LMFDB - Embedded newform 693.4.a.j.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:	$1$
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)

$f(q)$	$=$	$q-0.561553 q^{2} -7.68466 q^{4} -18.6847 q^{5} +7.00000 q^{7} +8.80776 q^{8} +O(q^{10})$	\(q-0.561553 q^{2} -7.68466 q^{4} -18.6847 q^{5} +7.00000 q^{7} +8.80776 q^{8} +10.4924 q^{10} +11.0000 q^{11} +36.4384 q^{13} -3.93087 q^{14} +56.5312 q^{16} -41.1231 q^{17} -23.6998 q^{19} +143.585 q^{20} -6.17708 q^{22} +140.047 q^{23} +224.116 q^{25} -20.4621 q^{26} -53.7926 q^{28} -278.455 q^{29} +191.447 q^{31} -102.207 q^{32} +23.0928 q^{34} -130.793 q^{35} +196.115 q^{37} +13.3087 q^{38} -164.570 q^{40} +322.695 q^{41} -3.67615 q^{43} -84.5312 q^{44} -78.6440 q^{46} +397.596 q^{47} +49.0000 q^{49} -125.853 q^{50} -280.017 q^{52} -597.508 q^{53} -205.531 q^{55} +61.6543 q^{56} +156.367 q^{58} -668.779 q^{59} -667.788 q^{61} -107.508 q^{62} -394.855 q^{64} -680.840 q^{65} -730.762 q^{67} +316.017 q^{68} +73.4470 q^{70} +31.2651 q^{71} -434.408 q^{73} -110.129 q^{74} +182.125 q^{76} +77.0000 q^{77} -782.004 q^{79} -1056.27 q^{80} -181.210 q^{82} +426.705 q^{83} +768.371 q^{85} +2.06435 q^{86} +96.8854 q^{88} +899.693 q^{89} +255.069 q^{91} -1076.22 q^{92} -223.271 q^{94} +442.823 q^{95} -942.716 q^{97} -27.5161 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - 3 q^{4} - 25 q^{5} + 14 q^{7} - 3 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 - 3 * q^4 - 25 * q^5 + 14 * q^7 - 3 * q^8	$ 2 q + 3 q^{2} - 3 q^{4} - 25 q^{5} + 14 q^{7} - 3 q^{8} - 12 q^{10} + 22 q^{11} + 77 q^{13} + 21 q^{14} - 23 q^{16} - 74 q^{17} - 101 q^{19} + 114 q^{20} + 33 q^{22} - 58 q^{23} + 139 q^{25} + 124 q^{26} - 21 q^{28} - 91 q^{29} + 152 q^{31} - 291 q^{32} - 94 q^{34} - 175 q^{35} + 619 q^{37} - 262 q^{38} - 90 q^{40} - 138 q^{41} - 230 q^{43} - 33 q^{44} - 784 q^{46} - 149 q^{47} + 98 q^{49} - 429 q^{50} - 90 q^{52} - 1228 q^{53} - 275 q^{55} - 21 q^{56} + 824 q^{58} - 649 q^{59} - 412 q^{61} - 248 q^{62} - 431 q^{64} - 937 q^{65} - 1243 q^{67} + 162 q^{68} - 84 q^{70} - 960 q^{71} - 741 q^{73} + 1396 q^{74} - 180 q^{76} + 154 q^{77} - 492 q^{79} - 554 q^{80} - 1822 q^{82} + 1744 q^{83} + 976 q^{85} - 804 q^{86} - 33 q^{88} + 1552 q^{89} + 539 q^{91} - 2004 q^{92} - 2170 q^{94} + 931 q^{95} + 440 q^{97} + 147 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 - 3 * q^4 - 25 * q^5 + 14 * q^7 - 3 * q^8 - 12 * q^10 + 22 * q^11 + 77 * q^13 + 21 * q^14 - 23 * q^16 - 74 * q^17 - 101 * q^19 + 114 * q^20 + 33 * q^22 - 58 * q^23 + 139 * q^25 + 124 * q^26 - 21 * q^28 - 91 * q^29 + 152 * q^31 - 291 * q^32 - 94 * q^34 - 175 * q^35 + 619 * q^37 - 262 * q^38 - 90 * q^40 - 138 * q^41 - 230 * q^43 - 33 * q^44 - 784 * q^46 - 149 * q^47 + 98 * q^49 - 429 * q^50 - 90 * q^52 - 1228 * q^53 - 275 * q^55 - 21 * q^56 + 824 * q^58 - 649 * q^59 - 412 * q^61 - 248 * q^62 - 431 * q^64 - 937 * q^65 - 1243 * q^67 + 162 * q^68 - 84 * q^70 - 960 * q^71 - 741 * q^73 + 1396 * q^74 - 180 * q^76 + 154 * q^77 - 492 * q^79 - 554 * q^80 - 1822 * q^82 + 1744 * q^83 + 976 * q^85 - 804 * q^86 - 33 * q^88 + 1552 * q^89 + 539 * q^91 - 2004 * q^92 - 2170 * q^94 + 931 * q^95 + 440 * q^97 + 147 * q^98

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 $

$2$	−0.561553	−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
$5$	−18.6847	−1.67121	−0.835603	−	0.549333i	$-0.814882\pi$
$5$	−18.6847	−1.67121	−0.835603	+	0.549333i	$0.814882\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.80776	0.389252
$9$	0	0
$10$	10.4924	0.331800
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	36.4384	0.777401	0.388700	−	0.921364i	$-0.372924\pi$
$13$	36.4384	0.777401	0.388700	+	0.921364i	$0.372924\pi$
$14$	−3.93087	−0.0750407
$15$	0	0
$16$	56.5312	0.883301
$17$	−41.1231	−0.586695	−0.293348	−	0.956006i	$-0.594769\pi$
$17$	−41.1231	−0.586695	−0.293348	+	0.956006i	$0.594769\pi$
$18$	0	0
$19$	−23.6998	−0.286164	−0.143082	−	0.989711i	$-0.545701\pi$
$19$	−23.6998	−0.286164	−0.143082	+	0.989711i	$0.545701\pi$
$20$	143.585	1.60533
$21$	0	0
$22$	−6.17708	−0.0598617
$23$	140.047	1.26965	0.634824	−	0.772657i	$-0.281073\pi$
$23$	140.047	1.26965	0.634824	+	0.772657i	$0.281073\pi$
$24$	0	0
$25$	224.116	1.79293
$26$	−20.4621	−0.154344
$27$	0	0
$28$	−53.7926	−0.363066
$29$	−278.455	−1.78303	−0.891515	−	0.452991i	$-0.850357\pi$
$29$	−278.455	−1.78303	−0.891515	+	0.452991i	$0.850357\pi$
$30$	0	0
$31$	191.447	1.10919	0.554595	−	0.832120i	$-0.312873\pi$
$31$	191.447	1.10919	0.554595	+	0.832120i	$0.312873\pi$
$32$	−102.207	−0.564621
$33$	0	0
$34$	23.0928	0.116482
$35$	−130.793	−0.631657
$36$	0	0
$37$	196.115	0.871379	0.435690	−	0.900097i	$-0.356504\pi$
$37$	196.115	0.871379	0.435690	+	0.900097i	$0.356504\pi$
$38$	13.3087	0.0568146
$39$	0	0
$40$	−164.570	−0.650520
$41$	322.695	1.22918	0.614591	−	0.788846i	$-0.289321\pi$
$41$	322.695	1.22918	0.614591	+	0.788846i	$0.289321\pi$
$42$	0	0
$43$	−3.67615	−0.0130374	−0.00651869	−	0.999979i	$-0.502075\pi$
$43$	−3.67615	−0.0130374	−0.00651869	+	0.999979i	$0.502075\pi$
$44$	−84.5312	−0.289626
$45$	0	0
$46$	−78.6440	−0.252074
$47$	397.596	1.23394	0.616971	−	0.786986i	$-0.288360\pi$
$47$	397.596	1.23394	0.616971	+	0.786986i	$0.288360\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−125.853	−0.355967
$51$	0	0
$52$	−280.017	−0.746757
$53$	−597.508	−1.54857	−0.774283	−	0.632840i	$-0.781889\pi$
$53$	−597.508	−1.54857	−0.774283	+	0.632840i	$0.781889\pi$
$54$	0	0
$55$	−205.531	−0.503888
$56$	61.6543	0.147123
$57$	0	0
$58$	156.367	0.354001
$59$	−668.779	−1.47572	−0.737861	−	0.674952i	$-0.764164\pi$
$59$	−668.779	−1.47572	−0.737861	+	0.674952i	$0.764164\pi$
$60$	0	0
$61$	−667.788	−1.40166	−0.700832	−	0.713327i	$-0.747188\pi$
$61$	−667.788	−1.40166	−0.700832	+	0.713327i	$0.747188\pi$
$62$	−107.508	−0.220217
$63$	0	0
$64$	−394.855	−0.771201
$65$	−680.840	−1.29920
$66$	0	0
$67$	−730.762	−1.33249	−0.666245	−	0.745733i	$-0.732099\pi$
$67$	−730.762	−1.33249	−0.666245	+	0.745733i	$0.732099\pi$
$68$	316.017	0.563569
$69$	0	0
$70$	73.4470	0.125408
$71$	31.2651	0.0522603	0.0261302	−	0.999659i	$-0.491682\pi$
$71$	31.2651	0.0522603	0.0261302	+	0.999659i	$0.491682\pi$
$72$	0	0
$73$	−434.408	−0.696488	−0.348244	−	0.937404i	$-0.613222\pi$
$73$	−434.408	−0.696488	−0.348244	+	0.937404i	$0.613222\pi$
$74$	−110.129	−0.173003
$75$	0	0
$76$	182.125	0.274884
$77$	77.0000	0.113961
$78$	0	0
$79$	−782.004	−1.11370	−0.556850	−	0.830613i	$-0.687990\pi$
$79$	−782.004	−1.11370	−0.556850	+	0.830613i	$0.687990\pi$
$80$	−1056.27	−1.47618
$81$	0	0
$82$	−181.210	−0.244041
$83$	426.705	0.564300	0.282150	−	0.959370i	$-0.408952\pi$
$83$	426.705	0.564300	0.282150	+	0.959370i	$0.408952\pi$
$84$	0	0
$85$	768.371	0.980489
$86$	2.06435	0.00258843
$87$	0	0
$88$	96.8854	0.117364
$89$	899.693	1.07154	0.535771	−	0.844363i	$-0.320021\pi$
$89$	899.693	1.07154	0.535771	+	0.844363i	$0.320021\pi$
$90$	0	0
$91$	255.069	0.293830
$92$	−1076.22	−1.21960
$93$	0	0
$94$	−223.271	−0.244986
$95$	442.823	0.478239
$96$	0	0
$97$	−942.716	−0.986786	−0.493393	−	0.869806i	$-0.664244\pi$
$97$	−942.716	−0.986786	−0.493393	+	0.869806i	$0.664244\pi$
$98$	−27.5161	−0.0283627
$99$	0	0
$100$	−1722.26	−1.72226
$101$	−366.945	−0.361509	−0.180754	−	0.983528i	$-0.557854\pi$
$101$	−366.945	−0.361509	−0.180754	+	0.983528i	$0.557854\pi$
$102$	0	0
$103$	1007.35	0.963658	0.481829	−	0.876265i	$-0.339973\pi$
$103$	1007.35	0.963658	0.481829	+	0.876265i	$0.339973\pi$
$104$	320.941	0.302605
$105$	0	0
$106$	335.532	0.307451
$107$	1690.80	1.52763	0.763814	−	0.645437i	$-0.223325\pi$
$107$	1690.80	1.52763	0.763814	+	0.645437i	$0.223325\pi$
$108$	0	0
$109$	−1808.57	−1.58926	−0.794631	−	0.607093i	$-0.792336\pi$
$109$	−1808.57	−1.58926	−0.794631	+	0.607093i	$0.792336\pi$
$110$	115.417	0.100041
$111$	0	0
$112$	395.719	0.333856
$113$	−952.557	−0.793000	−0.396500	−	0.918035i	$-0.629775\pi$
$113$	−952.557	−0.793000	−0.396500	+	0.918035i	$0.629775\pi$
$114$	0	0
$115$	−2616.74	−2.12184
$116$	2139.84	1.71275
$117$	0	0
$118$	375.555	0.292988
$119$	−287.862	−0.221750
$120$	0	0
$121$	121.000	0.0909091
$122$	374.998	0.278285
$123$	0	0
$124$	−1471.20	−1.06547
$125$	−1851.96	−1.32515
$126$	0	0
$127$	252.951	0.176738	0.0883691	−	0.996088i	$-0.471834\pi$
$127$	252.951	0.176738	0.0883691	+	0.996088i	$0.471834\pi$
$128$	1039.39	0.717735
$129$	0	0
$130$	382.328	0.257941
$131$	1497.38	0.998679	0.499339	−	0.866406i	$-0.333576\pi$
$131$	1497.38	0.998679	0.499339	+	0.866406i	$0.333576\pi$
$132$	0	0
$133$	−165.899	−0.108160
$134$	410.362	0.264551
$135$	0	0
$136$	−362.203	−0.228372
$137$	1207.56	0.753056	0.376528	−	0.926405i	$-0.377118\pi$
$137$	1207.56	0.753056	0.376528	+	0.926405i	$0.377118\pi$
$138$	0	0
$139$	212.519	0.129681	0.0648404	−	0.997896i	$-0.479346\pi$
$139$	212.519	0.129681	0.0648404	+	0.997896i	$0.479346\pi$
$140$	1005.10	0.606758
$141$	0	0
$142$	−17.5570	−0.0103757
$143$	400.823	0.234395
$144$	0	0
$145$	5202.85	2.97981
$146$	243.943	0.138280
$147$	0	0
$148$	−1507.07	−0.837032
$149$	−879.656	−0.483653	−0.241826	−	0.970320i	$-0.577746\pi$
$149$	−879.656	−0.483653	−0.241826	+	0.970320i	$0.577746\pi$
$150$	0	0
$151$	−1215.18	−0.654899	−0.327450	−	0.944869i	$-0.606189\pi$
$151$	−1215.18	−0.654899	−0.327450	+	0.944869i	$0.606189\pi$
$152$	−208.742	−0.111390
$153$	0	0
$154$	−43.2396	−0.0226256
$155$	−3577.12	−1.85369
$156$	0	0
$157$	1226.57	0.623507	0.311754	−	0.950163i	$-0.399084\pi$
$157$	1226.57	0.623507	0.311754	+	0.950163i	$0.399084\pi$
$158$	439.136	0.221113
$159$	0	0
$160$	1909.71	0.943599
$161$	980.331	0.479882
$162$	0	0
$163$	−441.224	−0.212021	−0.106010	−	0.994365i	$-0.533808\pi$
$163$	−441.224	−0.212021	−0.106010	+	0.994365i	$0.533808\pi$
$164$	−2479.80	−1.18073
$165$	0	0
$166$	−239.617	−0.112036
$167$	1793.53	0.831064	0.415532	−	0.909579i	$-0.363595\pi$
$167$	1793.53	0.831064	0.415532	+	0.909579i	$0.363595\pi$
$168$	0	0
$169$	−869.240	−0.395648
$170$	−431.481	−0.194665
$171$	0	0
$172$	28.2499	0.0125235
$173$	−3825.04	−1.68099	−0.840497	−	0.541816i	$-0.817737\pi$
$173$	−3825.04	−1.68099	−0.840497	+	0.541816i	$0.817737\pi$
$174$	0	0
$175$	1568.82	0.677664
$176$	621.844	0.266325
$177$	0	0
$178$	−505.225	−0.212743
$179$	−1315.65	−0.549365	−0.274683	−	0.961535i	$-0.588573\pi$
$179$	−1315.65	−0.549365	−0.274683	+	0.961535i	$0.588573\pi$
$180$	0	0
$181$	674.682	0.277065	0.138532	−	0.990358i	$-0.455762\pi$
$181$	674.682	0.277065	0.138532	+	0.990358i	$0.455762\pi$
$182$	−143.235	−0.0583366
$183$	0	0
$184$	1233.50	0.494213
$185$	−3664.33	−1.45626
$186$	0	0
$187$	−452.354	−0.176895
$188$	−3055.39	−1.18530
$189$	0	0
$190$	−248.668	−0.0949490
$191$	−1915.69	−0.725730	−0.362865	−	0.931842i	$-0.618201\pi$
$191$	−1915.69	−0.725730	−0.362865	+	0.931842i	$0.618201\pi$
$192$	0	0
$193$	−2394.71	−0.893134	−0.446567	−	0.894750i	$-0.647354\pi$
$193$	−2394.71	−0.893134	−0.446567	+	0.894750i	$0.647354\pi$
$194$	529.385	0.195915
$195$	0	0
$196$	−376.548	−0.137226
$197$	−1356.81	−0.490703	−0.245351	−	0.969434i	$-0.578903\pi$
$197$	−1356.81	−0.490703	−0.245351	+	0.969434i	$0.578903\pi$
$198$	0	0
$199$	2121.45	0.755707	0.377854	−	0.925865i	$-0.376662\pi$
$199$	2121.45	0.755707	0.377854	+	0.925865i	$0.376662\pi$
$200$	1973.96	0.697902
$201$	0	0
$202$	206.059	0.0717736
$203$	−1949.19	−0.673922
$204$	0	0
$205$	−6029.45	−2.05422
$206$	−565.678	−0.191324
$207$	0	0
$208$	2059.91	0.686678
$209$	−260.698	−0.0862816
$210$	0	0
$211$	4492.12	1.46564	0.732821	−	0.680421i	$-0.238203\pi$
$211$	4492.12	1.46564	0.732821	+	0.680421i	$0.238203\pi$
$212$	4591.64	1.48752
$213$	0	0
$214$	−949.476	−0.303294
$215$	68.6876	0.0217882
$216$	0	0
$217$	1340.13	0.419234
$218$	1015.61	0.315530
$219$	0	0
$220$	1579.44	0.484026
$221$	−1498.46	−0.456097
$222$	0	0
$223$	5142.35	1.54420	0.772101	−	0.635499i	$-0.219206\pi$
$223$	5142.35	1.54420	0.772101	+	0.635499i	$0.219206\pi$
$224$	−715.452	−0.213407
$225$	0	0
$226$	534.911	0.157441
$227$	−2987.86	−0.873618	−0.436809	−	0.899554i	$-0.643891\pi$
$227$	−2987.86	−0.873618	−0.436809	+	0.899554i	$0.643891\pi$
$228$	0	0
$229$	−5121.38	−1.47786	−0.738931	−	0.673781i	$-0.764669\pi$
$229$	−5121.38	−1.47786	−0.738931	+	0.673781i	$0.764669\pi$
$230$	1469.44	0.421268
$231$	0	0
$232$	−2452.57	−0.694048
$233$	−1103.58	−0.310291	−0.155146	−	0.987892i	$-0.549585\pi$
$233$	−1103.58	−0.310291	−0.155146	+	0.987892i	$0.549585\pi$
$234$	0	0
$235$	−7428.94	−2.06217
$236$	5139.34	1.41755
$237$	0	0
$238$	161.650	0.0440260
$239$	1798.18	0.486671	0.243336	−	0.969942i	$-0.421758\pi$
$239$	1798.18	0.486671	0.243336	+	0.969942i	$0.421758\pi$
$240$	0	0
$241$	−7342.09	−1.96243	−0.981215	−	0.192919i	$-0.938204\pi$
$241$	−7342.09	−1.96243	−0.981215	+	0.192919i	$0.938204\pi$
$242$	−67.9479	−0.0180490
$243$	0	0
$244$	5131.72	1.34641
$245$	−915.548	−0.238744
$246$	0	0
$247$	−863.584	−0.222464
$248$	1686.22	0.431754
$249$	0	0
$250$	1039.97	0.263094
$251$	−1799.05	−0.452411	−0.226206	−	0.974080i	$-0.572632\pi$
$251$	−1799.05	−0.452411	−0.226206	+	0.974080i	$0.572632\pi$
$252$	0	0
$253$	1540.52	0.382813
$254$	−142.045	−0.0350894
$255$	0	0
$256$	2575.17	0.628703
$257$	5282.71	1.28220	0.641102	−	0.767455i	$-0.278477\pi$
$257$	5282.71	1.28220	0.641102	+	0.767455i	$0.278477\pi$
$258$	0	0
$259$	1372.80	0.329350
$260$	5232.02	1.24799
$261$	0	0
$262$	−840.859	−0.198277
$263$	−3275.55	−0.767981	−0.383990	−	0.923337i	$-0.625450\pi$
$263$	−3275.55	−0.767981	−0.383990	+	0.923337i	$0.625450\pi$
$264$	0	0
$265$	11164.2	2.58797
$266$	93.1609	0.0214739
$267$	0	0
$268$	5615.66	1.27997
$269$	−3682.35	−0.834636	−0.417318	−	0.908761i	$-0.637030\pi$
$269$	−3682.35	−0.834636	−0.417318	+	0.908761i	$0.637030\pi$
$270$	0	0
$271$	−7301.19	−1.63659	−0.818295	−	0.574799i	$-0.805080\pi$
$271$	−7301.19	−1.63659	−0.818295	+	0.574799i	$0.805080\pi$
$272$	−2324.74	−0.518228
$273$	0	0
$274$	−678.108	−0.149511
$275$	2465.28	0.540589
$276$	0	0
$277$	9035.64	1.95993	0.979963	−	0.199182i	$-0.0638283\pi$
$277$	9035.64	1.95993	0.979963	+	0.199182i	$0.0638283\pi$
$278$	−119.341	−0.0257467
$279$	0	0
$280$	−1151.99	−0.245874
$281$	−1651.82	−0.350674	−0.175337	−	0.984509i	$-0.556101\pi$
$281$	−1651.82	−0.350674	−0.175337	+	0.984509i	$0.556101\pi$
$282$	0	0
$283$	3091.62	0.649391	0.324695	−	0.945819i	$-0.394738\pi$
$283$	3091.62	0.649391	0.324695	+	0.945819i	$0.394738\pi$
$284$	−240.262	−0.0502004
$285$	0	0
$286$	−225.083	−0.0465365
$287$	2258.87	0.464587
$288$	0	0
$289$	−3221.89	−0.655789
$290$	−2921.67	−0.591609
$291$	0	0
$292$	3338.28	0.669034
$293$	−2231.84	−0.445001	−0.222500	−	0.974933i	$-0.571422\pi$
$293$	−2231.84	−0.445001	−0.222500	+	0.974933i	$0.571422\pi$
$294$	0	0
$295$	12495.9	2.46624
$296$	1727.33	0.339186
$297$	0	0
$298$	493.973	0.0960239
$299$	5103.11	0.987024
$300$	0	0
$301$	−25.7330	−0.00492767
$302$	682.387	0.130023
$303$	0	0
$304$	−1339.78	−0.252769
$305$	12477.4	2.34247
$306$	0	0
$307$	−9667.03	−1.79715	−0.898577	−	0.438816i	$-0.855398\pi$
$307$	−9667.03	−1.79715	−0.898577	+	0.438816i	$0.855398\pi$
$308$	−591.719	−0.109469
$309$	0	0
$310$	2008.74	0.368029
$311$	7170.97	1.30749	0.653743	−	0.756717i	$-0.273198\pi$
$311$	7170.97	1.30749	0.653743	+	0.756717i	$0.273198\pi$
$312$	0	0
$313$	295.578	0.0533771	0.0266886	−	0.999644i	$-0.491504\pi$
$313$	295.578	0.0533771	0.0266886	+	0.999644i	$0.491504\pi$
$314$	−688.782	−0.123790
$315$	0	0
$316$	6009.43	1.06980
$317$	−7189.81	−1.27388	−0.636940	−	0.770914i	$-0.719800\pi$
$317$	−7189.81	−1.27388	−0.636940	+	0.770914i	$0.719800\pi$
$318$	0	0
$319$	−3063.01	−0.537604
$320$	7377.73	1.28884
$321$	0	0
$322$	−550.508	−0.0952752
$323$	974.610	0.167891
$324$	0	0
$325$	8166.46	1.39383
$326$	247.771	0.0420943
$327$	0	0
$328$	2842.22	0.478462
$329$	2783.17	0.466386
$330$	0	0
$331$	−2878.87	−0.478058	−0.239029	−	0.971012i	$-0.576829\pi$
$331$	−2878.87	−0.478058	−0.239029	+	0.971012i	$0.576829\pi$
$332$	−3279.08	−0.542057
$333$	0	0
$334$	−1007.16	−0.164999
$335$	13654.0	2.22687
$336$	0	0
$337$	−11639.7	−1.88146	−0.940731	−	0.339153i	$-0.889860\pi$
$337$	−11639.7	−1.88146	−0.940731	+	0.339153i	$0.889860\pi$
$338$	488.124	0.0785516
$339$	0	0
$340$	−5904.67	−0.941840
$341$	2105.92	0.334433
$342$	0	0
$343$	343.000	0.0539949
$344$	−32.3786	−0.00507482
$345$	0	0
$346$	2147.96	0.333743
$347$	4752.91	0.735302	0.367651	−	0.929964i	$-0.380162\pi$
$347$	4752.91	0.735302	0.367651	+	0.929964i	$0.380162\pi$
$348$	0	0
$349$	−7420.08	−1.13807	−0.569037	−	0.822312i	$-0.692684\pi$
$349$	−7420.08	−1.13807	−0.569037	+	0.822312i	$0.692684\pi$
$350$	−880.973	−0.134543
$351$	0	0
$352$	−1124.28	−0.170240
$353$	517.851	0.0780805	0.0390403	−	0.999238i	$-0.487570\pi$
$353$	517.851	0.0780805	0.0390403	+	0.999238i	$0.487570\pi$
$354$	0	0
$355$	−584.178	−0.0873378
$356$	−6913.83	−1.02930
$357$	0	0
$358$	738.808	0.109070
$359$	9874.35	1.45167	0.725833	−	0.687871i	$-0.241454\pi$
$359$	9874.35	1.45167	0.725833	+	0.687871i	$0.241454\pi$
$360$	0	0
$361$	−6297.32	−0.918110
$362$	−378.869	−0.0550081
$363$	0	0
$364$	−1960.12	−0.282248
$365$	8116.77	1.16398
$366$	0	0
$367$	2098.51	0.298477	0.149239	−	0.988801i	$-0.452318\pi$
$367$	2098.51	0.298477	0.149239	+	0.988801i	$0.452318\pi$
$368$	7917.05	1.12148
$369$	0	0
$370$	2057.72	0.289123
$371$	−4182.55	−0.585303
$372$	0	0
$373$	−19.4150	−0.00269510	−0.00134755	−	0.999999i	$-0.500429\pi$
$373$	−19.4150	−0.00269510	−0.00134755	+	0.999999i	$0.500429\pi$
$374$	254.021	0.0351206
$375$	0	0
$376$	3501.93	0.480314
$377$	−10146.5	−1.38613
$378$	0	0
$379$	12541.3	1.69974	0.849869	−	0.526994i	$-0.176681\pi$
$379$	12541.3	1.69974	0.849869	+	0.526994i	$0.176681\pi$
$380$	−3402.94	−0.459388
$381$	0	0
$382$	1075.76	0.144086
$383$	−9131.73	−1.21830	−0.609151	−	0.793054i	$-0.708490\pi$
$383$	−9131.73	−1.21830	−0.609151	+	0.793054i	$0.708490\pi$
$384$	0	0
$385$	−1438.72	−0.190452
$386$	1344.76	0.177322
$387$	0	0
$388$	7244.45	0.947890
$389$	−9542.57	−1.24377	−0.621886	−	0.783108i	$-0.713633\pi$
$389$	−9542.57	−1.24377	−0.621886	+	0.783108i	$0.713633\pi$
$390$	0	0
$391$	−5759.18	−0.744896
$392$	431.580	0.0556074
$393$	0	0
$394$	761.919	0.0974236
$395$	14611.5	1.86122
$396$	0	0
$397$	5322.85	0.672912	0.336456	−	0.941699i	$-0.390772\pi$
$397$	5322.85	0.672912	0.336456	+	0.941699i	$0.390772\pi$
$398$	−1191.31	−0.150037
$399$	0	0
$400$	12669.6	1.58370
$401$	−5875.92	−0.731744	−0.365872	−	0.930665i	$-0.619229\pi$
$401$	−5875.92	−0.731744	−0.365872	+	0.930665i	$0.619229\pi$
$402$	0	0
$403$	6976.03	0.862285
$404$	2819.85	0.347259
$405$	0	0
$406$	1094.57	0.133800
$407$	2157.26	0.262731
$408$	0	0
$409$	−6658.52	−0.804994	−0.402497	−	0.915421i	$-0.631858\pi$
$409$	−6658.52	−0.804994	−0.402497	+	0.915421i	$0.631858\pi$
$410$	3385.85	0.407842
$411$	0	0
$412$	−7741.11	−0.925673
$413$	−4681.46	−0.557771
$414$	0	0
$415$	−7972.83	−0.943062
$416$	−3724.28	−0.438937
$417$	0	0
$418$	146.396	0.0171303
$419$	−10913.7	−1.27248	−0.636238	−	0.771493i	$-0.719510\pi$
$419$	−10913.7	−1.27248	−0.636238	+	0.771493i	$0.719510\pi$
$420$	0	0
$421$	3936.75	0.455737	0.227868	−	0.973692i	$-0.426824\pi$
$421$	3936.75	0.455737	0.227868	+	0.973692i	$0.426824\pi$
$422$	−2522.56	−0.290987
$423$	0	0
$424$	−5262.71	−0.602782
$425$	−9216.36	−1.05190
$426$	0	0
$427$	−4674.51	−0.529779
$428$	−12993.2	−1.46741
$429$	0	0
$430$	−38.5717	−0.00432580
$431$	−13769.3	−1.53885	−0.769424	−	0.638738i	$-0.779457\pi$
$431$	−13769.3	−1.53885	−0.769424	+	0.638738i	$0.779457\pi$
$432$	0	0
$433$	2858.51	0.317254	0.158627	−	0.987339i	$-0.449293\pi$
$433$	2858.51	0.317254	0.158627	+	0.987339i	$0.449293\pi$
$434$	−752.553	−0.0832343
$435$	0	0
$436$	13898.2	1.52662
$437$	−3319.10	−0.363327
$438$	0	0
$439$	8408.98	0.914211	0.457106	−	0.889412i	$-0.348886\pi$
$439$	8408.98	0.914211	0.457106	+	0.889412i	$0.348886\pi$
$440$	−1810.27	−0.196139
$441$	0	0
$442$	841.466	0.0905530
$443$	3537.39	0.379383	0.189692	−	0.981844i	$-0.439251\pi$
$443$	3537.39	0.379383	0.189692	+	0.981844i	$0.439251\pi$
$444$	0	0
$445$	−16810.5	−1.79077
$446$	−2887.70	−0.306584
$447$	0	0
$448$	−2763.99	−0.291487
$449$	−473.820	−0.0498016	−0.0249008	−	0.999690i	$-0.507927\pi$
$449$	−473.820	−0.0498016	−0.0249008	+	0.999690i	$0.507927\pi$
$450$	0	0
$451$	3549.65	0.370613
$452$	7320.07	0.761742
$453$	0	0
$454$	1677.84	0.173447
$455$	−4765.88	−0.491050
$456$	0	0
$457$	535.534	0.0548167	0.0274083	−	0.999624i	$-0.491275\pi$
$457$	535.534	0.0548167	0.0274083	+	0.999624i	$0.491275\pi$
$458$	2875.93	0.293413
$459$	0	0
$460$	20108.7	2.03820
$461$	−1075.05	−0.108612	−0.0543058	−	0.998524i	$-0.517295\pi$
$461$	−1075.05	−0.108612	−0.0543058	+	0.998524i	$0.517295\pi$
$462$	0	0
$463$	−11373.8	−1.14165	−0.570826	−	0.821071i	$-0.693377\pi$
$463$	−11373.8	−1.14165	−0.570826	+	0.821071i	$0.693377\pi$
$464$	−15741.4	−1.57495
$465$	0	0
$466$	619.718	0.0616049
$467$	15171.1	1.50328	0.751642	−	0.659572i	$-0.229262\pi$
$467$	15171.1	1.50328	0.751642	+	0.659572i	$0.229262\pi$
$468$	0	0
$469$	−5115.34	−0.503634
$470$	4171.74	0.409421
$471$	0	0
$472$	−5890.45	−0.574428
$473$	−40.4376	−0.00393092
$474$	0	0
$475$	−5311.52	−0.513072
$476$	2212.12	0.213009
$477$	0	0
$478$	−1009.77	−0.0966231
$479$	3045.24	0.290482	0.145241	−	0.989396i	$-0.453604\pi$
$479$	3045.24	0.290482	0.145241	+	0.989396i	$0.453604\pi$
$480$	0	0
$481$	7146.11	0.677411
$482$	4122.97	0.389619
$483$	0	0
$484$	−929.844	−0.0873257
$485$	17614.3	1.64912
$486$	0	0
$487$	−16956.1	−1.57773	−0.788866	−	0.614565i	$-0.789331\pi$
$487$	−16956.1	−1.57773	−0.788866	+	0.614565i	$0.789331\pi$
$488$	−5881.72	−0.545600
$489$	0	0
$490$	514.129	0.0473999
$491$	13667.0	1.25618	0.628088	−	0.778142i	$-0.283838\pi$
$491$	13667.0	1.25618	0.628088	+	0.778142i	$0.283838\pi$
$492$	0	0
$493$	11451.0	1.04610
$494$	484.948	0.0441677
$495$	0	0
$496$	10822.7	0.979748
$497$	218.856	0.0197526
$498$	0	0
$499$	−10939.4	−0.981394	−0.490697	−	0.871330i	$-0.663258\pi$
$499$	−10939.4	−0.981394	−0.490697	+	0.871330i	$0.663258\pi$
$500$	14231.7	1.27292
$501$	0	0
$502$	1010.26	0.0898213
$503$	6194.02	0.549061	0.274530	−	0.961578i	$-0.411478\pi$
$503$	6194.02	0.549061	0.274530	+	0.961578i	$0.411478\pi$
$504$	0	0
$505$	6856.24	0.604156
$506$	−865.084	−0.0760033
$507$	0	0
$508$	−1943.84	−0.169772
$509$	−16742.7	−1.45797	−0.728987	−	0.684528i	$-0.760008\pi$
$509$	−16742.7	−1.45797	−0.728987	+	0.684528i	$0.760008\pi$
$510$	0	0
$511$	−3040.86	−0.263248
$512$	−9761.22	−0.842557
$513$	0	0
$514$	−2966.52	−0.254567
$515$	−18821.9	−1.61047
$516$	0	0
$517$	4373.55	0.372048
$518$	−770.901	−0.0653889
$519$	0	0
$520$	−5996.68	−0.505715
$521$	−21184.3	−1.78138	−0.890691	−	0.454609i	$-0.849779\pi$
$521$	−21184.3	−1.78138	−0.890691	+	0.454609i	$0.849779\pi$
$522$	0	0
$523$	7737.42	0.646910	0.323455	−	0.946244i	$-0.395156\pi$
$523$	7737.42	0.646910	0.323455	+	0.946244i	$0.395156\pi$
$524$	−11506.9	−0.959313
$525$	0	0
$526$	1839.39	0.152474
$527$	−7872.89	−0.650756
$528$	0	0
$529$	7446.25	0.612004
$530$	−6269.30	−0.513813
$531$	0	0
$532$	1274.87	0.103896
$533$	11758.5	0.955567
$534$	0	0
$535$	−31592.1	−2.55298
$536$	−6436.38	−0.518674
$537$	0	0
$538$	2067.84	0.165708
$539$	539.000	0.0430730
$540$	0	0
$541$	−12732.7	−1.01187	−0.505937	−	0.862571i	$-0.668853\pi$
$541$	−12732.7	−1.01187	−0.505937	+	0.862571i	$0.668853\pi$
$542$	4100.00	0.324927
$543$	0	0
$544$	4203.09	0.331261
$545$	33792.5	2.65599
$546$	0	0
$547$	4429.06	0.346202	0.173101	−	0.984904i	$-0.444621\pi$
$547$	4429.06	0.346202	0.173101	+	0.984904i	$0.444621\pi$
$548$	−9279.68	−0.723372
$549$	0	0
$550$	−1384.39	−0.107328
$551$	6599.34	0.510239
$552$	0	0
$553$	−5474.03	−0.420939
$554$	−5073.99	−0.389121
$555$	0	0
$556$	−1633.14	−0.124569
$557$	−6080.68	−0.462562	−0.231281	−	0.972887i	$-0.574292\pi$
$557$	−6080.68	−0.462562	−0.231281	+	0.972887i	$0.574292\pi$
$558$	0	0
$559$	−133.953	−0.0101353
$560$	−7393.87	−0.557943
$561$	0	0
$562$	927.584	0.0696223
$563$	12141.8	0.908907	0.454453	−	0.890771i	$-0.349835\pi$
$563$	12141.8	0.908907	0.454453	+	0.890771i	$0.349835\pi$
$564$	0	0
$565$	17798.2	1.32527
$566$	−1736.11	−0.128929
$567$	0	0
$568$	275.376	0.0203424
$569$	−22369.0	−1.64808	−0.824039	−	0.566533i	$-0.808284\pi$
$569$	−22369.0	−1.64808	−0.824039	+	0.566533i	$0.808284\pi$
$570$	0	0
$571$	−14445.2	−1.05869	−0.529346	−	0.848406i	$-0.677563\pi$
$571$	−14445.2	−1.05869	−0.529346	+	0.848406i	$0.677563\pi$
$572$	−3080.19	−0.225156
$573$	0	0
$574$	−1268.47	−0.0922387
$575$	31386.9	2.27639
$576$	0	0
$577$	1782.41	0.128601	0.0643004	−	0.997931i	$-0.479518\pi$
$577$	1782.41	0.128601	0.0643004	+	0.997931i	$0.479518\pi$
$578$	1809.26	0.130200
$579$	0	0
$580$	−39982.1	−2.86235
$581$	2986.93	0.213285
$582$	0	0
$583$	−6572.58	−0.466910
$584$	−3826.16	−0.271109
$585$	0	0
$586$	1253.29	0.0883499
$587$	−23646.2	−1.66266	−0.831330	−	0.555779i	$-0.812420\pi$
$587$	−23646.2	−1.66266	−0.831330	+	0.555779i	$0.812420\pi$
$588$	0	0
$589$	−4537.26	−0.317410
$590$	−7017.12	−0.489644
$591$	0	0
$592$	11086.6	0.769690
$593$	2871.03	0.198818	0.0994089	−	0.995047i	$-0.468305\pi$
$593$	2871.03	0.198818	0.0994089	+	0.995047i	$0.468305\pi$
$594$	0	0
$595$	5378.60	0.370590
$596$	6759.86	0.464588
$597$	0	0
$598$	−2865.66	−0.195963
$599$	−6685.11	−0.456004	−0.228002	−	0.973661i	$-0.573219\pi$
$599$	−6685.11	−0.456004	−0.228002	+	0.973661i	$0.573219\pi$
$600$	0	0
$601$	8875.42	0.602389	0.301195	−	0.953563i	$-0.402615\pi$
$601$	8875.42	0.602389	0.301195	+	0.953563i	$0.402615\pi$
$602$	14.4505	0.000978333 0
$603$	0	0
$604$	9338.23	0.629085
$605$	−2260.84	−0.151928
$606$	0	0
$607$	379.477	0.0253748	0.0126874	−	0.999920i	$-0.495961\pi$
$607$	379.477	0.0253748	0.0126874	+	0.999920i	$0.495961\pi$
$608$	2422.30	0.161574
$609$	0	0
$610$	−7006.71	−0.465071
$611$	14487.8	0.959267
$612$	0	0
$613$	17480.5	1.15177	0.575883	−	0.817532i	$-0.304658\pi$
$613$	17480.5	1.15177	0.575883	+	0.817532i	$0.304658\pi$
$614$	5428.55	0.356805
$615$	0	0
$616$	678.198	0.0443594
$617$	9901.31	0.646048	0.323024	−	0.946391i	$-0.395301\pi$
$617$	9901.31	0.646048	0.323024	+	0.946391i	$0.395301\pi$
$618$	0	0
$619$	−152.433	−0.00989788	−0.00494894	−	0.999988i	$-0.501575\pi$
$619$	−152.433	−0.00989788	−0.00494894	+	0.999988i	$0.501575\pi$
$620$	27489.0	1.78062
$621$	0	0
$622$	−4026.88	−0.259587
$623$	6297.85	0.405005
$624$	0	0
$625$	6588.63	0.421672
$626$	−165.983	−0.0105974
$627$	0	0
$628$	−9425.74	−0.598930
$629$	−8064.84	−0.511234
$630$	0	0
$631$	−23661.7	−1.49280	−0.746399	−	0.665499i	$-0.768219\pi$
$631$	−23661.7	−1.49280	−0.746399	+	0.665499i	$0.768219\pi$
$632$	−6887.70	−0.433510
$633$	0	0
$634$	4037.46	0.252915
$635$	−4726.30	−0.295366
$636$	0	0
$637$	1785.48	0.111057
$638$	1720.04	0.106735
$639$	0	0
$640$	−19420.7	−1.19948
$641$	−25340.3	−1.56144	−0.780719	−	0.624882i	$-0.785147\pi$
$641$	−25340.3	−1.56144	−0.780719	+	0.624882i	$0.785147\pi$
$642$	0	0
$643$	−1774.90	−0.108858	−0.0544288	−	0.998518i	$-0.517334\pi$
$643$	−1774.90	−0.108858	−0.0544288	+	0.998518i	$0.517334\pi$
$644$	−7533.51	−0.460966
$645$	0	0
$646$	−547.295	−0.0333329
$647$	23685.1	1.43919	0.719595	−	0.694394i	$-0.244328\pi$
$647$	23685.1	1.43919	0.719595	+	0.694394i	$0.244328\pi$
$648$	0	0
$649$	−7356.57	−0.444947
$650$	−4585.90	−0.276729
$651$	0	0
$652$	3390.66	0.203663
$653$	1969.99	0.118058	0.0590288	−	0.998256i	$-0.481200\pi$
$653$	1969.99	0.118058	0.0590288	+	0.998256i	$0.481200\pi$
$654$	0	0
$655$	−27978.1	−1.66900
$656$	18242.4	1.08574
$657$	0	0
$658$	−1562.90	−0.0925958
$659$	−15314.6	−0.905270	−0.452635	−	0.891696i	$-0.649516\pi$
$659$	−15314.6	−0.905270	−0.452635	+	0.891696i	$0.649516\pi$
$660$	0	0
$661$	31443.3	1.85023	0.925115	−	0.379687i	$-0.123968\pi$
$661$	31443.3	1.85023	0.925115	+	0.379687i	$0.123968\pi$
$662$	1616.64	0.0949130
$663$	0	0
$664$	3758.31	0.219655
$665$	3099.76	0.180757
$666$	0	0
$667$	−38996.9	−2.26382
$668$	−13782.7	−0.798305
$669$	0	0
$670$	−7667.47	−0.442120
$671$	−7345.67	−0.422617
$672$	0	0
$673$	−8079.34	−0.462757	−0.231379	−	0.972864i	$-0.574324\pi$
$673$	−8079.34	−0.462757	−0.231379	+	0.972864i	$0.574324\pi$
$674$	6536.29	0.373543
$675$	0	0
$676$	6679.81	0.380053
$677$	15701.7	0.891383	0.445692	−	0.895187i	$-0.352958\pi$
$677$	15701.7	0.891383	0.445692	+	0.895187i	$0.352958\pi$
$678$	0	0
$679$	−6599.01	−0.372970
$680$	6767.63	0.381657
$681$	0	0
$682$	−1182.58	−0.0663980
$683$	−29752.9	−1.66686	−0.833430	−	0.552626i	$-0.813626\pi$
$683$	−29752.9	−1.66686	−0.833430	+	0.552626i	$0.813626\pi$
$684$	0	0
$685$	−22562.8	−1.25851
$686$	−192.613	−0.0107201
$687$	0	0
$688$	−207.817	−0.0115159
$689$	−21772.2	−1.20386
$690$	0	0
$691$	3082.57	0.169706	0.0848528	−	0.996393i	$-0.472958\pi$
$691$	3082.57	0.169706	0.0848528	+	0.996393i	$0.472958\pi$
$692$	29394.1	1.61473
$693$	0	0
$694$	−2669.01	−0.145986
$695$	−3970.84	−0.216723
$696$	0	0
$697$	−13270.2	−0.721156
$698$	4166.77	0.225952
$699$	0	0
$700$	−12055.8	−0.650953
$701$	−19301.0	−1.03993	−0.519963	−	0.854189i	$-0.674054\pi$
$701$	−19301.0	−1.03993	−0.519963	+	0.854189i	$0.674054\pi$
$702$	0	0
$703$	−4647.88	−0.249357
$704$	−4343.41	−0.232526
$705$	0	0
$706$	−290.801	−0.0155020
$707$	−2568.61	−0.136637
$708$	0	0
$709$	28531.3	1.51130	0.755652	−	0.654973i	$-0.227320\pi$
$709$	28531.3	1.51130	0.755652	+	0.654973i	$0.227320\pi$
$710$	328.047	0.0173400
$711$	0	0
$712$	7924.29	0.417100
$713$	26811.6	1.40828
$714$	0	0
$715$	−7489.24	−0.391723
$716$	10110.3	0.527711
$717$	0	0
$718$	−5544.97	−0.288212
$719$	−3884.65	−0.201492	−0.100746	−	0.994912i	$-0.532123\pi$
$719$	−3884.65	−0.201492	−0.100746	+	0.994912i	$0.532123\pi$
$720$	0	0
$721$	7051.43	0.364229
$722$	3536.28	0.182281
$723$	0	0
$724$	−5184.70	−0.266143
$725$	−62406.5	−3.19685
$726$	0	0
$727$	23489.8	1.19833	0.599167	−	0.800624i	$-0.295499\pi$
$727$	23489.8	1.19833	0.599167	+	0.800624i	$0.295499\pi$
$728$	2246.59	0.114374
$729$	0	0
$730$	−4557.99	−0.231094
$731$	151.175	0.00764897
$732$	0	0
$733$	30288.1	1.52622	0.763109	−	0.646270i	$-0.223672\pi$
$733$	30288.1	1.52622	0.763109	+	0.646270i	$0.223672\pi$
$734$	−1178.42	−0.0592593
$735$	0	0
$736$	−14313.9	−0.716870
$737$	−8038.39	−0.401761
$738$	0	0
$739$	29765.0	1.48163	0.740815	−	0.671709i	$-0.234440\pi$
$739$	29765.0	1.48163	0.740815	+	0.671709i	$0.234440\pi$
$740$	28159.2	1.39885
$741$	0	0
$742$	2348.72	0.116205
$743$	1200.21	0.0592617	0.0296309	−	0.999561i	$-0.490567\pi$
$743$	1200.21	0.0592617	0.0296309	+	0.999561i	$0.490567\pi$
$744$	0	0
$745$	16436.1	0.808284
$746$	10.9026	0.000535082 0
$747$	0	0
$748$	3476.19	0.169922
$749$	11835.6	0.577389
$750$	0	0
$751$	441.096	0.0214325	0.0107163	−	0.999943i	$-0.496589\pi$
$751$	441.096	0.0214325	0.0107163	+	0.999943i	$0.496589\pi$
$752$	22476.6	1.08994
$753$	0	0
$754$	5697.79	0.275200
$755$	22705.2	1.09447
$756$	0	0
$757$	−37080.3	−1.78032	−0.890162	−	0.455644i	$-0.849409\pi$
$757$	−37080.3	−1.78032	−0.890162	+	0.455644i	$0.849409\pi$
$758$	−7042.58	−0.337464
$759$	0	0
$760$	3900.28	0.186155
$761$	3056.78	0.145609	0.0728044	−	0.997346i	$-0.476805\pi$
$761$	3056.78	0.145609	0.0728044	+	0.997346i	$0.476805\pi$
$762$	0	0
$763$	−12660.0	−0.600685
$764$	14721.4	0.697123
$765$	0	0
$766$	5127.95	0.241880
$767$	−24369.3	−1.14723
$768$	0	0
$769$	31834.7	1.49283	0.746417	−	0.665479i	$-0.231773\pi$
$769$	31834.7	1.49283	0.746417	+	0.665479i	$0.231773\pi$
$770$	807.917	0.0378121
$771$	0	0
$772$	18402.5	0.857929
$773$	2351.46	0.109413	0.0547064	−	0.998502i	$-0.482578\pi$
$773$	2351.46	0.109413	0.0547064	+	0.998502i	$0.482578\pi$
$774$	0	0
$775$	42906.4	1.98870
$776$	−8303.22	−0.384108
$777$	0	0
$778$	5358.65	0.246937
$779$	−7647.81	−0.351748
$780$	0	0
$781$	343.916	0.0157571
$782$	3234.08	0.147891
$783$	0	0
$784$	2770.03	0.126186
$785$	−22918.0	−1.04201
$786$	0	0
$787$	−41575.3	−1.88310	−0.941550	−	0.336874i	$-0.890630\pi$
$787$	−41575.3	−1.88310	−0.941550	+	0.336874i	$0.890630\pi$
$788$	10426.6	0.471361
$789$	0	0
$790$	−8205.11	−0.369525
$791$	−6667.90	−0.299726
$792$	0	0
$793$	−24333.2	−1.08965
$794$	−2989.06	−0.133599
$795$	0	0
$796$	−16302.6	−0.725919
$797$	26199.7	1.16442	0.582208	−	0.813040i	$-0.302189\pi$
$797$	26199.7	1.16442	0.582208	+	0.813040i	$0.302189\pi$
$798$	0	0
$799$	−16350.4	−0.723948
$800$	−22906.4	−1.01233
$801$	0	0
$802$	3299.64	0.145280
$803$	−4778.49	−0.209999
$804$	0	0
$805$	−18317.2	−0.801981
$806$	−3917.41	−0.171197
$807$	0	0
$808$	−3231.96	−0.140718
$809$	2338.60	0.101633	0.0508164	−	0.998708i	$-0.483818\pi$
$809$	2338.60	0.101633	0.0508164	+	0.998708i	$0.483818\pi$
$810$	0	0
$811$	−3032.48	−0.131301	−0.0656504	−	0.997843i	$-0.520912\pi$
$811$	−3032.48	−0.131301	−0.0656504	+	0.997843i	$0.520912\pi$
$812$	14978.8	0.647358
$813$	0	0
$814$	−1211.42	−0.0521623
$815$	8244.13	0.354330
$816$	0	0
$817$	87.1240	0.00373082
$818$	3739.11	0.159823
$819$	0	0
$820$	46334.2	1.97325
$821$	28900.3	1.22854	0.614268	−	0.789098i	$-0.289451\pi$
$821$	28900.3	1.22854	0.614268	+	0.789098i	$0.289451\pi$
$822$	0	0
$823$	−19810.2	−0.839054	−0.419527	−	0.907743i	$-0.637804\pi$
$823$	−19810.2	−0.839054	−0.419527	+	0.907743i	$0.637804\pi$
$824$	8872.47	0.375106
$825$	0	0
$826$	2628.88	0.110739
$827$	−12280.7	−0.516374	−0.258187	−	0.966095i	$-0.583125\pi$
$827$	−12280.7	−0.516374	−0.258187	+	0.966095i	$0.583125\pi$
$828$	0	0
$829$	−3499.72	−0.146623	−0.0733114	−	0.997309i	$-0.523357\pi$
$829$	−3499.72	−0.146623	−0.0733114	+	0.997309i	$0.523357\pi$
$830$	4477.16	0.187235
$831$	0	0
$832$	−14387.9	−0.599532
$833$	−2015.03	−0.0838136
$834$	0	0
$835$	−33511.5	−1.38888
$836$	2003.37	0.0828806
$837$	0	0
$838$	6128.60	0.252636
$839$	11751.6	0.483562	0.241781	−	0.970331i	$-0.422268\pi$
$839$	11751.6	0.483562	0.241781	+	0.970331i	$0.422268\pi$
$840$	0	0
$841$	53148.4	2.17920
$842$	−2210.69	−0.0904815
$843$	0	0
$844$	−34520.4	−1.40787
$845$	16241.4	0.661210
$846$	0	0
$847$	847.000	0.0343604
$848$	−33777.8	−1.36785
$849$	0	0
$850$	5175.48	0.208844
$851$	27465.3	1.10634
$852$	0	0
$853$	1403.01	0.0563166	0.0281583	−	0.999603i	$-0.491036\pi$
$853$	1403.01	0.0563166	0.0281583	+	0.999603i	$0.491036\pi$
$854$	2624.99	0.105182
$855$	0	0
$856$	14892.2	0.594632
$857$	−15111.0	−0.602311	−0.301156	−	0.953575i	$-0.597372\pi$
$857$	−15111.0	−0.602311	−0.301156	+	0.953575i	$0.597372\pi$
$858$	0	0
$859$	18620.9	0.739623	0.369811	−	0.929107i	$-0.379422\pi$
$859$	18620.9	0.739623	0.369811	+	0.929107i	$0.379422\pi$
$860$	−527.841	−0.0209293
$861$	0	0
$862$	7732.19	0.305521
$863$	33659.0	1.32765	0.663827	−	0.747886i	$-0.268931\pi$
$863$	33659.0	1.32765	0.663827	+	0.747886i	$0.268931\pi$
$864$	0	0
$865$	71469.5	2.80929
$866$	−1605.20	−0.0629873
$867$	0	0
$868$	−10298.4	−0.402709
$869$	−8602.04	−0.335793
$870$	0	0
$871$	−26627.8	−1.03588
$872$	−15929.5	−0.618623
$873$	0	0
$874$	1863.85	0.0721345
$875$	−12963.7	−0.500861
$876$	0	0
$877$	−2730.45	−0.105132	−0.0525659	−	0.998617i	$-0.516740\pi$
$877$	−2730.45	−0.105132	−0.0525659	+	0.998617i	$0.516740\pi$
$878$	−4722.09	−0.181507
$879$	0	0
$880$	−11618.9	−0.445084
$881$	16625.5	0.635787	0.317894	−	0.948126i	$-0.397025\pi$
$881$	16625.5	0.635787	0.317894	+	0.948126i	$0.397025\pi$
$882$	0	0
$883$	10063.4	0.383536	0.191768	−	0.981440i	$-0.438578\pi$
$883$	10063.4	0.383536	0.191768	+	0.981440i	$0.438578\pi$
$884$	11515.2	0.438119
$885$	0	0
$886$	−1986.43	−0.0753223
$887$	−29939.9	−1.13335	−0.566676	−	0.823940i	$-0.691771\pi$
$887$	−29939.9	−1.13335	−0.566676	+	0.823940i	$0.691771\pi$
$888$	0	0
$889$	1770.66	0.0668008
$890$	9439.96	0.355537
$891$	0	0
$892$	−39517.2	−1.48333
$893$	−9422.94	−0.353109
$894$	0	0
$895$	24582.5	0.918103
$896$	7275.74	0.271278
$897$	0	0
$898$	266.075	0.00988756
$899$	−53309.5	−1.97772
$900$	0	0
$901$	24571.4	0.908536
$902$	−1993.31	−0.0735810
$903$	0	0
$904$	−8389.90	−0.308677
$905$	−12606.2	−0.463032
$906$	0	0
$907$	296.554	0.0108566	0.00542829	−	0.999985i	$-0.498272\pi$
$907$	296.554	0.0108566	0.00542829	+	0.999985i	$0.498272\pi$
$908$	22960.7	0.839182
$909$	0	0
$910$	2676.29	0.0974926
$911$	3552.96	0.129215	0.0646074	−	0.997911i	$-0.479420\pi$
$911$	3552.96	0.129215	0.0646074	+	0.997911i	$0.479420\pi$
$912$	0	0
$913$	4693.75	0.170143
$914$	−300.730	−0.0108832
$915$	0	0
$916$	39356.1	1.41961
$917$	10481.7	0.377465
$918$	0	0
$919$	−32047.3	−1.15032	−0.575159	−	0.818042i	$-0.695060\pi$
$919$	−32047.3	−1.15032	−0.575159	+	0.818042i	$0.695060\pi$
$920$	−23047.6	−0.825931
$921$	0	0
$922$	603.696	0.0215636
$923$	1139.25	0.0406272
$924$	0	0
$925$	43952.5	1.56232
$926$	6386.98	0.226662
$927$	0	0
$928$	28460.2	1.00674
$929$	−5628.82	−0.198790	−0.0993949	−	0.995048i	$-0.531691\pi$
$929$	−5628.82	−0.198790	−0.0993949	+	0.995048i	$0.531691\pi$
$930$	0	0
$931$	−1161.29	−0.0408805
$932$	8480.63	0.298060
$933$	0	0
$934$	−8519.36	−0.298460
$935$	8452.08	0.295629
$936$	0	0
$937$	−41269.1	−1.43885	−0.719425	−	0.694570i	$-0.755595\pi$
$937$	−41269.1	−1.43885	−0.719425	+	0.694570i	$0.755595\pi$
$938$	2872.53	0.0999909
$939$	0	0
$940$	57088.9	1.98089
$941$	−2086.74	−0.0722909	−0.0361454	−	0.999347i	$-0.511508\pi$
$941$	−2086.74	−0.0722909	−0.0361454	+	0.999347i	$0.511508\pi$
$942$	0	0
$943$	45192.6	1.56063
$944$	−37806.9	−1.30351
$945$	0	0
$946$	22.7079	0.000780440 0
$947$	21040.7	0.721995	0.360998	−	0.932567i	$-0.382436\pi$
$947$	21040.7	0.721995	0.360998	+	0.932567i	$0.382436\pi$
$948$	0	0
$949$	−15829.2	−0.541450
$950$	2982.70	0.101865
$951$	0	0
$952$	−2535.42	−0.0863166
$953$	−25644.0	−0.871658	−0.435829	−	0.900030i	$-0.643545\pi$
$953$	−25644.0	−0.871658	−0.435829	+	0.900030i	$0.643545\pi$
$954$	0	0
$955$	35794.0	1.21284
$956$	−13818.4	−0.467488
$957$	0	0
$958$	−1710.06	−0.0576719
$959$	8452.91	0.284628
$960$	0	0
$961$	6860.94	0.230302
$962$	−4012.92	−0.134492
$963$	0	0
$964$	56421.4	1.88507
$965$	44744.3	1.49261
$966$	0	0
$967$	28262.5	0.939878	0.469939	−	0.882699i	$-0.344276\pi$
$967$	28262.5	0.939878	0.469939	+	0.882699i	$0.344276\pi$
$968$	1065.74	0.0353865
$969$	0	0
$970$	−9891.37	−0.327415
$971$	−19180.0	−0.633899	−0.316950	−	0.948442i	$-0.602659\pi$
$971$	−19180.0	−0.633899	−0.316950	+	0.948442i	$0.602659\pi$
$972$	0	0
$973$	1487.63	0.0490147
$974$	9521.76	0.313241
$975$	0	0
$976$	−37750.9	−1.23809
$977$	35269.8	1.15495	0.577473	−	0.816410i	$-0.304039\pi$
$977$	35269.8	1.15495	0.577473	+	0.816410i	$0.304039\pi$
$978$	0	0
$979$	9896.62	0.323082
$980$	7035.68	0.229333
$981$	0	0
$982$	−7674.73	−0.249400
$983$	−40100.1	−1.30111	−0.650557	−	0.759458i	$-0.725464\pi$
$983$	−40100.1	−1.30111	−0.650557	+	0.759458i	$0.725464\pi$
$984$	0	0
$985$	25351.5	0.820066
$986$	−6430.32	−0.207691
$987$	0	0
$988$	6636.35	0.213695
$989$	−514.835	−0.0165529
$990$	0	0
$991$	3191.81	0.102312	0.0511560	−	0.998691i	$-0.483709\pi$
$991$	3191.81	0.102312	0.0511560	+	0.998691i	$0.483709\pi$
$992$	−19567.3	−0.626272
$993$	0	0
$994$	−122.899	−0.00392165
$995$	−39638.6	−1.26294
$996$	0	0
$997$	8275.27	0.262869	0.131435	−	0.991325i	$-0.458042\pi$
$997$	8275.27	0.262869	0.131435	+	0.991325i	$0.458042\pi$
$998$	6143.06	0.194845
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.g.1.2	✓	2	3.2	odd	2
693.4.a.j.1.1		2	1.1	even	1	trivial
1617.4.a.h.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)

\(f(q)\)	\(=\)	\(q-0.561553 q^{2} -7.68466 q^{4} -18.6847 q^{5} +7.00000 q^{7} +8.80776 q^{8} +O(q^{10})\)	\(q-0.561553 q^{2} -7.68466 q^{4} -18.6847 q^{5} +7.00000 q^{7} +8.80776 q^{8} +10.4924 q^{10} +11.0000 q^{11} +36.4384 q^{13} -3.93087 q^{14} +56.5312 q^{16} -41.1231 q^{17} -23.6998 q^{19} +143.585 q^{20} -6.17708 q^{22} +140.047 q^{23} +224.116 q^{25} -20.4621 q^{26} -53.7926 q^{28} -278.455 q^{29} +191.447 q^{31} -102.207 q^{32} +23.0928 q^{34} -130.793 q^{35} +196.115 q^{37} +13.3087 q^{38} -164.570 q^{40} +322.695 q^{41} -3.67615 q^{43} -84.5312 q^{44} -78.6440 q^{46} +397.596 q^{47} +49.0000 q^{49} -125.853 q^{50} -280.017 q^{52} -597.508 q^{53} -205.531 q^{55} +61.6543 q^{56} +156.367 q^{58} -668.779 q^{59} -667.788 q^{61} -107.508 q^{62} -394.855 q^{64} -680.840 q^{65} -730.762 q^{67} +316.017 q^{68} +73.4470 q^{70} +31.2651 q^{71} -434.408 q^{73} -110.129 q^{74} +182.125 q^{76} +77.0000 q^{77} -782.004 q^{79} -1056.27 q^{80} -181.210 q^{82} +426.705 q^{83} +768.371 q^{85} +2.06435 q^{86} +96.8854 q^{88} +899.693 q^{89} +255.069 q^{91} -1076.22 q^{92} -223.271 q^{94} +442.823 q^{95} -942.716 q^{97} -27.5161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 3 q^{4} - 25 q^{5} + 14 q^{7} - 3 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 - 3 * q^4 - 25 * q^5 + 14 * q^7 - 3 * q^8	\( 2 q + 3 q^{2} - 3 q^{4} - 25 q^{5} + 14 q^{7} - 3 q^{8} - 12 q^{10} + 22 q^{11} + 77 q^{13} + 21 q^{14} - 23 q^{16} - 74 q^{17} - 101 q^{19} + 114 q^{20} + 33 q^{22} - 58 q^{23} + 139 q^{25} + 124 q^{26} - 21 q^{28} - 91 q^{29} + 152 q^{31} - 291 q^{32} - 94 q^{34} - 175 q^{35} + 619 q^{37} - 262 q^{38} - 90 q^{40} - 138 q^{41} - 230 q^{43} - 33 q^{44} - 784 q^{46} - 149 q^{47} + 98 q^{49} - 429 q^{50} - 90 q^{52} - 1228 q^{53} - 275 q^{55} - 21 q^{56} + 824 q^{58} - 649 q^{59} - 412 q^{61} - 248 q^{62} - 431 q^{64} - 937 q^{65} - 1243 q^{67} + 162 q^{68} - 84 q^{70} - 960 q^{71} - 741 q^{73} + 1396 q^{74} - 180 q^{76} + 154 q^{77} - 492 q^{79} - 554 q^{80} - 1822 q^{82} + 1744 q^{83} + 976 q^{85} - 804 q^{86} - 33 q^{88} + 1552 q^{89} + 539 q^{91} - 2004 q^{92} - 2170 q^{94} + 931 q^{95} + 440 q^{97} + 147 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 - 3 * q^4 - 25 * q^5 + 14 * q^7 - 3 * q^8 - 12 * q^10 + 22 * q^11 + 77 * q^13 + 21 * q^14 - 23 * q^16 - 74 * q^17 - 101 * q^19 + 114 * q^20 + 33 * q^22 - 58 * q^23 + 139 * q^25 + 124 * q^26 - 21 * q^28 - 91 * q^29 + 152 * q^31 - 291 * q^32 - 94 * q^34 - 175 * q^35 + 619 * q^37 - 262 * q^38 - 90 * q^40 - 138 * q^41 - 230 * q^43 - 33 * q^44 - 784 * q^46 - 149 * q^47 + 98 * q^49 - 429 * q^50 - 90 * q^52 - 1228 * q^53 - 275 * q^55 - 21 * q^56 + 824 * q^58 - 649 * q^59 - 412 * q^61 - 248 * q^62 - 431 * q^64 - 937 * q^65 - 1243 * q^67 + 162 * q^68 - 84 * q^70 - 960 * q^71 - 741 * q^73 + 1396 * q^74 - 180 * q^76 + 154 * q^77 - 492 * q^79 - 554 * q^80 - 1822 * q^82 + 1744 * q^83 + 976 * q^85 - 804 * q^86 - 33 * q^88 + 1552 * q^89 + 539 * q^91 - 2004 * q^92 - 2170 * q^94 + 931 * q^95 + 440 * q^97 + 147 * q^98

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