LMFDB - Embedded newform 693.4.a.q.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 $ x^5 - x^4 - 30x^3 + 21x^2 + 103*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.48454$ 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)

$f(q)$	$=$	$q+2.48454 q^{2} -1.82705 q^{4} -13.9229 q^{5} -7.00000 q^{7} -24.4157 q^{8} +O(q^{10})$	\(q+2.48454 q^{2} -1.82705 q^{4} -13.9229 q^{5} -7.00000 q^{7} -24.4157 q^{8} -34.5919 q^{10} +11.0000 q^{11} -53.9968 q^{13} -17.3918 q^{14} -46.0455 q^{16} +21.8891 q^{17} +73.6854 q^{19} +25.4377 q^{20} +27.3300 q^{22} +16.7425 q^{23} +68.8459 q^{25} -134.157 q^{26} +12.7893 q^{28} -153.954 q^{29} +224.363 q^{31} +80.9237 q^{32} +54.3845 q^{34} +97.4600 q^{35} +369.604 q^{37} +183.074 q^{38} +339.937 q^{40} +61.3701 q^{41} +37.2526 q^{43} -20.0975 q^{44} +41.5974 q^{46} -279.867 q^{47} +49.0000 q^{49} +171.051 q^{50} +98.6548 q^{52} -235.547 q^{53} -153.151 q^{55} +170.910 q^{56} -382.506 q^{58} -19.1584 q^{59} +713.262 q^{61} +557.439 q^{62} +569.423 q^{64} +751.790 q^{65} -198.118 q^{67} -39.9925 q^{68} +242.144 q^{70} +76.3922 q^{71} +984.250 q^{73} +918.297 q^{74} -134.627 q^{76} -77.0000 q^{77} -697.458 q^{79} +641.085 q^{80} +152.477 q^{82} +388.757 q^{83} -304.759 q^{85} +92.5557 q^{86} -268.573 q^{88} -33.0992 q^{89} +377.978 q^{91} -30.5893 q^{92} -695.341 q^{94} -1025.91 q^{95} +1384.34 q^{97} +121.743 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8}+O(q^{10}) $ 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8	$ 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26} - 147 q^{28} - 591 q^{29} + 366 q^{31} + 87 q^{32} + 594 q^{34} + 49 q^{35} + 259 q^{37} - 175 q^{38} + 1890 q^{40} + 104 q^{41} + 224 q^{43} + 231 q^{44} + 1416 q^{46} + 453 q^{47} + 245 q^{49} - 1350 q^{50} + 815 q^{52} - 1032 q^{53} - 77 q^{55} - 84 q^{56} + 293 q^{58} + 517 q^{59} + 958 q^{61} + 3030 q^{62} + 516 q^{64} + 197 q^{65} + 361 q^{67} + 1680 q^{68} - 791 q^{70} - 548 q^{71} - 1035 q^{73} + 2555 q^{74} - 5201 q^{76} - 385 q^{77} + 776 q^{79} + 1273 q^{80} + 376 q^{82} + 1974 q^{83} + 1838 q^{85} - 1224 q^{86} + 132 q^{88} - 2710 q^{89} - 161 q^{91} + 2834 q^{92} - 563 q^{94} + 355 q^{95} + 1988 q^{97} + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * 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 $

$2$	2.48454	0.878418	0.439209	−	0.898385i	$-0.355259\pi$
$2$	2.48454	0.878418	0.439209	+	0.898385i	$0.355259\pi$
$3$	0	0
$3$	0	0
$4$	−1.82705	−0.228381
$5$	−13.9229	−1.24530	−0.622649	−	0.782501i	$-0.713944\pi$
$5$	−13.9229	−1.24530	−0.622649	+	0.782501i	$0.713944\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−24.4157	−1.07903
$9$	0	0
$10$	−34.5919	−1.09389
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−53.9968	−1.15200	−0.576001	−	0.817449i	$-0.695388\pi$
$13$	−53.9968	−1.15200	−0.576001	+	0.817449i	$0.695388\pi$
$14$	−17.3918	−0.332011
$15$	0	0
$16$	−46.0455	−0.719461
$17$	21.8891	0.312288	0.156144	−	0.987734i	$-0.450094\pi$
$17$	21.8891	0.312288	0.156144	+	0.987734i	$0.450094\pi$
$18$	0	0
$19$	73.6854	0.889715	0.444858	−	0.895601i	$-0.353254\pi$
$19$	73.6854	0.889715	0.444858	+	0.895601i	$0.353254\pi$
$20$	25.4377	0.284402
$21$	0	0
$22$	27.3300	0.264853
$23$	16.7425	0.151785	0.0758923	−	0.997116i	$-0.475819\pi$
$23$	16.7425	0.151785	0.0758923	+	0.997116i	$0.475819\pi$
$24$	0	0
$25$	68.8459	0.550768
$26$	−134.157	−1.01194
$27$	0	0
$28$	12.7893	0.0863199
$29$	−153.954	−0.985813	−0.492906	−	0.870082i	$-0.664066\pi$
$29$	−153.954	−0.985813	−0.492906	+	0.870082i	$0.664066\pi$
$30$	0	0
$31$	224.363	1.29990	0.649948	−	0.759979i	$-0.274791\pi$
$31$	224.363	1.29990	0.649948	+	0.759979i	$0.274791\pi$
$32$	80.9237	0.447045
$33$	0	0
$34$	54.3845	0.274320
$35$	97.4600	0.470678
$36$	0	0
$37$	369.604	1.64223	0.821115	−	0.570763i	$-0.193352\pi$
$37$	369.604	1.64223	0.821115	+	0.570763i	$0.193352\pi$
$38$	183.074	0.781542
$39$	0	0
$40$	339.937	1.34372
$41$	61.3701	0.233766	0.116883	−	0.993146i	$-0.462710\pi$
$41$	61.3701	0.233766	0.116883	+	0.993146i	$0.462710\pi$
$42$	0	0
$43$	37.2526	0.132116	0.0660578	−	0.997816i	$-0.478958\pi$
$43$	37.2526	0.132116	0.0660578	+	0.997816i	$0.478958\pi$
$44$	−20.0975	−0.0688595
$45$	0	0
$46$	41.5974	0.133330
$47$	−279.867	−0.868570	−0.434285	−	0.900776i	$-0.642999\pi$
$47$	−279.867	−0.868570	−0.434285	+	0.900776i	$0.642999\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	171.051	0.483804
$51$	0	0
$52$	98.6548	0.263095
$53$	−235.547	−0.610470	−0.305235	−	0.952277i	$-0.598735\pi$
$53$	−235.547	−0.610470	−0.305235	+	0.952277i	$0.598735\pi$
$54$	0	0
$55$	−153.151	−0.375472
$56$	170.910	0.407836
$57$	0	0
$58$	−382.506	−0.865956
$59$	−19.1584	−0.0422748	−0.0211374	−	0.999777i	$-0.506729\pi$
$59$	−19.1584	−0.0422748	−0.0211374	+	0.999777i	$0.506729\pi$
$60$	0	0
$61$	713.262	1.49711	0.748556	−	0.663071i	$-0.230747\pi$
$61$	713.262	1.49711	0.748556	+	0.663071i	$0.230747\pi$
$62$	557.439	1.14185
$63$	0	0
$64$	569.423	1.11215
$65$	751.790	1.43459
$66$	0	0
$67$	−198.118	−0.361254	−0.180627	−	0.983552i	$-0.557813\pi$
$67$	−198.118	−0.361254	−0.180627	+	0.983552i	$0.557813\pi$
$68$	−39.9925	−0.0713207
$69$	0	0
$70$	242.144	0.413453
$71$	76.3922	0.127691	0.0638457	−	0.997960i	$-0.479663\pi$
$71$	76.3922	0.127691	0.0638457	+	0.997960i	$0.479663\pi$
$72$	0	0
$73$	984.250	1.57805	0.789026	−	0.614360i	$-0.210586\pi$
$73$	984.250	1.57805	0.789026	+	0.614360i	$0.210586\pi$
$74$	918.297	1.44257
$75$	0	0
$76$	−134.627	−0.203194
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−697.458	−0.993292	−0.496646	−	0.867953i	$-0.665435\pi$
$79$	−697.458	−0.993292	−0.496646	+	0.867953i	$0.665435\pi$
$80$	641.085	0.895944
$81$	0	0
$82$	152.477	0.205344
$83$	388.757	0.514116	0.257058	−	0.966396i	$-0.417247\pi$
$83$	388.757	0.514116	0.257058	+	0.966396i	$0.417247\pi$
$84$	0	0
$85$	−304.759	−0.388892
$86$	92.5557	0.116053
$87$	0	0
$88$	−268.573	−0.325341
$89$	−33.0992	−0.0394214	−0.0197107	−	0.999806i	$-0.506275\pi$
$89$	−33.0992	−0.0394214	−0.0197107	+	0.999806i	$0.506275\pi$
$90$	0	0
$91$	377.978	0.435416
$92$	−30.5893	−0.0346647
$93$	0	0
$94$	−695.341	−0.762968
$95$	−1025.91	−1.10796
$96$	0	0
$97$	1384.34	1.44906	0.724530	−	0.689243i	$-0.242057\pi$
$97$	1384.34	1.44906	0.724530	+	0.689243i	$0.242057\pi$
$98$	121.743	0.125488
$99$	0	0
$100$	−125.785	−0.125785
$101$	−1061.43	−1.04571	−0.522853	−	0.852423i	$-0.675133\pi$
$101$	−1061.43	−1.04571	−0.522853	+	0.852423i	$0.675133\pi$
$102$	0	0
$103$	−867.093	−0.829488	−0.414744	−	0.909938i	$-0.636129\pi$
$103$	−867.093	−0.829488	−0.414744	+	0.909938i	$0.636129\pi$
$104$	1318.37	1.24305
$105$	0	0
$106$	−585.228	−0.536248
$107$	826.383	0.746630	0.373315	−	0.927705i	$-0.378221\pi$
$107$	826.383	0.746630	0.373315	+	0.927705i	$0.378221\pi$
$108$	0	0
$109$	1021.07	0.897253	0.448626	−	0.893719i	$-0.351913\pi$
$109$	1021.07	0.897253	0.448626	+	0.893719i	$0.351913\pi$
$110$	−380.511	−0.329821
$111$	0	0
$112$	322.319	0.271931
$113$	−1174.94	−0.978131	−0.489066	−	0.872247i	$-0.662662\pi$
$113$	−1174.94	−0.978131	−0.489066	+	0.872247i	$0.662662\pi$
$114$	0	0
$115$	−233.103	−0.189017
$116$	281.282	0.225141
$117$	0	0
$118$	−47.5999	−0.0371350
$119$	−153.224	−0.118034
$120$	0	0
$121$	121.000	0.0909091
$122$	1772.13	1.31509
$123$	0	0
$124$	−409.922	−0.296871
$125$	781.825	0.559428
$126$	0	0
$127$	−1756.11	−1.22700	−0.613502	−	0.789693i	$-0.710240\pi$
$127$	−1756.11	−1.22700	−0.613502	+	0.789693i	$0.710240\pi$
$128$	767.365	0.529891
$129$	0	0
$130$	1867.85	1.26017
$131$	1306.21	0.871174	0.435587	−	0.900147i	$-0.356541\pi$
$131$	1306.21	0.871174	0.435587	+	0.900147i	$0.356541\pi$
$132$	0	0
$133$	−515.798	−0.336281
$134$	−492.234	−0.317332
$135$	0	0
$136$	−534.439	−0.336969
$137$	−756.981	−0.472067	−0.236034	−	0.971745i	$-0.575848\pi$
$137$	−756.981	−0.472067	−0.236034	+	0.971745i	$0.575848\pi$
$138$	0	0
$139$	−1958.12	−1.19486	−0.597431	−	0.801920i	$-0.703812\pi$
$139$	−1958.12	−1.19486	−0.597431	+	0.801920i	$0.703812\pi$
$140$	−178.064	−0.107494
$141$	0	0
$142$	189.800	0.112166
$143$	−593.965	−0.347342
$144$	0	0
$145$	2143.48	1.22763
$146$	2445.41	1.38619
$147$	0	0
$148$	−675.284	−0.375054
$149$	−1442.32	−0.793016	−0.396508	−	0.918031i	$-0.629778\pi$
$149$	−1442.32	−0.793016	−0.396508	+	0.918031i	$0.629778\pi$
$150$	0	0
$151$	2626.40	1.41545	0.707727	−	0.706486i	$-0.249721\pi$
$151$	2626.40	1.41545	0.707727	+	0.706486i	$0.249721\pi$
$152$	−1799.08	−0.960031
$153$	0	0
$154$	−191.310	−0.100105
$155$	−3123.77	−1.61876
$156$	0	0
$157$	−1812.72	−0.921471	−0.460735	−	0.887538i	$-0.652414\pi$
$157$	−1812.72	−0.921471	−0.460735	+	0.887538i	$0.652414\pi$
$158$	−1732.86	−0.872526
$159$	0	0
$160$	−1126.69	−0.556704
$161$	−117.197	−0.0573692
$162$	0	0
$163$	3552.18	1.70692	0.853460	−	0.521158i	$-0.174500\pi$
$163$	3552.18	1.70692	0.853460	+	0.521158i	$0.174500\pi$
$164$	−112.126	−0.0533877
$165$	0	0
$166$	965.883	0.451609
$167$	3217.80	1.49102	0.745511	−	0.666493i	$-0.232205\pi$
$167$	3217.80	1.49102	0.745511	+	0.666493i	$0.232205\pi$
$168$	0	0
$169$	718.657	0.327108
$170$	−757.188	−0.341610
$171$	0	0
$172$	−68.0623	−0.0301727
$173$	−2512.65	−1.10424	−0.552120	−	0.833765i	$-0.686181\pi$
$173$	−2512.65	−1.10424	−0.552120	+	0.833765i	$0.686181\pi$
$174$	0	0
$175$	−481.922	−0.208171
$176$	−506.501	−0.216926
$177$	0	0
$178$	−82.2363	−0.0346285
$179$	−4436.03	−1.85231	−0.926157	−	0.377139i	$-0.876908\pi$
$179$	−4436.03	−1.85231	−0.926157	+	0.377139i	$0.876908\pi$
$180$	0	0
$181$	4431.76	1.81994	0.909972	−	0.414670i	$-0.136103\pi$
$181$	4431.76	1.81994	0.909972	+	0.414670i	$0.136103\pi$
$182$	939.102	0.382477
$183$	0	0
$184$	−408.779	−0.163780
$185$	−5145.94	−2.04507
$186$	0	0
$187$	240.781	0.0941584
$188$	511.330	0.198365
$189$	0	0
$190$	−2548.92	−0.973253
$191$	−1612.21	−0.610760	−0.305380	−	0.952231i	$-0.598783\pi$
$191$	−1612.21	−0.610760	−0.305380	+	0.952231i	$0.598783\pi$
$192$	0	0
$193$	930.282	0.346960	0.173480	−	0.984837i	$-0.444499\pi$
$193$	930.282	0.346960	0.173480	+	0.984837i	$0.444499\pi$
$194$	3439.46	1.27288
$195$	0	0
$196$	−89.5253	−0.0326259
$197$	4707.36	1.70246	0.851231	−	0.524791i	$-0.175856\pi$
$197$	4707.36	1.70246	0.851231	+	0.524791i	$0.175856\pi$
$198$	0	0
$199$	−143.956	−0.0512801	−0.0256400	−	0.999671i	$-0.508162\pi$
$199$	−143.956	−0.0512801	−0.0256400	+	0.999671i	$0.508162\pi$
$200$	−1680.92	−0.594296
$201$	0	0
$202$	−2637.17	−0.918568
$203$	1077.68	0.372602
$204$	0	0
$205$	−854.448	−0.291108
$206$	−2154.33	−0.728637
$207$	0	0
$208$	2486.31	0.828821
$209$	810.539	0.268259
$210$	0	0
$211$	−451.368	−0.147268	−0.0736338	−	0.997285i	$-0.523460\pi$
$211$	−451.368	−0.147268	−0.0736338	+	0.997285i	$0.523460\pi$
$212$	430.356	0.139420
$213$	0	0
$214$	2053.18	0.655854
$215$	−518.663	−0.164523
$216$	0	0
$217$	−1570.54	−0.491314
$218$	2536.89	0.788164
$219$	0	0
$220$	279.815	0.0857506
$221$	−1181.94	−0.359756
$222$	0	0
$223$	377.140	0.113252	0.0566260	−	0.998395i	$-0.481966\pi$
$223$	377.140	0.113252	0.0566260	+	0.998395i	$0.481966\pi$
$224$	−566.466	−0.168967
$225$	0	0
$226$	−2919.18	−0.859209
$227$	5441.73	1.59110	0.795551	−	0.605886i	$-0.207181\pi$
$227$	5441.73	1.59110	0.795551	+	0.605886i	$0.207181\pi$
$228$	0	0
$229$	2191.27	0.632328	0.316164	−	0.948705i	$-0.397605\pi$
$229$	2191.27	0.632328	0.316164	+	0.948705i	$0.397605\pi$
$230$	−579.154	−0.166036
$231$	0	0
$232$	3758.90	1.06372
$233$	2379.94	0.669163	0.334582	−	0.942367i	$-0.391405\pi$
$233$	2379.94	0.669163	0.334582	+	0.942367i	$0.391405\pi$
$234$	0	0
$235$	3896.55	1.08163
$236$	35.0033	0.00965476
$237$	0	0
$238$	−380.692	−0.103683
$239$	−4504.56	−1.21914	−0.609572	−	0.792731i	$-0.708659\pi$
$239$	−4504.56	−1.21914	−0.609572	+	0.792731i	$0.708659\pi$
$240$	0	0
$241$	−5414.93	−1.44733	−0.723665	−	0.690152i	$-0.757544\pi$
$241$	−5414.93	−1.44733	−0.723665	+	0.690152i	$0.757544\pi$
$242$	300.630	0.0798562
$243$	0	0
$244$	−1303.16	−0.341912
$245$	−682.220	−0.177900
$246$	0	0
$247$	−3978.78	−1.02495
$248$	−5477.98	−1.40263
$249$	0	0
$250$	1942.48	0.491412
$251$	1594.94	0.401083	0.200541	−	0.979685i	$-0.435730\pi$
$251$	1594.94	0.401083	0.200541	+	0.979685i	$0.435730\pi$
$252$	0	0
$253$	184.167	0.0457648
$254$	−4363.13	−1.07782
$255$	0	0
$256$	−2648.83	−0.646687
$257$	6585.39	1.59839	0.799194	−	0.601073i	$-0.205260\pi$
$257$	6585.39	1.59839	0.799194	+	0.601073i	$0.205260\pi$
$258$	0	0
$259$	−2587.23	−0.620705
$260$	−1373.56	−0.327632
$261$	0	0
$262$	3245.33	0.765256
$263$	5615.25	1.31654	0.658272	−	0.752780i	$-0.271288\pi$
$263$	5615.25	1.31654	0.658272	+	0.752780i	$0.271288\pi$
$264$	0	0
$265$	3279.49	0.760218
$266$	−1281.52	−0.295395
$267$	0	0
$268$	361.972	0.0825035
$269$	5443.31	1.23377	0.616886	−	0.787053i	$-0.288394\pi$
$269$	5443.31	1.23377	0.616886	+	0.787053i	$0.288394\pi$
$270$	0	0
$271$	−370.300	−0.0830041	−0.0415020	−	0.999138i	$-0.513214\pi$
$271$	−370.300	−0.0830041	−0.0415020	+	0.999138i	$0.513214\pi$
$272$	−1007.90	−0.224679
$273$	0	0
$274$	−1880.75	−0.414673
$275$	757.305	0.166063
$276$	0	0
$277$	1308.68	0.283867	0.141933	−	0.989876i	$-0.454668\pi$
$277$	1308.68	0.283867	0.141933	+	0.989876i	$0.454668\pi$
$278$	−4865.04	−1.04959
$279$	0	0
$280$	−2379.56	−0.507877
$281$	2680.45	0.569046	0.284523	−	0.958669i	$-0.408165\pi$
$281$	2680.45	0.569046	0.284523	+	0.958669i	$0.408165\pi$
$282$	0	0
$283$	−6780.63	−1.42426	−0.712132	−	0.702046i	$-0.752270\pi$
$283$	−6780.63	−1.42426	−0.712132	+	0.702046i	$0.752270\pi$
$284$	−139.572	−0.0291623
$285$	0	0
$286$	−1475.73	−0.305111
$287$	−429.591	−0.0883552
$288$	0	0
$289$	−4433.87	−0.902476
$290$	5325.57	1.07837
$291$	0	0
$292$	−1798.27	−0.360397
$293$	4596.85	0.916557	0.458278	−	0.888809i	$-0.348466\pi$
$293$	4596.85	0.916557	0.458278	+	0.888809i	$0.348466\pi$
$294$	0	0
$295$	266.740	0.0526447
$296$	−9024.15	−1.77202
$297$	0	0
$298$	−3583.50	−0.696600
$299$	−904.040	−0.174856
$300$	0	0
$301$	−260.768	−0.0499350
$302$	6525.41	1.24336
$303$	0	0
$304$	−3392.88	−0.640115
$305$	−9930.65	−1.86435
$306$	0	0
$307$	1875.22	0.348615	0.174307	−	0.984691i	$-0.444231\pi$
$307$	1875.22	0.348615	0.174307	+	0.984691i	$0.444231\pi$
$308$	140.683	0.0260264
$309$	0	0
$310$	−7761.15	−1.42195
$311$	−621.628	−0.113342	−0.0566709	−	0.998393i	$-0.518049\pi$
$311$	−621.628	−0.113342	−0.0566709	+	0.998393i	$0.518049\pi$
$312$	0	0
$313$	2587.60	0.467283	0.233642	−	0.972323i	$-0.424936\pi$
$313$	2587.60	0.467283	0.233642	+	0.972323i	$0.424936\pi$
$314$	−4503.78	−0.809437
$315$	0	0
$316$	1274.29	0.226849
$317$	−4330.43	−0.767260	−0.383630	−	0.923487i	$-0.625326\pi$
$317$	−4330.43	−0.767260	−0.383630	+	0.923487i	$0.625326\pi$
$318$	0	0
$319$	−1693.50	−0.297234
$320$	−7927.99	−1.38496
$321$	0	0
$322$	−291.182	−0.0503941
$323$	1612.91	0.277847
$324$	0	0
$325$	−3717.46	−0.634485
$326$	8825.54	1.49939
$327$	0	0
$328$	−1498.40	−0.252241
$329$	1959.07	0.328289
$330$	0	0
$331$	6117.61	1.01587	0.507937	−	0.861394i	$-0.330408\pi$
$331$	6117.61	1.01587	0.507937	+	0.861394i	$0.330408\pi$
$332$	−710.278	−0.117414
$333$	0	0
$334$	7994.76	1.30974
$335$	2758.37	0.449869
$336$	0	0
$337$	−1504.23	−0.243148	−0.121574	−	0.992582i	$-0.538794\pi$
$337$	−1504.23	−0.243148	−0.121574	+	0.992582i	$0.538794\pi$
$338$	1785.53	0.287338
$339$	0	0
$340$	556.810	0.0888155
$341$	2467.99	0.391933
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−909.550	−0.142557
$345$	0	0
$346$	−6242.79	−0.969985
$347$	3421.45	0.529317	0.264658	−	0.964342i	$-0.414741\pi$
$347$	3421.45	0.529317	0.264658	+	0.964342i	$0.414741\pi$
$348$	0	0
$349$	2614.76	0.401045	0.200523	−	0.979689i	$-0.435736\pi$
$349$	2614.76	0.401045	0.200523	+	0.979689i	$0.435736\pi$
$350$	−1197.35	−0.182861
$351$	0	0
$352$	890.161	0.134789
$353$	11775.3	1.77546	0.887728	−	0.460369i	$-0.152283\pi$
$353$	11775.3	1.77546	0.887728	+	0.460369i	$0.152283\pi$
$354$	0	0
$355$	−1063.60	−0.159014
$356$	60.4738	0.00900310
$357$	0	0
$358$	−11021.5	−1.62711
$359$	−12034.1	−1.76918	−0.884590	−	0.466369i	$-0.845562\pi$
$359$	−12034.1	−1.76918	−0.884590	+	0.466369i	$0.845562\pi$
$360$	0	0
$361$	−1429.46	−0.208407
$362$	11010.9	1.59867
$363$	0	0
$364$	−690.583	−0.0994407
$365$	−13703.6	−1.96514
$366$	0	0
$367$	5824.07	0.828376	0.414188	−	0.910191i	$-0.364066\pi$
$367$	5824.07	0.828376	0.414188	+	0.910191i	$0.364066\pi$
$368$	−770.915	−0.109203
$369$	0	0
$370$	−12785.3	−1.79642
$371$	1648.83	0.230736
$372$	0	0
$373$	−3814.65	−0.529531	−0.264765	−	0.964313i	$-0.585295\pi$
$373$	−3814.65	−0.529531	−0.264765	+	0.964313i	$0.585295\pi$
$374$	598.230	0.0827105
$375$	0	0
$376$	6833.15	0.937215
$377$	8313.04	1.13566
$378$	0	0
$379$	8307.70	1.12596	0.562979	−	0.826471i	$-0.309655\pi$
$379$	8307.70	1.12596	0.562979	+	0.826471i	$0.309655\pi$
$380$	1874.39	0.253037
$381$	0	0
$382$	−4005.59	−0.536503
$383$	−5360.45	−0.715159	−0.357580	−	0.933883i	$-0.616398\pi$
$383$	−5360.45	−0.715159	−0.357580	+	0.933883i	$0.616398\pi$
$384$	0	0
$385$	1072.06	0.141915
$386$	2311.33	0.304776
$387$	0	0
$388$	−2529.26	−0.330938
$389$	−4791.44	−0.624513	−0.312256	−	0.949998i	$-0.601085\pi$
$389$	−4791.44	−0.624513	−0.312256	+	0.949998i	$0.601085\pi$
$390$	0	0
$391$	366.478	0.0474005
$392$	−1196.37	−0.154148
$393$	0	0
$394$	11695.6	1.49547
$395$	9710.60	1.23695
$396$	0	0
$397$	−2754.58	−0.348233	−0.174117	−	0.984725i	$-0.555707\pi$
$397$	−2754.58	−0.348233	−0.174117	+	0.984725i	$0.555707\pi$
$398$	−357.664	−0.0450454
$399$	0	0
$400$	−3170.05	−0.396256
$401$	−8180.69	−1.01876	−0.509382	−	0.860540i	$-0.670126\pi$
$401$	−8180.69	−1.01876	−0.509382	+	0.860540i	$0.670126\pi$
$402$	0	0
$403$	−12114.9	−1.49748
$404$	1939.29	0.238820
$405$	0	0
$406$	2677.54	0.327301
$407$	4065.64	0.495151
$408$	0	0
$409$	6869.60	0.830514	0.415257	−	0.909704i	$-0.363692\pi$
$409$	6869.60	0.830514	0.415257	+	0.909704i	$0.363692\pi$
$410$	−2122.91	−0.255715
$411$	0	0
$412$	1584.22	0.189439
$413$	134.109	0.0159784
$414$	0	0
$415$	−5412.61	−0.640228
$416$	−4369.62	−0.514996
$417$	0	0
$418$	2013.82	0.235644
$419$	13045.4	1.52102	0.760511	−	0.649325i	$-0.224949\pi$
$419$	13045.4	1.52102	0.760511	+	0.649325i	$0.224949\pi$
$420$	0	0
$421$	805.294	0.0932248	0.0466124	−	0.998913i	$-0.485157\pi$
$421$	805.294	0.0932248	0.0466124	+	0.998913i	$0.485157\pi$
$422$	−1121.44	−0.129363
$423$	0	0
$424$	5751.06	0.658717
$425$	1506.98	0.171998
$426$	0	0
$427$	−4992.84	−0.565855
$428$	−1509.84	−0.170516
$429$	0	0
$430$	−1288.64	−0.144520
$431$	−12020.6	−1.34341	−0.671707	−	0.740817i	$-0.734439\pi$
$431$	−12020.6	−1.34341	−0.671707	+	0.740817i	$0.734439\pi$
$432$	0	0
$433$	17240.9	1.91350	0.956749	−	0.290914i	$-0.0939592\pi$
$433$	17240.9	1.91350	0.956749	+	0.290914i	$0.0939592\pi$
$434$	−3902.07	−0.431580
$435$	0	0
$436$	−1865.54	−0.204915
$437$	1233.67	0.135045
$438$	0	0
$439$	−14280.0	−1.55250	−0.776250	−	0.630425i	$-0.782881\pi$
$439$	−14280.0	−1.55250	−0.776250	+	0.630425i	$0.782881\pi$
$440$	3739.30	0.405146
$441$	0	0
$442$	−2936.59	−0.316017
$443$	−11832.6	−1.26904	−0.634518	−	0.772908i	$-0.718801\pi$
$443$	−11832.6	−1.26904	−0.634518	+	0.772908i	$0.718801\pi$
$444$	0	0
$445$	460.835	0.0490914
$446$	937.021	0.0994826
$447$	0	0
$448$	−3985.96	−0.420354
$449$	11061.3	1.16261	0.581306	−	0.813685i	$-0.302542\pi$
$449$	11061.3	1.16261	0.581306	+	0.813685i	$0.302542\pi$
$450$	0	0
$451$	675.072	0.0704831
$452$	2146.67	0.223387
$453$	0	0
$454$	13520.2	1.39765
$455$	−5262.53	−0.542222
$456$	0	0
$457$	−7359.11	−0.753270	−0.376635	−	0.926362i	$-0.622919\pi$
$457$	−7359.11	−0.753270	−0.376635	+	0.926362i	$0.622919\pi$
$458$	5444.30	0.555449
$459$	0	0
$460$	425.890	0.0431679
$461$	−335.036	−0.0338486	−0.0169243	−	0.999857i	$-0.505387\pi$
$461$	−335.036	−0.0338486	−0.0169243	+	0.999857i	$0.505387\pi$
$462$	0	0
$463$	18973.8	1.90451	0.952255	−	0.305305i	$-0.0987584\pi$
$463$	18973.8	1.90451	0.952255	+	0.305305i	$0.0987584\pi$
$464$	7088.90	0.709254
$465$	0	0
$466$	5913.06	0.587806
$467$	3687.35	0.365375	0.182687	−	0.983171i	$-0.441520\pi$
$467$	3687.35	0.365375	0.182687	+	0.983171i	$0.441520\pi$
$468$	0	0
$469$	1386.83	0.136541
$470$	9681.14	0.950123
$471$	0	0
$472$	467.766	0.0456159
$473$	409.779	0.0398344
$474$	0	0
$475$	5072.94	0.490026
$476$	279.948	0.0269567
$477$	0	0
$478$	−11191.8	−1.07092
$479$	340.873	0.0325154	0.0162577	−	0.999868i	$-0.494825\pi$
$479$	340.873	0.0325154	0.0162577	+	0.999868i	$0.494825\pi$
$480$	0	0
$481$	−19957.4	−1.89185
$482$	−13453.6	−1.27136
$483$	0	0
$484$	−221.073	−0.0207619
$485$	−19274.0	−1.80451
$486$	0	0
$487$	−8353.29	−0.777256	−0.388628	−	0.921395i	$-0.627051\pi$
$487$	−8353.29	−0.777256	−0.388628	+	0.921395i	$0.627051\pi$
$488$	−17414.8	−1.61543
$489$	0	0
$490$	−1695.00	−0.156270
$491$	1511.64	0.138940	0.0694700	−	0.997584i	$-0.477869\pi$
$491$	1511.64	0.138940	0.0694700	+	0.997584i	$0.477869\pi$
$492$	0	0
$493$	−3369.93	−0.307858
$494$	−9885.44	−0.900338
$495$	0	0
$496$	−10330.9	−0.935225
$497$	−534.746	−0.0482628
$498$	0	0
$499$	15011.3	1.34669	0.673343	−	0.739330i	$-0.264858\pi$
$499$	15011.3	1.34669	0.673343	+	0.739330i	$0.264858\pi$
$500$	−1428.43	−0.127763
$501$	0	0
$502$	3962.70	0.352318
$503$	7209.83	0.639106	0.319553	−	0.947568i	$-0.396467\pi$
$503$	7209.83	0.639106	0.319553	+	0.947568i	$0.396467\pi$
$504$	0	0
$505$	14778.2	1.30222
$506$	457.571	0.0402006
$507$	0	0
$508$	3208.50	0.280224
$509$	−11403.4	−0.993016	−0.496508	−	0.868032i	$-0.665385\pi$
$509$	−11403.4	−0.993016	−0.496508	+	0.868032i	$0.665385\pi$
$510$	0	0
$511$	−6889.75	−0.596447
$512$	−12720.0	−1.09795
$513$	0	0
$514$	16361.7	1.40405
$515$	12072.4	1.03296
$516$	0	0
$517$	−3078.54	−0.261884
$518$	−6428.08	−0.545238
$519$	0	0
$520$	−18355.5	−1.54796
$521$	−552.992	−0.0465010	−0.0232505	−	0.999730i	$-0.507402\pi$
$521$	−552.992	−0.0465010	−0.0232505	+	0.999730i	$0.507402\pi$
$522$	0	0
$523$	14141.9	1.18238	0.591188	−	0.806534i	$-0.298659\pi$
$523$	14141.9	1.18238	0.591188	+	0.806534i	$0.298659\pi$
$524$	−2386.50	−0.198960
$525$	0	0
$526$	13951.3	1.15648
$527$	4911.11	0.405942
$528$	0	0
$529$	−11886.7	−0.976961
$530$	8148.04	0.667789
$531$	0	0
$532$	942.387	0.0768001
$533$	−3313.79	−0.269299
$534$	0	0
$535$	−11505.6	−0.929777
$536$	4837.20	0.389805
$537$	0	0
$538$	13524.1	1.08377
$539$	539.000	0.0430730
$540$	0	0
$541$	24148.2	1.91906	0.959530	−	0.281605i	$-0.0908669\pi$
$541$	24148.2	1.91906	0.959530	+	0.281605i	$0.0908669\pi$
$542$	−920.026	−0.0729123
$543$	0	0
$544$	1771.35	0.139607
$545$	−14216.2	−1.11735
$546$	0	0
$547$	−2578.89	−0.201582	−0.100791	−	0.994908i	$-0.532137\pi$
$547$	−2578.89	−0.201582	−0.100791	+	0.994908i	$0.532137\pi$
$548$	1383.04	0.107811
$549$	0	0
$550$	1881.56	0.145873
$551$	−11344.2	−0.877093
$552$	0	0
$553$	4882.20	0.375429
$554$	3251.48	0.249354
$555$	0	0
$556$	3577.58	0.272884
$557$	18640.9	1.41803	0.709014	−	0.705195i	$-0.249140\pi$
$557$	18640.9	1.41803	0.709014	+	0.705195i	$0.249140\pi$
$558$	0	0
$559$	−2011.52	−0.152197
$560$	−4487.60	−0.338635
$561$	0	0
$562$	6659.68	0.499861
$563$	−17900.9	−1.34002	−0.670010	−	0.742352i	$-0.733710\pi$
$563$	−17900.9	−1.34002	−0.670010	+	0.742352i	$0.733710\pi$
$564$	0	0
$565$	16358.5	1.21807
$566$	−16846.8	−1.25110
$567$	0	0
$568$	−1865.17	−0.137783
$569$	−6350.42	−0.467880	−0.233940	−	0.972251i	$-0.575162\pi$
$569$	−6350.42	−0.467880	−0.233940	+	0.972251i	$0.575162\pi$
$570$	0	0
$571$	5623.70	0.412162	0.206081	−	0.978535i	$-0.433929\pi$
$571$	5623.70	0.412162	0.206081	+	0.978535i	$0.433929\pi$
$572$	1085.20	0.0793262
$573$	0	0
$574$	−1067.34	−0.0776129
$575$	1152.65	0.0835980
$576$	0	0
$577$	23322.7	1.68273	0.841366	−	0.540466i	$-0.181752\pi$
$577$	23322.7	1.68273	0.841366	+	0.540466i	$0.181752\pi$
$578$	−11016.1	−0.792752
$579$	0	0
$580$	−3916.24	−0.280368
$581$	−2721.30	−0.194318
$582$	0	0
$583$	−2591.02	−0.184064
$584$	−24031.2	−1.70277
$585$	0	0
$586$	11421.1	0.805120
$587$	23553.9	1.65617	0.828086	−	0.560602i	$-0.189430\pi$
$587$	23553.9	1.65617	0.828086	+	0.560602i	$0.189430\pi$
$588$	0	0
$589$	16532.3	1.15654
$590$	662.726	0.0462441
$591$	0	0
$592$	−17018.6	−1.18152
$593$	−1765.18	−0.122238	−0.0611192	−	0.998130i	$-0.519467\pi$
$593$	−1765.18	−0.122238	−0.0611192	+	0.998130i	$0.519467\pi$
$594$	0	0
$595$	2133.32	0.146987
$596$	2635.18	0.181110
$597$	0	0
$598$	−2246.13	−0.153597
$599$	17470.9	1.19172	0.595860	−	0.803088i	$-0.296811\pi$
$599$	17470.9	1.19172	0.595860	+	0.803088i	$0.296811\pi$
$600$	0	0
$601$	−160.251	−0.0108765	−0.00543824	−	0.999985i	$-0.501731\pi$
$601$	−160.251	−0.0108765	−0.00543824	+	0.999985i	$0.501731\pi$
$602$	−647.890	−0.0438638
$603$	0	0
$604$	−4798.56	−0.323263
$605$	−1684.67	−0.113209
$606$	0	0
$607$	9184.30	0.614134	0.307067	−	0.951688i	$-0.400652\pi$
$607$	9184.30	0.614134	0.307067	+	0.951688i	$0.400652\pi$
$608$	5962.89	0.397742
$609$	0	0
$610$	−24673.1	−1.63768
$611$	15111.9	1.00059
$612$	0	0
$613$	−2847.11	−0.187592	−0.0937960	−	0.995591i	$-0.529900\pi$
$613$	−2847.11	−0.187592	−0.0937960	+	0.995591i	$0.529900\pi$
$614$	4659.07	0.306229
$615$	0	0
$616$	1880.01	0.122967
$617$	−18483.3	−1.20601	−0.603007	−	0.797736i	$-0.706031\pi$
$617$	−18483.3	−1.20601	−0.603007	+	0.797736i	$0.706031\pi$
$618$	0	0
$619$	6463.20	0.419674	0.209837	−	0.977736i	$-0.432707\pi$
$619$	6463.20	0.419674	0.209837	+	0.977736i	$0.432707\pi$
$620$	5707.28	0.369694
$621$	0	0
$622$	−1544.46	−0.0995615
$623$	231.694	0.0148999
$624$	0	0
$625$	−19491.0	−1.24742
$626$	6429.00	0.410470
$627$	0	0
$628$	3311.93	0.210446
$629$	8090.31	0.512849
$630$	0	0
$631$	7523.28	0.474639	0.237319	−	0.971432i	$-0.423731\pi$
$631$	7523.28	0.474639	0.237319	+	0.971432i	$0.423731\pi$
$632$	17028.9	1.07179
$633$	0	0
$634$	−10759.1	−0.673975
$635$	24450.1	1.52799
$636$	0	0
$637$	−2645.84	−0.164572
$638$	−4207.56	−0.261096
$639$	0	0
$640$	−10683.9	−0.659873
$641$	25452.2	1.56833	0.784165	−	0.620552i	$-0.213091\pi$
$641$	25452.2	1.56833	0.784165	+	0.620552i	$0.213091\pi$
$642$	0	0
$643$	−2227.13	−0.136593	−0.0682967	−	0.997665i	$-0.521756\pi$
$643$	−2227.13	−0.136593	−0.0682967	+	0.997665i	$0.521756\pi$
$644$	214.125	0.0131020
$645$	0	0
$646$	4007.34	0.244066
$647$	1204.60	0.0731960	0.0365980	−	0.999330i	$-0.488348\pi$
$647$	1204.60	0.0731960	0.0365980	+	0.999330i	$0.488348\pi$
$648$	0	0
$649$	−210.743	−0.0127463
$650$	−9236.19	−0.557344
$651$	0	0
$652$	−6490.00	−0.389828
$653$	21289.4	1.27584	0.637918	−	0.770105i	$-0.279796\pi$
$653$	21289.4	1.27584	0.637918	+	0.770105i	$0.279796\pi$
$654$	0	0
$655$	−18186.1	−1.08487
$656$	−2825.82	−0.168186
$657$	0	0
$658$	4867.39	0.288375
$659$	30617.8	1.80986	0.904931	−	0.425558i	$-0.139922\pi$
$659$	30617.8	1.80986	0.904931	+	0.425558i	$0.139922\pi$
$660$	0	0
$661$	−27593.1	−1.62367	−0.811837	−	0.583884i	$-0.801532\pi$
$661$	−27593.1	−1.62367	−0.811837	+	0.583884i	$0.801532\pi$
$662$	15199.5	0.892363
$663$	0	0
$664$	−9491.78	−0.554748
$665$	7181.38	0.418770
$666$	0	0
$667$	−2577.57	−0.149631
$668$	−5879.07	−0.340521
$669$	0	0
$670$	6853.30	0.395173
$671$	7845.88	0.451396
$672$	0	0
$673$	24287.2	1.39109	0.695546	−	0.718482i	$-0.255163\pi$
$673$	24287.2	1.39109	0.695546	+	0.718482i	$0.255163\pi$
$674$	−3737.33	−0.213585
$675$	0	0
$676$	−1313.02	−0.0747053
$677$	30433.9	1.72773	0.863863	−	0.503727i	$-0.168038\pi$
$677$	30433.9	1.72773	0.863863	+	0.503727i	$0.168038\pi$
$678$	0	0
$679$	−9690.40	−0.547693
$680$	7440.92	0.419627
$681$	0	0
$682$	6131.83	0.344281
$683$	848.117	0.0475143	0.0237572	−	0.999718i	$-0.492437\pi$
$683$	848.117	0.0475143	0.0237572	+	0.999718i	$0.492437\pi$
$684$	0	0
$685$	10539.3	0.587864
$686$	−852.198	−0.0474301
$687$	0	0
$688$	−1715.32	−0.0950521
$689$	12718.8	0.703263
$690$	0	0
$691$	21464.2	1.18167	0.590836	−	0.806792i	$-0.298798\pi$
$691$	21464.2	1.18167	0.590836	+	0.806792i	$0.298798\pi$
$692$	4590.74	0.252187
$693$	0	0
$694$	8500.73	0.464961
$695$	27262.7	1.48796
$696$	0	0
$697$	1343.34	0.0730023
$698$	6496.47	0.352285
$699$	0	0
$700$	880.494	0.0475422
$701$	−20127.9	−1.08448	−0.542240	−	0.840223i	$-0.682424\pi$
$701$	−20127.9	−1.08448	−0.542240	+	0.840223i	$0.682424\pi$
$702$	0	0
$703$	27234.4	1.46112
$704$	6263.65	0.335327
$705$	0	0
$706$	29256.2	1.55959
$707$	7430.02	0.395240
$708$	0	0
$709$	−27572.3	−1.46051	−0.730253	−	0.683177i	$-0.760598\pi$
$709$	−27572.3	−1.46051	−0.730253	+	0.683177i	$0.760598\pi$
$710$	−2642.55	−0.139681
$711$	0	0
$712$	808.140	0.0425370
$713$	3756.39	0.197304
$714$	0	0
$715$	8269.69	0.432544
$716$	8104.83	0.423033
$717$	0	0
$718$	−29899.3	−1.55408
$719$	781.592	0.0405403	0.0202701	−	0.999795i	$-0.493547\pi$
$719$	781.592	0.0405403	0.0202701	+	0.999795i	$0.493547\pi$
$720$	0	0
$721$	6069.65	0.313517
$722$	−3551.57	−0.183069
$723$	0	0
$724$	−8097.03	−0.415640
$725$	−10599.1	−0.542954
$726$	0	0
$727$	−26327.3	−1.34309	−0.671544	−	0.740964i	$-0.734369\pi$
$727$	−26327.3	−1.34309	−0.671544	+	0.740964i	$0.734369\pi$
$728$	−9228.60	−0.469828
$729$	0	0
$730$	−34047.1	−1.72622
$731$	815.428	0.0412581
$732$	0	0
$733$	30990.7	1.56162	0.780811	−	0.624767i	$-0.214806\pi$
$733$	30990.7	1.56162	0.780811	+	0.624767i	$0.214806\pi$
$734$	14470.1	0.727660
$735$	0	0
$736$	1354.86	0.0678545
$737$	−2179.30	−0.108922
$738$	0	0
$739$	18965.5	0.944053	0.472027	−	0.881584i	$-0.343523\pi$
$739$	18965.5	0.944053	0.472027	+	0.881584i	$0.343523\pi$
$740$	9401.88	0.467054
$741$	0	0
$742$	4096.59	0.202683
$743$	−7955.01	−0.392788	−0.196394	−	0.980525i	$-0.562923\pi$
$743$	−7955.01	−0.392788	−0.196394	+	0.980525i	$0.562923\pi$
$744$	0	0
$745$	20081.2	0.987541
$746$	−9477.66	−0.465150
$747$	0	0
$748$	−439.918	−0.0215040
$749$	−5784.68	−0.282200
$750$	0	0
$751$	16225.6	0.788388	0.394194	−	0.919027i	$-0.371024\pi$
$751$	16225.6	0.788388	0.394194	+	0.919027i	$0.371024\pi$
$752$	12886.6	0.624902
$753$	0	0
$754$	20654.1	0.997583
$755$	−36567.0	−1.76266
$756$	0	0
$757$	−4814.19	−0.231142	−0.115571	−	0.993299i	$-0.536870\pi$
$757$	−4814.19	−0.231142	−0.115571	+	0.993299i	$0.536870\pi$
$758$	20640.8	0.989062
$759$	0	0
$760$	25048.4	1.19553
$761$	22741.4	1.08328	0.541639	−	0.840612i	$-0.317804\pi$
$761$	22741.4	1.08328	0.541639	+	0.840612i	$0.317804\pi$
$762$	0	0
$763$	−7147.48	−0.339130
$764$	2945.58	0.139486
$765$	0	0
$766$	−13318.3	−0.628209
$767$	1034.49	0.0487006
$768$	0	0
$769$	42230.5	1.98033	0.990163	−	0.139917i	$-0.0446834\pi$
$769$	42230.5	1.98033	0.990163	+	0.139917i	$0.0446834\pi$
$770$	2663.58	0.124661
$771$	0	0
$772$	−1699.67	−0.0792390
$773$	−14738.1	−0.685759	−0.342879	−	0.939379i	$-0.611402\pi$
$773$	−14738.1	−0.685759	−0.342879	+	0.939379i	$0.611402\pi$
$774$	0	0
$775$	15446.5	0.715941
$776$	−33799.7	−1.56358
$777$	0	0
$778$	−11904.5	−0.548584
$779$	4522.08	0.207985
$780$	0	0
$781$	840.314	0.0385004
$782$	910.531	0.0416375
$783$	0	0
$784$	−2256.23	−0.102780
$785$	25238.3	1.14751
$786$	0	0
$787$	−40440.5	−1.83170	−0.915851	−	0.401518i	$-0.868483\pi$
$787$	−40440.5	−1.83170	−0.915851	+	0.401518i	$0.868483\pi$
$788$	−8600.56	−0.388810
$789$	0	0
$790$	24126.4	1.08656
$791$	8224.56	0.369699
$792$	0	0
$793$	−38513.9	−1.72468
$794$	−6843.88	−0.305895
$795$	0	0
$796$	263.014	0.0117114
$797$	−6768.37	−0.300813	−0.150406	−	0.988624i	$-0.548058\pi$
$797$	−6768.37	−0.300813	−0.150406	+	0.988624i	$0.548058\pi$
$798$	0	0
$799$	−6126.05	−0.271244
$800$	5571.27	0.246218
$801$	0	0
$802$	−20325.3	−0.894901
$803$	10826.7	0.475800
$804$	0	0
$805$	1631.72	0.0714417
$806$	−30099.9	−1.31542
$807$	0	0
$808$	25915.6	1.12835
$809$	−40401.4	−1.75579	−0.877896	−	0.478851i	$-0.841053\pi$
$809$	−40401.4	−1.75579	−0.877896	+	0.478851i	$0.841053\pi$
$810$	0	0
$811$	16309.7	0.706178	0.353089	−	0.935590i	$-0.385131\pi$
$811$	16309.7	0.706178	0.353089	+	0.935590i	$0.385131\pi$
$812$	−1968.97	−0.0850953
$813$	0	0
$814$	10101.3	0.434950
$815$	−49456.5	−2.12562
$816$	0	0
$817$	2744.97	0.117545
$818$	17067.8	0.729538
$819$	0	0
$820$	1561.12	0.0664836
$821$	−37753.3	−1.60487	−0.802436	−	0.596738i	$-0.796463\pi$
$821$	−37753.3	−1.60487	−0.802436	+	0.596738i	$0.796463\pi$
$822$	0	0
$823$	−37291.1	−1.57945	−0.789725	−	0.613461i	$-0.789777\pi$
$823$	−37291.1	−1.57945	−0.789725	+	0.613461i	$0.789777\pi$
$824$	21170.7	0.895044
$825$	0	0
$826$	333.199	0.0140357
$827$	−16814.3	−0.707000	−0.353500	−	0.935435i	$-0.615009\pi$
$827$	−16814.3	−0.707000	−0.353500	+	0.935435i	$0.615009\pi$
$828$	0	0
$829$	−13095.5	−0.548642	−0.274321	−	0.961638i	$-0.588453\pi$
$829$	−13095.5	−0.548642	−0.274321	+	0.961638i	$0.588453\pi$
$830$	−13447.9	−0.562388
$831$	0	0
$832$	−30747.0	−1.28120
$833$	1072.57	0.0446126
$834$	0	0
$835$	−44801.0	−1.85677
$836$	−1480.89	−0.0612653
$837$	0	0
$838$	32411.8	1.33609
$839$	11341.3	0.466680	0.233340	−	0.972395i	$-0.425034\pi$
$839$	11341.3	0.466680	0.233340	+	0.972395i	$0.425034\pi$
$840$	0	0
$841$	−687.109	−0.0281729
$842$	2000.79	0.0818904
$843$	0	0
$844$	824.671	0.0336331
$845$	−10005.8	−0.407347
$846$	0	0
$847$	−847.000	−0.0343604
$848$	10845.9	0.439210
$849$	0	0
$850$	3744.15	0.151086
$851$	6188.08	0.249265
$852$	0	0
$853$	3075.53	0.123451	0.0617257	−	0.998093i	$-0.480340\pi$
$853$	3075.53	0.123451	0.0617257	+	0.998093i	$0.480340\pi$
$854$	−12404.9	−0.497058
$855$	0	0
$856$	−20176.7	−0.805638
$857$	5615.24	0.223819	0.111910	−	0.993718i	$-0.464303\pi$
$857$	5615.24	0.223819	0.111910	+	0.993718i	$0.464303\pi$
$858$	0	0
$859$	−15825.9	−0.628606	−0.314303	−	0.949323i	$-0.601771\pi$
$859$	−15825.9	−0.628606	−0.314303	+	0.949323i	$0.601771\pi$
$860$	947.622	0.0375740
$861$	0	0
$862$	−29865.7	−1.18008
$863$	−43324.7	−1.70891	−0.854456	−	0.519523i	$-0.826110\pi$
$863$	−43324.7	−1.70891	−0.854456	+	0.519523i	$0.826110\pi$
$864$	0	0
$865$	34983.3	1.37511
$866$	42835.8	1.68085
$867$	0	0
$868$	2869.45	0.112207
$869$	−7672.03	−0.299489
$870$	0	0
$871$	10697.8	0.416165
$872$	−24930.1	−0.968165
$873$	0	0
$874$	3065.12	0.118626
$875$	−5472.77	−0.211444
$876$	0	0
$877$	−3754.22	−0.144551	−0.0722753	−	0.997385i	$-0.523026\pi$
$877$	−3754.22	−0.144551	−0.0722753	+	0.997385i	$0.523026\pi$
$878$	−35479.3	−1.36375
$879$	0	0
$880$	7051.94	0.270137
$881$	−4061.45	−0.155316	−0.0776581	−	0.996980i	$-0.524744\pi$
$881$	−4061.45	−0.155316	−0.0776581	+	0.996980i	$0.524744\pi$
$882$	0	0
$883$	−30774.8	−1.17288	−0.586441	−	0.809992i	$-0.699472\pi$
$883$	−30774.8	−1.17288	−0.586441	+	0.809992i	$0.699472\pi$
$884$	2159.47	0.0821615
$885$	0	0
$886$	−29398.6	−1.11474
$887$	33107.7	1.25327	0.626634	−	0.779314i	$-0.284432\pi$
$887$	33107.7	1.25327	0.626634	+	0.779314i	$0.284432\pi$
$888$	0	0
$889$	12292.8	0.463764
$890$	1144.96	0.0431228
$891$	0	0
$892$	−689.054	−0.0258646
$893$	−20622.1	−0.772780
$894$	0	0
$895$	61762.1	2.30668
$896$	−5371.55	−0.200280
$897$	0	0
$898$	27482.2	1.02126
$899$	−34541.6	−1.28145
$900$	0	0
$901$	−5155.93	−0.190643
$902$	1677.24	0.0619137
$903$	0	0
$904$	28687.0	1.05544
$905$	−61702.7	−2.26637
$906$	0	0
$907$	10605.6	0.388261	0.194130	−	0.980976i	$-0.437812\pi$
$907$	10605.6	0.388261	0.194130	+	0.980976i	$0.437812\pi$
$908$	−9942.30	−0.363378
$909$	0	0
$910$	−13075.0	−0.476298
$911$	45584.4	1.65783	0.828913	−	0.559377i	$-0.188960\pi$
$911$	45584.4	1.65783	0.828913	+	0.559377i	$0.188960\pi$
$912$	0	0
$913$	4276.33	0.155012
$914$	−18284.0	−0.661687
$915$	0	0
$916$	−4003.55	−0.144412
$917$	−9143.45	−0.329273
$918$	0	0
$919$	32114.9	1.15275	0.576373	−	0.817187i	$-0.304468\pi$
$919$	32114.9	1.15275	0.576373	+	0.817187i	$0.304468\pi$
$920$	5691.37	0.203956
$921$	0	0
$922$	−832.411	−0.0297332
$923$	−4124.94	−0.147101
$924$	0	0
$925$	25445.7	0.904487
$926$	47141.2	1.67296
$927$	0	0
$928$	−12458.5	−0.440702
$929$	−26956.8	−0.952015	−0.476008	−	0.879441i	$-0.657917\pi$
$929$	−26956.8	−0.952015	−0.476008	+	0.879441i	$0.657917\pi$
$930$	0	0
$931$	3610.58	0.127102
$932$	−4348.26	−0.152824
$933$	0	0
$934$	9161.37	0.320952
$935$	−3352.35	−0.117255
$936$	0	0
$937$	−47322.5	−1.64990	−0.824952	−	0.565203i	$-0.808798\pi$
$937$	−47322.5	−1.64990	−0.824952	+	0.565203i	$0.808798\pi$
$938$	3445.63	0.119940
$939$	0	0
$940$	−7119.18	−0.247023
$941$	28861.8	0.999859	0.499930	−	0.866066i	$-0.333359\pi$
$941$	28861.8	0.999859	0.499930	+	0.866066i	$0.333359\pi$
$942$	0	0
$943$	1027.49	0.0354821
$944$	882.159	0.0304151
$945$	0	0
$946$	1018.11	0.0349912
$947$	48160.7	1.65260	0.826300	−	0.563230i	$-0.190442\pi$
$947$	48160.7	1.65260	0.826300	+	0.563230i	$0.190442\pi$
$948$	0	0
$949$	−53146.4	−1.81792
$950$	12603.9	0.430448
$951$	0	0
$952$	3741.07	0.127362
$953$	−15956.2	−0.542363	−0.271181	−	0.962528i	$-0.587414\pi$
$953$	−15956.2	−0.542363	−0.271181	+	0.962528i	$0.587414\pi$
$954$	0	0
$955$	22446.5	0.760578
$956$	8230.04	0.278429
$957$	0	0
$958$	846.914	0.0285621
$959$	5298.86	0.178425
$960$	0	0
$961$	20547.7	0.689729
$962$	−49585.1	−1.66184
$963$	0	0
$964$	9893.34	0.330543
$965$	−12952.2	−0.432068
$966$	0	0
$967$	41672.9	1.38584	0.692921	−	0.721013i	$-0.256323\pi$
$967$	41672.9	1.38584	0.692921	+	0.721013i	$0.256323\pi$
$968$	−2954.30	−0.0980939
$969$	0	0
$970$	−47887.1	−1.58512
$971$	10718.0	0.354229	0.177115	−	0.984190i	$-0.443324\pi$
$971$	10718.0	0.354229	0.177115	+	0.984190i	$0.443324\pi$
$972$	0	0
$973$	13706.9	0.451615
$974$	−20754.1	−0.682756
$975$	0	0
$976$	−32842.5	−1.07711
$977$	−9552.87	−0.312818	−0.156409	−	0.987692i	$-0.549992\pi$
$977$	−9552.87	−0.312818	−0.156409	+	0.987692i	$0.549992\pi$
$978$	0	0
$979$	−364.091	−0.0118860
$980$	1246.45	0.0406289
$981$	0	0
$982$	3755.75	0.122047
$983$	−21150.8	−0.686272	−0.343136	−	0.939286i	$-0.611489\pi$
$983$	−21150.8	−0.686272	−0.343136	+	0.939286i	$0.611489\pi$
$984$	0	0
$985$	−65539.8	−2.12007
$986$	−8372.72	−0.270428
$987$	0	0
$988$	7269.41	0.234080
$989$	623.701	0.0200531
$990$	0	0
$991$	−42985.5	−1.37788	−0.688940	−	0.724818i	$-0.741924\pi$
$991$	−42985.5	−1.37788	−0.688940	+	0.724818i	$0.741924\pi$
$992$	18156.3	0.581111
$993$	0	0
$994$	−1328.60	−0.0423949
$995$	2004.27	0.0638590
$996$	0	0
$997$	−10090.8	−0.320542	−0.160271	−	0.987073i	$-0.551237\pi$
$997$	−10090.8	−0.320542	−0.160271	+	0.987073i	$0.551237\pi$
$998$	37296.1	1.18295
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.q.1.4		5
3.2	odd	2		231.4.a.j.1.2	✓	5
21.20	even	2		1617.4.a.o.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.j.1.2	✓	5	3.2	odd	2
693.4.a.q.1.4		5	1.1	even	1	trivial
1617.4.a.o.1.2		5	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.48454 q^{2} -1.82705 q^{4} -13.9229 q^{5} -7.00000 q^{7} -24.4157 q^{8} +O(q^{10})\)	\(q+2.48454 q^{2} -1.82705 q^{4} -13.9229 q^{5} -7.00000 q^{7} -24.4157 q^{8} -34.5919 q^{10} +11.0000 q^{11} -53.9968 q^{13} -17.3918 q^{14} -46.0455 q^{16} +21.8891 q^{17} +73.6854 q^{19} +25.4377 q^{20} +27.3300 q^{22} +16.7425 q^{23} +68.8459 q^{25} -134.157 q^{26} +12.7893 q^{28} -153.954 q^{29} +224.363 q^{31} +80.9237 q^{32} +54.3845 q^{34} +97.4600 q^{35} +369.604 q^{37} +183.074 q^{38} +339.937 q^{40} +61.3701 q^{41} +37.2526 q^{43} -20.0975 q^{44} +41.5974 q^{46} -279.867 q^{47} +49.0000 q^{49} +171.051 q^{50} +98.6548 q^{52} -235.547 q^{53} -153.151 q^{55} +170.910 q^{56} -382.506 q^{58} -19.1584 q^{59} +713.262 q^{61} +557.439 q^{62} +569.423 q^{64} +751.790 q^{65} -198.118 q^{67} -39.9925 q^{68} +242.144 q^{70} +76.3922 q^{71} +984.250 q^{73} +918.297 q^{74} -134.627 q^{76} -77.0000 q^{77} -697.458 q^{79} +641.085 q^{80} +152.477 q^{82} +388.757 q^{83} -304.759 q^{85} +92.5557 q^{86} -268.573 q^{88} -33.0992 q^{89} +377.978 q^{91} -30.5893 q^{92} -695.341 q^{94} -1025.91 q^{95} +1384.34 q^{97} +121.743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8}+O(q^{10}) \) 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8	\( 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26} - 147 q^{28} - 591 q^{29} + 366 q^{31} + 87 q^{32} + 594 q^{34} + 49 q^{35} + 259 q^{37} - 175 q^{38} + 1890 q^{40} + 104 q^{41} + 224 q^{43} + 231 q^{44} + 1416 q^{46} + 453 q^{47} + 245 q^{49} - 1350 q^{50} + 815 q^{52} - 1032 q^{53} - 77 q^{55} - 84 q^{56} + 293 q^{58} + 517 q^{59} + 958 q^{61} + 3030 q^{62} + 516 q^{64} + 197 q^{65} + 361 q^{67} + 1680 q^{68} - 791 q^{70} - 548 q^{71} - 1035 q^{73} + 2555 q^{74} - 5201 q^{76} - 385 q^{77} + 776 q^{79} + 1273 q^{80} + 376 q^{82} + 1974 q^{83} + 1838 q^{85} - 1224 q^{86} + 132 q^{88} - 2710 q^{89} - 161 q^{91} + 2834 q^{92} - 563 q^{94} + 355 q^{95} + 1988 q^{97} + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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