LMFDB - Embedded newform 693.4.a.t.1.2

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} - 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			1.2
Root			$-3.85529$ 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)

$f(q)$	$=$	\(q-3.85529 q^{2} +6.86327 q^{4} -5.82801 q^{5} -7.00000 q^{7} +4.38244 q^{8} +22.4687 q^{10} -11.0000 q^{11} -87.1386 q^{13} +26.9870 q^{14} -71.8017 q^{16} +119.043 q^{17} +23.9715 q^{19} -39.9992 q^{20} +42.4082 q^{22} -119.203 q^{23} -91.0343 q^{25} +335.945 q^{26} -48.0429 q^{28} +53.7711 q^{29} -69.0226 q^{31} +241.757 q^{32} -458.946 q^{34} +40.7961 q^{35} +28.6224 q^{37} -92.4171 q^{38} -25.5409 q^{40} -419.188 q^{41} -40.0463 q^{43} -75.4959 q^{44} +459.562 q^{46} -74.1512 q^{47} +49.0000 q^{49} +350.964 q^{50} -598.055 q^{52} +244.861 q^{53} +64.1081 q^{55} -30.6770 q^{56} -207.303 q^{58} +365.224 q^{59} -456.885 q^{61} +266.102 q^{62} -357.630 q^{64} +507.844 q^{65} -470.272 q^{67} +817.025 q^{68} -157.281 q^{70} -359.954 q^{71} -902.375 q^{73} -110.348 q^{74} +164.523 q^{76} +77.0000 q^{77} +707.463 q^{79} +418.461 q^{80} +1616.09 q^{82} -541.450 q^{83} -693.784 q^{85} +154.390 q^{86} -48.2068 q^{88} +559.797 q^{89} +609.970 q^{91} -818.122 q^{92} +285.875 q^{94} -139.706 q^{95} +1524.76 q^{97} -188.909 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 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}+ \cdots + 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

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 $

$2$	−3.85529	−1.36305	−0.681526	−	0.731794i	$-0.738683\pi$
$2$	−3.85529	−1.36305	−0.681526	+	0.731794i	$0.738683\pi$
$3$	0	0
$3$	0	0
$4$	6.86327	0.857908
$5$	−5.82801	−0.521273	−0.260636	−	0.965437i	$-0.583932\pi$
$5$	−5.82801	−0.521273	−0.260636	+	0.965437i	$0.583932\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	4.38244	0.193678
$9$	0	0
$10$	22.4687	0.710522
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−87.1386	−1.85907	−0.929535	−	0.368735i	$-0.879791\pi$
$13$	−87.1386	−1.85907	−0.929535	+	0.368735i	$0.879791\pi$
$14$	26.9870	0.515185
$15$	0	0
$16$	−71.8017	−1.12190
$17$	119.043	1.69836	0.849182	−	0.528100i	$-0.177095\pi$
$17$	119.043	1.69836	0.849182	+	0.528100i	$0.177095\pi$
$18$	0	0
$19$	23.9715	0.289444	0.144722	−	0.989472i	$-0.453771\pi$
$19$	23.9715	0.289444	0.144722	+	0.989472i	$0.453771\pi$
$20$	−39.9992	−0.447204
$21$	0	0
$22$	42.4082	0.410975
$23$	−119.203	−1.08068	−0.540338	−	0.841448i	$-0.681704\pi$
$23$	−119.203	−1.08068	−0.540338	+	0.841448i	$0.681704\pi$
$24$	0	0
$25$	−91.0343	−0.728275
$26$	335.945	2.53401
$27$	0	0
$28$	−48.0429	−0.324259
$29$	53.7711	0.344312	0.172156	−	0.985070i	$-0.444927\pi$
$29$	53.7711	0.344312	0.172156	+	0.985070i	$0.444927\pi$
$30$	0	0
$31$	−69.0226	−0.399898	−0.199949	−	0.979806i	$-0.564078\pi$
$31$	−69.0226	−0.399898	−0.199949	+	0.979806i	$0.564078\pi$
$32$	241.757	1.33553
$33$	0	0
$34$	−458.946	−2.31496
$35$	40.7961	0.197023
$36$	0	0
$37$	28.6224	0.127176	0.0635878	−	0.997976i	$-0.479746\pi$
$37$	28.6224	0.127176	0.0635878	+	0.997976i	$0.479746\pi$
$38$	−92.4171	−0.394527
$39$	0	0
$40$	−25.5409	−0.100959
$41$	−419.188	−1.59674	−0.798369	−	0.602169i	$-0.794303\pi$
$41$	−419.188	−1.59674	−0.798369	+	0.602169i	$0.794303\pi$
$42$	0	0
$43$	−40.0463	−0.142023	−0.0710117	−	0.997475i	$-0.522623\pi$
$43$	−40.0463	−0.142023	−0.0710117	+	0.997475i	$0.522623\pi$
$44$	−75.4959	−0.258669
$45$	0	0
$46$	459.562	1.47302
$47$	−74.1512	−0.230129	−0.115065	−	0.993358i	$-0.536707\pi$
$47$	−74.1512	−0.230129	−0.115065	+	0.993358i	$0.536707\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	350.964	0.992676
$51$	0	0
$52$	−598.055	−1.59491
$53$	244.861	0.634608	0.317304	−	0.948324i	$-0.397223\pi$
$53$	244.861	0.634608	0.317304	+	0.948324i	$0.397223\pi$
$54$	0	0
$55$	64.1081	0.157170
$56$	−30.6770	−0.0732034
$57$	0	0
$58$	−207.303	−0.469315
$59$	365.224	0.805900	0.402950	−	0.915222i	$-0.367985\pi$
$59$	365.224	0.805900	0.402950	+	0.915222i	$0.367985\pi$
$60$	0	0
$61$	−456.885	−0.958986	−0.479493	−	0.877546i	$-0.659179\pi$
$61$	−456.885	−0.958986	−0.479493	+	0.877546i	$0.659179\pi$
$62$	266.102	0.545081
$63$	0	0
$64$	−357.630	−0.698496
$65$	507.844	0.969082
$66$	0	0
$67$	−470.272	−0.857506	−0.428753	−	0.903422i	$-0.641047\pi$
$67$	−470.272	−0.857506	−0.428753	+	0.903422i	$0.641047\pi$
$68$	817.025	1.45704
$69$	0	0
$70$	−157.281	−0.268552
$71$	−359.954	−0.601671	−0.300835	−	0.953676i	$-0.597266\pi$
$71$	−359.954	−0.601671	−0.300835	+	0.953676i	$0.597266\pi$
$72$	0	0
$73$	−902.375	−1.44678	−0.723391	−	0.690439i	$-0.757417\pi$
$73$	−902.375	−1.44678	−0.723391	+	0.690439i	$0.757417\pi$
$74$	−110.348	−0.173347
$75$	0	0
$76$	164.523	0.248317
$77$	77.0000	0.113961
$78$	0	0
$79$	707.463	1.00754	0.503771	−	0.863837i	$-0.331945\pi$
$79$	707.463	1.00754	0.503771	+	0.863837i	$0.331945\pi$
$80$	418.461	0.584817
$81$	0	0
$82$	1616.09	2.17643
$83$	−541.450	−0.716047	−0.358024	−	0.933713i	$-0.616549\pi$
$83$	−541.450	−0.716047	−0.358024	+	0.933713i	$0.616549\pi$
$84$	0	0
$85$	−693.784	−0.885311
$86$	154.390	0.193585
$87$	0	0
$88$	−48.2068	−0.0583961
$89$	559.797	0.666723	0.333362	−	0.942799i	$-0.391817\pi$
$89$	559.797	0.666723	0.333362	+	0.942799i	$0.391817\pi$
$90$	0	0
$91$	609.970	0.702662
$92$	−818.122	−0.927121
$93$	0	0
$94$	285.875	0.313678
$95$	−139.706	−0.150879
$96$	0	0
$97$	1524.76	1.59604	0.798021	−	0.602629i	$-0.205880\pi$
$97$	1524.76	1.59604	0.798021	+	0.602629i	$0.205880\pi$
$98$	−188.909	−0.194722
$99$	0	0
$100$	−624.793	−0.624793
$101$	1053.14	1.03754	0.518768	−	0.854915i	$-0.326391\pi$
$101$	1053.14	1.03754	0.518768	+	0.854915i	$0.326391\pi$
$102$	0	0
$103$	43.8955	0.0419917	0.0209959	−	0.999780i	$-0.493316\pi$
$103$	43.8955	0.0419917	0.0209959	+	0.999780i	$0.493316\pi$
$104$	−381.879	−0.360061
$105$	0	0
$106$	−944.009	−0.865003
$107$	449.939	0.406516	0.203258	−	0.979125i	$-0.434847\pi$
$107$	449.939	0.406516	0.203258	+	0.979125i	$0.434847\pi$
$108$	0	0
$109$	−1315.63	−1.15610	−0.578048	−	0.816003i	$-0.696185\pi$
$109$	−1315.63	−1.15610	−0.578048	+	0.816003i	$0.696185\pi$
$110$	−247.155	−0.214230
$111$	0	0
$112$	502.612	0.424039
$113$	−725.136	−0.603673	−0.301837	−	0.953360i	$-0.597600\pi$
$113$	−725.136	−0.603673	−0.301837	+	0.953360i	$0.597600\pi$
$114$	0	0
$115$	694.716	0.563327
$116$	369.046	0.295388
$117$	0	0
$118$	−1408.04	−1.09848
$119$	−833.302	−0.641921
$120$	0	0
$121$	121.000	0.0909091
$122$	1761.42	1.30715
$123$	0	0
$124$	−473.720	−0.343075
$125$	1259.05	0.900903
$126$	0	0
$127$	2440.43	1.70514	0.852572	−	0.522610i	$-0.175042\pi$
$127$	2440.43	1.70514	0.852572	+	0.522610i	$0.175042\pi$
$128$	−555.289	−0.383446
$129$	0	0
$130$	−1957.89	−1.32091
$131$	1258.33	0.839242	0.419621	−	0.907699i	$-0.362163\pi$
$131$	1258.33	0.839242	0.419621	+	0.907699i	$0.362163\pi$
$132$	0	0
$133$	−167.801	−0.109400
$134$	1813.04	1.16882
$135$	0	0
$136$	521.699	0.328936
$137$	2790.08	1.73995	0.869974	−	0.493097i	$-0.164135\pi$
$137$	2790.08	1.73995	0.869974	+	0.493097i	$0.164135\pi$
$138$	0	0
$139$	539.573	0.329251	0.164626	−	0.986356i	$-0.447358\pi$
$139$	539.573	0.329251	0.164626	+	0.986356i	$0.447358\pi$
$140$	279.994	0.169027
$141$	0	0
$142$	1387.73	0.820108
$143$	958.525	0.560530
$144$	0	0
$145$	−313.379	−0.179481
$146$	3478.92	1.97204
$147$	0	0
$148$	196.443	0.109105
$149$	873.101	0.480049	0.240024	−	0.970767i	$-0.422845\pi$
$149$	873.101	0.480049	0.240024	+	0.970767i	$0.422845\pi$
$150$	0	0
$151$	1116.95	0.601963	0.300981	−	0.953630i	$-0.402686\pi$
$151$	1116.95	0.601963	0.300981	+	0.953630i	$0.402686\pi$
$152$	105.054	0.0560590
$153$	0	0
$154$	−296.857	−0.155334
$155$	402.264	0.208456
$156$	0	0
$157$	3064.79	1.55794	0.778970	−	0.627061i	$-0.215742\pi$
$157$	3064.79	1.55794	0.778970	+	0.627061i	$0.215742\pi$
$158$	−2727.48	−1.37333
$159$	0	0
$160$	−1408.96	−0.696176
$161$	834.421	0.408457
$162$	0	0
$163$	4051.58	1.94690	0.973449	−	0.228902i	$-0.0735136\pi$
$163$	4051.58	1.94690	0.973449	+	0.228902i	$0.0735136\pi$
$164$	−2877.00	−1.36985
$165$	0	0
$166$	2087.45	0.976009
$167$	1148.16	0.532021	0.266011	−	0.963970i	$-0.414294\pi$
$167$	1148.16	0.532021	0.266011	+	0.963970i	$0.414294\pi$
$168$	0	0
$169$	5396.14	2.45614
$170$	2674.74	1.20672
$171$	0	0
$172$	−274.848	−0.121843
$173$	2605.86	1.14520	0.572601	−	0.819834i	$-0.305934\pi$
$173$	2605.86	1.14520	0.572601	+	0.819834i	$0.305934\pi$
$174$	0	0
$175$	637.240	0.275262
$176$	789.819	0.338266
$177$	0	0
$178$	−2158.18	−0.908778
$179$	−3411.18	−1.42438	−0.712188	−	0.701988i	$-0.752296\pi$
$179$	−3411.18	−1.42438	−0.712188	+	0.701988i	$0.752296\pi$
$180$	0	0
$181$	−641.028	−0.263244	−0.131622	−	0.991300i	$-0.542019\pi$
$181$	−641.028	−0.263244	−0.131622	+	0.991300i	$0.542019\pi$
$182$	−2351.61	−0.957764
$183$	0	0
$184$	−522.399	−0.209303
$185$	−166.812	−0.0662932
$186$	0	0
$187$	−1309.47	−0.512076
$188$	−508.920	−0.197430
$189$	0	0
$190$	538.608	0.205656
$191$	3007.51	1.13935	0.569676	−	0.821870i	$-0.307069\pi$
$191$	3007.51	1.13935	0.569676	+	0.821870i	$0.307069\pi$
$192$	0	0
$193$	294.968	0.110012	0.0550058	−	0.998486i	$-0.482482\pi$
$193$	294.968	0.110012	0.0550058	+	0.998486i	$0.482482\pi$
$194$	−5878.40	−2.17549
$195$	0	0
$196$	336.300	0.122558
$197$	−2299.84	−0.831761	−0.415880	−	0.909419i	$-0.636527\pi$
$197$	−2299.84	−0.831761	−0.415880	+	0.909419i	$0.636527\pi$
$198$	0	0
$199$	−2375.19	−0.846095	−0.423048	−	0.906107i	$-0.639040\pi$
$199$	−2375.19	−0.846095	−0.423048	+	0.906107i	$0.639040\pi$
$200$	−398.952	−0.141051
$201$	0	0
$202$	−4060.15	−1.41422
$203$	−376.398	−0.130138
$204$	0	0
$205$	2443.03	0.832336
$206$	−169.230	−0.0572369
$207$	0	0
$208$	6256.70	2.08569
$209$	−263.687	−0.0872707
$210$	0	0
$211$	−2712.37	−0.884964	−0.442482	−	0.896777i	$-0.645902\pi$
$211$	−2712.37	−0.884964	−0.442482	+	0.896777i	$0.645902\pi$
$212$	1680.54	0.544435
$213$	0	0
$214$	−1734.64	−0.554102
$215$	233.390	0.0740329
$216$	0	0
$217$	483.158	0.151147
$218$	5072.13	1.57582
$219$	0	0
$220$	439.991	0.134837
$221$	−10373.3	−3.15738
$222$	0	0
$223$	−4334.42	−1.30159	−0.650794	−	0.759255i	$-0.725564\pi$
$223$	−4334.42	−1.30159	−0.650794	+	0.759255i	$0.725564\pi$
$224$	−1692.30	−0.504783
$225$	0	0
$226$	2795.61	0.822837
$227$	3140.12	0.918138	0.459069	−	0.888401i	$-0.348183\pi$
$227$	3140.12	0.918138	0.459069	+	0.888401i	$0.348183\pi$
$228$	0	0
$229$	5091.07	1.46912	0.734558	−	0.678546i	$-0.237390\pi$
$229$	5091.07	1.46912	0.734558	+	0.678546i	$0.237390\pi$
$230$	−2678.33	−0.767843
$231$	0	0
$232$	235.649	0.0666857
$233$	−1892.55	−0.532125	−0.266062	−	0.963956i	$-0.585723\pi$
$233$	−1892.55	−0.532125	−0.266062	+	0.963956i	$0.585723\pi$
$234$	0	0
$235$	432.154	0.119960
$236$	2506.63	0.691389
$237$	0	0
$238$	3212.62	0.874972
$239$	−5139.77	−1.39106	−0.695531	−	0.718496i	$-0.744831\pi$
$239$	−5139.77	−1.39106	−0.695531	+	0.718496i	$0.744831\pi$
$240$	0	0
$241$	−1898.64	−0.507477	−0.253738	−	0.967273i	$-0.581660\pi$
$241$	−1898.64	−0.507477	−0.253738	+	0.967273i	$0.581660\pi$
$242$	−466.490	−0.123914
$243$	0	0
$244$	−3135.72	−0.822722
$245$	−285.572	−0.0744675
$246$	0	0
$247$	−2088.84	−0.538097
$248$	−302.487	−0.0774514
$249$	0	0
$250$	−4854.00	−1.22798
$251$	−306.680	−0.0771215	−0.0385607	−	0.999256i	$-0.512277\pi$
$251$	−306.680	−0.0771215	−0.0385607	+	0.999256i	$0.512277\pi$
$252$	0	0
$253$	1311.23	0.325836
$254$	−9408.57	−2.32420
$255$	0	0
$256$	5001.84	1.22115
$257$	3742.75	0.908428	0.454214	−	0.890893i	$-0.349920\pi$
$257$	3742.75	0.908428	0.454214	+	0.890893i	$0.349920\pi$
$258$	0	0
$259$	−200.357	−0.0480679
$260$	3485.47	0.831384
$261$	0	0
$262$	−4851.23	−1.14393
$263$	3529.68	0.827564	0.413782	−	0.910376i	$-0.364208\pi$
$263$	3529.68	0.827564	0.413782	+	0.910376i	$0.364208\pi$
$264$	0	0
$265$	−1427.05	−0.330804
$266$	646.920	0.149117
$267$	0	0
$268$	−3227.60	−0.735662
$269$	1625.18	0.368361	0.184181	−	0.982892i	$-0.441037\pi$
$269$	1625.18	0.368361	0.184181	+	0.982892i	$0.441037\pi$
$270$	0	0
$271$	2770.60	0.621041	0.310521	−	0.950567i	$-0.399497\pi$
$271$	2770.60	0.621041	0.310521	+	0.950567i	$0.399497\pi$
$272$	−8547.50	−1.90540
$273$	0	0
$274$	−10756.6	−2.37164
$275$	1001.38	0.219583
$276$	0	0
$277$	−6395.47	−1.38724	−0.693622	−	0.720339i	$-0.743986\pi$
$277$	−6395.47	−1.38724	−0.693622	+	0.720339i	$0.743986\pi$
$278$	−2080.21	−0.448787
$279$	0	0
$280$	178.786	0.0381590
$281$	−5580.81	−1.18478	−0.592390	−	0.805652i	$-0.701815\pi$
$281$	−5580.81	−1.18478	−0.592390	+	0.805652i	$0.701815\pi$
$282$	0	0
$283$	−3176.32	−0.667182	−0.333591	−	0.942718i	$-0.608261\pi$
$283$	−3176.32	−0.667182	−0.333591	+	0.942718i	$0.608261\pi$
$284$	−2470.46	−0.516179
$285$	0	0
$286$	−3695.39	−0.764032
$287$	2934.32	0.603510
$288$	0	0
$289$	9258.26	1.88444
$290$	1208.17	0.244641
$291$	0	0
$292$	−6193.24	−1.24121
$293$	3683.92	0.734528	0.367264	−	0.930117i	$-0.380295\pi$
$293$	3683.92	0.734528	0.367264	+	0.930117i	$0.380295\pi$
$294$	0	0
$295$	−2128.53	−0.420094
$296$	125.436	0.0246311
$297$	0	0
$298$	−3366.06	−0.654331
$299$	10387.2	2.00905
$300$	0	0
$301$	280.324	0.0536798
$302$	−4306.18	−0.820506
$303$	0	0
$304$	−1721.20	−0.324728
$305$	2662.73	0.499893
$306$	0	0
$307$	−6180.56	−1.14900	−0.574501	−	0.818504i	$-0.694804\pi$
$307$	−6180.56	−1.14900	−0.574501	+	0.818504i	$0.694804\pi$
$308$	528.472	0.0977677
$309$	0	0
$310$	−1550.85	−0.284136
$311$	−5980.82	−1.09049	−0.545243	−	0.838278i	$-0.683563\pi$
$311$	−5980.82	−1.09049	−0.545243	+	0.838278i	$0.683563\pi$
$312$	0	0
$313$	4757.98	0.859224	0.429612	−	0.903014i	$-0.358650\pi$
$313$	4757.98	0.859224	0.429612	+	0.903014i	$0.358650\pi$
$314$	−11815.6	−2.12355
$315$	0	0
$316$	4855.51	0.864379
$317$	−4391.93	−0.778155	−0.389078	−	0.921205i	$-0.627206\pi$
$317$	−4391.93	−0.778155	−0.389078	+	0.921205i	$0.627206\pi$
$318$	0	0
$319$	−591.483	−0.103814
$320$	2084.27	0.364107
$321$	0	0
$322$	−3216.93	−0.556748
$323$	2853.64	0.491582
$324$	0	0
$325$	7932.60	1.35391
$326$	−15620.0	−2.65372
$327$	0	0
$328$	−1837.07	−0.309253
$329$	519.059	0.0869806
$330$	0	0
$331$	8105.82	1.34603	0.673015	−	0.739628i	$-0.264999\pi$
$331$	8105.82	1.34603	0.673015	+	0.739628i	$0.264999\pi$
$332$	−3716.12	−0.614303
$333$	0	0
$334$	−4426.50	−0.725172
$335$	2740.75	0.446995
$336$	0	0
$337$	4039.57	0.652965	0.326482	−	0.945203i	$-0.394137\pi$
$337$	4039.57	0.652965	0.326482	+	0.945203i	$0.394137\pi$
$338$	−20803.7	−3.34784
$339$	0	0
$340$	−4761.63	−0.759516
$341$	759.248	0.120574
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−175.500	−0.0275068
$345$	0	0
$346$	−10046.4	−1.56097
$347$	8412.95	1.30153	0.650765	−	0.759279i	$-0.274448\pi$
$347$	8412.95	1.30153	0.650765	+	0.759279i	$0.274448\pi$
$348$	0	0
$349$	−7559.52	−1.15946	−0.579730	−	0.814808i	$-0.696842\pi$
$349$	−7559.52	−1.15946	−0.579730	+	0.814808i	$0.696842\pi$
$350$	−2456.75	−0.375196
$351$	0	0
$352$	−2659.33	−0.402678
$353$	7724.92	1.16475	0.582374	−	0.812921i	$-0.302124\pi$
$353$	7724.92	1.16475	0.582374	+	0.812921i	$0.302124\pi$
$354$	0	0
$355$	2097.81	0.313635
$356$	3842.04	0.571987
$357$	0	0
$358$	13151.1	1.94150
$359$	−920.274	−0.135293	−0.0676465	−	0.997709i	$-0.521549\pi$
$359$	−920.274	−0.135293	−0.0676465	+	0.997709i	$0.521549\pi$
$360$	0	0
$361$	−6284.37	−0.916222
$362$	2471.35	0.358816
$363$	0	0
$364$	4186.39	0.602820
$365$	5259.05	0.754168
$366$	0	0
$367$	−2615.68	−0.372037	−0.186018	−	0.982546i	$-0.559558\pi$
$367$	−2615.68	−0.372037	−0.186018	+	0.982546i	$0.559558\pi$
$368$	8558.98	1.21241
$369$	0	0
$370$	643.108	0.0903610
$371$	−1714.02	−0.239859
$372$	0	0
$373$	−4037.37	−0.560449	−0.280224	−	0.959935i	$-0.590409\pi$
$373$	−4037.37	−0.560449	−0.280224	+	0.959935i	$0.590409\pi$
$374$	5048.40	0.697986
$375$	0	0
$376$	−324.963	−0.0445710
$377$	−4685.54	−0.640100
$378$	0	0
$379$	4939.84	0.669506	0.334753	−	0.942306i	$-0.391347\pi$
$379$	4939.84	0.669506	0.334753	+	0.942306i	$0.391347\pi$
$380$	−958.840	−0.129441
$381$	0	0
$382$	−11594.8	−1.55299
$383$	245.085	0.0326978	0.0163489	−	0.999866i	$-0.494796\pi$
$383$	245.085	0.0326978	0.0163489	+	0.999866i	$0.494796\pi$
$384$	0	0
$385$	−448.757	−0.0594046
$386$	−1137.19	−0.149951
$387$	0	0
$388$	10464.9	1.36926
$389$	−9961.20	−1.29834	−0.649168	−	0.760645i	$-0.724883\pi$
$389$	−9961.20	−1.29834	−0.649168	+	0.760645i	$0.724883\pi$
$390$	0	0
$391$	−14190.3	−1.83538
$392$	214.739	0.0276683
$393$	0	0
$394$	8866.56	1.13373
$395$	−4123.10	−0.525204
$396$	0	0
$397$	−4848.35	−0.612926	−0.306463	−	0.951883i	$-0.599146\pi$
$397$	−4848.35	−0.612926	−0.306463	+	0.951883i	$0.599146\pi$
$398$	9157.06	1.15327
$399$	0	0
$400$	6536.42	0.817052
$401$	954.553	0.118873	0.0594365	−	0.998232i	$-0.481070\pi$
$401$	954.553	0.118873	0.0594365	+	0.998232i	$0.481070\pi$
$402$	0	0
$403$	6014.53	0.743437
$404$	7227.97	0.890111
$405$	0	0
$406$	1451.12	0.177384
$407$	−314.847	−0.0383449
$408$	0	0
$409$	−199.139	−0.0240753	−0.0120376	−	0.999928i	$-0.503832\pi$
$409$	−199.139	−0.0240753	−0.0120376	+	0.999928i	$0.503832\pi$
$410$	−9418.60	−1.13452
$411$	0	0
$412$	301.266	0.0360251
$413$	−2556.57	−0.304602
$414$	0	0
$415$	3155.58	0.373256
$416$	−21066.4	−2.48284
$417$	0	0
$418$	1016.59	0.118954
$419$	16428.4	1.91547	0.957734	−	0.287654i	$-0.0928752\pi$
$419$	16428.4	1.91547	0.957734	+	0.287654i	$0.0928752\pi$
$420$	0	0
$421$	11222.8	1.29920	0.649602	−	0.760274i	$-0.274936\pi$
$421$	11222.8	1.29920	0.649602	+	0.760274i	$0.274936\pi$
$422$	10457.0	1.20625
$423$	0	0
$424$	1073.09	0.122910
$425$	−10837.0	−1.23688
$426$	0	0
$427$	3198.19	0.362463
$428$	3088.05	0.348753
$429$	0	0
$430$	−899.787	−0.100911
$431$	16786.0	1.87599	0.937996	−	0.346645i	$-0.112679\pi$
$431$	16786.0	1.87599	0.937996	+	0.346645i	$0.112679\pi$
$432$	0	0
$433$	−1619.31	−0.179721	−0.0898604	−	0.995954i	$-0.528642\pi$
$433$	−1619.31	−0.179721	−0.0898604	+	0.995954i	$0.528642\pi$
$434$	−1862.72	−0.206021
$435$	0	0
$436$	−9029.52	−0.991824
$437$	−2857.48	−0.312795
$438$	0	0
$439$	−581.616	−0.0632324	−0.0316162	−	0.999500i	$-0.510065\pi$
$439$	−581.616	−0.0632324	−0.0316162	+	0.999500i	$0.510065\pi$
$440$	280.950	0.0304403
$441$	0	0
$442$	39991.9	4.30367
$443$	−1171.84	−0.125679	−0.0628397	−	0.998024i	$-0.520016\pi$
$443$	−1171.84	−0.125679	−0.0628397	+	0.998024i	$0.520016\pi$
$444$	0	0
$445$	−3262.50	−0.347545
$446$	16710.4	1.77413
$447$	0	0
$448$	2503.41	0.264007
$449$	11428.6	1.20122	0.600609	−	0.799543i	$-0.294925\pi$
$449$	11428.6	1.20122	0.600609	+	0.799543i	$0.294925\pi$
$450$	0	0
$451$	4611.07	0.481434
$452$	−4976.80	−0.517896
$453$	0	0
$454$	−12106.1	−1.25147
$455$	−3554.91	−0.366279
$456$	0	0
$457$	−10643.0	−1.08940	−0.544701	−	0.838630i	$-0.683357\pi$
$457$	−10643.0	−1.08940	−0.544701	+	0.838630i	$0.683357\pi$
$458$	−19627.6	−2.00248
$459$	0	0
$460$	4768.02	0.483283
$461$	−2911.16	−0.294114	−0.147057	−	0.989128i	$-0.546980\pi$
$461$	−2911.16	−0.294114	−0.147057	+	0.989128i	$0.546980\pi$
$462$	0	0
$463$	−15206.9	−1.52640	−0.763202	−	0.646160i	$-0.776374\pi$
$463$	−15206.9	−1.52640	−0.763202	+	0.646160i	$0.776374\pi$
$464$	−3860.86	−0.386284
$465$	0	0
$466$	7296.33	0.725313
$467$	14031.2	1.39034	0.695169	−	0.718847i	$-0.255330\pi$
$467$	14031.2	1.39034	0.695169	+	0.718847i	$0.255330\pi$
$468$	0	0
$469$	3291.91	0.324107
$470$	−1666.08	−0.163512
$471$	0	0
$472$	1600.57	0.156085
$473$	440.509	0.0428216
$474$	0	0
$475$	−2182.23	−0.210795
$476$	−5719.17	−0.550710
$477$	0	0
$478$	19815.3	1.89609
$479$	−14647.2	−1.39718	−0.698590	−	0.715523i	$-0.746189\pi$
$479$	−14647.2	−1.39718	−0.698590	+	0.715523i	$0.746189\pi$
$480$	0	0
$481$	−2494.12	−0.236428
$482$	7319.79	0.691716
$483$	0	0
$484$	830.455	0.0779917
$485$	−8886.33	−0.831974
$486$	0	0
$487$	−11869.8	−1.10446	−0.552230	−	0.833692i	$-0.686223\pi$
$487$	−11869.8	−1.10446	−0.552230	+	0.833692i	$0.686223\pi$
$488$	−2002.27	−0.185735
$489$	0	0
$490$	1100.96	0.101503
$491$	−959.814	−0.0882195	−0.0441098	−	0.999027i	$-0.514045\pi$
$491$	−959.814	−0.0882195	−0.0441098	+	0.999027i	$0.514045\pi$
$492$	0	0
$493$	6401.08	0.584767
$494$	8053.10	0.733454
$495$	0	0
$496$	4955.94	0.448646
$497$	2519.68	0.227410
$498$	0	0
$499$	18146.9	1.62799	0.813997	−	0.580869i	$-0.197287\pi$
$499$	18146.9	1.62799	0.813997	+	0.580869i	$0.197287\pi$
$500$	8641.19	0.772892
$501$	0	0
$502$	1182.34	0.105121
$503$	−10934.7	−0.969294	−0.484647	−	0.874710i	$-0.661052\pi$
$503$	−10934.7	−0.969294	−0.484647	+	0.874710i	$0.661052\pi$
$504$	0	0
$505$	−6137.70	−0.540840
$506$	−5055.18	−0.444131
$507$	0	0
$508$	16749.3	1.46286
$509$	−3876.11	−0.337535	−0.168768	−	0.985656i	$-0.553979\pi$
$509$	−3876.11	−0.337535	−0.168768	+	0.985656i	$0.553979\pi$
$510$	0	0
$511$	6316.63	0.546832
$512$	−14841.2	−1.28105
$513$	0	0
$514$	−14429.4	−1.23823
$515$	−255.823	−0.0218892
$516$	0	0
$517$	815.664	0.0693865
$518$	772.435	0.0655190
$519$	0	0
$520$	2225.60	0.187690
$521$	−2971.13	−0.249842	−0.124921	−	0.992167i	$-0.539868\pi$
$521$	−2971.13	−0.249842	−0.124921	+	0.992167i	$0.539868\pi$
$522$	0	0
$523$	20391.8	1.70491	0.852457	−	0.522797i	$-0.175112\pi$
$523$	20391.8	1.70491	0.852457	+	0.522797i	$0.175112\pi$
$524$	8636.25	0.719993
$525$	0	0
$526$	−13607.9	−1.12801
$527$	−8216.66	−0.679172
$528$	0	0
$529$	2042.35	0.167860
$530$	5501.69	0.450902
$531$	0	0
$532$	−1151.66	−0.0938549
$533$	36527.5	2.96844
$534$	0	0
$535$	−2622.25	−0.211906
$536$	−2060.94	−0.166080
$537$	0	0
$538$	−6265.55	−0.502095
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−13628.9	−1.08309	−0.541544	−	0.840672i	$-0.682160\pi$
$541$	−13628.9	−1.08309	−0.541544	+	0.840672i	$0.682160\pi$
$542$	−10681.5	−0.846511
$543$	0	0
$544$	28779.5	2.26822
$545$	7667.50	0.602641
$546$	0	0
$547$	−4077.34	−0.318710	−0.159355	−	0.987221i	$-0.550941\pi$
$547$	−4077.34	−0.318710	−0.159355	+	0.987221i	$0.550941\pi$
$548$	19149.1	1.49272
$549$	0	0
$550$	−3860.60	−0.299303
$551$	1288.98	0.0996591
$552$	0	0
$553$	−4952.24	−0.380815
$554$	24656.4	1.89088
$555$	0	0
$556$	3703.23	0.282468
$557$	−7792.93	−0.592813	−0.296407	−	0.955062i	$-0.595788\pi$
$557$	−7792.93	−0.592813	−0.296407	+	0.955062i	$0.595788\pi$
$558$	0	0
$559$	3489.58	0.264031
$560$	−2929.23	−0.221040
$561$	0	0
$562$	21515.6	1.61491
$563$	18076.2	1.35315	0.676575	−	0.736374i	$-0.263464\pi$
$563$	18076.2	1.35315	0.676575	+	0.736374i	$0.263464\pi$
$564$	0	0
$565$	4226.10	0.314678
$566$	12245.6	0.909404
$567$	0	0
$568$	−1577.47	−0.116530
$569$	−23137.8	−1.70472	−0.852362	−	0.522951i	$-0.824831\pi$
$569$	−23137.8	−1.70472	−0.852362	+	0.522951i	$0.824831\pi$
$570$	0	0
$571$	−25202.6	−1.84711	−0.923553	−	0.383470i	$-0.874729\pi$
$571$	−25202.6	−1.84711	−0.923553	+	0.383470i	$0.874729\pi$
$572$	6578.61	0.480884
$573$	0	0
$574$	−11312.7	−0.822615
$575$	10851.6	0.787029
$576$	0	0
$577$	18777.6	1.35480	0.677402	−	0.735613i	$-0.263106\pi$
$577$	18777.6	1.35480	0.677402	+	0.735613i	$0.263106\pi$
$578$	−35693.3	−2.56859
$579$	0	0
$580$	−2150.80	−0.153978
$581$	3790.15	0.270640
$582$	0	0
$583$	−2693.47	−0.191341
$584$	−3954.60	−0.280210
$585$	0	0
$586$	−14202.6	−1.00120
$587$	17123.4	1.20402	0.602008	−	0.798490i	$-0.294367\pi$
$587$	17123.4	1.20402	0.602008	+	0.798490i	$0.294367\pi$
$588$	0	0
$589$	−1654.58	−0.115748
$590$	8206.09	0.572609
$591$	0	0
$592$	−2055.14	−0.142679
$593$	22319.3	1.54561	0.772803	−	0.634647i	$-0.218854\pi$
$593$	22319.3	1.54561	0.772803	+	0.634647i	$0.218854\pi$
$594$	0	0
$595$	4856.49	0.334616
$596$	5992.33	0.411838
$597$	0	0
$598$	−40045.6	−2.73844
$599$	5844.36	0.398655	0.199327	−	0.979933i	$-0.436124\pi$
$599$	5844.36	0.398655	0.199327	+	0.979933i	$0.436124\pi$
$600$	0	0
$601$	−27815.8	−1.88791	−0.943953	−	0.330081i	$-0.892924\pi$
$601$	−27815.8	−1.88791	−0.943953	+	0.330081i	$0.892924\pi$
$602$	−1080.73	−0.0731683
$603$	0	0
$604$	7665.95	0.516429
$605$	−705.189	−0.0473884
$606$	0	0
$607$	−15980.5	−1.06858	−0.534290	−	0.845301i	$-0.679421\pi$
$607$	−15980.5	−1.06858	−0.534290	+	0.845301i	$0.679421\pi$
$608$	5795.28	0.386562
$609$	0	0
$610$	−10265.6	−0.681380
$611$	6461.43	0.427826
$612$	0	0
$613$	22639.1	1.49166	0.745829	−	0.666138i	$-0.232054\pi$
$613$	22639.1	1.49166	0.745829	+	0.666138i	$0.232054\pi$
$614$	23827.9	1.56615
$615$	0	0
$616$	337.448	0.0220717
$617$	6134.28	0.400254	0.200127	−	0.979770i	$-0.435864\pi$
$617$	6134.28	0.400254	0.200127	+	0.979770i	$0.435864\pi$
$618$	0	0
$619$	13953.0	0.906005	0.453003	−	0.891509i	$-0.350353\pi$
$619$	13953.0	0.906005	0.453003	+	0.891509i	$0.350353\pi$
$620$	2760.85	0.178836
$621$	0	0
$622$	23057.8	1.48639
$623$	−3918.58	−0.251998
$624$	0	0
$625$	4041.54	0.258659
$626$	−18343.4	−1.17117
$627$	0	0
$628$	21034.4	1.33657
$629$	3407.30	0.215991
$630$	0	0
$631$	−15765.0	−0.994602	−0.497301	−	0.867578i	$-0.665676\pi$
$631$	−15765.0	−0.994602	−0.497301	+	0.867578i	$0.665676\pi$
$632$	3100.41	0.195139
$633$	0	0
$634$	16932.2	1.06067
$635$	−14222.9	−0.888845
$636$	0	0
$637$	−4269.79	−0.265581
$638$	2280.34	0.141504
$639$	0	0
$640$	3236.23	0.199880
$641$	1189.73	0.0733097	0.0366549	−	0.999328i	$-0.488330\pi$
$641$	1189.73	0.0733097	0.0366549	+	0.999328i	$0.488330\pi$
$642$	0	0
$643$	13633.3	0.836149	0.418075	−	0.908413i	$-0.362705\pi$
$643$	13633.3	0.836149	0.418075	+	0.908413i	$0.362705\pi$
$644$	5726.85	0.350419
$645$	0	0
$646$	−11001.6	−0.670051
$647$	11326.2	0.688221	0.344110	−	0.938929i	$-0.388181\pi$
$647$	11326.2	0.688221	0.344110	+	0.938929i	$0.388181\pi$
$648$	0	0
$649$	−4017.46	−0.242988
$650$	−30582.5	−1.84545
$651$	0	0
$652$	27807.1	1.67026
$653$	28224.5	1.69144	0.845719	−	0.533629i	$-0.179172\pi$
$653$	28224.5	1.69144	0.845719	+	0.533629i	$0.179172\pi$
$654$	0	0
$655$	−7333.55	−0.437474
$656$	30098.4	1.79138
$657$	0	0
$658$	−2001.12	−0.118559
$659$	5538.71	0.327402	0.163701	−	0.986510i	$-0.447657\pi$
$659$	5538.71	0.327402	0.163701	+	0.986510i	$0.447657\pi$
$660$	0	0
$661$	3176.03	0.186888	0.0934441	−	0.995625i	$-0.470212\pi$
$661$	3176.03	0.186888	0.0934441	+	0.995625i	$0.470212\pi$
$662$	−31250.3	−1.83471
$663$	0	0
$664$	−2372.87	−0.138683
$665$	977.943	0.0570271
$666$	0	0
$667$	−6409.68	−0.372090
$668$	7880.15	0.456425
$669$	0	0
$670$	−10566.4	−0.609277
$671$	5025.73	0.289145
$672$	0	0
$673$	32576.2	1.86586	0.932928	−	0.360062i	$-0.117245\pi$
$673$	32576.2	1.86586	0.932928	+	0.360062i	$0.117245\pi$
$674$	−15573.7	−0.890025
$675$	0	0
$676$	37035.1	2.10714
$677$	−18529.8	−1.05193	−0.525967	−	0.850505i	$-0.676296\pi$
$677$	−18529.8	−1.05193	−0.525967	+	0.850505i	$0.676296\pi$
$678$	0	0
$679$	−10673.3	−0.603248
$680$	−3040.46	−0.171465
$681$	0	0
$682$	−2927.12	−0.164348
$683$	34880.4	1.95412	0.977058	−	0.212972i	$-0.0683143\pi$
$683$	34880.4	1.95412	0.977058	+	0.212972i	$0.0683143\pi$
$684$	0	0
$685$	−16260.6	−0.906988
$686$	1322.36	0.0735978
$687$	0	0
$688$	2875.39	0.159336
$689$	−21336.8	−1.17978
$690$	0	0
$691$	−21942.1	−1.20798	−0.603992	−	0.796991i	$-0.706424\pi$
$691$	−21942.1	−1.20798	−0.603992	+	0.796991i	$0.706424\pi$
$692$	17884.7	0.982479
$693$	0	0
$694$	−32434.4	−1.77405
$695$	−3144.63	−0.171630
$696$	0	0
$697$	−49901.5	−2.71184
$698$	29144.1	1.58040
$699$	0	0
$700$	4373.55	0.236150
$701$	8513.96	0.458727	0.229364	−	0.973341i	$-0.426335\pi$
$701$	8513.96	0.458727	0.229364	+	0.973341i	$0.426335\pi$
$702$	0	0
$703$	686.123	0.0368103
$704$	3933.93	0.210604
$705$	0	0
$706$	−29781.8	−1.58761
$707$	−7371.97	−0.392152
$708$	0	0
$709$	−27813.5	−1.47329	−0.736643	−	0.676282i	$-0.763590\pi$
$709$	−27813.5	−1.47329	−0.736643	+	0.676282i	$0.763590\pi$
$710$	−8087.68	−0.427500
$711$	0	0
$712$	2453.27	0.129130
$713$	8227.70	0.432159
$714$	0	0
$715$	−5586.29	−0.292189
$716$	−23411.8	−1.22198
$717$	0	0
$718$	3547.92	0.184411
$719$	65.2658	0.00338526	0.00169263	−	0.999999i	$-0.499461\pi$
$719$	65.2658	0.00338526	0.00169263	+	0.999999i	$0.499461\pi$
$720$	0	0
$721$	−307.268	−0.0158714
$722$	24228.1	1.24886
$723$	0	0
$724$	−4399.55	−0.225840
$725$	−4895.02	−0.250754
$726$	0	0
$727$	24711.7	1.26067	0.630334	−	0.776324i	$-0.282918\pi$
$727$	24711.7	1.26067	0.630334	+	0.776324i	$0.282918\pi$
$728$	2673.16	0.136090
$729$	0	0
$730$	−20275.2	−1.02797
$731$	−4767.24	−0.241207
$732$	0	0
$733$	−27286.3	−1.37496	−0.687478	−	0.726205i	$-0.741282\pi$
$733$	−27286.3	−1.37496	−0.687478	+	0.726205i	$0.741282\pi$
$734$	10084.2	0.507105
$735$	0	0
$736$	−28818.1	−1.44328
$737$	5173.00	0.258548
$738$	0	0
$739$	1330.19	0.0662138	0.0331069	−	0.999452i	$-0.489460\pi$
$739$	1330.19	0.0662138	0.0331069	+	0.999452i	$0.489460\pi$
$740$	−1144.87	−0.0568735
$741$	0	0
$742$	6608.06	0.326940
$743$	14905.2	0.735963	0.367982	−	0.929833i	$-0.380049\pi$
$743$	14905.2	0.735963	0.367982	+	0.929833i	$0.380049\pi$
$744$	0	0
$745$	−5088.44	−0.250236
$746$	15565.3	0.763920
$747$	0	0
$748$	−8987.27	−0.439314
$749$	−3149.57	−0.153649
$750$	0	0
$751$	−21448.2	−1.04215	−0.521076	−	0.853510i	$-0.674469\pi$
$751$	−21448.2	−1.04215	−0.521076	+	0.853510i	$0.674469\pi$
$752$	5324.18	0.258182
$753$	0	0
$754$	18064.1	0.872489
$755$	−6509.61	−0.313787
$756$	0	0
$757$	5266.07	0.252838	0.126419	−	0.991977i	$-0.459652\pi$
$757$	5266.07	0.252838	0.126419	+	0.991977i	$0.459652\pi$
$758$	−19044.5	−0.912571
$759$	0	0
$760$	−612.253	−0.0292220
$761$	24314.9	1.15823	0.579117	−	0.815244i	$-0.303397\pi$
$761$	24314.9	1.15823	0.579117	+	0.815244i	$0.303397\pi$
$762$	0	0
$763$	9209.41	0.436963
$764$	20641.4	0.977459
$765$	0	0
$766$	−944.873	−0.0445687
$767$	−31825.1	−1.49822
$768$	0	0
$769$	−6799.74	−0.318862	−0.159431	−	0.987209i	$-0.550966\pi$
$769$	−6799.74	−0.318862	−0.159431	+	0.987209i	$0.550966\pi$
$770$	1730.09	0.0809714
$771$	0	0
$772$	2024.44	0.0943798
$773$	−2665.83	−0.124040	−0.0620202	−	0.998075i	$-0.519754\pi$
$773$	−2665.83	−0.124040	−0.0620202	+	0.998075i	$0.519754\pi$
$774$	0	0
$775$	6283.43	0.291235
$776$	6682.17	0.309119
$777$	0	0
$778$	38403.3	1.76970
$779$	−10048.6	−0.462166
$780$	0	0
$781$	3959.49	0.181411
$782$	54707.7	2.50172
$783$	0	0
$784$	−3518.28	−0.160272
$785$	−17861.6	−0.812112
$786$	0	0
$787$	−19896.6	−0.901191	−0.450596	−	0.892728i	$-0.648788\pi$
$787$	−19896.6	−0.901191	−0.450596	+	0.892728i	$0.648788\pi$
$788$	−15784.4	−0.713575
$789$	0	0
$790$	15895.8	0.715880
$791$	5075.95	0.228167
$792$	0	0
$793$	39812.3	1.78282
$794$	18691.8	0.835449
$795$	0	0
$796$	−16301.6	−0.725872
$797$	30665.5	1.36289	0.681447	−	0.731868i	$-0.261351\pi$
$797$	30665.5	1.36289	0.681447	+	0.731868i	$0.261351\pi$
$798$	0	0
$799$	−8827.19	−0.390843
$800$	−22008.2	−0.972633
$801$	0	0
$802$	−3680.08	−0.162030
$803$	9926.13	0.436221
$804$	0	0
$805$	−4863.01	−0.212918
$806$	−23187.8	−1.01334
$807$	0	0
$808$	4615.31	0.200948
$809$	−26189.0	−1.13814	−0.569071	−	0.822289i	$-0.692697\pi$
$809$	−26189.0	−1.13814	−0.569071	+	0.822289i	$0.692697\pi$
$810$	0	0
$811$	19230.2	0.832631	0.416315	−	0.909220i	$-0.363321\pi$
$811$	19230.2	0.832631	0.416315	+	0.909220i	$0.363321\pi$
$812$	−2583.32	−0.111646
$813$	0	0
$814$	1213.83	0.0522661
$815$	−23612.7	−1.01487
$816$	0	0
$817$	−959.970	−0.0411078
$818$	767.739	0.0328158
$819$	0	0
$820$	16767.2	0.714068
$821$	−42674.2	−1.81406	−0.907028	−	0.421070i	$-0.861655\pi$
$821$	−42674.2	−1.81406	−0.907028	+	0.421070i	$0.861655\pi$
$822$	0	0
$823$	−21127.7	−0.894856	−0.447428	−	0.894320i	$-0.647660\pi$
$823$	−21127.7	−0.894856	−0.447428	+	0.894320i	$0.647660\pi$
$824$	192.369	0.00813288
$825$	0	0
$826$	9856.31	0.415188
$827$	−3011.35	−0.126620	−0.0633101	−	0.997994i	$-0.520166\pi$
$827$	−3011.35	−0.126620	−0.0633101	+	0.997994i	$0.520166\pi$
$828$	0	0
$829$	−5221.80	−0.218770	−0.109385	−	0.993999i	$-0.534888\pi$
$829$	−5221.80	−0.218770	−0.109385	+	0.993999i	$0.534888\pi$
$830$	−12165.7	−0.508767
$831$	0	0
$832$	31163.4	1.29855
$833$	5833.11	0.242623
$834$	0	0
$835$	−6691.50	−0.277328
$836$	−1809.75	−0.0748703
$837$	0	0
$838$	−63336.4	−2.61088
$839$	−46130.3	−1.89821	−0.949104	−	0.314963i	$-0.898008\pi$
$839$	−46130.3	−1.89821	−0.949104	+	0.314963i	$0.898008\pi$
$840$	0	0
$841$	−21497.7	−0.881449
$842$	−43267.1	−1.77088
$843$	0	0
$844$	−18615.7	−0.759218
$845$	−31448.7	−1.28032
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−17581.4	−0.711967
$849$	0	0
$850$	41779.8	1.68592
$851$	−3411.88	−0.137436
$852$	0	0
$853$	15030.9	0.603338	0.301669	−	0.953413i	$-0.402456\pi$
$853$	15030.9	0.603338	0.301669	+	0.953413i	$0.402456\pi$
$854$	−12330.0	−0.494055
$855$	0	0
$856$	1971.83	0.0787332
$857$	33506.6	1.33555	0.667773	−	0.744365i	$-0.267248\pi$
$857$	33506.6	1.33555	0.667773	+	0.744365i	$0.267248\pi$
$858$	0	0
$859$	18085.0	0.718338	0.359169	−	0.933273i	$-0.383060\pi$
$859$	18085.0	0.718338	0.359169	+	0.933273i	$0.383060\pi$
$860$	1601.82	0.0635135
$861$	0	0
$862$	−64714.9	−2.55707
$863$	−23344.1	−0.920791	−0.460396	−	0.887714i	$-0.652292\pi$
$863$	−23344.1	−0.920791	−0.460396	+	0.887714i	$0.652292\pi$
$864$	0	0
$865$	−15187.0	−0.596963
$866$	6242.91	0.244969
$867$	0	0
$868$	3316.04	0.129670
$869$	−7782.10	−0.303785
$870$	0	0
$871$	40978.9	1.59416
$872$	−5765.66	−0.223910
$873$	0	0
$874$	11016.4	0.426356
$875$	−8813.35	−0.340509
$876$	0	0
$877$	−16403.0	−0.631573	−0.315786	−	0.948830i	$-0.602268\pi$
$877$	−16403.0	−0.631573	−0.315786	+	0.948830i	$0.602268\pi$
$878$	2242.30	0.0861890
$879$	0	0
$880$	−4603.07	−0.176329
$881$	−19795.6	−0.757017	−0.378509	−	0.925598i	$-0.623563\pi$
$881$	−19795.6	−0.757017	−0.378509	+	0.925598i	$0.623563\pi$
$882$	0	0
$883$	−26755.7	−1.01971	−0.509854	−	0.860261i	$-0.670301\pi$
$883$	−26755.7	−1.01971	−0.509854	+	0.860261i	$0.670301\pi$
$884$	−71194.4	−2.70874
$885$	0	0
$886$	4517.80	0.171307
$887$	26296.7	0.995442	0.497721	−	0.867337i	$-0.334170\pi$
$887$	26296.7	0.995442	0.497721	+	0.867337i	$0.334170\pi$
$888$	0	0
$889$	−17083.0	−0.644484
$890$	12577.9	0.473721
$891$	0	0
$892$	−29748.3	−1.11664
$893$	−1777.52	−0.0666096
$894$	0	0
$895$	19880.4	0.742489
$896$	3887.02	0.144929
$897$	0	0
$898$	−44060.4	−1.63732
$899$	−3711.42	−0.137690
$900$	0	0
$901$	29149.0	1.07780
$902$	−17777.0	−0.656220
$903$	0	0
$904$	−3177.86	−0.116918
$905$	3735.92	0.137222
$906$	0	0
$907$	−41082.5	−1.50399	−0.751996	−	0.659167i	$-0.770909\pi$
$907$	−41082.5	−1.50399	−0.751996	+	0.659167i	$0.770909\pi$
$908$	21551.5	0.787679
$909$	0	0
$910$	13705.2	0.499257
$911$	−34300.4	−1.24745	−0.623723	−	0.781645i	$-0.714381\pi$
$911$	−34300.4	−1.24745	−0.623723	+	0.781645i	$0.714381\pi$
$912$	0	0
$913$	5955.95	0.215896
$914$	41031.7	1.48491
$915$	0	0
$916$	34941.4	1.26037
$917$	−8808.31	−0.317204
$918$	0	0
$919$	40816.1	1.46507	0.732535	−	0.680730i	$-0.238337\pi$
$919$	40816.1	1.46507	0.732535	+	0.680730i	$0.238337\pi$
$920$	3044.55	0.109104
$921$	0	0
$922$	11223.4	0.400892
$923$	31365.9	1.11855
$924$	0	0
$925$	−2605.62	−0.0926188
$926$	58627.0	2.08057
$927$	0	0
$928$	12999.5	0.459839
$929$	−11293.8	−0.398857	−0.199429	−	0.979912i	$-0.563909\pi$
$929$	−11293.8	−0.398857	−0.199429	+	0.979912i	$0.563909\pi$
$930$	0	0
$931$	1174.60	0.0413492
$932$	−12989.1	−0.456514
$933$	0	0
$934$	−54094.5	−1.89510
$935$	7631.63	0.266931
$936$	0	0
$937$	27145.4	0.946428	0.473214	−	0.880948i	$-0.343094\pi$
$937$	27145.4	0.946428	0.473214	+	0.880948i	$0.343094\pi$
$938$	−12691.3	−0.441774
$939$	0	0
$940$	2965.99	0.102915
$941$	17834.4	0.617838	0.308919	−	0.951088i	$-0.400033\pi$
$941$	17834.4	0.617838	0.308919	+	0.951088i	$0.400033\pi$
$942$	0	0
$943$	49968.5	1.72555
$944$	−26223.7	−0.904141
$945$	0	0
$946$	−1698.29	−0.0583681
$947$	36957.3	1.26817	0.634083	−	0.773265i	$-0.281378\pi$
$947$	36957.3	1.26817	0.634083	+	0.773265i	$0.281378\pi$
$948$	0	0
$949$	78631.7	2.68967
$950$	8413.13	0.287324
$951$	0	0
$952$	−3651.89	−0.124326
$953$	−42725.6	−1.45227	−0.726137	−	0.687550i	$-0.758686\pi$
$953$	−42725.6	−1.45227	−0.726137	+	0.687550i	$0.758686\pi$
$954$	0	0
$955$	−17527.8	−0.593913
$956$	−35275.6	−1.19340
$957$	0	0
$958$	56469.3	1.90443
$959$	−19530.6	−0.657639
$960$	0	0
$961$	−25026.9	−0.840082
$962$	9615.55	0.322264
$963$	0	0
$964$	−13030.8	−0.435368
$965$	−1719.07	−0.0573460
$966$	0	0
$967$	−18274.2	−0.607713	−0.303856	−	0.952718i	$-0.598274\pi$
$967$	−18274.2	−0.607713	−0.303856	+	0.952718i	$0.598274\pi$
$968$	530.275	0.0176071
$969$	0	0
$970$	34259.4	1.13402
$971$	39923.1	1.31946	0.659729	−	0.751503i	$-0.270671\pi$
$971$	39923.1	1.31946	0.659729	+	0.751503i	$0.270671\pi$
$972$	0	0
$973$	−3777.01	−0.124445
$974$	45761.5	1.50543
$975$	0	0
$976$	32805.1	1.07589
$977$	−26746.2	−0.875832	−0.437916	−	0.899016i	$-0.644283\pi$
$977$	−26746.2	−0.875832	−0.437916	+	0.899016i	$0.644283\pi$
$978$	0	0
$979$	−6157.77	−0.201025
$980$	−1959.96	−0.0638863
$981$	0	0
$982$	3700.36	0.120248
$983$	−38975.0	−1.26461	−0.632303	−	0.774721i	$-0.717890\pi$
$983$	−38975.0	−1.26461	−0.632303	+	0.774721i	$0.717890\pi$
$984$	0	0
$985$	13403.5	0.433574
$986$	−24678.0	−0.797068
$987$	0	0
$988$	−14336.3	−0.461638
$989$	4773.64	0.153481
$990$	0	0
$991$	59856.3	1.91866	0.959332	−	0.282279i	$-0.0910903\pi$
$991$	59856.3	1.91866	0.959332	+	0.282279i	$0.0910903\pi$
$992$	−16686.7	−0.534076
$993$	0	0
$994$	−9714.08	−0.309972
$995$	13842.6	0.441047
$996$	0	0
$997$	42609.3	1.35351	0.676755	−	0.736208i	$-0.263386\pi$
$997$	42609.3	1.35351	0.676755	+	0.736208i	$0.263386\pi$
$998$	−69961.8	−2.21904
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.t.1.2	yes	8
3.2	odd	2		693.4.a.s.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.s.1.7	✓	8	3.2	odd	2
693.4.a.t.1.2	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)

\(f(q)\)	\(=\)	\(q-3.85529 q^{2} +6.86327 q^{4} -5.82801 q^{5} -7.00000 q^{7} +4.38244 q^{8} +22.4687 q^{10} -11.0000 q^{11} -87.1386 q^{13} +26.9870 q^{14} -71.8017 q^{16} +119.043 q^{17} +23.9715 q^{19} -39.9992 q^{20} +42.4082 q^{22} -119.203 q^{23} -91.0343 q^{25} +335.945 q^{26} -48.0429 q^{28} +53.7711 q^{29} -69.0226 q^{31} +241.757 q^{32} -458.946 q^{34} +40.7961 q^{35} +28.6224 q^{37} -92.4171 q^{38} -25.5409 q^{40} -419.188 q^{41} -40.0463 q^{43} -75.4959 q^{44} +459.562 q^{46} -74.1512 q^{47} +49.0000 q^{49} +350.964 q^{50} -598.055 q^{52} +244.861 q^{53} +64.1081 q^{55} -30.6770 q^{56} -207.303 q^{58} +365.224 q^{59} -456.885 q^{61} +266.102 q^{62} -357.630 q^{64} +507.844 q^{65} -470.272 q^{67} +817.025 q^{68} -157.281 q^{70} -359.954 q^{71} -902.375 q^{73} -110.348 q^{74} +164.523 q^{76} +77.0000 q^{77} +707.463 q^{79} +418.461 q^{80} +1616.09 q^{82} -541.450 q^{83} -693.784 q^{85} +154.390 q^{86} -48.2068 q^{88} +559.797 q^{89} +609.970 q^{91} -818.122 q^{92} +285.875 q^{94} -139.706 q^{95} +1524.76 q^{97} -188.909 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 30 q^{4} + 10 q^{5} - 56 q^{7} + 15 q^{8} - 13 q^{10} - 88 q^{11} - 148 q^{13} - 14 q^{14} + 266 q^{16} + 114 q^{17} + 58 q^{19} + 291 q^{20} - 22 q^{22} + 246 q^{23} + 244 q^{25} + 305 q^{26}+ \cdots + 98 q^{98}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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