LMFDB - Embedded newform 693.4.a.u.1.4

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:	$8$
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} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 $ x^8 - 2x^7 - 43x^6 + 57x^5 + 560x^4 - 439x^3 - 2246x^2 + 384*x + 1056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.778078$ 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)

$f(q)$	$=$	\(q+0.221922 q^{2} -7.95075 q^{4} +15.1615 q^{5} +7.00000 q^{7} -3.53982 q^{8} +3.36467 q^{10} +11.0000 q^{11} +37.8013 q^{13} +1.55345 q^{14} +62.8204 q^{16} -61.7617 q^{17} +54.5657 q^{19} -120.545 q^{20} +2.44114 q^{22} +24.4988 q^{23} +104.871 q^{25} +8.38893 q^{26} -55.6553 q^{28} -16.2692 q^{29} -190.027 q^{31} +42.2597 q^{32} -13.7063 q^{34} +106.131 q^{35} +170.813 q^{37} +12.1093 q^{38} -53.6690 q^{40} +78.0123 q^{41} +45.5205 q^{43} -87.4583 q^{44} +5.43681 q^{46} +273.595 q^{47} +49.0000 q^{49} +23.2732 q^{50} -300.549 q^{52} -163.247 q^{53} +166.777 q^{55} -24.7787 q^{56} -3.61049 q^{58} +650.249 q^{59} +257.506 q^{61} -42.1710 q^{62} -493.185 q^{64} +573.125 q^{65} -399.767 q^{67} +491.052 q^{68} +23.5527 q^{70} +198.479 q^{71} -226.479 q^{73} +37.9071 q^{74} -433.839 q^{76} +77.0000 q^{77} +138.641 q^{79} +952.453 q^{80} +17.3126 q^{82} +490.784 q^{83} -936.401 q^{85} +10.1020 q^{86} -38.9380 q^{88} +221.276 q^{89} +264.609 q^{91} -194.784 q^{92} +60.7167 q^{94} +827.299 q^{95} +1598.94 q^{97} +10.8742 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8} + 9 q^{10} + 88 q^{11} + 16 q^{13} + 42 q^{14} + 122 q^{16} + 90 q^{17} - 42 q^{19} + 291 q^{20} + 66 q^{22} + 338 q^{23} + 244 q^{25} + 209 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) $ 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8 + 9 * q^10 + 88 * q^11 + 16 * q^13 + 42 * q^14 + 122 * q^16 + 90 * q^17 - 42 * q^19 + 291 * q^20 + 66 * q^22 + 338 * q^23 + 244 * q^25 + 209 * q^26 + 210 * q^28 + 496 * q^29 - 8 * q^31 + 524 * q^32 - 302 * q^34 + 70 * q^35 - 360 * q^37 + 45 * q^38 - 6 * q^40 + 242 * q^41 - 66 * q^43 + 330 * q^44 + 344 * q^46 + 540 * q^47 + 392 * q^49 + 1171 * q^50 + 465 * q^52 + 906 * q^53 + 110 * q^55 + 567 * q^56 + 977 * q^58 + 1242 * q^59 - 318 * q^61 - 110 * q^62 + 525 * q^64 + 1258 * q^65 + 522 * q^67 + 678 * q^68 + 63 * q^70 + 858 * q^71 - 78 * q^73 + 1651 * q^74 + 1775 * q^76 + 616 * q^77 + 516 * q^79 + 567 * q^80 - 1212 * q^82 + 3192 * q^83 + 720 * q^85 + 1322 * q^86 + 891 * q^88 + 2356 * q^89 + 112 * q^91 + 4504 * q^92 - 423 * q^94 + 3308 * q^95 - 1556 * q^97 + 294 * 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 $

$2$	0.221922	0.0784611	0.0392306	−	0.999230i	$-0.487509\pi$
$2$	0.221922	0.0784611	0.0392306	+	0.999230i	$0.487509\pi$
$3$	0	0
$3$	0	0
$4$	−7.95075	−0.993844
$5$	15.1615	1.35609	0.678043	−	0.735022i	$-0.262828\pi$
$5$	15.1615	1.35609	0.678043	+	0.735022i	$0.262828\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−3.53982	−0.156439
$9$	0	0
$10$	3.36467	0.106400
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	37.8013	0.806477	0.403239	−	0.915095i	$-0.367884\pi$
$13$	37.8013	0.806477	0.403239	+	0.915095i	$0.367884\pi$
$14$	1.55345	0.0296555
$15$	0	0
$16$	62.8204	0.981569
$17$	−61.7617	−0.881142	−0.440571	−	0.897718i	$-0.645224\pi$
$17$	−61.7617	−0.881142	−0.440571	+	0.897718i	$0.645224\pi$
$18$	0	0
$19$	54.5657	0.658855	0.329427	−	0.944181i	$-0.393144\pi$
$19$	54.5657	0.658855	0.329427	+	0.944181i	$0.393144\pi$
$20$	−120.545	−1.34774
$21$	0	0
$22$	2.44114	0.0236569
$23$	24.4988	0.222102	0.111051	−	0.993815i	$-0.464578\pi$
$23$	24.4988	0.222102	0.111051	+	0.993815i	$0.464578\pi$
$24$	0	0
$25$	104.871	0.838972
$26$	8.38893	0.0632771
$27$	0	0
$28$	−55.6553	−0.375638
$29$	−16.2692	−0.104176	−0.0520882	−	0.998642i	$-0.516588\pi$
$29$	−16.2692	−0.104176	−0.0520882	+	0.998642i	$0.516588\pi$
$30$	0	0
$31$	−190.027	−1.10096	−0.550481	−	0.834848i	$-0.685556\pi$
$31$	−190.027	−1.10096	−0.550481	+	0.834848i	$0.685556\pi$
$32$	42.2597	0.233454
$33$	0	0
$34$	−13.7063	−0.0691354
$35$	106.131	0.512553
$36$	0	0
$37$	170.813	0.758958	0.379479	−	0.925200i	$-0.376103\pi$
$37$	170.813	0.758958	0.379479	+	0.925200i	$0.376103\pi$
$38$	12.1093	0.0516945
$39$	0	0
$40$	−53.6690	−0.212145
$41$	78.0123	0.297158	0.148579	−	0.988901i	$-0.452530\pi$
$41$	78.0123	0.297158	0.148579	+	0.988901i	$0.452530\pi$
$42$	0	0
$43$	45.5205	0.161437	0.0807186	−	0.996737i	$-0.474278\pi$
$43$	45.5205	0.161437	0.0807186	+	0.996737i	$0.474278\pi$
$44$	−87.4583	−0.299655
$45$	0	0
$46$	5.43681	0.0174264
$47$	273.595	0.849105	0.424553	−	0.905403i	$-0.360431\pi$
$47$	273.595	0.849105	0.424553	+	0.905403i	$0.360431\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	23.2732	0.0658267
$51$	0	0
$52$	−300.549	−0.801512
$53$	−163.247	−0.423088	−0.211544	−	0.977368i	$-0.567849\pi$
$53$	−163.247	−0.423088	−0.211544	+	0.977368i	$0.567849\pi$
$54$	0	0
$55$	166.777	0.408876
$56$	−24.7787	−0.0591285
$57$	0	0
$58$	−3.61049	−0.00817379
$59$	650.249	1.43483	0.717417	−	0.696644i	$-0.245324\pi$
$59$	650.249	1.43483	0.717417	+	0.696644i	$0.245324\pi$
$60$	0	0
$61$	257.506	0.540496	0.270248	−	0.962791i	$-0.412894\pi$
$61$	257.506	0.540496	0.270248	+	0.962791i	$0.412894\pi$
$62$	−42.1710	−0.0863827
$63$	0	0
$64$	−493.185	−0.963252
$65$	573.125	1.09365
$66$	0	0
$67$	−399.767	−0.728945	−0.364473	−	0.931214i	$-0.618751\pi$
$67$	−399.767	−0.728945	−0.364473	+	0.931214i	$0.618751\pi$
$68$	491.052	0.875717
$69$	0	0
$70$	23.5527	0.0402155
$71$	198.479	0.331763	0.165881	−	0.986146i	$-0.446953\pi$
$71$	198.479	0.331763	0.165881	+	0.986146i	$0.446953\pi$
$72$	0	0
$73$	−226.479	−0.363114	−0.181557	−	0.983380i	$-0.558114\pi$
$73$	−226.479	−0.363114	−0.181557	+	0.983380i	$0.558114\pi$
$74$	37.9071	0.0595487
$75$	0	0
$76$	−433.839	−0.654799
$77$	77.0000	0.113961
$78$	0	0
$79$	138.641	0.197447	0.0987234	−	0.995115i	$-0.468524\pi$
$79$	138.641	0.197447	0.0987234	+	0.995115i	$0.468524\pi$
$80$	952.453	1.33109
$81$	0	0
$82$	17.3126	0.0233154
$83$	490.784	0.649043	0.324521	−	0.945878i	$-0.394797\pi$
$83$	490.784	0.649043	0.324521	+	0.945878i	$0.394797\pi$
$84$	0	0
$85$	−936.401	−1.19491
$86$	10.1020	0.0126666
$87$	0	0
$88$	−38.9380	−0.0471682
$89$	221.276	0.263542	0.131771	−	0.991280i	$-0.457934\pi$
$89$	221.276	0.263542	0.131771	+	0.991280i	$0.457934\pi$
$90$	0	0
$91$	264.609	0.304820
$92$	−194.784	−0.220735
$93$	0	0
$94$	60.7167	0.0666218
$95$	827.299	0.893464
$96$	0	0
$97$	1598.94	1.67369	0.836846	−	0.547438i	$-0.184397\pi$
$97$	1598.94	1.67369	0.836846	+	0.547438i	$0.184397\pi$
$98$	10.8742	0.0112087
$99$	0	0
$100$	−833.807	−0.833807
$101$	1770.89	1.74465	0.872326	−	0.488924i	$-0.162611\pi$
$101$	1770.89	1.74465	0.872326	+	0.488924i	$0.162611\pi$
$102$	0	0
$103$	1159.21	1.10893	0.554466	−	0.832206i	$-0.312923\pi$
$103$	1159.21	1.10893	0.554466	+	0.832206i	$0.312923\pi$
$104$	−133.810	−0.126165
$105$	0	0
$106$	−36.2280	−0.0331960
$107$	1334.93	1.20610	0.603050	−	0.797703i	$-0.293952\pi$
$107$	1334.93	1.20610	0.603050	+	0.797703i	$0.293952\pi$
$108$	0	0
$109$	382.778	0.336362	0.168181	−	0.985756i	$-0.446211\pi$
$109$	382.778	0.336362	0.168181	+	0.985756i	$0.446211\pi$
$110$	37.0113	0.0320808
$111$	0	0
$112$	439.743	0.370998
$113$	641.007	0.533636	0.266818	−	0.963747i	$-0.414028\pi$
$113$	641.007	0.533636	0.266818	+	0.963747i	$0.414028\pi$
$114$	0	0
$115$	371.438	0.301190
$116$	129.352	0.103535
$117$	0	0
$118$	144.304	0.112579
$119$	−432.332	−0.333040
$120$	0	0
$121$	121.000	0.0909091
$122$	57.1462	0.0424080
$123$	0	0
$124$	1510.86	1.09418
$125$	−305.179	−0.218368
$126$	0	0
$127$	−574.823	−0.401632	−0.200816	−	0.979629i	$-0.564359\pi$
$127$	−574.823	−0.401632	−0.200816	+	0.979629i	$0.564359\pi$
$128$	−447.526	−0.309032
$129$	0	0
$130$	127.189	0.0858093
$131$	−733.130	−0.488961	−0.244480	−	0.969654i	$-0.578617\pi$
$131$	−733.130	−0.488961	−0.244480	+	0.969654i	$0.578617\pi$
$132$	0	0
$133$	381.960	0.249024
$134$	−88.7170	−0.0571939
$135$	0	0
$136$	218.625	0.137845
$137$	−700.501	−0.436846	−0.218423	−	0.975854i	$-0.570091\pi$
$137$	−700.501	−0.436846	−0.218423	+	0.975854i	$0.570091\pi$
$138$	0	0
$139$	619.481	0.378012	0.189006	−	0.981976i	$-0.439473\pi$
$139$	619.481	0.378012	0.189006	+	0.981976i	$0.439473\pi$
$140$	−843.818	−0.509397
$141$	0	0
$142$	44.0469	0.0260305
$143$	415.815	0.243162
$144$	0	0
$145$	−246.666	−0.141272
$146$	−50.2605	−0.0284904
$147$	0	0
$148$	−1358.09	−0.754286
$149$	2598.45	1.42868	0.714339	−	0.699800i	$-0.246727\pi$
$149$	2598.45	1.42868	0.714339	+	0.699800i	$0.246727\pi$
$150$	0	0
$151$	−2195.71	−1.18334	−0.591669	−	0.806181i	$-0.701531\pi$
$151$	−2195.71	−1.18334	−0.591669	+	0.806181i	$0.701531\pi$
$152$	−193.153	−0.103071
$153$	0	0
$154$	17.0880	0.00894148
$155$	−2881.09	−1.49300
$156$	0	0
$157$	−1585.74	−0.806090	−0.403045	−	0.915180i	$-0.632048\pi$
$157$	−1585.74	−0.806090	−0.403045	+	0.915180i	$0.632048\pi$
$158$	30.7674	0.0154919
$159$	0	0
$160$	640.722	0.316584
$161$	171.491	0.0839467
$162$	0	0
$163$	2688.22	1.29176	0.645882	−	0.763437i	$-0.276490\pi$
$163$	2688.22	1.29176	0.645882	+	0.763437i	$0.276490\pi$
$164$	−620.256	−0.295329
$165$	0	0
$166$	108.916	0.0509246
$167$	2366.77	1.09668	0.548342	−	0.836254i	$-0.315259\pi$
$167$	2366.77	1.09668	0.548342	+	0.836254i	$0.315259\pi$
$168$	0	0
$169$	−768.059	−0.349594
$170$	−207.808	−0.0937536
$171$	0	0
$172$	−361.922	−0.160443
$173$	−960.924	−0.422299	−0.211149	−	0.977454i	$-0.567721\pi$
$173$	−960.924	−0.422299	−0.211149	+	0.977454i	$0.567721\pi$
$174$	0	0
$175$	734.100	0.317102
$176$	691.025	0.295954
$177$	0	0
$178$	49.1060	0.0206778
$179$	1270.38	0.530464	0.265232	−	0.964185i	$-0.414552\pi$
$179$	1270.38	0.530464	0.265232	+	0.964185i	$0.414552\pi$
$180$	0	0
$181$	−1066.57	−0.437999	−0.218999	−	0.975725i	$-0.570279\pi$
$181$	−1066.57	−0.437999	−0.218999	+	0.975725i	$0.570279\pi$
$182$	58.7225	0.0239165
$183$	0	0
$184$	−86.7212	−0.0347455
$185$	2589.78	1.02921
$186$	0	0
$187$	−679.379	−0.265674
$188$	−2175.29	−0.843878
$189$	0	0
$190$	183.596	0.0701022
$191$	−946.790	−0.358677	−0.179339	−	0.983787i	$-0.557396\pi$
$191$	−946.790	−0.358677	−0.179339	+	0.983787i	$0.557396\pi$
$192$	0	0
$193$	−3405.75	−1.27022	−0.635108	−	0.772424i	$-0.719044\pi$
$193$	−3405.75	−1.27022	−0.635108	+	0.772424i	$0.719044\pi$
$194$	354.840	0.131320
$195$	0	0
$196$	−389.587	−0.141978
$197$	−3923.90	−1.41912	−0.709558	−	0.704647i	$-0.751106\pi$
$197$	−3923.90	−1.41912	−0.709558	+	0.704647i	$0.751106\pi$
$198$	0	0
$199$	−2560.58	−0.912136	−0.456068	−	0.889945i	$-0.650743\pi$
$199$	−2560.58	−0.912136	−0.456068	+	0.889945i	$0.650743\pi$
$200$	−371.226	−0.131248
$201$	0	0
$202$	392.998	0.136887
$203$	−113.884	−0.0393749
$204$	0	0
$205$	1182.78	0.402972
$206$	257.253	0.0870081
$207$	0	0
$208$	2374.70	0.791613
$209$	600.223	0.198652
$210$	0	0
$211$	−5739.00	−1.87246	−0.936230	−	0.351387i	$-0.885710\pi$
$211$	−5739.00	−1.87246	−0.936230	+	0.351387i	$0.885710\pi$
$212$	1297.93	0.420483
$213$	0	0
$214$	296.250	0.0946321
$215$	690.159	0.218923
$216$	0	0
$217$	−1330.19	−0.416124
$218$	84.9468	0.0263914
$219$	0	0
$220$	−1326.00	−0.406359
$221$	−2334.67	−0.710621
$222$	0	0
$223$	2730.77	0.820025	0.410012	−	0.912080i	$-0.365524\pi$
$223$	2730.77	0.820025	0.410012	+	0.912080i	$0.365524\pi$
$224$	295.818	0.0882374
$225$	0	0
$226$	142.253	0.0418697
$227$	2118.19	0.619336	0.309668	−	0.950845i	$-0.399782\pi$
$227$	2118.19	0.619336	0.309668	+	0.950845i	$0.399782\pi$
$228$	0	0
$229$	−2821.55	−0.814207	−0.407104	−	0.913382i	$-0.633461\pi$
$229$	−2821.55	−0.814207	−0.407104	+	0.913382i	$0.633461\pi$
$230$	82.4302	0.0236317
$231$	0	0
$232$	57.5900	0.0162973
$233$	6838.94	1.92289	0.961445	−	0.274996i	$-0.0886765\pi$
$233$	6838.94	1.92289	0.961445	+	0.274996i	$0.0886765\pi$
$234$	0	0
$235$	4148.12	1.15146
$236$	−5169.96	−1.42600
$237$	0	0
$238$	−95.9438	−0.0261307
$239$	3473.80	0.940174	0.470087	−	0.882620i	$-0.344223\pi$
$239$	3473.80	0.940174	0.470087	+	0.882620i	$0.344223\pi$
$240$	0	0
$241$	1672.96	0.447156	0.223578	−	0.974686i	$-0.428226\pi$
$241$	1672.96	0.447156	0.223578	+	0.974686i	$0.428226\pi$
$242$	26.8525	0.00713283
$243$	0	0
$244$	−2047.37	−0.537169
$245$	742.914	0.193727
$246$	0	0
$247$	2062.66	0.531351
$248$	672.660	0.172234
$249$	0	0
$250$	−67.7258	−0.0171334
$251$	−4468.56	−1.12372	−0.561859	−	0.827233i	$-0.689914\pi$
$251$	−4468.56	−1.12372	−0.561859	+	0.827233i	$0.689914\pi$
$252$	0	0
$253$	269.487	0.0669663
$254$	−127.566	−0.0315125
$255$	0	0
$256$	3846.17	0.939005
$257$	643.728	0.156244	0.0781219	−	0.996944i	$-0.475108\pi$
$257$	643.728	0.156244	0.0781219	+	0.996944i	$0.475108\pi$
$258$	0	0
$259$	1195.69	0.286859
$260$	−4556.78	−1.08692
$261$	0	0
$262$	−162.697	−0.0383644
$263$	−4100.67	−0.961438	−0.480719	−	0.876875i	$-0.659624\pi$
$263$	−4100.67	−0.961438	−0.480719	+	0.876875i	$0.659624\pi$
$264$	0	0
$265$	−2475.07	−0.573744
$266$	84.7652	0.0195387
$267$	0	0
$268$	3178.45	0.724458
$269$	1869.88	0.423823	0.211912	−	0.977289i	$-0.432031\pi$
$269$	1869.88	0.423823	0.211912	+	0.977289i	$0.432031\pi$
$270$	0	0
$271$	5446.72	1.22090	0.610451	−	0.792054i	$-0.290988\pi$
$271$	5446.72	1.22090	0.610451	+	0.792054i	$0.290988\pi$
$272$	−3879.90	−0.864902
$273$	0	0
$274$	−155.456	−0.0342754
$275$	1153.59	0.252960
$276$	0	0
$277$	−2570.87	−0.557648	−0.278824	−	0.960342i	$-0.589945\pi$
$277$	−2570.87	−0.557648	−0.278824	+	0.960342i	$0.589945\pi$
$278$	137.476	0.0296593
$279$	0	0
$280$	−375.683	−0.0801834
$281$	−6127.68	−1.30088	−0.650439	−	0.759559i	$-0.725415\pi$
$281$	−6127.68	−1.30088	−0.650439	+	0.759559i	$0.725415\pi$
$282$	0	0
$283$	−1634.83	−0.343395	−0.171698	−	0.985150i	$-0.554925\pi$
$283$	−1634.83	−0.343395	−0.171698	+	0.985150i	$0.554925\pi$
$284$	−1578.06	−0.329720
$285$	0	0
$286$	92.2783	0.0190788
$287$	546.086	0.112315
$288$	0	0
$289$	−1098.49	−0.223589
$290$	−54.7404	−0.0110844
$291$	0	0
$292$	1800.68	0.360879
$293$	−5802.05	−1.15686	−0.578429	−	0.815733i	$-0.696334\pi$
$293$	−5802.05	−1.15686	−0.578429	+	0.815733i	$0.696334\pi$
$294$	0	0
$295$	9858.75	1.94576
$296$	−604.646	−0.118731
$297$	0	0
$298$	576.652	0.112096
$299$	926.086	0.179120
$300$	0	0
$301$	318.643	0.0610176
$302$	−487.275	−0.0928461
$303$	0	0
$304$	3427.84	0.646712
$305$	3904.18	0.732960
$306$	0	0
$307$	1012.73	0.188273	0.0941365	−	0.995559i	$-0.469991\pi$
$307$	1012.73	0.188273	0.0941365	+	0.995559i	$0.469991\pi$
$308$	−612.208	−0.113259
$309$	0	0
$310$	−639.377	−0.117142
$311$	5612.02	1.02324	0.511621	−	0.859211i	$-0.329045\pi$
$311$	5612.02	1.02324	0.511621	+	0.859211i	$0.329045\pi$
$312$	0	0
$313$	−9363.85	−1.69098	−0.845489	−	0.533994i	$-0.820691\pi$
$313$	−9363.85	−1.69098	−0.845489	+	0.533994i	$0.820691\pi$
$314$	−351.911	−0.0632467
$315$	0	0
$316$	−1102.30	−0.196231
$317$	−7870.51	−1.39449	−0.697243	−	0.716835i	$-0.745590\pi$
$317$	−7870.51	−1.39449	−0.697243	+	0.716835i	$0.745590\pi$
$318$	0	0
$319$	−178.961	−0.0314103
$320$	−7477.43	−1.30625
$321$	0	0
$322$	38.0577	0.00658655
$323$	−3370.07	−0.580544
$324$	0	0
$325$	3964.28	0.676612
$326$	596.574	0.101353
$327$	0	0
$328$	−276.149	−0.0464872
$329$	1915.17	0.320932
$330$	0	0
$331$	−9672.26	−1.60615	−0.803075	−	0.595878i	$-0.796804\pi$
$331$	−9672.26	−1.60615	−0.803075	+	0.595878i	$0.796804\pi$
$332$	−3902.10	−0.645047
$333$	0	0
$334$	525.238	0.0860471
$335$	−6061.08	−0.988513
$336$	0	0
$337$	3705.97	0.599042	0.299521	−	0.954090i	$-0.403173\pi$
$337$	3705.97	0.599042	0.299521	+	0.954090i	$0.403173\pi$
$338$	−170.449	−0.0274296
$339$	0	0
$340$	7445.09	1.18755
$341$	−2090.29	−0.331952
$342$	0	0
$343$	343.000	0.0539949
$344$	−161.134	−0.0252551
$345$	0	0
$346$	−213.250	−0.0331340
$347$	4360.40	0.674578	0.337289	−	0.941401i	$-0.390490\pi$
$347$	4360.40	0.674578	0.337289	+	0.941401i	$0.390490\pi$
$348$	0	0
$349$	−5327.66	−0.817144	−0.408572	−	0.912726i	$-0.633973\pi$
$349$	−5327.66	−0.817144	−0.408572	+	0.912726i	$0.633973\pi$
$350$	162.913	0.0248802
$351$	0	0
$352$	464.857	0.0703891
$353$	−8697.75	−1.31143	−0.655715	−	0.755009i	$-0.727633\pi$
$353$	−8697.75	−1.31143	−0.655715	+	0.755009i	$0.727633\pi$
$354$	0	0
$355$	3009.25	0.449899
$356$	−1759.31	−0.261920
$357$	0	0
$358$	281.926	0.0416208
$359$	1904.47	0.279984	0.139992	−	0.990153i	$-0.455292\pi$
$359$	1904.47	0.279984	0.139992	+	0.990153i	$0.455292\pi$
$360$	0	0
$361$	−3881.58	−0.565910
$362$	−236.696	−0.0343659
$363$	0	0
$364$	−2103.84	−0.302943
$365$	−3433.76	−0.492414
$366$	0	0
$367$	11523.2	1.63898	0.819492	−	0.573090i	$-0.194256\pi$
$367$	11523.2	1.63898	0.819492	+	0.573090i	$0.194256\pi$
$368$	1539.02	0.218009
$369$	0	0
$370$	574.729	0.0807533
$371$	−1142.73	−0.159912
$372$	0	0
$373$	−8410.72	−1.16753	−0.583767	−	0.811921i	$-0.698422\pi$
$373$	−8410.72	−1.16753	−0.583767	+	0.811921i	$0.698422\pi$
$374$	−150.769	−0.0208451
$375$	0	0
$376$	−968.477	−0.132833
$377$	−614.997	−0.0840158
$378$	0	0
$379$	−1474.83	−0.199886	−0.0999429	−	0.994993i	$-0.531866\pi$
$379$	−1474.83	−0.199886	−0.0999429	+	0.994993i	$0.531866\pi$
$380$	−6577.65	−0.887964
$381$	0	0
$382$	−210.113	−0.0281422
$383$	−14807.8	−1.97558	−0.987788	−	0.155805i	$-0.950203\pi$
$383$	−14807.8	−1.97558	−0.987788	+	0.155805i	$0.950203\pi$
$384$	0	0
$385$	1167.44	0.154540
$386$	−755.811	−0.0996625
$387$	0	0
$388$	−12712.8	−1.66339
$389$	7886.38	1.02791	0.513953	−	0.857818i	$-0.328181\pi$
$389$	7886.38	1.02791	0.513953	+	0.857818i	$0.328181\pi$
$390$	0	0
$391$	−1513.09	−0.195703
$392$	−173.451	−0.0223485
$393$	0	0
$394$	−870.797	−0.111346
$395$	2102.00	0.267755
$396$	0	0
$397$	−9140.36	−1.15552	−0.577760	−	0.816207i	$-0.696073\pi$
$397$	−9140.36	−1.15552	−0.577760	+	0.816207i	$0.696073\pi$
$398$	−568.249	−0.0715672
$399$	0	0
$400$	6588.07	0.823509
$401$	3110.82	0.387399	0.193700	−	0.981061i	$-0.437951\pi$
$401$	3110.82	0.387399	0.193700	+	0.981061i	$0.437951\pi$
$402$	0	0
$403$	−7183.27	−0.887901
$404$	−14079.9	−1.73391
$405$	0	0
$406$	−25.2734	−0.00308940
$407$	1878.94	0.228835
$408$	0	0
$409$	4395.82	0.531441	0.265721	−	0.964050i	$-0.414390\pi$
$409$	4395.82	0.531441	0.265721	+	0.964050i	$0.414390\pi$
$410$	262.485	0.0316176
$411$	0	0
$412$	−9216.56	−1.10211
$413$	4551.74	0.542316
$414$	0	0
$415$	7441.03	0.880158
$416$	1597.48	0.188276
$417$	0	0
$418$	133.202	0.0155865
$419$	13680.4	1.59506	0.797532	−	0.603277i	$-0.206139\pi$
$419$	13680.4	1.59506	0.797532	+	0.603277i	$0.206139\pi$
$420$	0	0
$421$	4972.96	0.575694	0.287847	−	0.957676i	$-0.407060\pi$
$421$	4972.96	0.575694	0.287847	+	0.957676i	$0.407060\pi$
$422$	−1273.61	−0.146915
$423$	0	0
$424$	577.863	0.0661876
$425$	−6477.04	−0.739253
$426$	0	0
$427$	1802.54	0.204288
$428$	−10613.7	−1.19868
$429$	0	0
$430$	153.161	0.0171770
$431$	11287.7	1.26151	0.630753	−	0.775984i	$-0.282746\pi$
$431$	11287.7	1.26151	0.630753	+	0.775984i	$0.282746\pi$
$432$	0	0
$433$	1261.97	0.140061	0.0700303	−	0.997545i	$-0.477690\pi$
$433$	1261.97	0.140061	0.0700303	+	0.997545i	$0.477690\pi$
$434$	−295.197	−0.0326496
$435$	0	0
$436$	−3043.37	−0.334292
$437$	1336.79	0.146333
$438$	0	0
$439$	−7735.83	−0.841027	−0.420514	−	0.907286i	$-0.638150\pi$
$439$	−7735.83	−0.841027	−0.420514	+	0.907286i	$0.638150\pi$
$440$	−590.359	−0.0639642
$441$	0	0
$442$	−518.115	−0.0557561
$443$	−6696.73	−0.718219	−0.359110	−	0.933295i	$-0.616920\pi$
$443$	−6696.73	−0.718219	−0.359110	+	0.933295i	$0.616920\pi$
$444$	0	0
$445$	3354.88	0.357386
$446$	606.016	0.0643401
$447$	0	0
$448$	−3452.30	−0.364075
$449$	11839.5	1.24442	0.622208	−	0.782852i	$-0.286236\pi$
$449$	11839.5	1.24442	0.622208	+	0.782852i	$0.286236\pi$
$450$	0	0
$451$	858.135	0.0895965
$452$	−5096.49	−0.530351
$453$	0	0
$454$	470.073	0.0485938
$455$	4011.88	0.413362
$456$	0	0
$457$	−9116.78	−0.933183	−0.466592	−	0.884473i	$-0.654518\pi$
$457$	−9116.78	−0.933183	−0.466592	+	0.884473i	$0.654518\pi$
$458$	−626.163	−0.0638836
$459$	0	0
$460$	−2953.21	−0.299336
$461$	9601.92	0.970078	0.485039	−	0.874492i	$-0.338805\pi$
$461$	9601.92	0.970078	0.485039	+	0.874492i	$0.338805\pi$
$462$	0	0
$463$	10049.0	1.00867	0.504337	−	0.863507i	$-0.331737\pi$
$463$	10049.0	1.00867	0.504337	+	0.863507i	$0.331737\pi$
$464$	−1022.04	−0.102256
$465$	0	0
$466$	1517.71	0.150872
$467$	53.2684	0.00527831	0.00263915	−	0.999997i	$-0.499160\pi$
$467$	53.2684	0.00527831	0.00263915	+	0.999997i	$0.499160\pi$
$468$	0	0
$469$	−2798.37	−0.275515
$470$	920.557	0.0903449
$471$	0	0
$472$	−2301.76	−0.224464
$473$	500.725	0.0486752
$474$	0	0
$475$	5722.39	0.552761
$476$	3437.36	0.330990
$477$	0	0
$478$	770.912	0.0737671
$479$	−1553.49	−0.148185	−0.0740926	−	0.997251i	$-0.523606\pi$
$479$	−1553.49	−0.148185	−0.0740926	+	0.997251i	$0.523606\pi$
$480$	0	0
$481$	6456.95	0.612083
$482$	371.266	0.0350844
$483$	0	0
$484$	−962.041	−0.0903494
$485$	24242.4	2.26967
$486$	0	0
$487$	−4340.19	−0.403845	−0.201923	−	0.979401i	$-0.564719\pi$
$487$	−4340.19	−0.403845	−0.201923	+	0.979401i	$0.564719\pi$
$488$	−911.525	−0.0845549
$489$	0	0
$490$	164.869	0.0152000
$491$	−304.844	−0.0280192	−0.0140096	−	0.999902i	$-0.504460\pi$
$491$	−304.844	−0.0280192	−0.0140096	+	0.999902i	$0.504460\pi$
$492$	0	0
$493$	1004.81	0.0917941
$494$	457.748	0.0416904
$495$	0	0
$496$	−11937.6	−1.08067
$497$	1389.36	0.125395
$498$	0	0
$499$	14238.0	1.27732	0.638658	−	0.769491i	$-0.279490\pi$
$499$	14238.0	1.27732	0.638658	+	0.769491i	$0.279490\pi$
$500$	2426.40	0.217024
$501$	0	0
$502$	−991.671	−0.0881682
$503$	−863.867	−0.0765764	−0.0382882	−	0.999267i	$-0.512190\pi$
$503$	−863.867	−0.0765764	−0.0382882	+	0.999267i	$0.512190\pi$
$504$	0	0
$505$	26849.3	2.36590
$506$	59.8049	0.00525425
$507$	0	0
$508$	4570.28	0.399160
$509$	−15882.5	−1.38307	−0.691533	−	0.722345i	$-0.743064\pi$
$509$	−15882.5	−1.38307	−0.691533	+	0.722345i	$0.743064\pi$
$510$	0	0
$511$	−1585.35	−0.137244
$512$	4433.76	0.382708
$513$	0	0
$514$	142.857	0.0122591
$515$	17575.3	1.50381
$516$	0	0
$517$	3009.55	0.256015
$518$	265.349	0.0225073
$519$	0	0
$520$	−2028.76	−0.171090
$521$	−3222.11	−0.270946	−0.135473	−	0.990781i	$-0.543255\pi$
$521$	−3222.11	−0.270946	−0.135473	+	0.990781i	$0.543255\pi$
$522$	0	0
$523$	−15897.9	−1.32919	−0.664595	−	0.747203i	$-0.731396\pi$
$523$	−15897.9	−1.32919	−0.664595	+	0.747203i	$0.731396\pi$
$524$	5828.94	0.485951
$525$	0	0
$526$	−910.028	−0.0754355
$527$	11736.4	0.970104
$528$	0	0
$529$	−11566.8	−0.950671
$530$	−549.271	−0.0450166
$531$	0	0
$532$	−3036.87	−0.247491
$533$	2948.97	0.239651
$534$	0	0
$535$	20239.6	1.63558
$536$	1415.10	0.114036
$537$	0	0
$538$	414.966	0.0332537
$539$	539.000	0.0430730
$540$	0	0
$541$	5790.34	0.460159	0.230080	−	0.973172i	$-0.426101\pi$
$541$	5790.34	0.460159	0.230080	+	0.973172i	$0.426101\pi$
$542$	1208.74	0.0957934
$543$	0	0
$544$	−2610.03	−0.205706
$545$	5803.50	0.456137
$546$	0	0
$547$	−394.656	−0.0308487	−0.0154244	−	0.999881i	$-0.504910\pi$
$547$	−394.656	−0.0308487	−0.0154244	+	0.999881i	$0.504910\pi$
$548$	5569.51	0.434156
$549$	0	0
$550$	256.006	0.0198475
$551$	−887.740	−0.0686370
$552$	0	0
$553$	970.485	0.0746279
$554$	−570.531	−0.0437537
$555$	0	0
$556$	−4925.34	−0.375685
$557$	20388.5	1.55097	0.775484	−	0.631367i	$-0.217506\pi$
$557$	20388.5	1.55097	0.775484	+	0.631367i	$0.217506\pi$
$558$	0	0
$559$	1720.73	0.130196
$560$	6667.17	0.503106
$561$	0	0
$562$	−1359.86	−0.102068
$563$	−9186.83	−0.687707	−0.343853	−	0.939023i	$-0.611732\pi$
$563$	−9186.83	−0.687707	−0.343853	+	0.939023i	$0.611732\pi$
$564$	0	0
$565$	9718.64	0.723657
$566$	−362.805	−0.0269432
$567$	0	0
$568$	−702.580	−0.0519007
$569$	20287.6	1.49473	0.747364	−	0.664415i	$-0.231319\pi$
$569$	20287.6	1.49473	0.747364	+	0.664415i	$0.231319\pi$
$570$	0	0
$571$	16177.3	1.18564	0.592818	−	0.805337i	$-0.298015\pi$
$571$	16177.3	1.18564	0.592818	+	0.805337i	$0.298015\pi$
$572$	−3306.04	−0.241665
$573$	0	0
$574$	121.188	0.00881237
$575$	2569.22	0.186337
$576$	0	0
$577$	−3080.98	−0.222293	−0.111146	−	0.993804i	$-0.535452\pi$
$577$	−3080.98	−0.222293	−0.111146	+	0.993804i	$0.535452\pi$
$578$	−243.779	−0.0175430
$579$	0	0
$580$	1961.18	0.140402
$581$	3435.49	0.245315
$582$	0	0
$583$	−1795.71	−0.127566
$584$	801.693	0.0568053
$585$	0	0
$586$	−1287.60	−0.0907684
$587$	−13470.3	−0.947154	−0.473577	−	0.880752i	$-0.657037\pi$
$587$	−13470.3	−0.947154	−0.473577	+	0.880752i	$0.657037\pi$
$588$	0	0
$589$	−10369.0	−0.725374
$590$	2187.87	0.152666
$591$	0	0
$592$	10730.5	0.744970
$593$	−9759.61	−0.675850	−0.337925	−	0.941173i	$-0.609725\pi$
$593$	−9759.61	−0.675850	−0.337925	+	0.941173i	$0.609725\pi$
$594$	0	0
$595$	−6554.81	−0.451632
$596$	−20659.6	−1.41988
$597$	0	0
$598$	205.519	0.0140540
$599$	26993.6	1.84128	0.920642	−	0.390407i	$-0.127666\pi$
$599$	26993.6	1.84128	0.920642	+	0.390407i	$0.127666\pi$
$600$	0	0
$601$	921.111	0.0625173	0.0312587	−	0.999511i	$-0.490048\pi$
$601$	921.111	0.0625173	0.0312587	+	0.999511i	$0.490048\pi$
$602$	70.7138	0.00478751
$603$	0	0
$604$	17457.5	1.17605
$605$	1834.54	0.123281
$606$	0	0
$607$	8954.12	0.598742	0.299371	−	0.954137i	$-0.403223\pi$
$607$	8954.12	0.598742	0.299371	+	0.954137i	$0.403223\pi$
$608$	2305.93	0.153812
$609$	0	0
$610$	866.423	0.0575089
$611$	10342.3	0.684784
$612$	0	0
$613$	3495.30	0.230300	0.115150	−	0.993348i	$-0.463265\pi$
$613$	3495.30	0.230300	0.115150	+	0.993348i	$0.463265\pi$
$614$	224.748	0.0147721
$615$	0	0
$616$	−272.566	−0.0178279
$617$	703.452	0.0458994	0.0229497	−	0.999737i	$-0.492694\pi$
$617$	703.452	0.0458994	0.0229497	+	0.999737i	$0.492694\pi$
$618$	0	0
$619$	−22343.8	−1.45084	−0.725421	−	0.688305i	$-0.758355\pi$
$619$	−22343.8	−1.45084	−0.725421	+	0.688305i	$0.758355\pi$
$620$	22906.9	1.48381
$621$	0	0
$622$	1245.43	0.0802848
$623$	1548.93	0.0996095
$624$	0	0
$625$	−17735.9	−1.13510
$626$	−2078.04	−0.132676
$627$	0	0
$628$	12607.8	0.801127
$629$	−10549.7	−0.668750
$630$	0	0
$631$	−2517.59	−0.158833	−0.0794166	−	0.996842i	$-0.525306\pi$
$631$	−2517.59	−0.158833	−0.0794166	+	0.996842i	$0.525306\pi$
$632$	−490.762	−0.0308884
$633$	0	0
$634$	−1746.64	−0.109413
$635$	−8715.19	−0.544649
$636$	0	0
$637$	1852.27	0.115211
$638$	−39.7153	−0.00246449
$639$	0	0
$640$	−6785.18	−0.419075
$641$	−27413.1	−1.68916	−0.844581	−	0.535427i	$-0.820151\pi$
$641$	−27413.1	−1.68916	−0.844581	+	0.535427i	$0.820151\pi$
$642$	0	0
$643$	−14594.3	−0.895090	−0.447545	−	0.894261i	$-0.647702\pi$
$643$	−14594.3	−0.895090	−0.447545	+	0.894261i	$0.647702\pi$
$644$	−1363.49	−0.0834299
$645$	0	0
$646$	−747.892	−0.0455502
$647$	30744.2	1.86813	0.934065	−	0.357102i	$-0.116235\pi$
$647$	30744.2	1.86813	0.934065	+	0.357102i	$0.116235\pi$
$648$	0	0
$649$	7152.73	0.432618
$650$	879.760	0.0530877
$651$	0	0
$652$	−21373.4	−1.28381
$653$	−963.185	−0.0577218	−0.0288609	−	0.999583i	$-0.509188\pi$
$653$	−963.185	−0.0577218	−0.0288609	+	0.999583i	$0.509188\pi$
$654$	0	0
$655$	−11115.4	−0.663074
$656$	4900.77	0.291681
$657$	0	0
$658$	425.017	0.0251807
$659$	19165.9	1.13292	0.566461	−	0.824088i	$-0.308312\pi$
$659$	19165.9	1.13292	0.566461	+	0.824088i	$0.308312\pi$
$660$	0	0
$661$	1337.15	0.0786825	0.0393412	−	0.999226i	$-0.487474\pi$
$661$	1337.15	0.0786825	0.0393412	+	0.999226i	$0.487474\pi$
$662$	−2146.48	−0.126020
$663$	0	0
$664$	−1737.29	−0.101536
$665$	5791.09	0.337698
$666$	0	0
$667$	−398.575	−0.0231378
$668$	−18817.6	−1.08993
$669$	0	0
$670$	−1345.08	−0.0775599
$671$	2832.57	0.162966
$672$	0	0
$673$	−25541.2	−1.46291	−0.731456	−	0.681889i	$-0.761159\pi$
$673$	−25541.2	−1.46291	−0.731456	+	0.681889i	$0.761159\pi$
$674$	822.436	0.0470016
$675$	0	0
$676$	6106.64	0.347442
$677$	−28339.6	−1.60883	−0.804417	−	0.594065i	$-0.797522\pi$
$677$	−28339.6	−1.60883	−0.804417	+	0.594065i	$0.797522\pi$
$678$	0	0
$679$	11192.6	0.632596
$680$	3314.69	0.186930
$681$	0	0
$682$	−463.882	−0.0260454
$683$	29608.1	1.65875	0.829373	−	0.558696i	$-0.188698\pi$
$683$	29608.1	1.65875	0.829373	+	0.558696i	$0.188698\pi$
$684$	0	0
$685$	−10620.7	−0.592401
$686$	76.1191	0.00423650
$687$	0	0
$688$	2859.61	0.158462
$689$	−6170.94	−0.341211
$690$	0	0
$691$	−9526.18	−0.524447	−0.262224	−	0.965007i	$-0.584456\pi$
$691$	−9526.18	−0.524447	−0.262224	+	0.965007i	$0.584456\pi$
$692$	7640.07	0.419699
$693$	0	0
$694$	967.667	0.0529282
$695$	9392.27	0.512617
$696$	0	0
$697$	−4818.17	−0.261838
$698$	−1182.32	−0.0641141
$699$	0	0
$700$	−5836.65	−0.315149
$701$	9000.24	0.484928	0.242464	−	0.970160i	$-0.422044\pi$
$701$	9000.24	0.484928	0.242464	+	0.970160i	$0.422044\pi$
$702$	0	0
$703$	9320.53	0.500043
$704$	−5425.04	−0.290432
$705$	0	0
$706$	−1930.22	−0.102896
$707$	12396.2	0.659417
$708$	0	0
$709$	−26507.1	−1.40409	−0.702043	−	0.712135i	$-0.747728\pi$
$709$	−26507.1	−1.40409	−0.702043	+	0.712135i	$0.747728\pi$
$710$	667.817	0.0352996
$711$	0	0
$712$	−783.278	−0.0412283
$713$	−4655.42	−0.244526
$714$	0	0
$715$	6304.38	0.329749
$716$	−10100.5	−0.527198
$717$	0	0
$718$	422.643	0.0219678
$719$	−1142.82	−0.0592766	−0.0296383	−	0.999561i	$-0.509436\pi$
$719$	−1142.82	−0.0592766	−0.0296383	+	0.999561i	$0.509436\pi$
$720$	0	0
$721$	8114.44	0.419137
$722$	−861.407	−0.0444020
$723$	0	0
$724$	8480.06	0.435302
$725$	−1706.17	−0.0874010
$726$	0	0
$727$	8756.51	0.446714	0.223357	−	0.974737i	$-0.428298\pi$
$727$	8756.51	0.446714	0.223357	+	0.974737i	$0.428298\pi$
$728$	−936.669	−0.0476858
$729$	0	0
$730$	−762.026	−0.0386354
$731$	−2811.42	−0.142249
$732$	0	0
$733$	21531.5	1.08497	0.542485	−	0.840065i	$-0.317483\pi$
$733$	21531.5	1.08497	0.542485	+	0.840065i	$0.317483\pi$
$734$	2557.25	0.128597
$735$	0	0
$736$	1035.31	0.0518507
$737$	−4397.44	−0.219785
$738$	0	0
$739$	−29052.5	−1.44616	−0.723082	−	0.690762i	$-0.757275\pi$
$739$	−29052.5	−1.44616	−0.723082	+	0.690762i	$0.757275\pi$
$740$	−20590.7	−1.02288
$741$	0	0
$742$	−253.596	−0.0125469
$743$	−21077.0	−1.04070	−0.520349	−	0.853954i	$-0.674198\pi$
$743$	−21077.0	−1.04070	−0.520349	+	0.853954i	$0.674198\pi$
$744$	0	0
$745$	39396.4	1.93741
$746$	−1866.52	−0.0916061
$747$	0	0
$748$	5401.57	0.264039
$749$	9344.53	0.455863
$750$	0	0
$751$	−20376.3	−0.990068	−0.495034	−	0.868874i	$-0.664844\pi$
$751$	−20376.3	−0.990068	−0.495034	+	0.868874i	$0.664844\pi$
$752$	17187.4	0.833456
$753$	0	0
$754$	−136.481	−0.00659198
$755$	−33290.2	−1.60471
$756$	0	0
$757$	4174.74	0.200441	0.100220	−	0.994965i	$-0.468045\pi$
$757$	4174.74	0.200441	0.100220	+	0.994965i	$0.468045\pi$
$758$	−327.296	−0.0156833
$759$	0	0
$760$	−2928.49	−0.139773
$761$	−33842.9	−1.61210	−0.806048	−	0.591850i	$-0.798398\pi$
$761$	−33842.9	−1.61210	−0.806048	+	0.591850i	$0.798398\pi$
$762$	0	0
$763$	2679.45	0.127133
$764$	7527.69	0.356469
$765$	0	0
$766$	−3286.18	−0.155006
$767$	24580.3	1.15716
$768$	0	0
$769$	35288.8	1.65481	0.827403	−	0.561608i	$-0.189817\pi$
$769$	35288.8	1.65481	0.827403	+	0.561608i	$0.189817\pi$
$770$	259.079	0.0121254
$771$	0	0
$772$	27078.3	1.26240
$773$	7889.49	0.367096	0.183548	−	0.983011i	$-0.441242\pi$
$773$	7889.49	0.367096	0.183548	+	0.983011i	$0.441242\pi$
$774$	0	0
$775$	−19928.4	−0.923676
$776$	−5659.97	−0.261831
$777$	0	0
$778$	1750.16	0.0806507
$779$	4256.80	0.195784
$780$	0	0
$781$	2183.27	0.100030
$782$	−335.787	−0.0153551
$783$	0	0
$784$	3078.20	0.140224
$785$	−24042.3	−1.09313
$786$	0	0
$787$	−29792.1	−1.34940	−0.674698	−	0.738094i	$-0.735726\pi$
$787$	−29792.1	−1.34940	−0.674698	+	0.738094i	$0.735726\pi$
$788$	31197.9	1.41038
$789$	0	0
$790$	466.480	0.0210084
$791$	4487.05	0.201696
$792$	0	0
$793$	9734.08	0.435898
$794$	−2028.44	−0.0906634
$795$	0	0
$796$	20358.6	0.906520
$797$	9484.68	0.421537	0.210768	−	0.977536i	$-0.432403\pi$
$797$	9484.68	0.421537	0.210768	+	0.977536i	$0.432403\pi$
$798$	0	0
$799$	−16897.7	−0.748182
$800$	4431.84	0.195862
$801$	0	0
$802$	690.359	0.0303958
$803$	−2491.27	−0.109483
$804$	0	0
$805$	2600.07	0.113839
$806$	−1594.12	−0.0696657
$807$	0	0
$808$	−6268.62	−0.272932
$809$	−4355.71	−0.189294	−0.0946469	−	0.995511i	$-0.530172\pi$
$809$	−4355.71	−0.189294	−0.0946469	+	0.995511i	$0.530172\pi$
$810$	0	0
$811$	24779.5	1.07290	0.536452	−	0.843931i	$-0.319764\pi$
$811$	24779.5	1.07290	0.536452	+	0.843931i	$0.319764\pi$
$812$	905.466	0.0391325
$813$	0	0
$814$	416.978	0.0179546
$815$	40757.5	1.75174
$816$	0	0
$817$	2483.86	0.106364
$818$	975.528	0.0416975
$819$	0	0
$820$	−9404.02	−0.400491
$821$	−11432.8	−0.486001	−0.243001	−	0.970026i	$-0.578132\pi$
$821$	−11432.8	−0.486001	−0.243001	+	0.970026i	$0.578132\pi$
$822$	0	0
$823$	−294.772	−0.0124850	−0.00624248	−	0.999981i	$-0.501987\pi$
$823$	−294.772	−0.0124850	−0.00624248	+	0.999981i	$0.501987\pi$
$824$	−4103.38	−0.173480
$825$	0	0
$826$	1010.13	0.0425507
$827$	5247.79	0.220657	0.110329	−	0.993895i	$-0.464810\pi$
$827$	5247.79	0.220657	0.110329	+	0.993895i	$0.464810\pi$
$828$	0	0
$829$	−7639.88	−0.320077	−0.160039	−	0.987111i	$-0.551162\pi$
$829$	−7639.88	−0.320077	−0.160039	+	0.987111i	$0.551162\pi$
$830$	1651.33	0.0690582
$831$	0	0
$832$	−18643.1	−0.776841
$833$	−3026.32	−0.125877
$834$	0	0
$835$	35883.8	1.48720
$836$	−4772.22	−0.197429
$837$	0	0
$838$	3035.98	0.125150
$839$	6424.79	0.264372	0.132186	−	0.991225i	$-0.457800\pi$
$839$	6424.79	0.264372	0.132186	+	0.991225i	$0.457800\pi$
$840$	0	0
$841$	−24124.3	−0.989147
$842$	1103.61	0.0451696
$843$	0	0
$844$	45629.4	1.86093
$845$	−11644.9	−0.474080
$846$	0	0
$847$	847.000	0.0343604
$848$	−10255.2	−0.415290
$849$	0	0
$850$	−1437.40	−0.0580027
$851$	4184.71	0.168566
$852$	0	0
$853$	−8879.93	−0.356440	−0.178220	−	0.983991i	$-0.557034\pi$
$853$	−8879.93	−0.356440	−0.178220	+	0.983991i	$0.557034\pi$
$854$	400.023	0.0160287
$855$	0	0
$856$	−4725.42	−0.188682
$857$	33495.3	1.33510	0.667549	−	0.744566i	$-0.267344\pi$
$857$	33495.3	1.33510	0.667549	+	0.744566i	$0.267344\pi$
$858$	0	0
$859$	8626.96	0.342664	0.171332	−	0.985213i	$-0.445193\pi$
$859$	8626.96	0.342664	0.171332	+	0.985213i	$0.445193\pi$
$860$	−5487.28	−0.217575
$861$	0	0
$862$	2504.98	0.0989792
$863$	31254.6	1.23281	0.616407	−	0.787428i	$-0.288588\pi$
$863$	31254.6	1.23281	0.616407	+	0.787428i	$0.288588\pi$
$864$	0	0
$865$	−14569.1	−0.572674
$866$	280.058	0.0109893
$867$	0	0
$868$	10576.0	0.413563
$869$	1525.05	0.0595324
$870$	0	0
$871$	−15111.7	−0.587878
$872$	−1354.96	−0.0526203
$873$	0	0
$874$	296.663	0.0114815
$875$	−2136.25	−0.0825354
$876$	0	0
$877$	17993.1	0.692798	0.346399	−	0.938087i	$-0.387404\pi$
$877$	17993.1	0.692798	0.346399	+	0.938087i	$0.387404\pi$
$878$	−1716.75	−0.0659880
$879$	0	0
$880$	10477.0	0.401340
$881$	−21216.5	−0.811353	−0.405677	−	0.914017i	$-0.632964\pi$
$881$	−21216.5	−0.811353	−0.405677	+	0.914017i	$0.632964\pi$
$882$	0	0
$883$	−25621.2	−0.976470	−0.488235	−	0.872712i	$-0.662359\pi$
$883$	−25621.2	−0.976470	−0.488235	+	0.872712i	$0.662359\pi$
$884$	18562.4	0.706246
$885$	0	0
$886$	−1486.15	−0.0563523
$887$	9821.44	0.371783	0.185892	−	0.982570i	$-0.440483\pi$
$887$	9821.44	0.371783	0.185892	+	0.982570i	$0.440483\pi$
$888$	0	0
$889$	−4023.76	−0.151803
$890$	744.521	0.0280409
$891$	0	0
$892$	−21711.6	−0.814977
$893$	14928.9	0.559437
$894$	0	0
$895$	19260.9	0.719355
$896$	−3132.69	−0.116803
$897$	0	0
$898$	2627.45	0.0976382
$899$	3091.58	0.114694
$900$	0	0
$901$	10082.4	0.372801
$902$	190.439	0.00702984
$903$	0	0
$904$	−2269.05	−0.0834817
$905$	−16170.9	−0.593964
$906$	0	0
$907$	−42843.2	−1.56845	−0.784226	−	0.620475i	$-0.786940\pi$
$907$	−42843.2	−1.56845	−0.784226	+	0.620475i	$0.786940\pi$
$908$	−16841.2	−0.615524
$909$	0	0
$910$	890.323	0.0324329
$911$	−41062.3	−1.49336	−0.746681	−	0.665182i	$-0.768354\pi$
$911$	−41062.3	−1.49336	−0.746681	+	0.665182i	$0.768354\pi$
$912$	0	0
$913$	5398.62	0.195694
$914$	−2023.21	−0.0732186
$915$	0	0
$916$	22433.5	0.809195
$917$	−5131.91	−0.184810
$918$	0	0
$919$	48794.4	1.75144	0.875722	−	0.482815i	$-0.160386\pi$
$919$	48794.4	1.75144	0.875722	+	0.482815i	$0.160386\pi$
$920$	−1314.82	−0.0471179
$921$	0	0
$922$	2130.87	0.0761134
$923$	7502.78	0.267559
$924$	0	0
$925$	17913.4	0.636745
$926$	2230.09	0.0791417
$927$	0	0
$928$	−687.532	−0.0243204
$929$	−20643.0	−0.729038	−0.364519	−	0.931196i	$-0.618767\pi$
$929$	−20643.0	−0.729038	−0.364519	+	0.931196i	$0.618767\pi$
$930$	0	0
$931$	2673.72	0.0941221
$932$	−54374.7	−1.91105
$933$	0	0
$934$	11.8214	0.000414142 0
$935$	−10300.4	−0.360277
$936$	0	0
$937$	42485.1	1.48124	0.740622	−	0.671921i	$-0.234531\pi$
$937$	42485.1	1.48124	0.740622	+	0.671921i	$0.234531\pi$
$938$	−621.019	−0.0216173
$939$	0	0
$940$	−32980.6	−1.14437
$941$	1787.96	0.0619404	0.0309702	−	0.999520i	$-0.490140\pi$
$941$	1787.96	0.0619404	0.0309702	+	0.999520i	$0.490140\pi$
$942$	0	0
$943$	1911.21	0.0659994
$944$	40848.9	1.40839
$945$	0	0
$946$	111.122	0.00381911
$947$	4409.90	0.151323	0.0756613	−	0.997134i	$-0.475893\pi$
$947$	4409.90	0.151323	0.0756613	+	0.997134i	$0.475893\pi$
$948$	0	0
$949$	−8561.20	−0.292843
$950$	1269.92	0.0433702
$951$	0	0
$952$	1530.38	0.0521006
$953$	−50160.0	−1.70497	−0.852487	−	0.522748i	$-0.824907\pi$
$953$	−50160.0	−1.70497	−0.852487	+	0.522748i	$0.824907\pi$
$954$	0	0
$955$	−14354.8	−0.486397
$956$	−27619.3	−0.934386
$957$	0	0
$958$	−344.753	−0.0116268
$959$	−4903.51	−0.165112
$960$	0	0
$961$	6319.17	0.212117
$962$	1432.94	0.0480247
$963$	0	0
$964$	−13301.3	−0.444404
$965$	−51636.4	−1.72252
$966$	0	0
$967$	15711.7	0.522497	0.261248	−	0.965272i	$-0.415866\pi$
$967$	15711.7	0.522497	0.261248	+	0.965272i	$0.415866\pi$
$968$	−428.318	−0.0142218
$969$	0	0
$970$	5379.92	0.178081
$971$	−21930.3	−0.724795	−0.362397	−	0.932024i	$-0.618042\pi$
$971$	−21930.3	−0.724795	−0.362397	+	0.932024i	$0.618042\pi$
$972$	0	0
$973$	4336.37	0.142875
$974$	−963.181	−0.0316862
$975$	0	0
$976$	16176.7	0.530535
$977$	−46329.0	−1.51709	−0.758544	−	0.651621i	$-0.774089\pi$
$977$	−46329.0	−1.51709	−0.758544	+	0.651621i	$0.774089\pi$
$978$	0	0
$979$	2434.04	0.0794609
$980$	−5906.73	−0.192534
$981$	0	0
$982$	−67.6515	−0.00219842
$983$	−29563.1	−0.959224	−0.479612	−	0.877481i	$-0.659223\pi$
$983$	−29563.1	−0.959224	−0.479612	+	0.877481i	$0.659223\pi$
$984$	0	0
$985$	−59492.2	−1.92445
$986$	222.990	0.00720227
$987$	0	0
$988$	−16399.7	−0.528080
$989$	1115.20	0.0358556
$990$	0	0
$991$	−39722.4	−1.27328	−0.636642	−	0.771160i	$-0.719677\pi$
$991$	−39722.4	−1.27328	−0.636642	+	0.771160i	$0.719677\pi$
$992$	−8030.48	−0.257024
$993$	0	0
$994$	308.328	0.00983860
$995$	−38822.3	−1.23694
$996$	0	0
$997$	−38762.1	−1.23130	−0.615652	−	0.788018i	$-0.711107\pi$
$997$	−38762.1	−1.23130	−0.615652	+	0.788018i	$0.711107\pi$
$998$	3159.72	0.100220
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.u.1.4	yes	8
3.2	odd	2		693.4.a.r.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.r.1.5	✓	8	3.2	odd	2
693.4.a.u.1.4	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.221922 q^{2} -7.95075 q^{4} +15.1615 q^{5} +7.00000 q^{7} -3.53982 q^{8} +3.36467 q^{10} +11.0000 q^{11} +37.8013 q^{13} +1.55345 q^{14} +62.8204 q^{16} -61.7617 q^{17} +54.5657 q^{19} -120.545 q^{20} +2.44114 q^{22} +24.4988 q^{23} +104.871 q^{25} +8.38893 q^{26} -55.6553 q^{28} -16.2692 q^{29} -190.027 q^{31} +42.2597 q^{32} -13.7063 q^{34} +106.131 q^{35} +170.813 q^{37} +12.1093 q^{38} -53.6690 q^{40} +78.0123 q^{41} +45.5205 q^{43} -87.4583 q^{44} +5.43681 q^{46} +273.595 q^{47} +49.0000 q^{49} +23.2732 q^{50} -300.549 q^{52} -163.247 q^{53} +166.777 q^{55} -24.7787 q^{56} -3.61049 q^{58} +650.249 q^{59} +257.506 q^{61} -42.1710 q^{62} -493.185 q^{64} +573.125 q^{65} -399.767 q^{67} +491.052 q^{68} +23.5527 q^{70} +198.479 q^{71} -226.479 q^{73} +37.9071 q^{74} -433.839 q^{76} +77.0000 q^{77} +138.641 q^{79} +952.453 q^{80} +17.3126 q^{82} +490.784 q^{83} -936.401 q^{85} +10.1020 q^{86} -38.9380 q^{88} +221.276 q^{89} +264.609 q^{91} -194.784 q^{92} +60.7167 q^{94} +827.299 q^{95} +1598.94 q^{97} +10.8742 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8} + 9 q^{10} + 88 q^{11} + 16 q^{13} + 42 q^{14} + 122 q^{16} + 90 q^{17} - 42 q^{19} + 291 q^{20} + 66 q^{22} + 338 q^{23} + 244 q^{25} + 209 q^{26}+ \cdots + 294 q^{98}+O(q^{100}) \) 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8 + 9 * q^10 + 88 * q^11 + 16 * q^13 + 42 * q^14 + 122 * q^16 + 90 * q^17 - 42 * q^19 + 291 * q^20 + 66 * q^22 + 338 * q^23 + 244 * q^25 + 209 * q^26 + 210 * q^28 + 496 * q^29 - 8 * q^31 + 524 * q^32 - 302 * q^34 + 70 * q^35 - 360 * q^37 + 45 * q^38 - 6 * q^40 + 242 * q^41 - 66 * q^43 + 330 * q^44 + 344 * q^46 + 540 * q^47 + 392 * q^49 + 1171 * q^50 + 465 * q^52 + 906 * q^53 + 110 * q^55 + 567 * q^56 + 977 * q^58 + 1242 * q^59 - 318 * q^61 - 110 * q^62 + 525 * q^64 + 1258 * q^65 + 522 * q^67 + 678 * q^68 + 63 * q^70 + 858 * q^71 - 78 * q^73 + 1651 * q^74 + 1775 * q^76 + 616 * q^77 + 516 * q^79 + 567 * q^80 - 1212 * q^82 + 3192 * q^83 + 720 * q^85 + 1322 * q^86 + 891 * q^88 + 2356 * q^89 + 112 * q^91 + 4504 * q^92 - 423 * q^94 + 3308 * q^95 - 1556 * q^97 + 294 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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