LMFDB - Embedded newform 693.4.a.l.1.3

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			1.3
Root			$0.555307$ 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)

$f(q)$	$=$	\(q+3.24550 q^{2} +2.53327 q^{4} +16.0955 q^{5} -7.00000 q^{7} -17.7423 q^{8} +52.2379 q^{10} +11.0000 q^{11} +35.3712 q^{13} -22.7185 q^{14} -77.8487 q^{16} -40.4757 q^{17} +118.159 q^{19} +40.7743 q^{20} +35.7005 q^{22} +174.510 q^{23} +134.065 q^{25} +114.797 q^{26} -17.7329 q^{28} +262.725 q^{29} -36.1894 q^{31} -110.720 q^{32} -131.364 q^{34} -112.668 q^{35} +19.0464 q^{37} +383.487 q^{38} -285.571 q^{40} -156.996 q^{41} +287.182 q^{43} +27.8660 q^{44} +566.371 q^{46} -397.244 q^{47} +49.0000 q^{49} +435.108 q^{50} +89.6049 q^{52} -272.483 q^{53} +177.050 q^{55} +124.196 q^{56} +852.674 q^{58} +507.466 q^{59} +35.5608 q^{61} -117.453 q^{62} +263.448 q^{64} +569.317 q^{65} +979.229 q^{67} -102.536 q^{68} -365.666 q^{70} -750.404 q^{71} +395.594 q^{73} +61.8152 q^{74} +299.330 q^{76} -77.0000 q^{77} -736.516 q^{79} -1253.01 q^{80} -509.531 q^{82} -582.975 q^{83} -651.477 q^{85} +932.050 q^{86} -195.165 q^{88} +806.201 q^{89} -247.598 q^{91} +442.081 q^{92} -1289.26 q^{94} +1901.84 q^{95} -957.232 q^{97} +159.030 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.24550	1.14746	0.573729	−	0.819045i	$-0.305496\pi$
$2$	3.24550	1.14746	0.573729	+	0.819045i	$0.305496\pi$
$3$	0	0
$3$	0	0
$4$	2.53327	0.316659
$5$	16.0955	1.43962	0.719812	−	0.694169i	$-0.244228\pi$
$5$	16.0955	1.43962	0.719812	+	0.694169i	$0.244228\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−17.7423	−0.784105
$9$	0	0
$10$	52.2379	1.65191
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	35.3712	0.754631	0.377316	−	0.926085i	$-0.376847\pi$
$13$	35.3712	0.754631	0.377316	+	0.926085i	$0.376847\pi$
$14$	−22.7185	−0.433698
$15$	0	0
$16$	−77.8487	−1.21639
$17$	−40.4757	−0.577459	−0.288730	−	0.957411i	$-0.593233\pi$
$17$	−40.4757	−0.577459	−0.288730	+	0.957411i	$0.593233\pi$
$18$	0	0
$19$	118.159	1.42672	0.713359	−	0.700799i	$-0.247173\pi$
$19$	118.159	1.42672	0.713359	+	0.700799i	$0.247173\pi$
$20$	40.7743	0.455870
$21$	0	0
$22$	35.7005	0.345972
$23$	174.510	1.58208	0.791039	−	0.611766i	$-0.209541\pi$
$23$	174.510	1.58208	0.791039	+	0.611766i	$0.209541\pi$
$24$	0	0
$25$	134.065	1.07252
$26$	114.797	0.865907
$27$	0	0
$28$	−17.7329	−0.119686
$29$	262.725	1.68230	0.841152	−	0.540799i	$-0.181878\pi$
$29$	262.725	1.68230	0.841152	+	0.540799i	$0.181878\pi$
$30$	0	0
$31$	−36.1894	−0.209671	−0.104836	−	0.994490i	$-0.533432\pi$
$31$	−36.1894	−0.209671	−0.104836	+	0.994490i	$0.533432\pi$
$32$	−110.720	−0.611647
$33$	0	0
$34$	−131.364	−0.662610
$35$	−112.668	−0.544127
$36$	0	0
$37$	19.0464	0.0846274	0.0423137	−	0.999104i	$-0.486527\pi$
$37$	19.0464	0.0846274	0.0423137	+	0.999104i	$0.486527\pi$
$38$	383.487	1.63710
$39$	0	0
$40$	−285.571	−1.12882
$41$	−156.996	−0.598017	−0.299008	−	0.954250i	$-0.596656\pi$
$41$	−156.996	−0.598017	−0.299008	+	0.954250i	$0.596656\pi$
$42$	0	0
$43$	287.182	1.01849	0.509243	−	0.860623i	$-0.329926\pi$
$43$	287.182	1.01849	0.509243	+	0.860623i	$0.329926\pi$
$44$	27.8660	0.0954763
$45$	0	0
$46$	566.371	1.81537
$47$	−397.244	−1.23285	−0.616425	−	0.787413i	$-0.711420\pi$
$47$	−397.244	−1.23285	−0.616425	+	0.787413i	$0.711420\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	435.108	1.23067
$51$	0	0
$52$	89.6049	0.238961
$53$	−272.483	−0.706196	−0.353098	−	0.935586i	$-0.614872\pi$
$53$	−272.483	−0.706196	−0.353098	+	0.935586i	$0.614872\pi$
$54$	0	0
$55$	177.050	0.434063
$56$	124.196	0.296364
$57$	0	0
$58$	852.674	1.93037
$59$	507.466	1.11977	0.559885	−	0.828570i	$-0.310845\pi$
$59$	507.466	1.11977	0.559885	+	0.828570i	$0.310845\pi$
$60$	0	0
$61$	35.5608	0.0746409	0.0373205	−	0.999303i	$-0.488118\pi$
$61$	35.5608	0.0746409	0.0373205	+	0.999303i	$0.488118\pi$
$62$	−117.453	−0.240589
$63$	0	0
$64$	263.448	0.514547
$65$	569.317	1.08639
$66$	0	0
$67$	979.229	1.78555	0.892775	−	0.450502i	$-0.148755\pi$
$67$	979.229	1.78555	0.892775	+	0.450502i	$0.148755\pi$
$68$	−102.536	−0.182858
$69$	0	0
$70$	−365.666	−0.624363
$71$	−750.404	−1.25432	−0.627159	−	0.778891i	$-0.715782\pi$
$71$	−750.404	−1.25432	−0.627159	+	0.778891i	$0.715782\pi$
$72$	0	0
$73$	395.594	0.634257	0.317129	−	0.948383i	$-0.397281\pi$
$73$	395.594	0.634257	0.317129	+	0.948383i	$0.397281\pi$
$74$	61.8152	0.0971063
$75$	0	0
$76$	299.330	0.451783
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−736.516	−1.04892	−0.524459	−	0.851436i	$-0.675732\pi$
$79$	−736.516	−1.04892	−0.524459	+	0.851436i	$0.675732\pi$
$80$	−1253.01	−1.75114
$81$	0	0
$82$	−509.531	−0.686199
$83$	−582.975	−0.770962	−0.385481	−	0.922716i	$-0.625964\pi$
$83$	−582.975	−0.770962	−0.385481	+	0.922716i	$0.625964\pi$
$84$	0	0
$85$	−651.477	−0.831325
$86$	932.050	1.16867
$87$	0	0
$88$	−195.165	−0.236416
$89$	806.201	0.960192	0.480096	−	0.877216i	$-0.340602\pi$
$89$	806.201	0.960192	0.480096	+	0.877216i	$0.340602\pi$
$90$	0	0
$91$	−247.598	−0.285224
$92$	442.081	0.500979
$93$	0	0
$94$	−1289.26	−1.41464
$95$	1901.84	2.05394
$96$	0	0
$97$	−957.232	−1.00198	−0.500991	−	0.865453i	$-0.667031\pi$
$97$	−957.232	−1.00198	−0.500991	+	0.865453i	$0.667031\pi$
$98$	159.030	0.163923
$99$	0	0
$100$	339.623	0.339623
$101$	996.143	0.981386	0.490693	−	0.871333i	$-0.336744\pi$
$101$	996.143	0.981386	0.490693	+	0.871333i	$0.336744\pi$
$102$	0	0
$103$	1338.55	1.28050	0.640248	−	0.768169i	$-0.278832\pi$
$103$	1338.55	1.28050	0.640248	+	0.768169i	$0.278832\pi$
$104$	−627.565	−0.591710
$105$	0	0
$106$	−884.343	−0.810330
$107$	−1449.25	−1.30939	−0.654693	−	0.755895i	$-0.727202\pi$
$107$	−1449.25	−1.30939	−0.654693	+	0.755895i	$0.727202\pi$
$108$	0	0
$109$	−654.535	−0.575166	−0.287583	−	0.957756i	$-0.592852\pi$
$109$	−654.535	−0.575166	−0.287583	+	0.957756i	$0.592852\pi$
$110$	574.617	0.498069
$111$	0	0
$112$	544.941	0.459751
$113$	1160.63	0.966223	0.483111	−	0.875559i	$-0.339507\pi$
$113$	1160.63	0.966223	0.483111	+	0.875559i	$0.339507\pi$
$114$	0	0
$115$	2808.82	2.27760
$116$	665.554	0.532717
$117$	0	0
$118$	1646.98	1.28489
$119$	283.330	0.218259
$120$	0	0
$121$	121.000	0.0909091
$122$	115.413	0.0856473
$123$	0	0
$124$	−91.6776	−0.0663943
$125$	145.906	0.104402
$126$	0	0
$127$	−1055.41	−0.737420	−0.368710	−	0.929545i	$-0.620200\pi$
$127$	−1055.41	−0.737420	−0.368710	+	0.929545i	$0.620200\pi$
$128$	1740.78	1.20207
$129$	0	0
$130$	1847.72	1.24658
$131$	−2657.40	−1.77235	−0.886175	−	0.463351i	$-0.846647\pi$
$131$	−2657.40	−1.77235	−0.886175	+	0.463351i	$0.846647\pi$
$132$	0	0
$133$	−827.116	−0.539249
$134$	3178.09	2.04884
$135$	0	0
$136$	718.131	0.452788
$137$	147.314	0.0918676	0.0459338	−	0.998944i	$-0.485374\pi$
$137$	147.314	0.0918676	0.0459338	+	0.998944i	$0.485374\pi$
$138$	0	0
$139$	902.634	0.550794	0.275397	−	0.961331i	$-0.411191\pi$
$139$	902.634	0.550794	0.275397	+	0.961331i	$0.411191\pi$
$140$	−285.420	−0.172303
$141$	0	0
$142$	−2435.44	−1.43928
$143$	389.083	0.227530
$144$	0	0
$145$	4228.69	2.42189
$146$	1283.90	0.727783
$147$	0	0
$148$	48.2498	0.0267980
$149$	−1212.63	−0.666727	−0.333363	−	0.942798i	$-0.608184\pi$
$149$	−1212.63	−0.666727	−0.333363	+	0.942798i	$0.608184\pi$
$150$	0	0
$151$	2565.33	1.38254	0.691270	−	0.722597i	$-0.257052\pi$
$151$	2565.33	1.38254	0.691270	+	0.722597i	$0.257052\pi$
$152$	−2096.42	−1.11870
$153$	0	0
$154$	−249.904	−0.130765
$155$	−582.486	−0.301848
$156$	0	0
$157$	702.237	0.356972	0.178486	−	0.983942i	$-0.442880\pi$
$157$	702.237	0.356972	0.178486	+	0.983942i	$0.442880\pi$
$158$	−2390.36	−1.20359
$159$	0	0
$160$	−1782.09	−0.880542
$161$	−1221.57	−0.597969
$162$	0	0
$163$	−1146.27	−0.550814	−0.275407	−	0.961328i	$-0.588813\pi$
$163$	−1146.27	−0.550814	−0.275407	+	0.961328i	$0.588813\pi$
$164$	−397.714	−0.189368
$165$	0	0
$166$	−1892.05	−0.884646
$167$	3255.04	1.50828	0.754140	−	0.656714i	$-0.228054\pi$
$167$	3255.04	1.50828	0.754140	+	0.656714i	$0.228054\pi$
$168$	0	0
$169$	−945.879	−0.430532
$170$	−2114.37	−0.953910
$171$	0	0
$172$	727.511	0.322513
$173$	−4024.24	−1.76854	−0.884269	−	0.466977i	$-0.845343\pi$
$173$	−4024.24	−1.76854	−0.884269	+	0.466977i	$0.845343\pi$
$174$	0	0
$175$	−938.455	−0.405374
$176$	−856.336	−0.366754
$177$	0	0
$178$	2616.52	1.10178
$179$	−706.090	−0.294836	−0.147418	−	0.989074i	$-0.547096\pi$
$179$	−706.090	−0.294836	−0.147418	+	0.989074i	$0.547096\pi$
$180$	0	0
$181$	−1268.90	−0.521087	−0.260544	−	0.965462i	$-0.583902\pi$
$181$	−1268.90	−0.521087	−0.260544	+	0.965462i	$0.583902\pi$
$182$	−803.581	−0.327282
$183$	0	0
$184$	−3096.20	−1.24051
$185$	306.562	0.121832
$186$	0	0
$187$	−445.233	−0.174110
$188$	−1006.33	−0.390393
$189$	0	0
$190$	6172.41	2.35681
$191$	−4864.58	−1.84287	−0.921436	−	0.388529i	$-0.872983\pi$
$191$	−4864.58	−1.84287	−0.921436	+	0.388529i	$0.872983\pi$
$192$	0	0
$193$	−2675.49	−0.997855	−0.498928	−	0.866644i	$-0.666273\pi$
$193$	−2675.49	−0.997855	−0.498928	+	0.866644i	$0.666273\pi$
$194$	−3106.70	−1.14973
$195$	0	0
$196$	124.130	0.0452370
$197$	−1627.73	−0.588684	−0.294342	−	0.955700i	$-0.595101\pi$
$197$	−1627.73	−0.588684	−0.294342	+	0.955700i	$0.595101\pi$
$198$	0	0
$199$	−2254.07	−0.802947	−0.401474	−	0.915871i	$-0.631502\pi$
$199$	−2254.07	−0.802947	−0.401474	+	0.915871i	$0.631502\pi$
$200$	−2378.62	−0.840968
$201$	0	0
$202$	3232.98	1.12610
$203$	−1839.08	−0.635851
$204$	0	0
$205$	−2526.93	−0.860920
$206$	4344.26	1.46931
$207$	0	0
$208$	−2753.60	−0.917923
$209$	1299.75	0.430172
$210$	0	0
$211$	3112.07	1.01537	0.507687	−	0.861542i	$-0.330501\pi$
$211$	3112.07	1.01537	0.507687	+	0.861542i	$0.330501\pi$
$212$	−690.273	−0.223623
$213$	0	0
$214$	−4703.54	−1.50247
$215$	4622.34	1.46624
$216$	0	0
$217$	253.326	0.0792482
$218$	−2124.30	−0.659979
$219$	0	0
$220$	448.517	0.137450
$221$	−1431.67	−0.435769
$222$	0	0
$223$	−3558.38	−1.06855	−0.534275	−	0.845311i	$-0.679415\pi$
$223$	−3558.38	−1.06855	−0.534275	+	0.845311i	$0.679415\pi$
$224$	775.039	0.231181
$225$	0	0
$226$	3766.84	1.10870
$227$	2330.51	0.681416	0.340708	−	0.940169i	$-0.389333\pi$
$227$	2330.51	0.681416	0.340708	+	0.940169i	$0.389333\pi$
$228$	0	0
$229$	676.106	0.195102	0.0975509	−	0.995231i	$-0.468899\pi$
$229$	676.106	0.195102	0.0975509	+	0.995231i	$0.468899\pi$
$230$	9116.02	2.61345
$231$	0	0
$232$	−4661.34	−1.31910
$233$	−1620.20	−0.455548	−0.227774	−	0.973714i	$-0.573145\pi$
$233$	−1620.20	−0.455548	−0.227774	+	0.973714i	$0.573145\pi$
$234$	0	0
$235$	−6393.84	−1.77484
$236$	1285.55	0.354586
$237$	0	0
$238$	919.548	0.250443
$239$	2001.39	0.541669	0.270835	−	0.962626i	$-0.412700\pi$
$239$	2001.39	0.541669	0.270835	+	0.962626i	$0.412700\pi$
$240$	0	0
$241$	−1586.39	−0.424018	−0.212009	−	0.977268i	$-0.568001\pi$
$241$	−1586.39	−0.424018	−0.212009	+	0.977268i	$0.568001\pi$
$242$	392.706	0.104314
$243$	0	0
$244$	90.0853	0.0236357
$245$	788.679	0.205661
$246$	0	0
$247$	4179.44	1.07665
$248$	642.081	0.164404
$249$	0	0
$250$	473.537	0.119796
$251$	−2612.23	−0.656902	−0.328451	−	0.944521i	$-0.606527\pi$
$251$	−2612.23	−0.656902	−0.328451	+	0.944521i	$0.606527\pi$
$252$	0	0
$253$	1919.61	0.477014
$254$	−3425.33	−0.846158
$255$	0	0
$256$	3542.12	0.864775
$257$	−4198.23	−1.01898	−0.509491	−	0.860476i	$-0.670166\pi$
$257$	−4198.23	−1.01898	−0.509491	+	0.860476i	$0.670166\pi$
$258$	0	0
$259$	−133.325	−0.0319861
$260$	1442.24	0.344014
$261$	0	0
$262$	−8624.58	−2.03370
$263$	5170.31	1.21222	0.606112	−	0.795380i	$-0.292728\pi$
$263$	5170.31	1.21222	0.606112	+	0.795380i	$0.292728\pi$
$264$	0	0
$265$	−4385.74	−1.01666
$266$	−2684.41	−0.618765
$267$	0	0
$268$	2480.66	0.565411
$269$	−1648.20	−0.373578	−0.186789	−	0.982400i	$-0.559808\pi$
$269$	−1648.20	−0.373578	−0.186789	+	0.982400i	$0.559808\pi$
$270$	0	0
$271$	2562.79	0.574459	0.287230	−	0.957862i	$-0.407266\pi$
$271$	2562.79	0.574459	0.287230	+	0.957862i	$0.407266\pi$
$272$	3150.98	0.702413
$273$	0	0
$274$	478.107	0.105414
$275$	1474.72	0.323377
$276$	0	0
$277$	2762.94	0.599310	0.299655	−	0.954048i	$-0.403128\pi$
$277$	2762.94	0.599310	0.299655	+	0.954048i	$0.403128\pi$
$278$	2929.50	0.632013
$279$	0	0
$280$	1998.99	0.426653
$281$	−6453.68	−1.37009	−0.685044	−	0.728502i	$-0.740217\pi$
$281$	−6453.68	−1.37009	−0.685044	+	0.728502i	$0.740217\pi$
$282$	0	0
$283$	4540.78	0.953787	0.476893	−	0.878961i	$-0.341763\pi$
$283$	4540.78	0.953787	0.476893	+	0.878961i	$0.341763\pi$
$284$	−1900.98	−0.397191
$285$	0	0
$286$	1262.77	0.261081
$287$	1098.97	0.226029
$288$	0	0
$289$	−3274.72	−0.666541
$290$	13724.2	2.77901
$291$	0	0
$292$	1002.15	0.200843
$293$	−1916.34	−0.382094	−0.191047	−	0.981581i	$-0.561188\pi$
$293$	−1916.34	−0.382094	−0.191047	+	0.981581i	$0.561188\pi$
$294$	0	0
$295$	8167.92	1.61205
$296$	−337.927	−0.0663567
$297$	0	0
$298$	−3935.58	−0.765040
$299$	6172.61	1.19388
$300$	0	0
$301$	−2010.28	−0.384951
$302$	8325.78	1.58641
$303$	0	0
$304$	−9198.56	−1.73544
$305$	572.369	0.107455
$306$	0	0
$307$	6876.30	1.27834	0.639171	−	0.769064i	$-0.279278\pi$
$307$	6876.30	1.27834	0.639171	+	0.769064i	$0.279278\pi$
$308$	−195.062	−0.0360867
$309$	0	0
$310$	−1890.46	−0.346357
$311$	−8469.32	−1.54422	−0.772108	−	0.635492i	$-0.780797\pi$
$311$	−8469.32	−1.54422	−0.772108	+	0.635492i	$0.780797\pi$
$312$	0	0
$313$	−2882.28	−0.520498	−0.260249	−	0.965542i	$-0.583805\pi$
$313$	−2882.28	−0.520498	−0.260249	+	0.965542i	$0.583805\pi$
$314$	2279.11	0.409610
$315$	0	0
$316$	−1865.80	−0.332150
$317$	−9795.31	−1.73552	−0.867759	−	0.496985i	$-0.834440\pi$
$317$	−9795.31	−1.73552	−0.867759	+	0.496985i	$0.834440\pi$
$318$	0	0
$319$	2889.98	0.507234
$320$	4240.33	0.740755
$321$	0	0
$322$	−3964.60	−0.686144
$323$	−4782.59	−0.823871
$324$	0	0
$325$	4742.04	0.809357
$326$	−3720.22	−0.632036
$327$	0	0
$328$	2785.47	0.468908
$329$	2780.71	0.465974
$330$	0	0
$331$	1813.70	0.301178	0.150589	−	0.988596i	$-0.451883\pi$
$331$	1813.70	0.301178	0.150589	+	0.988596i	$0.451883\pi$
$332$	−1476.84	−0.244132
$333$	0	0
$334$	10564.2	1.73069
$335$	15761.2	2.57052
$336$	0	0
$337$	−11964.2	−1.93393	−0.966964	−	0.254913i	$-0.917953\pi$
$337$	−11964.2	−1.93393	−0.966964	+	0.254913i	$0.917953\pi$
$338$	−3069.85	−0.494017
$339$	0	0
$340$	−1650.37	−0.263247
$341$	−398.083	−0.0632182
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−5095.26	−0.798599
$345$	0	0
$346$	−13060.7	−2.02932
$347$	2283.89	0.353330	0.176665	−	0.984271i	$-0.443469\pi$
$347$	2283.89	0.353330	0.176665	+	0.984271i	$0.443469\pi$
$348$	0	0
$349$	−2472.48	−0.379223	−0.189612	−	0.981859i	$-0.560723\pi$
$349$	−2472.48	−0.379223	−0.189612	+	0.981859i	$0.560723\pi$
$350$	−3045.76	−0.465150
$351$	0	0
$352$	−1217.92	−0.184418
$353$	−10190.0	−1.53642	−0.768211	−	0.640197i	$-0.778853\pi$
$353$	−10190.0	−1.53642	−0.768211	+	0.640197i	$0.778853\pi$
$354$	0	0
$355$	−12078.1	−1.80575
$356$	2042.33	0.304054
$357$	0	0
$358$	−2291.62	−0.338312
$359$	43.4294	0.00638473	0.00319236	−	0.999995i	$-0.498984\pi$
$359$	43.4294	0.00638473	0.00319236	+	0.999995i	$0.498984\pi$
$360$	0	0
$361$	7102.66	1.03552
$362$	−4118.23	−0.597926
$363$	0	0
$364$	−627.234	−0.0903187
$365$	6367.28	0.913093
$366$	0	0
$367$	8775.78	1.24821	0.624104	−	0.781342i	$-0.285464\pi$
$367$	8775.78	1.24821	0.624104	+	0.781342i	$0.285464\pi$
$368$	−13585.4	−1.92442
$369$	0	0
$370$	994.946	0.139797
$371$	1907.38	0.266917
$372$	0	0
$373$	−2751.89	−0.382004	−0.191002	−	0.981590i	$-0.561174\pi$
$373$	−2751.89	−0.382004	−0.191002	+	0.981590i	$0.561174\pi$
$374$	−1445.00	−0.199784
$375$	0	0
$376$	7048.01	0.966684
$377$	9292.90	1.26952
$378$	0	0
$379$	2605.59	0.353141	0.176570	−	0.984288i	$-0.443500\pi$
$379$	2605.59	0.353141	0.176570	+	0.984288i	$0.443500\pi$
$380$	4817.87	0.650399
$381$	0	0
$382$	−15788.0	−2.11462
$383$	14360.5	1.91590	0.957949	−	0.286940i	$-0.0926380\pi$
$383$	14360.5	1.91590	0.957949	+	0.286940i	$0.0926380\pi$
$384$	0	0
$385$	−1239.35	−0.164060
$386$	−8683.31	−1.14500
$387$	0	0
$388$	−2424.93	−0.317287
$389$	10607.9	1.38262	0.691311	−	0.722557i	$-0.257033\pi$
$389$	10607.9	1.38262	0.691311	+	0.722557i	$0.257033\pi$
$390$	0	0
$391$	−7063.40	−0.913585
$392$	−869.371	−0.112015
$393$	0	0
$394$	−5282.79	−0.675490
$395$	−11854.6	−1.51005
$396$	0	0
$397$	7362.35	0.930745	0.465373	−	0.885115i	$-0.345920\pi$
$397$	7362.35	0.930745	0.465373	+	0.885115i	$0.345920\pi$
$398$	−7315.57	−0.921348
$399$	0	0
$400$	−10436.8	−1.30460
$401$	264.514	0.0329406	0.0164703	−	0.999864i	$-0.494757\pi$
$401$	264.514	0.0329406	0.0164703	+	0.999864i	$0.494757\pi$
$402$	0	0
$403$	−1280.06	−0.158224
$404$	2523.50	0.310765
$405$	0	0
$406$	−5968.72	−0.729612
$407$	209.511	0.0255161
$408$	0	0
$409$	−1244.03	−0.150400	−0.0751999	−	0.997168i	$-0.523959\pi$
$409$	−1244.03	−0.150400	−0.0751999	+	0.997168i	$0.523959\pi$
$410$	−8201.16	−0.987869
$411$	0	0
$412$	3390.91	0.405481
$413$	−3552.26	−0.423233
$414$	0	0
$415$	−9383.27	−1.10990
$416$	−3916.30	−0.461568
$417$	0	0
$418$	4218.35	0.493604
$419$	−3974.76	−0.463436	−0.231718	−	0.972783i	$-0.574435\pi$
$419$	−3974.76	−0.463436	−0.231718	+	0.972783i	$0.574435\pi$
$420$	0	0
$421$	−14910.2	−1.72608	−0.863041	−	0.505134i	$-0.831443\pi$
$421$	−14910.2	−1.72608	−0.863041	+	0.505134i	$0.831443\pi$
$422$	10100.2	1.16510
$423$	0	0
$424$	4834.46	0.553731
$425$	−5426.38	−0.619336
$426$	0	0
$427$	−248.926	−0.0282116
$428$	−3671.35	−0.414629
$429$	0	0
$430$	15001.8	1.68244
$431$	10622.4	1.18715	0.593574	−	0.804779i	$-0.297716\pi$
$431$	10622.4	1.18715	0.593574	+	0.804779i	$0.297716\pi$
$432$	0	0
$433$	17440.5	1.93565	0.967823	−	0.251630i	$-0.0809667\pi$
$433$	17440.5	1.93565	0.967823	+	0.251630i	$0.0809667\pi$
$434$	822.168	0.0909340
$435$	0	0
$436$	−1658.12	−0.182132
$437$	20620.0	2.25718
$438$	0	0
$439$	13627.1	1.48151	0.740756	−	0.671774i	$-0.234467\pi$
$439$	13627.1	1.48151	0.740756	+	0.671774i	$0.234467\pi$
$440$	−3141.28	−0.340351
$441$	0	0
$442$	−4646.50	−0.500026
$443$	−2135.29	−0.229008	−0.114504	−	0.993423i	$-0.536528\pi$
$443$	−2135.29	−0.229008	−0.114504	+	0.993423i	$0.536528\pi$
$444$	0	0
$445$	12976.2	1.38232
$446$	−11548.7	−1.22612
$447$	0	0
$448$	−1844.14	−0.194481
$449$	−17780.8	−1.86889	−0.934443	−	0.356113i	$-0.884102\pi$
$449$	−17780.8	−1.86889	−0.934443	+	0.356113i	$0.884102\pi$
$450$	0	0
$451$	−1726.96	−0.180309
$452$	2940.20	0.305963
$453$	0	0
$454$	7563.67	0.781896
$455$	−3985.22	−0.410615
$456$	0	0
$457$	10357.0	1.06013	0.530064	−	0.847957i	$-0.322168\pi$
$457$	10357.0	1.06013	0.530064	+	0.847957i	$0.322168\pi$
$458$	2194.30	0.223871
$459$	0	0
$460$	7115.51	0.721222
$461$	19679.5	1.98821	0.994106	−	0.108410i	$-0.0345760\pi$
$461$	19679.5	1.98821	0.994106	+	0.108410i	$0.0345760\pi$
$462$	0	0
$463$	−7171.43	−0.719838	−0.359919	−	0.932984i	$-0.617196\pi$
$463$	−7171.43	−0.719838	−0.359919	+	0.932984i	$0.617196\pi$
$464$	−20452.8	−2.04633
$465$	0	0
$466$	−5258.36	−0.522722
$467$	12192.8	1.20817	0.604085	−	0.796920i	$-0.293539\pi$
$467$	12192.8	1.20817	0.604085	+	0.796920i	$0.293539\pi$
$468$	0	0
$469$	−6854.61	−0.674875
$470$	−20751.2	−2.03656
$471$	0	0
$472$	−9003.60	−0.878017
$473$	3159.00	0.307085
$474$	0	0
$475$	15841.1	1.53018
$476$	717.752	0.0691137
$477$	0	0
$478$	6495.50	0.621542
$479$	−8475.11	−0.808429	−0.404215	−	0.914664i	$-0.632455\pi$
$479$	−8475.11	−0.808429	−0.404215	+	0.914664i	$0.632455\pi$
$480$	0	0
$481$	673.695	0.0638624
$482$	−5148.63	−0.486543
$483$	0	0
$484$	306.526	0.0287872
$485$	−15407.1	−1.44248
$486$	0	0
$487$	2735.29	0.254513	0.127257	−	0.991870i	$-0.459383\pi$
$487$	2735.29	0.254513	0.127257	+	0.991870i	$0.459383\pi$
$488$	−630.929	−0.0585263
$489$	0	0
$490$	2559.66	0.235987
$491$	2233.82	0.205317	0.102659	−	0.994717i	$-0.467265\pi$
$491$	2233.82	0.205317	0.102659	+	0.994717i	$0.467265\pi$
$492$	0	0
$493$	−10634.0	−0.971462
$494$	13564.4	1.23541
$495$	0	0
$496$	2817.30	0.255041
$497$	5252.83	0.474088
$498$	0	0
$499$	−18100.3	−1.62381	−0.811904	−	0.583791i	$-0.801569\pi$
$499$	−18100.3	−1.62381	−0.811904	+	0.583791i	$0.801569\pi$
$500$	369.619	0.0330597
$501$	0	0
$502$	−8477.99	−0.753768
$503$	−6149.06	−0.545076	−0.272538	−	0.962145i	$-0.587863\pi$
$503$	−6149.06	−0.545076	−0.272538	+	0.962145i	$0.587863\pi$
$504$	0	0
$505$	16033.4	1.41283
$506$	6230.08	0.547354
$507$	0	0
$508$	−2673.64	−0.233511
$509$	−14193.9	−1.23602	−0.618008	−	0.786172i	$-0.712060\pi$
$509$	−14193.9	−1.23602	−0.618008	+	0.786172i	$0.712060\pi$
$510$	0	0
$511$	−2769.16	−0.239727
$512$	−2430.30	−0.209775
$513$	0	0
$514$	−13625.4	−1.16924
$515$	21544.6	1.84343
$516$	0	0
$517$	−4369.68	−0.371718
$518$	−432.706	−0.0367027
$519$	0	0
$520$	−10101.0	−0.851840
$521$	−10371.8	−0.872163	−0.436082	−	0.899907i	$-0.643634\pi$
$521$	−10371.8	−0.872163	−0.436082	+	0.899907i	$0.643634\pi$
$522$	0	0
$523$	−11369.4	−0.950569	−0.475285	−	0.879832i	$-0.657655\pi$
$523$	−11369.4	−0.950569	−0.475285	+	0.879832i	$0.657655\pi$
$524$	−6731.91	−0.561231
$525$	0	0
$526$	16780.2	1.39098
$527$	1464.79	0.121076
$528$	0	0
$529$	18286.6	1.50297
$530$	−14233.9	−1.16657
$531$	0	0
$532$	−2095.31	−0.170758
$533$	−5553.14	−0.451282
$534$	0	0
$535$	−23326.4	−1.88503
$536$	−17373.7	−1.40006
$537$	0	0
$538$	−5349.24	−0.428665
$539$	539.000	0.0430730
$540$	0	0
$541$	5386.88	0.428096	0.214048	−	0.976823i	$-0.431335\pi$
$541$	5386.88	0.428096	0.214048	+	0.976823i	$0.431335\pi$
$542$	8317.54	0.659167
$543$	0	0
$544$	4481.47	0.353201
$545$	−10535.1	−0.828024
$546$	0	0
$547$	13890.8	1.08579	0.542894	−	0.839801i	$-0.317328\pi$
$547$	13890.8	1.08579	0.542894	+	0.839801i	$0.317328\pi$
$548$	373.186	0.0290907
$549$	0	0
$550$	4786.19	0.371061
$551$	31043.5	2.40017
$552$	0	0
$553$	5155.61	0.396454
$554$	8967.12	0.687683
$555$	0	0
$556$	2286.62	0.174414
$557$	−17498.3	−1.33111	−0.665553	−	0.746351i	$-0.731804\pi$
$557$	−17498.3	−1.33111	−0.665553	+	0.746351i	$0.731804\pi$
$558$	0	0
$559$	10158.0	0.768581
$560$	8771.10	0.661869
$561$	0	0
$562$	−20945.4	−1.57212
$563$	−147.373	−0.0110320	−0.00551600	−	0.999985i	$-0.501756\pi$
$563$	−147.373	−0.0110320	−0.00551600	+	0.999985i	$0.501756\pi$
$564$	0	0
$565$	18681.0	1.39100
$566$	14737.1	1.09443
$567$	0	0
$568$	13313.9	0.983517
$569$	14155.3	1.04292	0.521459	−	0.853276i	$-0.325388\pi$
$569$	14155.3	1.04292	0.521459	+	0.853276i	$0.325388\pi$
$570$	0	0
$571$	−248.361	−0.0182025	−0.00910123	−	0.999959i	$-0.502897\pi$
$571$	−248.361	−0.0182025	−0.00910123	+	0.999959i	$0.502897\pi$
$572$	985.654	0.0720494
$573$	0	0
$574$	3566.72	0.259359
$575$	23395.6	1.69681
$576$	0	0
$577$	19364.6	1.39715	0.698576	−	0.715536i	$-0.253817\pi$
$577$	19364.6	1.39715	0.698576	+	0.715536i	$0.253817\pi$
$578$	−10628.1	−0.764828
$579$	0	0
$580$	10712.4	0.766913
$581$	4080.83	0.291396
$582$	0	0
$583$	−2997.31	−0.212926
$584$	−7018.73	−0.497324
$585$	0	0
$586$	−6219.47	−0.438437
$587$	9135.18	0.642332	0.321166	−	0.947023i	$-0.395925\pi$
$587$	9135.18	0.642332	0.321166	+	0.947023i	$0.395925\pi$
$588$	0	0
$589$	−4276.12	−0.299141
$590$	26509.0	1.84976
$591$	0	0
$592$	−1482.74	−0.102940
$593$	12287.6	0.850916	0.425458	−	0.904978i	$-0.360113\pi$
$593$	12287.6	0.850916	0.425458	+	0.904978i	$0.360113\pi$
$594$	0	0
$595$	4560.34	0.314211
$596$	−3071.91	−0.211125
$597$	0	0
$598$	20033.2	1.36993
$599$	−12351.9	−0.842549	−0.421275	−	0.906933i	$-0.638417\pi$
$599$	−12351.9	−0.842549	−0.421275	+	0.906933i	$0.638417\pi$
$600$	0	0
$601$	21624.2	1.46767	0.733836	−	0.679327i	$-0.237728\pi$
$601$	21624.2	1.46767	0.733836	+	0.679327i	$0.237728\pi$
$602$	−6524.35	−0.441715
$603$	0	0
$604$	6498.68	0.437794
$605$	1947.56	0.130875
$606$	0	0
$607$	−2086.03	−0.139488	−0.0697442	−	0.997565i	$-0.522218\pi$
$607$	−2086.03	−0.139488	−0.0697442	+	0.997565i	$0.522218\pi$
$608$	−13082.6	−0.872648
$609$	0	0
$610$	1857.62	0.123300
$611$	−14051.0	−0.930347
$612$	0	0
$613$	−8338.46	−0.549408	−0.274704	−	0.961529i	$-0.588580\pi$
$613$	−8338.46	−0.549408	−0.274704	+	0.961529i	$0.588580\pi$
$614$	22317.0	1.46684
$615$	0	0
$616$	1366.15	0.0893570
$617$	4771.20	0.311315	0.155657	−	0.987811i	$-0.450250\pi$
$617$	4771.20	0.311315	0.155657	+	0.987811i	$0.450250\pi$
$618$	0	0
$619$	−16609.4	−1.07850	−0.539248	−	0.842147i	$-0.681292\pi$
$619$	−16609.4	−1.07850	−0.539248	+	0.842147i	$0.681292\pi$
$620$	−1475.60	−0.0955828
$621$	0	0
$622$	−27487.2	−1.77192
$623$	−5643.40	−0.362919
$624$	0	0
$625$	−14409.7	−0.922221
$626$	−9354.43	−0.597250
$627$	0	0
$628$	1778.96	0.113038
$629$	−770.918	−0.0488688
$630$	0	0
$631$	17254.9	1.08860	0.544299	−	0.838891i	$-0.316796\pi$
$631$	17254.9	1.08860	0.544299	+	0.838891i	$0.316796\pi$
$632$	13067.5	0.822462
$633$	0	0
$634$	−31790.7	−1.99143
$635$	−16987.3	−1.06161
$636$	0	0
$637$	1733.19	0.107804
$638$	9379.42	0.582029
$639$	0	0
$640$	28018.7	1.73053
$641$	7350.77	0.452945	0.226473	−	0.974018i	$-0.427281\pi$
$641$	7350.77	0.452945	0.226473	+	0.974018i	$0.427281\pi$
$642$	0	0
$643$	10117.8	0.620542	0.310271	−	0.950648i	$-0.399580\pi$
$643$	10117.8	0.620542	0.310271	+	0.950648i	$0.399580\pi$
$644$	−3094.56	−0.189352
$645$	0	0
$646$	−15521.9	−0.945358
$647$	24590.9	1.49423	0.747116	−	0.664693i	$-0.231438\pi$
$647$	24590.9	1.49423	0.747116	+	0.664693i	$0.231438\pi$
$648$	0	0
$649$	5582.13	0.337624
$650$	15390.3	0.928703
$651$	0	0
$652$	−2903.81	−0.174420
$653$	−2339.03	−0.140173	−0.0700867	−	0.997541i	$-0.522328\pi$
$653$	−2339.03	−0.140173	−0.0700867	+	0.997541i	$0.522328\pi$
$654$	0	0
$655$	−42772.1	−2.55152
$656$	12222.0	0.727420
$657$	0	0
$658$	9024.79	0.534685
$659$	15735.7	0.930162	0.465081	−	0.885268i	$-0.346025\pi$
$659$	15735.7	0.930162	0.465081	+	0.885268i	$0.346025\pi$
$660$	0	0
$661$	4846.75	0.285199	0.142600	−	0.989780i	$-0.454454\pi$
$661$	4846.75	0.285199	0.142600	+	0.989780i	$0.454454\pi$
$662$	5886.36	0.345589
$663$	0	0
$664$	10343.3	0.604515
$665$	−13312.8	−0.776316
$666$	0	0
$667$	45848.1	2.66154
$668$	8245.91	0.477611
$669$	0	0
$670$	51152.9	2.94957
$671$	391.169	0.0225051
$672$	0	0
$673$	15716.8	0.900205	0.450103	−	0.892977i	$-0.351387\pi$
$673$	15716.8	0.900205	0.450103	+	0.892977i	$0.351387\pi$
$674$	−38830.0	−2.21910
$675$	0	0
$676$	−2396.17	−0.136332
$677$	856.968	0.0486499	0.0243249	−	0.999704i	$-0.492256\pi$
$677$	856.968	0.0486499	0.0243249	+	0.999704i	$0.492256\pi$
$678$	0	0
$679$	6700.63	0.378713
$680$	11558.7	0.651846
$681$	0	0
$682$	−1291.98	−0.0725402
$683$	−24804.0	−1.38960	−0.694801	−	0.719202i	$-0.744508\pi$
$683$	−24804.0	−1.38960	−0.694801	+	0.719202i	$0.744508\pi$
$684$	0	0
$685$	2371.09	0.132255
$686$	−1113.21	−0.0619569
$687$	0	0
$688$	−22356.8	−1.23887
$689$	−9638.04	−0.532917
$690$	0	0
$691$	3475.97	0.191364	0.0956818	−	0.995412i	$-0.469497\pi$
$691$	3475.97	0.191364	0.0956818	+	0.995412i	$0.469497\pi$
$692$	−10194.5	−0.560024
$693$	0	0
$694$	7412.36	0.405431
$695$	14528.3	0.792937
$696$	0	0
$697$	6354.54	0.345330
$698$	−8024.44	−0.435143
$699$	0	0
$700$	−2377.36	−0.128366
$701$	−17461.0	−0.940790	−0.470395	−	0.882456i	$-0.655889\pi$
$701$	−17461.0	−0.940790	−0.470395	+	0.882456i	$0.655889\pi$
$702$	0	0
$703$	2250.52	0.120739
$704$	2897.93	0.155142
$705$	0	0
$706$	−33071.5	−1.76298
$707$	−6973.00	−0.370929
$708$	0	0
$709$	−18405.7	−0.974950	−0.487475	−	0.873137i	$-0.662082\pi$
$709$	−18405.7	−0.974950	−0.487475	+	0.873137i	$0.662082\pi$
$710$	−39199.6	−2.07202
$711$	0	0
$712$	−14303.8	−0.752891
$713$	−6315.39	−0.331716
$714$	0	0
$715$	6262.49	0.327558
$716$	−1788.72	−0.0933626
$717$	0	0
$718$	140.950	0.00732621
$719$	892.380	0.0462867	0.0231434	−	0.999732i	$-0.492633\pi$
$719$	892.380	0.0462867	0.0231434	+	0.999732i	$0.492633\pi$
$720$	0	0
$721$	−9369.83	−0.483982
$722$	23051.7	1.18822
$723$	0	0
$724$	−3214.48	−0.165007
$725$	35222.2	1.80431
$726$	0	0
$727$	27919.2	1.42430	0.712149	−	0.702028i	$-0.247722\pi$
$727$	27919.2	1.42430	0.712149	+	0.702028i	$0.247722\pi$
$728$	4392.96	0.223645
$729$	0	0
$730$	20665.0	1.04774
$731$	−11623.9	−0.588134
$732$	0	0
$733$	−4769.38	−0.240329	−0.120164	−	0.992754i	$-0.538342\pi$
$733$	−4769.38	−0.240329	−0.120164	+	0.992754i	$0.538342\pi$
$734$	28481.8	1.43227
$735$	0	0
$736$	−19321.7	−0.967673
$737$	10771.5	0.538364
$738$	0	0
$739$	−5170.63	−0.257381	−0.128691	−	0.991685i	$-0.541077\pi$
$739$	−5170.63	−0.257381	−0.128691	+	0.991685i	$0.541077\pi$
$740$	776.604	0.0385791
$741$	0	0
$742$	6190.40	0.306276
$743$	29407.9	1.45205	0.726024	−	0.687669i	$-0.241366\pi$
$743$	29407.9	1.45205	0.726024	+	0.687669i	$0.241366\pi$
$744$	0	0
$745$	−19517.8	−0.959836
$746$	−8931.26	−0.438333
$747$	0	0
$748$	−1127.90	−0.0551337
$749$	10144.8	0.494902
$750$	0	0
$751$	−16956.5	−0.823905	−0.411952	−	0.911205i	$-0.635153\pi$
$751$	−16956.5	−0.823905	−0.411952	+	0.911205i	$0.635153\pi$
$752$	30924.9	1.49962
$753$	0	0
$754$	30160.1	1.45672
$755$	41290.3	1.99034
$756$	0	0
$757$	−21322.0	−1.02373	−0.511864	−	0.859067i	$-0.671045\pi$
$757$	−21322.0	−1.02373	−0.511864	+	0.859067i	$0.671045\pi$
$758$	8456.45	0.405214
$759$	0	0
$760$	−33742.9	−1.61050
$761$	23548.0	1.12170	0.560851	−	0.827917i	$-0.310474\pi$
$761$	23548.0	1.12170	0.560851	+	0.827917i	$0.310474\pi$
$762$	0	0
$763$	4581.75	0.217392
$764$	−12323.3	−0.583563
$765$	0	0
$766$	46607.1	2.19841
$767$	17949.7	0.845014
$768$	0	0
$769$	−17230.9	−0.808015	−0.404007	−	0.914756i	$-0.632383\pi$
$769$	−17230.9	−0.808015	−0.404007	+	0.914756i	$0.632383\pi$
$770$	−4022.32	−0.188252
$771$	0	0
$772$	−6777.75	−0.315980
$773$	−12285.8	−0.571655	−0.285828	−	0.958281i	$-0.592268\pi$
$773$	−12285.8	−0.571655	−0.285828	+	0.958281i	$0.592268\pi$
$774$	0	0
$775$	−4851.73	−0.224876
$776$	16983.5	0.785658
$777$	0	0
$778$	34427.8	1.58650
$779$	−18550.6	−0.853202
$780$	0	0
$781$	−8254.45	−0.378191
$782$	−22924.3	−1.04830
$783$	0	0
$784$	−3814.59	−0.173769
$785$	11302.8	0.513906
$786$	0	0
$787$	663.152	0.0300366	0.0150183	−	0.999887i	$-0.495219\pi$
$787$	663.152	0.0300366	0.0150183	+	0.999887i	$0.495219\pi$
$788$	−4123.48	−0.186412
$789$	0	0
$790$	−38474.1	−1.73272
$791$	−8124.43	−0.365198
$792$	0	0
$793$	1257.83	0.0563264
$794$	23894.5	1.06799
$795$	0	0
$796$	−5710.17	−0.254261
$797$	−19216.3	−0.854050	−0.427025	−	0.904240i	$-0.640438\pi$
$797$	−19216.3	−0.854050	−0.427025	+	0.904240i	$0.640438\pi$
$798$	0	0
$799$	16078.7	0.711921
$800$	−14843.7	−0.656004
$801$	0	0
$802$	858.480	0.0377980
$803$	4351.53	0.191236
$804$	0	0
$805$	−19661.7	−0.860851
$806$	−4154.44	−0.181556
$807$	0	0
$808$	−17673.8	−0.769509
$809$	42881.8	1.86359	0.931794	−	0.362989i	$-0.118244\pi$
$809$	42881.8	1.86359	0.931794	+	0.362989i	$0.118244\pi$
$810$	0	0
$811$	−1205.73	−0.0522058	−0.0261029	−	0.999659i	$-0.508310\pi$
$811$	−1205.73	−0.0522058	−0.0261029	+	0.999659i	$0.508310\pi$
$812$	−4658.88	−0.201348
$813$	0	0
$814$	679.967	0.0292787
$815$	−18449.8	−0.792966
$816$	0	0
$817$	33933.3	1.45309
$818$	−4037.51	−0.172577
$819$	0	0
$820$	−6401.41	−0.272618
$821$	28577.6	1.21482	0.607408	−	0.794390i	$-0.292209\pi$
$821$	28577.6	1.21482	0.607408	+	0.794390i	$0.292209\pi$
$822$	0	0
$823$	42524.8	1.80112	0.900561	−	0.434730i	$-0.143156\pi$
$823$	42524.8	1.80112	0.900561	+	0.434730i	$0.143156\pi$
$824$	−23748.9	−1.00404
$825$	0	0
$826$	−11528.9	−0.485643
$827$	−30768.7	−1.29375	−0.646876	−	0.762595i	$-0.723925\pi$
$827$	−30768.7	−1.29375	−0.646876	+	0.762595i	$0.723925\pi$
$828$	0	0
$829$	−17583.3	−0.736661	−0.368330	−	0.929695i	$-0.620070\pi$
$829$	−17583.3	−0.736661	−0.368330	+	0.929695i	$0.620070\pi$
$830$	−30453.4	−1.27356
$831$	0	0
$832$	9318.48	0.388293
$833$	−1983.31	−0.0824942
$834$	0	0
$835$	52391.5	2.17136
$836$	3292.63	0.136218
$837$	0	0
$838$	−12900.1	−0.531773
$839$	−19552.8	−0.804573	−0.402287	−	0.915514i	$-0.631784\pi$
$839$	−19552.8	−0.804573	−0.402287	+	0.915514i	$0.631784\pi$
$840$	0	0
$841$	44635.5	1.83015
$842$	−48391.2	−1.98061
$843$	0	0
$844$	7883.72	0.321527
$845$	−15224.4	−0.619805
$846$	0	0
$847$	−847.000	−0.0343604
$848$	21212.4	0.859007
$849$	0	0
$850$	−17611.3	−0.710662
$851$	3323.78	0.133887
$852$	0	0
$853$	−18524.1	−0.743557	−0.371779	−	0.928321i	$-0.621252\pi$
$853$	−18524.1	−0.743557	−0.371779	+	0.928321i	$0.621252\pi$
$854$	−807.889	−0.0323716
$855$	0	0
$856$	25713.0	1.02670
$857$	−24439.0	−0.974118	−0.487059	−	0.873369i	$-0.661930\pi$
$857$	−24439.0	−0.974118	−0.487059	+	0.873369i	$0.661930\pi$
$858$	0	0
$859$	11301.4	0.448893	0.224447	−	0.974486i	$-0.427943\pi$
$859$	11301.4	0.448893	0.224447	+	0.974486i	$0.427943\pi$
$860$	11709.6	0.464297
$861$	0	0
$862$	34474.9	1.36220
$863$	26377.3	1.04043	0.520217	−	0.854034i	$-0.325851\pi$
$863$	26377.3	1.04043	0.520217	+	0.854034i	$0.325851\pi$
$864$	0	0
$865$	−64772.1	−2.54603
$866$	56603.0	2.22107
$867$	0	0
$868$	641.743	0.0250947
$869$	−8101.68	−0.316261
$870$	0	0
$871$	34636.5	1.34743
$872$	11612.9	0.450991
$873$	0	0
$874$	66922.1	2.59002
$875$	−1021.34	−0.0394601
$876$	0	0
$877$	−28425.0	−1.09446	−0.547232	−	0.836981i	$-0.684319\pi$
$877$	−28425.0	−1.09446	−0.547232	+	0.836981i	$0.684319\pi$
$878$	44226.6	1.69997
$879$	0	0
$880$	−13783.2	−0.527989
$881$	−16897.1	−0.646171	−0.323086	−	0.946370i	$-0.604720\pi$
$881$	−16897.1	−0.646171	−0.323086	+	0.946370i	$0.604720\pi$
$882$	0	0
$883$	−25538.9	−0.973331	−0.486665	−	0.873588i	$-0.661787\pi$
$883$	−25538.9	−0.973331	−0.486665	+	0.873588i	$0.661787\pi$
$884$	−3626.82	−0.137990
$885$	0	0
$886$	−6930.07	−0.262777
$887$	−6478.48	−0.245238	−0.122619	−	0.992454i	$-0.539129\pi$
$887$	−6478.48	−0.245238	−0.122619	+	0.992454i	$0.539129\pi$
$888$	0	0
$889$	7387.85	0.278718
$890$	42114.3	1.58615
$891$	0	0
$892$	−9014.35	−0.338366
$893$	−46938.1	−1.75893
$894$	0	0
$895$	−11364.9	−0.424453
$896$	−12185.5	−0.454339
$897$	0	0
$898$	−57707.7	−2.14447
$899$	−9507.86	−0.352731
$900$	0	0
$901$	11028.9	0.407799
$902$	−5604.85	−0.206897
$903$	0	0
$904$	−20592.3	−0.757620
$905$	−20423.6	−0.750171
$906$	0	0
$907$	−9356.17	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$907$	−9356.17	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$908$	5903.82	0.215777
$909$	0	0
$910$	−12934.0	−0.471164
$911$	16574.0	0.602768	0.301384	−	0.953503i	$-0.402551\pi$
$911$	16574.0	0.602768	0.301384	+	0.953503i	$0.402551\pi$
$912$	0	0
$913$	−6412.73	−0.232454
$914$	33613.6	1.21645
$915$	0	0
$916$	1712.76	0.0617808
$917$	18601.8	0.669885
$918$	0	0
$919$	8214.92	0.294870	0.147435	−	0.989072i	$-0.452898\pi$
$919$	8214.92	0.294870	0.147435	+	0.989072i	$0.452898\pi$
$920$	−49834.8	−1.78588
$921$	0	0
$922$	63869.8	2.28139
$923$	−26542.7	−0.946548
$924$	0	0
$925$	2553.46	0.0907646
$926$	−23274.9	−0.825983
$927$	0	0
$928$	−29088.9	−1.02898
$929$	−42653.9	−1.50638	−0.753192	−	0.657801i	$-0.771487\pi$
$929$	−42653.9	−1.50638	−0.753192	+	0.657801i	$0.771487\pi$
$930$	0	0
$931$	5789.81	0.203817
$932$	−4104.41	−0.144254
$933$	0	0
$934$	39571.7	1.38632
$935$	−7166.25	−0.250654
$936$	0	0
$937$	−18484.8	−0.644473	−0.322237	−	0.946659i	$-0.604435\pi$
$937$	−18484.8	−0.644473	−0.322237	+	0.946659i	$0.604435\pi$
$938$	−22246.6	−0.774390
$939$	0	0
$940$	−16197.3	−0.562020
$941$	7183.03	0.248842	0.124421	−	0.992230i	$-0.460293\pi$
$941$	7183.03	0.248842	0.124421	+	0.992230i	$0.460293\pi$
$942$	0	0
$943$	−27397.4	−0.946109
$944$	−39505.6	−1.36207
$945$	0	0
$946$	10252.5	0.352367
$947$	−41443.3	−1.42210	−0.711049	−	0.703143i	$-0.751779\pi$
$947$	−41443.3	−1.42210	−0.711049	+	0.703143i	$0.751779\pi$
$948$	0	0
$949$	13992.6	0.478630
$950$	51412.1	1.75582
$951$	0	0
$952$	−5026.92	−0.171138
$953$	−7981.30	−0.271290	−0.135645	−	0.990757i	$-0.543311\pi$
$953$	−7981.30	−0.271290	−0.135645	+	0.990757i	$0.543311\pi$
$954$	0	0
$955$	−78297.8	−2.65305
$956$	5070.06	0.171524
$957$	0	0
$958$	−27506.0	−0.927638
$959$	−1031.20	−0.0347227
$960$	0	0
$961$	−28481.3	−0.956038
$962$	2186.48	0.0732795
$963$	0	0
$964$	−4018.76	−0.134269
$965$	−43063.3	−1.43654
$966$	0	0
$967$	−18745.7	−0.623394	−0.311697	−	0.950182i	$-0.600897\pi$
$967$	−18745.7	−0.623394	−0.311697	+	0.950182i	$0.600897\pi$
$968$	−2146.81	−0.0712822
$969$	0	0
$970$	−50003.8	−1.65518
$971$	−3096.87	−0.102351	−0.0511757	−	0.998690i	$-0.516297\pi$
$971$	−3096.87	−0.102351	−0.0511757	+	0.998690i	$0.516297\pi$
$972$	0	0
$973$	−6318.44	−0.208181
$974$	8877.40	0.292043
$975$	0	0
$976$	−2768.36	−0.0907922
$977$	−19960.2	−0.653618	−0.326809	−	0.945090i	$-0.605973\pi$
$977$	−19960.2	−0.653618	−0.326809	+	0.945090i	$0.605973\pi$
$978$	0	0
$979$	8868.21	0.289509
$980$	1997.94	0.0651243
$981$	0	0
$982$	7249.85	0.235593
$983$	33434.5	1.08484	0.542419	−	0.840108i	$-0.317509\pi$
$983$	33434.5	1.08484	0.542419	+	0.840108i	$0.317509\pi$
$984$	0	0
$985$	−26199.1	−0.847485
$986$	−34512.6	−1.11471
$987$	0	0
$988$	10587.7	0.340930
$989$	50116.1	1.61132
$990$	0	0
$991$	24855.2	0.796722	0.398361	−	0.917229i	$-0.369579\pi$
$991$	24855.2	0.796722	0.398361	+	0.917229i	$0.369579\pi$
$992$	4006.88	0.128245
$993$	0	0
$994$	17048.1	0.543996
$995$	−36280.3	−1.15594
$996$	0	0
$997$	−7810.38	−0.248102	−0.124051	−	0.992276i	$-0.539589\pi$
$997$	−7810.38	−0.248102	−0.124051	+	0.992276i	$0.539589\pi$
$998$	−58744.5	−1.86325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.l.1.3		4
3.2	odd	2		77.4.a.d.1.2	✓	4
12.11	even	2		1232.4.a.s.1.4		4
15.14	odd	2		1925.4.a.p.1.3		4
21.20	even	2		539.4.a.g.1.2		4
33.32	even	2		847.4.a.d.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.2	✓	4	3.2	odd	2
539.4.a.g.1.2		4	21.20	even	2
693.4.a.l.1.3		4	1.1	even	1	trivial
847.4.a.d.1.3		4	33.32	even	2
1232.4.a.s.1.4		4	12.11	even	2
1925.4.a.p.1.3		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)

\(f(q)\)	\(=\)	\(q+3.24550 q^{2} +2.53327 q^{4} +16.0955 q^{5} -7.00000 q^{7} -17.7423 q^{8} +52.2379 q^{10} +11.0000 q^{11} +35.3712 q^{13} -22.7185 q^{14} -77.8487 q^{16} -40.4757 q^{17} +118.159 q^{19} +40.7743 q^{20} +35.7005 q^{22} +174.510 q^{23} +134.065 q^{25} +114.797 q^{26} -17.7329 q^{28} +262.725 q^{29} -36.1894 q^{31} -110.720 q^{32} -131.364 q^{34} -112.668 q^{35} +19.0464 q^{37} +383.487 q^{38} -285.571 q^{40} -156.996 q^{41} +287.182 q^{43} +27.8660 q^{44} +566.371 q^{46} -397.244 q^{47} +49.0000 q^{49} +435.108 q^{50} +89.6049 q^{52} -272.483 q^{53} +177.050 q^{55} +124.196 q^{56} +852.674 q^{58} +507.466 q^{59} +35.5608 q^{61} -117.453 q^{62} +263.448 q^{64} +569.317 q^{65} +979.229 q^{67} -102.536 q^{68} -365.666 q^{70} -750.404 q^{71} +395.594 q^{73} +61.8152 q^{74} +299.330 q^{76} -77.0000 q^{77} -736.516 q^{79} -1253.01 q^{80} -509.531 q^{82} -582.975 q^{83} -651.477 q^{85} +932.050 q^{86} -195.165 q^{88} +806.201 q^{89} -247.598 q^{91} +442.081 q^{92} -1289.26 q^{94} +1901.84 q^{95} -957.232 q^{97} +159.030 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 26 q^{4} - 10 q^{5} - 28 q^{7} + 18 q^{8} - 2 q^{10} + 44 q^{11} + 58 q^{13} - 14 q^{14} + 2 q^{16} - 4 q^{17} + 258 q^{19} - 182 q^{20} + 22 q^{22} - 8 q^{23} + 80 q^{25} + 482 q^{26} - 182 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 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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