LMFDB - Embedded newform 1323.4.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 $ x^7 - x^6 - 43x^5 + 10x^4 + 513x^3 + 258x^2 - 936*x - 504
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.62964$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.62964 q^{2} +5.17430 q^{4} +4.84163 q^{5} +10.2563 q^{8} +O(q^{10})$	$q-3.62964 q^{2} +5.17430 q^{4} +4.84163 q^{5} +10.2563 q^{8} -17.5734 q^{10} -20.1323 q^{11} +72.2571 q^{13} -78.6210 q^{16} +132.363 q^{17} +76.9369 q^{19} +25.0520 q^{20} +73.0732 q^{22} -22.4717 q^{23} -101.559 q^{25} -262.268 q^{26} +193.429 q^{29} +89.7705 q^{31} +203.316 q^{32} -480.430 q^{34} +47.9286 q^{37} -279.254 q^{38} +49.6571 q^{40} +3.41948 q^{41} -168.358 q^{43} -104.171 q^{44} +81.5644 q^{46} +163.178 q^{47} +368.622 q^{50} +373.880 q^{52} +337.375 q^{53} -97.4733 q^{55} -702.077 q^{58} +517.788 q^{59} +424.828 q^{61} -325.835 q^{62} -108.996 q^{64} +349.842 q^{65} -978.338 q^{67} +684.885 q^{68} -40.4250 q^{71} +482.240 q^{73} -173.964 q^{74} +398.095 q^{76} -1074.97 q^{79} -380.654 q^{80} -12.4115 q^{82} -811.700 q^{83} +640.851 q^{85} +611.078 q^{86} -206.483 q^{88} +997.039 q^{89} -116.276 q^{92} -592.279 q^{94} +372.500 q^{95} -1452.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10}) $ 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8	$ 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8} + 12 q^{10} + 98 q^{11} - 124 q^{13} + 139 q^{16} + 30 q^{17} + 182 q^{19} - 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} - 245 q^{26} + 323 q^{29} + 26 q^{31} + 398 q^{32} + 114 q^{34} - 112 q^{37} + 1015 q^{38} - 147 q^{40} - 524 q^{41} + 8 q^{43} + 937 q^{44} - 339 q^{46} + 288 q^{47} + 2576 q^{50} - 1075 q^{52} + 1353 q^{53} + 156 q^{55} - 81 q^{58} + 165 q^{59} + 56 q^{61} - 1215 q^{62} - 1706 q^{64} + 1694 q^{65} - 988 q^{67} + 2625 q^{68} + 792 q^{71} + 1487 q^{73} + 2736 q^{74} + 1952 q^{76} - 1273 q^{79} - 2501 q^{80} - 2049 q^{82} - 1170 q^{83} + 216 q^{85} - 160 q^{86} + 9 q^{88} + 1058 q^{89} + 3834 q^{92} + 1653 q^{94} + 3260 q^{95} - 3730 q^{97}+O(q^{100}) $ 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8 + 12 * q^10 + 98 * q^11 - 124 * q^13 + 139 * q^16 + 30 * q^17 + 182 * q^19 - 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 - 245 * q^26 + 323 * q^29 + 26 * q^31 + 398 * q^32 + 114 * q^34 - 112 * q^37 + 1015 * q^38 - 147 * q^40 - 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 + 288 * q^47 + 2576 * q^50 - 1075 * q^52 + 1353 * q^53 + 156 * q^55 - 81 * q^58 + 165 * q^59 + 56 * q^61 - 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 + 2625 * q^68 + 792 * q^71 + 1487 * q^73 + 2736 * q^74 + 1952 * q^76 - 1273 * q^79 - 2501 * q^80 - 2049 * q^82 - 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 + 1058 * q^89 + 3834 * q^92 + 1653 * q^94 + 3260 * q^95 - 3730 * q^97

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 $

$2$	−3.62964	−1.28327	−0.641636	−	0.767009i	$-0.721744\pi$
$2$	−3.62964	−1.28327	−0.641636	+	0.767009i	$0.721744\pi$
$3$	0	0
$3$	0	0
$4$	5.17430	0.646788
$5$	4.84163	0.433048	0.216524	−	0.976277i	$-0.430528\pi$
$5$	4.84163	0.433048	0.216524	+	0.976277i	$0.430528\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	10.2563	0.453268
$9$	0	0
$10$	−17.5734	−0.555719
$11$	−20.1323	−0.551830	−0.275915	−	0.961182i	$-0.588981\pi$
$11$	−20.1323	−0.551830	−0.275915	+	0.961182i	$0.588981\pi$
$12$	0	0
$13$	72.2571	1.54158	0.770789	−	0.637090i	$-0.219862\pi$
$13$	72.2571	1.54158	0.770789	+	0.637090i	$0.219862\pi$
$14$	0	0
$15$	0	0
$16$	−78.6210	−1.22845
$17$	132.363	1.88839	0.944197	−	0.329382i	$-0.106840\pi$
$17$	132.363	1.88839	0.944197	+	0.329382i	$0.106840\pi$
$18$	0	0
$19$	76.9369	0.928976	0.464488	−	0.885579i	$-0.346238\pi$
$19$	76.9369	0.928976	0.464488	+	0.885579i	$0.346238\pi$
$20$	25.0520	0.280090
$21$	0	0
$22$	73.0732	0.708148
$23$	−22.4717	−0.203725	−0.101863	−	0.994798i	$-0.532480\pi$
$23$	−22.4717	−0.203725	−0.101863	+	0.994798i	$0.532480\pi$
$24$	0	0
$25$	−101.559	−0.812469
$26$	−262.268	−1.97827
$27$	0	0
$28$	0	0
$29$	193.429	1.23858	0.619290	−	0.785163i	$-0.287421\pi$
$29$	193.429	1.23858	0.619290	+	0.785163i	$0.287421\pi$
$30$	0	0
$31$	89.7705	0.520105	0.260053	−	0.965594i	$-0.416260\pi$
$31$	89.7705	0.520105	0.260053	+	0.965594i	$0.416260\pi$
$32$	203.316	1.12317
$33$	0	0
$34$	−480.430	−2.42332
$35$	0	0
$36$	0	0
$37$	47.9286	0.212957	0.106479	−	0.994315i	$-0.466042\pi$
$37$	47.9286	0.212957	0.106479	+	0.994315i	$0.466042\pi$
$38$	−279.254	−1.19213
$39$	0	0
$40$	49.6571	0.196287
$41$	3.41948	0.0130252	0.00651260	−	0.999979i	$-0.497927\pi$
$41$	3.41948	0.0130252	0.00651260	+	0.999979i	$0.497927\pi$
$42$	0	0
$43$	−168.358	−0.597076	−0.298538	−	0.954398i	$-0.596499\pi$
$43$	−168.358	−0.597076	−0.298538	+	0.954398i	$0.596499\pi$
$44$	−104.171	−0.356917
$45$	0	0
$46$	81.5644	0.261435
$47$	163.178	0.506426	0.253213	−	0.967411i	$-0.418513\pi$
$47$	163.178	0.506426	0.253213	+	0.967411i	$0.418513\pi$
$48$	0	0
$49$	0	0
$50$	368.622	1.04262
$51$	0	0
$52$	373.880	0.997074
$53$	337.375	0.874377	0.437188	−	0.899370i	$-0.355974\pi$
$53$	337.375	0.874377	0.437188	+	0.899370i	$0.355974\pi$
$54$	0	0
$55$	−97.4733	−0.238969
$56$	0	0
$57$	0	0
$58$	−702.077	−1.58943
$59$	517.788	1.14255	0.571274	−	0.820760i	$-0.306449\pi$
$59$	517.788	1.14255	0.571274	+	0.820760i	$0.306449\pi$
$60$	0	0
$61$	424.828	0.891698	0.445849	−	0.895108i	$-0.352902\pi$
$61$	424.828	0.891698	0.445849	+	0.895108i	$0.352902\pi$
$62$	−325.835	−0.667436
$63$	0	0
$64$	−108.996	−0.212883
$65$	349.842	0.667578
$66$	0	0
$67$	−978.338	−1.78393	−0.891963	−	0.452109i	$-0.850672\pi$
$67$	−978.338	−1.78393	−0.891963	+	0.452109i	$0.850672\pi$
$68$	684.885	1.22139
$69$	0	0
$70$	0	0
$71$	−40.4250	−0.0675713	−0.0337856	−	0.999429i	$-0.510756\pi$
$71$	−40.4250	−0.0675713	−0.0337856	+	0.999429i	$0.510756\pi$
$72$	0	0
$73$	482.240	0.773176	0.386588	−	0.922252i	$-0.373653\pi$
$73$	482.240	0.773176	0.386588	+	0.922252i	$0.373653\pi$
$74$	−173.964	−0.273282
$75$	0	0
$76$	398.095	0.600850
$77$	0	0
$78$	0	0
$79$	−1074.97	−1.53093	−0.765466	−	0.643476i	$-0.777492\pi$
$79$	−1074.97	−1.53093	−0.765466	+	0.643476i	$0.777492\pi$
$80$	−380.654	−0.531980
$81$	0	0
$82$	−12.4115	−0.0167149
$83$	−811.700	−1.07344	−0.536721	−	0.843760i	$-0.680337\pi$
$83$	−811.700	−1.07344	−0.536721	+	0.843760i	$0.680337\pi$
$84$	0	0
$85$	640.851	0.817766
$86$	611.078	0.766212
$87$	0	0
$88$	−206.483	−0.250127
$89$	997.039	1.18748	0.593741	−	0.804656i	$-0.297650\pi$
$89$	997.039	1.18748	0.593741	+	0.804656i	$0.297650\pi$
$90$	0	0
$91$	0	0
$92$	−116.276	−0.131767
$93$	0	0
$94$	−592.279	−0.649882
$95$	372.500	0.402291
$96$	0	0
$97$	−1452.82	−1.52074	−0.760369	−	0.649491i	$-0.774982\pi$
$97$	−1452.82	−1.52074	−0.760369	+	0.649491i	$0.774982\pi$
$98$	0	0
$99$	0	0
$100$	−525.495	−0.525495
$101$	−223.725	−0.220411	−0.110205	−	0.993909i	$-0.535151\pi$
$101$	−223.725	−0.220411	−0.110205	+	0.993909i	$0.535151\pi$
$102$	0	0
$103$	−538.467	−0.515114	−0.257557	−	0.966263i	$-0.582918\pi$
$103$	−538.467	−0.515114	−0.257557	+	0.966263i	$0.582918\pi$
$104$	741.089	0.698748
$105$	0	0
$106$	−1224.55	−1.12206
$107$	1898.05	1.71488	0.857438	−	0.514588i	$-0.172055\pi$
$107$	1898.05	1.71488	0.857438	+	0.514588i	$0.172055\pi$
$108$	0	0
$109$	−854.503	−0.750886	−0.375443	−	0.926846i	$-0.622509\pi$
$109$	−854.503	−0.750886	−0.375443	+	0.926846i	$0.622509\pi$
$110$	353.793	0.306662
$111$	0	0
$112$	0	0
$113$	−802.087	−0.667734	−0.333867	−	0.942620i	$-0.608354\pi$
$113$	−802.087	−0.667734	−0.333867	+	0.942620i	$0.608354\pi$
$114$	0	0
$115$	−108.800	−0.0882229
$116$	1000.86	0.801098
$117$	0	0
$118$	−1879.39	−1.46620
$119$	0	0
$120$	0	0
$121$	−925.689	−0.695484
$122$	−1541.97	−1.14429
$123$	0	0
$124$	464.500	0.336398
$125$	−1096.91	−0.784887
$126$	0	0
$127$	−1723.69	−1.20435	−0.602176	−	0.798364i	$-0.705699\pi$
$127$	−1723.69	−1.20435	−0.602176	+	0.798364i	$0.705699\pi$
$128$	−1230.91	−0.849986
$129$	0	0
$130$	−1269.80	−0.856685
$131$	1018.94	0.679585	0.339792	−	0.940500i	$-0.389643\pi$
$131$	1018.94	0.679585	0.339792	+	0.940500i	$0.389643\pi$
$132$	0	0
$133$	0	0
$134$	3551.02	2.28926
$135$	0	0
$136$	1357.55	0.855948
$137$	1838.22	1.14635	0.573173	−	0.819434i	$-0.305712\pi$
$137$	1838.22	1.14635	0.573173	+	0.819434i	$0.305712\pi$
$138$	0	0
$139$	−1467.29	−0.895352	−0.447676	−	0.894196i	$-0.647748\pi$
$139$	−1467.29	−0.895352	−0.447676	+	0.894196i	$0.647748\pi$
$140$	0	0
$141$	0	0
$142$	146.728	0.0867124
$143$	−1454.70	−0.850689
$144$	0	0
$145$	936.509	0.536365
$146$	−1750.36	−0.992196
$147$	0	0
$148$	247.997	0.137738
$149$	−3510.13	−1.92994	−0.964971	−	0.262357i	$-0.915500\pi$
$149$	−3510.13	−1.92994	−0.964971	+	0.262357i	$0.915500\pi$
$150$	0	0
$151$	3237.59	1.74484	0.872421	−	0.488756i	$-0.162549\pi$
$151$	3237.59	1.74484	0.872421	+	0.488756i	$0.162549\pi$
$152$	789.086	0.421075
$153$	0	0
$154$	0	0
$155$	434.635	0.225231
$156$	0	0
$157$	2865.29	1.45653	0.728265	−	0.685296i	$-0.240327\pi$
$157$	2865.29	1.45653	0.728265	+	0.685296i	$0.240327\pi$
$158$	3901.76	1.96460
$159$	0	0
$160$	984.380	0.486388
$161$	0	0
$162$	0	0
$163$	−612.714	−0.294426	−0.147213	−	0.989105i	$-0.547030\pi$
$163$	−612.714	−0.294426	−0.147213	+	0.989105i	$0.547030\pi$
$164$	17.6934	0.00842453
$165$	0	0
$166$	2946.18	1.37752
$167$	1633.90	0.757096	0.378548	−	0.925582i	$-0.376423\pi$
$167$	1633.90	0.757096	0.378548	+	0.925582i	$0.376423\pi$
$168$	0	0
$169$	3024.09	1.37647
$170$	−2326.06	−1.04942
$171$	0	0
$172$	−871.133	−0.386182
$173$	−1036.77	−0.455632	−0.227816	−	0.973704i	$-0.573158\pi$
$173$	−1036.77	−0.455632	−0.227816	+	0.973704i	$0.573158\pi$
$174$	0	0
$175$	0	0
$176$	1582.82	0.677897
$177$	0	0
$178$	−3618.89	−1.52386
$179$	−522.258	−0.218075	−0.109037	−	0.994038i	$-0.534777\pi$
$179$	−522.258	−0.218075	−0.109037	+	0.994038i	$0.534777\pi$
$180$	0	0
$181$	2657.83	1.09147	0.545733	−	0.837959i	$-0.316251\pi$
$181$	2657.83	1.09147	0.545733	+	0.837959i	$0.316251\pi$
$182$	0	0
$183$	0	0
$184$	−230.476	−0.0923421
$185$	232.052	0.0922207
$186$	0	0
$187$	−2664.77	−1.04207
$188$	844.334	0.327550
$189$	0	0
$190$	−1352.04	−0.516249
$191$	2915.42	1.10446	0.552231	−	0.833691i	$-0.313777\pi$
$191$	2915.42	1.10446	0.552231	+	0.833691i	$0.313777\pi$
$192$	0	0
$193$	1852.38	0.690866	0.345433	−	0.938443i	$-0.387732\pi$
$193$	1852.38	0.690866	0.345433	+	0.938443i	$0.387732\pi$
$194$	5273.22	1.95152
$195$	0	0
$196$	0	0
$197$	−2926.04	−1.05823	−0.529116	−	0.848549i	$-0.677477\pi$
$197$	−2926.04	−1.05823	−0.529116	+	0.848549i	$0.677477\pi$
$198$	0	0
$199$	−1629.32	−0.580398	−0.290199	−	0.956966i	$-0.593722\pi$
$199$	−1629.32	−0.580398	−0.290199	+	0.956966i	$0.593722\pi$
$200$	−1041.61	−0.368266
$201$	0	0
$202$	812.041	0.282847
$203$	0	0
$204$	0	0
$205$	16.5558	0.00564054
$206$	1954.44	0.661031
$207$	0	0
$208$	−5680.93	−1.89376
$209$	−1548.92	−0.512637
$210$	0	0
$211$	1691.47	0.551874	0.275937	−	0.961176i	$-0.411012\pi$
$211$	1691.47	0.551874	0.275937	+	0.961176i	$0.411012\pi$
$212$	1745.68	0.565536
$213$	0	0
$214$	−6889.25	−2.20065
$215$	−815.125	−0.258563
$216$	0	0
$217$	0	0
$218$	3101.54	0.963591
$219$	0	0
$220$	−504.356	−0.154562
$221$	9564.16	2.91111
$222$	0	0
$223$	−749.266	−0.224998	−0.112499	−	0.993652i	$-0.535886\pi$
$223$	−749.266	−0.224998	−0.112499	+	0.993652i	$0.535886\pi$
$224$	0	0
$225$	0	0
$226$	2911.29	0.856885
$227$	−4707.28	−1.37636	−0.688179	−	0.725541i	$-0.741589\pi$
$227$	−4707.28	−1.37636	−0.688179	+	0.725541i	$0.741589\pi$
$228$	0	0
$229$	2430.11	0.701249	0.350625	−	0.936516i	$-0.385969\pi$
$229$	2430.11	0.701249	0.350625	+	0.936516i	$0.385969\pi$
$230$	394.904	0.113214
$231$	0	0
$232$	1983.86	0.561408
$233$	1789.19	0.503063	0.251531	−	0.967849i	$-0.419066\pi$
$233$	1789.19	0.503063	0.251531	+	0.967849i	$0.419066\pi$
$234$	0	0
$235$	790.048	0.219307
$236$	2679.19	0.738986
$237$	0	0
$238$	0	0
$239$	2506.96	0.678500	0.339250	−	0.940696i	$-0.389827\pi$
$239$	2506.96	0.678500	0.339250	+	0.940696i	$0.389827\pi$
$240$	0	0
$241$	3978.30	1.06334	0.531669	−	0.846952i	$-0.321565\pi$
$241$	3978.30	1.06334	0.531669	+	0.846952i	$0.321565\pi$
$242$	3359.92	0.892495
$243$	0	0
$244$	2198.19	0.576739
$245$	0	0
$246$	0	0
$247$	5559.24	1.43209
$248$	920.711	0.235747
$249$	0	0
$250$	3981.40	1.00722
$251$	−2907.80	−0.731231	−0.365615	−	0.930766i	$-0.619141\pi$
$251$	−2907.80	−0.731231	−0.365615	+	0.930766i	$0.619141\pi$
$252$	0	0
$253$	452.409	0.112422
$254$	6256.37	1.54551
$255$	0	0
$256$	5339.73	1.30365
$257$	3820.55	0.927313	0.463657	−	0.886015i	$-0.346537\pi$
$257$	3820.55	0.927313	0.463657	+	0.886015i	$0.346537\pi$
$258$	0	0
$259$	0	0
$260$	1810.19	0.431781
$261$	0	0
$262$	−3698.41	−0.872092
$263$	−6420.45	−1.50533	−0.752665	−	0.658404i	$-0.771232\pi$
$263$	−6420.45	−1.50533	−0.752665	+	0.658404i	$0.771232\pi$
$264$	0	0
$265$	1633.44	0.378647
$266$	0	0
$267$	0	0
$268$	−5062.22	−1.15382
$269$	1120.42	0.253952	0.126976	−	0.991906i	$-0.459473\pi$
$269$	1120.42	0.253952	0.126976	+	0.991906i	$0.459473\pi$
$270$	0	0
$271$	−2238.17	−0.501694	−0.250847	−	0.968027i	$-0.580709\pi$
$271$	−2238.17	−0.501694	−0.250847	+	0.968027i	$0.580709\pi$
$272$	−10406.5	−2.31980
$273$	0	0
$274$	−6672.07	−1.47107
$275$	2044.61	0.448345
$276$	0	0
$277$	1895.27	0.411104	0.205552	−	0.978646i	$-0.434101\pi$
$277$	1895.27	0.411104	0.205552	+	0.978646i	$0.434101\pi$
$278$	5325.74	1.14898
$279$	0	0
$280$	0	0
$281$	8622.88	1.83060	0.915298	−	0.402777i	$-0.131955\pi$
$281$	8622.88	1.83060	0.915298	+	0.402777i	$0.131955\pi$
$282$	0	0
$283$	3047.26	0.640074	0.320037	−	0.947405i	$-0.396305\pi$
$283$	3047.26	0.640074	0.320037	+	0.947405i	$0.396305\pi$
$284$	−209.171	−0.0437043
$285$	0	0
$286$	5280.06	1.09167
$287$	0	0
$288$	0	0
$289$	12606.9	2.56603
$290$	−3399.19	−0.688302
$291$	0	0
$292$	2495.25	0.500081
$293$	8032.29	1.60154	0.800770	−	0.598972i	$-0.204424\pi$
$293$	8032.29	1.60154	0.800770	+	0.598972i	$0.204424\pi$
$294$	0	0
$295$	2506.94	0.494778
$296$	491.569	0.0965266
$297$	0	0
$298$	12740.5	2.47664
$299$	−1623.74	−0.314059
$300$	0	0
$301$	0	0
$302$	−11751.3	−2.23911
$303$	0	0
$304$	−6048.86	−1.14120
$305$	2056.86	0.386148
$306$	0	0
$307$	1367.50	0.254226	0.127113	−	0.991888i	$-0.459429\pi$
$307$	1367.50	0.254226	0.127113	+	0.991888i	$0.459429\pi$
$308$	0	0
$309$	0	0
$310$	−1577.57	−0.289032
$311$	−1219.58	−0.222367	−0.111184	−	0.993800i	$-0.535464\pi$
$311$	−1219.58	−0.222367	−0.111184	+	0.993800i	$0.535464\pi$
$312$	0	0
$313$	−4724.22	−0.853127	−0.426564	−	0.904458i	$-0.640276\pi$
$313$	−4724.22	−0.853127	−0.426564	+	0.904458i	$0.640276\pi$
$314$	−10400.0	−1.86912
$315$	0	0
$316$	−5562.22	−0.990188
$317$	−1316.49	−0.233253	−0.116627	−	0.993176i	$-0.537208\pi$
$317$	−1316.49	−0.233253	−0.116627	+	0.993176i	$0.537208\pi$
$318$	0	0
$319$	−3894.17	−0.683485
$320$	−527.718	−0.0921885
$321$	0	0
$322$	0	0
$323$	10183.6	1.75427
$324$	0	0
$325$	−7338.34	−1.25249
$326$	2223.93	0.377829
$327$	0	0
$328$	35.0711	0.00590390
$329$	0	0
$330$	0	0
$331$	7636.14	1.26804	0.634019	−	0.773318i	$-0.281404\pi$
$331$	7636.14	1.26804	0.634019	+	0.773318i	$0.281404\pi$
$332$	−4199.98	−0.694289
$333$	0	0
$334$	−5930.48	−0.971561
$335$	−4736.75	−0.772526
$336$	0	0
$337$	9852.08	1.59251	0.796257	−	0.604959i	$-0.206810\pi$
$337$	9852.08	1.59251	0.796257	+	0.604959i	$0.206810\pi$
$338$	−10976.4	−1.76638
$339$	0	0
$340$	3315.96	0.528921
$341$	−1807.29	−0.287009
$342$	0	0
$343$	0	0
$344$	−1726.72	−0.270635
$345$	0	0
$346$	3763.11	0.584700
$347$	3583.86	0.554442	0.277221	−	0.960806i	$-0.410587\pi$
$347$	3583.86	0.554442	0.277221	+	0.960806i	$0.410587\pi$
$348$	0	0
$349$	9485.94	1.45493	0.727465	−	0.686145i	$-0.240698\pi$
$349$	9485.94	1.45493	0.727465	+	0.686145i	$0.240698\pi$
$350$	0	0
$351$	0	0
$352$	−4093.22	−0.619800
$353$	6507.74	0.981223	0.490612	−	0.871378i	$-0.336773\pi$
$353$	6507.74	0.981223	0.490612	+	0.871378i	$0.336773\pi$
$354$	0	0
$355$	−195.723	−0.0292616
$356$	5158.98	0.768049
$357$	0	0
$358$	1895.61	0.279849
$359$	4133.74	0.607718	0.303859	−	0.952717i	$-0.401725\pi$
$359$	4133.74	0.607718	0.303859	+	0.952717i	$0.401725\pi$
$360$	0	0
$361$	−939.709	−0.137004
$362$	−9646.99	−1.40065
$363$	0	0
$364$	0	0
$365$	2334.82	0.334823
$366$	0	0
$367$	3294.87	0.468640	0.234320	−	0.972160i	$-0.424714\pi$
$367$	3294.87	0.468640	0.234320	+	0.972160i	$0.424714\pi$
$368$	1766.75	0.250267
$369$	0	0
$370$	−842.267	−0.118344
$371$	0	0
$372$	0	0
$373$	2363.30	0.328062	0.164031	−	0.986455i	$-0.447550\pi$
$373$	2363.30	0.328062	0.164031	+	0.986455i	$0.447550\pi$
$374$	9672.17	1.33726
$375$	0	0
$376$	1673.60	0.229546
$377$	13976.6	1.90937
$378$	0	0
$379$	11071.4	1.50053	0.750266	−	0.661136i	$-0.229925\pi$
$379$	11071.4	1.50053	0.750266	+	0.661136i	$0.229925\pi$
$380$	1927.43	0.260197
$381$	0	0
$382$	−10581.9	−1.41732
$383$	7122.40	0.950228	0.475114	−	0.879924i	$-0.342407\pi$
$383$	7122.40	0.950228	0.475114	+	0.879924i	$0.342407\pi$
$384$	0	0
$385$	0	0
$386$	−6723.47	−0.886569
$387$	0	0
$388$	−7517.33	−0.983595
$389$	−259.070	−0.0337670	−0.0168835	−	0.999857i	$-0.505374\pi$
$389$	−259.070	−0.0337670	−0.0168835	+	0.999857i	$0.505374\pi$
$390$	0	0
$391$	−2974.42	−0.384714
$392$	0	0
$393$	0	0
$394$	10620.5	1.35800
$395$	−5204.61	−0.662968
$396$	0	0
$397$	−10222.8	−1.29236	−0.646181	−	0.763185i	$-0.723635\pi$
$397$	−10222.8	−1.29236	−0.646181	+	0.763185i	$0.723635\pi$
$398$	5913.84	0.744809
$399$	0	0
$400$	7984.64	0.998080
$401$	5164.36	0.643131	0.321566	−	0.946887i	$-0.395791\pi$
$401$	5164.36	0.643131	0.321566	+	0.946887i	$0.395791\pi$
$402$	0	0
$403$	6486.56	0.801783
$404$	−1157.62	−0.142559
$405$	0	0
$406$	0	0
$407$	−964.914	−0.117516
$408$	0	0
$409$	−4264.09	−0.515516	−0.257758	−	0.966210i	$-0.582984\pi$
$409$	−4264.09	−0.515516	−0.257758	+	0.966210i	$0.582984\pi$
$410$	−60.0918	−0.00723835
$411$	0	0
$412$	−2786.19	−0.333169
$413$	0	0
$414$	0	0
$415$	−3929.95	−0.464852
$416$	14691.0	1.73146
$417$	0	0
$418$	5622.02	0.657852
$419$	16613.9	1.93710	0.968548	−	0.248829i	$-0.0800456\pi$
$419$	16613.9	1.93710	0.968548	+	0.248829i	$0.0800456\pi$
$420$	0	0
$421$	−11821.1	−1.36847	−0.684235	−	0.729262i	$-0.739864\pi$
$421$	−11821.1	−1.36847	−0.684235	+	0.729262i	$0.739864\pi$
$422$	−6139.43	−0.708205
$423$	0	0
$424$	3460.21	0.396327
$425$	−13442.6	−1.53426
$426$	0	0
$427$	0	0
$428$	9821.10	1.10916
$429$	0	0
$430$	2958.61	0.331807
$431$	−13124.5	−1.46679	−0.733395	−	0.679803i	$-0.762065\pi$
$431$	−13124.5	−1.46679	−0.733395	+	0.679803i	$0.762065\pi$
$432$	0	0
$433$	−12962.7	−1.43868	−0.719341	−	0.694657i	$-0.755556\pi$
$433$	−12962.7	−1.43868	−0.719341	+	0.694657i	$0.755556\pi$
$434$	0	0
$435$	0	0
$436$	−4421.46	−0.485664
$437$	−1728.91	−0.189256
$438$	0	0
$439$	3842.58	0.417759	0.208880	−	0.977941i	$-0.433018\pi$
$439$	3842.58	0.417759	0.208880	+	0.977941i	$0.433018\pi$
$440$	−999.713	−0.108317
$441$	0	0
$442$	−34714.5	−3.73574
$443$	−5143.88	−0.551677	−0.275839	−	0.961204i	$-0.588956\pi$
$443$	−5143.88	−0.551677	−0.275839	+	0.961204i	$0.588956\pi$
$444$	0	0
$445$	4827.29	0.514237
$446$	2719.57	0.288734
$447$	0	0
$448$	0	0
$449$	−14627.7	−1.53747	−0.768733	−	0.639570i	$-0.779113\pi$
$449$	−14627.7	−1.53747	−0.768733	+	0.639570i	$0.779113\pi$
$450$	0	0
$451$	−68.8421	−0.00718769
$452$	−4150.24	−0.431882
$453$	0	0
$454$	17085.7	1.76624
$455$	0	0
$456$	0	0
$457$	−3104.00	−0.317722	−0.158861	−	0.987301i	$-0.550782\pi$
$457$	−3104.00	−0.317722	−0.158861	+	0.987301i	$0.550782\pi$
$458$	−8820.42	−0.899894
$459$	0	0
$460$	−562.963	−0.0570615
$461$	−13751.8	−1.38934	−0.694671	−	0.719327i	$-0.744450\pi$
$461$	−13751.8	−1.38934	−0.694671	+	0.719327i	$0.744450\pi$
$462$	0	0
$463$	5910.83	0.593303	0.296652	−	0.954986i	$-0.404130\pi$
$463$	5910.83	0.593303	0.296652	+	0.954986i	$0.404130\pi$
$464$	−15207.6	−1.52154
$465$	0	0
$466$	−6494.11	−0.645567
$467$	−9331.46	−0.924643	−0.462322	−	0.886712i	$-0.652983\pi$
$467$	−9331.46	−0.924643	−0.462322	+	0.886712i	$0.652983\pi$
$468$	0	0
$469$	0	0
$470$	−2867.59	−0.281430
$471$	0	0
$472$	5310.58	0.517880
$473$	3389.43	0.329484
$474$	0	0
$475$	−7813.61	−0.754764
$476$	0	0
$477$	0	0
$478$	−9099.35	−0.870700
$479$	−19656.5	−1.87501	−0.937504	−	0.347975i	$-0.886869\pi$
$479$	−19656.5	−1.87501	−0.937504	+	0.347975i	$0.886869\pi$
$480$	0	0
$481$	3463.18	0.328290
$482$	−14439.8	−1.36455
$483$	0	0
$484$	−4789.79	−0.449830
$485$	−7034.02	−0.658553
$486$	0	0
$487$	2272.06	0.211410	0.105705	−	0.994398i	$-0.466290\pi$
$487$	2272.06	0.211410	0.105705	+	0.994398i	$0.466290\pi$
$488$	4357.15	0.404178
$489$	0	0
$490$	0	0
$491$	−14877.7	−1.36746	−0.683730	−	0.729735i	$-0.739643\pi$
$491$	−14877.7	−1.36746	−0.683730	+	0.729735i	$0.739643\pi$
$492$	0	0
$493$	25602.8	2.33893
$494$	−20178.1	−1.83776
$495$	0	0
$496$	−7057.85	−0.638925
$497$	0	0
$498$	0	0
$499$	−16536.8	−1.48354	−0.741771	−	0.670653i	$-0.766014\pi$
$499$	−16536.8	−1.48354	−0.741771	+	0.670653i	$0.766014\pi$
$500$	−5675.76	−0.507655
$501$	0	0
$502$	10554.3	0.938368
$503$	7642.71	0.677478	0.338739	−	0.940880i	$-0.390000\pi$
$503$	7642.71	0.677478	0.338739	+	0.940880i	$0.390000\pi$
$504$	0	0
$505$	−1083.19	−0.0954484
$506$	−1642.08	−0.144268
$507$	0	0
$508$	−8918.88	−0.778959
$509$	−4706.47	−0.409844	−0.204922	−	0.978778i	$-0.565694\pi$
$509$	−4706.47	−0.409844	−0.204922	+	0.978778i	$0.565694\pi$
$510$	0	0
$511$	0	0
$512$	−9534.04	−0.822947
$513$	0	0
$514$	−13867.2	−1.19000
$515$	−2607.06	−0.223069
$516$	0	0
$517$	−3285.16	−0.279461
$518$	0	0
$519$	0	0
$520$	3588.08	0.302592
$521$	18520.7	1.55740	0.778701	−	0.627395i	$-0.215879\pi$
$521$	18520.7	1.55740	0.778701	+	0.627395i	$0.215879\pi$
$522$	0	0
$523$	−6993.69	−0.584728	−0.292364	−	0.956307i	$-0.594442\pi$
$523$	−6993.69	−0.584728	−0.292364	+	0.956307i	$0.594442\pi$
$524$	5272.33	0.439547
$525$	0	0
$526$	23303.9	1.93175
$527$	11882.3	0.982163
$528$	0	0
$529$	−11662.0	−0.958496
$530$	−5928.81	−0.485908
$531$	0	0
$532$	0	0
$533$	247.082	0.0200794
$534$	0	0
$535$	9189.66	0.742624
$536$	−10034.1	−0.808596
$537$	0	0
$538$	−4066.71	−0.325889
$539$	0	0
$540$	0	0
$541$	1378.85	0.109577	0.0547886	−	0.998498i	$-0.482552\pi$
$541$	1378.85	0.109577	0.0547886	+	0.998498i	$0.482552\pi$
$542$	8123.75	0.643810
$543$	0	0
$544$	26911.5	2.12099
$545$	−4137.19	−0.325170
$546$	0	0
$547$	10336.8	0.807985	0.403992	−	0.914762i	$-0.367622\pi$
$547$	10336.8	0.807985	0.403992	+	0.914762i	$0.367622\pi$
$548$	9511.49	0.741443
$549$	0	0
$550$	−7421.21	−0.575348
$551$	14881.8	1.15061
$552$	0	0
$553$	0	0
$554$	−6879.16	−0.527559
$555$	0	0
$556$	−7592.20	−0.579103
$557$	10149.6	0.772087	0.386043	−	0.922481i	$-0.373841\pi$
$557$	10149.6	0.772087	0.386043	+	0.922481i	$0.373841\pi$
$558$	0	0
$559$	−12165.0	−0.920440
$560$	0	0
$561$	0	0
$562$	−31298.0	−2.34915
$563$	−11168.3	−0.836034	−0.418017	−	0.908439i	$-0.637275\pi$
$563$	−11168.3	−0.836034	−0.418017	+	0.908439i	$0.637275\pi$
$564$	0	0
$565$	−3883.41	−0.289161
$566$	−11060.5	−0.821389
$567$	0	0
$568$	−414.610	−0.0306279
$569$	18776.6	1.38340	0.691700	−	0.722185i	$-0.256862\pi$
$569$	18776.6	1.38340	0.691700	+	0.722185i	$0.256862\pi$
$570$	0	0
$571$	−23499.3	−1.72227	−0.861133	−	0.508380i	$-0.830245\pi$
$571$	−23499.3	−1.72227	−0.861133	+	0.508380i	$0.830245\pi$
$572$	−7527.08	−0.550215
$573$	0	0
$574$	0	0
$575$	2282.20	0.165521
$576$	0	0
$577$	2612.83	0.188516	0.0942578	−	0.995548i	$-0.469952\pi$
$577$	2612.83	0.188516	0.0942578	+	0.995548i	$0.469952\pi$
$578$	−45758.6	−3.29292
$579$	0	0
$580$	4845.78	0.346914
$581$	0	0
$582$	0	0
$583$	−6792.14	−0.482507
$584$	4945.98	0.350456
$585$	0	0
$586$	−29154.3	−2.05521
$587$	18759.2	1.31904	0.659520	−	0.751687i	$-0.270759\pi$
$587$	18759.2	1.31904	0.659520	+	0.751687i	$0.270759\pi$
$588$	0	0
$589$	6906.67	0.483165
$590$	−9099.29	−0.634935
$591$	0	0
$592$	−3768.20	−0.261608
$593$	−8103.13	−0.561139	−0.280570	−	0.959834i	$-0.590523\pi$
$593$	−8103.13	−0.561139	−0.280570	+	0.959834i	$0.590523\pi$
$594$	0	0
$595$	0	0
$596$	−18162.5	−1.24826
$597$	0	0
$598$	5893.61	0.403023
$599$	−5552.86	−0.378771	−0.189385	−	0.981903i	$-0.560650\pi$
$599$	−5552.86	−0.378771	−0.189385	+	0.981903i	$0.560650\pi$
$600$	0	0
$601$	−21357.1	−1.44954	−0.724771	−	0.688990i	$-0.758054\pi$
$601$	−21357.1	−1.44954	−0.724771	+	0.688990i	$0.758054\pi$
$602$	0	0
$603$	0	0
$604$	16752.2	1.12854
$605$	−4481.84	−0.301178
$606$	0	0
$607$	5098.24	0.340908	0.170454	−	0.985366i	$-0.445477\pi$
$607$	5098.24	0.340908	0.170454	+	0.985366i	$0.445477\pi$
$608$	15642.5	1.04340
$609$	0	0
$610$	−7465.65	−0.495534
$611$	11790.8	0.780695
$612$	0	0
$613$	3247.67	0.213984	0.106992	−	0.994260i	$-0.465878\pi$
$613$	3247.67	0.213984	0.106992	+	0.994260i	$0.465878\pi$
$614$	−4963.54	−0.326241
$615$	0	0
$616$	0	0
$617$	−8366.93	−0.545932	−0.272966	−	0.962024i	$-0.588005\pi$
$617$	−8366.93	−0.545932	−0.272966	+	0.962024i	$0.588005\pi$
$618$	0	0
$619$	7396.28	0.480261	0.240130	−	0.970741i	$-0.422810\pi$
$619$	7396.28	0.480261	0.240130	+	0.970741i	$0.422810\pi$
$620$	2248.93	0.145676
$621$	0	0
$622$	4426.65	0.285357
$623$	0	0
$624$	0	0
$625$	7383.99	0.472575
$626$	17147.2	1.09479
$627$	0	0
$628$	14825.9	0.942066
$629$	6343.96	0.402147
$630$	0	0
$631$	−8786.48	−0.554333	−0.277167	−	0.960822i	$-0.589395\pi$
$631$	−8786.48	−0.554333	−0.277167	+	0.960822i	$0.589395\pi$
$632$	−11025.2	−0.693922
$633$	0	0
$634$	4778.38	0.299327
$635$	−8345.46	−0.521542
$636$	0	0
$637$	0	0
$638$	14134.4	0.877097
$639$	0	0
$640$	−5959.61	−0.368085
$641$	19571.4	1.20597	0.602984	−	0.797754i	$-0.293978\pi$
$641$	19571.4	1.20597	0.602984	+	0.797754i	$0.293978\pi$
$642$	0	0
$643$	−4317.58	−0.264804	−0.132402	−	0.991196i	$-0.542269\pi$
$643$	−4317.58	−0.264804	−0.132402	+	0.991196i	$0.542269\pi$
$644$	0	0
$645$	0	0
$646$	−36962.8	−2.25121
$647$	21785.5	1.32376	0.661882	−	0.749608i	$-0.269758\pi$
$647$	21785.5	1.32376	0.661882	+	0.749608i	$0.269758\pi$
$648$	0	0
$649$	−10424.3	−0.630492
$650$	26635.5	1.60728
$651$	0	0
$652$	−3170.37	−0.190431
$653$	15028.4	0.900620	0.450310	−	0.892872i	$-0.351313\pi$
$653$	15028.4	0.900620	0.450310	+	0.892872i	$0.351313\pi$
$654$	0	0
$655$	4933.35	0.294293
$656$	−268.843	−0.0160008
$657$	0	0
$658$	0	0
$659$	−12160.9	−0.718851	−0.359426	−	0.933174i	$-0.617027\pi$
$659$	−12160.9	−0.718851	−0.359426	+	0.933174i	$0.617027\pi$
$660$	0	0
$661$	9160.84	0.539055	0.269527	−	0.962993i	$-0.413133\pi$
$661$	9160.84	0.539055	0.269527	+	0.962993i	$0.413133\pi$
$662$	−27716.5	−1.62724
$663$	0	0
$664$	−8325.02	−0.486556
$665$	0	0
$666$	0	0
$667$	−4346.68	−0.252330
$668$	8454.30	0.489681
$669$	0	0
$670$	17192.7	0.991361
$671$	−8552.77	−0.492066
$672$	0	0
$673$	3430.71	0.196499	0.0982496	−	0.995162i	$-0.468676\pi$
$673$	3430.71	0.196499	0.0982496	+	0.995162i	$0.468676\pi$
$674$	−35759.5	−2.04363
$675$	0	0
$676$	15647.6	0.890281
$677$	−17001.5	−0.965169	−0.482585	−	0.875849i	$-0.660302\pi$
$677$	−17001.5	−0.965169	−0.482585	+	0.875849i	$0.660302\pi$
$678$	0	0
$679$	0	0
$680$	6572.75	0.370667
$681$	0	0
$682$	6559.82	0.368311
$683$	13444.5	0.753203	0.376602	−	0.926375i	$-0.377093\pi$
$683$	13444.5	0.753203	0.376602	+	0.926375i	$0.377093\pi$
$684$	0	0
$685$	8899.96	0.496424
$686$	0	0
$687$	0	0
$688$	13236.4	0.733480
$689$	24377.7	1.34792
$690$	0	0
$691$	31057.5	1.70982	0.854908	−	0.518780i	$-0.173614\pi$
$691$	31057.5	1.70982	0.854908	+	0.518780i	$0.173614\pi$
$692$	−5364.57	−0.294697
$693$	0	0
$694$	−13008.1	−0.711500
$695$	−7104.07	−0.387731
$696$	0	0
$697$	452.612	0.0245967
$698$	−34430.6	−1.86707
$699$	0	0
$700$	0	0
$701$	−2445.90	−0.131784	−0.0658919	−	0.997827i	$-0.520989\pi$
$701$	−2445.90	−0.131784	−0.0658919	+	0.997827i	$0.520989\pi$
$702$	0	0
$703$	3687.48	0.197832
$704$	2194.34	0.117475
$705$	0	0
$706$	−23620.8	−1.25918
$707$	0	0
$708$	0	0
$709$	−9660.67	−0.511726	−0.255863	−	0.966713i	$-0.582360\pi$
$709$	−9660.67	−0.511726	−0.255863	+	0.966713i	$0.582360\pi$
$710$	710.403	0.0375506
$711$	0	0
$712$	10225.9	0.538247
$713$	−2017.30	−0.105959
$714$	0	0
$715$	−7043.14	−0.368389
$716$	−2702.32	−0.141048
$717$	0	0
$718$	−15004.0	−0.779867
$719$	−1519.36	−0.0788074	−0.0394037	−	0.999223i	$-0.512546\pi$
$719$	−1519.36	−0.0788074	−0.0394037	+	0.999223i	$0.512546\pi$
$720$	0	0
$721$	0	0
$722$	3410.81	0.175813
$723$	0	0
$724$	13752.4	0.705946
$725$	−19644.3	−1.00631
$726$	0	0
$727$	19126.9	0.975760	0.487880	−	0.872911i	$-0.337770\pi$
$727$	19126.9	0.975760	0.487880	+	0.872911i	$0.337770\pi$
$728$	0	0
$729$	0	0
$730$	−8474.58	−0.429669
$731$	−22284.3	−1.12752
$732$	0	0
$733$	7231.51	0.364395	0.182198	−	0.983262i	$-0.441679\pi$
$733$	7231.51	0.364395	0.182198	+	0.983262i	$0.441679\pi$
$734$	−11959.2	−0.601392
$735$	0	0
$736$	−4568.86	−0.228819
$737$	19696.2	0.984423
$738$	0	0
$739$	17847.4	0.888400	0.444200	−	0.895928i	$-0.353488\pi$
$739$	17847.4	0.888400	0.444200	+	0.895928i	$0.353488\pi$
$740$	1200.71	0.0596472
$741$	0	0
$742$	0	0
$743$	26545.3	1.31070	0.655352	−	0.755323i	$-0.272520\pi$
$743$	26545.3	1.31070	0.655352	+	0.755323i	$0.272520\pi$
$744$	0	0
$745$	−16994.8	−0.835758
$746$	−8577.94	−0.420993
$747$	0	0
$748$	−13788.3	−0.673999
$749$	0	0
$750$	0	0
$751$	9560.13	0.464519	0.232260	−	0.972654i	$-0.425388\pi$
$751$	9560.13	0.464519	0.232260	+	0.972654i	$0.425388\pi$
$752$	−12829.2	−0.622120
$753$	0	0
$754$	−50730.1	−2.45024
$755$	15675.2	0.755601
$756$	0	0
$757$	−23995.4	−1.15209	−0.576043	−	0.817420i	$-0.695404\pi$
$757$	−23995.4	−1.15209	−0.576043	+	0.817420i	$0.695404\pi$
$758$	−40185.4	−1.92559
$759$	0	0
$760$	3820.46	0.182346
$761$	−33708.2	−1.60568	−0.802840	−	0.596195i	$-0.796679\pi$
$761$	−33708.2	−1.60568	−0.802840	+	0.596195i	$0.796679\pi$
$762$	0	0
$763$	0	0
$764$	15085.2	0.714352
$765$	0	0
$766$	−25851.7	−1.21940
$767$	37413.9	1.76133
$768$	0	0
$769$	−1030.27	−0.0483127	−0.0241563	−	0.999708i	$-0.507690\pi$
$769$	−1030.27	−0.0483127	−0.0241563	+	0.999708i	$0.507690\pi$
$770$	0	0
$771$	0	0
$772$	9584.76	0.446844
$773$	−22464.3	−1.04526	−0.522629	−	0.852560i	$-0.675049\pi$
$773$	−22464.3	−1.04526	−0.522629	+	0.852560i	$0.675049\pi$
$774$	0	0
$775$	−9116.97	−0.422569
$776$	−14900.5	−0.689301
$777$	0	0
$778$	940.331	0.0433322
$779$	263.084	0.0121001
$780$	0	0
$781$	813.849	0.0372879
$782$	10796.1	0.493692
$783$	0	0
$784$	0	0
$785$	13872.7	0.630748
$786$	0	0
$787$	−14149.4	−0.640878	−0.320439	−	0.947269i	$-0.603830\pi$
$787$	−14149.4	−0.640878	−0.320439	+	0.947269i	$0.603830\pi$
$788$	−15140.2	−0.684452
$789$	0	0
$790$	18890.9	0.850768
$791$	0	0
$792$	0	0
$793$	30696.8	1.37462
$794$	37105.1	1.65845
$795$	0	0
$796$	−8430.58	−0.375394
$797$	−38562.7	−1.71388	−0.856939	−	0.515418i	$-0.827637\pi$
$797$	−38562.7	−1.71388	−0.856939	+	0.515418i	$0.827637\pi$
$798$	0	0
$799$	21598.7	0.956331
$800$	−20648.5	−0.912543
$801$	0	0
$802$	−18744.8	−0.825313
$803$	−9708.61	−0.426662
$804$	0	0
$805$	0	0
$806$	−23543.9	−1.02891
$807$	0	0
$808$	−2294.58	−0.0999050
$809$	−27134.6	−1.17924	−0.589618	−	0.807682i	$-0.700722\pi$
$809$	−27134.6	−1.17924	−0.589618	+	0.807682i	$0.700722\pi$
$810$	0	0
$811$	23383.8	1.01247	0.506237	−	0.862394i	$-0.331036\pi$
$811$	23383.8	1.01247	0.506237	+	0.862394i	$0.331036\pi$
$812$	0	0
$813$	0	0
$814$	3502.29	0.150805
$815$	−2966.53	−0.127501
$816$	0	0
$817$	−12952.9	−0.554670
$818$	15477.1	0.661547
$819$	0	0
$820$	85.6649	0.00364823
$821$	−20243.9	−0.860558	−0.430279	−	0.902696i	$-0.641585\pi$
$821$	−20243.9	−0.860558	−0.430279	+	0.902696i	$0.641585\pi$
$822$	0	0
$823$	−21196.2	−0.897757	−0.448878	−	0.893593i	$-0.648176\pi$
$823$	−21196.2	−0.897757	−0.448878	+	0.893593i	$0.648176\pi$
$824$	−5522.67	−0.233484
$825$	0	0
$826$	0	0
$827$	−10826.0	−0.455207	−0.227604	−	0.973754i	$-0.573089\pi$
$827$	−10826.0	−0.455207	−0.227604	+	0.973754i	$0.573089\pi$
$828$	0	0
$829$	−9033.46	−0.378462	−0.189231	−	0.981933i	$-0.560600\pi$
$829$	−9033.46	−0.378462	−0.189231	+	0.981933i	$0.560600\pi$
$830$	14264.3	0.596532
$831$	0	0
$832$	−7875.74	−0.328176
$833$	0	0
$834$	0	0
$835$	7910.74	0.327859
$836$	−8014.58	−0.331567
$837$	0	0
$838$	−60302.6	−2.48582
$839$	43749.5	1.80024	0.900120	−	0.435642i	$-0.143478\pi$
$839$	43749.5	1.80024	0.900120	+	0.435642i	$0.143478\pi$
$840$	0	0
$841$	13025.6	0.534078
$842$	42906.4	1.75612
$843$	0	0
$844$	8752.17	0.356946
$845$	14641.5	0.596076
$846$	0	0
$847$	0	0
$848$	−26524.7	−1.07413
$849$	0	0
$850$	48791.8	1.96888
$851$	−1077.04	−0.0433848
$852$	0	0
$853$	39350.1	1.57951	0.789754	−	0.613423i	$-0.210208\pi$
$853$	39350.1	1.57951	0.789754	+	0.613423i	$0.210208\pi$
$854$	0	0
$855$	0	0
$856$	19467.0	0.777298
$857$	15285.5	0.609268	0.304634	−	0.952470i	$-0.401466\pi$
$857$	15285.5	0.609268	0.304634	+	0.952470i	$0.401466\pi$
$858$	0	0
$859$	−8166.69	−0.324382	−0.162191	−	0.986759i	$-0.551856\pi$
$859$	−8166.69	−0.324382	−0.162191	+	0.986759i	$0.551856\pi$
$860$	−4217.70	−0.167235
$861$	0	0
$862$	47637.3	1.88229
$863$	49204.1	1.94082	0.970409	−	0.241467i	$-0.0776284\pi$
$863$	49204.1	1.94082	0.970409	+	0.241467i	$0.0776284\pi$
$864$	0	0
$865$	−5019.67	−0.197311
$866$	47050.1	1.84622
$867$	0	0
$868$	0	0
$869$	21641.7	0.844814
$870$	0	0
$871$	−70691.9	−2.75006
$872$	−8764.02	−0.340352
$873$	0	0
$874$	6275.31	0.242867
$875$	0	0
$876$	0	0
$877$	−15821.4	−0.609180	−0.304590	−	0.952484i	$-0.598519\pi$
$877$	−15821.4	−0.609180	−0.304590	+	0.952484i	$0.598519\pi$
$878$	−13947.2	−0.536099
$879$	0	0
$880$	7663.45	0.293562
$881$	14538.7	0.555982	0.277991	−	0.960584i	$-0.410331\pi$
$881$	14538.7	0.555982	0.277991	+	0.960584i	$0.410331\pi$
$882$	0	0
$883$	3842.17	0.146432	0.0732160	−	0.997316i	$-0.476674\pi$
$883$	3842.17	0.146432	0.0732160	+	0.997316i	$0.476674\pi$
$884$	49487.8	1.88287
$885$	0	0
$886$	18670.4	0.707952
$887$	37771.5	1.42981	0.714906	−	0.699221i	$-0.246470\pi$
$887$	37771.5	1.42981	0.714906	+	0.699221i	$0.246470\pi$
$888$	0	0
$889$	0	0
$890$	−17521.3	−0.659906
$891$	0	0
$892$	−3876.93	−0.145526
$893$	12554.4	0.470457
$894$	0	0
$895$	−2528.58	−0.0944370
$896$	0	0
$897$	0	0
$898$	53093.2	1.97299
$899$	17364.2	0.644191
$900$	0	0
$901$	44655.9	1.65117
$902$	249.872	0.00922376
$903$	0	0
$904$	−8226.42	−0.302662
$905$	12868.2	0.472657
$906$	0	0
$907$	25275.6	0.925316	0.462658	−	0.886537i	$-0.346896\pi$
$907$	25275.6	0.925316	0.462658	+	0.886537i	$0.346896\pi$
$908$	−24356.9	−0.890211
$909$	0	0
$910$	0	0
$911$	−33060.3	−1.20234	−0.601172	−	0.799120i	$-0.705299\pi$
$911$	−33060.3	−1.20234	−0.601172	+	0.799120i	$0.705299\pi$
$912$	0	0
$913$	16341.4	0.592357
$914$	11266.4	0.407724
$915$	0	0
$916$	12574.1	0.453559
$917$	0	0
$918$	0	0
$919$	−17305.1	−0.621158	−0.310579	−	0.950548i	$-0.600523\pi$
$919$	−17305.1	−0.621158	−0.310579	+	0.950548i	$0.600523\pi$
$920$	−1115.88	−0.0399886
$921$	0	0
$922$	49914.2	1.78290
$923$	−2920.99	−0.104166
$924$	0	0
$925$	−4867.56	−0.173021
$926$	−21454.2	−0.761370
$927$	0	0
$928$	39327.1	1.39114
$929$	26353.3	0.930703	0.465352	−	0.885126i	$-0.345928\pi$
$929$	26353.3	0.930703	0.465352	+	0.885126i	$0.345928\pi$
$930$	0	0
$931$	0	0
$932$	9257.80	0.325375
$933$	0	0
$934$	33869.8	1.18657
$935$	−12901.8	−0.451267
$936$	0	0
$937$	−6967.42	−0.242920	−0.121460	−	0.992596i	$-0.538758\pi$
$937$	−6967.42	−0.242920	−0.121460	+	0.992596i	$0.538758\pi$
$938$	0	0
$939$	0	0
$940$	4087.95	0.141845
$941$	−44284.2	−1.53414	−0.767069	−	0.641565i	$-0.778285\pi$
$941$	−44284.2	−1.53414	−0.767069	+	0.641565i	$0.778285\pi$
$942$	0	0
$943$	−76.8417	−0.00265356
$944$	−40709.1	−1.40357
$945$	0	0
$946$	−12302.4	−0.422818
$947$	−29772.8	−1.02163	−0.510816	−	0.859690i	$-0.670657\pi$
$947$	−29772.8	−1.02163	−0.510816	+	0.859690i	$0.670657\pi$
$948$	0	0
$949$	34845.3	1.19191
$950$	28360.6	0.968568
$951$	0	0
$952$	0	0
$953$	17971.9	0.610878	0.305439	−	0.952212i	$-0.401197\pi$
$953$	17971.9	0.610878	0.305439	+	0.952212i	$0.401197\pi$
$954$	0	0
$955$	14115.4	0.478285
$956$	12971.7	0.438845
$957$	0	0
$958$	71346.1	2.40614
$959$	0	0
$960$	0	0
$961$	−21732.3	−0.729491
$962$	−12570.1	−0.421286
$963$	0	0
$964$	20584.9	0.687754
$965$	8968.53	0.299178
$966$	0	0
$967$	−34100.9	−1.13403	−0.567017	−	0.823706i	$-0.691903\pi$
$967$	−34100.9	−1.13403	−0.567017	+	0.823706i	$0.691903\pi$
$968$	−9494.12	−0.315240
$969$	0	0
$970$	25531.0	0.845103
$971$	22606.2	0.747134	0.373567	−	0.927603i	$-0.378135\pi$
$971$	22606.2	0.747134	0.373567	+	0.927603i	$0.378135\pi$
$972$	0	0
$973$	0	0
$974$	−8246.77	−0.271297
$975$	0	0
$976$	−33400.4	−1.09541
$977$	−8740.62	−0.286220	−0.143110	−	0.989707i	$-0.545710\pi$
$977$	−8740.62	−0.286220	−0.143110	+	0.989707i	$0.545710\pi$
$978$	0	0
$979$	−20072.7	−0.655288
$980$	0	0
$981$	0	0
$982$	54000.8	1.75482
$983$	−44704.8	−1.45052	−0.725260	−	0.688475i	$-0.758281\pi$
$983$	−44704.8	−1.45052	−0.725260	+	0.688475i	$0.758281\pi$
$984$	0	0
$985$	−14166.8	−0.458266
$986$	−92928.8	−3.00148
$987$	0	0
$988$	28765.2	0.926258
$989$	3783.29	0.121640
$990$	0	0
$991$	17035.5	0.546064	0.273032	−	0.962005i	$-0.411973\pi$
$991$	17035.5	0.546064	0.273032	+	0.962005i	$0.411973\pi$
$992$	18251.8	0.584168
$993$	0	0
$994$	0	0
$995$	−7888.55	−0.251341
$996$	0	0
$997$	−29071.6	−0.923478	−0.461739	−	0.887016i	$-0.652774\pi$
$997$	−29071.6	−0.923478	−0.461739	+	0.887016i	$0.652774\pi$
$998$	60022.6	1.90379
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bj.1.2		7
3.2	odd	2		1323.4.a.bi.1.6		7
7.2	even	3		189.4.e.f.109.6	✓	14
7.4	even	3		189.4.e.f.163.6	yes	14
7.6	odd	2		1323.4.a.bk.1.2		7
21.2	odd	6		189.4.e.g.109.2	yes	14
21.11	odd	6		189.4.e.g.163.2	yes	14
21.20	even	2		1323.4.a.bh.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.6	✓	14	7.2	even	3
189.4.e.f.163.6	yes	14	7.4	even	3
189.4.e.g.109.2	yes	14	21.2	odd	6
189.4.e.g.163.2	yes	14	21.11	odd	6
1323.4.a.bh.1.6		7	21.20	even	2
1323.4.a.bi.1.6		7	3.2	odd	2
1323.4.a.bj.1.2		7	1.1	even	1	trivial
1323.4.a.bk.1.2		7	7.6	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-3.62964 q^{2} +5.17430 q^{4} +4.84163 q^{5} +10.2563 q^{8} +O(q^{10})\)	\(q-3.62964 q^{2} +5.17430 q^{4} +4.84163 q^{5} +10.2563 q^{8} -17.5734 q^{10} -20.1323 q^{11} +72.2571 q^{13} -78.6210 q^{16} +132.363 q^{17} +76.9369 q^{19} +25.0520 q^{20} +73.0732 q^{22} -22.4717 q^{23} -101.559 q^{25} -262.268 q^{26} +193.429 q^{29} +89.7705 q^{31} +203.316 q^{32} -480.430 q^{34} +47.9286 q^{37} -279.254 q^{38} +49.6571 q^{40} +3.41948 q^{41} -168.358 q^{43} -104.171 q^{44} +81.5644 q^{46} +163.178 q^{47} +368.622 q^{50} +373.880 q^{52} +337.375 q^{53} -97.4733 q^{55} -702.077 q^{58} +517.788 q^{59} +424.828 q^{61} -325.835 q^{62} -108.996 q^{64} +349.842 q^{65} -978.338 q^{67} +684.885 q^{68} -40.4250 q^{71} +482.240 q^{73} -173.964 q^{74} +398.095 q^{76} -1074.97 q^{79} -380.654 q^{80} -12.4115 q^{82} -811.700 q^{83} +640.851 q^{85} +611.078 q^{86} -206.483 q^{88} +997.039 q^{89} -116.276 q^{92} -592.279 q^{94} +372.500 q^{95} -1452.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8}+O(q^{10}) \) 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8	\( 7 q + q^{2} + 31 q^{4} - q^{5} + 84 q^{8} + 12 q^{10} + 98 q^{11} - 124 q^{13} + 139 q^{16} + 30 q^{17} + 182 q^{19} - 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} - 245 q^{26} + 323 q^{29} + 26 q^{31} + 398 q^{32} + 114 q^{34} - 112 q^{37} + 1015 q^{38} - 147 q^{40} - 524 q^{41} + 8 q^{43} + 937 q^{44} - 339 q^{46} + 288 q^{47} + 2576 q^{50} - 1075 q^{52} + 1353 q^{53} + 156 q^{55} - 81 q^{58} + 165 q^{59} + 56 q^{61} - 1215 q^{62} - 1706 q^{64} + 1694 q^{65} - 988 q^{67} + 2625 q^{68} + 792 q^{71} + 1487 q^{73} + 2736 q^{74} + 1952 q^{76} - 1273 q^{79} - 2501 q^{80} - 2049 q^{82} - 1170 q^{83} + 216 q^{85} - 160 q^{86} + 9 q^{88} + 1058 q^{89} + 3834 q^{92} + 1653 q^{94} + 3260 q^{95} - 3730 q^{97}+O(q^{100}) \) 7 * q + q^2 + 31 * q^4 - q^5 + 84 * q^8 + 12 * q^10 + 98 * q^11 - 124 * q^13 + 139 * q^16 + 30 * q^17 + 182 * q^19 - 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 - 245 * q^26 + 323 * q^29 + 26 * q^31 + 398 * q^32 + 114 * q^34 - 112 * q^37 + 1015 * q^38 - 147 * q^40 - 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 + 288 * q^47 + 2576 * q^50 - 1075 * q^52 + 1353 * q^53 + 156 * q^55 - 81 * q^58 + 165 * q^59 + 56 * q^61 - 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 + 2625 * q^68 + 792 * q^71 + 1487 * q^73 + 2736 * q^74 + 1952 * q^76 - 1273 * q^79 - 2501 * q^80 - 2049 * q^82 - 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 + 1058 * q^89 + 3834 * q^92 + 1653 * q^94 + 3260 * q^95 - 3730 * q^97

Introduction

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