LMFDB - Embedded newform 693.4.a.s.1.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 $ x^8 - 2x^7 - 45x^6 + 77x^5 + 540x^4 - 915x^3 - 1452x^2 + 2660*x - 672
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.21621$ 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)

$f(q)$	$=$	$q-2.21621 q^{2} -3.08840 q^{4} -22.0394 q^{5} -7.00000 q^{7} +24.5743 q^{8} +O(q^{10})$	\(q-2.21621 q^{2} -3.08840 q^{4} -22.0394 q^{5} -7.00000 q^{7} +24.5743 q^{8} +48.8440 q^{10} +11.0000 q^{11} +2.76840 q^{13} +15.5135 q^{14} -29.7545 q^{16} -50.6354 q^{17} +56.5022 q^{19} +68.0666 q^{20} -24.3783 q^{22} -172.339 q^{23} +360.735 q^{25} -6.13535 q^{26} +21.6188 q^{28} +282.011 q^{29} +237.587 q^{31} -130.652 q^{32} +112.219 q^{34} +154.276 q^{35} +207.061 q^{37} -125.221 q^{38} -541.602 q^{40} +386.514 q^{41} -18.3909 q^{43} -33.9725 q^{44} +381.939 q^{46} -309.957 q^{47} +49.0000 q^{49} -799.465 q^{50} -8.54993 q^{52} -480.364 q^{53} -242.433 q^{55} -172.020 q^{56} -624.996 q^{58} +114.221 q^{59} +109.928 q^{61} -526.543 q^{62} +527.588 q^{64} -61.0138 q^{65} -567.534 q^{67} +156.383 q^{68} -341.908 q^{70} -780.713 q^{71} -605.836 q^{73} -458.891 q^{74} -174.502 q^{76} -77.0000 q^{77} +686.614 q^{79} +655.772 q^{80} -856.597 q^{82} +316.962 q^{83} +1115.97 q^{85} +40.7582 q^{86} +270.317 q^{88} -1611.41 q^{89} -19.3788 q^{91} +532.252 q^{92} +686.931 q^{94} -1245.27 q^{95} -1231.47 q^{97} -108.594 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8	$ 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26} - 210 q^{28} + 72 q^{29} + 252 q^{31} - 1272 q^{32} + 630 q^{34} + 70 q^{35} - 80 q^{37} - 1885 q^{38} - 342 q^{40} - 682 q^{41} - 106 q^{43} + 330 q^{44} + 120 q^{46} - 828 q^{47} + 392 q^{49} - 801 q^{50} - 1681 q^{52} - 462 q^{53} - 110 q^{55} + 105 q^{56} - 1087 q^{58} - 626 q^{59} - 854 q^{61} - 1350 q^{62} + 2997 q^{64} - 22 q^{65} + 130 q^{67} - 2202 q^{68} + 91 q^{70} - 326 q^{71} - 390 q^{73} + 359 q^{74} + 2041 q^{76} - 616 q^{77} - 508 q^{79} - 4391 q^{80} + 1528 q^{82} - 1596 q^{83} - 880 q^{85} + 414 q^{86} - 165 q^{88} - 4324 q^{89} + 1036 q^{91} - 2092 q^{92} - 1685 q^{94} - 1076 q^{95} - 964 q^{97} - 98 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8 - 13 * q^10 + 88 * q^11 - 148 * q^13 + 14 * q^14 + 266 * q^16 - 114 * q^17 + 58 * q^19 - 291 * q^20 - 22 * q^22 - 246 * q^23 + 244 * q^25 - 305 * q^26 - 210 * q^28 + 72 * q^29 + 252 * q^31 - 1272 * q^32 + 630 * q^34 + 70 * q^35 - 80 * q^37 - 1885 * q^38 - 342 * q^40 - 682 * q^41 - 106 * q^43 + 330 * q^44 + 120 * q^46 - 828 * q^47 + 392 * q^49 - 801 * q^50 - 1681 * q^52 - 462 * q^53 - 110 * q^55 + 105 * q^56 - 1087 * q^58 - 626 * q^59 - 854 * q^61 - 1350 * q^62 + 2997 * q^64 - 22 * q^65 + 130 * q^67 - 2202 * q^68 + 91 * q^70 - 326 * q^71 - 390 * q^73 + 359 * q^74 + 2041 * q^76 - 616 * q^77 - 508 * q^79 - 4391 * q^80 + 1528 * q^82 - 1596 * q^83 - 880 * q^85 + 414 * q^86 - 165 * q^88 - 4324 * q^89 + 1036 * q^91 - 2092 * q^92 - 1685 * q^94 - 1076 * q^95 - 964 * q^97 - 98 * 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 $

$2$	−2.21621	−0.783549	−0.391775	−	0.920061i	$-0.628139\pi$
$2$	−2.21621	−0.783549	−0.391775	+	0.920061i	$0.628139\pi$
$3$	0	0
$3$	0	0
$4$	−3.08840	−0.386051
$5$	−22.0394	−1.97126	−0.985632	−	0.168908i	$-0.945976\pi$
$5$	−22.0394	−1.97126	−0.985632	+	0.168908i	$0.945976\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	24.5743	1.08604
$9$	0	0
$10$	48.8440	1.54458
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	2.76840	0.0590627	0.0295313	−	0.999564i	$-0.490599\pi$
$13$	2.76840	0.0590627	0.0295313	+	0.999564i	$0.490599\pi$
$14$	15.5135	0.296154
$15$	0	0
$16$	−29.7545	−0.464914
$17$	−50.6354	−0.722405	−0.361203	−	0.932487i	$-0.617634\pi$
$17$	−50.6354	−0.722405	−0.361203	+	0.932487i	$0.617634\pi$
$18$	0	0
$19$	56.5022	0.682236	0.341118	−	0.940020i	$-0.389194\pi$
$19$	56.5022	0.682236	0.341118	+	0.940020i	$0.389194\pi$
$20$	68.0666	0.761007
$21$	0	0
$22$	−24.3783	−0.236249
$23$	−172.339	−1.56240	−0.781198	−	0.624283i	$-0.785391\pi$
$23$	−172.339	−1.56240	−0.781198	+	0.624283i	$0.785391\pi$
$24$	0	0
$25$	360.735	2.88588
$26$	−6.13535	−0.0462785
$27$	0	0
$28$	21.6188	0.145913
$29$	282.011	1.80580	0.902899	−	0.429853i	$-0.141434\pi$
$29$	282.011	1.80580	0.902899	+	0.429853i	$0.141434\pi$
$30$	0	0
$31$	237.587	1.37651	0.688256	−	0.725468i	$-0.258377\pi$
$31$	237.587	1.37651	0.688256	+	0.725468i	$0.258377\pi$
$32$	−130.652	−0.721756
$33$	0	0
$34$	112.219	0.566040
$35$	154.276	0.745068
$36$	0	0
$37$	207.061	0.920017	0.460009	−	0.887914i	$-0.347846\pi$
$37$	207.061	0.920017	0.460009	+	0.887914i	$0.347846\pi$
$38$	−125.221	−0.534566
$39$	0	0
$40$	−541.602	−2.14087
$41$	386.514	1.47228	0.736138	−	0.676831i	$-0.236647\pi$
$41$	386.514	1.47228	0.736138	+	0.676831i	$0.236647\pi$
$42$	0	0
$43$	−18.3909	−0.0652230	−0.0326115	−	0.999468i	$-0.510382\pi$
$43$	−18.3909	−0.0652230	−0.0326115	+	0.999468i	$0.510382\pi$
$44$	−33.9725	−0.116399
$45$	0	0
$46$	381.939	1.22421
$47$	−309.957	−0.961955	−0.480978	−	0.876733i	$-0.659718\pi$
$47$	−309.957	−0.961955	−0.480978	+	0.876733i	$0.659718\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−799.465	−2.26123
$51$	0	0
$52$	−8.54993	−0.0228012
$53$	−480.364	−1.24496	−0.622482	−	0.782634i	$-0.713876\pi$
$53$	−480.364	−1.24496	−0.622482	+	0.782634i	$0.713876\pi$
$54$	0	0
$55$	−242.433	−0.594358
$56$	−172.020	−0.410484
$57$	0	0
$58$	−624.996	−1.41493
$59$	114.221	0.252039	0.126020	−	0.992028i	$-0.459780\pi$
$59$	114.221	0.252039	0.126020	+	0.992028i	$0.459780\pi$
$60$	0	0
$61$	109.928	0.230734	0.115367	−	0.993323i	$-0.463196\pi$
$61$	109.928	0.230734	0.115367	+	0.993323i	$0.463196\pi$
$62$	−526.543	−1.07856
$63$	0	0
$64$	527.588	1.03045
$65$	−61.0138	−0.116428
$66$	0	0
$67$	−567.534	−1.03485	−0.517427	−	0.855727i	$-0.673110\pi$
$67$	−567.534	−1.03485	−0.517427	+	0.855727i	$0.673110\pi$
$68$	156.383	0.278885
$69$	0	0
$70$	−341.908	−0.583797
$71$	−780.713	−1.30498	−0.652490	−	0.757797i	$-0.726276\pi$
$71$	−780.713	−1.30498	−0.652490	+	0.757797i	$0.726276\pi$
$72$	0	0
$73$	−605.836	−0.971340	−0.485670	−	0.874142i	$-0.661424\pi$
$73$	−605.836	−0.971340	−0.485670	+	0.874142i	$0.661424\pi$
$74$	−458.891	−0.720879
$75$	0	0
$76$	−174.502	−0.263378
$77$	−77.0000	−0.113961
$78$	0	0
$79$	686.614	0.977850	0.488925	−	0.872326i	$-0.337389\pi$
$79$	686.614	0.977850	0.488925	+	0.872326i	$0.337389\pi$
$80$	655.772	0.916469
$81$	0	0
$82$	−856.597	−1.15360
$83$	316.962	0.419170	0.209585	−	0.977790i	$-0.432789\pi$
$83$	316.962	0.419170	0.209585	+	0.977790i	$0.432789\pi$
$84$	0	0
$85$	1115.97	1.42405
$86$	40.7582	0.0511054
$87$	0	0
$88$	270.317	0.327453
$89$	−1611.41	−1.91920	−0.959601	−	0.281365i	$-0.909213\pi$
$89$	−1611.41	−1.91920	−0.959601	+	0.281365i	$0.909213\pi$
$90$	0	0
$91$	−19.3788	−0.0223236
$92$	532.252	0.603164
$93$	0	0
$94$	686.931	0.753739
$95$	−1245.27	−1.34487
$96$	0	0
$97$	−1231.47	−1.28904	−0.644520	−	0.764587i	$-0.722943\pi$
$97$	−1231.47	−1.28904	−0.644520	+	0.764587i	$0.722943\pi$
$98$	−108.594	−0.111936
$99$	0	0
$100$	−1114.10	−1.11410
$101$	1184.54	1.16699	0.583494	−	0.812117i	$-0.301685\pi$
$101$	1184.54	1.16699	0.583494	+	0.812117i	$0.301685\pi$
$102$	0	0
$103$	−654.836	−0.626436	−0.313218	−	0.949681i	$-0.601407\pi$
$103$	−654.836	−0.626436	−0.313218	+	0.949681i	$0.601407\pi$
$104$	68.0313	0.0641444
$105$	0	0
$106$	1064.59	0.975490
$107$	−654.327	−0.591179	−0.295589	−	0.955315i	$-0.595516\pi$
$107$	−654.327	−0.591179	−0.295589	+	0.955315i	$0.595516\pi$
$108$	0	0
$109$	1787.78	1.57099	0.785497	−	0.618866i	$-0.212408\pi$
$109$	1787.78	1.57099	0.785497	+	0.618866i	$0.212408\pi$
$110$	537.284	0.465709
$111$	0	0
$112$	208.282	0.175721
$113$	824.970	0.686784	0.343392	−	0.939192i	$-0.388424\pi$
$113$	824.970	0.686784	0.343392	+	0.939192i	$0.388424\pi$
$114$	0	0
$115$	3798.24	3.07989
$116$	−870.964	−0.697129
$117$	0	0
$118$	−253.138	−0.197485
$119$	354.448	0.273043
$120$	0	0
$121$	121.000	0.0909091
$122$	−243.623	−0.180792
$123$	0	0
$124$	−733.764	−0.531403
$125$	−5195.46	−3.71757
$126$	0	0
$127$	1754.34	1.22577	0.612885	−	0.790172i	$-0.290009\pi$
$127$	1754.34	1.22577	0.612885	+	0.790172i	$0.290009\pi$
$128$	−124.033	−0.0856491
$129$	0	0
$130$	135.219	0.0912272
$131$	−2065.18	−1.37737	−0.688687	−	0.725059i	$-0.741812\pi$
$131$	−2065.18	−1.37737	−0.688687	+	0.725059i	$0.741812\pi$
$132$	0	0
$133$	−395.515	−0.257861
$134$	1257.77	0.810859
$135$	0	0
$136$	−1244.33	−0.784560
$137$	658.269	0.410509	0.205255	−	0.978709i	$-0.434198\pi$
$137$	658.269	0.410509	0.205255	+	0.978709i	$0.434198\pi$
$138$	0	0
$139$	−704.321	−0.429782	−0.214891	−	0.976638i	$-0.568940\pi$
$139$	−704.321	−0.429782	−0.214891	+	0.976638i	$0.568940\pi$
$140$	−476.466	−0.287634
$141$	0	0
$142$	1730.22	1.02252
$143$	30.4524	0.0178081
$144$	0	0
$145$	−6215.35	−3.55970
$146$	1342.66	0.761092
$147$	0	0
$148$	−639.489	−0.355173
$149$	1246.54	0.685375	0.342687	−	0.939450i	$-0.388663\pi$
$149$	1246.54	0.685375	0.342687	+	0.939450i	$0.388663\pi$
$150$	0	0
$151$	1137.10	0.612818	0.306409	−	0.951900i	$-0.400872\pi$
$151$	1137.10	0.612818	0.306409	+	0.951900i	$0.400872\pi$
$152$	1388.50	0.740935
$153$	0	0
$154$	170.648	0.0892937
$155$	−5236.27	−2.71347
$156$	0	0
$157$	−130.308	−0.0662400	−0.0331200	−	0.999451i	$-0.510544\pi$
$157$	−130.308	−0.0662400	−0.0331200	+	0.999451i	$0.510544\pi$
$158$	−1521.68	−0.766193
$159$	0	0
$160$	2879.49	1.42277
$161$	1206.37	0.590530
$162$	0	0
$163$	−907.833	−0.436239	−0.218120	−	0.975922i	$-0.569992\pi$
$163$	−907.833	−0.436239	−0.218120	+	0.975922i	$0.569992\pi$
$164$	−1193.71	−0.568373
$165$	0	0
$166$	−702.456	−0.328441
$167$	3709.74	1.71897	0.859486	−	0.511159i	$-0.170784\pi$
$167$	3709.74	1.71897	0.859486	+	0.511159i	$0.170784\pi$
$168$	0	0
$169$	−2189.34	−0.996512
$170$	−2473.23	−1.11581
$171$	0	0
$172$	56.7986	0.0251794
$173$	−2172.32	−0.954673	−0.477336	−	0.878721i	$-0.658398\pi$
$173$	−2172.32	−0.954673	−0.477336	+	0.878721i	$0.658398\pi$
$174$	0	0
$175$	−2525.15	−1.09076
$176$	−327.300	−0.140177
$177$	0	0
$178$	3571.22	1.50379
$179$	2149.14	0.897398	0.448699	−	0.893683i	$-0.351888\pi$
$179$	2149.14	0.897398	0.448699	+	0.893683i	$0.351888\pi$
$180$	0	0
$181$	−1083.16	−0.444810	−0.222405	−	0.974954i	$-0.571391\pi$
$181$	−1083.16	−0.444810	−0.222405	+	0.974954i	$0.571391\pi$
$182$	42.9475	0.0174916
$183$	0	0
$184$	−4235.10	−1.69682
$185$	−4563.50	−1.81360
$186$	0	0
$187$	−556.989	−0.217813
$188$	957.273	0.371363
$189$	0	0
$190$	2759.79	1.05377
$191$	2847.74	1.07882	0.539412	−	0.842042i	$-0.318647\pi$
$191$	2847.74	1.07882	0.539412	+	0.842042i	$0.318647\pi$
$192$	0	0
$193$	−4265.96	−1.59104	−0.795519	−	0.605929i	$-0.792802\pi$
$193$	−4265.96	−1.59104	−0.795519	+	0.605929i	$0.792802\pi$
$194$	2729.20	1.01003
$195$	0	0
$196$	−151.332	−0.0551501
$197$	3081.31	1.11439	0.557194	−	0.830382i	$-0.311878\pi$
$197$	3081.31	1.11439	0.557194	+	0.830382i	$0.311878\pi$
$198$	0	0
$199$	−1880.41	−0.669843	−0.334921	−	0.942246i	$-0.608710\pi$
$199$	−1880.41	−0.669843	−0.334921	+	0.942246i	$0.608710\pi$
$200$	8864.79	3.13418
$201$	0	0
$202$	−2625.18	−0.914392
$203$	−1974.08	−0.682528
$204$	0	0
$205$	−8518.54	−2.90225
$206$	1451.26	0.490844
$207$	0	0
$208$	−82.3723	−0.0274591
$209$	621.524	0.205702
$210$	0	0
$211$	−1383.75	−0.451475	−0.225738	−	0.974188i	$-0.572479\pi$
$211$	−1383.75	−0.451475	−0.225738	+	0.974188i	$0.572479\pi$
$212$	1483.56	0.480619
$213$	0	0
$214$	1450.13	0.463218
$215$	405.325	0.128572
$216$	0	0
$217$	−1663.11	−0.520272
$218$	−3962.10	−1.23095
$219$	0	0
$220$	748.732	0.229452
$221$	−140.179	−0.0426672
$222$	0	0
$223$	1565.56	0.470123	0.235062	−	0.971980i	$-0.424471\pi$
$223$	1565.56	0.470123	0.235062	+	0.971980i	$0.424471\pi$
$224$	914.562	0.272798
$225$	0	0
$226$	−1828.31	−0.538129
$227$	−2467.42	−0.721448	−0.360724	−	0.932673i	$-0.617470\pi$
$227$	−2467.42	−0.721448	−0.360724	+	0.932673i	$0.617470\pi$
$228$	0	0
$229$	−1294.61	−0.373581	−0.186791	−	0.982400i	$-0.559809\pi$
$229$	−1294.61	−0.373581	−0.186791	+	0.982400i	$0.559809\pi$
$230$	−8417.71	−2.41325
$231$	0	0
$232$	6930.21	1.96117
$233$	5470.03	1.53800	0.768999	−	0.639249i	$-0.220755\pi$
$233$	5470.03	1.53800	0.768999	+	0.639249i	$0.220755\pi$
$234$	0	0
$235$	6831.27	1.89627
$236$	−352.761	−0.0972999
$237$	0	0
$238$	−785.531	−0.213943
$239$	−370.090	−0.100164	−0.0500819	−	0.998745i	$-0.515948\pi$
$239$	−370.090	−0.100164	−0.0500819	+	0.998745i	$0.515948\pi$
$240$	0	0
$241$	−1364.30	−0.364658	−0.182329	−	0.983238i	$-0.558364\pi$
$241$	−1364.30	−0.364658	−0.182329	+	0.983238i	$0.558364\pi$
$242$	−268.162	−0.0712317
$243$	0	0
$244$	−339.501	−0.0890751
$245$	−1079.93	−0.281609
$246$	0	0
$247$	156.420	0.0402947
$248$	5838.52	1.49495
$249$	0	0
$250$	11514.2	2.91290
$251$	−3715.37	−0.934311	−0.467155	−	0.884175i	$-0.654721\pi$
$251$	−3715.37	−0.934311	−0.467155	+	0.884175i	$0.654721\pi$
$252$	0	0
$253$	−1895.73	−0.471080
$254$	−3888.00	−0.960451
$255$	0	0
$256$	−3945.82	−0.963335
$257$	450.250	0.109283	0.0546417	−	0.998506i	$-0.482598\pi$
$257$	450.250	0.109283	0.0546417	+	0.998506i	$0.482598\pi$
$258$	0	0
$259$	−1449.43	−0.347734
$260$	188.435	0.0449471
$261$	0	0
$262$	4576.88	1.07924
$263$	1842.16	0.431910	0.215955	−	0.976403i	$-0.430714\pi$
$263$	1842.16	0.431910	0.215955	+	0.976403i	$0.430714\pi$
$264$	0	0
$265$	10586.9	2.45415
$266$	876.546	0.202047
$267$	0	0
$268$	1752.77	0.399506
$269$	2039.63	0.462298	0.231149	−	0.972918i	$-0.425751\pi$
$269$	2039.63	0.462298	0.231149	+	0.972918i	$0.425751\pi$
$270$	0	0
$271$	7810.34	1.75072	0.875359	−	0.483473i	$-0.160625\pi$
$271$	7810.34	1.75072	0.875359	+	0.483473i	$0.160625\pi$
$272$	1506.63	0.335857
$273$	0	0
$274$	−1458.86	−0.321654
$275$	3968.09	0.870126
$276$	0	0
$277$	6066.15	1.31581	0.657905	−	0.753101i	$-0.271443\pi$
$277$	6066.15	1.31581	0.657905	+	0.753101i	$0.271443\pi$
$278$	1560.93	0.336756
$279$	0	0
$280$	3791.21	0.809172
$281$	−4527.28	−0.961121	−0.480560	−	0.876962i	$-0.659567\pi$
$281$	−4527.28	−0.961121	−0.480560	+	0.876962i	$0.659567\pi$
$282$	0	0
$283$	−3694.07	−0.775935	−0.387968	−	0.921673i	$-0.626823\pi$
$283$	−3694.07	−0.775935	−0.387968	+	0.921673i	$0.626823\pi$
$284$	2411.16	0.503788
$285$	0	0
$286$	−67.4889	−0.0139535
$287$	−2705.60	−0.556468
$288$	0	0
$289$	−2349.06	−0.478131
$290$	13774.5	2.78920
$291$	0	0
$292$	1871.07	0.374986
$293$	−902.927	−0.180033	−0.0900163	−	0.995940i	$-0.528692\pi$
$293$	−902.927	−0.180033	−0.0900163	+	0.995940i	$0.528692\pi$
$294$	0	0
$295$	−2517.36	−0.496836
$296$	5088.37	0.999175
$297$	0	0
$298$	−2762.60	−0.537025
$299$	−477.102	−0.0922793
$300$	0	0
$301$	128.736	0.0246520
$302$	−2520.05	−0.480173
$303$	0	0
$304$	−1681.20	−0.317181
$305$	−2422.74	−0.454838
$306$	0	0
$307$	−3119.31	−0.579898	−0.289949	−	0.957042i	$-0.593638\pi$
$307$	−3119.31	−0.579898	−0.289949	+	0.957042i	$0.593638\pi$
$308$	237.807	0.0439945
$309$	0	0
$310$	11604.7	2.12614
$311$	3520.52	0.641898	0.320949	−	0.947097i	$-0.395998\pi$
$311$	3520.52	0.641898	0.320949	+	0.947097i	$0.395998\pi$
$312$	0	0
$313$	−10174.8	−1.83742	−0.918709	−	0.394935i	$-0.870767\pi$
$313$	−10174.8	−1.83742	−0.918709	+	0.394935i	$0.870767\pi$
$314$	288.789	0.0519023
$315$	0	0
$316$	−2120.54	−0.377499
$317$	3907.40	0.692307	0.346153	−	0.938178i	$-0.387488\pi$
$317$	3907.40	0.692307	0.346153	+	0.938178i	$0.387488\pi$
$318$	0	0
$319$	3102.12	0.544469
$320$	−11627.7	−2.03128
$321$	0	0
$322$	−2673.57	−0.462710
$323$	−2861.01	−0.492851
$324$	0	0
$325$	998.657	0.170448
$326$	2011.95	0.341815
$327$	0	0
$328$	9498.29	1.59895
$329$	2169.70	0.363585
$330$	0	0
$331$	9015.12	1.49703	0.748513	−	0.663120i	$-0.230768\pi$
$331$	9015.12	1.49703	0.748513	+	0.663120i	$0.230768\pi$
$332$	−978.908	−0.161821
$333$	0	0
$334$	−8221.57	−1.34690
$335$	12508.1	2.03997
$336$	0	0
$337$	2727.61	0.440898	0.220449	−	0.975399i	$-0.429248\pi$
$337$	2727.61	0.440898	0.220449	+	0.975399i	$0.429248\pi$
$338$	4852.03	0.780816
$339$	0	0
$340$	−3446.58	−0.549756
$341$	2613.46	0.415034
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−451.943	−0.0708347
$345$	0	0
$346$	4814.32	0.748033
$347$	−919.041	−0.142181	−0.0710903	−	0.997470i	$-0.522648\pi$
$347$	−919.041	−0.142181	−0.0710903	+	0.997470i	$0.522648\pi$
$348$	0	0
$349$	1634.89	0.250755	0.125378	−	0.992109i	$-0.459986\pi$
$349$	1634.89	0.250755	0.125378	+	0.992109i	$0.459986\pi$
$350$	5596.26	0.854664
$351$	0	0
$352$	−1437.17	−0.217618
$353$	4994.77	0.753101	0.376551	−	0.926396i	$-0.377110\pi$
$353$	4994.77	0.753101	0.376551	+	0.926396i	$0.377110\pi$
$354$	0	0
$355$	17206.4	2.57246
$356$	4976.68	0.740909
$357$	0	0
$358$	−4762.95	−0.703156
$359$	−9765.04	−1.43560	−0.717798	−	0.696251i	$-0.754850\pi$
$359$	−9765.04	−1.43560	−0.717798	+	0.696251i	$0.754850\pi$
$360$	0	0
$361$	−3666.50	−0.534554
$362$	2400.51	0.348531
$363$	0	0
$364$	59.8495	0.00861804
$365$	13352.3	1.91477
$366$	0	0
$367$	−9537.33	−1.35652	−0.678262	−	0.734820i	$-0.737267\pi$
$367$	−9537.33	−1.35652	−0.678262	+	0.734820i	$0.737267\pi$
$368$	5127.86	0.726380
$369$	0	0
$370$	10113.7	1.42104
$371$	3362.55	0.470552
$372$	0	0
$373$	−8576.29	−1.19052	−0.595259	−	0.803534i	$-0.702951\pi$
$373$	−8576.29	−1.19052	−0.595259	+	0.803534i	$0.702951\pi$
$374$	1234.41	0.170667
$375$	0	0
$376$	−7616.97	−1.04472
$377$	780.718	0.106655
$378$	0	0
$379$	−13710.9	−1.85826	−0.929131	−	0.369752i	$-0.879443\pi$
$379$	−13710.9	−1.85826	−0.929131	+	0.369752i	$0.879443\pi$
$380$	3845.91	0.519187
$381$	0	0
$382$	−6311.20	−0.845311
$383$	−8428.82	−1.12452	−0.562262	−	0.826959i	$-0.690069\pi$
$383$	−8428.82	−1.12452	−0.562262	+	0.826959i	$0.690069\pi$
$384$	0	0
$385$	1697.03	0.224646
$386$	9454.26	1.24666
$387$	0	0
$388$	3803.28	0.497635
$389$	−1757.23	−0.229036	−0.114518	−	0.993421i	$-0.536532\pi$
$389$	−1757.23	−0.229036	−0.114518	+	0.993421i	$0.536532\pi$
$390$	0	0
$391$	8726.44	1.12868
$392$	1204.14	0.155148
$393$	0	0
$394$	−6828.84	−0.873178
$395$	−15132.6	−1.92760
$396$	0	0
$397$	4018.53	0.508021	0.254010	−	0.967201i	$-0.418250\pi$
$397$	4018.53	0.508021	0.254010	+	0.967201i	$0.418250\pi$
$398$	4167.39	0.524855
$399$	0	0
$400$	−10733.5	−1.34169
$401$	3344.16	0.416458	0.208229	−	0.978080i	$-0.433230\pi$
$401$	3344.16	0.416458	0.208229	+	0.978080i	$0.433230\pi$
$402$	0	0
$403$	657.734	0.0813005
$404$	−3658.33	−0.450516
$405$	0	0
$406$	4374.97	0.534794
$407$	2277.67	0.277396
$408$	0	0
$409$	−13540.8	−1.63704	−0.818521	−	0.574477i	$-0.805206\pi$
$409$	−13540.8	−1.63704	−0.818521	+	0.574477i	$0.805206\pi$
$410$	18878.9	2.27405
$411$	0	0
$412$	2022.40	0.241836
$413$	−799.547	−0.0952619
$414$	0	0
$415$	−6985.66	−0.826295
$416$	−361.696	−0.0426288
$417$	0	0
$418$	−1377.43	−0.161178
$419$	−7286.55	−0.849573	−0.424786	−	0.905294i	$-0.639651\pi$
$419$	−7286.55	−0.849573	−0.424786	+	0.905294i	$0.639651\pi$
$420$	0	0
$421$	−10778.5	−1.24777	−0.623886	−	0.781515i	$-0.714447\pi$
$421$	−10778.5	−1.24777	−0.623886	+	0.781515i	$0.714447\pi$
$422$	3066.68	0.353753
$423$	0	0
$424$	−11804.6	−1.35208
$425$	−18266.0	−2.08477
$426$	0	0
$427$	−769.494	−0.0872094
$428$	2020.83	0.228225
$429$	0	0
$430$	−898.285	−0.100742
$431$	−8547.58	−0.955272	−0.477636	−	0.878558i	$-0.658506\pi$
$431$	−8547.58	−0.955272	−0.477636	+	0.878558i	$0.658506\pi$
$432$	0	0
$433$	7140.86	0.792535	0.396267	−	0.918135i	$-0.370305\pi$
$433$	7140.86	0.792535	0.396267	+	0.918135i	$0.370305\pi$
$434$	3685.80	0.407659
$435$	0	0
$436$	−5521.39	−0.606483
$437$	−9737.52	−1.06592
$438$	0	0
$439$	−2411.65	−0.262191	−0.131095	−	0.991370i	$-0.541849\pi$
$439$	−2411.65	−0.262191	−0.131095	+	0.991370i	$0.541849\pi$
$440$	−5957.62	−0.645496
$441$	0	0
$442$	310.666	0.0334318
$443$	−12210.1	−1.30953	−0.654764	−	0.755834i	$-0.727232\pi$
$443$	−12210.1	−1.30953	−0.654764	+	0.755834i	$0.727232\pi$
$444$	0	0
$445$	35514.5	3.78325
$446$	−3469.61	−0.368365
$447$	0	0
$448$	−3693.12	−0.389472
$449$	−4214.35	−0.442956	−0.221478	−	0.975165i	$-0.571088\pi$
$449$	−4214.35	−0.442956	−0.221478	+	0.975165i	$0.571088\pi$
$450$	0	0
$451$	4251.65	0.443908
$452$	−2547.84	−0.265134
$453$	0	0
$454$	5468.33	0.565290
$455$	427.096	0.0440057
$456$	0	0
$457$	169.614	0.0173615	0.00868075	−	0.999962i	$-0.497237\pi$
$457$	169.614	0.0173615	0.00868075	+	0.999962i	$0.497237\pi$
$458$	2869.13	0.292719
$459$	0	0
$460$	−11730.5	−1.18900
$461$	−813.238	−0.0821611	−0.0410805	−	0.999156i	$-0.513080\pi$
$461$	−813.238	−0.0821611	−0.0410805	+	0.999156i	$0.513080\pi$
$462$	0	0
$463$	−12681.2	−1.27289	−0.636443	−	0.771324i	$-0.719595\pi$
$463$	−12681.2	−1.27289	−0.636443	+	0.771324i	$0.719595\pi$
$464$	−8391.10	−0.839541
$465$	0	0
$466$	−12122.8	−1.20510
$467$	−7297.16	−0.723067	−0.361534	−	0.932359i	$-0.617747\pi$
$467$	−7297.16	−0.723067	−0.361534	+	0.932359i	$0.617747\pi$
$468$	0	0
$469$	3972.73	0.391138
$470$	−15139.5	−1.48582
$471$	0	0
$472$	2806.90	0.273724
$473$	−202.300	−0.0196655
$474$	0	0
$475$	20382.3	1.96885
$476$	−1094.68	−0.105409
$477$	0	0
$478$	820.199	0.0784833
$479$	9874.50	0.941915	0.470958	−	0.882156i	$-0.343908\pi$
$479$	9874.50	0.941915	0.470958	+	0.882156i	$0.343908\pi$
$480$	0	0
$481$	573.227	0.0543387
$482$	3023.59	0.285727
$483$	0	0
$484$	−373.697	−0.0350955
$485$	27140.9	2.54104
$486$	0	0
$487$	4995.82	0.464851	0.232425	−	0.972614i	$-0.425334\pi$
$487$	4995.82	0.464851	0.232425	+	0.972614i	$0.425334\pi$
$488$	2701.39	0.250586
$489$	0	0
$490$	2393.35	0.220655
$491$	900.456	0.0827638	0.0413819	−	0.999143i	$-0.486824\pi$
$491$	900.456	0.0827638	0.0413819	+	0.999143i	$0.486824\pi$
$492$	0	0
$493$	−14279.7	−1.30452
$494$	−346.661	−0.0315729
$495$	0	0
$496$	−7069.28	−0.639960
$497$	5464.99	0.493236
$498$	0	0
$499$	11471.6	1.02914	0.514568	−	0.857450i	$-0.327952\pi$
$499$	11471.6	1.02914	0.514568	+	0.857450i	$0.327952\pi$
$500$	16045.7	1.43517
$501$	0	0
$502$	8234.04	0.732078
$503$	−4206.65	−0.372893	−0.186447	−	0.982465i	$-0.559697\pi$
$503$	−4206.65	−0.372893	−0.186447	+	0.982465i	$0.559697\pi$
$504$	0	0
$505$	−26106.5	−2.30044
$506$	4201.33	0.369114
$507$	0	0
$508$	−5418.12	−0.473209
$509$	−16162.9	−1.40748	−0.703742	−	0.710456i	$-0.748489\pi$
$509$	−16162.9	−1.40748	−0.703742	+	0.710456i	$0.748489\pi$
$510$	0	0
$511$	4240.85	0.367132
$512$	9737.04	0.840470
$513$	0	0
$514$	−997.849	−0.0856289
$515$	14432.2	1.23487
$516$	0	0
$517$	−3409.53	−0.290040
$518$	3212.24	0.272467
$519$	0	0
$520$	−1499.37	−0.126445
$521$	11332.7	0.952964	0.476482	−	0.879184i	$-0.341912\pi$
$521$	11332.7	0.952964	0.476482	+	0.879184i	$0.341912\pi$
$522$	0	0
$523$	14087.3	1.17781	0.588904	−	0.808203i	$-0.299560\pi$
$523$	14087.3	1.17781	0.588904	+	0.808203i	$0.299560\pi$
$524$	6378.12	0.531736
$525$	0	0
$526$	−4082.61	−0.338423
$527$	−12030.3	−0.994399
$528$	0	0
$529$	17533.6	1.44108
$530$	−23462.9	−1.92295
$531$	0	0
$532$	1221.51	0.0995474
$533$	1070.02	0.0869566
$534$	0	0
$535$	14421.0	1.16537
$536$	−13946.7	−1.12389
$537$	0	0
$538$	−4520.25	−0.362234
$539$	539.000	0.0430730
$540$	0	0
$541$	−6339.91	−0.503833	−0.251917	−	0.967749i	$-0.581061\pi$
$541$	−6339.91	−0.503833	−0.251917	+	0.967749i	$0.581061\pi$
$542$	−17309.4	−1.37177
$543$	0	0
$544$	6615.60	0.521400
$545$	−39401.6	−3.09684
$546$	0	0
$547$	959.649	0.0750121	0.0375061	−	0.999296i	$-0.488059\pi$
$547$	959.649	0.0750121	0.0375061	+	0.999296i	$0.488059\pi$
$548$	−2033.00	−0.158477
$549$	0	0
$550$	−8794.12	−0.681786
$551$	15934.2	1.23198
$552$	0	0
$553$	−4806.30	−0.369592
$554$	−13443.9	−1.03100
$555$	0	0
$556$	2175.23	0.165918
$557$	−17184.0	−1.30720	−0.653598	−	0.756842i	$-0.726741\pi$
$557$	−17184.0	−1.30720	−0.653598	+	0.756842i	$0.726741\pi$
$558$	0	0
$559$	−50.9133	−0.00385224
$560$	−4590.40	−0.346393
$561$	0	0
$562$	10033.4	0.753086
$563$	6598.36	0.493939	0.246969	−	0.969023i	$-0.420565\pi$
$563$	6598.36	0.493939	0.246969	+	0.969023i	$0.420565\pi$
$564$	0	0
$565$	−18181.8	−1.35383
$566$	8186.84	0.607984
$567$	0	0
$568$	−19185.4	−1.41726
$569$	17565.4	1.29416	0.647081	−	0.762422i	$-0.275990\pi$
$569$	17565.4	1.29416	0.647081	+	0.762422i	$0.275990\pi$
$570$	0	0
$571$	−10179.7	−0.746074	−0.373037	−	0.927817i	$-0.621684\pi$
$571$	−10179.7	−0.746074	−0.373037	+	0.927817i	$0.621684\pi$
$572$	−94.0492	−0.00687482
$573$	0	0
$574$	5996.18	0.436020
$575$	−62168.6	−4.50889
$576$	0	0
$577$	6287.76	0.453662	0.226831	−	0.973934i	$-0.427164\pi$
$577$	6287.76	0.453662	0.226831	+	0.973934i	$0.427164\pi$
$578$	5206.01	0.374639
$579$	0	0
$580$	19195.5	1.37423
$581$	−2218.74	−0.158432
$582$	0	0
$583$	−5284.01	−0.375371
$584$	−14888.0	−1.05491
$585$	0	0
$586$	2001.08	0.141064
$587$	18601.0	1.30791	0.653957	−	0.756532i	$-0.273108\pi$
$587$	18601.0	1.30791	0.653957	+	0.756532i	$0.273108\pi$
$588$	0	0
$589$	13424.2	0.939106
$590$	5579.01	0.389295
$591$	0	0
$592$	−6161.00	−0.427729
$593$	−6362.24	−0.440583	−0.220292	−	0.975434i	$-0.570701\pi$
$593$	−6362.24	−0.440583	−0.220292	+	0.975434i	$0.570701\pi$
$594$	0	0
$595$	−7811.82	−0.538241
$596$	−3849.83	−0.264589
$597$	0	0
$598$	1057.36	0.0723054
$599$	12238.1	0.834787	0.417393	−	0.908726i	$-0.362944\pi$
$599$	12238.1	0.834787	0.417393	+	0.908726i	$0.362944\pi$
$600$	0	0
$601$	−25394.1	−1.72354	−0.861770	−	0.507300i	$-0.830644\pi$
$601$	−25394.1	−1.72354	−0.861770	+	0.507300i	$0.830644\pi$
$602$	−285.307	−0.0193160
$603$	0	0
$604$	−3511.81	−0.236579
$605$	−2666.77	−0.179206
$606$	0	0
$607$	5680.37	0.379834	0.189917	−	0.981800i	$-0.439178\pi$
$607$	5680.37	0.379834	0.189917	+	0.981800i	$0.439178\pi$
$608$	−7382.11	−0.492408
$609$	0	0
$610$	5369.30	0.356388
$611$	−858.084	−0.0568157
$612$	0	0
$613$	682.502	0.0449690	0.0224845	−	0.999747i	$-0.492842\pi$
$613$	682.502	0.0449690	0.0224845	+	0.999747i	$0.492842\pi$
$614$	6913.06	0.454378
$615$	0	0
$616$	−1892.22	−0.123766
$617$	−4304.44	−0.280859	−0.140430	−	0.990091i	$-0.544848\pi$
$617$	−4304.44	−0.280859	−0.140430	+	0.990091i	$0.544848\pi$
$618$	0	0
$619$	−13022.7	−0.845602	−0.422801	−	0.906223i	$-0.638953\pi$
$619$	−13022.7	−0.845602	−0.422801	+	0.906223i	$0.638953\pi$
$620$	16171.7	1.04754
$621$	0	0
$622$	−7802.21	−0.502958
$623$	11279.9	0.725390
$624$	0	0
$625$	69412.9	4.44242
$626$	22549.4	1.43971
$627$	0	0
$628$	402.443	0.0255720
$629$	−10484.6	−0.664625
$630$	0	0
$631$	18156.4	1.14548	0.572739	−	0.819738i	$-0.305881\pi$
$631$	18156.4	1.14548	0.572739	+	0.819738i	$0.305881\pi$
$632$	16873.0	1.06198
$633$	0	0
$634$	−8659.62	−0.542456
$635$	−38664.7	−2.41632
$636$	0	0
$637$	135.651	0.00843753
$638$	−6874.96	−0.426618
$639$	0	0
$640$	2733.62	0.168837
$641$	10368.7	0.638905	0.319452	−	0.947602i	$-0.396501\pi$
$641$	10368.7	0.638905	0.319452	+	0.947602i	$0.396501\pi$
$642$	0	0
$643$	−17679.6	−1.08431	−0.542157	−	0.840277i	$-0.682392\pi$
$643$	−17679.6	−1.08431	−0.542157	+	0.840277i	$0.682392\pi$
$644$	−3725.76	−0.227975
$645$	0	0
$646$	6340.61	0.386173
$647$	−1774.38	−0.107817	−0.0539087	−	0.998546i	$-0.517168\pi$
$647$	−1774.38	−0.107817	−0.0539087	+	0.998546i	$0.517168\pi$
$648$	0	0
$649$	1256.43	0.0759927
$650$	−2213.24	−0.133554
$651$	0	0
$652$	2803.76	0.168410
$653$	−25514.1	−1.52901	−0.764506	−	0.644616i	$-0.777017\pi$
$653$	−25514.1	−1.52901	−0.764506	+	0.644616i	$0.777017\pi$
$654$	0	0
$655$	45515.4	2.71517
$656$	−11500.5	−0.684483
$657$	0	0
$658$	−4808.52	−0.284887
$659$	−557.327	−0.0329444	−0.0164722	−	0.999864i	$-0.505244\pi$
$659$	−557.327	−0.0329444	−0.0164722	+	0.999864i	$0.505244\pi$
$660$	0	0
$661$	11263.2	0.662768	0.331384	−	0.943496i	$-0.392484\pi$
$661$	11263.2	0.662768	0.331384	+	0.943496i	$0.392484\pi$
$662$	−19979.4	−1.17299
$663$	0	0
$664$	7789.11	0.455235
$665$	8716.92	0.508312
$666$	0	0
$667$	−48601.4	−2.82137
$668$	−11457.2	−0.663610
$669$	0	0
$670$	−27720.6	−1.59842
$671$	1209.20	0.0695690
$672$	0	0
$673$	−3615.18	−0.207065	−0.103533	−	0.994626i	$-0.533015\pi$
$673$	−3615.18	−0.207065	−0.103533	+	0.994626i	$0.533015\pi$
$674$	−6044.97	−0.345465
$675$	0	0
$676$	6761.56	0.384704
$677$	−23421.6	−1.32964	−0.664820	−	0.747004i	$-0.731492\pi$
$677$	−23421.6	−1.32964	−0.664820	+	0.747004i	$0.731492\pi$
$678$	0	0
$679$	8620.30	0.487212
$680$	27424.2	1.54657
$681$	0	0
$682$	−5791.97	−0.325199
$683$	29364.9	1.64512	0.822559	−	0.568679i	$-0.192545\pi$
$683$	29364.9	1.64512	0.822559	+	0.568679i	$0.192545\pi$
$684$	0	0
$685$	−14507.9	−0.809222
$686$	760.161	0.0423077
$687$	0	0
$688$	547.213	0.0303231
$689$	−1329.84	−0.0735309
$690$	0	0
$691$	−20430.9	−1.12479	−0.562394	−	0.826869i	$-0.690120\pi$
$691$	−20430.9	−1.12479	−0.562394	+	0.826869i	$0.690120\pi$
$692$	6709.00	0.368552
$693$	0	0
$694$	2036.79	0.111406
$695$	15522.8	0.847215
$696$	0	0
$697$	−19571.3	−1.06358
$698$	−3623.26	−0.196479
$699$	0	0
$700$	7798.67	0.421089
$701$	−6673.16	−0.359546	−0.179773	−	0.983708i	$-0.557536\pi$
$701$	−6673.16	−0.359546	−0.179773	+	0.983708i	$0.557536\pi$
$702$	0	0
$703$	11699.4	0.627669
$704$	5803.47	0.310691
$705$	0	0
$706$	−11069.5	−0.590092
$707$	−8291.75	−0.441080
$708$	0	0
$709$	−14706.4	−0.778999	−0.389500	−	0.921027i	$-0.627352\pi$
$709$	−14706.4	−0.778999	−0.389500	+	0.921027i	$0.627352\pi$
$710$	−38133.1	−2.01565
$711$	0	0
$712$	−39599.2	−2.08433
$713$	−40945.4	−2.15066
$714$	0	0
$715$	−671.152	−0.0351044
$716$	−6637.41	−0.346441
$717$	0	0
$718$	21641.4	1.12486
$719$	−35481.4	−1.84038	−0.920189	−	0.391473i	$-0.871965\pi$
$719$	−35481.4	−1.84038	−0.920189	+	0.391473i	$0.871965\pi$
$720$	0	0
$721$	4583.85	0.236771
$722$	8125.75	0.418849
$723$	0	0
$724$	3345.24	0.171719
$725$	101731.	5.21132
$726$	0	0
$727$	8818.93	0.449898	0.224949	−	0.974371i	$-0.427778\pi$
$727$	8818.93	0.449898	0.224949	+	0.974371i	$0.427778\pi$
$728$	−476.219	−0.0242443
$729$	0	0
$730$	−29591.5	−1.50031
$731$	931.231	0.0471174
$732$	0	0
$733$	−645.865	−0.0325451	−0.0162726	−	0.999868i	$-0.505180\pi$
$733$	−645.865	−0.0325451	−0.0162726	+	0.999868i	$0.505180\pi$
$734$	21136.7	1.06290
$735$	0	0
$736$	22516.4	1.12767
$737$	−6242.87	−0.312020
$738$	0	0
$739$	6264.49	0.311831	0.155915	−	0.987770i	$-0.450167\pi$
$739$	6264.49	0.311831	0.155915	+	0.987770i	$0.450167\pi$
$740$	14093.9	0.700140
$741$	0	0
$742$	−7452.12	−0.368701
$743$	4320.14	0.213312	0.106656	−	0.994296i	$-0.465986\pi$
$743$	4320.14	0.213312	0.106656	+	0.994296i	$0.465986\pi$
$744$	0	0
$745$	−27473.1	−1.35105
$746$	19006.9	0.932830
$747$	0	0
$748$	1720.21	0.0840870
$749$	4580.29	0.223445
$750$	0	0
$751$	32107.5	1.56008	0.780041	−	0.625729i	$-0.215198\pi$
$751$	32107.5	1.56008	0.780041	+	0.625729i	$0.215198\pi$
$752$	9222.63	0.447227
$753$	0	0
$754$	−1730.24	−0.0835697
$755$	−25060.9	−1.20803
$756$	0	0
$757$	25385.0	1.21880	0.609401	−	0.792862i	$-0.291410\pi$
$757$	25385.0	1.21880	0.609401	+	0.792862i	$0.291410\pi$
$758$	30386.2	1.45604
$759$	0	0
$760$	−30601.7	−1.46058
$761$	5092.57	0.242583	0.121291	−	0.992617i	$-0.461296\pi$
$761$	5092.57	0.242583	0.121291	+	0.992617i	$0.461296\pi$
$762$	0	0
$763$	−12514.5	−0.593780
$764$	−8794.98	−0.416480
$765$	0	0
$766$	18680.0	0.881119
$767$	316.209	0.0148861
$768$	0	0
$769$	10162.4	0.476547	0.238273	−	0.971198i	$-0.423419\pi$
$769$	10162.4	0.476547	0.238273	+	0.971198i	$0.423419\pi$
$770$	−3760.99	−0.176021
$771$	0	0
$772$	13175.0	0.614221
$773$	−14229.8	−0.662108	−0.331054	−	0.943612i	$-0.607404\pi$
$773$	−14229.8	−0.662108	−0.331054	+	0.943612i	$0.607404\pi$
$774$	0	0
$775$	85705.9	3.97245
$776$	−30262.5	−1.39995
$777$	0	0
$778$	3894.40	0.179461
$779$	21838.9	1.00444
$780$	0	0
$781$	−8587.84	−0.393466
$782$	−19339.6	−0.884379
$783$	0	0
$784$	−1457.97	−0.0664163
$785$	2871.90	0.130576
$786$	0	0
$787$	13391.2	0.606537	0.303269	−	0.952905i	$-0.401922\pi$
$787$	13391.2	0.606537	0.303269	+	0.952905i	$0.401922\pi$
$788$	−9516.34	−0.430210
$789$	0	0
$790$	33537.0	1.51037
$791$	−5774.79	−0.259580
$792$	0	0
$793$	304.323	0.0136278
$794$	−8905.91	−0.398059
$795$	0	0
$796$	5807.46	0.258593
$797$	8830.58	0.392466	0.196233	−	0.980557i	$-0.437129\pi$
$797$	8830.58	0.392466	0.196233	+	0.980557i	$0.437129\pi$
$798$	0	0
$799$	15694.8	0.694922
$800$	−47130.7	−2.08290
$801$	0	0
$802$	−7411.38	−0.326315
$803$	−6664.20	−0.292870
$804$	0	0
$805$	−26587.7	−1.16409
$806$	−1457.68	−0.0637029
$807$	0	0
$808$	29109.1	1.26739
$809$	−44283.6	−1.92451	−0.962255	−	0.272149i	$-0.912266\pi$
$809$	−44283.6	−1.92451	−0.962255	+	0.272149i	$0.912266\pi$
$810$	0	0
$811$	−42502.7	−1.84029	−0.920143	−	0.391582i	$-0.871928\pi$
$811$	−42502.7	−1.84029	−0.920143	+	0.391582i	$0.871928\pi$
$812$	6096.75	0.263490
$813$	0	0
$814$	−5047.81	−0.217353
$815$	20008.1	0.859942
$816$	0	0
$817$	−1039.13	−0.0444975
$818$	30009.3	1.28270
$819$	0	0
$820$	26308.7	1.12041
$821$	20018.3	0.850968	0.425484	−	0.904966i	$-0.360104\pi$
$821$	20018.3	0.850968	0.425484	+	0.904966i	$0.360104\pi$
$822$	0	0
$823$	−31778.1	−1.34595	−0.672973	−	0.739667i	$-0.734983\pi$
$823$	−31778.1	−1.34595	−0.672973	+	0.739667i	$0.734983\pi$
$824$	−16092.1	−0.680334
$825$	0	0
$826$	1771.97	0.0746424
$827$	−9416.88	−0.395958	−0.197979	−	0.980206i	$-0.563438\pi$
$827$	−9416.88	−0.395958	−0.197979	+	0.980206i	$0.563438\pi$
$828$	0	0
$829$	−30106.3	−1.26132	−0.630661	−	0.776058i	$-0.717216\pi$
$829$	−30106.3	−1.26132	−0.630661	+	0.776058i	$0.717216\pi$
$830$	15481.7	0.647443
$831$	0	0
$832$	1460.57	0.0608609
$833$	−2481.13	−0.103201
$834$	0	0
$835$	−81760.4	−3.38855
$836$	−1919.52	−0.0794114
$837$	0	0
$838$	16148.5	0.665682
$839$	−45186.6	−1.85937	−0.929687	−	0.368350i	$-0.879923\pi$
$839$	−45186.6	−1.85937	−0.929687	+	0.368350i	$0.879923\pi$
$840$	0	0
$841$	55141.3	2.26091
$842$	23887.4	0.977691
$843$	0	0
$844$	4273.58	0.174292
$845$	48251.6	1.96439
$846$	0	0
$847$	−847.000	−0.0343604
$848$	14293.0	0.578802
$849$	0	0
$850$	40481.2	1.63352
$851$	−35684.7	−1.43743
$852$	0	0
$853$	−2401.55	−0.0963980	−0.0481990	−	0.998838i	$-0.515348\pi$
$853$	−2401.55	−0.0963980	−0.0481990	+	0.998838i	$0.515348\pi$
$854$	1705.36	0.0683328
$855$	0	0
$856$	−16079.6	−0.642043
$857$	−4218.83	−0.168159	−0.0840797	−	0.996459i	$-0.526795\pi$
$857$	−4218.83	−0.168159	−0.0840797	+	0.996459i	$0.526795\pi$
$858$	0	0
$859$	−15508.3	−0.615992	−0.307996	−	0.951388i	$-0.599658\pi$
$859$	−15508.3	−0.615992	−0.307996	+	0.951388i	$0.599658\pi$
$860$	−1251.81	−0.0496352
$861$	0	0
$862$	18943.2	0.748503
$863$	18753.2	0.739706	0.369853	−	0.929090i	$-0.379408\pi$
$863$	18753.2	0.739706	0.369853	+	0.929090i	$0.379408\pi$
$864$	0	0
$865$	47876.6	1.88191
$866$	−15825.6	−0.620990
$867$	0	0
$868$	5136.35	0.200851
$869$	7552.75	0.294833
$870$	0	0
$871$	−1571.16	−0.0611213
$872$	43933.4	1.70616
$873$	0	0
$874$	21580.4	0.835203
$875$	36368.2	1.40511
$876$	0	0
$877$	45435.5	1.74943	0.874713	−	0.484641i	$-0.161050\pi$
$877$	45435.5	1.74943	0.874713	+	0.484641i	$0.161050\pi$
$878$	5344.73	0.205439
$879$	0	0
$880$	7213.49	0.276326
$881$	20328.9	0.777410	0.388705	−	0.921362i	$-0.372922\pi$
$881$	20328.9	0.777410	0.388705	+	0.921362i	$0.372922\pi$
$882$	0	0
$883$	18438.6	0.702728	0.351364	−	0.936239i	$-0.385718\pi$
$883$	18438.6	0.702728	0.351364	+	0.936239i	$0.385718\pi$
$884$	432.929	0.0164717
$885$	0	0
$886$	27060.2	1.02608
$887$	−35240.4	−1.33400	−0.667000	−	0.745058i	$-0.732422\pi$
$887$	−35240.4	−1.33400	−0.667000	+	0.745058i	$0.732422\pi$
$888$	0	0
$889$	−12280.4	−0.463298
$890$	−78707.6	−2.96436
$891$	0	0
$892$	−4835.08	−0.181491
$893$	−17513.3	−0.656281
$894$	0	0
$895$	−47365.7	−1.76901
$896$	868.232	0.0323723
$897$	0	0
$898$	9339.89	0.347078
$899$	67002.1	2.48570
$900$	0	0
$901$	24323.4	0.899368
$902$	−9422.57	−0.347824
$903$	0	0
$904$	20273.0	0.745875
$905$	23872.2	0.876839
$906$	0	0
$907$	−27735.7	−1.01538	−0.507690	−	0.861540i	$-0.669500\pi$
$907$	−27735.7	−1.01538	−0.507690	+	0.861540i	$0.669500\pi$
$908$	7620.40	0.278515
$909$	0	0
$910$	−946.536	−0.0344806
$911$	−12663.6	−0.460554	−0.230277	−	0.973125i	$-0.573963\pi$
$911$	−12663.6	−0.460554	−0.230277	+	0.973125i	$0.573963\pi$
$912$	0	0
$913$	3486.59	0.126385
$914$	−375.901	−0.0136036
$915$	0	0
$916$	3998.28	0.144221
$917$	14456.3	0.520598
$918$	0	0
$919$	−16398.4	−0.588612	−0.294306	−	0.955711i	$-0.595088\pi$
$919$	−16398.4	−0.588612	−0.294306	+	0.955711i	$0.595088\pi$
$920$	93339.0	3.34489
$921$	0	0
$922$	1802.31	0.0643773
$923$	−2161.32	−0.0770756
$924$	0	0
$925$	74694.2	2.65506
$926$	28104.3	0.997369
$927$	0	0
$928$	−36845.2	−1.30335
$929$	42073.8	1.48589	0.742947	−	0.669350i	$-0.233427\pi$
$929$	42073.8	1.48589	0.742947	+	0.669350i	$0.233427\pi$
$930$	0	0
$931$	2768.61	0.0974623
$932$	−16893.7	−0.593745
$933$	0	0
$934$	16172.1	0.566559
$935$	12275.7	0.429368
$936$	0	0
$937$	11771.2	0.410402	0.205201	−	0.978720i	$-0.434215\pi$
$937$	11771.2	0.410402	0.205201	+	0.978720i	$0.434215\pi$
$938$	−8804.42	−0.306476
$939$	0	0
$940$	−21097.7	−0.732055
$941$	−6056.53	−0.209817	−0.104908	−	0.994482i	$-0.533455\pi$
$941$	−6056.53	−0.209817	−0.104908	+	0.994482i	$0.533455\pi$
$942$	0	0
$943$	−66611.3	−2.30028
$944$	−3398.59	−0.117177
$945$	0	0
$946$	448.340	0.0154089
$947$	2545.73	0.0873551	0.0436775	−	0.999046i	$-0.486093\pi$
$947$	2545.73	0.0873551	0.0436775	+	0.999046i	$0.486093\pi$
$948$	0	0
$949$	−1677.19	−0.0573699
$950$	−45171.5	−1.54269
$951$	0	0
$952$	8710.29	0.296536
$953$	17699.0	0.601602	0.300801	−	0.953687i	$-0.402746\pi$
$953$	17699.0	0.601602	0.300801	+	0.953687i	$0.402746\pi$
$954$	0	0
$955$	−62762.5	−2.12665
$956$	1142.99	0.0386683
$957$	0	0
$958$	−21884.0	−0.738037
$959$	−4607.89	−0.155158
$960$	0	0
$961$	26656.5	0.894784
$962$	−1270.39	−0.0425770
$963$	0	0
$964$	4213.52	0.140776
$965$	94019.1	3.13635
$966$	0	0
$967$	−48183.5	−1.60235	−0.801177	−	0.598427i	$-0.795793\pi$
$967$	−48183.5	−1.60235	−0.801177	+	0.598427i	$0.795793\pi$
$968$	2973.48	0.0987308
$969$	0	0
$970$	−60149.9	−1.99103
$971$	−9670.55	−0.319612	−0.159806	−	0.987148i	$-0.551087\pi$
$971$	−9670.55	−0.319612	−0.159806	+	0.987148i	$0.551087\pi$
$972$	0	0
$973$	4930.25	0.162443
$974$	−11071.8	−0.364234
$975$	0	0
$976$	−3270.84	−0.107272
$977$	−14624.6	−0.478897	−0.239449	−	0.970909i	$-0.576967\pi$
$977$	−14624.6	−0.478897	−0.239449	+	0.970909i	$0.576967\pi$
$978$	0	0
$979$	−17725.5	−0.578661
$980$	3335.26	0.108715
$981$	0	0
$982$	−1995.60	−0.0648495
$983$	−11183.6	−0.362871	−0.181436	−	0.983403i	$-0.558074\pi$
$983$	−11183.6	−0.362871	−0.181436	+	0.983403i	$0.558074\pi$
$984$	0	0
$985$	−67910.3	−2.19675
$986$	31646.9	1.02215
$987$	0	0
$988$	−483.090	−0.0155558
$989$	3169.47	0.101904
$990$	0	0
$991$	18858.3	0.604493	0.302247	−	0.953230i	$-0.402263\pi$
$991$	18858.3	0.604493	0.302247	+	0.953230i	$0.402263\pi$
$992$	−31041.1	−0.993505
$993$	0	0
$994$	−12111.6	−0.386475
$995$	41443.1	1.32044
$996$	0	0
$997$	156.252	0.00496345	0.00248172	−	0.999997i	$-0.499210\pi$
$997$	156.252	0.00496345	0.00248172	+	0.999997i	$0.499210\pi$
$998$	−25423.5	−0.806379
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.s.1.3	✓	8
3.2	odd	2		693.4.a.t.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.3	✓	8	1.1	even	1	trivial
693.4.a.t.1.6	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-2.21621 q^{2} -3.08840 q^{4} -22.0394 q^{5} -7.00000 q^{7} +24.5743 q^{8} +O(q^{10})\)	\(q-2.21621 q^{2} -3.08840 q^{4} -22.0394 q^{5} -7.00000 q^{7} +24.5743 q^{8} +48.8440 q^{10} +11.0000 q^{11} +2.76840 q^{13} +15.5135 q^{14} -29.7545 q^{16} -50.6354 q^{17} +56.5022 q^{19} +68.0666 q^{20} -24.3783 q^{22} -172.339 q^{23} +360.735 q^{25} -6.13535 q^{26} +21.6188 q^{28} +282.011 q^{29} +237.587 q^{31} -130.652 q^{32} +112.219 q^{34} +154.276 q^{35} +207.061 q^{37} -125.221 q^{38} -541.602 q^{40} +386.514 q^{41} -18.3909 q^{43} -33.9725 q^{44} +381.939 q^{46} -309.957 q^{47} +49.0000 q^{49} -799.465 q^{50} -8.54993 q^{52} -480.364 q^{53} -242.433 q^{55} -172.020 q^{56} -624.996 q^{58} +114.221 q^{59} +109.928 q^{61} -526.543 q^{62} +527.588 q^{64} -61.0138 q^{65} -567.534 q^{67} +156.383 q^{68} -341.908 q^{70} -780.713 q^{71} -605.836 q^{73} -458.891 q^{74} -174.502 q^{76} -77.0000 q^{77} +686.614 q^{79} +655.772 q^{80} -856.597 q^{82} +316.962 q^{83} +1115.97 q^{85} +40.7582 q^{86} +270.317 q^{88} -1611.41 q^{89} -19.3788 q^{91} +532.252 q^{92} +686.931 q^{94} -1245.27 q^{95} -1231.47 q^{97} -108.594 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8	\( 8 q - 2 q^{2} + 30 q^{4} - 10 q^{5} - 56 q^{7} - 15 q^{8} - 13 q^{10} + 88 q^{11} - 148 q^{13} + 14 q^{14} + 266 q^{16} - 114 q^{17} + 58 q^{19} - 291 q^{20} - 22 q^{22} - 246 q^{23} + 244 q^{25} - 305 q^{26} - 210 q^{28} + 72 q^{29} + 252 q^{31} - 1272 q^{32} + 630 q^{34} + 70 q^{35} - 80 q^{37} - 1885 q^{38} - 342 q^{40} - 682 q^{41} - 106 q^{43} + 330 q^{44} + 120 q^{46} - 828 q^{47} + 392 q^{49} - 801 q^{50} - 1681 q^{52} - 462 q^{53} - 110 q^{55} + 105 q^{56} - 1087 q^{58} - 626 q^{59} - 854 q^{61} - 1350 q^{62} + 2997 q^{64} - 22 q^{65} + 130 q^{67} - 2202 q^{68} + 91 q^{70} - 326 q^{71} - 390 q^{73} + 359 q^{74} + 2041 q^{76} - 616 q^{77} - 508 q^{79} - 4391 q^{80} + 1528 q^{82} - 1596 q^{83} - 880 q^{85} + 414 q^{86} - 165 q^{88} - 4324 q^{89} + 1036 q^{91} - 2092 q^{92} - 1685 q^{94} - 1076 q^{95} - 964 q^{97} - 98 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 30 * q^4 - 10 * q^5 - 56 * q^7 - 15 * q^8 - 13 * q^10 + 88 * q^11 - 148 * q^13 + 14 * q^14 + 266 * q^16 - 114 * q^17 + 58 * q^19 - 291 * q^20 - 22 * q^22 - 246 * q^23 + 244 * q^25 - 305 * q^26 - 210 * q^28 + 72 * q^29 + 252 * q^31 - 1272 * q^32 + 630 * q^34 + 70 * q^35 - 80 * q^37 - 1885 * q^38 - 342 * q^40 - 682 * q^41 - 106 * q^43 + 330 * q^44 + 120 * q^46 - 828 * q^47 + 392 * q^49 - 801 * q^50 - 1681 * q^52 - 462 * q^53 - 110 * q^55 + 105 * q^56 - 1087 * q^58 - 626 * q^59 - 854 * q^61 - 1350 * q^62 + 2997 * q^64 - 22 * q^65 + 130 * q^67 - 2202 * q^68 + 91 * q^70 - 326 * q^71 - 390 * q^73 + 359 * q^74 + 2041 * q^76 - 616 * q^77 - 508 * q^79 - 4391 * q^80 + 1528 * q^82 - 1596 * q^83 - 880 * q^85 + 414 * q^86 - 165 * q^88 - 4324 * q^89 + 1036 * q^91 - 2092 * q^92 - 1685 * q^94 - 1076 * q^95 - 964 * q^97 - 98 * q^98

Introduction

L-functions

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