LMFDB - Embedded newform 1323.4.a.bh.1.5

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:	$1$
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			1.5
Root			$-1.63185$ 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)

$f(q)$	$=$	$q+1.63185 q^{2} -5.33705 q^{4} +12.3790 q^{5} -21.7641 q^{8} +O(q^{10})$	$q+1.63185 q^{2} -5.33705 q^{4} +12.3790 q^{5} -21.7641 q^{8} +20.2007 q^{10} +29.0051 q^{11} +52.9884 q^{13} +7.18051 q^{16} -122.047 q^{17} -141.105 q^{19} -66.0672 q^{20} +47.3321 q^{22} -60.2992 q^{23} +28.2389 q^{25} +86.4693 q^{26} +126.997 q^{29} -150.849 q^{31} +185.831 q^{32} -199.163 q^{34} -341.581 q^{37} -230.262 q^{38} -269.418 q^{40} +292.082 q^{41} +290.696 q^{43} -154.802 q^{44} -98.3995 q^{46} +284.082 q^{47} +46.0818 q^{50} -282.802 q^{52} -387.993 q^{53} +359.053 q^{55} +207.241 q^{58} +269.312 q^{59} +239.606 q^{61} -246.163 q^{62} +245.804 q^{64} +655.941 q^{65} -712.250 q^{67} +651.370 q^{68} -270.507 q^{71} -146.083 q^{73} -557.411 q^{74} +753.083 q^{76} -652.792 q^{79} +88.8874 q^{80} +476.636 q^{82} -35.0239 q^{83} -1510.81 q^{85} +474.373 q^{86} -631.271 q^{88} -1394.56 q^{89} +321.820 q^{92} +463.581 q^{94} -1746.73 q^{95} -805.822 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.63185	0.576948	0.288474	−	0.957488i	$-0.406852\pi$
$2$	1.63185	0.576948	0.288474	+	0.957488i	$0.406852\pi$
$3$	0	0
$3$	0	0
$4$	−5.33705	−0.667131
$5$	12.3790	1.10721	0.553604	−	0.832780i	$-0.313252\pi$
$5$	12.3790	1.10721	0.553604	+	0.832780i	$0.313252\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−21.7641	−0.961848
$9$	0	0
$10$	20.2007	0.638802
$11$	29.0051	0.795033	0.397517	−	0.917595i	$-0.369872\pi$
$11$	29.0051	0.795033	0.397517	+	0.917595i	$0.369872\pi$
$12$	0	0
$13$	52.9884	1.13049	0.565243	−	0.824924i	$-0.308782\pi$
$13$	52.9884	1.13049	0.565243	+	0.824924i	$0.308782\pi$
$14$	0	0
$15$	0	0
$16$	7.18051	0.112196
$17$	−122.047	−1.74122	−0.870609	−	0.491975i	$-0.836275\pi$
$17$	−122.047	−1.74122	−0.870609	+	0.491975i	$0.836275\pi$
$18$	0	0
$19$	−141.105	−1.70377	−0.851885	−	0.523728i	$-0.824541\pi$
$19$	−141.105	−1.70377	−0.851885	+	0.523728i	$0.824541\pi$
$20$	−66.0672	−0.738654
$21$	0	0
$22$	47.3321	0.458693
$23$	−60.2992	−0.546663	−0.273332	−	0.961920i	$-0.588126\pi$
$23$	−60.2992	−0.546663	−0.273332	+	0.961920i	$0.588126\pi$
$24$	0	0
$25$	28.2389	0.225912
$26$	86.4693	0.652232
$27$	0	0
$28$	0	0
$29$	126.997	0.813201	0.406601	−	0.913606i	$-0.366714\pi$
$29$	126.997	0.813201	0.406601	+	0.913606i	$0.366714\pi$
$30$	0	0
$31$	−150.849	−0.873975	−0.436988	−	0.899468i	$-0.643955\pi$
$31$	−150.849	−0.873975	−0.436988	+	0.899468i	$0.643955\pi$
$32$	185.831	1.02658
$33$	0	0
$34$	−199.163	−1.00459
$35$	0	0
$36$	0	0
$37$	−341.581	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$37$	−341.581	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$38$	−230.262	−0.982987
$39$	0	0
$40$	−269.418	−1.06497
$41$	292.082	1.11258	0.556288	−	0.830990i	$-0.312225\pi$
$41$	292.082	1.11258	0.556288	+	0.830990i	$0.312225\pi$
$42$	0	0
$43$	290.696	1.03095	0.515473	−	0.856906i	$-0.327616\pi$
$43$	290.696	1.03095	0.515473	+	0.856906i	$0.327616\pi$
$44$	−154.802	−0.530392
$45$	0	0
$46$	−98.3995	−0.315396
$47$	284.082	0.881652	0.440826	−	0.897593i	$-0.354686\pi$
$47$	284.082	0.881652	0.440826	+	0.897593i	$0.354686\pi$
$48$	0	0
$49$	0	0
$50$	46.0818	0.130339
$51$	0	0
$52$	−282.802	−0.754183
$53$	−387.993	−1.00556	−0.502782	−	0.864413i	$-0.667690\pi$
$53$	−387.993	−1.00556	−0.502782	+	0.864413i	$0.667690\pi$
$54$	0	0
$55$	359.053	0.880268
$56$	0	0
$57$	0	0
$58$	207.241	0.469175
$59$	269.312	0.594261	0.297131	−	0.954837i	$-0.403970\pi$
$59$	269.312	0.594261	0.297131	+	0.954837i	$0.403970\pi$
$60$	0	0
$61$	239.606	0.502925	0.251462	−	0.967867i	$-0.419089\pi$
$61$	239.606	0.502925	0.251462	+	0.967867i	$0.419089\pi$
$62$	−246.163	−0.504238
$63$	0	0
$64$	245.804	0.480087
$65$	655.941	1.25168
$66$	0	0
$67$	−712.250	−1.29873	−0.649367	−	0.760475i	$-0.724966\pi$
$67$	−712.250	−1.29873	−0.649367	+	0.760475i	$0.724966\pi$
$68$	651.370	1.16162
$69$	0	0
$70$	0	0
$71$	−270.507	−0.452160	−0.226080	−	0.974109i	$-0.572591\pi$
$71$	−270.507	−0.452160	−0.226080	+	0.974109i	$0.572591\pi$
$72$	0	0
$73$	−146.083	−0.234215	−0.117108	−	0.993119i	$-0.537362\pi$
$73$	−146.083	−0.234215	−0.117108	+	0.993119i	$0.537362\pi$
$74$	−557.411	−0.875645
$75$	0	0
$76$	753.083	1.13664
$77$	0	0
$78$	0	0
$79$	−652.792	−0.929681	−0.464841	−	0.885394i	$-0.653888\pi$
$79$	−652.792	−0.929681	−0.464841	+	0.885394i	$0.653888\pi$
$80$	88.8874	0.124224
$81$	0	0
$82$	476.636	0.641898
$83$	−35.0239	−0.0463178	−0.0231589	−	0.999732i	$-0.507372\pi$
$83$	−35.0239	−0.0463178	−0.0231589	+	0.999732i	$0.507372\pi$
$84$	0	0
$85$	−1510.81	−1.92789
$86$	474.373	0.594802
$87$	0	0
$88$	−631.271	−0.764701
$89$	−1394.56	−1.66093	−0.830465	−	0.557071i	$-0.811925\pi$
$89$	−1394.56	−1.66093	−0.830465	+	0.557071i	$0.811925\pi$
$90$	0	0
$91$	0	0
$92$	321.820	0.364696
$93$	0	0
$94$	463.581	0.508667
$95$	−1746.73	−1.88643
$96$	0	0
$97$	−805.822	−0.843493	−0.421746	−	0.906714i	$-0.638583\pi$
$97$	−805.822	−0.843493	−0.421746	+	0.906714i	$0.638583\pi$
$98$	0	0
$99$	0	0
$100$	−150.713	−0.150713
$101$	−1014.81	−0.999778	−0.499889	−	0.866089i	$-0.666626\pi$
$101$	−1014.81	−0.999778	−0.499889	+	0.866089i	$0.666626\pi$
$102$	0	0
$103$	−1039.39	−0.994310	−0.497155	−	0.867662i	$-0.665622\pi$
$103$	−1039.39	−0.994310	−0.497155	+	0.867662i	$0.665622\pi$
$104$	−1153.25	−1.08736
$105$	0	0
$106$	−633.147	−0.580158
$107$	−608.338	−0.549628	−0.274814	−	0.961497i	$-0.588616\pi$
$107$	−608.338	−0.549628	−0.274814	+	0.961497i	$0.588616\pi$
$108$	0	0
$109$	1077.71	0.947029	0.473514	−	0.880786i	$-0.342985\pi$
$109$	1077.71	0.947029	0.473514	+	0.880786i	$0.342985\pi$
$110$	585.923	0.507869
$111$	0	0
$112$	0	0
$113$	−2064.78	−1.71893	−0.859463	−	0.511199i	$-0.829202\pi$
$113$	−2064.78	−1.71893	−0.859463	+	0.511199i	$0.829202\pi$
$114$	0	0
$115$	−746.442	−0.605270
$116$	−677.792	−0.542512
$117$	0	0
$118$	439.478	0.342858
$119$	0	0
$120$	0	0
$121$	−489.704	−0.367922
$122$	391.002	0.290161
$123$	0	0
$124$	805.087	0.583056
$125$	−1197.80	−0.857078
$126$	0	0
$127$	−130.430	−0.0911320	−0.0455660	−	0.998961i	$-0.514509\pi$
$127$	−130.430	−0.0911320	−0.0455660	+	0.998961i	$0.514509\pi$
$128$	−1085.53	−0.749594
$129$	0	0
$130$	1070.40	0.722157
$131$	1905.07	1.27058	0.635292	−	0.772272i	$-0.280880\pi$
$131$	1905.07	1.27058	0.635292	+	0.772272i	$0.280880\pi$
$132$	0	0
$133$	0	0
$134$	−1162.29	−0.749302
$135$	0	0
$136$	2656.24	1.67479
$137$	559.298	0.348789	0.174394	−	0.984676i	$-0.444203\pi$
$137$	559.298	0.348789	0.174394	+	0.984676i	$0.444203\pi$
$138$	0	0
$139$	699.079	0.426583	0.213292	−	0.976989i	$-0.431582\pi$
$139$	699.079	0.426583	0.213292	+	0.976989i	$0.431582\pi$
$140$	0	0
$141$	0	0
$142$	−441.429	−0.260872
$143$	1536.93	0.898775
$144$	0	0
$145$	1572.10	0.900384
$146$	−238.386	−0.135130
$147$	0	0
$148$	1823.04	1.01252
$149$	−1614.90	−0.887904	−0.443952	−	0.896050i	$-0.646424\pi$
$149$	−1614.90	−0.887904	−0.443952	+	0.896050i	$0.646424\pi$
$150$	0	0
$151$	−3.59271	−0.00193623	−0.000968116	−	1.00000i	$-0.500308\pi$
$151$	−3.59271	−0.00193623	−0.000968116		1.00000i	$0.500308\pi$
$152$	3071.02	1.63877
$153$	0	0
$154$	0	0
$155$	−1867.35	−0.967673
$156$	0	0
$157$	−697.746	−0.354689	−0.177345	−	0.984149i	$-0.556751\pi$
$157$	−697.746	−0.354689	−0.177345	+	0.984149i	$0.556751\pi$
$158$	−1065.26	−0.536377
$159$	0	0
$160$	2300.39	1.13664
$161$	0	0
$162$	0	0
$163$	145.908	0.0701130	0.0350565	−	0.999385i	$-0.488839\pi$
$163$	145.908	0.0701130	0.0350565	+	0.999385i	$0.488839\pi$
$164$	−1558.86	−0.742234
$165$	0	0
$166$	−57.1539	−0.0267229
$167$	1816.18	0.841556	0.420778	−	0.907164i	$-0.361757\pi$
$167$	1816.18	0.841556	0.420778	+	0.907164i	$0.361757\pi$
$168$	0	0
$169$	610.766	0.278000
$170$	−2465.43	−1.11229
$171$	0	0
$172$	−1551.46	−0.687776
$173$	2424.68	1.06558	0.532790	−	0.846247i	$-0.321144\pi$
$173$	2424.68	1.06558	0.532790	+	0.846247i	$0.321144\pi$
$174$	0	0
$175$	0	0
$176$	208.272	0.0891992
$177$	0	0
$178$	−2275.71	−0.958270
$179$	−2835.36	−1.18394	−0.591969	−	0.805961i	$-0.701649\pi$
$179$	−2835.36	−1.18394	−0.591969	+	0.805961i	$0.701649\pi$
$180$	0	0
$181$	219.212	0.0900214	0.0450107	−	0.998987i	$-0.485668\pi$
$181$	219.212	0.0900214	0.0450107	+	0.998987i	$0.485668\pi$
$182$	0	0
$183$	0	0
$184$	1312.36	0.525807
$185$	−4228.43	−1.68043
$186$	0	0
$187$	−3539.98	−1.38433
$188$	−1516.16	−0.588178
$189$	0	0
$190$	−2850.41	−1.08837
$191$	−223.424	−0.0846410	−0.0423205	−	0.999104i	$-0.513475\pi$
$191$	−223.424	−0.0846410	−0.0423205	+	0.999104i	$0.513475\pi$
$192$	0	0
$193$	−352.158	−0.131342	−0.0656708	−	0.997841i	$-0.520919\pi$
$193$	−352.158	−0.131342	−0.0656708	+	0.997841i	$0.520919\pi$
$194$	−1314.98	−0.486651
$195$	0	0
$196$	0	0
$197$	−5347.45	−1.93396	−0.966981	−	0.254850i	$-0.917974\pi$
$197$	−5347.45	−1.93396	−0.966981	+	0.254850i	$0.917974\pi$
$198$	0	0
$199$	−582.228	−0.207402	−0.103701	−	0.994609i	$-0.533069\pi$
$199$	−582.228	−0.207402	−0.103701	+	0.994609i	$0.533069\pi$
$200$	−614.596	−0.217292
$201$	0	0
$202$	−1656.03	−0.576820
$203$	0	0
$204$	0	0
$205$	3615.68	1.23185
$206$	−1696.13	−0.573665
$207$	0	0
$208$	380.484	0.126836
$209$	−4092.76	−1.35455
$210$	0	0
$211$	5467.96	1.78403	0.892014	−	0.452008i	$-0.149292\pi$
$211$	5467.96	1.78403	0.892014	+	0.452008i	$0.149292\pi$
$212$	2070.74	0.670843
$213$	0	0
$214$	−992.719	−0.317107
$215$	3598.51	1.14147
$216$	0	0
$217$	0	0
$218$	1758.67	0.546386
$219$	0	0
$220$	−1916.29	−0.587254
$221$	−6467.06	−1.96842
$222$	0	0
$223$	−4368.05	−1.31169	−0.655844	−	0.754897i	$-0.727687\pi$
$223$	−4368.05	−1.31169	−0.655844	+	0.754897i	$0.727687\pi$
$224$	0	0
$225$	0	0
$226$	−3369.43	−0.991730
$227$	636.952	0.186238	0.0931189	−	0.995655i	$-0.470316\pi$
$227$	636.952	0.186238	0.0931189	+	0.995655i	$0.470316\pi$
$228$	0	0
$229$	−983.495	−0.283804	−0.141902	−	0.989881i	$-0.545322\pi$
$229$	−983.495	−0.283804	−0.141902	+	0.989881i	$0.545322\pi$
$230$	−1218.09	−0.349209
$231$	0	0
$232$	−2763.99	−0.782176
$233$	1677.37	0.471623	0.235811	−	0.971799i	$-0.424225\pi$
$233$	1677.37	0.471623	0.235811	+	0.971799i	$0.424225\pi$
$234$	0	0
$235$	3516.64	0.976173
$236$	−1437.33	−0.396450
$237$	0	0
$238$	0	0
$239$	2705.18	0.732148	0.366074	−	0.930586i	$-0.380702\pi$
$239$	2705.18	0.732148	0.366074	+	0.930586i	$0.380702\pi$
$240$	0	0
$241$	2140.73	0.572185	0.286092	−	0.958202i	$-0.407644\pi$
$241$	2140.73	0.572185	0.286092	+	0.958202i	$0.407644\pi$
$242$	−799.126	−0.212272
$243$	0	0
$244$	−1278.79	−0.335517
$245$	0	0
$246$	0	0
$247$	−7476.91	−1.92609
$248$	3283.09	0.840631
$249$	0	0
$250$	−1954.64	−0.494489
$251$	918.484	0.230973	0.115486	−	0.993309i	$-0.463157\pi$
$251$	918.484	0.230973	0.115486	+	0.993309i	$0.463157\pi$
$252$	0	0
$253$	−1748.98	−0.434616
$254$	−212.842	−0.0525784
$255$	0	0
$256$	−3737.86	−0.912563
$257$	4232.18	1.02722	0.513611	−	0.858023i	$-0.328308\pi$
$257$	4232.18	1.02722	0.513611	+	0.858023i	$0.328308\pi$
$258$	0	0
$259$	0	0
$260$	−3500.79	−0.835038
$261$	0	0
$262$	3108.79	0.733060
$263$	−5825.05	−1.36573	−0.682867	−	0.730542i	$-0.739267\pi$
$263$	−5825.05	−1.36573	−0.682867	+	0.730542i	$0.739267\pi$
$264$	0	0
$265$	−4802.95	−1.11337
$266$	0	0
$267$	0	0
$268$	3801.31	0.866426
$269$	−2592.23	−0.587550	−0.293775	−	0.955875i	$-0.594912\pi$
$269$	−2592.23	−0.587550	−0.293775	+	0.955875i	$0.594912\pi$
$270$	0	0
$271$	3709.89	0.831586	0.415793	−	0.909459i	$-0.363504\pi$
$271$	3709.89	0.831586	0.415793	+	0.909459i	$0.363504\pi$
$272$	−876.359	−0.195357
$273$	0	0
$274$	912.694	0.201233
$275$	819.073	0.179607
$276$	0	0
$277$	−323.380	−0.0701445	−0.0350722	−	0.999385i	$-0.511166\pi$
$277$	−323.380	−0.0701445	−0.0350722	+	0.999385i	$0.511166\pi$
$278$	1140.79	0.246116
$279$	0	0
$280$	0	0
$281$	−4990.53	−1.05947	−0.529733	−	0.848165i	$-0.677708\pi$
$281$	−4990.53	−1.05947	−0.529733	+	0.848165i	$0.677708\pi$
$282$	0	0
$283$	−2458.67	−0.516441	−0.258221	−	0.966086i	$-0.583136\pi$
$283$	−2458.67	−0.516441	−0.258221	+	0.966086i	$0.583136\pi$
$284$	1443.71	0.301650
$285$	0	0
$286$	2508.05	0.518546
$287$	0	0
$288$	0	0
$289$	9982.43	2.03184
$290$	2565.44	0.519474
$291$	0	0
$292$	779.651	0.156252
$293$	3028.06	0.603758	0.301879	−	0.953346i	$-0.402386\pi$
$293$	3028.06	0.603758	0.301879	+	0.953346i	$0.402386\pi$
$294$	0	0
$295$	3333.80	0.657972
$296$	7434.22	1.45982
$297$	0	0
$298$	−2635.28	−0.512274
$299$	−3195.16	−0.617995
$300$	0	0
$301$	0	0
$302$	−5.86279	−0.00111710
$303$	0	0
$304$	−1013.20	−0.191155
$305$	2966.08	0.556843
$306$	0	0
$307$	8277.86	1.53890	0.769450	−	0.638707i	$-0.220530\pi$
$307$	8277.86	1.53890	0.769450	+	0.638707i	$0.220530\pi$
$308$	0	0
$309$	0	0
$310$	−3047.25	−0.558297
$311$	824.071	0.150253	0.0751267	−	0.997174i	$-0.476064\pi$
$311$	824.071	0.150253	0.0751267	+	0.997174i	$0.476064\pi$
$312$	0	0
$313$	2427.80	0.438426	0.219213	−	0.975677i	$-0.429651\pi$
$313$	2427.80	0.438426	0.219213	+	0.975677i	$0.429651\pi$
$314$	−1138.62	−0.204637
$315$	0	0
$316$	3483.98	0.620219
$317$	−7970.35	−1.41218	−0.706088	−	0.708125i	$-0.749542\pi$
$317$	−7970.35	−1.41218	−0.706088	+	0.708125i	$0.749542\pi$
$318$	0	0
$319$	3683.58	0.646522
$320$	3042.81	0.531556
$321$	0	0
$322$	0	0
$323$	17221.4	2.96664
$324$	0	0
$325$	1496.34	0.255390
$326$	238.101	0.0404515
$327$	0	0
$328$	−6356.92	−1.07013
$329$	0	0
$330$	0	0
$331$	2273.66	0.377558	0.188779	−	0.982020i	$-0.439547\pi$
$331$	2273.66	0.377558	0.188779	+	0.982020i	$0.439547\pi$
$332$	186.924	0.0309000
$333$	0	0
$334$	2963.73	0.485534
$335$	−8816.92	−1.43797
$336$	0	0
$337$	3142.92	0.508030	0.254015	−	0.967200i	$-0.418249\pi$
$337$	3142.92	0.508030	0.254015	+	0.967200i	$0.418249\pi$
$338$	996.681	0.160391
$339$	0	0
$340$	8063.29	1.28616
$341$	−4375.38	−0.694839
$342$	0	0
$343$	0	0
$344$	−6326.74	−0.991613
$345$	0	0
$346$	3956.73	0.614784
$347$	−8966.74	−1.38720	−0.693602	−	0.720358i	$-0.743977\pi$
$347$	−8966.74	−1.38720	−0.693602	+	0.720358i	$0.743977\pi$
$348$	0	0
$349$	140.981	0.0216234	0.0108117	−	0.999942i	$-0.496558\pi$
$349$	140.981	0.0216234	0.0108117	+	0.999942i	$0.496558\pi$
$350$	0	0
$351$	0	0
$352$	5390.03	0.816164
$353$	−8208.98	−1.23773	−0.618867	−	0.785496i	$-0.712408\pi$
$353$	−8208.98	−1.23773	−0.618867	+	0.785496i	$0.712408\pi$
$354$	0	0
$355$	−3348.60	−0.500635
$356$	7442.82	1.10806
$357$	0	0
$358$	−4626.90	−0.683070
$359$	−8749.05	−1.28623	−0.643116	−	0.765769i	$-0.722359\pi$
$359$	−8749.05	−1.28623	−0.643116	+	0.765769i	$0.722359\pi$
$360$	0	0
$361$	13051.5	1.90284
$362$	357.722	0.0519376
$363$	0	0
$364$	0	0
$365$	−1808.36	−0.259325
$366$	0	0
$367$	5917.89	0.841721	0.420860	−	0.907125i	$-0.361728\pi$
$367$	5917.89	0.841721	0.420860	+	0.907125i	$0.361728\pi$
$368$	−432.979	−0.0613332
$369$	0	0
$370$	−6900.18	−0.969522
$371$	0	0
$372$	0	0
$373$	11660.7	1.61868	0.809342	−	0.587338i	$-0.199824\pi$
$373$	11660.7	1.61868	0.809342	+	0.587338i	$0.199824\pi$
$374$	−5776.73	−0.798684
$375$	0	0
$376$	−6182.80	−0.848015
$377$	6729.39	0.919313
$378$	0	0
$379$	−14306.7	−1.93902	−0.969508	−	0.245060i	$-0.921192\pi$
$379$	−14306.7	−1.93902	−0.969508	+	0.245060i	$0.921192\pi$
$380$	9322.39	1.25850
$381$	0	0
$382$	−364.596	−0.0488334
$383$	10840.5	1.44627	0.723136	−	0.690705i	$-0.242700\pi$
$383$	10840.5	1.44627	0.723136	+	0.690705i	$0.242700\pi$
$384$	0	0
$385$	0	0
$386$	−574.671	−0.0757772
$387$	0	0
$388$	4300.71	0.562721
$389$	11175.5	1.45660	0.728301	−	0.685257i	$-0.240310\pi$
$389$	11175.5	1.45660	0.728301	+	0.685257i	$0.240310\pi$
$390$	0	0
$391$	7359.33	0.951860
$392$	0	0
$393$	0	0
$394$	−8726.27	−1.11579
$395$	−8080.89	−1.02935
$396$	0	0
$397$	13077.9	1.65330	0.826650	−	0.562716i	$-0.190244\pi$
$397$	13077.9	1.65330	0.826650	+	0.562716i	$0.190244\pi$
$398$	−950.111	−0.119660
$399$	0	0
$400$	202.770	0.0253463
$401$	−2733.63	−0.340427	−0.170213	−	0.985407i	$-0.554446\pi$
$401$	−2733.63	−0.340427	−0.170213	+	0.985407i	$0.554446\pi$
$402$	0	0
$403$	−7993.22	−0.988017
$404$	5416.10	0.666983
$405$	0	0
$406$	0	0
$407$	−9907.60	−1.20664
$408$	0	0
$409$	−11521.1	−1.39286	−0.696431	−	0.717623i	$-0.745230\pi$
$409$	−11521.1	−1.39286	−0.696431	+	0.717623i	$0.745230\pi$
$410$	5900.26	0.710715
$411$	0	0
$412$	5547.27	0.663336
$413$	0	0
$414$	0	0
$415$	−433.560	−0.0512834
$416$	9846.86	1.16053
$417$	0	0
$418$	−6678.78	−0.781507
$419$	−2426.62	−0.282931	−0.141465	−	0.989943i	$-0.545181\pi$
$419$	−2426.62	−0.282931	−0.141465	+	0.989943i	$0.545181\pi$
$420$	0	0
$421$	10188.2	1.17943	0.589716	−	0.807611i	$-0.299240\pi$
$421$	10188.2	1.17943	0.589716	+	0.807611i	$0.299240\pi$
$422$	8922.91	1.02929
$423$	0	0
$424$	8444.32	0.967199
$425$	−3446.47	−0.393361
$426$	0	0
$427$	0	0
$428$	3246.73	0.366674
$429$	0	0
$430$	5872.25	0.658570
$431$	6780.40	0.757773	0.378887	−	0.925443i	$-0.376307\pi$
$431$	6780.40	0.757773	0.378887	+	0.925443i	$0.376307\pi$
$432$	0	0
$433$	−6122.85	−0.679550	−0.339775	−	0.940507i	$-0.610351\pi$
$433$	−6122.85	−0.679550	−0.339775	+	0.940507i	$0.610351\pi$
$434$	0	0
$435$	0	0
$436$	−5751.81	−0.631792
$437$	8508.50	0.931389
$438$	0	0
$439$	15747.5	1.71204	0.856022	−	0.516939i	$-0.172929\pi$
$439$	15747.5	1.71204	0.856022	+	0.516939i	$0.172929\pi$
$440$	−7814.48	−0.846684
$441$	0	0
$442$	−10553.3	−1.13568
$443$	−15063.8	−1.61558	−0.807789	−	0.589472i	$-0.799336\pi$
$443$	−15063.8	−1.61558	−0.807789	+	0.589472i	$0.799336\pi$
$444$	0	0
$445$	−17263.2	−1.83900
$446$	−7128.02	−0.756775
$447$	0	0
$448$	0	0
$449$	175.064	0.0184004	0.00920020	−	0.999958i	$-0.497071\pi$
$449$	175.064	0.0184004	0.00920020	+	0.999958i	$0.497071\pi$
$450$	0	0
$451$	8471.88	0.884535
$452$	11019.9	1.14675
$453$	0	0
$454$	1039.41	0.107450
$455$	0	0
$456$	0	0
$457$	−12593.5	−1.28905	−0.644527	−	0.764582i	$-0.722946\pi$
$457$	−12593.5	−1.28905	−0.644527	+	0.764582i	$0.722946\pi$
$458$	−1604.92	−0.163740
$459$	0	0
$460$	3983.80	0.403795
$461$	13781.7	1.39236	0.696182	−	0.717865i	$-0.254881\pi$
$461$	13781.7	1.39236	0.696182	+	0.717865i	$0.254881\pi$
$462$	0	0
$463$	8867.45	0.890076	0.445038	−	0.895512i	$-0.353190\pi$
$463$	8867.45	0.890076	0.445038	+	0.895512i	$0.353190\pi$
$464$	911.907	0.0912376
$465$	0	0
$466$	2737.22	0.272102
$467$	−12839.0	−1.27221	−0.636103	−	0.771604i	$-0.719455\pi$
$467$	−12839.0	−1.27221	−0.636103	+	0.771604i	$0.719455\pi$
$468$	0	0
$469$	0	0
$470$	5738.65	0.563201
$471$	0	0
$472$	−5861.34	−0.571589
$473$	8431.66	0.819636
$474$	0	0
$475$	−3984.65	−0.384902
$476$	0	0
$477$	0	0
$478$	4414.46	0.422411
$479$	9955.21	0.949614	0.474807	−	0.880090i	$-0.342518\pi$
$479$	9955.21	0.949614	0.474807	+	0.880090i	$0.342518\pi$
$480$	0	0
$481$	−18099.8	−1.71576
$482$	3493.36	0.330121
$483$	0	0
$484$	2613.58	0.245452
$485$	−9975.25	−0.933923
$486$	0	0
$487$	−16716.0	−1.55539	−0.777693	−	0.628644i	$-0.783610\pi$
$487$	−16716.0	−1.55539	−0.777693	+	0.628644i	$0.783610\pi$
$488$	−5214.82	−0.483737
$489$	0	0
$490$	0	0
$491$	2981.39	0.274029	0.137014	−	0.990569i	$-0.456249\pi$
$491$	2981.39	0.274029	0.137014	+	0.990569i	$0.456249\pi$
$492$	0	0
$493$	−15499.6	−1.41596
$494$	−12201.2	−1.11125
$495$	0	0
$496$	−1083.17	−0.0980561
$497$	0	0
$498$	0	0
$499$	8169.47	0.732897	0.366449	−	0.930438i	$-0.380574\pi$
$499$	8169.47	0.732897	0.366449	+	0.930438i	$0.380574\pi$
$500$	6392.73	0.571783
$501$	0	0
$502$	1498.83	0.133259
$503$	−1858.71	−0.164763	−0.0823814	−	0.996601i	$-0.526253\pi$
$503$	−1858.71	−0.164763	−0.0823814	+	0.996601i	$0.526253\pi$
$504$	0	0
$505$	−12562.3	−1.10696
$506$	−2854.09	−0.250750
$507$	0	0
$508$	696.110	0.0607970
$509$	−14749.0	−1.28436	−0.642179	−	0.766554i	$-0.721970\pi$
$509$	−14749.0	−1.28436	−0.642179	+	0.766554i	$0.721970\pi$
$510$	0	0
$511$	0	0
$512$	2584.58	0.223093
$513$	0	0
$514$	6906.30	0.592653
$515$	−12866.6	−1.10091
$516$	0	0
$517$	8239.83	0.700943
$518$	0	0
$519$	0	0
$520$	−14276.0	−1.20393
$521$	13232.1	1.11268	0.556342	−	0.830953i	$-0.312204\pi$
$521$	13232.1	1.11268	0.556342	+	0.830953i	$0.312204\pi$
$522$	0	0
$523$	16100.9	1.34617	0.673083	−	0.739567i	$-0.264969\pi$
$523$	16100.9	1.34617	0.673083	+	0.739567i	$0.264969\pi$
$524$	−10167.4	−0.847646
$525$	0	0
$526$	−9505.64	−0.787958
$527$	18410.6	1.52178
$528$	0	0
$529$	−8531.01	−0.701159
$530$	−7837.71	−0.642356
$531$	0	0
$532$	0	0
$533$	15477.0	1.25775
$534$	0	0
$535$	−7530.60	−0.608553
$536$	15501.5	1.24918
$537$	0	0
$538$	−4230.14	−0.338986
$539$	0	0
$540$	0	0
$541$	3622.62	0.287890	0.143945	−	0.989586i	$-0.454021\pi$
$541$	3622.62	0.287890	0.143945	+	0.989586i	$0.454021\pi$
$542$	6054.00	0.479782
$543$	0	0
$544$	−22680.0	−1.78750
$545$	13341.0	1.04856
$546$	0	0
$547$	14543.8	1.13683	0.568417	−	0.822741i	$-0.307556\pi$
$547$	14543.8	1.13683	0.568417	+	0.822741i	$0.307556\pi$
$548$	−2985.00	−0.232688
$549$	0	0
$550$	1336.61	0.103624
$551$	−17919.9	−1.38551
$552$	0	0
$553$	0	0
$554$	−527.709	−0.0404697
$555$	0	0
$556$	−3731.02	−0.284587
$557$	11533.3	0.877346	0.438673	−	0.898647i	$-0.355449\pi$
$557$	11533.3	0.877346	0.438673	+	0.898647i	$0.355449\pi$
$558$	0	0
$559$	15403.5	1.16547
$560$	0	0
$561$	0	0
$562$	−8143.81	−0.611256
$563$	−276.313	−0.0206842	−0.0103421	−	0.999947i	$-0.503292\pi$
$563$	−276.313	−0.0206842	−0.0103421	+	0.999947i	$0.503292\pi$
$564$	0	0
$565$	−25559.9	−1.90321
$566$	−4012.20	−0.297960
$567$	0	0
$568$	5887.36	0.434909
$569$	15729.5	1.15890	0.579452	−	0.815007i	$-0.303267\pi$
$569$	15729.5	1.15890	0.579452	+	0.815007i	$0.303267\pi$
$570$	0	0
$571$	7718.30	0.565676	0.282838	−	0.959168i	$-0.408724\pi$
$571$	7718.30	0.565676	0.282838	+	0.959168i	$0.408724\pi$
$572$	−8202.69	−0.599601
$573$	0	0
$574$	0	0
$575$	−1702.79	−0.123498
$576$	0	0
$577$	−15638.4	−1.12831	−0.564156	−	0.825668i	$-0.690798\pi$
$577$	−15638.4	−1.12831	−0.564156	+	0.825668i	$0.690798\pi$
$578$	16289.9	1.17227
$579$	0	0
$580$	−8390.37	−0.600674
$581$	0	0
$582$	0	0
$583$	−11253.8	−0.799457
$584$	3179.37	0.225279
$585$	0	0
$586$	4941.35	0.348337
$587$	14520.1	1.02097	0.510484	−	0.859888i	$-0.329466\pi$
$587$	14520.1	1.02097	0.510484	+	0.859888i	$0.329466\pi$
$588$	0	0
$589$	21285.5	1.48905
$590$	5440.28	0.379615
$591$	0	0
$592$	−2452.73	−0.170281
$593$	−5938.70	−0.411253	−0.205627	−	0.978631i	$-0.565923\pi$
$593$	−5938.70	−0.411253	−0.205627	+	0.978631i	$0.565923\pi$
$594$	0	0
$595$	0	0
$596$	8618.80	0.592349
$597$	0	0
$598$	−5214.03	−0.356551
$599$	−7264.82	−0.495547	−0.247774	−	0.968818i	$-0.579699\pi$
$599$	−7264.82	−0.495547	−0.247774	+	0.968818i	$0.579699\pi$
$600$	0	0
$601$	1691.27	0.114789	0.0573945	−	0.998352i	$-0.481721\pi$
$601$	1691.27	0.114789	0.0573945	+	0.998352i	$0.481721\pi$
$602$	0	0
$603$	0	0
$604$	19.1745	0.00129172
$605$	−6062.03	−0.407366
$606$	0	0
$607$	−17410.4	−1.16420	−0.582099	−	0.813118i	$-0.697768\pi$
$607$	−17410.4	−1.16420	−0.582099	+	0.813118i	$0.697768\pi$
$608$	−26221.6	−1.74905
$609$	0	0
$610$	4840.21	0.321269
$611$	15053.0	0.996696
$612$	0	0
$613$	10205.4	0.672418	0.336209	−	0.941787i	$-0.390855\pi$
$613$	10205.4	0.672418	0.336209	+	0.941787i	$0.390855\pi$
$614$	13508.3	0.887865
$615$	0	0
$616$	0	0
$617$	−8936.08	−0.583068	−0.291534	−	0.956560i	$-0.594166\pi$
$617$	−8936.08	−0.583068	−0.291534	+	0.956560i	$0.594166\pi$
$618$	0	0
$619$	18560.2	1.20517	0.602584	−	0.798056i	$-0.294138\pi$
$619$	18560.2	1.20517	0.602584	+	0.798056i	$0.294138\pi$
$620$	9966.15	0.645565
$621$	0	0
$622$	1344.76	0.0866883
$623$	0	0
$624$	0	0
$625$	−18357.4	−1.17488
$626$	3961.82	0.252949
$627$	0	0
$628$	3723.91	0.236624
$629$	41688.9	2.64268
$630$	0	0
$631$	−6191.13	−0.390594	−0.195297	−	0.980744i	$-0.562567\pi$
$631$	−6191.13	−0.390594	−0.195297	+	0.980744i	$0.562567\pi$
$632$	14207.4	0.894212
$633$	0	0
$634$	−13006.5	−0.814751
$635$	−1614.59	−0.100902
$636$	0	0
$637$	0	0
$638$	6011.06	0.373010
$639$	0	0
$640$	−13437.7	−0.829957
$641$	−1592.53	−0.0981298	−0.0490649	−	0.998796i	$-0.515624\pi$
$641$	−1592.53	−0.0981298	−0.0490649	+	0.998796i	$0.515624\pi$
$642$	0	0
$643$	20101.4	1.23285	0.616426	−	0.787413i	$-0.288580\pi$
$643$	20101.4	1.23285	0.616426	+	0.787413i	$0.288580\pi$
$644$	0	0
$645$	0	0
$646$	28102.8	1.71159
$647$	13953.3	0.847855	0.423927	−	0.905696i	$-0.360651\pi$
$647$	13953.3	0.847855	0.423927	+	0.905696i	$0.360651\pi$
$648$	0	0
$649$	7811.42	0.472458
$650$	2441.80	0.147347
$651$	0	0
$652$	−778.720	−0.0467746
$653$	−32040.0	−1.92010	−0.960049	−	0.279832i	$-0.909721\pi$
$653$	−32040.0	−1.92010	−0.960049	+	0.279832i	$0.909721\pi$
$654$	0	0
$655$	23582.8	1.40680
$656$	2097.30	0.124826
$657$	0	0
$658$	0	0
$659$	22240.2	1.31465	0.657325	−	0.753607i	$-0.271688\pi$
$659$	22240.2	1.31465	0.657325	+	0.753607i	$0.271688\pi$
$660$	0	0
$661$	−913.832	−0.0537730	−0.0268865	−	0.999638i	$-0.508559\pi$
$661$	−913.832	−0.0537730	−0.0268865	+	0.999638i	$0.508559\pi$
$662$	3710.28	0.217831
$663$	0	0
$664$	762.265	0.0445506
$665$	0	0
$666$	0	0
$667$	−7657.85	−0.444547
$668$	−9693.02	−0.561429
$669$	0	0
$670$	−14387.9	−0.829634
$671$	6949.80	0.399842
$672$	0	0
$673$	−7951.84	−0.455454	−0.227727	−	0.973725i	$-0.573129\pi$
$673$	−7951.84	−0.455454	−0.227727	+	0.973725i	$0.573129\pi$
$674$	5128.79	0.293106
$675$	0	0
$676$	−3259.69	−0.185462
$677$	−2633.40	−0.149498	−0.0747488	−	0.997202i	$-0.523816\pi$
$677$	−2633.40	−0.149498	−0.0747488	+	0.997202i	$0.523816\pi$
$678$	0	0
$679$	0	0
$680$	32881.6	1.85434
$681$	0	0
$682$	−7139.99	−0.400886
$683$	31241.6	1.75026	0.875129	−	0.483890i	$-0.160776\pi$
$683$	31241.6	1.75026	0.875129	+	0.483890i	$0.160776\pi$
$684$	0	0
$685$	6923.54	0.386182
$686$	0	0
$687$	0	0
$688$	2087.34	0.115668
$689$	−20559.1	−1.13678
$690$	0	0
$691$	5773.50	0.317850	0.158925	−	0.987291i	$-0.449197\pi$
$691$	5773.50	0.317850	0.158925	+	0.987291i	$0.449197\pi$
$692$	−12940.7	−0.710882
$693$	0	0
$694$	−14632.4	−0.800344
$695$	8653.87	0.472317
$696$	0	0
$697$	−35647.7	−1.93724
$698$	230.061	0.0124756
$699$	0	0
$700$	0	0
$701$	−18643.2	−1.00449	−0.502243	−	0.864726i	$-0.667492\pi$
$701$	−18643.2	−1.00449	−0.502243	+	0.864726i	$0.667492\pi$
$702$	0	0
$703$	48198.8	2.58585
$704$	7129.58	0.381685
$705$	0	0
$706$	−13395.9	−0.714107
$707$	0	0
$708$	0	0
$709$	10420.4	0.551970	0.275985	−	0.961162i	$-0.410996\pi$
$709$	10420.4	0.551970	0.275985	+	0.961162i	$0.410996\pi$
$710$	−5464.43	−0.288840
$711$	0	0
$712$	30351.3	1.59756
$713$	9096.06	0.477770
$714$	0	0
$715$	19025.6	0.995131
$716$	15132.5	0.789842
$717$	0	0
$718$	−14277.2	−0.742088
$719$	−8269.85	−0.428947	−0.214474	−	0.976730i	$-0.568804\pi$
$719$	−8269.85	−0.428947	−0.214474	+	0.976730i	$0.568804\pi$
$720$	0	0
$721$	0	0
$722$	21298.2	1.09784
$723$	0	0
$724$	−1169.94	−0.0600561
$725$	3586.27	0.183712
$726$	0	0
$727$	4112.34	0.209791	0.104896	−	0.994483i	$-0.466549\pi$
$727$	4112.34	0.209791	0.104896	+	0.994483i	$0.466549\pi$
$728$	0	0
$729$	0	0
$730$	−2950.97	−0.149617
$731$	−35478.5	−1.79510
$732$	0	0
$733$	28562.7	1.43927	0.719636	−	0.694352i	$-0.244309\pi$
$733$	28562.7	1.43927	0.719636	+	0.694352i	$0.244309\pi$
$734$	9657.14	0.485629
$735$	0	0
$736$	−11205.4	−0.561193
$737$	−20658.9	−1.03254
$738$	0	0
$739$	23183.9	1.15404	0.577019	−	0.816731i	$-0.304216\pi$
$739$	23183.9	1.15404	0.577019	+	0.816731i	$0.304216\pi$
$740$	22567.3	1.12107
$741$	0	0
$742$	0	0
$743$	7212.11	0.356106	0.178053	−	0.984021i	$-0.443020\pi$
$743$	7212.11	0.356106	0.178053	+	0.984021i	$0.443020\pi$
$744$	0	0
$745$	−19990.8	−0.983096
$746$	19028.6	0.933896
$747$	0	0
$748$	18893.1	0.923528
$749$	0	0
$750$	0	0
$751$	−21762.2	−1.05741	−0.528703	−	0.848807i	$-0.677322\pi$
$751$	−21762.2	−1.05741	−0.528703	+	0.848807i	$0.677322\pi$
$752$	2039.86	0.0989174
$753$	0	0
$754$	10981.4	0.530396
$755$	−44.4741	−0.00214381
$756$	0	0
$757$	−29737.8	−1.42779	−0.713896	−	0.700252i	$-0.753071\pi$
$757$	−29737.8	−1.42779	−0.713896	+	0.700252i	$0.753071\pi$
$758$	−23346.5	−1.11871
$759$	0	0
$760$	38016.1	1.81446
$761$	−30539.9	−1.45476	−0.727378	−	0.686237i	$-0.759261\pi$
$761$	−30539.9	−1.45476	−0.727378	+	0.686237i	$0.759261\pi$
$762$	0	0
$763$	0	0
$764$	1192.43	0.0564667
$765$	0	0
$766$	17690.1	0.834424
$767$	14270.4	0.671805
$768$	0	0
$769$	−39801.3	−1.86642	−0.933208	−	0.359338i	$-0.883003\pi$
$769$	−39801.3	−1.86642	−0.933208	+	0.359338i	$0.883003\pi$
$770$	0	0
$771$	0	0
$772$	1879.49	0.0876220
$773$	15163.9	0.705574	0.352787	−	0.935704i	$-0.385234\pi$
$773$	15163.9	0.705574	0.352787	+	0.935704i	$0.385234\pi$
$774$	0	0
$775$	−4259.81	−0.197441
$776$	17538.0	0.811312
$777$	0	0
$778$	18236.7	0.840383
$779$	−41214.2	−1.89557
$780$	0	0
$781$	−7846.10	−0.359482
$782$	12009.4	0.549173
$783$	0	0
$784$	0	0
$785$	−8637.38	−0.392715
$786$	0	0
$787$	−21917.3	−0.992717	−0.496359	−	0.868118i	$-0.665330\pi$
$787$	−21917.3	−0.992717	−0.496359	+	0.868118i	$0.665330\pi$
$788$	28539.6	1.29021
$789$	0	0
$790$	−13186.8	−0.593882
$791$	0	0
$792$	0	0
$793$	12696.3	0.568550
$794$	21341.2	0.953868
$795$	0	0
$796$	3107.38	0.138364
$797$	−3205.98	−0.142486	−0.0712431	−	0.997459i	$-0.522697\pi$
$797$	−3205.98	−0.142486	−0.0712431	+	0.997459i	$0.522697\pi$
$798$	0	0
$799$	−34671.3	−1.53515
$800$	5247.66	0.231916
$801$	0	0
$802$	−4460.89	−0.196408
$803$	−4237.15	−0.186209
$804$	0	0
$805$	0	0
$806$	−13043.8	−0.570034
$807$	0	0
$808$	22086.5	0.961634
$809$	−39211.8	−1.70410	−0.852048	−	0.523464i	$-0.824640\pi$
$809$	−39211.8	−1.70410	−0.852048	+	0.523464i	$0.824640\pi$
$810$	0	0
$811$	−14711.3	−0.636973	−0.318486	−	0.947927i	$-0.603175\pi$
$811$	−14711.3	−0.636973	−0.318486	+	0.947927i	$0.603175\pi$
$812$	0	0
$813$	0	0
$814$	−16167.8	−0.696167
$815$	1806.19	0.0776297
$816$	0	0
$817$	−41018.5	−1.75650
$818$	−18800.7	−0.803609
$819$	0	0
$820$	−19297.1	−0.821808
$821$	42916.2	1.82434	0.912171	−	0.409810i	$-0.134405\pi$
$821$	42916.2	1.82434	0.912171	+	0.409810i	$0.134405\pi$
$822$	0	0
$823$	6882.13	0.291490	0.145745	−	0.989322i	$-0.453442\pi$
$823$	6882.13	0.291490	0.145745	+	0.989322i	$0.453442\pi$
$824$	22621.4	0.956375
$825$	0	0
$826$	0	0
$827$	6598.74	0.277461	0.138731	−	0.990330i	$-0.455698\pi$
$827$	6598.74	0.277461	0.138731	+	0.990330i	$0.455698\pi$
$828$	0	0
$829$	23585.6	0.988134	0.494067	−	0.869424i	$-0.335510\pi$
$829$	23585.6	0.988134	0.494067	+	0.869424i	$0.335510\pi$
$830$	−707.507	−0.0295879
$831$	0	0
$832$	13024.8	0.542731
$833$	0	0
$834$	0	0
$835$	22482.4	0.931779
$836$	21843.3	0.903666
$837$	0	0
$838$	−3959.88	−0.163236
$839$	656.729	0.0270236	0.0135118	−	0.999909i	$-0.495699\pi$
$839$	656.729	0.0270236	0.0135118	+	0.999909i	$0.495699\pi$
$840$	0	0
$841$	−8260.64	−0.338703
$842$	16625.6	0.680470
$843$	0	0
$844$	−29182.8	−1.19018
$845$	7560.65	0.307804
$846$	0	0
$847$	0	0
$848$	−2785.99	−0.112820
$849$	0	0
$850$	−5624.14	−0.226949
$851$	20597.1	0.829682
$852$	0	0
$853$	26201.7	1.05173	0.525867	−	0.850567i	$-0.323741\pi$
$853$	26201.7	1.05173	0.525867	+	0.850567i	$0.323741\pi$
$854$	0	0
$855$	0	0
$856$	13239.9	0.528659
$857$	−23208.5	−0.925072	−0.462536	−	0.886601i	$-0.653060\pi$
$857$	−23208.5	−0.925072	−0.462536	+	0.886601i	$0.653060\pi$
$858$	0	0
$859$	26175.4	1.03969	0.519845	−	0.854260i	$-0.325990\pi$
$859$	26175.4	1.03969	0.519845	+	0.854260i	$0.325990\pi$
$860$	−19205.4	−0.761512
$861$	0	0
$862$	11064.6	0.437196
$863$	−5651.84	−0.222933	−0.111466	−	0.993768i	$-0.535555\pi$
$863$	−5651.84	−0.222933	−0.111466	+	0.993768i	$0.535555\pi$
$864$	0	0
$865$	30015.1	1.17982
$866$	−9991.60	−0.392065
$867$	0	0
$868$	0	0
$869$	−18934.3	−0.739128
$870$	0	0
$871$	−37741.0	−1.46820
$872$	−23455.5	−0.910897
$873$	0	0
$874$	13884.6	0.537363
$875$	0	0
$876$	0	0
$877$	37510.2	1.44428	0.722138	−	0.691749i	$-0.243160\pi$
$877$	37510.2	1.44428	0.722138	+	0.691749i	$0.243160\pi$
$878$	25697.6	0.987760
$879$	0	0
$880$	2578.19	0.0987622
$881$	−22133.8	−0.846434	−0.423217	−	0.906028i	$-0.639099\pi$
$881$	−22133.8	−0.846434	−0.423217	+	0.906028i	$0.639099\pi$
$882$	0	0
$883$	20678.1	0.788080	0.394040	−	0.919093i	$-0.371077\pi$
$883$	20678.1	0.788080	0.394040	+	0.919093i	$0.371077\pi$
$884$	34515.0	1.31320
$885$	0	0
$886$	−24581.9	−0.932104
$887$	−37469.8	−1.41839	−0.709196	−	0.705012i	$-0.750942\pi$
$887$	−37469.8	−1.41839	−0.709196	+	0.705012i	$0.750942\pi$
$888$	0	0
$889$	0	0
$890$	−28171.0	−1.06100
$891$	0	0
$892$	23312.5	0.875068
$893$	−40085.3	−1.50213
$894$	0	0
$895$	−35098.9	−1.31087
$896$	0	0
$897$	0	0
$898$	285.679	0.0106161
$899$	−19157.4	−0.710718
$900$	0	0
$901$	47353.3	1.75091
$902$	13824.9	0.510330
$903$	0	0
$904$	44938.2	1.65334
$905$	2713.61	0.0996725
$906$	0	0
$907$	20503.9	0.750630	0.375315	−	0.926897i	$-0.377535\pi$
$907$	20503.9	0.750630	0.375315	+	0.926897i	$0.377535\pi$
$908$	−3399.45	−0.124245
$909$	0	0
$910$	0	0
$911$	−3533.81	−0.128519	−0.0642593	−	0.997933i	$-0.520468\pi$
$911$	−3533.81	−0.128519	−0.0642593	+	0.997933i	$0.520468\pi$
$912$	0	0
$913$	−1015.87	−0.0368242
$914$	−20550.7	−0.743717
$915$	0	0
$916$	5248.96	0.189335
$917$	0	0
$918$	0	0
$919$	29803.8	1.06979	0.534895	−	0.844919i	$-0.320351\pi$
$919$	29803.8	1.06979	0.534895	+	0.844919i	$0.320351\pi$
$920$	16245.7	0.582178
$921$	0	0
$922$	22489.8	0.803321
$923$	−14333.7	−0.511160
$924$	0	0
$925$	−9645.90	−0.342870
$926$	14470.4	0.513528
$927$	0	0
$928$	23600.0	0.834815
$929$	14446.4	0.510194	0.255097	−	0.966915i	$-0.417893\pi$
$929$	14446.4	0.510194	0.255097	+	0.966915i	$0.417893\pi$
$930$	0	0
$931$	0	0
$932$	−8952.20	−0.314634
$933$	0	0
$934$	−20951.5	−0.733996
$935$	−43821.3	−1.53274
$936$	0	0
$937$	−29384.8	−1.02450	−0.512251	−	0.858836i	$-0.671188\pi$
$937$	−29384.8	−1.02450	−0.512251	+	0.858836i	$0.671188\pi$
$938$	0	0
$939$	0	0
$940$	−18768.5	−0.651235
$941$	4659.97	0.161435	0.0807176	−	0.996737i	$-0.474279\pi$
$941$	4659.97	0.161435	0.0807176	+	0.996737i	$0.474279\pi$
$942$	0	0
$943$	−17612.3	−0.608204
$944$	1933.80	0.0666735
$945$	0	0
$946$	13759.2	0.472887
$947$	−54321.9	−1.86402	−0.932008	−	0.362437i	$-0.881945\pi$
$947$	−54321.9	−1.86402	−0.932008	+	0.362437i	$0.881945\pi$
$948$	0	0
$949$	−7740.69	−0.264777
$950$	−6502.37	−0.222068
$951$	0	0
$952$	0	0
$953$	23606.5	0.802404	0.401202	−	0.915990i	$-0.368593\pi$
$953$	23606.5	0.802404	0.401202	+	0.915990i	$0.368593\pi$
$954$	0	0
$955$	−2765.77	−0.0937153
$956$	−14437.7	−0.488439
$957$	0	0
$958$	16245.5	0.547878
$959$	0	0
$960$	0	0
$961$	−7035.67	−0.236168
$962$	−29536.3	−0.989905
$963$	0	0
$964$	−11425.2	−0.381722
$965$	−4359.36	−0.145422
$966$	0	0
$967$	7021.95	0.233517	0.116758	−	0.993160i	$-0.462750\pi$
$967$	7021.95	0.233517	0.116758	+	0.993160i	$0.462750\pi$
$968$	10658.0	0.353885
$969$	0	0
$970$	−16278.2	−0.538825
$971$	−1483.48	−0.0490290	−0.0245145	−	0.999699i	$-0.507804\pi$
$971$	−1483.48	−0.0490290	−0.0245145	+	0.999699i	$0.507804\pi$
$972$	0	0
$973$	0	0
$974$	−27278.1	−0.897377
$975$	0	0
$976$	1720.49	0.0564259
$977$	−26152.1	−0.856375	−0.428187	−	0.903690i	$-0.640848\pi$
$977$	−26152.1	−0.856375	−0.428187	+	0.903690i	$0.640848\pi$
$978$	0	0
$979$	−40449.3	−1.32049
$980$	0	0
$981$	0	0
$982$	4865.19	0.158100
$983$	39185.3	1.27143	0.635716	−	0.771923i	$-0.280705\pi$
$983$	39185.3	1.27143	0.635716	+	0.771923i	$0.280705\pi$
$984$	0	0
$985$	−66196.0	−2.14130
$986$	−25293.2	−0.816936
$987$	0	0
$988$	39904.6	1.28496
$989$	−17528.7	−0.563580
$990$	0	0
$991$	−23009.7	−0.737564	−0.368782	−	0.929516i	$-0.620225\pi$
$991$	−23009.7	−0.737564	−0.368782	+	0.929516i	$0.620225\pi$
$992$	−28032.3	−0.897204
$993$	0	0
$994$	0	0
$995$	−7207.38	−0.229638
$996$	0	0
$997$	11154.2	0.354320	0.177160	−	0.984182i	$-0.443309\pi$
$997$	11154.2	0.354320	0.177160	+	0.984182i	$0.443309\pi$
$998$	13331.4	0.422843
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bh.1.5		7
3.2	odd	2		1323.4.a.bk.1.3		7
7.3	odd	6		189.4.e.g.163.3	yes	14
7.5	odd	6		189.4.e.g.109.3	yes	14
7.6	odd	2		1323.4.a.bi.1.5		7
21.5	even	6		189.4.e.f.109.5	✓	14
21.17	even	6		189.4.e.f.163.5	yes	14
21.20	even	2		1323.4.a.bj.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.5	✓	14	21.5	even	6
189.4.e.f.163.5	yes	14	21.17	even	6
189.4.e.g.109.3	yes	14	7.5	odd	6
189.4.e.g.163.3	yes	14	7.3	odd	6
1323.4.a.bh.1.5		7	1.1	even	1	trivial
1323.4.a.bi.1.5		7	7.6	odd	2
1323.4.a.bj.1.3		7	21.20	even	2
1323.4.a.bk.1.3		7	3.2	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)

\(f(q)\)	\(=\)	\(q+1.63185 q^{2} -5.33705 q^{4} +12.3790 q^{5} -21.7641 q^{8} +O(q^{10})\)	\(q+1.63185 q^{2} -5.33705 q^{4} +12.3790 q^{5} -21.7641 q^{8} +20.2007 q^{10} +29.0051 q^{11} +52.9884 q^{13} +7.18051 q^{16} -122.047 q^{17} -141.105 q^{19} -66.0672 q^{20} +47.3321 q^{22} -60.2992 q^{23} +28.2389 q^{25} +86.4693 q^{26} +126.997 q^{29} -150.849 q^{31} +185.831 q^{32} -199.163 q^{34} -341.581 q^{37} -230.262 q^{38} -269.418 q^{40} +292.082 q^{41} +290.696 q^{43} -154.802 q^{44} -98.3995 q^{46} +284.082 q^{47} +46.0818 q^{50} -282.802 q^{52} -387.993 q^{53} +359.053 q^{55} +207.241 q^{58} +269.312 q^{59} +239.606 q^{61} -246.163 q^{62} +245.804 q^{64} +655.941 q^{65} -712.250 q^{67} +651.370 q^{68} -270.507 q^{71} -146.083 q^{73} -557.411 q^{74} +753.083 q^{76} -652.792 q^{79} +88.8874 q^{80} +476.636 q^{82} -35.0239 q^{83} -1510.81 q^{85} +474.373 q^{86} -631.271 q^{88} -1394.56 q^{89} +321.820 q^{92} +463.581 q^{94} -1746.73 q^{95} -805.822 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 31 q^{4} - q^{5} - 84 q^{8}+O(q^{10}) \) 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

L-functions

Modular forms

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