064
$603$	0	0
$604$	−14334.5	−0.965668
$605$	401.156	0.0269576
$606$	0	0
$607$	−7862.63	−0.525756	−0.262878	−	0.964829i	$-0.584672\pi$
$607$	−7862.63	−0.525756	−0.262878	+	0.964829i	$0.584672\pi$
$608$	11693.4	0.779981
$609$	0	0
$610$	−353.505	−0.0234639
$611$	4643.39	0.307449
$612$	0	0
$613$	−16366.2	−1.07835	−0.539173	−	0.842195i	$-0.681263\pi$
$613$	−16366.2	−1.07835	−0.539173	+	0.842195i	$0.681263\pi$
$614$	2429.23	0.159668
$615$	0	0
$616$	−678.198	−0.0443594
$617$	−3749.07	−0.244622	−0.122311	−	0.992492i	$-0.539031\pi$
$617$	−3749.07	−0.244622	−0.122311	+	0.992492i	$0.539031\pi$
$618$	0	0
$619$	20036.9	1.30105	0.650525	−	0.759485i	$-0.274549\pi$
$619$	20036.9	1.30105	0.650525	+	0.759485i	$0.274549\pi$
$620$	1425.95	0.0923673
$621$	0	0
$622$	1036.19	0.0667965
$623$	5909.35	0.380021
$624$	0	0
$625$	11624.0	0.743936
$626$	2969.01	0.189562
$627$	0	0
$628$	−11539.2	−0.733222
$629$	11223.3	0.711453
$630$	0	0
$631$	5839.34	0.368400	0.184200	−	0.982889i	$-0.441031\pi$
$631$	5839.34	0.368400	0.184200	+	0.982889i	$0.441031\pi$
$632$	9600.80	0.604271
$633$	0	0
$634$	5252.62	0.329035
$635$	−5957.49	−0.372308
$636$	0	0
$637$	−2053.87	−0.127751
$638$	912.166	0.0566035
$639$	0	0
$640$	3445.94	0.212832
$641$	14682.2	0.904702	0.452351	−	0.891840i	$-0.350585\pi$
$641$	14682.2	0.904702	0.452351	+	0.891840i	$0.350585\pi$
$642$	0	0
$643$	30541.3	1.87314	0.936571	−	0.350477i	$-0.113981\pi$
$643$	30541.3	1.87314	0.936571	+	0.350477i	$0.113981\pi$
$644$	−6723.26	−0.411388
$645$	0	0
$646$	4425.08	0.269508
$647$	−7991.44	−0.485588	−0.242794	−	0.970078i	$-0.578064\pi$
$647$	−7991.44	−0.485588	−0.242794	+	0.970078i	$0.578064\pi$
$648$	0	0
$649$	−2005.11	−0.121275
$650$	−2683.52	−0.161933
$651$	0	0
$652$	−4029.84	−0.242056
$653$	−18400.0	−1.10268	−0.551338	−	0.834282i	$-0.685883\pi$
$653$	−18400.0	−1.10268	−0.551338	+	0.834282i	$0.685883\pi$
$654$	0	0
$655$	−3097.54	−0.184780
$656$	−14620.0	−0.870143
$657$	0	0
$658$	435.459	0.0257993
$659$	−15887.4	−0.939126	−0.469563	−	0.882899i	$-0.655589\pi$
$659$	−15887.4	−0.939126	−0.469563	+	0.882899i	$0.655589\pi$
$660$	0	0
$661$	−14947.8	−0.879582	−0.439791	−	0.898100i	$-0.644947\pi$
$661$	−14947.8	−0.879582	−0.439791	+	0.898100i	$0.644947\pi$
$662$	−6393.61	−0.375370
$663$	0	0
$664$	7705.29	0.450336
$665$	−2655.11	−0.154828
$666$	0	0
$667$	18456.4	1.07142
$668$	−25306.3	−1.46576
$669$	0	0
$670$	−1081.26	−0.0623473
$671$	−2088.67	−0.120167
$672$	0	0
$673$	1091.22	0.0625014	0.0312507	−	0.999512i	$-0.490051\pi$
$673$	1091.22	0.0625014	0.0312507	+	0.999512i	$0.490051\pi$
$674$	6000.79	0.342940
$675$	0	0
$676$	3381.81	0.192411
$677$	26564.2	1.50804	0.754022	−	0.656849i	$-0.228111\pi$
$677$	26564.2	1.50804	0.754022	+	0.656849i	$0.228111\pi$
$678$	0	0
$679$	6475.68	0.365999
$680$	2011.26	0.113424
$681$	0	0
$682$	−345.729	−0.0194115
$683$	20461.6	1.14633	0.573164	−	0.819441i	$-0.305716\pi$
$683$	20461.6	1.14633	0.573164	+	0.819441i	$0.305716\pi$
$684$	0	0
$685$	5274.28	0.294190
$686$	−192.613	−0.0107201
$687$	0	0
$688$	−6034.07	−0.334370
$689$	437.813	0.0242081
$690$	0	0
$691$	−12100.6	−0.666175	−0.333087	−	0.942896i	$-0.608090\pi$
$691$	−12100.6	−0.666175	−0.333087	+	0.942896i	$0.608090\pi$
$692$	−23338.0	−1.28205
$693$	0	0
$694$	3505.61	0.191745
$695$	−5135.47	−0.280287
$696$	0	0
$697$	−17812.8	−0.968015
$698$	−2537.11	−0.137580
$699$	0	0
$700$	6132.82	0.331141
$701$	−27360.5	−1.47417	−0.737083	−	0.675802i	$-0.763797\pi$
$701$	−27360.5	−1.47417	−0.737083	+	0.675802i	$0.763797\pi$
$702$	0	0
$703$	−18642.6	−1.00017
$704$	4343.41	0.232526
$705$	0	0
$706$	−5205.76	−0.277509
$707$	6194.07	0.329494
$708$	0	0
$709$	19314.2	1.02308	0.511538	−	0.859261i	$-0.329076\pi$
$709$	19314.2	1.02308	0.511538	+	0.859261i	$0.329076\pi$
$710$	−2162.21	−0.114290
$711$	0	0
$712$	7435.45	0.391370
$713$	−6995.36	−0.367431
$714$	0	0
$715$	1528.61	0.0799537
$716$	5303.58	0.276821
$717$	0	0
$718$	4615.70	0.239911
$719$	11811.2	0.612635	0.306317	−	0.951929i	$-0.400903\pi$
$719$	11811.2	0.612635	0.306317	+	0.951929i	$0.400903\pi$
$720$	0	0
$721$	7488.49	0.386804
$722$	−3498.60	−0.180338
$723$	0	0
$724$	19472.5	0.999571
$725$	−16835.6	−0.862424
$726$	0	0
$727$	17995.3	0.918034	0.459017	−	0.888428i	$-0.348202\pi$
$727$	17995.3	0.918034	0.459017	+	0.888428i	$0.348202\pi$
$728$	−2584.29	−0.131566
$729$	0	0
$730$	148.380	0.00752301
$731$	−7351.83	−0.371980
$732$	0	0
$733$	−28542.7	−1.43827	−0.719133	−	0.694873i	$-0.755461\pi$
$733$	−28542.7	−1.43827	−0.719133	+	0.694873i	$0.755461\pi$
$734$	−5054.86	−0.254194
$735$	0	0
$736$	−12774.4	−0.639769
$737$	−6388.57	−0.319303
$738$	0	0
$739$	−5283.32	−0.262991	−0.131495	−	0.991317i	$-0.541978\pi$
$739$	−5283.32	−0.262991	−0.131495	+	0.991317i	$0.541978\pi$
$740$	−4151.47	−0.206231
$741$	0	0
$742$	41.0583	0.00203140
$743$	8423.21	0.415905	0.207953	−	0.978139i	$-0.433320\pi$
$743$	8423.21	0.415905	0.207953	+	0.978139i	$0.433320\pi$
$744$	0	0
$745$	1551.89	0.0763180
$746$	−1104.10	−0.0541874
$747$	0	0
$748$	5822.25	0.284602
$749$	−1247.37	−0.0608518
$750$	0	0
$751$	26417.9	1.28363	0.641814	−	0.766861i	$-0.278182\pi$
$751$	26417.9	1.28363	0.641814	+	0.766861i	$0.278182\pi$
$752$	−6262.49	−0.303683
$753$	0	0
$754$	3475.83	0.167881
$755$	6184.25	0.298103
$756$	0	0
$757$	−24157.9	−1.15988	−0.579942	−	0.814658i	$-0.696925\pi$
$757$	−24157.9	−1.15988	−0.579942	+	0.814658i	$0.696925\pi$
$758$	−574.073	−0.0275083
$759$	0	0
$760$	−3340.80	−0.159452
$761$	9195.59	0.438029	0.219014	−	0.975722i	$-0.429716\pi$
$761$	9195.59	0.438029	0.219014	+	0.975722i	$0.429716\pi$
$762$	0	0
$763$	13815.0	0.655488
$764$	14179.2	0.671445
$765$	0	0
$766$	7202.43	0.339732
$767$	−7640.53	−0.359692
$768$	0	0
$769$	−15430.5	−0.723589	−0.361794	−	0.932258i	$-0.617836\pi$
$769$	−15430.5	−0.723589	−0.361794	+	0.932258i	$0.617836\pi$
$770$	143.354	0.00670924
$771$	0	0
$772$	20398.8	0.950994
$773$	14100.0	0.656068	0.328034	−	0.944666i	$-0.393614\pi$
$773$	14100.0	0.656068	0.328034	+	0.944666i	$0.393614\pi$
$774$	0	0
$775$	6381.02	0.295759
$776$	8148.03	0.376930
$777$	0	0
$778$	−6867.26	−0.316457
$779$	29587.9	1.36084
$780$	0	0
$781$	−12775.3	−0.585321
$782$	−4834.17	−0.221061
$783$	0	0
$784$	2770.03	0.126186
$785$	4978.27	0.226347
$786$	0	0
$787$	4428.96	0.200604	0.100302	−	0.994957i	$-0.468019\pi$
$787$	4428.96	0.200604	0.100302	+	0.994957i	$0.468019\pi$
$788$	1260.72	0.0569941
$789$	0	0
$790$	−2029.37	−0.0913944
$791$	−11498.8	−0.516876
$792$	0	0
$793$	−7958.90	−0.356405
$794$	1909.88	0.0853640
$795$	0	0
$796$	−22196.4	−0.988357
$797$	−32311.2	−1.43604	−0.718019	−	0.696024i	$-0.754951\pi$
$797$	−32311.2	−1.43604	−0.718019	+	0.696024i	$0.754951\pi$
$798$	0	0
$799$	−7630.14	−0.337841
$800$	11652.5	0.514973
$801$	0	0
$802$	−3500.94	−0.154143
$803$	876.698	0.0385280
$804$	0	0
$805$	2900.57	0.126996
$806$	−1317.41	−0.0575729
$807$	0	0
$808$	7793.70	0.339334
$809$	45404.2	1.97321	0.986605	−	0.163125i	$-0.0521574\pi$
$809$	45404.2	1.97321	0.986605	+	0.163125i	$0.0521574\pi$
$810$	0	0
$811$	−36636.7	−1.58630	−0.793150	−	0.609026i	$-0.791560\pi$
$811$	−36636.7	−1.58630	−0.793150	+	0.609026i	$0.791560\pi$
$812$	−7943.53	−0.343304
$813$	0	0
$814$	1006.54	0.0433407
$815$	1738.57	0.0747231
$816$	0	0
$817$	12211.8	0.522932
$818$	6418.17	0.274335
$819$	0	0
$820$	6588.86	0.280601
$821$	14967.9	0.636279	0.318139	−	0.948044i	$-0.396942\pi$
$821$	14967.9	0.636279	0.318139	+	0.948044i	$0.396942\pi$
$822$	0	0
$823$	−8353.81	−0.353822	−0.176911	−	0.984227i	$-0.556610\pi$
$823$	−8353.81	−0.353822	−0.176911	+	0.984227i	$0.556610\pi$
$824$	9422.41	0.398356
$825$	0	0
$826$	−716.531	−0.0301832
$827$	−26797.4	−1.12677	−0.563384	−	0.826195i	$-0.690501\pi$
$827$	−26797.4	−1.12677	−0.563384	+	0.826195i	$0.690501\pi$
$828$	0	0
$829$	−14645.0	−0.613562	−0.306781	−	0.951780i	$-0.599252\pi$
$829$	−14645.0	−0.613562	−0.306781	+	0.951780i	$0.599252\pi$
$830$	−1628.70	−0.0681122
$831$	0	0
$832$	16550.6	0.689651
$833$	3374.97	0.140379
$834$	0	0
$835$	10917.7	0.452484
$836$	−9671.06	−0.400097
$837$	0	0
$838$	2149.47	0.0886065
$839$	10093.3	0.415327	0.207664	−	0.978200i	$-0.433414\pi$
$839$	10093.3	0.415327	0.207664	+	0.978200i	$0.433414\pi$
$840$	0	0
$841$	−2582.72	−0.105897
$842$	6529.06	0.267228
$843$	0	0
$844$	−4848.68	−0.197747
$845$	−1458.99	−0.0593975
$846$	0	0
$847$	847.000	0.0343604
$848$	−590.474	−0.0239115
$849$	0	0
$850$	4409.62	0.177940
$851$	20366.0	0.820374
$852$	0	0
$853$	28743.0	1.15374	0.576872	−	0.816835i	$-0.304273\pi$
$853$	28743.0	1.15374	0.576872	+	0.816835i	$0.304273\pi$
$854$	−746.389	−0.0299074
$855$	0	0
$856$	−1569.51	−0.0626690
$857$	19741.2	0.786870	0.393435	−	0.919352i	$-0.371287\pi$
$857$	19741.2	0.786870	0.393435	+	0.919352i	$0.371287\pi$
$858$	0	0
$859$	−41550.0	−1.65037	−0.825185	−	0.564863i	$-0.808929\pi$
$859$	−41550.0	−1.65037	−0.825185	+	0.564863i	$0.808929\pi$
$860$	2719.41	0.107827
$861$	0	0
$862$	4194.69	0.165744
$863$	−45844.4	−1.80830	−0.904150	−	0.427215i	$-0.859495\pi$
$863$	−45844.4	−1.80830	−0.904150	+	0.427215i	$0.859495\pi$
$864$	0	0
$865$	10068.6	0.395771
$866$	−868.704	−0.0340875
$867$	0	0
$868$	3010.76	0.117732
$869$	−11990.4	−0.468063
$870$	0	0
$871$	−24343.8	−0.947024
$872$	17382.8	0.675064
$873$	0	0
$874$	8029.80	0.310769
$875$	−5546.76	−0.214303
$876$	0	0
$877$	2558.92	0.0985277	0.0492638	−	0.998786i	$-0.484312\pi$
$877$	2558.92	0.0985277	0.0492638	+	0.998786i	$0.484312\pi$
$878$	4538.91	0.174466
$879$	0	0
$880$	−2061.62	−0.0789742
$881$	−1119.16	−0.0427983	−0.0213992	−	0.999771i	$-0.506812\pi$
$881$	−1119.16	−0.0427983	−0.0213992	+	0.999771i	$0.506812\pi$
$882$	0	0
$883$	−1677.64	−0.0639377	−0.0319688	−	0.999489i	$-0.510178\pi$
$883$	−1677.64	−0.0639377	−0.0319688	+	0.999489i	$0.510178\pi$
$884$	22185.8	0.844105
$885$	0	0
$886$	−7300.27	−0.276814
$887$	−12127.5	−0.459076	−0.229538	−	0.973300i	$-0.573722\pi$
$887$	−12127.5	−0.459076	−0.229538	+	0.973300i	$0.573722\pi$
$888$	0	0
$889$	−12578.6	−0.474548
$890$	−1571.67	−0.0591937
$891$	0	0
$892$	48048.4	1.80356
$893$	12674.1	0.474940
$894$	0	0
$895$	−2288.09	−0.0854551
$896$	7275.74	0.271278
$897$	0	0
$898$	−2567.80	−0.0954216
$899$	−8265.02	−0.306623
$900$	0	0
$901$	−719.425	−0.0266010
$902$	−1597.50	−0.0589700
$903$	0	0
$904$	−14468.4	−0.532312
$905$	−8400.89	−0.308569
$906$	0	0
$907$	−13826.8	−0.506186	−0.253093	−	0.967442i	$-0.581448\pi$
$907$	−13826.8	−0.506186	−0.253093	+	0.967442i	$0.581448\pi$
$908$	−25483.9	−0.931401
$909$	0	0
$910$	546.253	0.0198990
$911$	41945.3	1.52548	0.762739	−	0.646706i	$-0.223854\pi$
$911$	41945.3	1.52548	0.762739	+	0.646706i	$0.223854\pi$
$912$	0	0
$913$	−9623.13	−0.348827
$914$	6090.90	0.220426
$915$	0	0
$916$	−23647.2	−0.852977
$917$	−6540.13	−0.235523
$918$	0	0
$919$	−8132.54	−0.291913	−0.145956	−	0.989291i	$-0.546626\pi$
$919$	−8132.54	−0.291913	−0.145956	+	0.989291i	$0.546626\pi$
$920$	3649.65	0.130789
$921$	0	0
$922$	−729.641	−0.0260623
$923$	−48680.5	−1.73601
$924$	0	0
$925$	−18577.4	−0.660349
$926$	−6804.27	−0.241471
$927$	0	0
$928$	−15092.9	−0.533889
$929$	48334.7	1.70701	0.853504	−	0.521086i	$-0.174473\pi$
$929$	48334.7	1.70701	0.853504	+	0.521086i	$0.174473\pi$
$930$	0	0
$931$	−5606.00	−0.197346
$932$	10436.4	0.366797
$933$	0	0
$934$	−7607.15	−0.266503
$935$	−2511.85	−0.0878571
$936$	0	0
$937$	−6068.39	−0.211575	−0.105787	−	0.994389i	$-0.533736\pi$
$937$	−6068.39	−0.211575	−0.105787	+	0.994389i	$0.533736\pi$
$938$	−2282.97	−0.0794686
$939$	0	0
$940$	2822.35	0.0979309
$941$	−15416.2	−0.534065	−0.267032	−	0.963688i	$-0.586043\pi$
$941$	−15416.2	−0.534065	−0.267032	+	0.963688i	$0.586043\pi$
$942$	0	0
$943$	−32323.3	−1.11621
$944$	10304.7	0.355285
$945$	0	0
$946$	−659.333	−0.0226604
$947$	45788.4	1.57120	0.785598	−	0.618737i	$-0.212355\pi$
$947$	45788.4	1.57120	0.785598	+	0.618737i	$0.212355\pi$
$948$	0	0
$949$	3340.67	0.114271
$950$	−7324.61	−0.250149
$951$	0	0
$952$	4246.56	0.144571
$953$	−36543.7	−1.24215	−0.621074	−	0.783752i	$-0.713303\pi$
$953$	−36543.7	−1.24215	−0.621074	+	0.783752i	$0.713303\pi$
$954$	0	0
$955$	−6117.23	−0.207276
$956$	−31900.4	−1.07922
$957$	0	0
$958$	−5987.94	−0.201943
$959$	11136.1	0.374977
$960$	0	0
$961$	−26658.4	−0.894847
$962$	3835.45	0.128545
$963$	0	0
$964$	26832.6	0.896494
$965$	−8800.50	−0.293573
$966$	0	0
$967$	3077.00	0.102327	0.0511633	−	0.998690i	$-0.483707\pi$
$967$	3077.00	0.102327	0.0511633	+	0.998690i	$0.483707\pi$
$968$	1065.74	0.0353865
$969$	0	0
$970$	−1722.29	−0.0570096
$971$	−41714.2	−1.37865	−0.689327	−	0.724451i	$-0.742094\pi$
$971$	−41714.2	−1.37865	−0.689327	+	0.724451i	$0.742094\pi$
$972$	0	0
$973$	−10843.0	−0.357257
$974$	−4576.61	−0.150558
$975$	0	0
$976$	10734.1	0.352039
$977$	22237.2	0.728178	0.364089	−	0.931364i	$-0.381380\pi$
$977$	22237.2	0.728178	0.364089	+	0.931364i	$0.381380\pi$
$978$	0	0
$979$	−9286.12	−0.303152
$980$	−1248.39	−0.0406921
$981$	0	0
$982$	−8503.23	−0.276323
$983$	26617.2	0.863639	0.431820	−	0.901960i	$-0.357872\pi$
$983$	26617.2	0.863639	0.431820	+	0.901960i	$0.357872\pi$
$984$	0	0
$985$	−543.905	−0.0175942
$986$	−5711.56	−0.184476
$987$	0	0
$988$	−36851.8	−1.18665
$989$	−13340.7	−0.428928
$990$	0	0
$991$	28839.4	0.924435	0.462217	−	0.886767i	$-0.347054\pi$
$991$	28839.4	0.924435	0.462217	+	0.886767i	$0.347054\pi$
$992$	5720.52	0.183091
$993$	0	0
$994$	−4565.27	−0.145676
$995$	9576.07	0.305107
$996$	0	0
$997$	35924.1	1.14115	0.570575	−	0.821246i	$-0.306720\pi$
$997$	35924.1	1.14115	0.570575	+	0.821246i	$0.306720\pi$
$998$	−5857.44	−0.185786
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.k.1.1		2
3.2	odd	2		231.4.a.f.1.2	✓	2
21.20	even	2		1617.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.f.1.2	✓	2	3.2	odd	2
693.4.a.k.1.1		2	1.1	even	1	trivial
1617.4.a.i.1.2		2	21.20	even	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-0.561553 q^{2} -7.68466 q^{4} +3.31534 q^{5} +7.00000 q^{7} +8.80776 q^{8} -1.86174 q^{10} -11.0000 q^{11} -41.9157 q^{13} -3.93087 q^{14} +56.5312 q^{16} +68.8769 q^{17} -114.408 q^{19} -25.4773 q^{20} +6.17708 q^{22} +124.985 q^{23} -114.009 q^{25} +23.5379 q^{26} -53.7926 q^{28} +147.670 q^{29} -55.9697 q^{31} -102.207 q^{32} -38.6780 q^{34} +23.2074 q^{35} +162.948 q^{37} +64.2462 q^{38} +29.2007 q^{40} -258.617 q^{41} -106.739 q^{43} +84.5312 q^{44} -70.1856 q^{46} -110.779 q^{47} +49.0000 q^{49} +64.0218 q^{50} +322.108 q^{52} -10.4451 q^{53} -36.4688 q^{55} +61.6543 q^{56} -82.9242 q^{58} +182.283 q^{59} +189.879 q^{61} +31.4299 q^{62} -394.855 q^{64} -138.965 q^{65} +580.779 q^{67} -529.295 q^{68} -13.0322 q^{70} +1161.39 q^{71} -79.6998 q^{73} -91.5038 q^{74} +879.187 q^{76} -77.0000 q^{77} +1090.04 q^{79} +187.420 q^{80} +145.227 q^{82} +874.830 q^{83} +228.350 q^{85} +59.9394 q^{86} -96.8854 q^{88} +844.193 q^{89} -293.410 q^{91} -960.466 q^{92} +62.2084 q^{94} -379.302 q^{95} +925.097 q^{97} -27.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 3 q^{4} + 19 q^{5} + 14 q^{7} - 3 q^{8} + 54 q^{10} - 22 q^{11} + 11 q^{13} + 21 q^{14} - 23 q^{16} + 146 q^{17} - 101 q^{19} + 48 q^{20} - 33 q^{22} + 184 q^{23} + 7 q^{25} + 212 q^{26}+ \cdots + 147 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 - 3 * q^4 + 19 * q^5 + 14 * q^7 - 3 * q^8 + 54 * q^10 - 22 * q^11 + 11 * q^13 + 21 * q^14 - 23 * q^16 + 146 * q^17 - 101 * q^19 + 48 * q^20 - 33 * q^22 + 184 * q^23 + 7 * q^25 + 212 * q^26 - 21 * q^28 + 217 * q^29 + 20 * q^31 - 291 * q^32 + 236 * q^34 + 133 * q^35 - 173 * q^37 + 112 * q^38 - 156 * q^40 + 60 * q^41 - 164 * q^43 + 33 * q^44 + 140 * q^46 + 467 * q^47 + 98 * q^49 + 495 * q^50 + 570 * q^52 - 326 * q^53 - 209 * q^55 - 21 * q^56 + 164 * q^58 + 781 * q^59 - 148 * q^61 + 302 * q^62 - 431 * q^64 + 691 * q^65 + 473 * q^67 - 168 * q^68 + 378 * q^70 + 756 * q^71 - 213 * q^73 - 1288 * q^74 + 942 * q^76 - 154 * q^77 + 168 * q^79 - 1060 * q^80 + 1280 * q^82 + 2096 * q^83 + 1438 * q^85 - 144 * q^86 + 33 * q^88 - 736 * q^89 + 77 * q^91 - 684 * q^92 + 2120 * q^94 - 169 * q^95 + 638 * q^97 + 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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