LMFDB - Embedded newform 693.4.a.l.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			$-0.148103$ 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)

$f(q)$	$=$	\(q-4.60395 q^{2} +13.1964 q^{4} -1.84418 q^{5} -7.00000 q^{7} -23.9238 q^{8} +8.49053 q^{10} +11.0000 q^{11} +24.6401 q^{13} +32.2277 q^{14} +4.57310 q^{16} -17.8800 q^{17} +32.1459 q^{19} -24.3365 q^{20} -50.6435 q^{22} -14.1248 q^{23} -121.599 q^{25} -113.442 q^{26} -92.3745 q^{28} +41.5471 q^{29} +175.766 q^{31} +170.336 q^{32} +82.3187 q^{34} +12.9093 q^{35} +292.877 q^{37} -147.998 q^{38} +44.1199 q^{40} -154.296 q^{41} -277.144 q^{43} +145.160 q^{44} +65.0300 q^{46} +52.1450 q^{47} +49.0000 q^{49} +559.836 q^{50} +325.160 q^{52} -82.3907 q^{53} -20.2860 q^{55} +167.467 q^{56} -191.281 q^{58} -712.816 q^{59} -647.078 q^{61} -809.217 q^{62} -820.804 q^{64} -45.4408 q^{65} +260.867 q^{67} -235.951 q^{68} -59.4337 q^{70} -369.025 q^{71} +1145.77 q^{73} -1348.39 q^{74} +424.209 q^{76} -77.0000 q^{77} +488.885 q^{79} -8.43362 q^{80} +710.372 q^{82} -548.982 q^{83} +32.9740 q^{85} +1275.96 q^{86} -263.162 q^{88} -105.039 q^{89} -172.481 q^{91} -186.396 q^{92} -240.073 q^{94} -59.2829 q^{95} -1361.91 q^{97} -225.594 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * q^97 + 98 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.60395	−1.62774	−0.813871	−	0.581045i	$-0.802644\pi$
$2$	−4.60395	−1.62774	−0.813871	+	0.581045i	$0.802644\pi$
$3$	0	0
$3$	0	0
$4$	13.1964	1.64955
$5$	−1.84418	−0.164949	−0.0824744	−	0.996593i	$-0.526282\pi$
$5$	−1.84418	−0.164949	−0.0824744	+	0.996593i	$0.526282\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−23.9238	−1.05729
$9$	0	0
$10$	8.49053	0.268494
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	24.6401	0.525687	0.262844	−	0.964838i	$-0.415340\pi$
$13$	24.6401	0.525687	0.262844	+	0.964838i	$0.415340\pi$
$14$	32.2277	0.615229
$15$	0	0
$16$	4.57310	0.0714546
$17$	−17.8800	−0.255090	−0.127545	−	0.991833i	$-0.540710\pi$
$17$	−17.8800	−0.255090	−0.127545	+	0.991833i	$0.540710\pi$
$18$	0	0
$19$	32.1459	0.388146	0.194073	−	0.980987i	$-0.437830\pi$
$19$	32.1459	0.388146	0.194073	+	0.980987i	$0.437830\pi$
$20$	−24.3365	−0.272090
$21$	0	0
$22$	−50.6435	−0.490783
$23$	−14.1248	−0.128054	−0.0640268	−	0.997948i	$-0.520394\pi$
$23$	−14.1248	−0.128054	−0.0640268	+	0.997948i	$0.520394\pi$
$24$	0	0
$25$	−121.599	−0.972792
$26$	−113.442	−0.855683
$27$	0	0
$28$	−92.3745	−0.623470
$29$	41.5471	0.266038	0.133019	−	0.991113i	$-0.457533\pi$
$29$	41.5471	0.266038	0.133019	+	0.991113i	$0.457533\pi$
$30$	0	0
$31$	175.766	1.01834	0.509169	−	0.860667i	$-0.329953\pi$
$31$	175.766	1.01834	0.509169	+	0.860667i	$0.329953\pi$
$32$	170.336	0.940983
$33$	0	0
$34$	82.3187	0.415222
$35$	12.9093	0.0623448
$36$	0	0
$37$	292.877	1.30131	0.650657	−	0.759372i	$-0.274494\pi$
$37$	292.877	1.30131	0.650657	+	0.759372i	$0.274494\pi$
$38$	−147.998	−0.631801
$39$	0	0
$40$	44.1199	0.174399
$41$	−154.296	−0.587732	−0.293866	−	0.955847i	$-0.594942\pi$
$41$	−154.296	−0.587732	−0.293866	+	0.955847i	$0.594942\pi$
$42$	0	0
$43$	−277.144	−0.982887	−0.491443	−	0.870910i	$-0.663530\pi$
$43$	−277.144	−0.982887	−0.491443	+	0.870910i	$0.663530\pi$
$44$	145.160	0.497357
$45$	0	0
$46$	65.0300	0.208438
$47$	52.1450	0.161832	0.0809162	−	0.996721i	$-0.474215\pi$
$47$	52.1450	0.161832	0.0809162	+	0.996721i	$0.474215\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	559.836	1.58345
$51$	0	0
$52$	325.160	0.867145
$53$	−82.3907	−0.213533	−0.106766	−	0.994284i	$-0.534050\pi$
$53$	−82.3907	−0.213533	−0.106766	+	0.994284i	$0.534050\pi$
$54$	0	0
$55$	−20.2860	−0.0497339
$56$	167.467	0.399619
$57$	0	0
$58$	−191.281	−0.433041
$59$	−712.816	−1.57289	−0.786447	−	0.617658i	$-0.788081\pi$
$59$	−712.816	−1.57289	−0.786447	+	0.617658i	$0.788081\pi$
$60$	0	0
$61$	−647.078	−1.35819	−0.679097	−	0.734048i	$-0.737629\pi$
$61$	−647.078	−1.35819	−0.679097	+	0.734048i	$0.737629\pi$
$62$	−809.217	−1.65759
$63$	0	0
$64$	−820.804	−1.60313
$65$	−45.4408	−0.0867114
$66$	0	0
$67$	260.867	0.475672	0.237836	−	0.971305i	$-0.423562\pi$
$67$	260.867	0.475672	0.237836	+	0.971305i	$0.423562\pi$
$68$	−235.951	−0.420783
$69$	0	0
$70$	−59.4337	−0.101481
$71$	−369.025	−0.616833	−0.308417	−	0.951251i	$-0.599799\pi$
$71$	−369.025	−0.616833	−0.308417	+	0.951251i	$0.599799\pi$
$72$	0	0
$73$	1145.77	1.83702	0.918509	−	0.395401i	$-0.129394\pi$
$73$	1145.77	1.83702	0.918509	+	0.395401i	$0.129394\pi$
$74$	−1348.39	−2.11820
$75$	0	0
$76$	424.209	0.640264
$77$	−77.0000	−0.113961
$78$	0	0
$79$	488.885	0.696251	0.348125	−	0.937448i	$-0.386818\pi$
$79$	488.885	0.696251	0.348125	+	0.937448i	$0.386818\pi$
$80$	−8.43362	−0.0117863
$81$	0	0
$82$	710.372	0.956676
$83$	−548.982	−0.726008	−0.363004	−	0.931788i	$-0.618249\pi$
$83$	−548.982	−0.726008	−0.363004	+	0.931788i	$0.618249\pi$
$84$	0	0
$85$	32.9740	0.0420769
$86$	1275.96	1.59989
$87$	0	0
$88$	−263.162	−0.318786
$89$	−105.039	−0.125102	−0.0625510	−	0.998042i	$-0.519924\pi$
$89$	−105.039	−0.125102	−0.0625510	+	0.998042i	$0.519924\pi$
$90$	0	0
$91$	−172.481	−0.198691
$92$	−186.396	−0.211230
$93$	0	0
$94$	−240.073	−0.263422
$95$	−59.2829	−0.0640242
$96$	0	0
$97$	−1361.91	−1.42557	−0.712787	−	0.701381i	$-0.752567\pi$
$97$	−1361.91	−1.42557	−0.712787	+	0.701381i	$0.752567\pi$
$98$	−225.594	−0.232535
$99$	0	0
$100$	−1604.66	−1.60466
$101$	1610.32	1.58647	0.793234	−	0.608917i	$-0.208396\pi$
$101$	1610.32	1.58647	0.793234	+	0.608917i	$0.208396\pi$
$102$	0	0
$103$	−123.044	−0.117708	−0.0588540	−	0.998267i	$-0.518745\pi$
$103$	−123.044	−0.117708	−0.0588540	+	0.998267i	$0.518745\pi$
$104$	−589.485	−0.555805
$105$	0	0
$106$	379.323	0.347576
$107$	1740.90	1.57289	0.786446	−	0.617660i	$-0.211919\pi$
$107$	1740.90	1.57289	0.786446	+	0.617660i	$0.211919\pi$
$108$	0	0
$109$	248.938	0.218752	0.109376	−	0.994000i	$-0.465115\pi$
$109$	248.938	0.218752	0.109376	+	0.994000i	$0.465115\pi$
$110$	93.3958	0.0809540
$111$	0	0
$112$	−32.0117	−0.0270073
$113$	494.465	0.411641	0.205820	−	0.978590i	$-0.434014\pi$
$113$	494.465	0.411641	0.205820	+	0.978590i	$0.434014\pi$
$114$	0	0
$115$	26.0488	0.0211223
$116$	548.271	0.438842
$117$	0	0
$118$	3281.77	2.56026
$119$	125.160	0.0964151
$120$	0	0
$121$	121.000	0.0909091
$122$	2979.12	2.21079
$123$	0	0
$124$	2319.47	1.67979
$125$	454.774	0.325410
$126$	0	0
$127$	−979.104	−0.684106	−0.342053	−	0.939681i	$-0.611122\pi$
$127$	−979.104	−0.684106	−0.342053	+	0.939681i	$0.611122\pi$
$128$	2416.25	1.66850
$129$	0	0
$130$	209.207	0.141144
$131$	1660.85	1.10770	0.553851	−	0.832616i	$-0.313158\pi$
$131$	1660.85	1.10770	0.553851	+	0.832616i	$0.313158\pi$
$132$	0	0
$133$	−225.021	−0.146705
$134$	−1201.02	−0.774271
$135$	0	0
$136$	427.758	0.269705
$137$	1618.17	1.00912	0.504559	−	0.863377i	$-0.331655\pi$
$137$	1618.17	1.00912	0.504559	+	0.863377i	$0.331655\pi$
$138$	0	0
$139$	695.736	0.424544	0.212272	−	0.977211i	$-0.431914\pi$
$139$	695.736	0.424544	0.212272	+	0.977211i	$0.431914\pi$
$140$	170.356	0.102841
$141$	0	0
$142$	1698.97	1.00405
$143$	271.041	0.158501
$144$	0	0
$145$	−76.6205	−0.0438826
$146$	−5275.07	−2.99019
$147$	0	0
$148$	3864.90	2.14658
$149$	2081.84	1.14464	0.572318	−	0.820032i	$-0.306044\pi$
$149$	2081.84	1.14464	0.572318	+	0.820032i	$0.306044\pi$
$150$	0	0
$151$	2679.28	1.44395	0.721975	−	0.691919i	$-0.243234\pi$
$151$	2679.28	1.44395	0.721975	+	0.691919i	$0.243234\pi$
$152$	−769.052	−0.410384
$153$	0	0
$154$	354.504	0.185498
$155$	−324.144	−0.167973
$156$	0	0
$157$	2410.32	1.22525	0.612627	−	0.790372i	$-0.290113\pi$
$157$	2410.32	1.22525	0.612627	+	0.790372i	$0.290113\pi$
$158$	−2250.80	−1.13332
$159$	0	0
$160$	−314.131	−0.155214
$161$	98.8738	0.0483997
$162$	0	0
$163$	3629.08	1.74388	0.871938	−	0.489616i	$-0.162863\pi$
$163$	3629.08	1.74388	0.871938	+	0.489616i	$0.162863\pi$
$164$	−2036.15	−0.969491
$165$	0	0
$166$	2527.49	1.18175
$167$	3826.04	1.77286	0.886432	−	0.462859i	$-0.153176\pi$
$167$	3826.04	1.77286	0.886432	+	0.462859i	$0.153176\pi$
$168$	0	0
$169$	−1589.87	−0.723653
$170$	−151.811	−0.0684903
$171$	0	0
$172$	−3657.30	−1.62132
$173$	4310.67	1.89442	0.947209	−	0.320617i	$-0.103890\pi$
$173$	4310.67	1.89442	0.947209	+	0.320617i	$0.103890\pi$
$174$	0	0
$175$	851.193	0.367681
$176$	50.3040	0.0215444
$177$	0	0
$178$	483.593	0.203634
$179$	−2491.12	−1.04019	−0.520097	−	0.854107i	$-0.674104\pi$
$179$	−2491.12	−1.04019	−0.520097	+	0.854107i	$0.674104\pi$
$180$	0	0
$181$	−4315.49	−1.77220	−0.886098	−	0.463498i	$-0.846594\pi$
$181$	−4315.49	−1.77220	−0.886098	+	0.463498i	$0.846594\pi$
$182$	794.093	0.323418
$183$	0	0
$184$	337.920	0.135390
$185$	−540.118	−0.214650
$186$	0	0
$187$	−196.680	−0.0769127
$188$	688.124	0.266950
$189$	0	0
$190$	272.936	0.104215
$191$	−2840.41	−1.07605	−0.538024	−	0.842930i	$-0.680829\pi$
$191$	−2840.41	−1.07605	−0.538024	+	0.842930i	$0.680829\pi$
$192$	0	0
$193$	−1734.68	−0.646969	−0.323485	−	0.946233i	$-0.604854\pi$
$193$	−1734.68	−0.646969	−0.323485	+	0.946233i	$0.604854\pi$
$194$	6270.15	2.32047
$195$	0	0
$196$	646.622	0.235649
$197$	3098.42	1.12057	0.560287	−	0.828298i	$-0.310691\pi$
$197$	3098.42	1.12057	0.560287	+	0.828298i	$0.310691\pi$
$198$	0	0
$199$	4497.38	1.60206	0.801032	−	0.598622i	$-0.204285\pi$
$199$	4497.38	1.60206	0.801032	+	0.598622i	$0.204285\pi$
$200$	2909.11	1.02853
$201$	0	0
$202$	−7413.85	−2.58236
$203$	−290.830	−0.100553
$204$	0	0
$205$	284.550	0.0969456
$206$	566.491	0.191598
$207$	0	0
$208$	112.682	0.0375628
$209$	353.605	0.117030
$210$	0	0
$211$	1262.32	0.411857	0.205929	−	0.978567i	$-0.433979\pi$
$211$	1262.32	0.411857	0.205929	+	0.978567i	$0.433979\pi$
$212$	−1087.26	−0.352232
$213$	0	0
$214$	−8015.03	−2.56026
$215$	511.105	0.162126
$216$	0	0
$217$	−1230.36	−0.384895
$218$	−1146.10	−0.356071
$219$	0	0
$220$	−267.702	−0.0820384
$221$	−440.565	−0.134098
$222$	0	0
$223$	2931.38	0.880268	0.440134	−	0.897932i	$-0.354931\pi$
$223$	2931.38	0.880268	0.440134	+	0.897932i	$0.354931\pi$
$224$	−1192.35	−0.355658
$225$	0	0
$226$	−2276.49	−0.670045
$227$	−4298.37	−1.25680	−0.628399	−	0.777891i	$-0.716289\pi$
$227$	−4298.37	−1.25680	−0.628399	+	0.777891i	$0.716289\pi$
$228$	0	0
$229$	698.500	0.201564	0.100782	−	0.994909i	$-0.467866\pi$
$229$	698.500	0.201564	0.100782	+	0.994909i	$0.467866\pi$
$230$	−119.927	−0.0343816
$231$	0	0
$232$	−993.965	−0.281280
$233$	−1887.78	−0.530783	−0.265391	−	0.964141i	$-0.585501\pi$
$233$	−1887.78	−0.530783	−0.265391	+	0.964141i	$0.585501\pi$
$234$	0	0
$235$	−96.1649	−0.0266941
$236$	−9406.57	−2.59456
$237$	0	0
$238$	−576.231	−0.156939
$239$	6449.93	1.74565	0.872827	−	0.488030i	$-0.162284\pi$
$239$	6449.93	1.74565	0.872827	+	0.488030i	$0.162284\pi$
$240$	0	0
$241$	4636.98	1.23939	0.619697	−	0.784841i	$-0.287256\pi$
$241$	4636.98	1.23939	0.619697	+	0.784841i	$0.287256\pi$
$242$	−557.078	−0.147977
$243$	0	0
$244$	−8539.08	−2.24040
$245$	−90.3650	−0.0235641
$246$	0	0
$247$	792.078	0.204043
$248$	−4204.98	−1.07668
$249$	0	0
$250$	−2093.76	−0.529683
$251$	−2194.11	−0.551758	−0.275879	−	0.961192i	$-0.588969\pi$
$251$	−2194.11	−0.551758	−0.275879	+	0.961192i	$0.588969\pi$
$252$	0	0
$253$	−155.373	−0.0386096
$254$	4507.75	1.11355
$255$	0	0
$256$	−4557.87	−1.11276
$257$	2966.97	0.720133	0.360067	−	0.932927i	$-0.382754\pi$
$257$	2966.97	0.720133	0.360067	+	0.932927i	$0.382754\pi$
$258$	0	0
$259$	−2050.14	−0.491850
$260$	−599.654	−0.143034
$261$	0	0
$262$	−7646.47	−1.80305
$263$	−915.810	−0.214720	−0.107360	−	0.994220i	$-0.534240\pi$
$263$	−915.810	−0.214720	−0.107360	+	0.994220i	$0.534240\pi$
$264$	0	0
$265$	151.944	0.0352220
$266$	1035.99	0.238799
$267$	0	0
$268$	3442.50	0.784642
$269$	164.462	0.0372768	0.0186384	−	0.999826i	$-0.494067\pi$
$269$	164.462	0.0372768	0.0186384	+	0.999826i	$0.494067\pi$
$270$	0	0
$271$	1502.60	0.336815	0.168407	−	0.985718i	$-0.446138\pi$
$271$	1502.60	0.336815	0.168407	+	0.985718i	$0.446138\pi$
$272$	−81.7670	−0.0182274
$273$	0	0
$274$	−7449.96	−1.64259
$275$	−1337.59	−0.293308
$276$	0	0
$277$	−7500.11	−1.62685	−0.813426	−	0.581669i	$-0.802400\pi$
$277$	−7500.11	−1.62685	−0.813426	+	0.581669i	$0.802400\pi$
$278$	−3203.14	−0.691048
$279$	0	0
$280$	−308.839	−0.0659167
$281$	2945.12	0.625235	0.312618	−	0.949879i	$-0.398794\pi$
$281$	2945.12	0.625235	0.312618	+	0.949879i	$0.398794\pi$
$282$	0	0
$283$	5215.29	1.09547	0.547733	−	0.836653i	$-0.315491\pi$
$283$	5215.29	1.09547	0.547733	+	0.836653i	$0.315491\pi$
$284$	−4869.78	−1.01749
$285$	0	0
$286$	−1247.86	−0.257998
$287$	1080.07	0.222142
$288$	0	0
$289$	−4593.31	−0.934929
$290$	352.757	0.0714296
$291$	0	0
$292$	15120.0	3.03024
$293$	−7407.99	−1.47706	−0.738531	−	0.674219i	$-0.764480\pi$
$293$	−7407.99	−1.47706	−0.738531	+	0.674219i	$0.764480\pi$
$294$	0	0
$295$	1314.56	0.259447
$296$	−7006.72	−1.37587
$297$	0	0
$298$	−9584.68	−1.86317
$299$	−348.037	−0.0673161
$300$	0	0
$301$	1940.01	0.371496
$302$	−12335.3	−2.35038
$303$	0	0
$304$	147.006	0.0277348
$305$	1193.33	0.224033
$306$	0	0
$307$	−6850.01	−1.27345	−0.636727	−	0.771089i	$-0.719712\pi$
$307$	−6850.01	−1.27345	−0.636727	+	0.771089i	$0.719712\pi$
$308$	−1016.12	−0.187983
$309$	0	0
$310$	1492.34	0.273417
$311$	5538.62	1.00986	0.504929	−	0.863161i	$-0.331519\pi$
$311$	5538.62	1.00986	0.504929	+	0.863161i	$0.331519\pi$
$312$	0	0
$313$	9361.13	1.69049	0.845243	−	0.534382i	$-0.179456\pi$
$313$	9361.13	1.69049	0.845243	+	0.534382i	$0.179456\pi$
$314$	−11097.0	−1.99440
$315$	0	0
$316$	6451.50	1.14850
$317$	−219.221	−0.0388413	−0.0194206	−	0.999811i	$-0.506182\pi$
$317$	−219.221	−0.0388413	−0.0194206	+	0.999811i	$0.506182\pi$
$318$	0	0
$319$	457.018	0.0802135
$320$	1513.71	0.264435
$321$	0	0
$322$	−455.210	−0.0787822
$323$	−574.769	−0.0990123
$324$	0	0
$325$	−2996.21	−0.511384
$326$	−16708.1	−2.83858
$327$	0	0
$328$	3691.35	0.621405
$329$	−365.015	−0.0611669
$330$	0	0
$331$	3377.63	0.560880	0.280440	−	0.959872i	$-0.409520\pi$
$331$	3377.63	0.560880	0.280440	+	0.959872i	$0.409520\pi$
$332$	−7244.57	−1.19758
$333$	0	0
$334$	−17614.9	−2.88577
$335$	−481.087	−0.0784615
$336$	0	0
$337$	7755.93	1.25369	0.626843	−	0.779145i	$-0.284347\pi$
$337$	7755.93	1.25369	0.626843	+	0.779145i	$0.284347\pi$
$338$	7319.66	1.17792
$339$	0	0
$340$	435.137	0.0694077
$341$	1933.42	0.307040
$342$	0	0
$343$	−343.000	−0.0539949
$344$	6630.35	1.03920
$345$	0	0
$346$	−19846.1	−3.08362
$347$	831.044	0.128567	0.0642835	−	0.997932i	$-0.479524\pi$
$347$	831.044	0.128567	0.0642835	+	0.997932i	$0.479524\pi$
$348$	0	0
$349$	1840.94	0.282359	0.141180	−	0.989984i	$-0.454911\pi$
$349$	1840.94	0.282359	0.141180	+	0.989984i	$0.454911\pi$
$350$	−3918.85	−0.598490
$351$	0	0
$352$	1873.70	0.283717
$353$	−3409.11	−0.514019	−0.257009	−	0.966409i	$-0.582737\pi$
$353$	−3409.11	−0.514019	−0.257009	+	0.966409i	$0.582737\pi$
$354$	0	0
$355$	680.549	0.101746
$356$	−1386.13	−0.206361
$357$	0	0
$358$	11469.0	1.69317
$359$	−2199.75	−0.323394	−0.161697	−	0.986840i	$-0.551697\pi$
$359$	−2199.75	−0.323394	−0.161697	+	0.986840i	$0.551697\pi$
$360$	0	0
$361$	−5825.64	−0.849343
$362$	19868.3	2.88468
$363$	0	0
$364$	−2276.12	−0.327750
$365$	−2113.01	−0.303014
$366$	0	0
$367$	−855.008	−0.121611	−0.0608053	−	0.998150i	$-0.519367\pi$
$367$	−855.008	−0.121611	−0.0608053	+	0.998150i	$0.519367\pi$
$368$	−64.5942	−0.00915002
$369$	0	0
$370$	2486.68	0.349395
$371$	576.735	0.0807078
$372$	0	0
$373$	3193.40	0.443293	0.221646	−	0.975127i	$-0.428857\pi$
$373$	3193.40	0.443293	0.221646	+	0.975127i	$0.428857\pi$
$374$	905.505	0.125194
$375$	0	0
$376$	−1247.51	−0.171104
$377$	1023.72	0.139853
$378$	0	0
$379$	−5614.48	−0.760940	−0.380470	−	0.924793i	$-0.624238\pi$
$379$	−5614.48	−0.760940	−0.380470	+	0.924793i	$0.624238\pi$
$380$	−782.319	−0.105611
$381$	0	0
$382$	13077.1	1.75153
$383$	1736.86	0.231721	0.115861	−	0.993265i	$-0.463037\pi$
$383$	1736.86	0.231721	0.115861	+	0.993265i	$0.463037\pi$
$384$	0	0
$385$	142.002	0.0187977
$386$	7986.38	1.05310
$387$	0	0
$388$	−17972.2	−2.35155
$389$	−8710.78	−1.13536	−0.567679	−	0.823250i	$-0.692159\pi$
$389$	−8710.78	−1.13536	−0.567679	+	0.823250i	$0.692159\pi$
$390$	0	0
$391$	252.552	0.0326652
$392$	−1172.27	−0.151042
$393$	0	0
$394$	−14265.0	−1.82401
$395$	−901.593	−0.114846
$396$	0	0
$397$	11731.6	1.48311	0.741553	−	0.670894i	$-0.234089\pi$
$397$	11731.6	1.48311	0.741553	+	0.670894i	$0.234089\pi$
$398$	−20705.7	−2.60775
$399$	0	0
$400$	−556.084	−0.0695105
$401$	14408.8	1.79437	0.897183	−	0.441659i	$-0.145610\pi$
$401$	14408.8	1.79437	0.897183	+	0.441659i	$0.145610\pi$
$402$	0	0
$403$	4330.88	0.535327
$404$	21250.4	2.61695
$405$	0	0
$406$	1338.97	0.163674
$407$	3221.64	0.392361
$408$	0	0
$409$	−2155.54	−0.260598	−0.130299	−	0.991475i	$-0.541594\pi$
$409$	−2155.54	−0.260598	−0.130299	+	0.991475i	$0.541594\pi$
$410$	−1310.06	−0.157803
$411$	0	0
$412$	−1623.74	−0.194165
$413$	4989.71	0.594498
$414$	0	0
$415$	1012.42	0.119754
$416$	4197.10	0.494663
$417$	0	0
$418$	−1627.98	−0.190495
$419$	−8443.41	−0.984458	−0.492229	−	0.870466i	$-0.663818\pi$
$419$	−8443.41	−0.984458	−0.492229	+	0.870466i	$0.663818\pi$
$420$	0	0
$421$	−3070.28	−0.355430	−0.177715	−	0.984082i	$-0.556871\pi$
$421$	−3070.28	−0.355430	−0.177715	+	0.984082i	$0.556871\pi$
$422$	−5811.67	−0.670398
$423$	0	0
$424$	1971.10	0.225767
$425$	2174.19	0.248150
$426$	0	0
$427$	4529.55	0.513349
$428$	22973.6	2.59456
$429$	0	0
$430$	−2353.10	−0.263899
$431$	7004.40	0.782808	0.391404	−	0.920219i	$-0.371990\pi$
$431$	7004.40	0.782808	0.391404	+	0.920219i	$0.371990\pi$
$432$	0	0
$433$	7486.00	0.830841	0.415421	−	0.909629i	$-0.363634\pi$
$433$	7486.00	0.830841	0.415421	+	0.909629i	$0.363634\pi$
$434$	5664.52	0.626510
$435$	0	0
$436$	3285.07	0.360841
$437$	−454.055	−0.0497034
$438$	0	0
$439$	−13046.2	−1.41836	−0.709179	−	0.705029i	$-0.750934\pi$
$439$	−13046.2	−1.41836	−0.709179	+	0.705029i	$0.750934\pi$
$440$	485.319	0.0525833
$441$	0	0
$442$	2028.34	0.218277
$443$	11781.3	1.26354	0.631768	−	0.775157i	$-0.282329\pi$
$443$	11781.3	1.26354	0.631768	+	0.775157i	$0.282329\pi$
$444$	0	0
$445$	193.711	0.0206354
$446$	−13495.9	−1.43285
$447$	0	0
$448$	5745.63	0.605927
$449$	−7576.58	−0.796349	−0.398175	−	0.917310i	$-0.630356\pi$
$449$	−7576.58	−0.796349	−0.398175	+	0.917310i	$0.630356\pi$
$450$	0	0
$451$	−1697.26	−0.177208
$452$	6525.14	0.679020
$453$	0	0
$454$	19789.5	2.04574
$455$	318.086	0.0327738
$456$	0	0
$457$	11793.0	1.20712	0.603560	−	0.797318i	$-0.293748\pi$
$457$	11793.0	1.20712	0.603560	+	0.797318i	$0.293748\pi$
$458$	−3215.86	−0.328095
$459$	0	0
$460$	343.749	0.0348421
$461$	−4228.32	−0.427185	−0.213592	−	0.976923i	$-0.568516\pi$
$461$	−4228.32	−0.427185	−0.213592	+	0.976923i	$0.568516\pi$
$462$	0	0
$463$	14448.8	1.45031	0.725154	−	0.688586i	$-0.241768\pi$
$463$	14448.8	1.45031	0.725154	+	0.688586i	$0.241768\pi$
$464$	189.999	0.0190096
$465$	0	0
$466$	8691.23	0.863977
$467$	−16547.5	−1.63967	−0.819836	−	0.572599i	$-0.805935\pi$
$467$	−16547.5	−1.63967	−0.819836	+	0.572599i	$0.805935\pi$
$468$	0	0
$469$	−1826.07	−0.179787
$470$	442.738	0.0434511
$471$	0	0
$472$	17053.3	1.66301
$473$	−3048.59	−0.296351
$474$	0	0
$475$	−3908.91	−0.377585
$476$	1651.66	0.159041
$477$	0	0
$478$	−29695.2	−2.84148
$479$	2989.34	0.285149	0.142575	−	0.989784i	$-0.454462\pi$
$479$	2989.34	0.285149	0.142575	+	0.989784i	$0.454462\pi$
$480$	0	0
$481$	7216.51	0.684084
$482$	−21348.4	−2.01741
$483$	0	0
$484$	1596.76	0.149959
$485$	2511.60	0.235147
$486$	0	0
$487$	−5549.61	−0.516379	−0.258190	−	0.966094i	$-0.583126\pi$
$487$	−5549.61	−0.516379	−0.258190	+	0.966094i	$0.583126\pi$
$488$	15480.6	1.43601
$489$	0	0
$490$	416.036	0.0383563
$491$	7751.20	0.712438	0.356219	−	0.934403i	$-0.384066\pi$
$491$	7751.20	0.712438	0.356219	+	0.934403i	$0.384066\pi$
$492$	0	0
$493$	−742.862	−0.0678638
$494$	−3646.69	−0.332130
$495$	0	0
$496$	803.793	0.0727649
$497$	2583.17	0.233141
$498$	0	0
$499$	−6841.83	−0.613792	−0.306896	−	0.951743i	$-0.599290\pi$
$499$	−6841.83	−0.613792	−0.306896	+	0.951743i	$0.599290\pi$
$500$	6001.36	0.536778
$501$	0	0
$502$	10101.6	0.898120
$503$	11518.5	1.02105	0.510523	−	0.859864i	$-0.329452\pi$
$503$	11518.5	1.02105	0.510523	+	0.859864i	$0.329452\pi$
$504$	0	0
$505$	−2969.73	−0.261686
$506$	715.330	0.0628465
$507$	0	0
$508$	−12920.6	−1.12846
$509$	−8292.40	−0.722110	−0.361055	−	0.932545i	$-0.617583\pi$
$509$	−8292.40	−0.722110	−0.361055	+	0.932545i	$0.617583\pi$
$510$	0	0
$511$	−8020.39	−0.694327
$512$	1654.21	0.142786
$513$	0	0
$514$	−13659.8	−1.17219
$515$	226.917	0.0194158
$516$	0	0
$517$	573.595	0.0487943
$518$	9438.72	0.800606
$519$	0	0
$520$	1087.12	0.0916794
$521$	−10891.6	−0.915868	−0.457934	−	0.888986i	$-0.651410\pi$
$521$	−10891.6	−0.915868	−0.457934	+	0.888986i	$0.651410\pi$
$522$	0	0
$523$	14662.0	1.22586	0.612928	−	0.790139i	$-0.289992\pi$
$523$	14662.0	1.22586	0.612928	+	0.790139i	$0.289992\pi$
$524$	21917.2	1.82721
$525$	0	0
$526$	4216.34	0.349508
$527$	−3142.69	−0.259768
$528$	0	0
$529$	−11967.5	−0.983602
$530$	−699.541	−0.0573323
$531$	0	0
$532$	−2969.46	−0.241997
$533$	−3801.87	−0.308963
$534$	0	0
$535$	−3210.54	−0.259446
$536$	−6240.94	−0.502924
$537$	0	0
$538$	−757.177	−0.0606770
$539$	539.000	0.0430730
$540$	0	0
$541$	−19825.8	−1.57556	−0.787778	−	0.615960i	$-0.788768\pi$
$541$	−19825.8	−1.57556	−0.787778	+	0.615960i	$0.788768\pi$
$542$	−6917.92	−0.548247
$543$	0	0
$544$	−3045.61	−0.240036
$545$	−459.087	−0.0360828
$546$	0	0
$547$	12706.1	0.993187	0.496593	−	0.867983i	$-0.334584\pi$
$547$	12706.1	0.993187	0.496593	+	0.867983i	$0.334584\pi$
$548$	21353.9	1.66459
$549$	0	0
$550$	6158.19	0.477430
$551$	1335.57	0.103262
$552$	0	0
$553$	−3422.19	−0.263158
$554$	34530.1	2.64809
$555$	0	0
$556$	9181.19	0.700304
$557$	12599.1	0.958421	0.479211	−	0.877700i	$-0.340923\pi$
$557$	12599.1	0.958421	0.479211	+	0.877700i	$0.340923\pi$
$558$	0	0
$559$	−6828.86	−0.516691
$560$	59.0354	0.00445482
$561$	0	0
$562$	−13559.2	−1.01772
$563$	−6004.47	−0.449482	−0.224741	−	0.974419i	$-0.572154\pi$
$563$	−6004.47	−0.449482	−0.224741	+	0.974419i	$0.572154\pi$
$564$	0	0
$565$	−911.884	−0.0678996
$566$	−24010.9	−1.78314
$567$	0	0
$568$	8828.47	0.652173
$569$	3145.89	0.231779	0.115890	−	0.993262i	$-0.463028\pi$
$569$	3145.89	0.231779	0.115890	+	0.993262i	$0.463028\pi$
$570$	0	0
$571$	23549.1	1.72592	0.862960	−	0.505273i	$-0.168608\pi$
$571$	23549.1	1.72592	0.862960	+	0.505273i	$0.168608\pi$
$572$	3576.76	0.261454
$573$	0	0
$574$	−4972.60	−0.361590
$575$	1717.57	0.124569
$576$	0	0
$577$	327.335	0.0236172	0.0118086	−	0.999930i	$-0.496241\pi$
$577$	327.335	0.0236172	0.0118086	+	0.999930i	$0.496241\pi$
$578$	21147.4	1.52182
$579$	0	0
$580$	−1011.11	−0.0723864
$581$	3842.88	0.274405
$582$	0	0
$583$	−906.298	−0.0643825
$584$	−27411.2	−1.94226
$585$	0	0
$586$	34106.0	2.40428
$587$	13270.8	0.933123	0.466562	−	0.884489i	$-0.345493\pi$
$587$	13270.8	0.933123	0.466562	+	0.884489i	$0.345493\pi$
$588$	0	0
$589$	5650.14	0.395263
$590$	−6052.18	−0.422312
$591$	0	0
$592$	1339.35	0.0929849
$593$	−5098.92	−0.353099	−0.176549	−	0.984292i	$-0.556494\pi$
$593$	−5098.92	−0.353099	−0.176549	+	0.984292i	$0.556494\pi$
$594$	0	0
$595$	−230.818	−0.0159036
$596$	27472.7	1.88813
$597$	0	0
$598$	1602.35	0.109573
$599$	−19358.2	−1.32046	−0.660230	−	0.751064i	$-0.729541\pi$
$599$	−19358.2	−1.32046	−0.660230	+	0.751064i	$0.729541\pi$
$600$	0	0
$601$	1238.87	0.0840841	0.0420420	−	0.999116i	$-0.486614\pi$
$601$	1238.87	0.0840841	0.0420420	+	0.999116i	$0.486614\pi$
$602$	−8931.71	−0.604700
$603$	0	0
$604$	35356.7	2.38186
$605$	−223.146	−0.0149953
$606$	0	0
$607$	−14175.6	−0.947888	−0.473944	−	0.880555i	$-0.657170\pi$
$607$	−14175.6	−0.947888	−0.473944	+	0.880555i	$0.657170\pi$
$608$	5475.60	0.365239
$609$	0	0
$610$	−5494.04	−0.364667
$611$	1284.86	0.0850732
$612$	0	0
$613$	6906.67	0.455070	0.227535	−	0.973770i	$-0.426933\pi$
$613$	6906.67	0.455070	0.227535	+	0.973770i	$0.426933\pi$
$614$	31537.1	2.07286
$615$	0	0
$616$	1842.13	0.120490
$617$	−12104.3	−0.789789	−0.394894	−	0.918727i	$-0.629219\pi$
$617$	−12104.3	−0.789789	−0.394894	+	0.918727i	$0.629219\pi$
$618$	0	0
$619$	14945.2	0.970437	0.485218	−	0.874393i	$-0.338740\pi$
$619$	14945.2	0.970437	0.485218	+	0.874393i	$0.338740\pi$
$620$	−4277.52	−0.277080
$621$	0	0
$622$	−25499.5	−1.64379
$623$	735.271	0.0472841
$624$	0	0
$625$	14361.2	0.919116
$626$	−43098.2	−2.75168
$627$	0	0
$628$	31807.5	2.02111
$629$	−5236.63	−0.331953
$630$	0	0
$631$	−17711.9	−1.11743	−0.558717	−	0.829358i	$-0.688706\pi$
$631$	−17711.9	−1.11743	−0.558717	+	0.829358i	$0.688706\pi$
$632$	−11696.0	−0.736141
$633$	0	0
$634$	1009.28	0.0632236
$635$	1805.65	0.112842
$636$	0	0
$637$	1207.36	0.0750982
$638$	−2104.09	−0.130567
$639$	0	0
$640$	−4456.01	−0.275218
$641$	17994.4	1.10879	0.554396	−	0.832253i	$-0.312949\pi$
$641$	17994.4	1.10879	0.554396	+	0.832253i	$0.312949\pi$
$642$	0	0
$643$	27947.8	1.71408	0.857039	−	0.515251i	$-0.172301\pi$
$643$	27947.8	1.71408	0.857039	+	0.515251i	$0.172301\pi$
$644$	1304.77	0.0798375
$645$	0	0
$646$	2646.21	0.161167
$647$	14336.2	0.871122	0.435561	−	0.900159i	$-0.356550\pi$
$647$	14336.2	0.871122	0.435561	+	0.900159i	$0.356550\pi$
$648$	0	0
$649$	−7840.97	−0.474245
$650$	13794.4	0.832402
$651$	0	0
$652$	47890.7	2.87660
$653$	−4315.79	−0.258637	−0.129318	−	0.991603i	$-0.541279\pi$
$653$	−4315.79	−0.258637	−0.129318	+	0.991603i	$0.541279\pi$
$654$	0	0
$655$	−3062.91	−0.182714
$656$	−705.611	−0.0419962
$657$	0	0
$658$	1680.51	0.0995640
$659$	−4002.23	−0.236578	−0.118289	−	0.992979i	$-0.537741\pi$
$659$	−4002.23	−0.236578	−0.118289	+	0.992979i	$0.537741\pi$
$660$	0	0
$661$	−12223.4	−0.719268	−0.359634	−	0.933094i	$-0.617098\pi$
$661$	−12223.4	−0.719268	−0.359634	+	0.933094i	$0.617098\pi$
$662$	−15550.4	−0.912968
$663$	0	0
$664$	13133.7	0.767603
$665$	414.980	0.0241989
$666$	0	0
$667$	−586.846	−0.0340671
$668$	50489.9	2.92442
$669$	0	0
$670$	2214.90	0.127715
$671$	−7117.86	−0.409511
$672$	0	0
$673$	−6121.53	−0.350621	−0.175310	−	0.984513i	$-0.556093\pi$
$673$	−6121.53	−0.350621	−0.175310	+	0.984513i	$0.556093\pi$
$674$	−35707.9	−2.04068
$675$	0	0
$676$	−20980.4	−1.19370
$677$	−9626.46	−0.546492	−0.273246	−	0.961944i	$-0.588097\pi$
$677$	−9626.46	−0.546492	−0.273246	+	0.961944i	$0.588097\pi$
$678$	0	0
$679$	9533.34	0.538816
$680$	−788.863	−0.0444875
$681$	0	0
$682$	−8901.38	−0.499782
$683$	−11554.4	−0.647315	−0.323658	−	0.946174i	$-0.604913\pi$
$683$	−11554.4	−0.647315	−0.323658	+	0.946174i	$0.604913\pi$
$684$	0	0
$685$	−2984.19	−0.166453
$686$	1579.16	0.0878898
$687$	0	0
$688$	−1267.41	−0.0702318
$689$	−2030.12	−0.112251
$690$	0	0
$691$	2991.45	0.164689	0.0823444	−	0.996604i	$-0.473759\pi$
$691$	2991.45	0.164689	0.0823444	+	0.996604i	$0.473759\pi$
$692$	56885.2	3.12493
$693$	0	0
$694$	−3826.08	−0.209274
$695$	−1283.06	−0.0700279
$696$	0	0
$697$	2758.82	0.149925
$698$	−8475.60	−0.459608
$699$	0	0
$700$	11232.7	0.606506
$701$	−30772.8	−1.65802	−0.829010	−	0.559234i	$-0.811095\pi$
$701$	−30772.8	−1.65802	−0.829010	+	0.559234i	$0.811095\pi$
$702$	0	0
$703$	9414.77	0.505099
$704$	−9028.84	−0.483363
$705$	0	0
$706$	15695.4	0.836690
$707$	−11272.3	−0.599628
$708$	0	0
$709$	−21698.3	−1.14936	−0.574681	−	0.818377i	$-0.694874\pi$
$709$	−21698.3	−1.14936	−0.574681	+	0.818377i	$0.694874\pi$
$710$	−3133.21	−0.165616
$711$	0	0
$712$	2512.93	0.132269
$713$	−2482.66	−0.130402
$714$	0	0
$715$	−499.849	−0.0261445
$716$	−32873.7	−1.71585
$717$	0	0
$718$	10127.5	0.526402
$719$	12364.2	0.641318	0.320659	−	0.947195i	$-0.396096\pi$
$719$	12364.2	0.641318	0.320659	+	0.947195i	$0.396096\pi$
$720$	0	0
$721$	861.311	0.0444895
$722$	26821.0	1.38251
$723$	0	0
$724$	−56948.7	−2.92332
$725$	−5052.09	−0.258800
$726$	0	0
$727$	−16017.2	−0.817119	−0.408559	−	0.912732i	$-0.633969\pi$
$727$	−16017.2	−0.817119	−0.408559	+	0.912732i	$0.633969\pi$
$728$	4126.39	0.210075
$729$	0	0
$730$	9728.19	0.493228
$731$	4955.34	0.250725
$732$	0	0
$733$	10564.4	0.532340	0.266170	−	0.963926i	$-0.414242\pi$
$733$	10564.4	0.532340	0.266170	+	0.963926i	$0.414242\pi$
$734$	3936.42	0.197951
$735$	0	0
$736$	−2405.97	−0.120496
$737$	2869.54	0.143420
$738$	0	0
$739$	1760.32	0.0876242	0.0438121	−	0.999040i	$-0.486050\pi$
$739$	1760.32	0.0876242	0.0438121	+	0.999040i	$0.486050\pi$
$740$	−7127.59	−0.354075
$741$	0	0
$742$	−2655.26	−0.131371
$743$	−11383.4	−0.562067	−0.281034	−	0.959698i	$-0.590677\pi$
$743$	−11383.4	−0.562067	−0.281034	+	0.959698i	$0.590677\pi$
$744$	0	0
$745$	−3839.29	−0.188806
$746$	−14702.3	−0.721566
$747$	0	0
$748$	−2595.46	−0.126871
$749$	−12186.3	−0.594497
$750$	0	0
$751$	−15058.3	−0.731670	−0.365835	−	0.930680i	$-0.619217\pi$
$751$	−15058.3	−0.731670	−0.365835	+	0.930680i	$0.619217\pi$
$752$	238.464	0.0115637
$753$	0	0
$754$	−4713.18	−0.227644
$755$	−4941.08	−0.238178
$756$	0	0
$757$	38073.4	1.82801	0.914004	−	0.405706i	$-0.132974\pi$
$757$	38073.4	1.82801	0.914004	+	0.405706i	$0.132974\pi$
$758$	25848.8	1.23861
$759$	0	0
$760$	1418.27	0.0676923
$761$	−15232.1	−0.725577	−0.362788	−	0.931872i	$-0.618175\pi$
$761$	−15232.1	−0.725577	−0.362788	+	0.931872i	$0.618175\pi$
$762$	0	0
$763$	−1742.57	−0.0826803
$764$	−37483.1	−1.77499
$765$	0	0
$766$	−7996.40	−0.377182
$767$	−17563.8	−0.826850
$768$	0	0
$769$	12013.1	0.563332	0.281666	−	0.959513i	$-0.409113\pi$
$769$	12013.1	0.563332	0.281666	+	0.959513i	$0.409113\pi$
$770$	−653.771	−0.0305977
$771$	0	0
$772$	−22891.5	−1.06720
$773$	14258.5	0.663443	0.331721	−	0.943377i	$-0.392371\pi$
$773$	14258.5	0.663443	0.331721	+	0.943377i	$0.392371\pi$
$774$	0	0
$775$	−21372.9	−0.990630
$776$	32582.0	1.50725
$777$	0	0
$778$	40104.0	1.84807
$779$	−4959.99	−0.228126
$780$	0	0
$781$	−4059.27	−0.185982
$782$	−1162.74	−0.0531706
$783$	0	0
$784$	224.082	0.0102078
$785$	−4445.08	−0.202104
$786$	0	0
$787$	−14377.9	−0.651227	−0.325613	−	0.945503i	$-0.605571\pi$
$787$	−14377.9	−0.651227	−0.325613	+	0.945503i	$0.605571\pi$
$788$	40887.9	1.84844
$789$	0	0
$790$	4150.89	0.186939
$791$	−3461.26	−0.155585
$792$	0	0
$793$	−15944.1	−0.713986
$794$	−54011.8	−2.41412
$795$	0	0
$796$	59349.0	2.64268
$797$	−38515.9	−1.71180	−0.855899	−	0.517144i	$-0.826995\pi$
$797$	−38515.9	−1.71180	−0.855899	+	0.517144i	$0.826995\pi$
$798$	0	0
$799$	−932.352	−0.0412819
$800$	−20712.7	−0.915381
$801$	0	0
$802$	−66337.4	−2.92076
$803$	12603.5	0.553882
$804$	0	0
$805$	−182.341	−0.00798347
$806$	−19939.2	−0.871374
$807$	0	0
$808$	−38525.1	−1.67736
$809$	−11438.6	−0.497106	−0.248553	−	0.968618i	$-0.579955\pi$
$809$	−11438.6	−0.497106	−0.248553	+	0.968618i	$0.579955\pi$
$810$	0	0
$811$	15872.9	0.687265	0.343632	−	0.939104i	$-0.388343\pi$
$811$	15872.9	0.687265	0.343632	+	0.939104i	$0.388343\pi$
$812$	−3837.89	−0.165867
$813$	0	0
$814$	−14832.3	−0.638662
$815$	−6692.70	−0.287650
$816$	0	0
$817$	−8909.05	−0.381503
$818$	9924.00	0.424186
$819$	0	0
$820$	3755.03	0.159916
$821$	11988.8	0.509638	0.254819	−	0.966989i	$-0.417984\pi$
$821$	11988.8	0.509638	0.254819	+	0.966989i	$0.417984\pi$
$822$	0	0
$823$	−222.224	−0.00941219	−0.00470610	−	0.999989i	$-0.501498\pi$
$823$	−222.224	−0.00941219	−0.00470610	+	0.999989i	$0.501498\pi$
$824$	2943.69	0.124452
$825$	0	0
$826$	−22972.4	−0.967689
$827$	−7524.36	−0.316382	−0.158191	−	0.987409i	$-0.550566\pi$
$827$	−7524.36	−0.316382	−0.158191	+	0.987409i	$0.550566\pi$
$828$	0	0
$829$	−11807.1	−0.494663	−0.247332	−	0.968931i	$-0.579554\pi$
$829$	−11807.1	−0.494663	−0.247332	+	0.968931i	$0.579554\pi$
$830$	−4661.15	−0.194929
$831$	0	0
$832$	−20224.7	−0.842746
$833$	−876.120	−0.0364415
$834$	0	0
$835$	−7055.93	−0.292432
$836$	4666.30	0.193047
$837$	0	0
$838$	38873.1	1.60244
$839$	29112.9	1.19796	0.598980	−	0.800764i	$-0.295573\pi$
$839$	29112.9	1.19796	0.598980	+	0.800764i	$0.295573\pi$
$840$	0	0
$841$	−22662.8	−0.929224
$842$	14135.4	0.578549
$843$	0	0
$844$	16658.1	0.679377
$845$	2932.00	0.119366
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−376.781	−0.0152579
$849$	0	0
$850$	−10009.9	−0.403924
$851$	−4136.83	−0.166638
$852$	0	0
$853$	−24892.1	−0.999169	−0.499584	−	0.866265i	$-0.666514\pi$
$853$	−24892.1	−0.999169	−0.499584	+	0.866265i	$0.666514\pi$
$854$	−20853.8	−0.835601
$855$	0	0
$856$	−41649.0	−1.66301
$857$	27299.2	1.08812	0.544062	−	0.839045i	$-0.316886\pi$
$857$	27299.2	1.08812	0.544062	+	0.839045i	$0.316886\pi$
$858$	0	0
$859$	−6975.13	−0.277053	−0.138526	−	0.990359i	$-0.544237\pi$
$859$	−6975.13	−0.277053	−0.138526	+	0.990359i	$0.544237\pi$
$860$	6744.73	0.267434
$861$	0	0
$862$	−32247.9	−1.27421
$863$	4481.56	0.176772	0.0883858	−	0.996086i	$-0.471829\pi$
$863$	4481.56	0.176772	0.0883858	+	0.996086i	$0.471829\pi$
$864$	0	0
$865$	−7949.67	−0.312482
$866$	−34465.2	−1.35240
$867$	0	0
$868$	−16236.3	−0.634902
$869$	5377.73	0.209928
$870$	0	0
$871$	6427.80	0.250055
$872$	−5955.54	−0.231284
$873$	0	0
$874$	2090.45	0.0809044
$875$	−3183.42	−0.122993
$876$	0	0
$877$	−7207.96	−0.277532	−0.138766	−	0.990325i	$-0.544314\pi$
$877$	−7207.96	−0.277532	−0.138766	+	0.990325i	$0.544314\pi$
$878$	60063.9	2.30872
$879$	0	0
$880$	−92.7699	−0.00355372
$881$	38413.7	1.46900	0.734502	−	0.678607i	$-0.237416\pi$
$881$	38413.7	1.46900	0.734502	+	0.678607i	$0.237416\pi$
$882$	0	0
$883$	−8705.04	−0.331764	−0.165882	−	0.986146i	$-0.553047\pi$
$883$	−8705.04	−0.331764	−0.165882	+	0.986146i	$0.553047\pi$
$884$	−5813.86	−0.221200
$885$	0	0
$886$	−54240.6	−2.05671
$887$	44100.0	1.66937	0.834687	−	0.550725i	$-0.185649\pi$
$887$	44100.0	1.66937	0.834687	+	0.550725i	$0.185649\pi$
$888$	0	0
$889$	6853.73	0.258568
$890$	−891.834	−0.0335892
$891$	0	0
$892$	38683.6	1.45204
$893$	1676.25	0.0628146
$894$	0	0
$895$	4594.08	0.171579
$896$	−16913.8	−0.630635
$897$	0	0
$898$	34882.2	1.29625
$899$	7302.55	0.270916
$900$	0	0
$901$	1473.15	0.0544702
$902$	7814.09	0.288449
$903$	0	0
$904$	−11829.5	−0.435224
$905$	7958.54	0.292322
$906$	0	0
$907$	19921.2	0.729296	0.364648	−	0.931145i	$-0.381189\pi$
$907$	19921.2	0.729296	0.364648	+	0.931145i	$0.381189\pi$
$908$	−56722.9	−2.07314
$909$	0	0
$910$	−1464.45	−0.0533474
$911$	14446.7	0.525402	0.262701	−	0.964877i	$-0.415387\pi$
$911$	14446.7	0.525402	0.262701	+	0.964877i	$0.415387\pi$
$912$	0	0
$913$	−6038.81	−0.218900
$914$	−54294.4	−1.96488
$915$	0	0
$916$	9217.66	0.332489
$917$	−11625.9	−0.418672
$918$	0	0
$919$	44381.1	1.59303	0.796516	−	0.604617i	$-0.206674\pi$
$919$	44381.1	1.59303	0.796516	+	0.604617i	$0.206674\pi$
$920$	−623.186	−0.0223324
$921$	0	0
$922$	19467.0	0.695347
$923$	−9092.80	−0.324261
$924$	0	0
$925$	−35613.5	−1.26591
$926$	−66521.6	−2.36073
$927$	0	0
$928$	7076.97	0.250337
$929$	3455.30	0.122029	0.0610145	−	0.998137i	$-0.480566\pi$
$929$	3455.30	0.122029	0.0610145	+	0.998137i	$0.480566\pi$
$930$	0	0
$931$	1575.15	0.0554494
$932$	−24911.8	−0.875550
$933$	0	0
$934$	76183.8	2.66896
$935$	362.714	0.0126866
$936$	0	0
$937$	−14962.8	−0.521678	−0.260839	−	0.965382i	$-0.583999\pi$
$937$	−14962.8	−0.521678	−0.260839	+	0.965382i	$0.583999\pi$
$938$	8407.14	0.292647
$939$	0	0
$940$	−1269.03	−0.0440331
$941$	6116.95	0.211909	0.105955	−	0.994371i	$-0.466210\pi$
$941$	6116.95	0.211909	0.105955	+	0.994371i	$0.466210\pi$
$942$	0	0
$943$	2179.41	0.0752611
$944$	−3259.77	−0.112390
$945$	0	0
$946$	14035.6	0.482384
$947$	−53343.7	−1.83045	−0.915225	−	0.402943i	$-0.867987\pi$
$947$	−53343.7	−1.83045	−0.915225	+	0.402943i	$0.867987\pi$
$948$	0	0
$949$	28231.9	0.965696
$950$	17996.4	0.614611
$951$	0	0
$952$	−2994.30	−0.101939
$953$	−1979.62	−0.0672887	−0.0336443	−	0.999434i	$-0.510711\pi$
$953$	−1979.62	−0.0672887	−0.0336443	+	0.999434i	$0.510711\pi$
$954$	0	0
$955$	5238.24	0.177493
$956$	85115.6	2.87954
$957$	0	0
$958$	−13762.8	−0.464149
$959$	−11327.2	−0.381411
$960$	0	0
$961$	1102.58	0.0370104
$962$	−33224.4	−1.11351
$963$	0	0
$964$	61191.2	2.04444
$965$	3199.07	0.106717
$966$	0	0
$967$	−38892.0	−1.29336	−0.646681	−	0.762761i	$-0.723843\pi$
$967$	−38892.0	−1.29336	−0.646681	+	0.762761i	$0.723843\pi$
$968$	−2894.78	−0.0961175
$969$	0	0
$970$	−11563.3	−0.382758
$971$	47826.1	1.58065	0.790325	−	0.612688i	$-0.209912\pi$
$971$	47826.1	1.58065	0.790325	+	0.612688i	$0.209912\pi$
$972$	0	0
$973$	−4870.15	−0.160462
$974$	25550.1	0.840533
$975$	0	0
$976$	−2959.15	−0.0970493
$977$	−20840.5	−0.682442	−0.341221	−	0.939983i	$-0.610840\pi$
$977$	−20840.5	−0.682442	−0.341221	+	0.939983i	$0.610840\pi$
$978$	0	0
$979$	−1155.43	−0.0377197
$980$	−1192.49	−0.0388701
$981$	0	0
$982$	−35686.2	−1.15966
$983$	54430.5	1.76609	0.883044	−	0.469290i	$-0.155490\pi$
$983$	54430.5	1.76609	0.883044	+	0.469290i	$0.155490\pi$
$984$	0	0
$985$	−5714.05	−0.184837
$986$	3420.10	0.110465
$987$	0	0
$988$	10452.5	0.336579
$989$	3914.62	0.125862
$990$	0	0
$991$	−35161.9	−1.12710	−0.563550	−	0.826082i	$-0.690565\pi$
$991$	−35161.9	−1.12710	−0.563550	+	0.826082i	$0.690565\pi$
$992$	29939.2	0.958238
$993$	0	0
$994$	−11892.8	−0.379494
$995$	−8293.99	−0.264258
$996$	0	0
$997$	26961.6	0.856451	0.428225	−	0.903672i	$-0.359139\pi$
$997$	26961.6	0.856451	0.428225	+	0.903672i	$0.359139\pi$
$998$	31499.4	0.999096
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.l.1.1		4
3.2	odd	2		77.4.a.d.1.4	✓	4
12.11	even	2		1232.4.a.s.1.3		4
15.14	odd	2		1925.4.a.p.1.1		4
21.20	even	2		539.4.a.g.1.4		4
33.32	even	2		847.4.a.d.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.4	✓	4	3.2	odd	2
539.4.a.g.1.4		4	21.20	even	2
693.4.a.l.1.1		4	1.1	even	1	trivial
847.4.a.d.1.1		4	33.32	even	2
1232.4.a.s.1.3		4	12.11	even	2
1925.4.a.p.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)

\(f(q)\)	\(=\)	\(q-4.60395 q^{2} +13.1964 q^{4} -1.84418 q^{5} -7.00000 q^{7} -23.9238 q^{8} +8.49053 q^{10} +11.0000 q^{11} +24.6401 q^{13} +32.2277 q^{14} +4.57310 q^{16} -17.8800 q^{17} +32.1459 q^{19} -24.3365 q^{20} -50.6435 q^{22} -14.1248 q^{23} -121.599 q^{25} -113.442 q^{26} -92.3745 q^{28} +41.5471 q^{29} +175.766 q^{31} +170.336 q^{32} +82.3187 q^{34} +12.9093 q^{35} +292.877 q^{37} -147.998 q^{38} +44.1199 q^{40} -154.296 q^{41} -277.144 q^{43} +145.160 q^{44} +65.0300 q^{46} +52.1450 q^{47} +49.0000 q^{49} +559.836 q^{50} +325.160 q^{52} -82.3907 q^{53} -20.2860 q^{55} +167.467 q^{56} -191.281 q^{58} -712.816 q^{59} -647.078 q^{61} -809.217 q^{62} -820.804 q^{64} -45.4408 q^{65} +260.867 q^{67} -235.951 q^{68} -59.4337 q^{70} -369.025 q^{71} +1145.77 q^{73} -1348.39 q^{74} +424.209 q^{76} -77.0000 q^{77} +488.885 q^{79} -8.43362 q^{80} +710.372 q^{82} -548.982 q^{83} +32.9740 q^{85} +1275.96 q^{86} -263.162 q^{88} -105.039 q^{89} -172.481 q^{91} -186.396 q^{92} -240.073 q^{94} -59.2829 q^{95} -1361.91 q^{97} -225.594 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 26 * q^4 - 10 * q^5 - 28 * q^7 + 18 * q^8 - 2 * q^10 + 44 * q^11 + 58 * q^13 - 14 * q^14 + 2 * q^16 - 4 * q^17 + 258 * q^19 - 182 * q^20 + 22 * q^22 - 8 * q^23 + 80 * q^25 + 482 * q^26 - 182 * q^28 + 396 * q^29 - 56 * q^31 - 134 * q^32 + 472 * q^34 + 70 * q^35 + 84 * q^37 + 942 * q^38 - 1026 * q^40 - 52 * q^41 + 408 * q^43 + 286 * q^44 + 368 * q^46 - 8 * q^47 + 196 * q^49 + 1642 * q^50 + 2030 * q^52 - 624 * q^53 - 110 * q^55 - 126 * q^56 + 864 * q^58 + 238 * q^59 - 162 * q^61 - 688 * q^62 - 902 * q^64 + 32 * q^65 + 1340 * q^67 + 1384 * q^68 + 14 * q^70 - 1788 * q^71 + 1456 * q^73 - 996 * q^74 + 3042 * q^76 - 308 * q^77 - 1324 * q^79 - 2342 * q^80 + 1984 * q^82 - 450 * q^83 - 1736 * q^85 + 4380 * q^86 + 198 * q^88 + 3072 * q^89 - 406 * q^91 - 544 * q^92 - 1696 * q^94 - 24 * q^95 - 652 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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