LMFDB - Embedded newform 693.4.a.m.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.509800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 5x + 15 $ x^4 - x^3 - 10x^2 + 5x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.66444$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.76366 q^{2} +6.16515 q^{4} +15.4926 q^{5} -7.00000 q^{7} +6.90574 q^{8} +O(q^{10})$	\(q-3.76366 q^{2} +6.16515 q^{4} +15.4926 q^{5} -7.00000 q^{7} +6.90574 q^{8} -58.3089 q^{10} -11.0000 q^{11} +49.0777 q^{13} +26.3456 q^{14} -75.3121 q^{16} +34.1261 q^{17} -144.114 q^{19} +95.5142 q^{20} +41.4003 q^{22} -118.906 q^{23} +115.020 q^{25} -184.712 q^{26} -43.1561 q^{28} +63.6830 q^{29} -212.912 q^{31} +228.203 q^{32} -128.439 q^{34} -108.448 q^{35} -200.224 q^{37} +542.396 q^{38} +106.988 q^{40} -451.267 q^{41} -130.664 q^{43} -67.8167 q^{44} +447.523 q^{46} +176.271 q^{47} +49.0000 q^{49} -432.898 q^{50} +302.571 q^{52} +629.988 q^{53} -170.419 q^{55} -48.3402 q^{56} -239.681 q^{58} +86.9318 q^{59} +644.248 q^{61} +801.330 q^{62} -256.384 q^{64} +760.340 q^{65} -400.974 q^{67} +210.393 q^{68} +408.162 q^{70} -507.611 q^{71} +176.392 q^{73} +753.575 q^{74} -888.485 q^{76} +77.0000 q^{77} -701.122 q^{79} -1166.78 q^{80} +1698.42 q^{82} +1259.27 q^{83} +528.702 q^{85} +491.775 q^{86} -75.9631 q^{88} -788.394 q^{89} -343.544 q^{91} -733.075 q^{92} -663.423 q^{94} -2232.70 q^{95} +185.039 q^{97} -184.419 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * 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 $

$2$	−3.76366	−1.33066	−0.665328	−	0.746551i	$-0.731708\pi$
$2$	−3.76366	−1.33066	−0.665328	+	0.746551i	$0.731708\pi$
$3$	0	0
$3$	0	0
$4$	6.16515	0.770644
$5$	15.4926	1.38570	0.692850	−	0.721082i	$-0.256355\pi$
$5$	15.4926	1.38570	0.692850	+	0.721082i	$0.256355\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	6.90574	0.305193
$9$	0	0
$10$	−58.3089	−1.84389
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	49.0777	1.04705	0.523527	−	0.852009i	$-0.324616\pi$
$13$	49.0777	1.04705	0.523527	+	0.852009i	$0.324616\pi$
$14$	26.3456	0.502941
$15$	0	0
$16$	−75.3121	−1.17675
$17$	34.1261	0.486871	0.243435	−	0.969917i	$-0.421726\pi$
$17$	34.1261	0.486871	0.243435	+	0.969917i	$0.421726\pi$
$18$	0	0
$19$	−144.114	−1.74011	−0.870053	−	0.492958i	$-0.835916\pi$
$19$	−144.114	−1.74011	−0.870053	+	0.492958i	$0.835916\pi$
$20$	95.5142	1.06788
$21$	0	0
$22$	41.4003	0.401208
$23$	−118.906	−1.07798	−0.538992	−	0.842311i	$-0.681195\pi$
$23$	−118.906	−1.07798	−0.538992	+	0.842311i	$0.681195\pi$
$24$	0	0
$25$	115.020	0.920164
$26$	−184.712	−1.39327
$27$	0	0
$28$	−43.1561	−0.291276
$29$	63.6830	0.407780	0.203890	−	0.978994i	$-0.434641\pi$
$29$	63.6830	0.407780	0.203890	+	0.978994i	$0.434641\pi$
$30$	0	0
$31$	−212.912	−1.23355	−0.616777	−	0.787138i	$-0.711562\pi$
$31$	−212.912	−1.23355	−0.616777	+	0.787138i	$0.711562\pi$
$32$	228.203	1.26066
$33$	0	0
$34$	−128.439	−0.647857
$35$	−108.448	−0.523745
$36$	0	0
$37$	−200.224	−0.889638	−0.444819	−	0.895620i	$-0.646732\pi$
$37$	−200.224	−0.889638	−0.444819	+	0.895620i	$0.646732\pi$
$38$	542.396	2.31548
$39$	0	0
$40$	106.988	0.422907
$41$	−451.267	−1.71893	−0.859464	−	0.511197i	$-0.829202\pi$
$41$	−451.267	−1.71893	−0.859464	+	0.511197i	$0.829202\pi$
$42$	0	0
$43$	−130.664	−0.463396	−0.231698	−	0.972788i	$-0.574428\pi$
$43$	−130.664	−0.463396	−0.231698	+	0.972788i	$0.574428\pi$
$44$	−67.8167	−0.232358
$45$	0	0
$46$	447.523	1.43443
$47$	176.271	0.547057	0.273529	−	0.961864i	$-0.411809\pi$
$47$	176.271	0.547057	0.273529	+	0.961864i	$0.411809\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−432.898	−1.22442
$51$	0	0
$52$	302.571	0.806906
$53$	629.988	1.63274	0.816372	−	0.577526i	$-0.195982\pi$
$53$	629.988	1.63274	0.816372	+	0.577526i	$0.195982\pi$
$54$	0	0
$55$	−170.419	−0.417804
$56$	−48.3402	−0.115352
$57$	0	0
$58$	−239.681	−0.542615
$59$	86.9318	0.191823	0.0959115	−	0.995390i	$-0.469423\pi$
$59$	86.9318	0.191823	0.0959115	+	0.995390i	$0.469423\pi$
$60$	0	0
$61$	644.248	1.35225	0.676127	−	0.736785i	$-0.263657\pi$
$61$	644.248	1.35225	0.676127	+	0.736785i	$0.263657\pi$
$62$	801.330	1.64143
$63$	0	0
$64$	−256.384	−0.500750
$65$	760.340	1.45090
$66$	0	0
$67$	−400.974	−0.731147	−0.365573	−	0.930783i	$-0.619127\pi$
$67$	−400.974	−0.731147	−0.365573	+	0.930783i	$0.619127\pi$
$68$	210.393	0.375204
$69$	0	0
$70$	408.162	0.696925
$71$	−507.611	−0.848484	−0.424242	−	0.905549i	$-0.639459\pi$
$71$	−507.611	−0.848484	−0.424242	+	0.905549i	$0.639459\pi$
$72$	0	0
$73$	176.392	0.282811	0.141405	−	0.989952i	$-0.454838\pi$
$73$	176.392	0.282811	0.141405	+	0.989952i	$0.454838\pi$
$74$	753.575	1.18380
$75$	0	0
$76$	−888.485	−1.34100
$77$	77.0000	0.113961
$78$	0	0
$79$	−701.122	−0.998512	−0.499256	−	0.866455i	$-0.666393\pi$
$79$	−701.122	−0.998512	−0.499256	+	0.866455i	$0.666393\pi$
$80$	−1166.78	−1.63062
$81$	0	0
$82$	1698.42	2.28730
$83$	1259.27	1.66534	0.832670	−	0.553770i	$-0.186811\pi$
$83$	1259.27	1.66534	0.832670	+	0.553770i	$0.186811\pi$
$84$	0	0
$85$	528.702	0.674656
$86$	491.775	0.616621
$87$	0	0
$88$	−75.9631	−0.0920193
$89$	−788.394	−0.938984	−0.469492	−	0.882937i	$-0.655563\pi$
$89$	−788.394	−0.938984	−0.469492	+	0.882937i	$0.655563\pi$
$90$	0	0
$91$	−343.544	−0.395749
$92$	−733.075	−0.830743
$93$	0	0
$94$	−663.423	−0.727945
$95$	−2232.70	−2.41126
$96$	0	0
$97$	185.039	0.193689	0.0968446	−	0.995300i	$-0.469125\pi$
$97$	185.039	0.193689	0.0968446	+	0.995300i	$0.469125\pi$
$98$	−184.419	−0.190094
$99$	0	0
$100$	709.119	0.709119
$101$	−1243.55	−1.22513	−0.612565	−	0.790420i	$-0.709862\pi$
$101$	−1243.55	−1.22513	−0.612565	+	0.790420i	$0.709862\pi$
$102$	0	0
$103$	−1555.21	−1.48776	−0.743879	−	0.668315i	$-0.767016\pi$
$103$	−1555.21	−1.48776	−0.743879	+	0.668315i	$0.767016\pi$
$104$	338.918	0.319554
$105$	0	0
$106$	−2371.06	−2.17262
$107$	247.062	0.223218	0.111609	−	0.993752i	$-0.464400\pi$
$107$	247.062	0.223218	0.111609	+	0.993752i	$0.464400\pi$
$108$	0	0
$109$	1160.58	1.01985	0.509923	−	0.860220i	$-0.329674\pi$
$109$	1160.58	1.01985	0.509923	+	0.860220i	$0.329674\pi$
$110$	641.398	0.555953
$111$	0	0
$112$	527.185	0.444770
$113$	−1738.50	−1.44729	−0.723646	−	0.690171i	$-0.757535\pi$
$113$	−1738.50	−1.44729	−0.723646	+	0.690171i	$0.757535\pi$
$114$	0	0
$115$	−1842.16	−1.49376
$116$	392.615	0.314254
$117$	0	0
$118$	−327.182	−0.255250
$119$	−238.883	−0.184020
$120$	0	0
$121$	121.000	0.0909091
$122$	−2424.73	−1.79938
$123$	0	0
$124$	−1312.64	−0.950631
$125$	−154.609	−0.110629
$126$	0	0
$127$	−1507.65	−1.05340	−0.526701	−	0.850051i	$-0.676571\pi$
$127$	−1507.65	−1.05340	−0.526701	+	0.850051i	$0.676571\pi$
$128$	−860.685	−0.594333
$129$	0	0
$130$	−2861.66	−1.93065
$131$	−1327.55	−0.885408	−0.442704	−	0.896668i	$-0.645981\pi$
$131$	−1327.55	−0.885408	−0.442704	+	0.896668i	$0.645981\pi$
$132$	0	0
$133$	1008.80	0.657698
$134$	1509.13	0.972904
$135$	0	0
$136$	235.666	0.148590
$137$	441.531	0.275347	0.137674	−	0.990478i	$-0.456038\pi$
$137$	441.531	0.275347	0.137674	+	0.990478i	$0.456038\pi$
$138$	0	0
$139$	179.162	0.109326	0.0546631	−	0.998505i	$-0.482592\pi$
$139$	179.162	0.109326	0.0546631	+	0.998505i	$0.482592\pi$
$140$	−668.600	−0.403621
$141$	0	0
$142$	1910.48	1.12904
$143$	−539.854	−0.315698
$144$	0	0
$145$	986.615	0.565061
$146$	−663.882	−0.376324
$147$	0	0
$148$	−1234.41	−0.685594
$149$	1147.07	0.630680	0.315340	−	0.948979i	$-0.397882\pi$
$149$	1147.07	0.630680	0.315340	+	0.948979i	$0.397882\pi$
$150$	0	0
$151$	1873.03	1.00944	0.504719	−	0.863284i	$-0.331596\pi$
$151$	1873.03	1.00944	0.504719	+	0.863284i	$0.331596\pi$
$152$	−995.214	−0.531069
$153$	0	0
$154$	−289.802	−0.151642
$155$	−3298.56	−1.70933
$156$	0	0
$157$	−643.927	−0.327331	−0.163665	−	0.986516i	$-0.552332\pi$
$157$	−643.927	−0.327331	−0.163665	+	0.986516i	$0.552332\pi$
$158$	2638.79	1.32868
$159$	0	0
$160$	3535.46	1.74689
$161$	832.343	0.407440
$162$	0	0
$163$	−3044.47	−1.46295	−0.731477	−	0.681866i	$-0.761169\pi$
$163$	−3044.47	−1.46295	−0.731477	+	0.681866i	$0.761169\pi$
$164$	−2782.13	−1.32468
$165$	0	0
$166$	−4739.48	−2.21599
$167$	337.038	0.156172	0.0780861	−	0.996947i	$-0.475119\pi$
$167$	337.038	0.156172	0.0780861	+	0.996947i	$0.475119\pi$
$168$	0	0
$169$	211.617	0.0963209
$170$	−1989.86	−0.897735
$171$	0	0
$172$	−805.563	−0.357114
$173$	3504.80	1.54026	0.770131	−	0.637886i	$-0.220191\pi$
$173$	3504.80	1.54026	0.770131	+	0.637886i	$0.220191\pi$
$174$	0	0
$175$	−805.143	−0.347789
$176$	828.433	0.354804
$177$	0	0
$178$	2967.25	1.24946
$179$	−221.078	−0.0923135	−0.0461568	−	0.998934i	$-0.514697\pi$
$179$	−221.078	−0.0923135	−0.0461568	+	0.998934i	$0.514697\pi$
$180$	0	0
$181$	−4561.63	−1.87328	−0.936640	−	0.350295i	$-0.886081\pi$
$181$	−4561.63	−1.87328	−0.936640	+	0.350295i	$0.886081\pi$
$182$	1292.98	0.526606
$183$	0	0
$184$	−821.135	−0.328994
$185$	−3101.99	−1.23277
$186$	0	0
$187$	−375.387	−0.146797
$188$	1086.74	0.421587
$189$	0	0
$190$	8403.13	3.20856
$191$	−4180.40	−1.58368	−0.791841	−	0.610727i	$-0.790877\pi$
$191$	−4180.40	−1.58368	−0.791841	+	0.610727i	$0.790877\pi$
$192$	0	0
$193$	−1942.59	−0.724510	−0.362255	−	0.932079i	$-0.617993\pi$
$193$	−1942.59	−0.724510	−0.362255	+	0.932079i	$0.617993\pi$
$194$	−696.424	−0.257734
$195$	0	0
$196$	302.093	0.110092
$197$	−3775.15	−1.36532	−0.682660	−	0.730736i	$-0.739177\pi$
$197$	−3775.15	−1.36532	−0.682660	+	0.730736i	$0.739177\pi$
$198$	0	0
$199$	−993.760	−0.353999	−0.176999	−	0.984211i	$-0.556639\pi$
$199$	−993.760	−0.353999	−0.176999	+	0.984211i	$0.556639\pi$
$200$	794.302	0.280828
$201$	0	0
$202$	4680.31	1.63023
$203$	−445.781	−0.154127
$204$	0	0
$205$	−6991.29	−2.38192
$206$	5853.27	1.97969
$207$	0	0
$208$	−3696.14	−1.23212
$209$	1585.25	0.524662
$210$	0	0
$211$	−2912.24	−0.950176	−0.475088	−	0.879938i	$-0.657584\pi$
$211$	−2912.24	−0.950176	−0.475088	+	0.879938i	$0.657584\pi$
$212$	3883.97	1.25827
$213$	0	0
$214$	−929.856	−0.297026
$215$	−2024.32	−0.642128
$216$	0	0
$217$	1490.39	0.466239
$218$	−4368.02	−1.35706
$219$	0	0
$220$	−1050.66	−0.321978
$221$	1674.83	0.509780
$222$	0	0
$223$	5069.26	1.52225	0.761127	−	0.648603i	$-0.224647\pi$
$223$	5069.26	1.52225	0.761127	+	0.648603i	$0.224647\pi$
$224$	−1597.42	−0.476484
$225$	0	0
$226$	6543.12	1.92585
$227$	5653.25	1.65295	0.826475	−	0.562974i	$-0.190343\pi$
$227$	5653.25	1.65295	0.826475	+	0.562974i	$0.190343\pi$
$228$	0	0
$229$	−5141.63	−1.48371	−0.741853	−	0.670563i	$-0.766053\pi$
$229$	−5141.63	−1.48371	−0.741853	+	0.670563i	$0.766053\pi$
$230$	6933.29	1.98768
$231$	0	0
$232$	439.778	0.124452
$233$	−312.296	−0.0878077	−0.0439039	−	0.999036i	$-0.513980\pi$
$233$	−312.296	−0.0878077	−0.0439039	+	0.999036i	$0.513980\pi$
$234$	0	0
$235$	2730.89	0.758057
$236$	535.948	0.147827
$237$	0	0
$238$	899.074	0.244867
$239$	4012.12	1.08587	0.542935	−	0.839775i	$-0.317313\pi$
$239$	4012.12	1.08587	0.542935	+	0.839775i	$0.317313\pi$
$240$	0	0
$241$	499.662	0.133552	0.0667761	−	0.997768i	$-0.478729\pi$
$241$	499.662	0.133552	0.0667761	+	0.997768i	$0.478729\pi$
$242$	−455.403	−0.120969
$243$	0	0
$244$	3971.89	1.04211
$245$	759.137	0.197957
$246$	0	0
$247$	−7072.78	−1.82198
$248$	−1470.32	−0.376472
$249$	0	0
$250$	581.895	0.147209
$251$	6112.34	1.53708	0.768541	−	0.639801i	$-0.220983\pi$
$251$	6112.34	1.53708	0.768541	+	0.639801i	$0.220983\pi$
$252$	0	0
$253$	1307.97	0.325025
$254$	5674.28	1.40172
$255$	0	0
$256$	5290.40	1.29160
$257$	1910.22	0.463642	0.231821	−	0.972758i	$-0.425532\pi$
$257$	1910.22	0.463642	0.231821	+	0.972758i	$0.425532\pi$
$258$	0	0
$259$	1401.57	0.336252
$260$	4687.62	1.11813
$261$	0	0
$262$	4996.44	1.17817
$263$	−1749.49	−0.410184	−0.205092	−	0.978743i	$-0.565749\pi$
$263$	−1749.49	−0.410184	−0.205092	+	0.978743i	$0.565749\pi$
$264$	0	0
$265$	9760.14	2.26249
$266$	−3796.77	−0.875170
$267$	0	0
$268$	−2472.07	−0.563454
$269$	2884.89	0.653883	0.326942	−	0.945045i	$-0.393982\pi$
$269$	2884.89	0.653883	0.326942	+	0.945045i	$0.393982\pi$
$270$	0	0
$271$	6065.04	1.35950	0.679751	−	0.733443i	$-0.262088\pi$
$271$	6065.04	1.35950	0.679751	+	0.733443i	$0.262088\pi$
$272$	−2570.11	−0.572926
$273$	0	0
$274$	−1661.77	−0.366392
$275$	−1265.23	−0.277440
$276$	0	0
$277$	−3331.31	−0.722595	−0.361297	−	0.932451i	$-0.617666\pi$
$277$	−3331.31	−0.722595	−0.361297	+	0.932451i	$0.617666\pi$
$278$	−674.307	−0.145476
$279$	0	0
$280$	−748.915	−0.159844
$281$	384.608	0.0816505	0.0408252	−	0.999166i	$-0.487001\pi$
$281$	384.608	0.0816505	0.0408252	+	0.999166i	$0.487001\pi$
$282$	0	0
$283$	1768.02	0.371371	0.185685	−	0.982609i	$-0.440549\pi$
$283$	1768.02	0.371371	0.185685	+	0.982609i	$0.440549\pi$
$284$	−3129.50	−0.653879
$285$	0	0
$286$	2031.83	0.420086
$287$	3158.87	0.649693
$288$	0	0
$289$	−3748.41	−0.762957
$290$	−3713.28	−0.751902
$291$	0	0
$292$	1087.49	0.217946
$293$	−3801.87	−0.758047	−0.379023	−	0.925387i	$-0.623740\pi$
$293$	−3801.87	−0.758047	−0.379023	+	0.925387i	$0.623740\pi$
$294$	0	0
$295$	1346.80	0.265809
$296$	−1382.69	−0.271512
$297$	0	0
$298$	−4317.17	−0.839217
$299$	−5835.64	−1.12871
$300$	0	0
$301$	914.647	0.175147
$302$	−7049.46	−1.34321
$303$	0	0
$304$	10853.5	2.04767
$305$	9981.07	1.87382
$306$	0	0
$307$	−687.619	−0.127832	−0.0639161	−	0.997955i	$-0.520359\pi$
$307$	−687.619	−0.127832	−0.0639161	+	0.997955i	$0.520359\pi$
$308$	474.717	0.0878231
$309$	0	0
$310$	12414.7	2.27454
$311$	2517.73	0.459059	0.229530	−	0.973302i	$-0.426281\pi$
$311$	2517.73	0.459059	0.229530	+	0.973302i	$0.426281\pi$
$312$	0	0
$313$	−4725.03	−0.853274	−0.426637	−	0.904423i	$-0.640302\pi$
$313$	−4725.03	−0.853274	−0.426637	+	0.904423i	$0.640302\pi$
$314$	2423.52	0.435565
$315$	0	0
$316$	−4322.53	−0.769497
$317$	−2870.09	−0.508519	−0.254259	−	0.967136i	$-0.581832\pi$
$317$	−2870.09	−0.508519	−0.254259	+	0.967136i	$0.581832\pi$
$318$	0	0
$319$	−700.513	−0.122950
$320$	−3972.05	−0.693888
$321$	0	0
$322$	−3132.66	−0.542162
$323$	−4918.05	−0.847207
$324$	0	0
$325$	5644.94	0.963461
$326$	11458.4	1.94669
$327$	0	0
$328$	−3116.33	−0.524605
$329$	−1233.89	−0.206768
$330$	0	0
$331$	365.467	0.0606885	0.0303443	−	0.999540i	$-0.490340\pi$
$331$	365.467	0.0606885	0.0303443	+	0.999540i	$0.490340\pi$
$332$	7763.61	1.28338
$333$	0	0
$334$	−1268.50	−0.207811
$335$	−6212.13	−1.01315
$336$	0	0
$337$	−4292.33	−0.693822	−0.346911	−	0.937898i	$-0.612769\pi$
$337$	−4292.33	−0.693822	−0.346911	+	0.937898i	$0.612769\pi$
$338$	−796.455	−0.128170
$339$	0	0
$340$	3259.53	0.519920
$341$	2342.03	0.371930
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−902.330	−0.141426
$345$	0	0
$346$	−13190.9	−2.04956
$347$	6090.40	0.942218	0.471109	−	0.882075i	$-0.343854\pi$
$347$	6090.40	0.942218	0.471109	+	0.882075i	$0.343854\pi$
$348$	0	0
$349$	6338.89	0.972244	0.486122	−	0.873891i	$-0.338411\pi$
$349$	6338.89	0.972244	0.486122	+	0.873891i	$0.338411\pi$
$350$	3030.29	0.462788
$351$	0	0
$352$	−2510.24	−0.380103
$353$	−5754.44	−0.867643	−0.433821	−	0.900999i	$-0.642835\pi$
$353$	−5754.44	−0.867643	−0.433821	+	0.900999i	$0.642835\pi$
$354$	0	0
$355$	−7864.21	−1.17574
$356$	−4860.57	−0.723622
$357$	0	0
$358$	832.062	0.122838
$359$	−7396.98	−1.08746	−0.543729	−	0.839261i	$-0.682988\pi$
$359$	−7396.98	−1.08746	−0.543729	+	0.839261i	$0.682988\pi$
$360$	0	0
$361$	13909.8	2.02797
$362$	17168.4	2.49269
$363$	0	0
$364$	−2118.00	−0.304982
$365$	2732.78	0.391891
$366$	0	0
$367$	−4613.59	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$367$	−4613.59	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$368$	8955.07	1.26852
$369$	0	0
$370$	11674.8	1.64039
$371$	−4409.91	−0.617119
$372$	0	0
$373$	4603.15	0.638986	0.319493	−	0.947589i	$-0.396487\pi$
$373$	4603.15	0.638986	0.319493	+	0.947589i	$0.396487\pi$
$374$	1412.83	0.195336
$375$	0	0
$376$	1217.28	0.166958
$377$	3125.41	0.426968
$378$	0	0
$379$	6982.50	0.946350	0.473175	−	0.880968i	$-0.343108\pi$
$379$	6982.50	0.946350	0.473175	+	0.880968i	$0.343108\pi$
$380$	−13764.9	−1.85823
$381$	0	0
$382$	15733.6	2.10734
$383$	−12743.5	−1.70016	−0.850079	−	0.526655i	$-0.823446\pi$
$383$	−12743.5	−1.70016	−0.850079	+	0.526655i	$0.823446\pi$
$384$	0	0
$385$	1192.93	0.157915
$386$	7311.24	0.964073
$387$	0	0
$388$	1140.79	0.149265
$389$	−1568.76	−0.204471	−0.102235	−	0.994760i	$-0.532600\pi$
$389$	−1568.76	−0.204471	−0.102235	+	0.994760i	$0.532600\pi$
$390$	0	0
$391$	−4057.81	−0.524839
$392$	338.381	0.0435991
$393$	0	0
$394$	14208.4	1.81677
$395$	−10862.2	−1.38364
$396$	0	0
$397$	9841.87	1.24420	0.622102	−	0.782936i	$-0.286279\pi$
$397$	9841.87	1.24420	0.622102	+	0.782936i	$0.286279\pi$
$398$	3740.18	0.471050
$399$	0	0
$400$	−8662.43	−1.08280
$401$	−5128.60	−0.638679	−0.319339	−	0.947640i	$-0.603461\pi$
$401$	−5128.60	−0.638679	−0.319339	+	0.947640i	$0.603461\pi$
$402$	0	0
$403$	−10449.2	−1.29160
$404$	−7666.69	−0.944139
$405$	0	0
$406$	1677.77	0.205089
$407$	2202.46	0.268236
$408$	0	0
$409$	−87.6006	−0.0105906	−0.00529532	−	0.999986i	$-0.501686\pi$
$409$	−87.6006	−0.0105906	−0.00529532	+	0.999986i	$0.501686\pi$
$410$	26312.9	3.16951
$411$	0	0
$412$	−9588.08	−1.14653
$413$	−608.523	−0.0725023
$414$	0	0
$415$	19509.4	2.30766
$416$	11199.7	1.31998
$417$	0	0
$418$	−5966.36	−0.698144
$419$	7130.86	0.831420	0.415710	−	0.909497i	$-0.363533\pi$
$419$	7130.86	0.831420	0.415710	+	0.909497i	$0.363533\pi$
$420$	0	0
$421$	6159.77	0.713085	0.356543	−	0.934279i	$-0.383955\pi$
$421$	6159.77	0.713085	0.356543	+	0.934279i	$0.383955\pi$
$422$	10960.7	1.26436
$423$	0	0
$424$	4350.53	0.498303
$425$	3925.20	0.448001
$426$	0	0
$427$	−4509.74	−0.511104
$428$	1523.17	0.172022
$429$	0	0
$430$	7618.86	0.854452
$431$	−12044.1	−1.34604	−0.673019	−	0.739625i	$-0.735003\pi$
$431$	−12044.1	−1.34604	−0.673019	+	0.739625i	$0.735003\pi$
$432$	0	0
$433$	−9609.67	−1.06654	−0.533270	−	0.845945i	$-0.679037\pi$
$433$	−9609.67	−1.06654	−0.533270	+	0.845945i	$0.679037\pi$
$434$	−5609.31	−0.620404
$435$	0	0
$436$	7155.14	0.785938
$437$	17136.0	1.87581
$438$	0	0
$439$	−3122.62	−0.339486	−0.169743	−	0.985488i	$-0.554294\pi$
$439$	−3122.62	−0.339486	−0.169743	+	0.985488i	$0.554294\pi$
$440$	−1176.87	−0.127511
$441$	0	0
$442$	−6303.50	−0.678341
$443$	−9381.76	−1.00619	−0.503093	−	0.864232i	$-0.667805\pi$
$443$	−9381.76	−1.00619	−0.503093	+	0.864232i	$0.667805\pi$
$444$	0	0
$445$	−12214.3	−1.30115
$446$	−19079.0	−2.02559
$447$	0	0
$448$	1794.69	0.189266
$449$	−3407.36	−0.358136	−0.179068	−	0.983837i	$-0.557308\pi$
$449$	−3407.36	−0.358136	−0.179068	+	0.983837i	$0.557308\pi$
$450$	0	0
$451$	4963.93	0.518276
$452$	−10718.1	−1.11535
$453$	0	0
$454$	−21276.9	−2.19951
$455$	−5322.38	−0.548389
$456$	0	0
$457$	−13714.0	−1.40375	−0.701876	−	0.712299i	$-0.747654\pi$
$457$	−13714.0	−1.40375	−0.701876	+	0.712299i	$0.747654\pi$
$458$	19351.4	1.97430
$459$	0	0
$460$	−11357.2	−1.15116
$461$	8864.54	0.895581	0.447790	−	0.894139i	$-0.352211\pi$
$461$	8864.54	0.895581	0.447790	+	0.894139i	$0.352211\pi$
$462$	0	0
$463$	−14753.3	−1.48087	−0.740437	−	0.672126i	$-0.765381\pi$
$463$	−14753.3	−1.48087	−0.740437	+	0.672126i	$0.765381\pi$
$464$	−4796.10	−0.479856
$465$	0	0
$466$	1175.38	0.116842
$467$	−488.428	−0.0483977	−0.0241989	−	0.999707i	$-0.507703\pi$
$467$	−488.428	−0.0483977	−0.0241989	+	0.999707i	$0.507703\pi$
$468$	0	0
$469$	2806.82	0.276347
$470$	−10278.1	−1.00871
$471$	0	0
$472$	600.329	0.0585431
$473$	1437.30	0.139719
$474$	0	0
$475$	−16576.1	−1.60118
$476$	−1472.75	−0.141814
$477$	0	0
$478$	−15100.3	−1.44492
$479$	1233.56	0.117668	0.0588338	−	0.998268i	$-0.481262\pi$
$479$	1233.56	0.117668	0.0588338	+	0.998268i	$0.481262\pi$
$480$	0	0
$481$	−9826.52	−0.931499
$482$	−1880.56	−0.177712
$483$	0	0
$484$	745.984	0.0700586
$485$	2866.73	0.268395
$486$	0	0
$487$	16700.6	1.55395	0.776977	−	0.629529i	$-0.216752\pi$
$487$	16700.6	1.55395	0.776977	+	0.629529i	$0.216752\pi$
$488$	4449.01	0.412699
$489$	0	0
$490$	−2857.14	−0.263413
$491$	2127.96	0.195587	0.0977937	−	0.995207i	$-0.468821\pi$
$491$	2127.96	0.195587	0.0977937	+	0.995207i	$0.468821\pi$
$492$	0	0
$493$	2173.25	0.198536
$494$	26619.5	2.42443
$495$	0	0
$496$	16034.9	1.45159
$497$	3553.28	0.320697
$498$	0	0
$499$	12341.5	1.10718	0.553590	−	0.832789i	$-0.313258\pi$
$499$	12341.5	1.10718	0.553590	+	0.832789i	$0.313258\pi$
$500$	−953.186	−0.0852556
$501$	0	0
$502$	−23004.8	−2.04533
$503$	3433.67	0.304374	0.152187	−	0.988352i	$-0.451368\pi$
$503$	3433.67	0.304374	0.152187	+	0.988352i	$0.451368\pi$
$504$	0	0
$505$	−19265.9	−1.69766
$506$	−4922.75	−0.432496
$507$	0	0
$508$	−9294.88	−0.811798
$509$	1842.48	0.160445	0.0802223	−	0.996777i	$-0.474437\pi$
$509$	1842.48	0.160445	0.0802223	+	0.996777i	$0.474437\pi$
$510$	0	0
$511$	−1234.75	−0.106892
$512$	−13025.8	−1.12434
$513$	0	0
$514$	−7189.41	−0.616948
$515$	−24094.2	−2.06158
$516$	0	0
$517$	−1938.98	−0.164944
$518$	−5275.03	−0.447435
$519$	0	0
$520$	5250.71	0.442806
$521$	−3532.65	−0.297060	−0.148530	−	0.988908i	$-0.547454\pi$
$521$	−3532.65	−0.297060	−0.148530	+	0.988908i	$0.547454\pi$
$522$	0	0
$523$	−4497.20	−0.376001	−0.188001	−	0.982169i	$-0.560201\pi$
$523$	−4497.20	−0.376001	−0.188001	+	0.982169i	$0.560201\pi$
$524$	−8184.54	−0.682335
$525$	0	0
$526$	6584.50	0.545814
$527$	−7265.87	−0.600581
$528$	0	0
$529$	1971.67	0.162050
$530$	−36733.9	−3.01060
$531$	0	0
$532$	6219.39	0.506851
$533$	−22147.1	−1.79981
$534$	0	0
$535$	3827.62	0.309313
$536$	−2769.02	−0.223141
$537$	0	0
$538$	−10857.7	−0.870093
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−11426.0	−0.908023	−0.454011	−	0.890996i	$-0.650007\pi$
$541$	−11426.0	−0.908023	−0.454011	+	0.890996i	$0.650007\pi$
$542$	−22826.8	−1.80903
$543$	0	0
$544$	7787.70	0.613777
$545$	17980.4	1.41320
$546$	0	0
$547$	17875.3	1.39725	0.698623	−	0.715490i	$-0.253796\pi$
$547$	17875.3	1.39725	0.698623	+	0.715490i	$0.253796\pi$
$548$	2722.11	0.212195
$549$	0	0
$550$	4761.88	0.369177
$551$	−9177.61	−0.709581
$552$	0	0
$553$	4907.86	0.377402
$554$	12537.9	0.961525
$555$	0	0
$556$	1104.56	0.0842517
$557$	−25343.4	−1.92789	−0.963946	−	0.266097i	$-0.914266\pi$
$557$	−25343.4	−1.92789	−0.963946	+	0.266097i	$0.914266\pi$
$558$	0	0
$559$	−6412.68	−0.485201
$560$	8167.46	0.616318
$561$	0	0
$562$	−1447.53	−0.108649
$563$	1597.92	0.119617	0.0598084	−	0.998210i	$-0.480951\pi$
$563$	1597.92	0.119617	0.0598084	+	0.998210i	$0.480951\pi$
$564$	0	0
$565$	−26933.8	−2.00551
$566$	−6654.23	−0.494166
$567$	0	0
$568$	−3505.43	−0.258952
$569$	11487.7	0.846377	0.423188	−	0.906042i	$-0.360911\pi$
$569$	11487.7	0.846377	0.423188	+	0.906042i	$0.360911\pi$
$570$	0	0
$571$	17027.0	1.24791	0.623957	−	0.781459i	$-0.285524\pi$
$571$	17027.0	1.24791	0.623957	+	0.781459i	$0.285524\pi$
$572$	−3328.28	−0.243291
$573$	0	0
$574$	−11888.9	−0.864518
$575$	−13676.6	−0.991922
$576$	0	0
$577$	19368.0	1.39740	0.698701	−	0.715414i	$-0.253762\pi$
$577$	19368.0	1.39740	0.698701	+	0.715414i	$0.253762\pi$
$578$	14107.7	1.01523
$579$	0	0
$580$	6082.63	0.435461
$581$	−8814.91	−0.629439
$582$	0	0
$583$	−6929.86	−0.492291
$584$	1218.12	0.0863120
$585$	0	0
$586$	14309.0	1.00870
$587$	27035.0	1.90094	0.950472	−	0.310809i	$-0.100600\pi$
$587$	27035.0	1.90094	0.950472	+	0.310809i	$0.100600\pi$
$588$	0	0
$589$	30683.6	2.14651
$590$	−5068.90	−0.353700
$591$	0	0
$592$	15079.3	1.04688
$593$	9717.62	0.672943	0.336471	−	0.941694i	$-0.390766\pi$
$593$	9717.62	0.672943	0.336471	+	0.941694i	$0.390766\pi$
$594$	0	0
$595$	−3700.92	−0.254996
$596$	7071.83	0.486030
$597$	0	0
$598$	21963.4	1.50192
$599$	18386.6	1.25418	0.627092	−	0.778945i	$-0.284245\pi$
$599$	18386.6	1.25418	0.627092	+	0.778945i	$0.284245\pi$
$600$	0	0
$601$	−3288.31	−0.223183	−0.111591	−	0.993754i	$-0.535595\pi$
$601$	−3288.31	−0.223183	−0.111591	+	0.993754i	$0.535595\pi$
$602$	−3442.42	−0.233061
$603$	0	0
$604$	11547.5	0.777918
$605$	1874.60	0.125973
$606$	0	0
$607$	−27356.2	−1.82925	−0.914623	−	0.404307i	$-0.867513\pi$
$607$	−27356.2	−1.82925	−0.914623	+	0.404307i	$0.867513\pi$
$608$	−32887.3	−2.19368
$609$	0	0
$610$	−37565.4	−2.49341
$611$	8650.95	0.572798
$612$	0	0
$613$	−9573.59	−0.630789	−0.315395	−	0.948961i	$-0.602137\pi$
$613$	−9573.59	−0.630789	−0.315395	+	0.948961i	$0.602137\pi$
$614$	2587.97	0.170101
$615$	0	0
$616$	531.742	0.0347800
$617$	7506.76	0.489807	0.244903	−	0.969547i	$-0.421244\pi$
$617$	7506.76	0.489807	0.244903	+	0.969547i	$0.421244\pi$
$618$	0	0
$619$	−9294.64	−0.603527	−0.301763	−	0.953383i	$-0.597575\pi$
$619$	−9294.64	−0.603527	−0.301763	+	0.953383i	$0.597575\pi$
$620$	−20336.1	−1.31729
$621$	0	0
$622$	−9475.89	−0.610850
$623$	5518.75	0.354902
$624$	0	0
$625$	−16772.8	−1.07346
$626$	17783.4	1.13541
$627$	0	0
$628$	−3969.91	−0.252256
$629$	−6832.87	−0.433139
$630$	0	0
$631$	4254.20	0.268395	0.134197	−	0.990955i	$-0.457154\pi$
$631$	4254.20	0.268395	0.134197	+	0.990955i	$0.457154\pi$
$632$	−4841.77	−0.304739
$633$	0	0
$634$	10802.1	0.676663
$635$	−23357.4	−1.45970
$636$	0	0
$637$	2404.81	0.149579
$638$	2636.49	0.163605
$639$	0	0
$640$	−13334.3	−0.823566
$641$	−5112.91	−0.315051	−0.157526	−	0.987515i	$-0.550352\pi$
$641$	−5112.91	−0.315051	−0.157526	+	0.987515i	$0.550352\pi$
$642$	0	0
$643$	7296.68	0.447516	0.223758	−	0.974645i	$-0.428167\pi$
$643$	7296.68	0.447516	0.223758	+	0.974645i	$0.428167\pi$
$644$	5131.52	0.313991
$645$	0	0
$646$	18509.9	1.12734
$647$	−15612.1	−0.948649	−0.474324	−	0.880350i	$-0.657308\pi$
$647$	−15612.1	−0.948649	−0.474324	+	0.880350i	$0.657308\pi$
$648$	0	0
$649$	−956.250	−0.0578368
$650$	−21245.6	−1.28203
$651$	0	0
$652$	−18769.6	−1.12742
$653$	23160.6	1.38797	0.693986	−	0.719988i	$-0.255853\pi$
$653$	23160.6	1.38797	0.693986	+	0.719988i	$0.255853\pi$
$654$	0	0
$655$	−20567.2	−1.22691
$656$	33985.8	2.02275
$657$	0	0
$658$	4643.96	0.275137
$659$	5707.65	0.337388	0.168694	−	0.985668i	$-0.446045\pi$
$659$	5707.65	0.337388	0.168694	+	0.985668i	$0.446045\pi$
$660$	0	0
$661$	−16202.5	−0.953411	−0.476706	−	0.879063i	$-0.658169\pi$
$661$	−16202.5	−0.953411	−0.476706	+	0.879063i	$0.658169\pi$
$662$	−1375.50	−0.0807555
$663$	0	0
$664$	8696.21	0.508251
$665$	15628.9	0.911372
$666$	0	0
$667$	−7572.30	−0.439581
$668$	2077.89	0.120353
$669$	0	0
$670$	23380.4	1.34815
$671$	−7086.73	−0.407720
$672$	0	0
$673$	−16956.2	−0.971192	−0.485596	−	0.874183i	$-0.661397\pi$
$673$	−16956.2	−0.971192	−0.485596	+	0.874183i	$0.661397\pi$
$674$	16154.9	0.923239
$675$	0	0
$676$	1304.65	0.0742292
$677$	−21259.0	−1.20687	−0.603433	−	0.797414i	$-0.706201\pi$
$677$	−21259.0	−1.20687	−0.603433	+	0.797414i	$0.706201\pi$
$678$	0	0
$679$	−1295.27	−0.0732076
$680$	3651.08	0.205901
$681$	0	0
$682$	−8814.62	−0.494911
$683$	−5867.98	−0.328744	−0.164372	−	0.986398i	$-0.552560\pi$
$683$	−5867.98	−0.328744	−0.164372	+	0.986398i	$0.552560\pi$
$684$	0	0
$685$	6840.46	0.381548
$686$	1290.94	0.0718486
$687$	0	0
$688$	9840.57	0.545303
$689$	30918.3	1.70957
$690$	0	0
$691$	15583.0	0.857892	0.428946	−	0.903330i	$-0.358885\pi$
$691$	15583.0	0.857892	0.428946	+	0.903330i	$0.358885\pi$
$692$	21607.7	1.18699
$693$	0	0
$694$	−22922.2	−1.25377
$695$	2775.69	0.151493
$696$	0	0
$697$	−15400.0	−0.836895
$698$	−23857.4	−1.29372
$699$	0	0
$700$	−4963.83	−0.268022
$701$	19210.5	1.03505	0.517525	−	0.855668i	$-0.326853\pi$
$701$	19210.5	1.03505	0.517525	+	0.855668i	$0.326853\pi$
$702$	0	0
$703$	28855.1	1.54806
$704$	2820.22	0.150982
$705$	0	0
$706$	21657.8	1.15453
$707$	8704.87	0.463055
$708$	0	0
$709$	−1760.77	−0.0932683	−0.0466342	−	0.998912i	$-0.514850\pi$
$709$	−1760.77	−0.0932683	−0.0466342	+	0.998912i	$0.514850\pi$
$710$	29598.2	1.56451
$711$	0	0
$712$	−5444.44	−0.286572
$713$	25316.6	1.32975
$714$	0	0
$715$	−8363.74	−0.437463
$716$	−1362.98	−0.0711409
$717$	0	0
$718$	27839.7	1.44703
$719$	−10517.7	−0.545541	−0.272770	−	0.962079i	$-0.587940\pi$
$719$	−10517.7	−0.545541	−0.272770	+	0.962079i	$0.587940\pi$
$720$	0	0
$721$	10886.4	0.562319
$722$	−52352.0	−2.69853
$723$	0	0
$724$	−28123.2	−1.44363
$725$	7324.85	0.375225
$726$	0	0
$727$	−10483.4	−0.534814	−0.267407	−	0.963584i	$-0.586167\pi$
$727$	−10483.4	−0.534814	−0.267407	+	0.963584i	$0.586167\pi$
$728$	−2372.42	−0.120780
$729$	0	0
$730$	−10285.2	−0.521471
$731$	−4459.05	−0.225614
$732$	0	0
$733$	−8568.83	−0.431783	−0.215892	−	0.976417i	$-0.569266\pi$
$733$	−8568.83	−0.431783	−0.215892	+	0.976417i	$0.569266\pi$
$734$	17364.0	0.873183
$735$	0	0
$736$	−27134.8	−1.35897
$737$	4410.72	0.220449
$738$	0	0
$739$	−3089.07	−0.153766	−0.0768832	−	0.997040i	$-0.524497\pi$
$739$	−3089.07	−0.153766	−0.0768832	+	0.997040i	$0.524497\pi$
$740$	−19124.2	−0.950028
$741$	0	0
$742$	16597.4	0.821173
$743$	28558.8	1.41012	0.705060	−	0.709148i	$-0.250920\pi$
$743$	28558.8	1.41012	0.705060	+	0.709148i	$0.250920\pi$
$744$	0	0
$745$	17771.0	0.873933
$746$	−17324.7	−0.850270
$747$	0	0
$748$	−2314.32	−0.113128
$749$	−1729.43	−0.0843685
$750$	0	0
$751$	−2062.51	−0.100216	−0.0501078	−	0.998744i	$-0.515956\pi$
$751$	−2062.51	−0.100216	−0.0501078	+	0.998744i	$0.515956\pi$
$752$	−13275.3	−0.643751
$753$	0	0
$754$	−11763.0	−0.568147
$755$	29018.1	1.39878
$756$	0	0
$757$	39089.1	1.87678	0.938388	−	0.345584i	$-0.112319\pi$
$757$	39089.1	1.87678	0.938388	+	0.345584i	$0.112319\pi$
$758$	−26279.8	−1.25927
$759$	0	0
$760$	−15418.4	−0.735902
$761$	13820.5	0.658337	0.329169	−	0.944271i	$-0.393232\pi$
$761$	13820.5	0.658337	0.329169	+	0.944271i	$0.393232\pi$
$762$	0	0
$763$	−8124.04	−0.385465
$764$	−25772.8	−1.22046
$765$	0	0
$766$	47962.1	2.26233
$767$	4266.41	0.200849
$768$	0	0
$769$	−28495.1	−1.33623	−0.668115	−	0.744058i	$-0.732899\pi$
$769$	−28495.1	−1.33623	−0.668115	+	0.744058i	$0.732899\pi$
$770$	−4489.78	−0.210131
$771$	0	0
$772$	−11976.3	−0.558339
$773$	−1874.14	−0.0872032	−0.0436016	−	0.999049i	$-0.513883\pi$
$773$	−1874.14	−0.0872032	−0.0436016	+	0.999049i	$0.513883\pi$
$774$	0	0
$775$	−24489.3	−1.13507
$776$	1277.83	0.0591127
$777$	0	0
$778$	5904.28	0.272080
$779$	65033.8	2.99112
$780$	0	0
$781$	5583.72	0.255828
$782$	15272.2	0.698380
$783$	0	0
$784$	−3690.29	−0.168107
$785$	−9976.10	−0.453582
$786$	0	0
$787$	16614.9	0.752551	0.376275	−	0.926508i	$-0.377205\pi$
$787$	16614.9	0.752551	0.376275	+	0.926508i	$0.377205\pi$
$788$	−23274.4	−1.05218
$789$	0	0
$790$	40881.7	1.84114
$791$	12169.5	0.547025
$792$	0	0
$793$	31618.2	1.41588
$794$	−37041.5	−1.65561
$795$	0	0
$796$	−6126.68	−0.272807
$797$	−19323.7	−0.858823	−0.429411	−	0.903109i	$-0.641279\pi$
$797$	−19323.7	−0.858823	−0.429411	+	0.903109i	$0.641279\pi$
$798$	0	0
$799$	6015.43	0.266346
$800$	26248.1	1.16001
$801$	0	0
$802$	19302.3	0.849861
$803$	−1940.32	−0.0852706
$804$	0	0
$805$	12895.2	0.564589
$806$	39327.4	1.71867
$807$	0	0
$808$	−8587.65	−0.373902
$809$	15363.0	0.667657	0.333829	−	0.942634i	$-0.391659\pi$
$809$	15363.0	0.667657	0.333829	+	0.942634i	$0.391659\pi$
$810$	0	0
$811$	11404.0	0.493771	0.246886	−	0.969045i	$-0.420593\pi$
$811$	11404.0	0.493771	0.246886	+	0.969045i	$0.420593\pi$
$812$	−2748.31	−0.118777
$813$	0	0
$814$	−8289.33	−0.356930
$815$	−47166.8	−2.02722
$816$	0	0
$817$	18830.5	0.806359
$818$	329.699	0.0140925
$819$	0	0
$820$	−43102.4	−1.83561
$821$	−5529.00	−0.235035	−0.117517	−	0.993071i	$-0.537494\pi$
$821$	−5529.00	−0.235035	−0.117517	+	0.993071i	$0.537494\pi$
$822$	0	0
$823$	−12216.1	−0.517407	−0.258704	−	0.965957i	$-0.583295\pi$
$823$	−12216.1	−0.517407	−0.258704	+	0.965957i	$0.583295\pi$
$824$	−10739.8	−0.454054
$825$	0	0
$826$	2290.27	0.0964756
$827$	29123.0	1.22455	0.612277	−	0.790643i	$-0.290254\pi$
$827$	29123.0	1.22455	0.612277	+	0.790643i	$0.290254\pi$
$828$	0	0
$829$	25679.6	1.07586	0.537930	−	0.842990i	$-0.319207\pi$
$829$	25679.6	1.07586	0.537930	+	0.842990i	$0.319207\pi$
$830$	−73426.8	−3.07070
$831$	0	0
$832$	−12582.7	−0.524312
$833$	1672.18	0.0695529
$834$	0	0
$835$	5221.59	0.216408
$836$	9773.33	0.404328
$837$	0	0
$838$	−26838.1	−1.10633
$839$	7009.97	0.288452	0.144226	−	0.989545i	$-0.453931\pi$
$839$	7009.97	0.288452	0.144226	+	0.989545i	$0.453931\pi$
$840$	0	0
$841$	−20333.5	−0.833715
$842$	−23183.3	−0.948871
$843$	0	0
$844$	−17954.4	−0.732247
$845$	3278.50	0.133472
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−47445.7	−1.92133
$849$	0	0
$850$	−14773.1	−0.596135
$851$	23807.9	0.959016
$852$	0	0
$853$	7430.54	0.298261	0.149131	−	0.988818i	$-0.452353\pi$
$853$	7430.54	0.298261	0.149131	+	0.988818i	$0.452353\pi$
$854$	16973.1	0.680103
$855$	0	0
$856$	1706.14	0.0681247
$857$	16112.3	0.642225	0.321113	−	0.947041i	$-0.395943\pi$
$857$	16112.3	0.642225	0.321113	+	0.947041i	$0.395943\pi$
$858$	0	0
$859$	46836.9	1.86037	0.930183	−	0.367097i	$-0.119648\pi$
$859$	46836.9	1.86037	0.930183	+	0.367097i	$0.119648\pi$
$860$	−12480.3	−0.494853
$861$	0	0
$862$	45329.8	1.79111
$863$	2453.00	0.0967568	0.0483784	−	0.998829i	$-0.484595\pi$
$863$	2453.00	0.0967568	0.0483784	+	0.998829i	$0.484595\pi$
$864$	0	0
$865$	54298.5	2.13434
$866$	36167.6	1.41920
$867$	0	0
$868$	9188.45	0.359305
$869$	7712.34	0.301063
$870$	0	0
$871$	−19678.9	−0.765549
$872$	8014.65	0.311250
$873$	0	0
$874$	−64494.3	−2.49605
$875$	1082.26	0.0418138
$876$	0	0
$877$	23733.6	0.913829	0.456914	−	0.889511i	$-0.348955\pi$
$877$	23733.6	0.913829	0.456914	+	0.889511i	$0.348955\pi$
$878$	11752.5	0.451740
$879$	0	0
$880$	12834.6	0.491652
$881$	−10868.6	−0.415634	−0.207817	−	0.978168i	$-0.566636\pi$
$881$	−10868.6	−0.415634	−0.207817	+	0.978168i	$0.566636\pi$
$882$	0	0
$883$	−43075.4	−1.64168	−0.820840	−	0.571158i	$-0.806495\pi$
$883$	−43075.4	−1.64168	−0.820840	+	0.571158i	$0.806495\pi$
$884$	10325.6	0.392859
$885$	0	0
$886$	35309.8	1.33889
$887$	16224.0	0.614149	0.307075	−	0.951685i	$-0.400650\pi$
$887$	16224.0	0.614149	0.307075	+	0.951685i	$0.400650\pi$
$888$	0	0
$889$	10553.5	0.398149
$890$	45970.4	1.73138
$891$	0	0
$892$	31252.7	1.17312
$893$	−25403.1	−0.951938
$894$	0	0
$895$	−3425.07	−0.127919
$896$	6024.80	0.224637
$897$	0	0
$898$	12824.1	0.476556
$899$	−13558.9	−0.503019
$900$	0	0
$901$	21499.0	0.794935
$902$	−18682.6	−0.689647
$903$	0	0
$904$	−12005.6	−0.441704
$905$	−70671.5	−2.59580
$906$	0	0
$907$	39985.6	1.46384	0.731918	−	0.681392i	$-0.238625\pi$
$907$	39985.6	1.46384	0.731918	+	0.681392i	$0.238625\pi$
$908$	34853.2	1.27384
$909$	0	0
$910$	20031.7	0.729717
$911$	−20330.9	−0.739398	−0.369699	−	0.929152i	$-0.620539\pi$
$911$	−20330.9	−0.739398	−0.369699	+	0.929152i	$0.620539\pi$
$912$	0	0
$913$	−13852.0	−0.502119
$914$	51614.9	1.86791
$915$	0	0
$916$	−31699.0	−1.14341
$917$	9292.84	0.334653
$918$	0	0
$919$	−3734.71	−0.134055	−0.0670275	−	0.997751i	$-0.521352\pi$
$919$	−3734.71	−0.134055	−0.0670275	+	0.997751i	$0.521352\pi$
$920$	−12721.5	−0.455887
$921$	0	0
$922$	−33363.1	−1.19171
$923$	−24912.4	−0.888408
$924$	0	0
$925$	−23029.9	−0.818613
$926$	55526.5	1.97053
$927$	0	0
$928$	14532.7	0.514072
$929$	−30122.2	−1.06381	−0.531904	−	0.846805i	$-0.678523\pi$
$929$	−30122.2	−1.06381	−0.531904	+	0.846805i	$0.678523\pi$
$930$	0	0
$931$	−7061.59	−0.248587
$932$	−1925.35	−0.0676685
$933$	0	0
$934$	1838.28	0.0644007
$935$	−5815.72	−0.203417
$936$	0	0
$937$	−33701.9	−1.17502	−0.587509	−	0.809217i	$-0.699891\pi$
$937$	−33701.9	−1.17502	−0.587509	+	0.809217i	$0.699891\pi$
$938$	−10563.9	−0.367723
$939$	0	0
$940$	16836.3	0.584193
$941$	−20897.3	−0.723944	−0.361972	−	0.932189i	$-0.617896\pi$
$941$	−20897.3	−0.723944	−0.361972	+	0.932189i	$0.617896\pi$
$942$	0	0
$943$	53658.4	1.85298
$944$	−6547.02	−0.225728
$945$	0	0
$946$	−5409.52	−0.185918
$947$	43378.9	1.48852	0.744258	−	0.667892i	$-0.232803\pi$
$947$	43378.9	1.48852	0.744258	+	0.667892i	$0.232803\pi$
$948$	0	0
$949$	8656.93	0.296118
$950$	62386.7	2.13062
$951$	0	0
$952$	−1649.66	−0.0561616
$953$	36362.5	1.23599	0.617994	−	0.786183i	$-0.287946\pi$
$953$	36362.5	1.23599	0.617994	+	0.786183i	$0.287946\pi$
$954$	0	0
$955$	−64765.3	−2.19451
$956$	24735.4	0.836819
$957$	0	0
$958$	−4642.70	−0.156575
$959$	−3090.72	−0.104071
$960$	0	0
$961$	15540.6	0.521654
$962$	36983.7	1.23950
$963$	0	0
$964$	3080.49	0.102921
$965$	−30095.7	−1.00395
$966$	0	0
$967$	23769.2	0.790451	0.395225	−	0.918584i	$-0.370667\pi$
$967$	23769.2	0.790451	0.395225	+	0.918584i	$0.370667\pi$
$968$	835.595	0.0277449
$969$	0	0
$970$	−10789.4	−0.357141
$971$	27125.4	0.896494	0.448247	−	0.893910i	$-0.352048\pi$
$971$	27125.4	0.896494	0.448247	+	0.893910i	$0.352048\pi$
$972$	0	0
$973$	−1254.14	−0.0413214
$974$	−62855.3	−2.06778
$975$	0	0
$976$	−48519.7	−1.59127
$977$	30145.5	0.987146	0.493573	−	0.869704i	$-0.335691\pi$
$977$	30145.5	0.987146	0.493573	+	0.869704i	$0.335691\pi$
$978$	0	0
$979$	8672.33	0.283114
$980$	4680.20	0.152555
$981$	0	0
$982$	−8008.92	−0.260259
$983$	−23037.5	−0.747489	−0.373745	−	0.927532i	$-0.621926\pi$
$983$	−23037.5	−0.747489	−0.373745	+	0.927532i	$0.621926\pi$
$984$	0	0
$985$	−58486.8	−1.89192
$986$	−8179.39	−0.264183
$987$	0	0
$988$	−43604.8	−1.40410
$989$	15536.7	0.499534
$990$	0	0
$991$	13580.1	0.435303	0.217651	−	0.976027i	$-0.430160\pi$
$991$	13580.1	0.435303	0.217651	+	0.976027i	$0.430160\pi$
$992$	−48587.3	−1.55509
$993$	0	0
$994$	−13373.3	−0.426737
$995$	−15395.9	−0.490536
$996$	0	0
$997$	−26627.3	−0.845832	−0.422916	−	0.906169i	$-0.638993\pi$
$997$	−26627.3	−0.845832	−0.422916	+	0.906169i	$0.638993\pi$
$998$	−46449.3	−1.47327
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.m.1.1		4
3.2	odd	2		77.4.a.c.1.4	✓	4
12.11	even	2		1232.4.a.w.1.3		4
15.14	odd	2		1925.4.a.q.1.1		4
21.20	even	2		539.4.a.f.1.4		4
33.32	even	2		847.4.a.e.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.c.1.4	✓	4	3.2	odd	2
539.4.a.f.1.4		4	21.20	even	2
693.4.a.m.1.1		4	1.1	even	1	trivial
847.4.a.e.1.1		4	33.32	even	2
1232.4.a.w.1.3		4	12.11	even	2
1925.4.a.q.1.1		4	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)

\(f(q)\)	\(=\)	\(q-3.76366 q^{2} +6.16515 q^{4} +15.4926 q^{5} -7.00000 q^{7} +6.90574 q^{8} +O(q^{10})\)	\(q-3.76366 q^{2} +6.16515 q^{4} +15.4926 q^{5} -7.00000 q^{7} +6.90574 q^{8} -58.3089 q^{10} -11.0000 q^{11} +49.0777 q^{13} +26.3456 q^{14} -75.3121 q^{16} +34.1261 q^{17} -144.114 q^{19} +95.5142 q^{20} +41.4003 q^{22} -118.906 q^{23} +115.020 q^{25} -184.712 q^{26} -43.1561 q^{28} +63.6830 q^{29} -212.912 q^{31} +228.203 q^{32} -128.439 q^{34} -108.448 q^{35} -200.224 q^{37} +542.396 q^{38} +106.988 q^{40} -451.267 q^{41} -130.664 q^{43} -67.8167 q^{44} +447.523 q^{46} +176.271 q^{47} +49.0000 q^{49} -432.898 q^{50} +302.571 q^{52} +629.988 q^{53} -170.419 q^{55} -48.3402 q^{56} -239.681 q^{58} +86.9318 q^{59} +644.248 q^{61} +801.330 q^{62} -256.384 q^{64} +760.340 q^{65} -400.974 q^{67} +210.393 q^{68} +408.162 q^{70} -507.611 q^{71} +176.392 q^{73} +753.575 q^{74} -888.485 q^{76} +77.0000 q^{77} -701.122 q^{79} -1166.78 q^{80} +1698.42 q^{82} +1259.27 q^{83} +528.702 q^{85} +491.775 q^{86} -75.9631 q^{88} -788.394 q^{89} -343.544 q^{91} -733.075 q^{92} -663.423 q^{94} -2232.70 q^{95} +185.039 q^{97} -184.419 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * q^98

Introduction

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