LMFDB - Embedded newform 693.4.a.p.1.5

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} + 11x^{2} + 185x + 162 $ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
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			1.5
Root			$4.79323$ 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)

$f(q)$	$=$	$q+4.79323 q^{2} +14.9751 q^{4} +6.25369 q^{5} +7.00000 q^{7} +33.4331 q^{8} +O(q^{10})$	\(q+4.79323 q^{2} +14.9751 q^{4} +6.25369 q^{5} +7.00000 q^{7} +33.4331 q^{8} +29.9754 q^{10} -11.0000 q^{11} -35.7637 q^{13} +33.5526 q^{14} +40.4519 q^{16} +133.102 q^{17} +161.508 q^{19} +93.6494 q^{20} -52.7255 q^{22} +66.2680 q^{23} -85.8913 q^{25} -171.424 q^{26} +104.825 q^{28} +208.993 q^{29} -39.3732 q^{31} -73.5692 q^{32} +637.987 q^{34} +43.7759 q^{35} +197.404 q^{37} +774.146 q^{38} +209.080 q^{40} -434.978 q^{41} +375.872 q^{43} -164.726 q^{44} +317.638 q^{46} -503.794 q^{47} +49.0000 q^{49} -411.697 q^{50} -535.563 q^{52} -44.8246 q^{53} -68.7906 q^{55} +234.031 q^{56} +1001.75 q^{58} -582.890 q^{59} +73.2219 q^{61} -188.725 q^{62} -676.249 q^{64} -223.655 q^{65} -928.317 q^{67} +1993.21 q^{68} +209.828 q^{70} +755.968 q^{71} +277.345 q^{73} +946.204 q^{74} +2418.60 q^{76} -77.0000 q^{77} +651.033 q^{79} +252.974 q^{80} -2084.95 q^{82} -282.009 q^{83} +832.378 q^{85} +1801.64 q^{86} -367.764 q^{88} -1040.08 q^{89} -250.346 q^{91} +992.367 q^{92} -2414.80 q^{94} +1010.02 q^{95} +1116.81 q^{97} +234.868 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8	$ 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * 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 $

$2$	4.79323	1.69466	0.847331	−	0.531064i	$-0.178208\pi$
$2$	4.79323	1.69466	0.847331	+	0.531064i	$0.178208\pi$
$3$	0	0
$3$	0	0
$4$	14.9751	1.87188
$5$	6.25369	0.559347	0.279674	−	0.960095i	$-0.409774\pi$
$5$	6.25369	0.559347	0.279674	+	0.960095i	$0.409774\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	33.4331	1.47755
$9$	0	0
$10$	29.9754	0.947905
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−35.7637	−0.763005	−0.381503	−	0.924368i	$-0.624593\pi$
$13$	−35.7637	−0.763005	−0.381503	+	0.924368i	$0.624593\pi$
$14$	33.5526	0.640522
$15$	0	0
$16$	40.4519	0.632061
$17$	133.102	1.89894	0.949468	−	0.313863i	$-0.101623\pi$
$17$	133.102	1.89894	0.949468	+	0.313863i	$0.101623\pi$
$18$	0	0
$19$	161.508	1.95013	0.975067	−	0.221912i	$-0.0712299\pi$
$19$	161.508	1.95013	0.975067	+	0.221912i	$0.0712299\pi$
$20$	93.6494	1.04703
$21$	0	0
$22$	−52.7255	−0.510960
$23$	66.2680	0.600775	0.300388	−	0.953817i	$-0.402884\pi$
$23$	66.2680	0.600775	0.300388	+	0.953817i	$0.402884\pi$
$24$	0	0
$25$	−85.8913	−0.687131
$26$	−171.424	−1.29304
$27$	0	0
$28$	104.825	0.707505
$29$	208.993	1.33824	0.669121	−	0.743153i	$-0.266671\pi$
$29$	208.993	1.33824	0.669121	+	0.743153i	$0.266671\pi$
$30$	0	0
$31$	−39.3732	−0.228117	−0.114059	−	0.993474i	$-0.536385\pi$
$31$	−39.3732	−0.228117	−0.114059	+	0.993474i	$0.536385\pi$
$32$	−73.5692	−0.406416
$33$	0	0
$34$	637.987	3.21806
$35$	43.7759	0.211413
$36$	0	0
$37$	197.404	0.877109	0.438555	−	0.898705i	$-0.355491\pi$
$37$	197.404	0.877109	0.438555	+	0.898705i	$0.355491\pi$
$38$	774.146	3.30482
$39$	0	0
$40$	209.080	0.826462
$41$	−434.978	−1.65688	−0.828440	−	0.560078i	$-0.810771\pi$
$41$	−434.978	−1.65688	−0.828440	+	0.560078i	$0.810771\pi$
$42$	0	0
$43$	375.872	1.33302	0.666511	−	0.745495i	$-0.267787\pi$
$43$	375.872	1.33302	0.666511	+	0.745495i	$0.267787\pi$
$44$	−164.726	−0.564394
$45$	0	0
$46$	317.638	1.01811
$47$	−503.794	−1.56353	−0.781765	−	0.623574i	$-0.785680\pi$
$47$	−503.794	−1.56353	−0.781765	+	0.623574i	$0.785680\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−411.697	−1.16445
$51$	0	0
$52$	−535.563	−1.42826
$53$	−44.8246	−0.116172	−0.0580861	−	0.998312i	$-0.518500\pi$
$53$	−44.8246	−0.116172	−0.0580861	+	0.998312i	$0.518500\pi$
$54$	0	0
$55$	−68.7906	−0.168650
$56$	234.031	0.558460
$57$	0	0
$58$	1001.75	2.26787
$59$	−582.890	−1.28620	−0.643100	−	0.765782i	$-0.722352\pi$
$59$	−582.890	−1.28620	−0.643100	+	0.765782i	$0.722352\pi$
$60$	0	0
$61$	73.2219	0.153690	0.0768451	−	0.997043i	$-0.475515\pi$
$61$	73.2219	0.153690	0.0768451	+	0.997043i	$0.475515\pi$
$62$	−188.725	−0.386582
$63$	0	0
$64$	−676.249	−1.32080
$65$	−223.655	−0.426785
$66$	0	0
$67$	−928.317	−1.69272	−0.846358	−	0.532614i	$-0.821210\pi$
$67$	−928.317	−1.69272	−0.846358	+	0.532614i	$0.821210\pi$
$68$	1993.21	3.55459
$69$	0	0
$70$	209.828	0.358274
$71$	755.968	1.26362	0.631809	−	0.775124i	$-0.282313\pi$
$71$	755.968	1.26362	0.631809	+	0.775124i	$0.282313\pi$
$72$	0	0
$73$	277.345	0.444669	0.222334	−	0.974970i	$-0.428632\pi$
$73$	277.345	0.444669	0.222334	+	0.974970i	$0.428632\pi$
$74$	946.204	1.48640
$75$	0	0
$76$	2418.60	3.65042
$77$	−77.0000	−0.113961
$78$	0	0
$79$	651.033	0.927177	0.463589	−	0.886051i	$-0.346562\pi$
$79$	651.033	0.927177	0.463589	+	0.886051i	$0.346562\pi$
$80$	252.974	0.353541
$81$	0	0
$82$	−2084.95	−2.80785
$83$	−282.009	−0.372946	−0.186473	−	0.982460i	$-0.559706\pi$
$83$	−282.009	−0.372946	−0.186473	+	0.982460i	$0.559706\pi$
$84$	0	0
$85$	832.378	1.06216
$86$	1801.64	2.25902
$87$	0	0
$88$	−367.764	−0.445497
$89$	−1040.08	−1.23875	−0.619373	−	0.785097i	$-0.712613\pi$
$89$	−1040.08	−1.23875	−0.619373	+	0.785097i	$0.712613\pi$
$90$	0	0
$91$	−250.346	−0.288389
$92$	992.367	1.12458
$93$	0	0
$94$	−2414.80	−2.64965
$95$	1010.02	1.09080
$96$	0	0
$97$	1116.81	1.16902	0.584510	−	0.811387i	$-0.301287\pi$
$97$	1116.81	1.16902	0.584510	+	0.811387i	$0.301287\pi$
$98$	234.868	0.242095
$99$	0	0
$100$	−1286.23	−1.28623
$101$	−368.860	−0.363396	−0.181698	−	0.983354i	$-0.558159\pi$
$101$	−368.860	−0.363396	−0.181698	+	0.983354i	$0.558159\pi$
$102$	0	0
$103$	−1473.16	−1.40927	−0.704636	−	0.709569i	$-0.748890\pi$
$103$	−1473.16	−1.40927	−0.704636	+	0.709569i	$0.748890\pi$
$104$	−1195.69	−1.12738
$105$	0	0
$106$	−214.854	−0.196873
$107$	−405.943	−0.366766	−0.183383	−	0.983042i	$-0.558705\pi$
$107$	−405.943	−0.366766	−0.183383	+	0.983042i	$0.558705\pi$
$108$	0	0
$109$	671.258	0.589861	0.294931	−	0.955519i	$-0.404703\pi$
$109$	671.258	0.589861	0.294931	+	0.955519i	$0.404703\pi$
$110$	−329.729	−0.285804
$111$	0	0
$112$	283.163	0.238897
$113$	686.797	0.571756	0.285878	−	0.958266i	$-0.407715\pi$
$113$	686.797	0.571756	0.285878	+	0.958266i	$0.407715\pi$
$114$	0	0
$115$	414.420	0.336042
$116$	3129.68	2.50503
$117$	0	0
$118$	−2793.93	−2.17968
$119$	931.712	0.717731
$120$	0	0
$121$	121.000	0.0909091
$122$	350.969	0.260453
$123$	0	0
$124$	−589.617	−0.427009
$125$	−1318.85	−0.943692
$126$	0	0
$127$	2058.06	1.43798	0.718988	−	0.695022i	$-0.244605\pi$
$127$	2058.06	1.43798	0.718988	+	0.695022i	$0.244605\pi$
$128$	−2652.87	−1.83189
$129$	0	0
$130$	−1072.03	−0.723256
$131$	−1805.09	−1.20391	−0.601953	−	0.798532i	$-0.705610\pi$
$131$	−1805.09	−1.20391	−0.601953	+	0.798532i	$0.705610\pi$
$132$	0	0
$133$	1130.56	0.737081
$134$	−4449.64	−2.86858
$135$	0	0
$136$	4450.00	2.80577
$137$	−1428.36	−0.890751	−0.445375	−	0.895344i	$-0.646930\pi$
$137$	−1428.36	−0.890751	−0.445375	+	0.895344i	$0.646930\pi$
$138$	0	0
$139$	1401.90	0.855453	0.427727	−	0.903908i	$-0.359315\pi$
$139$	1401.90	0.855453	0.427727	+	0.903908i	$0.359315\pi$
$140$	655.546	0.395741
$141$	0	0
$142$	3623.53	2.14141
$143$	393.401	0.230055
$144$	0	0
$145$	1306.98	0.748542
$146$	1329.38	0.753564
$147$	0	0
$148$	2956.14	1.64185
$149$	−1510.58	−0.830549	−0.415275	−	0.909696i	$-0.636315\pi$
$149$	−1510.58	−0.830549	−0.415275	+	0.909696i	$0.636315\pi$
$150$	0	0
$151$	−187.595	−0.101101	−0.0505505	−	0.998722i	$-0.516098\pi$
$151$	−187.595	−0.101101	−0.0505505	+	0.998722i	$0.516098\pi$
$152$	5399.71	2.88141
$153$	0	0
$154$	−369.079	−0.193125
$155$	−246.228	−0.127597
$156$	0	0
$157$	−3332.23	−1.69389	−0.846946	−	0.531678i	$-0.821561\pi$
$157$	−3332.23	−1.69389	−0.846946	+	0.531678i	$0.821561\pi$
$158$	3120.55	1.57125
$159$	0	0
$160$	−460.079	−0.227328
$161$	463.876	0.227072
$162$	0	0
$163$	−1643.24	−0.789621	−0.394810	−	0.918763i	$-0.629190\pi$
$163$	−1643.24	−0.789621	−0.394810	+	0.918763i	$0.629190\pi$
$164$	−6513.81	−3.10148
$165$	0	0
$166$	−1351.74	−0.632019
$167$	−1609.21	−0.745656	−0.372828	−	0.927900i	$-0.621612\pi$
$167$	−1609.21	−0.745656	−0.372828	+	0.927900i	$0.621612\pi$
$168$	0	0
$169$	−917.958	−0.417823
$170$	3989.78	1.80001
$171$	0	0
$172$	5628.70	2.49526
$173$	700.721	0.307947	0.153974	−	0.988075i	$-0.450793\pi$
$173$	700.721	0.307947	0.153974	+	0.988075i	$0.450793\pi$
$174$	0	0
$175$	−601.239	−0.259711
$176$	−444.971	−0.190573
$177$	0	0
$178$	−4985.35	−2.09926
$179$	435.174	0.181712	0.0908559	−	0.995864i	$-0.471040\pi$
$179$	435.174	0.181712	0.0908559	+	0.995864i	$0.471040\pi$
$180$	0	0
$181$	90.8432	0.0373056	0.0186528	−	0.999826i	$-0.494062\pi$
$181$	90.8432	0.0373056	0.0186528	+	0.999826i	$0.494062\pi$
$182$	−1199.97	−0.488722
$183$	0	0
$184$	2215.54	0.887673
$185$	1234.50	0.490609
$186$	0	0
$187$	−1464.12	−0.572551
$188$	−7544.34	−2.92674
$189$	0	0
$190$	4841.27	1.84854
$191$	−2335.80	−0.884883	−0.442441	−	0.896797i	$-0.645888\pi$
$191$	−2335.80	−0.884883	−0.442441	+	0.896797i	$0.645888\pi$
$192$	0	0
$193$	2319.35	0.865029	0.432514	−	0.901627i	$-0.357626\pi$
$193$	2319.35	0.865029	0.432514	+	0.901627i	$0.357626\pi$
$194$	5353.13	1.98109
$195$	0	0
$196$	733.778	0.267412
$197$	−2267.56	−0.820085	−0.410043	−	0.912066i	$-0.634486\pi$
$197$	−2267.56	−0.820085	−0.410043	+	0.912066i	$0.634486\pi$
$198$	0	0
$199$	−614.366	−0.218850	−0.109425	−	0.993995i	$-0.534901\pi$
$199$	−614.366	−0.218850	−0.109425	+	0.993995i	$0.534901\pi$
$200$	−2871.61	−1.01527
$201$	0	0
$202$	−1768.03	−0.615833
$203$	1462.95	0.505808
$204$	0	0
$205$	−2720.22	−0.926771
$206$	−7061.21	−2.38824
$207$	0	0
$208$	−1446.71	−0.482266
$209$	−1776.59	−0.587987
$210$	0	0
$211$	−506.531	−0.165266	−0.0826328	−	0.996580i	$-0.526333\pi$
$211$	−506.531	−0.165266	−0.0826328	+	0.996580i	$0.526333\pi$
$212$	−671.251	−0.217461
$213$	0	0
$214$	−1945.78	−0.621545
$215$	2350.59	0.745622
$216$	0	0
$217$	−275.613	−0.0862203
$218$	3217.49	0.999616
$219$	0	0
$220$	−1030.14	−0.315692
$221$	−4760.21	−1.44890
$222$	0	0
$223$	1555.55	0.467118	0.233559	−	0.972343i	$-0.424963\pi$
$223$	1555.55	0.467118	0.233559	+	0.972343i	$0.424963\pi$
$224$	−514.985	−0.153611
$225$	0	0
$226$	3291.97	0.968933
$227$	−131.215	−0.0383657	−0.0191829	−	0.999816i	$-0.506106\pi$
$227$	−131.215	−0.0383657	−0.0191829	+	0.999816i	$0.506106\pi$
$228$	0	0
$229$	−1605.65	−0.463339	−0.231670	−	0.972795i	$-0.574419\pi$
$229$	−1605.65	−0.463339	−0.231670	+	0.972795i	$0.574419\pi$
$230$	1986.41	0.569478
$231$	0	0
$232$	6987.28	1.97732
$233$	−2243.59	−0.630825	−0.315413	−	0.948955i	$-0.602143\pi$
$233$	−2243.59	−0.630825	−0.315413	+	0.948955i	$0.602143\pi$
$234$	0	0
$235$	−3150.57	−0.874556
$236$	−8728.81	−2.40762
$237$	0	0
$238$	4465.91	1.21631
$239$	3120.00	0.844419	0.422209	−	0.906498i	$-0.361255\pi$
$239$	3120.00	0.844419	0.422209	+	0.906498i	$0.361255\pi$
$240$	0	0
$241$	−695.537	−0.185907	−0.0929533	−	0.995670i	$-0.529631\pi$
$241$	−695.537	−0.185907	−0.0929533	+	0.995670i	$0.529631\pi$
$242$	579.981	0.154060
$243$	0	0
$244$	1096.50	0.287690
$245$	306.431	0.0799068
$246$	0	0
$247$	−5776.13	−1.48796
$248$	−1316.37	−0.337054
$249$	0	0
$250$	−6321.55	−1.59924
$251$	−1380.04	−0.347042	−0.173521	−	0.984830i	$-0.555515\pi$
$251$	−1380.04	−0.347042	−0.173521	+	0.984830i	$0.555515\pi$
$252$	0	0
$253$	−728.948	−0.181141
$254$	9864.74	2.43689
$255$	0	0
$256$	−7305.80	−1.78364
$257$	5776.54	1.40206	0.701032	−	0.713130i	$-0.252723\pi$
$257$	5776.54	1.40206	0.701032	+	0.713130i	$0.252723\pi$
$258$	0	0
$259$	1381.83	0.331516
$260$	−3349.25	−0.798891
$261$	0	0
$262$	−8652.22	−2.04021
$263$	−3276.42	−0.768186	−0.384093	−	0.923294i	$-0.625486\pi$
$263$	−3276.42	−0.768186	−0.384093	+	0.923294i	$0.625486\pi$
$264$	0	0
$265$	−280.319	−0.0649806
$266$	5419.02	1.24910
$267$	0	0
$268$	−13901.6	−3.16857
$269$	−5671.94	−1.28559	−0.642796	−	0.766037i	$-0.722226\pi$
$269$	−5671.94	−1.28559	−0.642796	+	0.766037i	$0.722226\pi$
$270$	0	0
$271$	7504.85	1.68224	0.841120	−	0.540849i	$-0.181897\pi$
$271$	7504.85	1.68224	0.841120	+	0.540849i	$0.181897\pi$
$272$	5384.22	1.20024
$273$	0	0
$274$	−6846.45	−1.50952
$275$	944.805	0.207178
$276$	0	0
$277$	−8841.46	−1.91780	−0.958902	−	0.283738i	$-0.908425\pi$
$277$	−8841.46	−1.91780	−0.958902	+	0.283738i	$0.908425\pi$
$278$	6719.65	1.44970
$279$	0	0
$280$	1463.56	0.312373
$281$	−3532.31	−0.749894	−0.374947	−	0.927046i	$-0.622339\pi$
$281$	−3532.31	−0.749894	−0.374947	+	0.927046i	$0.622339\pi$
$282$	0	0
$283$	−304.915	−0.0640469	−0.0320235	−	0.999487i	$-0.510195\pi$
$283$	−304.915	−0.0640469	−0.0320235	+	0.999487i	$0.510195\pi$
$284$	11320.7	2.36535
$285$	0	0
$286$	1885.66	0.389865
$287$	−3044.84	−0.626242
$288$	0	0
$289$	12803.1	2.60596
$290$	6264.65	1.26853
$291$	0	0
$292$	4153.26	0.832368
$293$	5776.94	1.15185	0.575926	−	0.817502i	$-0.304642\pi$
$293$	5776.94	1.15185	0.575926	+	0.817502i	$0.304642\pi$
$294$	0	0
$295$	−3645.22	−0.719433
$296$	6599.82	1.29597
$297$	0	0
$298$	−7240.58	−1.40750
$299$	−2369.99	−0.458395
$300$	0	0
$301$	2631.10	0.503835
$302$	−899.186	−0.171332
$303$	0	0
$304$	6533.31	1.23260
$305$	457.907	0.0859662
$306$	0	0
$307$	−5134.47	−0.954528	−0.477264	−	0.878760i	$-0.658371\pi$
$307$	−5134.47	−0.954528	−0.477264	+	0.878760i	$0.658371\pi$
$308$	−1153.08	−0.213321
$309$	0	0
$310$	−1180.23	−0.216234
$311$	4555.82	0.830665	0.415332	−	0.909670i	$-0.363665\pi$
$311$	4555.82	0.830665	0.415332	+	0.909670i	$0.363665\pi$
$312$	0	0
$313$	6114.99	1.10428	0.552140	−	0.833752i	$-0.313811\pi$
$313$	6114.99	1.10428	0.552140	+	0.833752i	$0.313811\pi$
$314$	−15972.2	−2.87058
$315$	0	0
$316$	9749.26	1.73557
$317$	5618.31	0.995444	0.497722	−	0.867337i	$-0.334170\pi$
$317$	5618.31	0.995444	0.497722	+	0.867337i	$0.334170\pi$
$318$	0	0
$319$	−2298.92	−0.403495
$320$	−4229.06	−0.738786
$321$	0	0
$322$	2223.46	0.384810
$323$	21497.0	3.70318
$324$	0	0
$325$	3071.79	0.524284
$326$	−7876.41	−1.33814
$327$	0	0
$328$	−14542.6	−2.44812
$329$	−3526.56	−0.590958
$330$	0	0
$331$	2766.08	0.459329	0.229664	−	0.973270i	$-0.426237\pi$
$331$	2766.08	0.459329	0.229664	+	0.973270i	$0.426237\pi$
$332$	−4223.11	−0.698112
$333$	0	0
$334$	−7713.32	−1.26364
$335$	−5805.41	−0.946817
$336$	0	0
$337$	−2423.16	−0.391686	−0.195843	−	0.980635i	$-0.562744\pi$
$337$	−2423.16	−0.391686	−0.195843	+	0.980635i	$0.562744\pi$
$338$	−4399.98	−0.708070
$339$	0	0
$340$	12464.9	1.98825
$341$	433.106	0.0687800
$342$	0	0
$343$	343.000	0.0539949
$344$	12566.5	1.96960
$345$	0	0
$346$	3358.72	0.521867
$347$	9699.44	1.50056	0.750278	−	0.661122i	$-0.229919\pi$
$347$	9699.44	1.50056	0.750278	+	0.661122i	$0.229919\pi$
$348$	0	0
$349$	−3970.28	−0.608952	−0.304476	−	0.952520i	$-0.598481\pi$
$349$	−3970.28	−0.608952	−0.304476	+	0.952520i	$0.598481\pi$
$350$	−2881.88	−0.440123
$351$	0	0
$352$	809.262	0.122539
$353$	−9933.97	−1.49782	−0.748912	−	0.662669i	$-0.769423\pi$
$353$	−9933.97	−1.49782	−0.748912	+	0.662669i	$0.769423\pi$
$354$	0	0
$355$	4727.59	0.706802
$356$	−15575.3	−2.31879
$357$	0	0
$358$	2085.89	0.307940
$359$	12895.5	1.89582	0.947911	−	0.318536i	$-0.103191\pi$
$359$	12895.5	1.89582	0.947911	+	0.318536i	$0.103191\pi$
$360$	0	0
$361$	19225.9	2.80302
$362$	435.432	0.0632205
$363$	0	0
$364$	−3748.94	−0.539830
$365$	1734.43	0.248724
$366$	0	0
$367$	9626.00	1.36914	0.684568	−	0.728949i	$-0.259991\pi$
$367$	9626.00	1.36914	0.684568	+	0.728949i	$0.259991\pi$
$368$	2680.67	0.379726
$369$	0	0
$370$	5917.27	0.831416
$371$	−313.772	−0.0439090
$372$	0	0
$373$	2947.54	0.409163	0.204581	−	0.978850i	$-0.434417\pi$
$373$	2947.54	0.409163	0.204581	+	0.978850i	$0.434417\pi$
$374$	−7017.86	−0.970281
$375$	0	0
$376$	−16843.4	−2.31019
$377$	−7474.36	−1.02109
$378$	0	0
$379$	−3086.33	−0.418296	−0.209148	−	0.977884i	$-0.567069\pi$
$379$	−3086.33	−0.418296	−0.209148	+	0.977884i	$0.567069\pi$
$380$	15125.2	2.04185
$381$	0	0
$382$	−11196.0	−1.49958
$383$	−8041.26	−1.07282	−0.536409	−	0.843958i	$-0.680219\pi$
$383$	−8041.26	−1.07282	−0.536409	+	0.843958i	$0.680219\pi$
$384$	0	0
$385$	−481.534	−0.0637435
$386$	11117.2	1.46593
$387$	0	0
$388$	16724.3	2.18827
$389$	6313.27	0.822868	0.411434	−	0.911440i	$-0.365028\pi$
$389$	6313.27	0.822868	0.411434	+	0.911440i	$0.365028\pi$
$390$	0	0
$391$	8820.39	1.14083
$392$	1638.22	0.211078
$393$	0	0
$394$	−10868.9	−1.38977
$395$	4071.36	0.518614
$396$	0	0
$397$	4285.76	0.541804	0.270902	−	0.962607i	$-0.412678\pi$
$397$	4285.76	0.541804	0.270902	+	0.962607i	$0.412678\pi$
$398$	−2944.80	−0.370878
$399$	0	0
$400$	−3474.47	−0.434308
$401$	951.273	0.118465	0.0592323	−	0.998244i	$-0.481135\pi$
$401$	951.273	0.118465	0.0592323	+	0.998244i	$0.481135\pi$
$402$	0	0
$403$	1408.13	0.174055
$404$	−5523.70	−0.680234
$405$	0	0
$406$	7012.26	0.857174
$407$	−2171.45	−0.264458
$408$	0	0
$409$	−15156.0	−1.83232	−0.916158	−	0.400818i	$-0.868726\pi$
$409$	−15156.0	−1.83232	−0.916158	+	0.400818i	$0.868726\pi$
$410$	−13038.6	−1.57056
$411$	0	0
$412$	−22060.7	−2.63799
$413$	−4080.23	−0.486138
$414$	0	0
$415$	−1763.60	−0.208607
$416$	2631.11	0.310098
$417$	0	0
$418$	−8515.61	−0.996440
$419$	9558.95	1.11452	0.557261	−	0.830337i	$-0.311852\pi$
$419$	9558.95	1.11452	0.557261	+	0.830337i	$0.311852\pi$
$420$	0	0
$421$	−5919.87	−0.685313	−0.342656	−	0.939461i	$-0.611327\pi$
$421$	−5919.87	−0.685313	−0.342656	+	0.939461i	$0.611327\pi$
$422$	−2427.92	−0.280069
$423$	0	0
$424$	−1498.62	−0.171650
$425$	−11432.3	−1.30482
$426$	0	0
$427$	512.553	0.0580894
$428$	−6079.01	−0.686543
$429$	0	0
$430$	11266.9	1.26358
$431$	−14272.8	−1.59512	−0.797559	−	0.603241i	$-0.793876\pi$
$431$	−14272.8	−1.59512	−0.797559	+	0.603241i	$0.793876\pi$
$432$	0	0
$433$	3731.52	0.414147	0.207073	−	0.978325i	$-0.433606\pi$
$433$	3731.52	0.414147	0.207073	+	0.978325i	$0.433606\pi$
$434$	−1321.08	−0.146114
$435$	0	0
$436$	10052.1	1.10415
$437$	10702.8	1.17159
$438$	0	0
$439$	−186.159	−0.0202389	−0.0101194	−	0.999949i	$-0.503221\pi$
$439$	−186.159	−0.0202389	−0.0101194	+	0.999949i	$0.503221\pi$
$440$	−2299.88	−0.249188
$441$	0	0
$442$	−22816.8	−2.45539
$443$	7883.12	0.845459	0.422730	−	0.906256i	$-0.361072\pi$
$443$	7883.12	0.845459	0.422730	+	0.906256i	$0.361072\pi$
$444$	0	0
$445$	−6504.35	−0.692889
$446$	7456.11	0.791608
$447$	0	0
$448$	−4733.75	−0.499215
$449$	4736.80	0.497869	0.248935	−	0.968520i	$-0.419920\pi$
$449$	4736.80	0.497869	0.248935	+	0.968520i	$0.419920\pi$
$450$	0	0
$451$	4784.75	0.499568
$452$	10284.8	1.07026
$453$	0	0
$454$	−628.942	−0.0650170
$455$	−1565.59	−0.161309
$456$	0	0
$457$	6351.28	0.650110	0.325055	−	0.945695i	$-0.394617\pi$
$457$	6351.28	0.650110	0.325055	+	0.945695i	$0.394617\pi$
$458$	−7696.27	−0.785203
$459$	0	0
$460$	6205.96	0.629031
$461$	−10834.9	−1.09464	−0.547320	−	0.836923i	$-0.684352\pi$
$461$	−10834.9	−1.09464	−0.547320	+	0.836923i	$0.684352\pi$
$462$	0	0
$463$	−6015.91	−0.603851	−0.301925	−	0.953332i	$-0.597629\pi$
$463$	−6015.91	−0.603851	−0.301925	+	0.953332i	$0.597629\pi$
$464$	8454.16	0.845850
$465$	0	0
$466$	−10754.0	−1.06904
$467$	−16477.9	−1.63278	−0.816389	−	0.577503i	$-0.804027\pi$
$467$	−16477.9	−1.63278	−0.816389	+	0.577503i	$0.804027\pi$
$468$	0	0
$469$	−6498.22	−0.639787
$470$	−15101.4	−1.48208
$471$	0	0
$472$	−19487.8	−1.90042
$473$	−4134.59	−0.401921
$474$	0	0
$475$	−13872.2	−1.34000
$476$	13952.4	1.34351
$477$	0	0
$478$	14954.9	1.43101
$479$	3761.95	0.358847	0.179424	−	0.983772i	$-0.442577\pi$
$479$	3761.95	0.358847	0.179424	+	0.983772i	$0.442577\pi$
$480$	0	0
$481$	−7059.90	−0.669239
$482$	−3333.87	−0.315049
$483$	0	0
$484$	1811.98	0.170171
$485$	6984.19	0.653888
$486$	0	0
$487$	−9632.57	−0.896290	−0.448145	−	0.893961i	$-0.647915\pi$
$487$	−9632.57	−0.896290	−0.448145	+	0.893961i	$0.647915\pi$
$488$	2448.03	0.227084
$489$	0	0
$490$	1468.79	0.135415
$491$	12411.6	1.14079	0.570393	−	0.821372i	$-0.306791\pi$
$491$	12411.6	1.14079	0.570393	+	0.821372i	$0.306791\pi$
$492$	0	0
$493$	27817.3	2.54124
$494$	−27686.3	−2.52159
$495$	0	0
$496$	−1592.72	−0.144184
$497$	5291.78	0.477603
$498$	0	0
$499$	9706.78	0.870811	0.435406	−	0.900234i	$-0.356605\pi$
$499$	9706.78	0.870811	0.435406	+	0.900234i	$0.356605\pi$
$500$	−19749.8	−1.76648
$501$	0	0
$502$	−6614.87	−0.588120
$503$	−14127.9	−1.25235	−0.626175	−	0.779682i	$-0.715381\pi$
$503$	−14127.9	−1.25235	−0.626175	+	0.779682i	$0.715381\pi$
$504$	0	0
$505$	−2306.74	−0.203264
$506$	−3494.01	−0.306972
$507$	0	0
$508$	30819.5	2.69172
$509$	−1981.02	−0.172509	−0.0862547	−	0.996273i	$-0.527490\pi$
$509$	−1981.02	−0.172509	−0.0862547	+	0.996273i	$0.527490\pi$
$510$	0	0
$511$	1941.42	0.168069
$512$	−13795.5	−1.19078
$513$	0	0
$514$	27688.3	2.37603
$515$	−9212.71	−0.788273
$516$	0	0
$517$	5541.73	0.471422
$518$	6623.42	0.561808
$519$	0	0
$520$	−7477.48	−0.630594
$521$	−7464.83	−0.627716	−0.313858	−	0.949470i	$-0.601622\pi$
$521$	−7464.83	−0.627716	−0.313858	+	0.949470i	$0.601622\pi$
$522$	0	0
$523$	−1139.06	−0.0952347	−0.0476173	−	0.998866i	$-0.515163\pi$
$523$	−1139.06	−0.0952347	−0.0476173	+	0.998866i	$0.515163\pi$
$524$	−27031.3	−2.25357
$525$	0	0
$526$	−15704.7	−1.30182
$527$	−5240.65	−0.433181
$528$	0	0
$529$	−7775.55	−0.639069
$530$	−1343.63	−0.110120
$531$	0	0
$532$	16930.2	1.37973
$533$	15556.4	1.26421
$534$	0	0
$535$	−2538.64	−0.205150
$536$	−31036.5	−2.50107
$537$	0	0
$538$	−27186.9	−2.17865
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−420.573	−0.0334230	−0.0167115	−	0.999860i	$-0.505320\pi$
$541$	−420.573	−0.0334230	−0.0167115	+	0.999860i	$0.505320\pi$
$542$	35972.5	2.85083
$543$	0	0
$544$	−9792.20	−0.771759
$545$	4197.84	0.329937
$546$	0	0
$547$	16177.8	1.26456	0.632280	−	0.774740i	$-0.282119\pi$
$547$	16177.8	1.26456	0.632280	+	0.774740i	$0.282119\pi$
$548$	−21389.7	−1.66738
$549$	0	0
$550$	4528.67	0.351096
$551$	33754.1	2.60975
$552$	0	0
$553$	4557.23	0.350440
$554$	−42379.1	−3.25003
$555$	0	0
$556$	20993.6	1.60131
$557$	−6605.27	−0.502467	−0.251234	−	0.967926i	$-0.580836\pi$
$557$	−6605.27	−0.502467	−0.251234	+	0.967926i	$0.580836\pi$
$558$	0	0
$559$	−13442.6	−1.01710
$560$	1770.82	0.133626
$561$	0	0
$562$	−16931.2	−1.27082
$563$	14912.2	1.11630	0.558149	−	0.829741i	$-0.311512\pi$
$563$	14912.2	1.11630	0.558149	+	0.829741i	$0.311512\pi$
$564$	0	0
$565$	4295.01	0.319810
$566$	−1461.53	−0.108538
$567$	0	0
$568$	25274.3	1.86706
$569$	21190.9	1.56128	0.780642	−	0.624978i	$-0.214892\pi$
$569$	21190.9	1.56128	0.780642	+	0.624978i	$0.214892\pi$
$570$	0	0
$571$	8669.22	0.635368	0.317684	−	0.948197i	$-0.397095\pi$
$571$	8669.22	0.635368	0.317684	+	0.948197i	$0.397095\pi$
$572$	5891.20	0.430635
$573$	0	0
$574$	−14594.6	−1.06127
$575$	−5691.85	−0.412811
$576$	0	0
$577$	−1289.88	−0.0930649	−0.0465325	−	0.998917i	$-0.514817\pi$
$577$	−1289.88	−0.0930649	−0.0465325	+	0.998917i	$0.514817\pi$
$578$	61368.1	4.41622
$579$	0	0
$580$	19572.1	1.40118
$581$	−1974.07	−0.140961
$582$	0	0
$583$	493.070	0.0350272
$584$	9272.50	0.657019
$585$	0	0
$586$	27690.2	1.95200
$587$	6734.31	0.473517	0.236759	−	0.971568i	$-0.423915\pi$
$587$	6734.31	0.473517	0.236759	+	0.971568i	$0.423915\pi$
$588$	0	0
$589$	−6359.10	−0.444860
$590$	−17472.4	−1.21920
$591$	0	0
$592$	7985.37	0.554386
$593$	−8224.97	−0.569577	−0.284789	−	0.958590i	$-0.591923\pi$
$593$	−8224.97	−0.569577	−0.284789	+	0.958590i	$0.591923\pi$
$594$	0	0
$595$	5826.64	0.401461
$596$	−22621.1	−1.55469
$597$	0	0
$598$	−11359.9	−0.776824
$599$	5061.47	0.345252	0.172626	−	0.984987i	$-0.444775\pi$
$599$	5061.47	0.345252	0.172626	+	0.984987i	$0.444775\pi$
$600$	0	0
$601$	7471.70	0.507117	0.253558	−	0.967320i	$-0.418399\pi$
$601$	7471.70	0.507117	0.253558	+	0.967320i	$0.418399\pi$
$602$	12611.5	0.853830
$603$	0	0
$604$	−2809.24	−0.189249
$605$	756.697	0.0508498
$606$	0	0
$607$	−6029.48	−0.403178	−0.201589	−	0.979470i	$-0.564611\pi$
$607$	−6029.48	−0.403178	−0.201589	+	0.979470i	$0.564611\pi$
$608$	−11882.0	−0.792566
$609$	0	0
$610$	2194.85	0.145684
$611$	18017.5	1.19298
$612$	0	0
$613$	29623.3	1.95183	0.975916	−	0.218148i	$-0.0700014\pi$
$613$	29623.3	1.95183	0.975916	+	0.218148i	$0.0700014\pi$
$614$	−24610.7	−1.61760
$615$	0	0
$616$	−2574.35	−0.168382
$617$	−26531.3	−1.73113	−0.865567	−	0.500793i	$-0.833042\pi$
$617$	−26531.3	−1.73113	−0.865567	+	0.500793i	$0.833042\pi$
$618$	0	0
$619$	16416.9	1.06599	0.532997	−	0.846117i	$-0.321066\pi$
$619$	16416.9	1.06599	0.532997	+	0.846117i	$0.321066\pi$
$620$	−3687.28	−0.238846
$621$	0	0
$622$	21837.1	1.40770
$623$	−7280.57	−0.468202
$624$	0	0
$625$	2488.74	0.159279
$626$	29310.5	1.87138
$627$	0	0
$628$	−49900.4	−3.17077
$629$	26274.8	1.66557
$630$	0	0
$631$	−19608.7	−1.23710	−0.618549	−	0.785746i	$-0.712279\pi$
$631$	−19608.7	−1.23710	−0.618549	+	0.785746i	$0.712279\pi$
$632$	21766.0	1.36995
$633$	0	0
$634$	26929.9	1.68694
$635$	12870.5	0.804329
$636$	0	0
$637$	−1752.42	−0.109001
$638$	−11019.3	−0.683788
$639$	0	0
$640$	−16590.2	−1.02466
$641$	5926.25	0.365168	0.182584	−	0.983190i	$-0.441554\pi$
$641$	5926.25	0.365168	0.182584	+	0.983190i	$0.441554\pi$
$642$	0	0
$643$	−21124.6	−1.29560	−0.647801	−	0.761810i	$-0.724311\pi$
$643$	−21124.6	−1.29560	−0.647801	+	0.761810i	$0.724311\pi$
$644$	6946.57	0.425051
$645$	0	0
$646$	103040.	6.27564
$647$	19495.9	1.18464	0.592321	−	0.805702i	$-0.298212\pi$
$647$	19495.9	1.18464	0.592321	+	0.805702i	$0.298212\pi$
$648$	0	0
$649$	6411.79	0.387804
$650$	14723.8	0.888485
$651$	0	0
$652$	−24607.6	−1.47808
$653$	6274.73	0.376032	0.188016	−	0.982166i	$-0.439794\pi$
$653$	6274.73	0.376032	0.188016	+	0.982166i	$0.439794\pi$
$654$	0	0
$655$	−11288.5	−0.673401
$656$	−17595.7	−1.04725
$657$	0	0
$658$	−16903.6	−1.00148
$659$	23654.7	1.39827	0.699133	−	0.714991i	$-0.253569\pi$
$659$	23654.7	1.39827	0.699133	+	0.714991i	$0.253569\pi$
$660$	0	0
$661$	−22414.0	−1.31891	−0.659457	−	0.751742i	$-0.729214\pi$
$661$	−22414.0	−1.31891	−0.659457	+	0.751742i	$0.729214\pi$
$662$	13258.5	0.778407
$663$	0	0
$664$	−9428.44	−0.551046
$665$	7070.16	0.412284
$666$	0	0
$667$	13849.5	0.803983
$668$	−24098.0	−1.39578
$669$	0	0
$670$	−27826.7	−1.60453
$671$	−805.441	−0.0463393
$672$	0	0
$673$	2434.10	0.139417	0.0697085	−	0.997567i	$-0.477793\pi$
$673$	2434.10	0.139417	0.0697085	+	0.997567i	$0.477793\pi$
$674$	−11614.8	−0.663775
$675$	0	0
$676$	−13746.5	−0.782116
$677$	−4575.10	−0.259728	−0.129864	−	0.991532i	$-0.541454\pi$
$677$	−4575.10	−0.259728	−0.129864	+	0.991532i	$0.541454\pi$
$678$	0	0
$679$	7817.67	0.441848
$680$	27828.9	1.56940
$681$	0	0
$682$	2075.98	0.116559
$683$	−5093.80	−0.285372	−0.142686	−	0.989768i	$-0.545574\pi$
$683$	−5093.80	−0.285372	−0.142686	+	0.989768i	$0.545574\pi$
$684$	0	0
$685$	−8932.51	−0.498239
$686$	1644.08	0.0915032
$687$	0	0
$688$	15204.7	0.842551
$689$	1603.09	0.0886400
$690$	0	0
$691$	20263.2	1.11556	0.557778	−	0.829990i	$-0.311654\pi$
$691$	20263.2	1.11556	0.557778	+	0.829990i	$0.311654\pi$
$692$	10493.3	0.576441
$693$	0	0
$694$	46491.6	2.54294
$695$	8767.08	0.478495
$696$	0	0
$697$	−57896.3	−3.14631
$698$	−19030.5	−1.03197
$699$	0	0
$700$	−9003.59	−0.486148
$701$	7972.79	0.429569	0.214785	−	0.976661i	$-0.431095\pi$
$701$	7972.79	0.429569	0.214785	+	0.976661i	$0.431095\pi$
$702$	0	0
$703$	31882.4	1.71048
$704$	7438.74	0.398236
$705$	0	0
$706$	−47615.8	−2.53831
$707$	−2582.02	−0.137351
$708$	0	0
$709$	13085.1	0.693120	0.346560	−	0.938028i	$-0.387350\pi$
$709$	13085.1	0.693120	0.346560	+	0.938028i	$0.387350\pi$
$710$	22660.4	1.19779
$711$	0	0
$712$	−34773.1	−1.83030
$713$	−2609.19	−0.137047
$714$	0	0
$715$	2460.21	0.128680
$716$	6516.76	0.340143
$717$	0	0
$718$	61811.2	3.21278
$719$	1593.98	0.0826780	0.0413390	−	0.999145i	$-0.486838\pi$
$719$	1593.98	0.0826780	0.0413390	+	0.999145i	$0.486838\pi$
$720$	0	0
$721$	−10312.1	−0.532655
$722$	92154.2	4.75017
$723$	0	0
$724$	1360.38	0.0698318
$725$	−17950.7	−0.919547
$726$	0	0
$727$	34397.6	1.75480	0.877398	−	0.479763i	$-0.159277\pi$
$727$	34397.6	1.75480	0.877398	+	0.479763i	$0.159277\pi$
$728$	−8369.83	−0.426108
$729$	0	0
$730$	8313.54	0.421504
$731$	50029.2	2.53132
$732$	0	0
$733$	16008.0	0.806645	0.403322	−	0.915058i	$-0.367855\pi$
$733$	16008.0	0.806645	0.403322	+	0.915058i	$0.367855\pi$
$734$	46139.6	2.32022
$735$	0	0
$736$	−4875.29	−0.244165
$737$	10211.5	0.510373
$738$	0	0
$739$	−21.8580	−0.00108804	−0.000544018	−	1.00000i	$-0.500173\pi$
$739$	−21.8580	−0.00108804	−0.000544018		1.00000i	$0.500173\pi$
$740$	18486.8	0.918362
$741$	0	0
$742$	−1503.98	−0.0744109
$743$	−23767.2	−1.17353	−0.586767	−	0.809756i	$-0.699600\pi$
$743$	−23767.2	−1.17353	−0.586767	+	0.809756i	$0.699600\pi$
$744$	0	0
$745$	−9446.73	−0.464566
$746$	14128.2	0.693393
$747$	0	0
$748$	−21925.3	−1.07175
$749$	−2841.60	−0.138624
$750$	0	0
$751$	15151.8	0.736215	0.368108	−	0.929783i	$-0.380006\pi$
$751$	15151.8	0.736215	0.368108	+	0.929783i	$0.380006\pi$
$752$	−20379.4	−0.988245
$753$	0	0
$754$	−35826.3	−1.73040
$755$	−1173.16	−0.0565506
$756$	0	0
$757$	−15635.8	−0.750716	−0.375358	−	0.926880i	$-0.622480\pi$
$757$	−15635.8	−0.750716	−0.375358	+	0.926880i	$0.622480\pi$
$758$	−14793.5	−0.708870
$759$	0	0
$760$	33768.2	1.61171
$761$	−11984.8	−0.570892	−0.285446	−	0.958395i	$-0.592142\pi$
$761$	−11984.8	−0.570892	−0.285446	+	0.958395i	$0.592142\pi$
$762$	0	0
$763$	4698.81	0.222947
$764$	−34978.8	−1.65640
$765$	0	0
$766$	−38543.6	−1.81806
$767$	20846.3	0.981378
$768$	0	0
$769$	−79.6326	−0.00373423	−0.00186712	−	0.999998i	$-0.500594\pi$
$769$	−79.6326	−0.00373423	−0.00186712	+	0.999998i	$0.500594\pi$
$770$	−2308.11	−0.108024
$771$	0	0
$772$	34732.4	1.61923
$773$	−6298.84	−0.293083	−0.146542	−	0.989205i	$-0.546814\pi$
$773$	−6298.84	−0.293083	−0.146542	+	0.989205i	$0.546814\pi$
$774$	0	0
$775$	3381.82	0.156747
$776$	37338.4	1.72728
$777$	0	0
$778$	30261.0	1.39448
$779$	−70252.5	−3.23114
$780$	0	0
$781$	−8315.65	−0.380995
$782$	42278.1	1.93333
$783$	0	0
$784$	1982.14	0.0902944
$785$	−20838.8	−0.947474
$786$	0	0
$787$	18770.7	0.850193	0.425096	−	0.905148i	$-0.360240\pi$
$787$	18770.7	0.850193	0.425096	+	0.905148i	$0.360240\pi$
$788$	−33956.8	−1.53510
$789$	0	0
$790$	19515.0	0.878876
$791$	4807.58	0.216103
$792$	0	0
$793$	−2618.69	−0.117266
$794$	20542.6	0.918175
$795$	0	0
$796$	−9200.17	−0.409662
$797$	7740.06	0.343999	0.171999	−	0.985097i	$-0.444977\pi$
$797$	7740.06	0.343999	0.171999	+	0.985097i	$0.444977\pi$
$798$	0	0
$799$	−67055.8	−2.96904
$800$	6318.96	0.279261
$801$	0	0
$802$	4559.67	0.200758
$803$	−3050.80	−0.134073
$804$	0	0
$805$	2900.94	0.127012
$806$	6749.50	0.294964
$807$	0	0
$808$	−12332.1	−0.536934
$809$	6261.69	0.272125	0.136063	−	0.990700i	$-0.456555\pi$
$809$	6261.69	0.272125	0.136063	+	0.990700i	$0.456555\pi$
$810$	0	0
$811$	−30638.5	−1.32659	−0.663294	−	0.748359i	$-0.730842\pi$
$811$	−30638.5	−1.32659	−0.663294	+	0.748359i	$0.730842\pi$
$812$	21907.8	0.946813
$813$	0	0
$814$	−10408.2	−0.448168
$815$	−10276.3	−0.441672
$816$	0	0
$817$	60706.4	2.59957
$818$	−72646.3	−3.10516
$819$	0	0
$820$	−40735.4	−1.73481
$821$	−19473.1	−0.827790	−0.413895	−	0.910325i	$-0.635832\pi$
$821$	−19473.1	−0.827790	−0.413895	+	0.910325i	$0.635832\pi$
$822$	0	0
$823$	32950.7	1.39561	0.697807	−	0.716286i	$-0.254159\pi$
$823$	32950.7	1.39561	0.697807	+	0.716286i	$0.254159\pi$
$824$	−49252.4	−2.08227
$825$	0	0
$826$	−19557.5	−0.823840
$827$	35537.8	1.49428	0.747140	−	0.664667i	$-0.231427\pi$
$827$	35537.8	1.49428	0.747140	+	0.664667i	$0.231427\pi$
$828$	0	0
$829$	−23814.6	−0.997727	−0.498863	−	0.866681i	$-0.666249\pi$
$829$	−23814.6	−0.997727	−0.498863	+	0.866681i	$0.666249\pi$
$830$	−8453.34	−0.353518
$831$	0	0
$832$	24185.2	1.00778
$833$	6521.99	0.271277
$834$	0	0
$835$	−10063.5	−0.417081
$836$	−26604.5	−1.10064
$837$	0	0
$838$	45818.2	1.88874
$839$	−2519.64	−0.103680	−0.0518402	−	0.998655i	$-0.516509\pi$
$839$	−2519.64	−0.103680	−0.0518402	+	0.998655i	$0.516509\pi$
$840$	0	0
$841$	19289.1	0.790892
$842$	−28375.3	−1.16137
$843$	0	0
$844$	−7585.33	−0.309358
$845$	−5740.63	−0.233708
$846$	0	0
$847$	847.000	0.0343604
$848$	−1813.24	−0.0734279
$849$	0	0
$850$	−54797.6	−2.21123
$851$	13081.6	0.526946
$852$	0	0
$853$	35395.1	1.42076	0.710379	−	0.703819i	$-0.248523\pi$
$853$	35395.1	1.42076	0.710379	+	0.703819i	$0.248523\pi$
$854$	2456.79	0.0984420
$855$	0	0
$856$	−13571.9	−0.541914
$857$	19373.0	0.772193	0.386096	−	0.922458i	$-0.373823\pi$
$857$	19373.0	0.772193	0.386096	+	0.922458i	$0.373823\pi$
$858$	0	0
$859$	20398.7	0.810237	0.405119	−	0.914264i	$-0.367230\pi$
$859$	20398.7	0.810237	0.405119	+	0.914264i	$0.367230\pi$
$860$	35200.2	1.39572
$861$	0	0
$862$	−68412.7	−2.70319
$863$	16248.7	0.640916	0.320458	−	0.947263i	$-0.396163\pi$
$863$	16248.7	0.640916	0.320458	+	0.947263i	$0.396163\pi$
$864$	0	0
$865$	4382.10	0.172249
$866$	17886.1	0.701839
$867$	0	0
$868$	−4127.32	−0.161394
$869$	−7161.37	−0.279554
$870$	0	0
$871$	33200.1	1.29155
$872$	22442.2	0.871547
$873$	0	0
$874$	51301.1	1.98545
$875$	−9231.95	−0.356682
$876$	0	0
$877$	−21500.2	−0.827833	−0.413916	−	0.910315i	$-0.635839\pi$
$877$	−21500.2	−0.827833	−0.413916	+	0.910315i	$0.635839\pi$
$878$	−892.301	−0.0342981
$879$	0	0
$880$	−2782.71	−0.106597
$881$	−28093.4	−1.07434	−0.537169	−	0.843475i	$-0.680506\pi$
$881$	−28093.4	−1.07434	−0.537169	+	0.843475i	$0.680506\pi$
$882$	0	0
$883$	17244.0	0.657198	0.328599	−	0.944469i	$-0.393423\pi$
$883$	17244.0	0.657198	0.328599	+	0.944469i	$0.393423\pi$
$884$	−71284.4	−2.71217
$885$	0	0
$886$	37785.6	1.43277
$887$	17179.4	0.650312	0.325156	−	0.945660i	$-0.394583\pi$
$887$	17179.4	0.650312	0.325156	+	0.945660i	$0.394583\pi$
$888$	0	0
$889$	14406.4	0.543504
$890$	−31176.8	−1.17421
$891$	0	0
$892$	23294.4	0.874390
$893$	−81366.8	−3.04909
$894$	0	0
$895$	2721.44	0.101640
$896$	−18570.1	−0.692391
$897$	0	0
$898$	22704.6	0.843721
$899$	−8228.73	−0.305276
$900$	0	0
$901$	−5966.23	−0.220604
$902$	22934.4	0.846600
$903$	0	0
$904$	22961.7	0.844795
$905$	568.105	0.0208668
$906$	0	0
$907$	−16174.7	−0.592141	−0.296070	−	0.955166i	$-0.595676\pi$
$907$	−16174.7	−0.592141	−0.296070	+	0.955166i	$0.595676\pi$
$908$	−1964.95	−0.0718161
$909$	0	0
$910$	−7504.22	−0.273365
$911$	−14781.8	−0.537590	−0.268795	−	0.963197i	$-0.586625\pi$
$911$	−14781.8	−0.537590	−0.268795	+	0.963197i	$0.586625\pi$
$912$	0	0
$913$	3102.10	0.112448
$914$	30443.1	1.10172
$915$	0	0
$916$	−24044.8	−0.867316
$917$	−12635.6	−0.455033
$918$	0	0
$919$	−36700.1	−1.31733	−0.658664	−	0.752438i	$-0.728878\pi$
$919$	−36700.1	−1.31733	−0.658664	+	0.752438i	$0.728878\pi$
$920$	13855.3	0.496518
$921$	0	0
$922$	−51933.9	−1.85505
$923$	−27036.2	−0.964148
$924$	0	0
$925$	−16955.3	−0.602689
$926$	−28835.6	−1.02332
$927$	0	0
$928$	−15375.5	−0.543884
$929$	−9845.54	−0.347709	−0.173855	−	0.984771i	$-0.555622\pi$
$929$	−9845.54	−0.347709	−0.173855	+	0.984771i	$0.555622\pi$
$930$	0	0
$931$	7913.90	0.278590
$932$	−33597.8	−1.18083
$933$	0	0
$934$	−78982.5	−2.76701
$935$	−9156.15	−0.320255
$936$	0	0
$937$	−2671.44	−0.0931401	−0.0465700	−	0.998915i	$-0.514829\pi$
$937$	−2671.44	−0.0931401	−0.0465700	+	0.998915i	$0.514829\pi$
$938$	−31147.5	−1.08422
$939$	0	0
$940$	−47180.0	−1.63707
$941$	−17228.1	−0.596834	−0.298417	−	0.954436i	$-0.596459\pi$
$941$	−17228.1	−0.596834	−0.298417	+	0.954436i	$0.596459\pi$
$942$	0	0
$943$	−28825.1	−0.995412
$944$	−23579.0	−0.812957
$945$	0	0
$946$	−19818.1	−0.681121
$947$	−40042.3	−1.37402	−0.687012	−	0.726646i	$-0.741078\pi$
$947$	−40042.3	−1.37402	−0.687012	+	0.726646i	$0.741078\pi$
$948$	0	0
$949$	−9918.90	−0.339284
$950$	−66492.4	−2.27084
$951$	0	0
$952$	31150.0	1.06048
$953$	−37714.4	−1.28194	−0.640970	−	0.767566i	$-0.721468\pi$
$953$	−37714.4	−1.28194	−0.640970	+	0.767566i	$0.721468\pi$
$954$	0	0
$955$	−14607.4	−0.494957
$956$	46722.2	1.58065
$957$	0	0
$958$	18031.9	0.608125
$959$	−9998.51	−0.336672
$960$	0	0
$961$	−28240.7	−0.947962
$962$	−33839.7	−1.13413
$963$	0	0
$964$	−10415.7	−0.347995
$965$	14504.5	0.483852
$966$	0	0
$967$	30455.1	1.01279	0.506397	−	0.862301i	$-0.330977\pi$
$967$	30455.1	1.01279	0.506397	+	0.862301i	$0.330977\pi$
$968$	4045.40	0.134322
$969$	0	0
$970$	33476.8	1.10812
$971$	−27850.4	−0.920456	−0.460228	−	0.887801i	$-0.652232\pi$
$971$	−27850.4	−0.920456	−0.460228	+	0.887801i	$0.652232\pi$
$972$	0	0
$973$	9813.33	0.323331
$974$	−46171.1	−1.51891
$975$	0	0
$976$	2961.96	0.0971416
$977$	18375.9	0.601736	0.300868	−	0.953666i	$-0.402724\pi$
$977$	18375.9	0.601736	0.300868	+	0.953666i	$0.402724\pi$
$978$	0	0
$979$	11440.9	0.373496
$980$	4588.82	0.149576
$981$	0	0
$982$	59491.5	1.93325
$983$	−20597.9	−0.668334	−0.334167	−	0.942514i	$-0.608455\pi$
$983$	−20597.9	−0.668334	−0.334167	+	0.942514i	$0.608455\pi$
$984$	0	0
$985$	−14180.6	−0.458713
$986$	133335.	4.30654
$987$	0	0
$988$	−86497.9	−2.78529
$989$	24908.3	0.800846
$990$	0	0
$991$	40687.6	1.30422	0.652111	−	0.758124i	$-0.273884\pi$
$991$	40687.6	1.30422	0.652111	+	0.758124i	$0.273884\pi$
$992$	2896.66	0.0927107
$993$	0	0
$994$	25364.7	0.809376
$995$	−3842.06	−0.122413
$996$	0	0
$997$	9152.42	0.290732	0.145366	−	0.989378i	$-0.453564\pi$
$997$	9152.42	0.290732	0.145366	+	0.989378i	$0.453564\pi$
$998$	46526.8	1.47573
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.p.1.5		5
3.2	odd	2		231.4.a.k.1.1	✓	5
21.20	even	2		1617.4.a.n.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.1	✓	5	3.2	odd	2
693.4.a.p.1.5		5	1.1	even	1	trivial
1617.4.a.n.1.1		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)

\(f(q)\)	\(=\)	\(q+4.79323 q^{2} +14.9751 q^{4} +6.25369 q^{5} +7.00000 q^{7} +33.4331 q^{8} +O(q^{10})\)	\(q+4.79323 q^{2} +14.9751 q^{4} +6.25369 q^{5} +7.00000 q^{7} +33.4331 q^{8} +29.9754 q^{10} -11.0000 q^{11} -35.7637 q^{13} +33.5526 q^{14} +40.4519 q^{16} +133.102 q^{17} +161.508 q^{19} +93.6494 q^{20} -52.7255 q^{22} +66.2680 q^{23} -85.8913 q^{25} -171.424 q^{26} +104.825 q^{28} +208.993 q^{29} -39.3732 q^{31} -73.5692 q^{32} +637.987 q^{34} +43.7759 q^{35} +197.404 q^{37} +774.146 q^{38} +209.080 q^{40} -434.978 q^{41} +375.872 q^{43} -164.726 q^{44} +317.638 q^{46} -503.794 q^{47} +49.0000 q^{49} -411.697 q^{50} -535.563 q^{52} -44.8246 q^{53} -68.7906 q^{55} +234.031 q^{56} +1001.75 q^{58} -582.890 q^{59} +73.2219 q^{61} -188.725 q^{62} -676.249 q^{64} -223.655 q^{65} -928.317 q^{67} +1993.21 q^{68} +209.828 q^{70} +755.968 q^{71} +277.345 q^{73} +946.204 q^{74} +2418.60 q^{76} -77.0000 q^{77} +651.033 q^{79} +252.974 q^{80} -2084.95 q^{82} -282.009 q^{83} +832.378 q^{85} +1801.64 q^{86} -367.764 q^{88} -1040.08 q^{89} -250.346 q^{91} +992.367 q^{92} -2414.80 q^{94} +1010.02 q^{95} +1116.81 q^{97} +234.868 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8}+O(q^{10}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8	\( 5 q + q^{2} + 21 q^{4} - 21 q^{5} + 35 q^{7} + 42 q^{8} - 23 q^{10} - 55 q^{11} + 101 q^{13} + 7 q^{14} - 7 q^{16} + 20 q^{17} + 237 q^{19} - 85 q^{20} - 11 q^{22} + 80 q^{23} + 486 q^{25} - 165 q^{26} + 147 q^{28} + 11 q^{29} + 316 q^{31} - 453 q^{32} + 936 q^{34} - 147 q^{35} + 319 q^{37} - 89 q^{38} + 624 q^{40} - 1190 q^{41} + 88 q^{43} - 231 q^{44} + 1000 q^{46} - 377 q^{47} + 245 q^{49} + 644 q^{50} + 1001 q^{52} + 992 q^{53} + 231 q^{55} + 294 q^{56} + 721 q^{58} - 71 q^{59} - 574 q^{61} - 272 q^{62} - 1380 q^{64} - 589 q^{65} - 527 q^{67} + 2974 q^{68} - 161 q^{70} + 1156 q^{71} + 1061 q^{73} + 1609 q^{74} + 2399 q^{76} - 385 q^{77} + 588 q^{79} + 1643 q^{80} - 2602 q^{82} + 212 q^{83} + 1918 q^{85} + 4760 q^{86} - 462 q^{88} - 1030 q^{89} + 707 q^{91} + 1174 q^{92} - 1799 q^{94} + 3593 q^{95} + 2488 q^{97} + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 21 * q^5 + 35 * q^7 + 42 * q^8 - 23 * q^10 - 55 * q^11 + 101 * q^13 + 7 * q^14 - 7 * q^16 + 20 * q^17 + 237 * q^19 - 85 * q^20 - 11 * q^22 + 80 * q^23 + 486 * q^25 - 165 * q^26 + 147 * q^28 + 11 * q^29 + 316 * q^31 - 453 * q^32 + 936 * q^34 - 147 * q^35 + 319 * q^37 - 89 * q^38 + 624 * q^40 - 1190 * q^41 + 88 * q^43 - 231 * q^44 + 1000 * q^46 - 377 * q^47 + 245 * q^49 + 644 * q^50 + 1001 * q^52 + 992 * q^53 + 231 * q^55 + 294 * q^56 + 721 * q^58 - 71 * q^59 - 574 * q^61 - 272 * q^62 - 1380 * q^64 - 589 * q^65 - 527 * q^67 + 2974 * q^68 - 161 * q^70 + 1156 * q^71 + 1061 * q^73 + 1609 * q^74 + 2399 * q^76 - 385 * q^77 + 588 * q^79 + 1643 * q^80 - 2602 * q^82 + 212 * q^83 + 1918 * q^85 + 4760 * q^86 - 462 * q^88 - 1030 * q^89 + 707 * q^91 + 1174 * q^92 - 1799 * q^94 + 3593 * q^95 + 2488 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more