LMFDB - Embedded newform 693.4.a.o.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:	$5$
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} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.44399$ 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)

$f(q)$	$=$	\(q-4.44399 q^{2} +11.7491 q^{4} +22.0150 q^{5} +7.00000 q^{7} -16.6609 q^{8} -97.8345 q^{10} -11.0000 q^{11} -51.5769 q^{13} -31.1080 q^{14} -19.9519 q^{16} +26.5590 q^{17} +99.6432 q^{19} +258.656 q^{20} +48.8839 q^{22} -28.1455 q^{23} +359.660 q^{25} +229.207 q^{26} +82.2435 q^{28} +43.9369 q^{29} -83.8402 q^{31} +221.953 q^{32} -118.028 q^{34} +154.105 q^{35} +306.353 q^{37} -442.814 q^{38} -366.789 q^{40} -200.991 q^{41} -13.7546 q^{43} -129.240 q^{44} +125.079 q^{46} +266.533 q^{47} +49.0000 q^{49} -1598.33 q^{50} -605.980 q^{52} -308.867 q^{53} -242.165 q^{55} -116.626 q^{56} -195.255 q^{58} +622.446 q^{59} -87.3303 q^{61} +372.585 q^{62} -826.742 q^{64} -1135.46 q^{65} +608.395 q^{67} +312.044 q^{68} -684.842 q^{70} +464.926 q^{71} -255.407 q^{73} -1361.43 q^{74} +1170.72 q^{76} -77.0000 q^{77} +261.237 q^{79} -439.240 q^{80} +893.204 q^{82} -953.986 q^{83} +584.696 q^{85} +61.1255 q^{86} +183.269 q^{88} +839.910 q^{89} -361.038 q^{91} -330.684 q^{92} -1184.47 q^{94} +2193.64 q^{95} -349.146 q^{97} -217.756 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8} - 10 q^{10} - 55 q^{11} - 50 q^{13} - 7 q^{14} + 433 q^{16} - 222 q^{17} + 160 q^{19} + 430 q^{20} + 11 q^{22} - 54 q^{23} + 125 q^{25} + 1026 q^{26}+ \cdots - 49 q^{98}+O(q^{100}) $ 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8 - 10 * q^10 - 55 * q^11 - 50 * q^13 - 7 * q^14 + 433 * q^16 - 222 * q^17 + 160 * q^19 + 430 * q^20 + 11 * q^22 - 54 * q^23 + 125 * q^25 + 1026 * q^26 + 315 * q^28 - 14 * q^29 - 34 * q^31 + 583 * q^32 - 750 * q^34 + 168 * q^35 + 1044 * q^37 + 156 * q^38 + 1158 * q^40 + 114 * q^41 + 672 * q^43 - 495 * q^44 - 1224 * q^46 + 292 * q^47 + 245 * q^49 - 1143 * q^50 - 914 * q^52 + 710 * q^53 - 264 * q^55 - 399 * q^56 - 810 * q^58 - 270 * q^59 + 138 * q^61 + 480 * q^62 + 3001 * q^64 + 196 * q^65 + 1942 * q^67 - 3130 * q^68 - 70 * q^70 + 278 * q^71 - 338 * q^73 - 462 * q^74 - 1332 * q^76 - 385 * q^77 + 576 * q^79 + 1602 * q^80 + 1386 * q^82 - 1644 * q^83 + 360 * q^85 - 2020 * q^86 + 627 * q^88 + 3656 * q^89 - 350 * q^91 + 748 * q^92 + 976 * q^94 + 1276 * q^95 + 692 * q^97 - 49 * 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 $

$2$	−4.44399	−1.57119	−0.785594	−	0.618742i	$-0.787643\pi$
$2$	−4.44399	−1.57119	−0.785594	+	0.618742i	$0.787643\pi$
$3$	0	0
$3$	0	0
$4$	11.7491	1.46863
$5$	22.0150	1.96908	0.984541	−	0.175155i	$-0.0560428\pi$
$5$	22.0150	1.96908	0.984541	+	0.175155i	$0.0560428\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−16.6609	−0.736313
$9$	0	0
$10$	−97.8345	−3.09380
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−51.5769	−1.10037	−0.550186	−	0.835042i	$-0.685443\pi$
$13$	−51.5769	−1.10037	−0.550186	+	0.835042i	$0.685443\pi$
$14$	−31.1080	−0.593854
$15$	0	0
$16$	−19.9519	−0.311748
$17$	26.5590	0.378912	0.189456	−	0.981889i	$-0.439328\pi$
$17$	26.5590	0.378912	0.189456	+	0.981889i	$0.439328\pi$
$18$	0	0
$19$	99.6432	1.20314	0.601571	−	0.798819i	$-0.294542\pi$
$19$	99.6432	1.20314	0.601571	+	0.798819i	$0.294542\pi$
$20$	258.656	2.89186
$21$	0	0
$22$	48.8839	0.473731
$23$	−28.1455	−0.255163	−0.127582	−	0.991828i	$-0.540721\pi$
$23$	−28.1455	−0.255163	−0.127582	+	0.991828i	$0.540721\pi$
$24$	0	0
$25$	359.660	2.87728
$26$	229.207	1.72889
$27$	0	0
$28$	82.2435	0.555092
$29$	43.9369	0.281340	0.140670	−	0.990057i	$-0.455074\pi$
$29$	43.9369	0.281340	0.140670	+	0.990057i	$0.455074\pi$
$30$	0	0
$31$	−83.8402	−0.485747	−0.242873	−	0.970058i	$-0.578090\pi$
$31$	−83.8402	−0.485747	−0.242873	+	0.970058i	$0.578090\pi$
$32$	221.953	1.22613
$33$	0	0
$34$	−118.028	−0.595342
$35$	154.105	0.744243
$36$	0	0
$37$	306.353	1.36119	0.680596	−	0.732659i	$-0.261721\pi$
$37$	306.353	1.36119	0.680596	+	0.732659i	$0.261721\pi$
$38$	−442.814	−1.89036
$39$	0	0
$40$	−366.789	−1.44986
$41$	−200.991	−0.765599	−0.382800	−	0.923831i	$-0.625040\pi$
$41$	−200.991	−0.765599	−0.382800	+	0.923831i	$0.625040\pi$
$42$	0	0
$43$	−13.7546	−0.0487805	−0.0243903	−	0.999703i	$-0.507764\pi$
$43$	−13.7546	−0.0487805	−0.0243903	+	0.999703i	$0.507764\pi$
$44$	−129.240	−0.442810
$45$	0	0
$46$	125.079	0.400909
$47$	266.533	0.827189	0.413594	−	0.910461i	$-0.364273\pi$
$47$	266.533	0.827189	0.413594	+	0.910461i	$0.364273\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−1598.33	−4.52075
$51$	0	0
$52$	−605.980	−1.61605
$53$	−308.867	−0.800493	−0.400247	−	0.916407i	$-0.631076\pi$
$53$	−308.867	−0.800493	−0.400247	+	0.916407i	$0.631076\pi$
$54$	0	0
$55$	−242.165	−0.593700
$56$	−116.626	−0.278300
$57$	0	0
$58$	−195.255	−0.442039
$59$	622.446	1.37348	0.686742	−	0.726901i	$-0.259040\pi$
$59$	622.446	1.37348	0.686742	+	0.726901i	$0.259040\pi$
$60$	0	0
$61$	−87.3303	−0.183303	−0.0916516	−	0.995791i	$-0.529215\pi$
$61$	−87.3303	−0.183303	−0.0916516	+	0.995791i	$0.529215\pi$
$62$	372.585	0.763200
$63$	0	0
$64$	−826.742	−1.61473
$65$	−1135.46	−2.16672
$66$	0	0
$67$	608.395	1.10936	0.554681	−	0.832063i	$-0.312840\pi$
$67$	608.395	1.10936	0.554681	+	0.832063i	$0.312840\pi$
$68$	312.044	0.556483
$69$	0	0
$70$	−684.842	−1.16935
$71$	464.926	0.777135	0.388567	−	0.921420i	$-0.372970\pi$
$71$	464.926	0.777135	0.388567	+	0.921420i	$0.372970\pi$
$72$	0	0
$73$	−255.407	−0.409495	−0.204747	−	0.978815i	$-0.565637\pi$
$73$	−255.407	−0.409495	−0.204747	+	0.978815i	$0.565637\pi$
$74$	−1361.43	−2.13869
$75$	0	0
$76$	1170.72	1.76698
$77$	−77.0000	−0.113961
$78$	0	0
$79$	261.237	0.372043	0.186022	−	0.982546i	$-0.440441\pi$
$79$	261.237	0.372043	0.186022	+	0.982546i	$0.440441\pi$
$80$	−439.240	−0.613857
$81$	0	0
$82$	893.204	1.20290
$83$	−953.986	−1.26161	−0.630804	−	0.775942i	$-0.717275\pi$
$83$	−953.986	−1.26161	−0.630804	+	0.775942i	$0.717275\pi$
$84$	0	0
$85$	584.696	0.746109
$86$	61.1255	0.0766434
$87$	0	0
$88$	183.269	0.222007
$89$	839.910	1.00034	0.500170	−	0.865927i	$-0.333271\pi$
$89$	839.910	1.00034	0.500170	+	0.865927i	$0.333271\pi$
$90$	0	0
$91$	−361.038	−0.415902
$92$	−330.684	−0.374741
$93$	0	0
$94$	−1184.47	−1.29967
$95$	2193.64	2.36909
$96$	0	0
$97$	−349.146	−0.365468	−0.182734	−	0.983162i	$-0.558495\pi$
$97$	−349.146	−0.365468	−0.182734	+	0.983162i	$0.558495\pi$
$98$	−217.756	−0.224456
$99$	0	0
$100$	4225.68	4.22568
$101$	−1492.44	−1.47033	−0.735163	−	0.677890i	$-0.762894\pi$
$101$	−1492.44	−1.47033	−0.735163	+	0.677890i	$0.762894\pi$
$102$	0	0
$103$	558.687	0.534457	0.267228	−	0.963633i	$-0.413892\pi$
$103$	558.687	0.534457	0.267228	+	0.963633i	$0.413892\pi$
$104$	859.314	0.810218
$105$	0	0
$106$	1372.60	1.25773
$107$	694.047	0.627066	0.313533	−	0.949577i	$-0.398487\pi$
$107$	694.047	0.627066	0.313533	+	0.949577i	$0.398487\pi$
$108$	0	0
$109$	−341.005	−0.299654	−0.149827	−	0.988712i	$-0.547872\pi$
$109$	−341.005	−0.299654	−0.149827	+	0.988712i	$0.547872\pi$
$110$	1076.18	0.932815
$111$	0	0
$112$	−139.663	−0.117830
$113$	990.910	0.824929	0.412464	−	0.910974i	$-0.364668\pi$
$113$	990.910	0.824929	0.412464	+	0.910974i	$0.364668\pi$
$114$	0	0
$115$	−619.624	−0.502437
$116$	516.218	0.413186
$117$	0	0
$118$	−2766.15	−2.15800
$119$	185.913	0.143215
$120$	0	0
$121$	121.000	0.0909091
$122$	388.095	0.288004
$123$	0	0
$124$	−985.045	−0.713384
$125$	5166.05	3.69652
$126$	0	0
$127$	−666.090	−0.465401	−0.232700	−	0.972548i	$-0.574756\pi$
$127$	−666.090	−0.465401	−0.232700	+	0.972548i	$0.574756\pi$
$128$	1898.41	1.31092
$129$	0	0
$130$	5046.00	3.40433
$131$	−30.4356	−0.0202990	−0.0101495	−	0.999948i	$-0.503231\pi$
$131$	−30.4356	−0.0202990	−0.0101495	+	0.999948i	$0.503231\pi$
$132$	0	0
$133$	697.502	0.454745
$134$	−2703.70	−1.74302
$135$	0	0
$136$	−442.496	−0.278998
$137$	2810.25	1.75252	0.876262	−	0.481836i	$-0.160030\pi$
$137$	2810.25	1.75252	0.876262	+	0.481836i	$0.160030\pi$
$138$	0	0
$139$	3110.49	1.89804	0.949021	−	0.315212i	$-0.102076\pi$
$139$	3110.49	1.89804	0.949021	+	0.315212i	$0.102076\pi$
$140$	1810.59	1.09302
$141$	0	0
$142$	−2066.13	−1.22103
$143$	567.345	0.331775
$144$	0	0
$145$	967.270	0.553982
$146$	1135.03	0.643394
$147$	0	0
$148$	3599.36	1.99909
$149$	−1916.92	−1.05396	−0.526979	−	0.849878i	$-0.676675\pi$
$149$	−1916.92	−1.05396	−0.526979	+	0.849878i	$0.676675\pi$
$150$	0	0
$151$	−2289.28	−1.23377	−0.616883	−	0.787055i	$-0.711605\pi$
$151$	−2289.28	−1.23377	−0.616883	+	0.787055i	$0.711605\pi$
$152$	−1660.14	−0.885889
$153$	0	0
$154$	342.187	0.179054
$155$	−1845.74	−0.956475
$156$	0	0
$157$	280.036	0.142352	0.0711762	−	0.997464i	$-0.477325\pi$
$157$	280.036	0.142352	0.0711762	+	0.997464i	$0.477325\pi$
$158$	−1160.93	−0.584550
$159$	0	0
$160$	4886.29	2.41435
$161$	−197.019	−0.0964426
$162$	0	0
$163$	−866.571	−0.416411	−0.208206	−	0.978085i	$-0.566762\pi$
$163$	−866.571	−0.416411	−0.208206	+	0.978085i	$0.566762\pi$
$164$	−2361.46	−1.12439
$165$	0	0
$166$	4239.51	1.98223
$167$	1965.18	0.910600	0.455300	−	0.890338i	$-0.349532\pi$
$167$	1965.18	0.910600	0.455300	+	0.890338i	$0.349532\pi$
$168$	0	0
$169$	463.173	0.210820
$170$	−2598.39	−1.17228
$171$	0	0
$172$	−161.604	−0.0716407
$173$	−3956.88	−1.73894	−0.869469	−	0.493988i	$-0.835539\pi$
$173$	−3956.88	−1.73894	−0.869469	+	0.493988i	$0.835539\pi$
$174$	0	0
$175$	2517.62	1.08751
$176$	219.471	0.0939956
$177$	0	0
$178$	−3732.55	−1.57172
$179$	3143.58	1.31264	0.656318	−	0.754484i	$-0.272113\pi$
$179$	3143.58	1.31264	0.656318	+	0.754484i	$0.272113\pi$
$180$	0	0
$181$	683.772	0.280798	0.140399	−	0.990095i	$-0.455162\pi$
$181$	683.772	0.280798	0.140399	+	0.990095i	$0.455162\pi$
$182$	1604.45	0.653460
$183$	0	0
$184$	468.929	0.187880
$185$	6744.36	2.68030
$186$	0	0
$187$	−292.149	−0.114246
$188$	3131.52	1.21484
$189$	0	0
$190$	−9748.54	−3.72228
$191$	−2739.68	−1.03789	−0.518944	−	0.854809i	$-0.673675\pi$
$191$	−2739.68	−1.03789	−0.518944	+	0.854809i	$0.673675\pi$
$192$	0	0
$193$	2651.93	0.989067	0.494534	−	0.869159i	$-0.335339\pi$
$193$	2651.93	0.989067	0.494534	+	0.869159i	$0.335339\pi$
$194$	1551.60	0.574219
$195$	0	0
$196$	575.705	0.209805
$197$	1879.52	0.679749	0.339874	−	0.940471i	$-0.389615\pi$
$197$	1879.52	0.679749	0.339874	+	0.940471i	$0.389615\pi$
$198$	0	0
$199$	−3119.39	−1.11119	−0.555597	−	0.831452i	$-0.687510\pi$
$199$	−3119.39	−1.11119	−0.555597	+	0.831452i	$0.687510\pi$
$200$	−5992.25	−2.11858
$201$	0	0
$202$	6632.37	2.31016
$203$	307.558	0.106337
$204$	0	0
$205$	−4424.82	−1.50753
$206$	−2482.80	−0.839733
$207$	0	0
$208$	1029.05	0.343039
$209$	−1096.08	−0.362761
$210$	0	0
$211$	−520.718	−0.169894	−0.0849472	−	0.996385i	$-0.527072\pi$
$211$	−520.718	−0.169894	−0.0849472	+	0.996385i	$0.527072\pi$
$212$	−3628.90	−1.17563
$213$	0	0
$214$	−3084.34	−0.985238
$215$	−302.808	−0.0960528
$216$	0	0
$217$	−586.881	−0.183595
$218$	1515.42	0.470814
$219$	0	0
$220$	−2845.21	−0.871929
$221$	−1369.83	−0.416944
$222$	0	0
$223$	2101.08	0.630935	0.315467	−	0.948936i	$-0.397839\pi$
$223$	2101.08	0.630935	0.315467	+	0.948936i	$0.397839\pi$
$224$	1553.67	0.463433
$225$	0	0
$226$	−4403.60	−1.29612
$227$	6051.96	1.76953	0.884764	−	0.466040i	$-0.154320\pi$
$227$	6051.96	1.76953	0.884764	+	0.466040i	$0.154320\pi$
$228$	0	0
$229$	−2995.73	−0.864470	−0.432235	−	0.901761i	$-0.642275\pi$
$229$	−2995.73	−0.864470	−0.432235	+	0.901761i	$0.642275\pi$
$230$	2753.61	0.789424
$231$	0	0
$232$	−732.026	−0.207155
$233$	65.3656	0.0183787	0.00918936	−	0.999958i	$-0.497075\pi$
$233$	65.3656	0.0183787	0.00918936	+	0.999958i	$0.497075\pi$
$234$	0	0
$235$	5867.73	1.62880
$236$	7313.17	2.01715
$237$	0	0
$238$	−826.196	−0.225018
$239$	1102.33	0.298343	0.149171	−	0.988811i	$-0.452339\pi$
$239$	1102.33	0.298343	0.149171	+	0.988811i	$0.452339\pi$
$240$	0	0
$241$	5297.43	1.41592	0.707962	−	0.706250i	$-0.249615\pi$
$241$	5297.43	1.41592	0.707962	+	0.706250i	$0.249615\pi$
$242$	−537.723	−0.142835
$243$	0	0
$244$	−1026.05	−0.269205
$245$	1078.74	0.281297
$246$	0	0
$247$	−5139.28	−1.32391
$248$	1396.85	0.357661
$249$	0	0
$250$	−22957.9	−5.80793
$251$	−177.964	−0.0447530	−0.0223765	−	0.999750i	$-0.507123\pi$
$251$	−177.964	−0.0447530	−0.0223765	+	0.999750i	$0.507123\pi$
$252$	0	0
$253$	309.601	0.0769346
$254$	2960.10	0.731232
$255$	0	0
$256$	−1822.59	−0.444969
$257$	−3496.69	−0.848707	−0.424354	−	0.905497i	$-0.639499\pi$
$257$	−3496.69	−0.848707	−0.424354	+	0.905497i	$0.639499\pi$
$258$	0	0
$259$	2144.47	0.514482
$260$	−13340.7	−3.18212
$261$	0	0
$262$	135.256	0.0318936
$263$	−5747.94	−1.34766	−0.673828	−	0.738889i	$-0.735351\pi$
$263$	−5747.94	−1.34766	−0.673828	+	0.738889i	$0.735351\pi$
$264$	0	0
$265$	−6799.71	−1.57624
$266$	−3099.70	−0.714491
$267$	0	0
$268$	7148.08	1.62925
$269$	−235.217	−0.0533140	−0.0266570	−	0.999645i	$-0.508486\pi$
$269$	−235.217	−0.0533140	−0.0266570	+	0.999645i	$0.508486\pi$
$270$	0	0
$271$	−1179.58	−0.264406	−0.132203	−	0.991223i	$-0.542205\pi$
$271$	−1179.58	−0.264406	−0.132203	+	0.991223i	$0.542205\pi$
$272$	−529.902	−0.118125
$273$	0	0
$274$	−12488.7	−2.75354
$275$	−3956.26	−0.867533
$276$	0	0
$277$	−3638.98	−0.789331	−0.394666	−	0.918825i	$-0.629140\pi$
$277$	−3638.98	−0.789331	−0.394666	+	0.918825i	$0.629140\pi$
$278$	−13823.0	−2.98218
$279$	0	0
$280$	−2567.52	−0.547995
$281$	−3236.81	−0.687160	−0.343580	−	0.939123i	$-0.611640\pi$
$281$	−3236.81	−0.687160	−0.343580	+	0.939123i	$0.611640\pi$
$282$	0	0
$283$	8303.78	1.74420	0.872100	−	0.489328i	$-0.162758\pi$
$283$	8303.78	1.74420	0.872100	+	0.489328i	$0.162758\pi$
$284$	5462.45	1.14133
$285$	0	0
$286$	−2521.28	−0.521281
$287$	−1406.94	−0.289369
$288$	0	0
$289$	−4207.62	−0.856426
$290$	−4298.54	−0.870411
$291$	0	0
$292$	−3000.80	−0.601398
$293$	1894.16	0.377672	0.188836	−	0.982009i	$-0.439529\pi$
$293$	1894.16	0.377672	0.188836	+	0.982009i	$0.439529\pi$
$294$	0	0
$295$	13703.2	2.70450
$296$	−5104.10	−1.00226
$297$	0	0
$298$	8518.76	1.65597
$299$	1451.66	0.280775
$300$	0	0
$301$	−96.2825	−0.0184373
$302$	10173.5	1.93848
$303$	0	0
$304$	−1988.07	−0.375077
$305$	−1922.58	−0.360939
$306$	0	0
$307$	6596.30	1.22629	0.613144	−	0.789971i	$-0.289904\pi$
$307$	6596.30	1.22629	0.613144	+	0.789971i	$0.289904\pi$
$308$	−904.679	−0.167366
$309$	0	0
$310$	8202.47	1.50280
$311$	5242.26	0.955824	0.477912	−	0.878408i	$-0.341394\pi$
$311$	5242.26	0.955824	0.477912	+	0.878408i	$0.341394\pi$
$312$	0	0
$313$	5338.75	0.964103	0.482051	−	0.876143i	$-0.339892\pi$
$313$	5338.75	0.964103	0.482051	+	0.876143i	$0.339892\pi$
$314$	−1244.48	−0.223662
$315$	0	0
$316$	3069.29	0.546395
$317$	5807.21	1.02891	0.514456	−	0.857517i	$-0.327994\pi$
$317$	5807.21	1.02891	0.514456	+	0.857517i	$0.327994\pi$
$318$	0	0
$319$	−483.306	−0.0848273
$320$	−18200.7	−3.17953
$321$	0	0
$322$	875.550	0.151530
$323$	2646.42	0.455885
$324$	0	0
$325$	−18550.1	−3.16608
$326$	3851.03	0.654261
$327$	0	0
$328$	3348.69	0.563720
$329$	1865.73	0.312648
$330$	0	0
$331$	−1366.51	−0.226919	−0.113460	−	0.993543i	$-0.536193\pi$
$331$	−1366.51	−0.226919	−0.113460	+	0.993543i	$0.536193\pi$
$332$	−11208.4	−1.85284
$333$	0	0
$334$	−8733.25	−1.43072
$335$	13393.8	2.18443
$336$	0	0
$337$	−3363.75	−0.543724	−0.271862	−	0.962336i	$-0.587639\pi$
$337$	−3363.75	−0.543724	−0.271862	+	0.962336i	$0.587639\pi$
$338$	−2058.34	−0.331239
$339$	0	0
$340$	6869.64	1.09576
$341$	922.242	0.146458
$342$	0	0
$343$	343.000	0.0539949
$344$	229.164	0.0359177
$345$	0	0
$346$	17584.4	2.73220
$347$	−2984.97	−0.461791	−0.230896	−	0.972979i	$-0.574166\pi$
$347$	−2984.97	−0.461791	−0.230896	+	0.972979i	$0.574166\pi$
$348$	0	0
$349$	1286.08	0.197255	0.0986276	−	0.995124i	$-0.468555\pi$
$349$	1286.08	0.197255	0.0986276	+	0.995124i	$0.468555\pi$
$350$	−11188.3	−1.70868
$351$	0	0
$352$	−2441.48	−0.369691
$353$	−8417.60	−1.26919	−0.634594	−	0.772846i	$-0.718833\pi$
$353$	−8417.60	−1.26919	−0.634594	+	0.772846i	$0.718833\pi$
$354$	0	0
$355$	10235.3	1.53024
$356$	9868.16	1.46913
$357$	0	0
$358$	−13970.0	−2.06240
$359$	−7483.47	−1.10017	−0.550087	−	0.835108i	$-0.685405\pi$
$359$	−7483.47	−1.10017	−0.550087	+	0.835108i	$0.685405\pi$
$360$	0	0
$361$	3069.76	0.447553
$362$	−3038.68	−0.441186
$363$	0	0
$364$	−4241.86	−0.610808
$365$	−5622.79	−0.806329
$366$	0	0
$367$	8588.73	1.22160	0.610801	−	0.791784i	$-0.290847\pi$
$367$	8588.73	1.22160	0.610801	+	0.791784i	$0.290847\pi$
$368$	561.556	0.0795466
$369$	0	0
$370$	−29971.9	−4.21125
$371$	−2162.07	−0.302558
$372$	0	0
$373$	11833.0	1.64260	0.821298	−	0.570500i	$-0.193251\pi$
$373$	11833.0	1.64260	0.821298	+	0.570500i	$0.193251\pi$
$374$	1298.31	0.179502
$375$	0	0
$376$	−4440.67	−0.609070
$377$	−2266.13	−0.309579
$378$	0	0
$379$	5056.39	0.685301	0.342651	−	0.939463i	$-0.388675\pi$
$379$	5056.39	0.685301	0.342651	+	0.939463i	$0.388675\pi$
$380$	25773.3	3.47932
$381$	0	0
$382$	12175.1	1.63072
$383$	−6457.09	−0.861467	−0.430733	−	0.902479i	$-0.641745\pi$
$383$	−6457.09	−0.861467	−0.430733	+	0.902479i	$0.641745\pi$
$384$	0	0
$385$	−1695.16	−0.224398
$386$	−11785.1	−1.55401
$387$	0	0
$388$	−4102.14	−0.536739
$389$	−12444.5	−1.62201	−0.811004	−	0.585040i	$-0.801079\pi$
$389$	−12444.5	−1.62201	−0.811004	+	0.585040i	$0.801079\pi$
$390$	0	0
$391$	−747.518	−0.0966844
$392$	−816.382	−0.105188
$393$	0	0
$394$	−8352.59	−1.06801
$395$	5751.12	0.732584
$396$	0	0
$397$	−619.207	−0.0782799	−0.0391400	−	0.999234i	$-0.512462\pi$
$397$	−619.207	−0.0782799	−0.0391400	+	0.999234i	$0.512462\pi$
$398$	13862.5	1.74589
$399$	0	0
$400$	−7175.90	−0.896987
$401$	−9731.89	−1.21194	−0.605969	−	0.795488i	$-0.707215\pi$
$401$	−9731.89	−1.21194	−0.605969	+	0.795488i	$0.707215\pi$
$402$	0	0
$403$	4324.22	0.534502
$404$	−17534.7	−2.15937
$405$	0	0
$406$	−1366.79	−0.167075
$407$	−3369.88	−0.410415
$408$	0	0
$409$	4621.43	0.558717	0.279358	−	0.960187i	$-0.409878\pi$
$409$	4621.43	0.558717	0.279358	+	0.960187i	$0.409878\pi$
$410$	19663.9	2.36861
$411$	0	0
$412$	6564.05	0.784922
$413$	4357.12	0.519128
$414$	0	0
$415$	−21002.0	−2.48421
$416$	−11447.6	−1.34920
$417$	0	0
$418$	4870.95	0.569966
$419$	−186.428	−0.0217365	−0.0108682	−	0.999941i	$-0.503460\pi$
$419$	−186.428	−0.0217365	−0.0108682	+	0.999941i	$0.503460\pi$
$420$	0	0
$421$	2670.29	0.309126	0.154563	−	0.987983i	$-0.450603\pi$
$421$	2670.29	0.309126	0.154563	+	0.987983i	$0.450603\pi$
$422$	2314.07	0.266936
$423$	0	0
$424$	5145.99	0.589413
$425$	9552.22	1.09024
$426$	0	0
$427$	−611.312	−0.0692821
$428$	8154.40	0.920930
$429$	0	0
$430$	1345.68	0.150917
$431$	12514.9	1.39866	0.699328	−	0.714801i	$-0.253483\pi$
$431$	12514.9	1.39866	0.699328	+	0.714801i	$0.253483\pi$
$432$	0	0
$433$	−16651.2	−1.84805	−0.924025	−	0.382332i	$-0.875121\pi$
$433$	−16651.2	−1.84805	−0.924025	+	0.382332i	$0.875121\pi$
$434$	2608.10	0.288462
$435$	0	0
$436$	−4006.49	−0.440083
$437$	−2804.51	−0.306998
$438$	0	0
$439$	6033.38	0.655940	0.327970	−	0.944688i	$-0.393636\pi$
$439$	6033.38	0.655940	0.327970	+	0.944688i	$0.393636\pi$
$440$	4034.68	0.437149
$441$	0	0
$442$	6087.51	0.655098
$443$	−6320.03	−0.677819	−0.338910	−	0.940819i	$-0.610058\pi$
$443$	−6320.03	−0.677819	−0.338910	+	0.940819i	$0.610058\pi$
$444$	0	0
$445$	18490.6	1.96975
$446$	−9337.17	−0.991318
$447$	0	0
$448$	−5787.19	−0.610311
$449$	17893.6	1.88074	0.940368	−	0.340159i	$-0.110481\pi$
$449$	17893.6	1.88074	0.940368	+	0.340159i	$0.110481\pi$
$450$	0	0
$451$	2210.90	0.230837
$452$	11642.3	1.21152
$453$	0	0
$454$	−26894.9	−2.78026
$455$	−7948.25	−0.818945
$456$	0	0
$457$	6208.00	0.635444	0.317722	−	0.948184i	$-0.397082\pi$
$457$	6208.00	0.635444	0.317722	+	0.948184i	$0.397082\pi$
$458$	13313.0	1.35825
$459$	0	0
$460$	−7280.01	−0.737896
$461$	−7981.28	−0.806346	−0.403173	−	0.915124i	$-0.632093\pi$
$461$	−7981.28	−0.806346	−0.403173	+	0.915124i	$0.632093\pi$
$462$	0	0
$463$	−7495.19	−0.752334	−0.376167	−	0.926552i	$-0.622758\pi$
$463$	−7495.19	−0.752334	−0.376167	+	0.926552i	$0.622758\pi$
$464$	−876.623	−0.0877073
$465$	0	0
$466$	−290.484	−0.0288765
$467$	−1519.39	−0.150555	−0.0752773	−	0.997163i	$-0.523984\pi$
$467$	−1519.39	−0.150555	−0.0752773	+	0.997163i	$0.523984\pi$
$468$	0	0
$469$	4258.77	0.419300
$470$	−26076.2	−2.55916
$471$	0	0
$472$	−10370.5	−1.01131
$473$	151.301	0.0147079
$474$	0	0
$475$	35837.7	3.46178
$476$	2184.31	0.210331
$477$	0	0
$478$	−4898.75	−0.468753
$479$	16394.2	1.56382	0.781909	−	0.623393i	$-0.214246\pi$
$479$	16394.2	1.56382	0.781909	+	0.623393i	$0.214246\pi$
$480$	0	0
$481$	−15800.7	−1.49782
$482$	−23541.8	−2.22468
$483$	0	0
$484$	1421.64	0.133512
$485$	−7686.45	−0.719636
$486$	0	0
$487$	2275.01	0.211685	0.105843	−	0.994383i	$-0.466246\pi$
$487$	2275.01	0.211685	0.105843	+	0.994383i	$0.466246\pi$
$488$	1455.00	0.134968
$489$	0	0
$490$	−4793.89	−0.441971
$491$	14629.2	1.34462	0.672308	−	0.740272i	$-0.265303\pi$
$491$	14629.2	1.34462	0.672308	+	0.740272i	$0.265303\pi$
$492$	0	0
$493$	1166.92	0.106603
$494$	22838.9	2.08011
$495$	0	0
$496$	1672.77	0.151431
$497$	3254.48	0.293729
$498$	0	0
$499$	3534.01	0.317042	0.158521	−	0.987356i	$-0.449327\pi$
$499$	3534.01	0.317042	0.158521	+	0.987356i	$0.449327\pi$
$500$	60696.3	5.42884
$501$	0	0
$502$	790.872	0.0703155
$503$	9233.35	0.818479	0.409239	−	0.912427i	$-0.365794\pi$
$503$	9233.35	0.818479	0.409239	+	0.912427i	$0.365794\pi$
$504$	0	0
$505$	−32856.0	−2.89519
$506$	−1375.86	−0.120879
$507$	0	0
$508$	−7825.93	−0.683503
$509$	−11565.3	−1.00712	−0.503560	−	0.863960i	$-0.667977\pi$
$509$	−11565.3	−1.00712	−0.503560	+	0.863960i	$0.667977\pi$
$510$	0	0
$511$	−1787.85	−0.154775
$512$	−7087.70	−0.611787
$513$	0	0
$514$	15539.3	1.33348
$515$	12299.5	1.05239
$516$	0	0
$517$	−2931.87	−0.249407
$518$	−9530.01	−0.808349
$519$	0	0
$520$	18917.8	1.59539
$521$	2440.24	0.205200	0.102600	−	0.994723i	$-0.467284\pi$
$521$	2440.24	0.205200	0.102600	+	0.994723i	$0.467284\pi$
$522$	0	0
$523$	−911.213	−0.0761847	−0.0380923	−	0.999274i	$-0.512128\pi$
$523$	−911.213	−0.0761847	−0.0380923	+	0.999274i	$0.512128\pi$
$524$	−357.590	−0.0298119
$525$	0	0
$526$	25543.8	2.11742
$527$	−2226.71	−0.184055
$528$	0	0
$529$	−11374.8	−0.934892
$530$	30217.9	2.47657
$531$	0	0
$532$	8195.01	0.667854
$533$	10366.5	0.842445
$534$	0	0
$535$	15279.4	1.23474
$536$	−10136.4	−0.816838
$537$	0	0
$538$	1045.30	0.0837663
$539$	−539.000	−0.0430730
$540$	0	0
$541$	12277.5	0.975692	0.487846	−	0.872930i	$-0.337783\pi$
$541$	12277.5	0.975692	0.487846	+	0.872930i	$0.337783\pi$
$542$	5242.02	0.415432
$543$	0	0
$544$	5894.84	0.464594
$545$	−7507.22	−0.590044
$546$	0	0
$547$	−12539.2	−0.980141	−0.490071	−	0.871683i	$-0.663029\pi$
$547$	−12539.2	−0.980141	−0.490071	+	0.871683i	$0.663029\pi$
$548$	33017.8	2.57382
$549$	0	0
$550$	17581.6	1.36306
$551$	4378.01	0.338493
$552$	0	0
$553$	1828.66	0.140619
$554$	16171.6	1.24019
$555$	0	0
$556$	36545.3	2.78753
$557$	14212.8	1.08118	0.540588	−	0.841287i	$-0.318202\pi$
$557$	14212.8	1.08118	0.540588	+	0.841287i	$0.318202\pi$
$558$	0	0
$559$	709.421	0.0536768
$560$	−3074.68	−0.232016
$561$	0	0
$562$	14384.4	1.07966
$563$	−4446.83	−0.332880	−0.166440	−	0.986052i	$-0.553227\pi$
$563$	−4446.83	−0.332880	−0.166440	+	0.986052i	$0.553227\pi$
$564$	0	0
$565$	21814.9	1.62435
$566$	−36901.9	−2.74047
$567$	0	0
$568$	−7746.07	−0.572214
$569$	−11258.8	−0.829511	−0.414756	−	0.909933i	$-0.636133\pi$
$569$	−11258.8	−0.829511	−0.414756	+	0.909933i	$0.636133\pi$
$570$	0	0
$571$	−16450.6	−1.20567	−0.602835	−	0.797866i	$-0.705962\pi$
$571$	−16450.6	−1.20567	−0.602835	+	0.797866i	$0.705962\pi$
$572$	6665.78	0.487256
$573$	0	0
$574$	6252.43	0.454654
$575$	−10122.8	−0.734176
$576$	0	0
$577$	10175.6	0.734173	0.367086	−	0.930187i	$-0.380355\pi$
$577$	10175.6	0.734173	0.367086	+	0.930187i	$0.380355\pi$
$578$	18698.6	1.34561
$579$	0	0
$580$	11364.5	0.813597
$581$	−6677.90	−0.476843
$582$	0	0
$583$	3397.54	0.241358
$584$	4255.30	0.301516
$585$	0	0
$586$	−8417.62	−0.593394
$587$	5123.98	0.360289	0.180144	−	0.983640i	$-0.442344\pi$
$587$	5123.98	0.360289	0.180144	+	0.983640i	$0.442344\pi$
$588$	0	0
$589$	−8354.11	−0.584423
$590$	−60896.7	−4.24929
$591$	0	0
$592$	−6112.31	−0.424349
$593$	23816.7	1.64930	0.824650	−	0.565643i	$-0.191372\pi$
$593$	23816.7	1.64930	0.824650	+	0.565643i	$0.191372\pi$
$594$	0	0
$595$	4092.88	0.282003
$596$	−22522.0	−1.54788
$597$	0	0
$598$	−6451.16	−0.441150
$599$	−11801.0	−0.804965	−0.402482	−	0.915428i	$-0.631853\pi$
$599$	−11801.0	−0.804965	−0.402482	+	0.915428i	$0.631853\pi$
$600$	0	0
$601$	−10944.5	−0.742820	−0.371410	−	0.928469i	$-0.621125\pi$
$601$	−10944.5	−0.742820	−0.371410	+	0.928469i	$0.621125\pi$
$602$	427.879	0.0289685
$603$	0	0
$604$	−26896.9	−1.81195
$605$	2663.82	0.179007
$606$	0	0
$607$	−1280.36	−0.0856150	−0.0428075	−	0.999083i	$-0.513630\pi$
$607$	−1280.36	−0.0856150	−0.0428075	+	0.999083i	$0.513630\pi$
$608$	22116.1	1.47521
$609$	0	0
$610$	8543.91	0.567103
$611$	−13747.0	−0.910216
$612$	0	0
$613$	−11029.9	−0.726744	−0.363372	−	0.931644i	$-0.618375\pi$
$613$	−11029.9	−0.726744	−0.363372	+	0.931644i	$0.618375\pi$
$614$	−29313.9	−1.92673
$615$	0	0
$616$	1282.89	0.0839106
$617$	20861.3	1.36117	0.680586	−	0.732668i	$-0.261725\pi$
$617$	20861.3	1.36117	0.680586	+	0.732668i	$0.261725\pi$
$618$	0	0
$619$	−16877.9	−1.09593	−0.547966	−	0.836501i	$-0.684598\pi$
$619$	−16877.9	−1.09593	−0.547966	+	0.836501i	$0.684598\pi$
$620$	−21685.8	−1.40471
$621$	0	0
$622$	−23296.6	−1.50178
$623$	5879.37	0.378093
$624$	0	0
$625$	68773.0	4.40147
$626$	−23725.4	−1.51479
$627$	0	0
$628$	3290.17	0.209064
$629$	8136.43	0.515772
$630$	0	0
$631$	−1332.58	−0.0840719	−0.0420359	−	0.999116i	$-0.513384\pi$
$631$	−1332.58	−0.0840719	−0.0420359	+	0.999116i	$0.513384\pi$
$632$	−4352.42	−0.273940
$633$	0	0
$634$	−25807.2	−1.61662
$635$	−14664.0	−0.916412
$636$	0	0
$637$	−2527.27	−0.157196
$638$	2147.81	0.133280
$639$	0	0
$640$	41793.5	2.58130
$641$	−20472.2	−1.26147	−0.630736	−	0.775998i	$-0.717247\pi$
$641$	−20472.2	−1.26147	−0.630736	+	0.775998i	$0.717247\pi$
$642$	0	0
$643$	27140.3	1.66455	0.832276	−	0.554361i	$-0.187037\pi$
$643$	27140.3	1.66455	0.832276	+	0.554361i	$0.187037\pi$
$644$	−2314.79	−0.141639
$645$	0	0
$646$	−11760.7	−0.716282
$647$	−13662.1	−0.830159	−0.415079	−	0.909785i	$-0.636246\pi$
$647$	−13662.1	−0.830159	−0.415079	+	0.909785i	$0.636246\pi$
$648$	0	0
$649$	−6846.91	−0.414121
$650$	82436.7	4.97451
$651$	0	0
$652$	−10181.4	−0.611556
$653$	−1607.56	−0.0963379	−0.0481689	−	0.998839i	$-0.515339\pi$
$653$	−1607.56	−0.0963379	−0.0481689	+	0.998839i	$0.515339\pi$
$654$	0	0
$655$	−670.040	−0.0399705
$656$	4010.15	0.238674
$657$	0	0
$658$	−8291.30	−0.491229
$659$	−27361.6	−1.61738	−0.808692	−	0.588233i	$-0.799824\pi$
$659$	−27361.6	−1.61738	−0.808692	+	0.588233i	$0.799824\pi$
$660$	0	0
$661$	−5117.29	−0.301119	−0.150559	−	0.988601i	$-0.548107\pi$
$661$	−5117.29	−0.301119	−0.150559	+	0.988601i	$0.548107\pi$
$662$	6072.76	0.356533
$663$	0	0
$664$	15894.2	0.928939
$665$	15355.5	0.895431
$666$	0	0
$667$	−1236.63	−0.0717877
$668$	23089.0	1.33734
$669$	0	0
$670$	−59522.1	−3.43215
$671$	960.633	0.0552680
$672$	0	0
$673$	11605.6	0.664729	0.332365	−	0.943151i	$-0.392154\pi$
$673$	11605.6	0.664729	0.332365	+	0.943151i	$0.392154\pi$
$674$	14948.5	0.854292
$675$	0	0
$676$	5441.85	0.309618
$677$	−32514.6	−1.84584	−0.922922	−	0.384986i	$-0.874206\pi$
$677$	−32514.6	−1.84584	−0.922922	+	0.384986i	$0.874206\pi$
$678$	0	0
$679$	−2444.02	−0.138134
$680$	−9741.54	−0.549369
$681$	0	0
$682$	−4098.44	−0.230113
$683$	−7201.06	−0.403427	−0.201714	−	0.979445i	$-0.564651\pi$
$683$	−7201.06	−0.403427	−0.201714	+	0.979445i	$0.564651\pi$
$684$	0	0
$685$	61867.6	3.45086
$686$	−1524.29	−0.0848362
$687$	0	0
$688$	274.431	0.0152072
$689$	15930.4	0.880841
$690$	0	0
$691$	−32357.7	−1.78140	−0.890698	−	0.454596i	$-0.849784\pi$
$691$	−32357.7	−1.78140	−0.890698	+	0.454596i	$0.849784\pi$
$692$	−46489.7	−2.55386
$693$	0	0
$694$	13265.2	0.725562
$695$	68477.3	3.73740
$696$	0	0
$697$	−5338.13	−0.290095
$698$	−5715.31	−0.309925
$699$	0	0
$700$	29579.7	1.59716
$701$	−11077.3	−0.596838	−0.298419	−	0.954435i	$-0.596459\pi$
$701$	−11077.3	−0.596838	−0.298419	+	0.954435i	$0.596459\pi$
$702$	0	0
$703$	30526.0	1.63771
$704$	9094.16	0.486859
$705$	0	0
$706$	37407.7	1.99413
$707$	−10447.1	−0.555731
$708$	0	0
$709$	−28594.0	−1.51463	−0.757314	−	0.653051i	$-0.773489\pi$
$709$	−28594.0	−1.51463	−0.757314	+	0.653051i	$0.773489\pi$
$710$	−45485.8	−2.40430
$711$	0	0
$712$	−13993.6	−0.736563
$713$	2359.73	0.123945
$714$	0	0
$715$	12490.1	0.653292
$716$	36934.1	1.92778
$717$	0	0
$718$	33256.5	1.72858
$719$	−18240.5	−0.946114	−0.473057	−	0.881032i	$-0.656850\pi$
$719$	−18240.5	−0.946114	−0.473057	+	0.881032i	$0.656850\pi$
$720$	0	0
$721$	3910.81	0.202006
$722$	−13642.0	−0.703190
$723$	0	0
$724$	8033.68	0.412389
$725$	15802.4	0.809496
$726$	0	0
$727$	−9792.53	−0.499566	−0.249783	−	0.968302i	$-0.580359\pi$
$727$	−9792.53	−0.499566	−0.249783	+	0.968302i	$0.580359\pi$
$728$	6015.20	0.306234
$729$	0	0
$730$	24987.6	1.26690
$731$	−365.309	−0.0184835
$732$	0	0
$733$	−22723.8	−1.14505	−0.572525	−	0.819887i	$-0.694036\pi$
$733$	−22723.8	−1.14505	−0.572525	+	0.819887i	$0.694036\pi$
$734$	−38168.3	−1.91937
$735$	0	0
$736$	−6246.98	−0.312863
$737$	−6692.35	−0.334485
$738$	0	0
$739$	−24063.6	−1.19783	−0.598913	−	0.800814i	$-0.704401\pi$
$739$	−24063.6	−1.19783	−0.598913	+	0.800814i	$0.704401\pi$
$740$	79240.0	3.93638
$741$	0	0
$742$	9608.22	0.475376
$743$	29211.6	1.44235	0.721177	−	0.692751i	$-0.243601\pi$
$743$	29211.6	1.44235	0.721177	+	0.692751i	$0.243601\pi$
$744$	0	0
$745$	−42200.9	−2.07533
$746$	−52585.6	−2.58083
$747$	0	0
$748$	−3432.48	−0.167786
$749$	4858.33	0.237008
$750$	0	0
$751$	1880.93	0.0913931	0.0456965	−	0.998955i	$-0.485449\pi$
$751$	1880.93	0.0913931	0.0456965	+	0.998955i	$0.485449\pi$
$752$	−5317.84	−0.257874
$753$	0	0
$754$	10070.6	0.486408
$755$	−50398.4	−2.42939
$756$	0	0
$757$	−36218.7	−1.73896	−0.869480	−	0.493968i	$-0.835546\pi$
$757$	−36218.7	−1.73896	−0.869480	+	0.493968i	$0.835546\pi$
$758$	−22470.5	−1.07674
$759$	0	0
$760$	−36548.0	−1.74439
$761$	−36966.4	−1.76088	−0.880441	−	0.474156i	$-0.842753\pi$
$761$	−36966.4	−1.76088	−0.880441	+	0.474156i	$0.842753\pi$
$762$	0	0
$763$	−2387.03	−0.113259
$764$	−32188.7	−1.52428
$765$	0	0
$766$	28695.2	1.35353
$767$	−32103.8	−1.51135
$768$	0	0
$769$	−38975.5	−1.82769	−0.913845	−	0.406062i	$-0.866902\pi$
$769$	−38975.5	−1.82769	−0.913845	+	0.406062i	$0.866902\pi$
$770$	7533.26	0.352571
$771$	0	0
$772$	31157.7	1.45258
$773$	−27341.9	−1.27221	−0.636105	−	0.771603i	$-0.719455\pi$
$773$	−27341.9	−1.27221	−0.636105	+	0.771603i	$0.719455\pi$
$774$	0	0
$775$	−30154.0	−1.39763
$776$	5817.07	0.269099
$777$	0	0
$778$	55303.3	2.54848
$779$	−20027.4	−0.921125
$780$	0	0
$781$	−5114.19	−0.234315
$782$	3321.96	0.151909
$783$	0	0
$784$	−977.642	−0.0445354
$785$	6165.00	0.280303
$786$	0	0
$787$	−18268.5	−0.827446	−0.413723	−	0.910403i	$-0.635772\pi$
$787$	−18268.5	−0.827446	−0.413723	+	0.910403i	$0.635772\pi$
$788$	22082.7	0.998302
$789$	0	0
$790$	−25558.0	−1.15103
$791$	6936.37	0.311794
$792$	0	0
$793$	4504.22	0.201702
$794$	2751.75	0.122993
$795$	0	0
$796$	−36649.9	−1.63194
$797$	−12717.6	−0.565219	−0.282610	−	0.959235i	$-0.591200\pi$
$797$	−12717.6	−0.565219	−0.282610	+	0.959235i	$0.591200\pi$
$798$	0	0
$799$	7078.86	0.313432
$800$	79827.6	3.52792
$801$	0	0
$802$	43248.4	1.90418
$803$	2809.48	0.123467
$804$	0	0
$805$	−4337.37	−0.189903
$806$	−19216.8	−0.839804
$807$	0	0
$808$	24865.3	1.08262
$809$	−12502.0	−0.543322	−0.271661	−	0.962393i	$-0.587573\pi$
$809$	−12502.0	−0.543322	−0.271661	+	0.962393i	$0.587573\pi$
$810$	0	0
$811$	−23431.3	−1.01453	−0.507264	−	0.861791i	$-0.669343\pi$
$811$	−23431.3	−1.01453	−0.507264	+	0.861791i	$0.669343\pi$
$812$	3613.52	0.156170
$813$	0	0
$814$	14975.7	0.644839
$815$	−19077.6	−0.819948
$816$	0	0
$817$	−1370.56	−0.0586899
$818$	−20537.6	−0.877850
$819$	0	0
$820$	−51987.6	−2.21401
$821$	−33116.5	−1.40776	−0.703881	−	0.710317i	$-0.748551\pi$
$821$	−33116.5	−1.40776	−0.703881	+	0.710317i	$0.748551\pi$
$822$	0	0
$823$	−6383.39	−0.270366	−0.135183	−	0.990821i	$-0.543162\pi$
$823$	−6383.39	−0.270366	−0.135183	+	0.990821i	$0.543162\pi$
$824$	−9308.20	−0.393527
$825$	0	0
$826$	−19363.0	−0.815649
$827$	27701.3	1.16477	0.582386	−	0.812912i	$-0.302119\pi$
$827$	27701.3	1.16477	0.582386	+	0.812912i	$0.302119\pi$
$828$	0	0
$829$	−13160.2	−0.551353	−0.275676	−	0.961251i	$-0.588902\pi$
$829$	−13160.2	−0.551353	−0.275676	+	0.961251i	$0.588902\pi$
$830$	93332.7	3.90316
$831$	0	0
$832$	42640.7	1.77680
$833$	1301.39	0.0541303
$834$	0	0
$835$	43263.4	1.79305
$836$	−12877.9	−0.532763
$837$	0	0
$838$	828.483	0.0341521
$839$	24842.9	1.02226	0.511128	−	0.859505i	$-0.329228\pi$
$839$	24842.9	1.02226	0.511128	+	0.859505i	$0.329228\pi$
$840$	0	0
$841$	−22458.6	−0.920848
$842$	−11866.8	−0.485696
$843$	0	0
$844$	−6117.95	−0.249513
$845$	10196.7	0.415123
$846$	0	0
$847$	847.000	0.0343604
$848$	6162.48	0.249552
$849$	0	0
$850$	−42450.0	−1.71297
$851$	−8622.47	−0.347326
$852$	0	0
$853$	−10131.6	−0.406681	−0.203340	−	0.979108i	$-0.565180\pi$
$853$	−10131.6	−0.406681	−0.203340	+	0.979108i	$0.565180\pi$
$854$	2716.67	0.108855
$855$	0	0
$856$	−11563.4	−0.461716
$857$	−10115.6	−0.403199	−0.201599	−	0.979468i	$-0.564614\pi$
$857$	−10115.6	−0.403199	−0.201599	+	0.979468i	$0.564614\pi$
$858$	0	0
$859$	−27491.4	−1.09196	−0.545980	−	0.837798i	$-0.683843\pi$
$859$	−27491.4	−1.09196	−0.545980	+	0.837798i	$0.683843\pi$
$860$	−3557.72	−0.141066
$861$	0	0
$862$	−55616.1	−2.19755
$863$	117.276	0.00462588	0.00231294	−	0.999997i	$-0.499264\pi$
$863$	117.276	0.00462588	0.00231294	+	0.999997i	$0.499264\pi$
$864$	0	0
$865$	−87110.8	−3.42411
$866$	73997.8	2.90364
$867$	0	0
$868$	−6895.31	−0.269634
$869$	−2873.60	−0.112175
$870$	0	0
$871$	−31379.1	−1.22071
$872$	5681.43	0.220639
$873$	0	0
$874$	12463.2	0.482351
$875$	36162.3	1.39715
$876$	0	0
$877$	11597.4	0.446543	0.223271	−	0.974756i	$-0.428326\pi$
$877$	11597.4	0.446543	0.223271	+	0.974756i	$0.428326\pi$
$878$	−26812.3	−1.03061
$879$	0	0
$880$	4831.65	0.185085
$881$	−7524.18	−0.287737	−0.143868	−	0.989597i	$-0.545954\pi$
$881$	−7524.18	−0.287737	−0.143868	+	0.989597i	$0.545954\pi$
$882$	0	0
$883$	13467.4	0.513266	0.256633	−	0.966509i	$-0.417387\pi$
$883$	13467.4	0.513266	0.256633	+	0.966509i	$0.417387\pi$
$884$	−16094.2	−0.612339
$885$	0	0
$886$	28086.2	1.06498
$887$	12955.7	0.490427	0.245214	−	0.969469i	$-0.421142\pi$
$887$	12955.7	0.490427	0.245214	+	0.969469i	$0.421142\pi$
$888$	0	0
$889$	−4662.63	−0.175905
$890$	−82172.1	−3.09485
$891$	0	0
$892$	24685.7	0.926612
$893$	26558.2	0.995226
$894$	0	0
$895$	69205.8	2.58469
$896$	13288.9	0.495480
$897$	0	0
$898$	−79518.9	−2.95499
$899$	−3683.68	−0.136660
$900$	0	0
$901$	−8203.20	−0.303317
$902$	−9825.25	−0.362688
$903$	0	0
$904$	−16509.4	−0.607405
$905$	15053.2	0.552913
$906$	0	0
$907$	47843.5	1.75151	0.875753	−	0.482759i	$-0.160365\pi$
$907$	47843.5	1.75151	0.875753	+	0.482759i	$0.160365\pi$
$908$	71104.9	2.59879
$909$	0	0
$910$	35322.0	1.28672
$911$	−16969.8	−0.617162	−0.308581	−	0.951198i	$-0.599854\pi$
$911$	−16969.8	−0.617162	−0.308581	+	0.951198i	$0.599854\pi$
$912$	0	0
$913$	10493.8	0.380389
$914$	−27588.3	−0.998402
$915$	0	0
$916$	−35197.1	−1.26959
$917$	−213.049	−0.00767231
$918$	0	0
$919$	12095.2	0.434149	0.217075	−	0.976155i	$-0.430348\pi$
$919$	12095.2	0.434149	0.217075	+	0.976155i	$0.430348\pi$
$920$	10323.5	0.369951
$921$	0	0
$922$	35468.8	1.26692
$923$	−23979.4	−0.855138
$924$	0	0
$925$	110183.	3.91653
$926$	33308.6	1.18206
$927$	0	0
$928$	9751.91	0.344959
$929$	44544.3	1.57314	0.786571	−	0.617499i	$-0.211854\pi$
$929$	44544.3	1.57314	0.786571	+	0.617499i	$0.211854\pi$
$930$	0	0
$931$	4882.52	0.171878
$932$	767.985	0.0269916
$933$	0	0
$934$	6752.16	0.236550
$935$	−6431.66	−0.224960
$936$	0	0
$937$	−49265.8	−1.71766	−0.858828	−	0.512264i	$-0.828807\pi$
$937$	−49265.8	−1.71766	−0.858828	+	0.512264i	$0.828807\pi$
$938$	−18925.9	−0.658799
$939$	0	0
$940$	68940.4	2.39211
$941$	18403.1	0.637538	0.318769	−	0.947832i	$-0.396731\pi$
$941$	18403.1	0.637538	0.318769	+	0.947832i	$0.396731\pi$
$942$	0	0
$943$	5657.01	0.195353
$944$	−12419.0	−0.428181
$945$	0	0
$946$	−672.381	−0.0231089
$947$	−17689.3	−0.606996	−0.303498	−	0.952832i	$-0.598155\pi$
$947$	−17689.3	−0.606996	−0.303498	+	0.952832i	$0.598155\pi$
$948$	0	0
$949$	13173.1	0.450597
$950$	−159262.	−5.43911
$951$	0	0
$952$	−3097.47	−0.105451
$953$	5298.19	0.180090	0.0900448	−	0.995938i	$-0.471299\pi$
$953$	5298.19	0.180090	0.0900448	+	0.995938i	$0.471299\pi$
$954$	0	0
$955$	−60314.1	−2.04368
$956$	12951.4	0.438156
$957$	0	0
$958$	−72855.6	−2.45705
$959$	19671.7	0.662392
$960$	0	0
$961$	−22761.8	−0.764050
$962$	70218.3	2.35336
$963$	0	0
$964$	62239.9	2.07947
$965$	58382.2	1.94755
$966$	0	0
$967$	33990.6	1.13037	0.565184	−	0.824965i	$-0.308805\pi$
$967$	33990.6	1.13037	0.565184	+	0.824965i	$0.308805\pi$
$968$	−2015.96	−0.0669375
$969$	0	0
$970$	34158.5	1.13068
$971$	−41991.0	−1.38780	−0.693900	−	0.720071i	$-0.744109\pi$
$971$	−41991.0	−1.38780	−0.693900	+	0.720071i	$0.744109\pi$
$972$	0	0
$973$	21773.4	0.717393
$974$	−10110.1	−0.332597
$975$	0	0
$976$	1742.40	0.0571444
$977$	31233.4	1.02277	0.511384	−	0.859352i	$-0.329133\pi$
$977$	31233.4	1.02277	0.511384	+	0.859352i	$0.329133\pi$
$978$	0	0
$979$	−9239.00	−0.301614
$980$	12674.1	0.413123
$981$	0	0
$982$	−65012.0	−2.11264
$983$	45702.5	1.48289	0.741447	−	0.671012i	$-0.234140\pi$
$983$	45702.5	1.48289	0.741447	+	0.671012i	$0.234140\pi$
$984$	0	0
$985$	41377.7	1.33848
$986$	−5185.78	−0.167494
$987$	0	0
$988$	−60381.8	−1.94433
$989$	387.132	0.0124470
$990$	0	0
$991$	−4310.72	−0.138178	−0.0690890	−	0.997611i	$-0.522009\pi$
$991$	−4310.72	−0.138178	−0.0690890	+	0.997611i	$0.522009\pi$
$992$	−18608.6	−0.595587
$993$	0	0
$994$	−14462.9	−0.461504
$995$	−68673.3	−2.18803
$996$	0	0
$997$	−6103.28	−0.193875	−0.0969373	−	0.995290i	$-0.530905\pi$
$997$	−6103.28	−0.193875	−0.0969373	+	0.995290i	$0.530905\pi$
$998$	−15705.1	−0.498133
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.o.1.2		5
3.2	odd	2		77.4.a.e.1.4	✓	5
12.11	even	2		1232.4.a.y.1.1		5
15.14	odd	2		1925.4.a.r.1.2		5
21.20	even	2		539.4.a.h.1.4		5
33.32	even	2		847.4.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.4	✓	5	3.2	odd	2
539.4.a.h.1.4		5	21.20	even	2
693.4.a.o.1.2		5	1.1	even	1	trivial
847.4.a.f.1.2		5	33.32	even	2
1232.4.a.y.1.1		5	12.11	even	2
1925.4.a.r.1.2		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.44399 q^{2} +11.7491 q^{4} +22.0150 q^{5} +7.00000 q^{7} -16.6609 q^{8} -97.8345 q^{10} -11.0000 q^{11} -51.5769 q^{13} -31.1080 q^{14} -19.9519 q^{16} +26.5590 q^{17} +99.6432 q^{19} +258.656 q^{20} +48.8839 q^{22} -28.1455 q^{23} +359.660 q^{25} +229.207 q^{26} +82.2435 q^{28} +43.9369 q^{29} -83.8402 q^{31} +221.953 q^{32} -118.028 q^{34} +154.105 q^{35} +306.353 q^{37} -442.814 q^{38} -366.789 q^{40} -200.991 q^{41} -13.7546 q^{43} -129.240 q^{44} +125.079 q^{46} +266.533 q^{47} +49.0000 q^{49} -1598.33 q^{50} -605.980 q^{52} -308.867 q^{53} -242.165 q^{55} -116.626 q^{56} -195.255 q^{58} +622.446 q^{59} -87.3303 q^{61} +372.585 q^{62} -826.742 q^{64} -1135.46 q^{65} +608.395 q^{67} +312.044 q^{68} -684.842 q^{70} +464.926 q^{71} -255.407 q^{73} -1361.43 q^{74} +1170.72 q^{76} -77.0000 q^{77} +261.237 q^{79} -439.240 q^{80} +893.204 q^{82} -953.986 q^{83} +584.696 q^{85} +61.1255 q^{86} +183.269 q^{88} +839.910 q^{89} -361.038 q^{91} -330.684 q^{92} -1184.47 q^{94} +2193.64 q^{95} -349.146 q^{97} -217.756 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 45 q^{4} + 24 q^{5} + 35 q^{7} - 57 q^{8} - 10 q^{10} - 55 q^{11} - 50 q^{13} - 7 q^{14} + 433 q^{16} - 222 q^{17} + 160 q^{19} + 430 q^{20} + 11 q^{22} - 54 q^{23} + 125 q^{25} + 1026 q^{26}+ \cdots - 49 q^{98}+O(q^{100}) \) 5 * q - q^2 + 45 * q^4 + 24 * q^5 + 35 * q^7 - 57 * q^8 - 10 * q^10 - 55 * q^11 - 50 * q^13 - 7 * q^14 + 433 * q^16 - 222 * q^17 + 160 * q^19 + 430 * q^20 + 11 * q^22 - 54 * q^23 + 125 * q^25 + 1026 * q^26 + 315 * q^28 - 14 * q^29 - 34 * q^31 + 583 * q^32 - 750 * q^34 + 168 * q^35 + 1044 * q^37 + 156 * q^38 + 1158 * q^40 + 114 * q^41 + 672 * q^43 - 495 * q^44 - 1224 * q^46 + 292 * q^47 + 245 * q^49 - 1143 * q^50 - 914 * q^52 + 710 * q^53 - 264 * q^55 - 399 * q^56 - 810 * q^58 - 270 * q^59 + 138 * q^61 + 480 * q^62 + 3001 * q^64 + 196 * q^65 + 1942 * q^67 - 3130 * q^68 - 70 * q^70 + 278 * q^71 - 338 * q^73 - 462 * q^74 - 1332 * q^76 - 385 * q^77 + 576 * q^79 + 1602 * q^80 + 1386 * q^82 - 1644 * q^83 + 360 * q^85 - 2020 * q^86 + 627 * q^88 + 3656 * q^89 - 350 * q^91 + 748 * q^92 + 976 * q^94 + 1276 * q^95 + 692 * q^97 - 49 * q^98

Introduction

L-functions

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