LMFDB - Embedded newform 5994.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5994,2,Mod(1,5994)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5994, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5994.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5994 = 2 \cdot 3^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5994.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:	$47.8623309716$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 26x^{8} + 49x^{7} + 236x^{6} - 420x^{5} - 860x^{4} + 1461x^{3} + 993x^{2} - 1638x + 99 $ x^10 - 2x^9 - 26x^8 + 49x^7 + 236x^6 - 420x^5 - 860x^4 + 1461x^3 + 993x^2 - 1638*x + 99
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 666)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.79044$ of defining polynomial
Character	$\chi$	$=$	5994.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.949474 q^{5} -4.04317 q^{7} +1.00000 q^{8} -0.949474 q^{10} -1.82314 q^{11} +2.65283 q^{13} -4.04317 q^{14} +1.00000 q^{16} +7.58372 q^{17} -5.40297 q^{19} -0.949474 q^{20} -1.82314 q^{22} +3.54317 q^{23} -4.09850 q^{25} +2.65283 q^{26} -4.04317 q^{28} +2.25294 q^{29} -5.23080 q^{31} +1.00000 q^{32} +7.58372 q^{34} +3.83888 q^{35} -1.00000 q^{37} -5.40297 q^{38} -0.949474 q^{40} -2.42733 q^{41} +9.79523 q^{43} -1.82314 q^{44} +3.54317 q^{46} +1.51051 q^{47} +9.34719 q^{49} -4.09850 q^{50} +2.65283 q^{52} -4.98930 q^{53} +1.73102 q^{55} -4.04317 q^{56} +2.25294 q^{58} -12.2070 q^{59} +1.61261 q^{61} -5.23080 q^{62} +1.00000 q^{64} -2.51879 q^{65} +9.03491 q^{67} +7.58372 q^{68} +3.83888 q^{70} -10.3584 q^{71} +2.93156 q^{73} -1.00000 q^{74} -5.40297 q^{76} +7.37124 q^{77} +9.64020 q^{79} -0.949474 q^{80} -2.42733 q^{82} +5.29091 q^{83} -7.20054 q^{85} +9.79523 q^{86} -1.82314 q^{88} +15.0956 q^{89} -10.7258 q^{91} +3.54317 q^{92} +1.51051 q^{94} +5.12997 q^{95} +9.33056 q^{97} +9.34719 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} + q^{5} + q^{7} + 10 q^{8} + q^{10} + 3 q^{11} + 12 q^{13} + q^{14} + 10 q^{16} + 12 q^{17} + 24 q^{19} + q^{20} + 3 q^{22} + 3 q^{23} + 21 q^{25} + 12 q^{26} + q^{28} + 4 q^{29}+ \cdots + 35 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 + q^5 + q^7 + 10 * q^8 + q^10 + 3 * q^11 + 12 * q^13 + q^14 + 10 * q^16 + 12 * q^17 + 24 * q^19 + q^20 + 3 * q^22 + 3 * q^23 + 21 * q^25 + 12 * q^26 + q^28 + 4 * q^29 + 11 * q^31 + 10 * q^32 + 12 * q^34 - 7 * q^35 - 10 * q^37 + 24 * q^38 + q^40 - 3 * q^41 + 6 * q^43 + 3 * q^44 + 3 * q^46 + 8 * q^47 + 35 * q^49 + 21 * q^50 + 12 * q^52 - 10 * q^53 + 2 * q^55 + q^56 + 4 * q^58 + 10 * q^59 + 2 * q^61 + 11 * q^62 + 10 * q^64 + 26 * q^65 + 7 * q^67 + 12 * q^68 - 7 * q^70 + 15 * q^71 + 3 * q^73 - 10 * q^74 + 24 * q^76 + 9 * q^77 + 16 * q^79 + q^80 - 3 * q^82 - 23 * q^83 + 15 * q^85 + 6 * q^86 + 3 * q^88 + 21 * q^89 + 59 * q^91 + 3 * q^92 + 8 * q^94 - 6 * q^95 + 24 * q^97 + 35 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.949474	−0.424617	−0.212309	−	0.977203i	$-0.568098\pi$
$5$	−0.949474	−0.424617	−0.212309	+	0.977203i	$0.568098\pi$
$6$	0	0
$7$	−4.04317	−1.52817	−0.764087	−	0.645114i	$-0.776810\pi$
$7$	−4.04317	−1.52817	−0.764087	+	0.645114i	$0.776810\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.949474	−0.300250
$11$	−1.82314	−0.549696	−0.274848	−	0.961488i	$-0.588628\pi$
$11$	−1.82314	−0.549696	−0.274848	+	0.961488i	$0.588628\pi$
$12$	0	0
$13$	2.65283	0.735762	0.367881	−	0.929873i	$-0.380083\pi$
$13$	2.65283	0.735762	0.367881	+	0.929873i	$0.380083\pi$
$14$	−4.04317	−1.08058
$15$	0	0
$16$	1.00000	0.250000
$17$	7.58372	1.83932	0.919661	−	0.392714i	$-0.128464\pi$
$17$	7.58372	1.83932	0.919661	+	0.392714i	$0.128464\pi$
$18$	0	0
$19$	−5.40297	−1.23953	−0.619763	−	0.784789i	$-0.712771\pi$
$19$	−5.40297	−1.23953	−0.619763	+	0.784789i	$0.712771\pi$
$20$	−0.949474	−0.212309
$21$	0	0
$22$	−1.82314	−0.388694
$23$	3.54317	0.738802	0.369401	−	0.929270i	$-0.379563\pi$
$23$	3.54317	0.738802	0.369401	+	0.929270i	$0.379563\pi$
$24$	0	0
$25$	−4.09850	−0.819700
$26$	2.65283	0.520262
$27$	0	0
$28$	−4.04317	−0.764087
$29$	2.25294	0.418360	0.209180	−	0.977877i	$-0.432921\pi$
$29$	2.25294	0.418360	0.209180	+	0.977877i	$0.432921\pi$
$30$	0	0
$31$	−5.23080	−0.939479	−0.469739	−	0.882805i	$-0.655652\pi$
$31$	−5.23080	−0.939479	−0.469739	+	0.882805i	$0.655652\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.58372	1.30060
$35$	3.83888	0.648889
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−5.40297	−0.876477
$39$	0	0
$40$	−0.949474	−0.150125
$41$	−2.42733	−0.379086	−0.189543	−	0.981872i	$-0.560701\pi$
$41$	−2.42733	−0.379086	−0.189543	+	0.981872i	$0.560701\pi$
$42$	0	0
$43$	9.79523	1.49376	0.746879	−	0.664959i	$-0.231551\pi$
$43$	9.79523	1.49376	0.746879	+	0.664959i	$0.231551\pi$
$44$	−1.82314	−0.274848
$45$	0	0
$46$	3.54317	0.522412
$47$	1.51051	0.220331	0.110166	−	0.993913i	$-0.464862\pi$
$47$	1.51051	0.220331	0.110166	+	0.993913i	$0.464862\pi$
$48$	0	0
$49$	9.34719	1.33531
$50$	−4.09850	−0.579615
$51$	0	0
$52$	2.65283	0.367881
$53$	−4.98930	−0.685333	−0.342666	−	0.939457i	$-0.611330\pi$
$53$	−4.98930	−0.685333	−0.342666	+	0.939457i	$0.611330\pi$
$54$	0	0
$55$	1.73102	0.233411
$56$	−4.04317	−0.540291
$57$	0	0
$58$	2.25294	0.295825
$59$	−12.2070	−1.58922	−0.794611	−	0.607119i	$-0.792325\pi$
$59$	−12.2070	−1.58922	−0.794611	+	0.607119i	$0.792325\pi$
$60$	0	0
$61$	1.61261	0.206473	0.103236	−	0.994657i	$-0.467080\pi$
$61$	1.61261	0.206473	0.103236	+	0.994657i	$0.467080\pi$
$62$	−5.23080	−0.664312
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.51879	−0.312417
$66$	0	0
$67$	9.03491	1.10379	0.551895	−	0.833913i	$-0.313905\pi$
$67$	9.03491	1.10379	0.551895	+	0.833913i	$0.313905\pi$
$68$	7.58372	0.919661
$69$	0	0
$70$	3.83888	0.458834
$71$	−10.3584	−1.22932	−0.614658	−	0.788793i	$-0.710706\pi$
$71$	−10.3584	−1.22932	−0.614658	+	0.788793i	$0.710706\pi$
$72$	0	0
$73$	2.93156	0.343113	0.171557	−	0.985174i	$-0.445120\pi$
$73$	2.93156	0.343113	0.171557	+	0.985174i	$0.445120\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−5.40297	−0.619763
$77$	7.37124	0.840031
$78$	0	0
$79$	9.64020	1.08461	0.542304	−	0.840182i	$-0.317552\pi$
$79$	9.64020	1.08461	0.542304	+	0.840182i	$0.317552\pi$
$80$	−0.949474	−0.106154
$81$	0	0
$82$	−2.42733	−0.268054
$83$	5.29091	0.580753	0.290376	−	0.956913i	$-0.406220\pi$
$83$	5.29091	0.580753	0.290376	+	0.956913i	$0.406220\pi$
$84$	0	0
$85$	−7.20054	−0.781008
$86$	9.79523	1.05625
$87$	0	0
$88$	−1.82314	−0.194347
$89$	15.0956	1.60013	0.800066	−	0.599912i	$-0.204798\pi$
$89$	15.0956	1.60013	0.800066	+	0.599912i	$0.204798\pi$
$90$	0	0
$91$	−10.7258	−1.12437
$92$	3.54317	0.369401
$93$	0	0
$94$	1.51051	0.155798
$95$	5.12997	0.526324
$96$	0	0
$97$	9.33056	0.947375	0.473688	−	0.880693i	$-0.342923\pi$
$97$	9.33056	0.947375	0.473688	+	0.880693i	$0.342923\pi$
$98$	9.34719	0.944209
$99$	0	0
$100$	−4.09850	−0.409850
$101$	7.69792	0.765972	0.382986	−	0.923754i	$-0.374896\pi$
$101$	7.69792	0.765972	0.382986	+	0.923754i	$0.374896\pi$
$102$	0	0
$103$	17.3840	1.71289	0.856446	−	0.516236i	$-0.172667\pi$
$103$	17.3840	1.71289	0.856446	+	0.516236i	$0.172667\pi$
$104$	2.65283	0.260131
$105$	0	0
$106$	−4.98930	−0.484603
$107$	9.97908	0.964714	0.482357	−	0.875975i	$-0.339781\pi$
$107$	9.97908	0.964714	0.482357	+	0.875975i	$0.339781\pi$
$108$	0	0
$109$	7.02860	0.673218	0.336609	−	0.941644i	$-0.390720\pi$
$109$	7.02860	0.673218	0.336609	+	0.941644i	$0.390720\pi$
$110$	1.73102	0.165046
$111$	0	0
$112$	−4.04317	−0.382043
$113$	12.7285	1.19740	0.598700	−	0.800973i	$-0.295684\pi$
$113$	12.7285	1.19740	0.598700	+	0.800973i	$0.295684\pi$
$114$	0	0
$115$	−3.36414	−0.313708
$116$	2.25294	0.209180
$117$	0	0
$118$	−12.2070	−1.12375
$119$	−30.6622	−2.81080
$120$	0	0
$121$	−7.67617	−0.697834
$122$	1.61261	0.145998
$123$	0	0
$124$	−5.23080	−0.469739
$125$	8.63878	0.772676
$126$	0	0
$127$	−2.00548	−0.177957	−0.0889786	−	0.996034i	$-0.528360\pi$
$127$	−2.00548	−0.177957	−0.0889786	+	0.996034i	$0.528360\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.51879	−0.220912
$131$	−5.60693	−0.489880	−0.244940	−	0.969538i	$-0.578768\pi$
$131$	−5.60693	−0.489880	−0.244940	+	0.969538i	$0.578768\pi$
$132$	0	0
$133$	21.8451	1.89421
$134$	9.03491	0.780498
$135$	0	0
$136$	7.58372	0.650298
$137$	−6.11247	−0.522224	−0.261112	−	0.965309i	$-0.584089\pi$
$137$	−6.11247	−0.522224	−0.261112	+	0.965309i	$0.584089\pi$
$138$	0	0
$139$	9.39268	0.796677	0.398338	−	0.917239i	$-0.369587\pi$
$139$	9.39268	0.796677	0.398338	+	0.917239i	$0.369587\pi$
$140$	3.83888	0.324444
$141$	0	0
$142$	−10.3584	−0.869258
$143$	−4.83647	−0.404446
$144$	0	0
$145$	−2.13911	−0.177643
$146$	2.93156	0.242618
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	5.31790	0.435659	0.217830	−	0.975987i	$-0.430102\pi$
$149$	5.31790	0.435659	0.217830	+	0.975987i	$0.430102\pi$
$150$	0	0
$151$	4.07823	0.331882	0.165941	−	0.986136i	$-0.446934\pi$
$151$	4.07823	0.331882	0.165941	+	0.986136i	$0.446934\pi$
$152$	−5.40297	−0.438238
$153$	0	0
$154$	7.37124	0.593992
$155$	4.96650	0.398919
$156$	0	0
$157$	22.3662	1.78502	0.892509	−	0.451029i	$-0.148943\pi$
$157$	22.3662	1.78502	0.892509	+	0.451029i	$0.148943\pi$
$158$	9.64020	0.766933
$159$	0	0
$160$	−0.949474	−0.0750625
$161$	−14.3256	−1.12902
$162$	0	0
$163$	−3.87053	−0.303164	−0.151582	−	0.988445i	$-0.548437\pi$
$163$	−3.87053	−0.303164	−0.151582	+	0.988445i	$0.548437\pi$
$164$	−2.42733	−0.189543
$165$	0	0
$166$	5.29091	0.410654
$167$	17.2998	1.33870	0.669349	−	0.742948i	$-0.266573\pi$
$167$	17.2998	1.33870	0.669349	+	0.742948i	$0.266573\pi$
$168$	0	0
$169$	−5.96251	−0.458654
$170$	−7.20054	−0.552256
$171$	0	0
$172$	9.79523	0.746879
$173$	10.2202	0.777029	0.388515	−	0.921443i	$-0.372988\pi$
$173$	10.2202	0.777029	0.388515	+	0.921443i	$0.372988\pi$
$174$	0	0
$175$	16.5709	1.25264
$176$	−1.82314	−0.137424
$177$	0	0
$178$	15.0956	1.13146
$179$	−21.1620	−1.58172	−0.790861	−	0.611995i	$-0.790367\pi$
$179$	−21.1620	−1.58172	−0.790861	+	0.611995i	$0.790367\pi$
$180$	0	0
$181$	−9.76447	−0.725787	−0.362894	−	0.931831i	$-0.618211\pi$
$181$	−9.76447	−0.725787	−0.362894	+	0.931831i	$0.618211\pi$
$182$	−10.7258	−0.795051
$183$	0	0
$184$	3.54317	0.261206
$185$	0.949474	0.0698067
$186$	0	0
$187$	−13.8262	−1.01107
$188$	1.51051	0.110166
$189$	0	0
$190$	5.12997	0.372167
$191$	8.02147	0.580413	0.290206	−	0.956964i	$-0.406276\pi$
$191$	8.02147	0.580413	0.290206	+	0.956964i	$0.406276\pi$
$192$	0	0
$193$	12.2955	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$193$	12.2955	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$194$	9.33056	0.669896
$195$	0	0
$196$	9.34719	0.667656
$197$	23.3938	1.66674	0.833370	−	0.552715i	$-0.186408\pi$
$197$	23.3938	1.66674	0.833370	+	0.552715i	$0.186408\pi$
$198$	0	0
$199$	9.00271	0.638185	0.319092	−	0.947724i	$-0.396622\pi$
$199$	9.00271	0.638185	0.319092	+	0.947724i	$0.396622\pi$
$200$	−4.09850	−0.289808
$201$	0	0
$202$	7.69792	0.541624
$203$	−9.10901	−0.639327
$204$	0	0
$205$	2.30469	0.160966
$206$	17.3840	1.21120
$207$	0	0
$208$	2.65283	0.183941
$209$	9.85034	0.681363
$210$	0	0
$211$	23.8522	1.64205	0.821025	−	0.570892i	$-0.193402\pi$
$211$	23.8522	1.64205	0.821025	+	0.570892i	$0.193402\pi$
$212$	−4.98930	−0.342666
$213$	0	0
$214$	9.97908	0.682156
$215$	−9.30031	−0.634276
$216$	0	0
$217$	21.1490	1.43569
$218$	7.02860	0.476037
$219$	0	0
$220$	1.73102	0.116705
$221$	20.1183	1.35330
$222$	0	0
$223$	−10.7527	−0.720052	−0.360026	−	0.932942i	$-0.617232\pi$
$223$	−10.7527	−0.720052	−0.360026	+	0.932942i	$0.617232\pi$
$224$	−4.04317	−0.270145
$225$	0	0
$226$	12.7285	0.846689
$227$	1.16889	0.0775819	0.0387910	−	0.999247i	$-0.487649\pi$
$227$	1.16889	0.0775819	0.0387910	+	0.999247i	$0.487649\pi$
$228$	0	0
$229$	17.5125	1.15726	0.578628	−	0.815592i	$-0.303588\pi$
$229$	17.5125	1.15726	0.578628	+	0.815592i	$0.303588\pi$
$230$	−3.36414	−0.221825
$231$	0	0
$232$	2.25294	0.147913
$233$	13.6563	0.894657	0.447328	−	0.894370i	$-0.352376\pi$
$233$	13.6563	0.894657	0.447328	+	0.894370i	$0.352376\pi$
$234$	0	0
$235$	−1.43419	−0.0935565
$236$	−12.2070	−0.794611
$237$	0	0
$238$	−30.6622	−1.98754
$239$	−20.0715	−1.29832	−0.649160	−	0.760652i	$-0.724879\pi$
$239$	−20.0715	−1.29832	−0.649160	+	0.760652i	$0.724879\pi$
$240$	0	0
$241$	3.04219	0.195964	0.0979822	−	0.995188i	$-0.468761\pi$
$241$	3.04219	0.195964	0.0979822	+	0.995188i	$0.468761\pi$
$242$	−7.67617	−0.493443
$243$	0	0
$244$	1.61261	0.103236
$245$	−8.87491	−0.566997
$246$	0	0
$247$	−14.3331	−0.911996
$248$	−5.23080	−0.332156
$249$	0	0
$250$	8.63878	0.546365
$251$	−30.0821	−1.89877	−0.949383	−	0.314121i	$-0.898290\pi$
$251$	−30.0821	−1.89877	−0.949383	+	0.314121i	$0.898290\pi$
$252$	0	0
$253$	−6.45968	−0.406117
$254$	−2.00548	−0.125835
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0228	−1.87277	−0.936384	−	0.350976i	$-0.885850\pi$
$257$	−30.0228	−1.87277	−0.936384	+	0.350976i	$0.885850\pi$
$258$	0	0
$259$	4.04317	0.251230
$260$	−2.51879	−0.156209
$261$	0	0
$262$	−5.60693	−0.346397
$263$	5.85068	0.360768	0.180384	−	0.983596i	$-0.442266\pi$
$263$	5.85068	0.360768	0.180384	+	0.983596i	$0.442266\pi$
$264$	0	0
$265$	4.73721	0.291004
$266$	21.8451	1.33941
$267$	0	0
$268$	9.03491	0.551895
$269$	−13.2488	−0.807794	−0.403897	−	0.914804i	$-0.632345\pi$
$269$	−13.2488	−0.807794	−0.403897	+	0.914804i	$0.632345\pi$
$270$	0	0
$271$	28.1980	1.71290	0.856452	−	0.516227i	$-0.172664\pi$
$271$	28.1980	1.71290	0.856452	+	0.516227i	$0.172664\pi$
$272$	7.58372	0.459830
$273$	0	0
$274$	−6.11247	−0.369268
$275$	7.47213	0.450586
$276$	0	0
$277$	16.5354	0.993513	0.496756	−	0.867890i	$-0.334524\pi$
$277$	16.5354	0.993513	0.496756	+	0.867890i	$0.334524\pi$
$278$	9.39268	0.563336
$279$	0	0
$280$	3.83888	0.229417
$281$	20.5622	1.22664	0.613319	−	0.789835i	$-0.289834\pi$
$281$	20.5622	1.22664	0.613319	+	0.789835i	$0.289834\pi$
$282$	0	0
$283$	−28.6982	−1.70593	−0.852966	−	0.521967i	$-0.825199\pi$
$283$	−28.6982	−1.70593	−0.852966	+	0.521967i	$0.825199\pi$
$284$	−10.3584	−0.614658
$285$	0	0
$286$	−4.83647	−0.285986
$287$	9.81411	0.579308
$288$	0	0
$289$	40.5127	2.38310
$290$	−2.13911	−0.125613
$291$	0	0
$292$	2.93156	0.171557
$293$	−12.4437	−0.726969	−0.363484	−	0.931600i	$-0.618413\pi$
$293$	−12.4437	−0.726969	−0.363484	+	0.931600i	$0.618413\pi$
$294$	0	0
$295$	11.5903	0.674811
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	5.31790	0.308058
$299$	9.39941	0.543582
$300$	0	0
$301$	−39.6037	−2.28272
$302$	4.07823	0.234676
$303$	0	0
$304$	−5.40297	−0.309881
$305$	−1.53113	−0.0876720
$306$	0	0
$307$	0.945546	0.0539652	0.0269826	−	0.999636i	$-0.491410\pi$
$307$	0.945546	0.0539652	0.0269826	+	0.999636i	$0.491410\pi$
$308$	7.37124	0.420016
$309$	0	0
$310$	4.96650	0.282078
$311$	−0.353447	−0.0200422	−0.0100211	−	0.999950i	$-0.503190\pi$
$311$	−0.353447	−0.0200422	−0.0100211	+	0.999950i	$0.503190\pi$
$312$	0	0
$313$	13.2578	0.749376	0.374688	−	0.927151i	$-0.377750\pi$
$313$	13.2578	0.749376	0.374688	+	0.927151i	$0.377750\pi$
$314$	22.3662	1.26220
$315$	0	0
$316$	9.64020	0.542304
$317$	11.3256	0.636111	0.318055	−	0.948072i	$-0.396970\pi$
$317$	11.3256	0.636111	0.318055	+	0.948072i	$0.396970\pi$
$318$	0	0
$319$	−4.10742	−0.229971
$320$	−0.949474	−0.0530772
$321$	0	0
$322$	−14.3256	−0.798335
$323$	−40.9746	−2.27989
$324$	0	0
$325$	−10.8726	−0.603104
$326$	−3.87053	−0.214369
$327$	0	0
$328$	−2.42733	−0.134027
$329$	−6.10726	−0.336704
$330$	0	0
$331$	−26.7870	−1.47235	−0.736174	−	0.676792i	$-0.763370\pi$
$331$	−26.7870	−1.47235	−0.736174	+	0.676792i	$0.763370\pi$
$332$	5.29091	0.290376
$333$	0	0
$334$	17.2998	0.946603
$335$	−8.57841	−0.468689
$336$	0	0
$337$	−20.0579	−1.09262	−0.546310	−	0.837583i	$-0.683968\pi$
$337$	−20.0579	−1.09262	−0.546310	+	0.837583i	$0.683968\pi$
$338$	−5.96251	−0.324318
$339$	0	0
$340$	−7.20054	−0.390504
$341$	9.53646	0.516428
$342$	0	0
$343$	−9.49008	−0.512416
$344$	9.79523	0.528124
$345$	0	0
$346$	10.2202	0.549443
$347$	−16.2126	−0.870335	−0.435168	−	0.900349i	$-0.643311\pi$
$347$	−16.2126	−0.870335	−0.435168	+	0.900349i	$0.643311\pi$
$348$	0	0
$349$	−17.3932	−0.931038	−0.465519	−	0.885038i	$-0.654132\pi$
$349$	−17.3932	−0.931038	−0.465519	+	0.885038i	$0.654132\pi$
$350$	16.5709	0.885753
$351$	0	0
$352$	−1.82314	−0.0971735
$353$	−12.6778	−0.674772	−0.337386	−	0.941366i	$-0.609543\pi$
$353$	−12.6778	−0.674772	−0.337386	+	0.941366i	$0.609543\pi$
$354$	0	0
$355$	9.83503	0.521989
$356$	15.0956	0.800066
$357$	0	0
$358$	−21.1620	−1.11845
$359$	−15.4716	−0.816560	−0.408280	−	0.912857i	$-0.633871\pi$
$359$	−15.4716	−0.816560	−0.408280	+	0.912857i	$0.633871\pi$
$360$	0	0
$361$	10.1920	0.536423
$362$	−9.76447	−0.513209
$363$	0	0
$364$	−10.7258	−0.562186
$365$	−2.78344	−0.145692
$366$	0	0
$367$	−17.9237	−0.935611	−0.467805	−	0.883832i	$-0.654955\pi$
$367$	−17.9237	−0.935611	−0.467805	+	0.883832i	$0.654955\pi$
$368$	3.54317	0.184700
$369$	0	0
$370$	0.949474	0.0493608
$371$	20.1726	1.04731
$372$	0	0
$373$	−20.8143	−1.07772	−0.538862	−	0.842394i	$-0.681146\pi$
$373$	−20.8143	−1.07772	−0.538862	+	0.842394i	$0.681146\pi$
$374$	−13.8262	−0.714933
$375$	0	0
$376$	1.51051	0.0778988
$377$	5.97666	0.307814
$378$	0	0
$379$	−24.4311	−1.25494	−0.627472	−	0.778639i	$-0.715910\pi$
$379$	−24.4311	−1.25494	−0.627472	+	0.778639i	$0.715910\pi$
$380$	5.12997	0.263162
$381$	0	0
$382$	8.02147	0.410414
$383$	17.3044	0.884216	0.442108	−	0.896962i	$-0.354231\pi$
$383$	17.3044	0.884216	0.442108	+	0.896962i	$0.354231\pi$
$384$	0	0
$385$	−6.99880	−0.356692
$386$	12.2955	0.625825
$387$	0	0
$388$	9.33056	0.473688
$389$	−5.49088	−0.278399	−0.139199	−	0.990264i	$-0.544453\pi$
$389$	−5.49088	−0.278399	−0.139199	+	0.990264i	$0.544453\pi$
$390$	0	0
$391$	26.8704	1.35889
$392$	9.34719	0.472104
$393$	0	0
$394$	23.3938	1.17856
$395$	−9.15312	−0.460543
$396$	0	0
$397$	26.3543	1.32269	0.661343	−	0.750084i	$-0.269987\pi$
$397$	26.3543	1.32269	0.661343	+	0.750084i	$0.269987\pi$
$398$	9.00271	0.451265
$399$	0	0
$400$	−4.09850	−0.204925
$401$	−30.3957	−1.51789	−0.758945	−	0.651154i	$-0.774285\pi$
$401$	−30.3957	−1.51789	−0.758945	+	0.651154i	$0.774285\pi$
$402$	0	0
$403$	−13.8764	−0.691233
$404$	7.69792	0.382986
$405$	0	0
$406$	−9.10901	−0.452072
$407$	1.82314	0.0903695
$408$	0	0
$409$	2.75587	0.136269	0.0681345	−	0.997676i	$-0.478295\pi$
$409$	2.75587	0.136269	0.0681345	+	0.997676i	$0.478295\pi$
$410$	2.30469	0.113820
$411$	0	0
$412$	17.3840	0.856446
$413$	49.3551	2.42861
$414$	0	0
$415$	−5.02358	−0.246598
$416$	2.65283	0.130066
$417$	0	0
$418$	9.85034	0.481796
$419$	−27.7422	−1.35529	−0.677647	−	0.735387i	$-0.737000\pi$
$419$	−27.7422	−1.35529	−0.677647	+	0.735387i	$0.737000\pi$
$420$	0	0
$421$	−18.9114	−0.921686	−0.460843	−	0.887482i	$-0.652453\pi$
$421$	−18.9114	−0.921686	−0.460843	+	0.887482i	$0.652453\pi$
$422$	23.8522	1.16111
$423$	0	0
$424$	−4.98930	−0.242302
$425$	−31.0819	−1.50769
$426$	0	0
$427$	−6.52003	−0.315526
$428$	9.97908	0.482357
$429$	0	0
$430$	−9.30031	−0.448501
$431$	−38.0148	−1.83111	−0.915555	−	0.402192i	$-0.868248\pi$
$431$	−38.0148	−1.83111	−0.915555	+	0.402192i	$0.868248\pi$
$432$	0	0
$433$	−36.4405	−1.75122	−0.875609	−	0.483021i	$-0.839539\pi$
$433$	−36.4405	−1.75122	−0.875609	+	0.483021i	$0.839539\pi$
$434$	21.1490	1.01518
$435$	0	0
$436$	7.02860	0.336609
$437$	−19.1436	−0.915763
$438$	0	0
$439$	21.3279	1.01793	0.508963	−	0.860789i	$-0.330029\pi$
$439$	21.3279	1.01793	0.508963	+	0.860789i	$0.330029\pi$
$440$	1.73102	0.0825231
$441$	0	0
$442$	20.1183	0.956930
$443$	8.11476	0.385544	0.192772	−	0.981244i	$-0.438252\pi$
$443$	8.11476	0.385544	0.192772	+	0.981244i	$0.438252\pi$
$444$	0	0
$445$	−14.3329	−0.679444
$446$	−10.7527	−0.509154
$447$	0	0
$448$	−4.04317	−0.191022
$449$	−11.0770	−0.522757	−0.261378	−	0.965236i	$-0.584177\pi$
$449$	−11.0770	−0.522757	−0.261378	+	0.965236i	$0.584177\pi$
$450$	0	0
$451$	4.42536	0.208382
$452$	12.7285	0.598700
$453$	0	0
$454$	1.16889	0.0548587
$455$	10.1839	0.477428
$456$	0	0
$457$	18.4871	0.864788	0.432394	−	0.901685i	$-0.357669\pi$
$457$	18.4871	0.864788	0.432394	+	0.901685i	$0.357669\pi$
$458$	17.5125	0.818304
$459$	0	0
$460$	−3.36414	−0.156854
$461$	13.6124	0.633991	0.316996	−	0.948427i	$-0.397326\pi$
$461$	13.6124	0.633991	0.316996	+	0.948427i	$0.397326\pi$
$462$	0	0
$463$	−7.87343	−0.365909	−0.182955	−	0.983121i	$-0.558566\pi$
$463$	−7.87343	−0.365909	−0.182955	+	0.983121i	$0.558566\pi$
$464$	2.25294	0.104590
$465$	0	0
$466$	13.6563	0.632618
$467$	7.42635	0.343650	0.171825	−	0.985127i	$-0.445034\pi$
$467$	7.42635	0.343650	0.171825	+	0.985127i	$0.445034\pi$
$468$	0	0
$469$	−36.5297	−1.68678
$470$	−1.43419	−0.0661544
$471$	0	0
$472$	−12.2070	−0.561875
$473$	−17.8580	−0.821114
$474$	0	0
$475$	22.1441	1.01604
$476$	−30.6622	−1.40540
$477$	0	0
$478$	−20.0715	−0.918050
$479$	−24.5363	−1.12109	−0.560546	−	0.828123i	$-0.689409\pi$
$479$	−24.5363	−1.12109	−0.560546	+	0.828123i	$0.689409\pi$
$480$	0	0
$481$	−2.65283	−0.120959
$482$	3.04219	0.138568
$483$	0	0
$484$	−7.67617	−0.348917
$485$	−8.85912	−0.402272
$486$	0	0
$487$	22.7588	1.03130	0.515650	−	0.856800i	$-0.327551\pi$
$487$	22.7588	1.03130	0.515650	+	0.856800i	$0.327551\pi$
$488$	1.61261	0.0729992
$489$	0	0
$490$	−8.87491	−0.400928
$491$	34.1327	1.54039	0.770193	−	0.637811i	$-0.220160\pi$
$491$	34.1327	1.54039	0.770193	+	0.637811i	$0.220160\pi$
$492$	0	0
$493$	17.0856	0.769499
$494$	−14.3331	−0.644878
$495$	0	0
$496$	−5.23080	−0.234870
$497$	41.8808	1.87861
$498$	0	0
$499$	40.5029	1.81316	0.906580	−	0.422033i	$-0.138683\pi$
$499$	40.5029	1.81316	0.906580	+	0.422033i	$0.138683\pi$
$500$	8.63878	0.386338
$501$	0	0
$502$	−30.0821	−1.34263
$503$	2.32706	0.103758	0.0518792	−	0.998653i	$-0.483479\pi$
$503$	2.32706	0.103758	0.0518792	+	0.998653i	$0.483479\pi$
$504$	0	0
$505$	−7.30897	−0.325245
$506$	−6.45968	−0.287168
$507$	0	0
$508$	−2.00548	−0.0889786
$509$	−1.67955	−0.0744448	−0.0372224	−	0.999307i	$-0.511851\pi$
$509$	−1.67955	−0.0744448	−0.0372224	+	0.999307i	$0.511851\pi$
$510$	0	0
$511$	−11.8528	−0.524337
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.0228	−1.32425
$515$	−16.5056	−0.727324
$516$	0	0
$517$	−2.75387	−0.121115
$518$	4.04317	0.177647
$519$	0	0
$520$	−2.51879	−0.110456
$521$	−14.0917	−0.617370	−0.308685	−	0.951164i	$-0.599889\pi$
$521$	−14.0917	−0.617370	−0.308685	+	0.951164i	$0.599889\pi$
$522$	0	0
$523$	14.8457	0.649159	0.324580	−	0.945858i	$-0.394777\pi$
$523$	14.8457	0.649159	0.324580	+	0.945858i	$0.394777\pi$
$524$	−5.60693	−0.244940
$525$	0	0
$526$	5.85068	0.255102
$527$	−39.6689	−1.72800
$528$	0	0
$529$	−10.4460	−0.454172
$530$	4.73721	0.205771
$531$	0	0
$532$	21.8451	0.947105
$533$	−6.43929	−0.278917
$534$	0	0
$535$	−9.47487	−0.409634
$536$	9.03491	0.390249
$537$	0	0
$538$	−13.2488	−0.571197
$539$	−17.0412	−0.734017
$540$	0	0
$541$	−25.4514	−1.09424	−0.547120	−	0.837054i	$-0.684276\pi$
$541$	−25.4514	−1.09424	−0.547120	+	0.837054i	$0.684276\pi$
$542$	28.1980	1.21121
$543$	0	0
$544$	7.58372	0.325149
$545$	−6.67347	−0.285860
$546$	0	0
$547$	−40.3821	−1.72662	−0.863308	−	0.504678i	$-0.831611\pi$
$547$	−40.3821	−1.72662	−0.863308	+	0.504678i	$0.831611\pi$
$548$	−6.11247	−0.261112
$549$	0	0
$550$	7.47213	0.318613
$551$	−12.1726	−0.518568
$552$	0	0
$553$	−38.9769	−1.65747
$554$	16.5354	0.702520
$555$	0	0
$556$	9.39268	0.398338
$557$	14.3391	0.607568	0.303784	−	0.952741i	$-0.401750\pi$
$557$	14.3391	0.607568	0.303784	+	0.952741i	$0.401750\pi$
$558$	0	0
$559$	25.9851	1.09905
$560$	3.83888	0.162222
$561$	0	0
$562$	20.5622	0.867364
$563$	4.16081	0.175357	0.0876785	−	0.996149i	$-0.472055\pi$
$563$	4.16081	0.175357	0.0876785	+	0.996149i	$0.472055\pi$
$564$	0	0
$565$	−12.0854	−0.508437
$566$	−28.6982	−1.20628
$567$	0	0
$568$	−10.3584	−0.434629
$569$	−14.1606	−0.593645	−0.296822	−	0.954933i	$-0.595927\pi$
$569$	−14.1606	−0.593645	−0.296822	+	0.954933i	$0.595927\pi$
$570$	0	0
$571$	−6.79736	−0.284461	−0.142230	−	0.989834i	$-0.545427\pi$
$571$	−6.79736	−0.284461	−0.142230	+	0.989834i	$0.545427\pi$
$572$	−4.83647	−0.202223
$573$	0	0
$574$	9.81411	0.409633
$575$	−14.5217	−0.605596
$576$	0	0
$577$	42.6431	1.77526	0.887628	−	0.460561i	$-0.152352\pi$
$577$	42.6431	1.77526	0.887628	+	0.460561i	$0.152352\pi$
$578$	40.5127	1.68511
$579$	0	0
$580$	−2.13911	−0.0888215
$581$	−21.3920	−0.887491
$582$	0	0
$583$	9.09617	0.376725
$584$	2.93156	0.121309
$585$	0	0
$586$	−12.4437	−0.514045
$587$	−2.93783	−0.121257	−0.0606286	−	0.998160i	$-0.519311\pi$
$587$	−2.93783	−0.121257	−0.0606286	+	0.998160i	$0.519311\pi$
$588$	0	0
$589$	28.2618	1.16451
$590$	11.5903	0.477164
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	7.58214	0.311361	0.155681	−	0.987807i	$-0.450243\pi$
$593$	7.58214	0.311361	0.155681	+	0.987807i	$0.450243\pi$
$594$	0	0
$595$	29.1130	1.19352
$596$	5.31790	0.217830
$597$	0	0
$598$	9.39941	0.384371
$599$	12.4825	0.510021	0.255011	−	0.966938i	$-0.417921\pi$
$599$	12.4825	0.510021	0.255011	+	0.966938i	$0.417921\pi$
$600$	0	0
$601$	25.7914	1.05205	0.526027	−	0.850468i	$-0.323681\pi$
$601$	25.7914	1.05205	0.526027	+	0.850468i	$0.323681\pi$
$602$	−39.6037	−1.61413
$603$	0	0
$604$	4.07823	0.165941
$605$	7.28832	0.296312
$606$	0	0
$607$	41.7117	1.69303	0.846513	−	0.532368i	$-0.178698\pi$
$607$	41.7117	1.69303	0.846513	+	0.532368i	$0.178698\pi$
$608$	−5.40297	−0.219119
$609$	0	0
$610$	−1.53113	−0.0619935
$611$	4.00713	0.162111
$612$	0	0
$613$	−17.2841	−0.698099	−0.349049	−	0.937104i	$-0.613495\pi$
$613$	−17.2841	−0.698099	−0.349049	+	0.937104i	$0.613495\pi$
$614$	0.945546	0.0381591
$615$	0	0
$616$	7.37124	0.296996
$617$	33.3670	1.34330	0.671651	−	0.740867i	$-0.265585\pi$
$617$	33.3670	1.34330	0.671651	+	0.740867i	$0.265585\pi$
$618$	0	0
$619$	29.6455	1.19155	0.595776	−	0.803150i	$-0.296844\pi$
$619$	29.6455	1.19155	0.595776	+	0.803150i	$0.296844\pi$
$620$	4.96650	0.199460
$621$	0	0
$622$	−0.353447	−0.0141719
$623$	−61.0341	−2.44528
$624$	0	0
$625$	12.2902	0.491608
$626$	13.2578	0.529889
$627$	0	0
$628$	22.3662	0.892509
$629$	−7.58372	−0.302383
$630$	0	0
$631$	1.45670	0.0579903	0.0289952	−	0.999580i	$-0.490769\pi$
$631$	1.45670	0.0579903	0.0289952	+	0.999580i	$0.490769\pi$
$632$	9.64020	0.383467
$633$	0	0
$634$	11.3256	0.449798
$635$	1.90415	0.0755637
$636$	0	0
$637$	24.7965	0.982473
$638$	−4.10742	−0.162614
$639$	0	0
$640$	−0.949474	−0.0375312
$641$	−3.92506	−0.155031	−0.0775154	−	0.996991i	$-0.524699\pi$
$641$	−3.92506	−0.155031	−0.0775154	+	0.996991i	$0.524699\pi$
$642$	0	0
$643$	−24.8996	−0.981943	−0.490971	−	0.871176i	$-0.663358\pi$
$643$	−24.8996	−0.981943	−0.490971	+	0.871176i	$0.663358\pi$
$644$	−14.3256	−0.564508
$645$	0	0
$646$	−40.9746	−1.61212
$647$	2.56819	0.100966	0.0504829	−	0.998725i	$-0.483924\pi$
$647$	2.56819	0.100966	0.0504829	+	0.998725i	$0.483924\pi$
$648$	0	0
$649$	22.2551	0.873590
$650$	−10.8726	−0.426459
$651$	0	0
$652$	−3.87053	−0.151582
$653$	0.768873	0.0300883	0.0150442	−	0.999887i	$-0.495211\pi$
$653$	0.768873	0.0300883	0.0150442	+	0.999887i	$0.495211\pi$
$654$	0	0
$655$	5.32363	0.208012
$656$	−2.42733	−0.0947714
$657$	0	0
$658$	−6.10726	−0.238086
$659$	9.64449	0.375696	0.187848	−	0.982198i	$-0.439849\pi$
$659$	9.64449	0.375696	0.187848	+	0.982198i	$0.439849\pi$
$660$	0	0
$661$	−3.86806	−0.150450	−0.0752250	−	0.997167i	$-0.523968\pi$
$661$	−3.86806	−0.150450	−0.0752250	+	0.997167i	$0.523968\pi$
$662$	−26.7870	−1.04111
$663$	0	0
$664$	5.29091	0.205327
$665$	−20.7413	−0.804314
$666$	0	0
$667$	7.98254	0.309085
$668$	17.2998	0.669349
$669$	0	0
$670$	−8.57841	−0.331413
$671$	−2.94000	−0.113497
$672$	0	0
$673$	40.2074	1.54988	0.774940	−	0.632035i	$-0.217780\pi$
$673$	40.2074	1.54988	0.774940	+	0.632035i	$0.217780\pi$
$674$	−20.0579	−0.772600
$675$	0	0
$676$	−5.96251	−0.229327
$677$	−17.4762	−0.671665	−0.335832	−	0.941922i	$-0.609018\pi$
$677$	−17.4762	−0.671665	−0.335832	+	0.941922i	$0.609018\pi$
$678$	0	0
$679$	−37.7250	−1.44775
$680$	−7.20054	−0.276128
$681$	0	0
$682$	9.53646	0.365170
$683$	7.05834	0.270080	0.135040	−	0.990840i	$-0.456884\pi$
$683$	7.05834	0.270080	0.135040	+	0.990840i	$0.456884\pi$
$684$	0	0
$685$	5.80363	0.221745
$686$	−9.49008	−0.362333
$687$	0	0
$688$	9.79523	0.373440
$689$	−13.2357	−0.504242
$690$	0	0
$691$	18.8279	0.716246	0.358123	−	0.933674i	$-0.383417\pi$
$691$	18.8279	0.716246	0.358123	+	0.933674i	$0.383417\pi$
$692$	10.2202	0.388515
$693$	0	0
$694$	−16.2126	−0.615420
$695$	−8.91810	−0.338283
$696$	0	0
$697$	−18.4082	−0.697260
$698$	−17.3932	−0.658344
$699$	0	0
$700$	16.5709	0.626322
$701$	35.9910	1.35936	0.679681	−	0.733507i	$-0.262118\pi$
$701$	35.9910	1.35936	0.679681	+	0.733507i	$0.262118\pi$
$702$	0	0
$703$	5.40297	0.203777
$704$	−1.82314	−0.0687121
$705$	0	0
$706$	−12.6778	−0.477136
$707$	−31.1240	−1.17054
$708$	0	0
$709$	−19.3572	−0.726976	−0.363488	−	0.931599i	$-0.618414\pi$
$709$	−19.3572	−0.726976	−0.363488	+	0.931599i	$0.618414\pi$
$710$	9.83503	0.369102
$711$	0	0
$712$	15.0956	0.565732
$713$	−18.5336	−0.694088
$714$	0	0
$715$	4.59210	0.171735
$716$	−21.1620	−0.790861
$717$	0	0
$718$	−15.4716	−0.577395
$719$	−0.921503	−0.0343663	−0.0171831	−	0.999852i	$-0.505470\pi$
$719$	−0.921503	−0.0343663	−0.0171831	+	0.999852i	$0.505470\pi$
$720$	0	0
$721$	−70.2862	−2.61760
$722$	10.1920	0.379308
$723$	0	0
$724$	−9.76447	−0.362894
$725$	−9.23367	−0.342930
$726$	0	0
$727$	−37.7887	−1.40151	−0.700753	−	0.713404i	$-0.747152\pi$
$727$	−37.7887	−1.40151	−0.700753	+	0.713404i	$0.747152\pi$
$728$	−10.7258	−0.397525
$729$	0	0
$730$	−2.78344	−0.103020
$731$	74.2843	2.74750
$732$	0	0
$733$	40.3737	1.49124	0.745619	−	0.666373i	$-0.232154\pi$
$733$	40.3737	1.49124	0.745619	+	0.666373i	$0.232154\pi$
$734$	−17.9237	−0.661577
$735$	0	0
$736$	3.54317	0.130603
$737$	−16.4719	−0.606750
$738$	0	0
$739$	40.9013	1.50458	0.752290	−	0.658832i	$-0.228949\pi$
$739$	40.9013	1.50458	0.752290	+	0.658832i	$0.228949\pi$
$740$	0.949474	0.0349033
$741$	0	0
$742$	20.1726	0.740558
$743$	−35.1813	−1.29068	−0.645338	−	0.763897i	$-0.723283\pi$
$743$	−35.1813	−1.29068	−0.645338	+	0.763897i	$0.723283\pi$
$744$	0	0
$745$	−5.04920	−0.184989
$746$	−20.8143	−0.762066
$747$	0	0
$748$	−13.8262	−0.505534
$749$	−40.3471	−1.47425
$750$	0	0
$751$	31.5323	1.15063	0.575315	−	0.817932i	$-0.304879\pi$
$751$	31.5323	1.15063	0.575315	+	0.817932i	$0.304879\pi$
$752$	1.51051	0.0550828
$753$	0	0
$754$	5.97666	0.217657
$755$	−3.87217	−0.140923
$756$	0	0
$757$	−21.9828	−0.798980	−0.399490	−	0.916738i	$-0.630813\pi$
$757$	−21.9828	−0.798980	−0.399490	+	0.916738i	$0.630813\pi$
$758$	−24.4311	−0.887379
$759$	0	0
$760$	5.12997	0.186084
$761$	0.722838	0.0262029	0.0131014	−	0.999914i	$-0.495830\pi$
$761$	0.722838	0.0262029	0.0131014	+	0.999914i	$0.495830\pi$
$762$	0	0
$763$	−28.4178	−1.02879
$764$	8.02147	0.290206
$765$	0	0
$766$	17.3044	0.625235
$767$	−32.3832	−1.16929
$768$	0	0
$769$	5.21208	0.187952	0.0939761	−	0.995574i	$-0.470042\pi$
$769$	5.21208	0.187952	0.0939761	+	0.995574i	$0.470042\pi$
$770$	−6.99880	−0.252219
$771$	0	0
$772$	12.2955	0.442525
$773$	−14.8217	−0.533100	−0.266550	−	0.963821i	$-0.585884\pi$
$773$	−14.8217	−0.533100	−0.266550	+	0.963821i	$0.585884\pi$
$774$	0	0
$775$	21.4384	0.770091
$776$	9.33056	0.334948
$777$	0	0
$778$	−5.49088	−0.196858
$779$	13.1148	0.469886
$780$	0	0
$781$	18.8848	0.675751
$782$	26.8704	0.960883
$783$	0	0
$784$	9.34719	0.333828
$785$	−21.2361	−0.757950
$786$	0	0
$787$	24.4344	0.870992	0.435496	−	0.900191i	$-0.356573\pi$
$787$	24.4344	0.870992	0.435496	+	0.900191i	$0.356573\pi$
$788$	23.3938	0.833370
$789$	0	0
$790$	−9.15312	−0.325653
$791$	−51.4636	−1.82983
$792$	0	0
$793$	4.27796	0.151915
$794$	26.3543	0.935280
$795$	0	0
$796$	9.00271	0.319092
$797$	−2.52142	−0.0893134	−0.0446567	−	0.999002i	$-0.514219\pi$
$797$	−2.52142	−0.0893134	−0.0446567	+	0.999002i	$0.514219\pi$
$798$	0	0
$799$	11.4553	0.405260
$800$	−4.09850	−0.144904
$801$	0	0
$802$	−30.3957	−1.07331
$803$	−5.34464	−0.188608
$804$	0	0
$805$	13.6018	0.479400
$806$	−13.8764	−0.488775
$807$	0	0
$808$	7.69792	0.270812
$809$	−15.8013	−0.555543	−0.277771	−	0.960647i	$-0.589596\pi$
$809$	−15.8013	−0.555543	−0.277771	+	0.960647i	$0.589596\pi$
$810$	0	0
$811$	−8.96155	−0.314683	−0.157341	−	0.987544i	$-0.550292\pi$
$811$	−8.96155	−0.314683	−0.157341	+	0.987544i	$0.550292\pi$
$812$	−9.10901	−0.319663
$813$	0	0
$814$	1.82314	0.0639009
$815$	3.67497	0.128729
$816$	0	0
$817$	−52.9233	−1.85155
$818$	2.75587	0.0963567
$819$	0	0
$820$	2.30469	0.0804832
$821$	−47.9438	−1.67325	−0.836625	−	0.547777i	$-0.815474\pi$
$821$	−47.9438	−1.67325	−0.836625	+	0.547777i	$0.815474\pi$
$822$	0	0
$823$	29.0381	1.01220	0.506102	−	0.862473i	$-0.331086\pi$
$823$	29.0381	1.01220	0.506102	+	0.862473i	$0.331086\pi$
$824$	17.3840	0.605599
$825$	0	0
$826$	49.3551	1.71728
$827$	21.6384	0.752442	0.376221	−	0.926530i	$-0.377223\pi$
$827$	21.6384	0.752442	0.376221	+	0.926530i	$0.377223\pi$
$828$	0	0
$829$	47.7197	1.65737	0.828687	−	0.559712i	$-0.189088\pi$
$829$	47.7197	1.65737	0.828687	+	0.559712i	$0.189088\pi$
$830$	−5.02358	−0.174371
$831$	0	0
$832$	2.65283	0.0919703
$833$	70.8864	2.45607
$834$	0	0
$835$	−16.4257	−0.568435
$836$	9.85034	0.340681
$837$	0	0
$838$	−27.7422	−0.958338
$839$	−24.4739	−0.844932	−0.422466	−	0.906379i	$-0.638835\pi$
$839$	−24.4739	−0.844932	−0.422466	+	0.906379i	$0.638835\pi$
$840$	0	0
$841$	−23.9243	−0.824975
$842$	−18.9114	−0.651730
$843$	0	0
$844$	23.8522	0.821025
$845$	5.66124	0.194753
$846$	0	0
$847$	31.0360	1.06641
$848$	−4.98930	−0.171333
$849$	0	0
$850$	−31.0819	−1.06610
$851$	−3.54317	−0.121458
$852$	0	0
$853$	−33.2211	−1.13747	−0.568735	−	0.822521i	$-0.692567\pi$
$853$	−33.2211	−1.13747	−0.568735	+	0.822521i	$0.692567\pi$
$854$	−6.52003	−0.223111
$855$	0	0
$856$	9.97908	0.341078
$857$	26.1000	0.891560	0.445780	−	0.895143i	$-0.352926\pi$
$857$	26.1000	0.891560	0.445780	+	0.895143i	$0.352926\pi$
$858$	0	0
$859$	−10.8040	−0.368627	−0.184314	−	0.982867i	$-0.559006\pi$
$859$	−10.8040	−0.368627	−0.184314	+	0.982867i	$0.559006\pi$
$860$	−9.30031	−0.317138
$861$	0	0
$862$	−38.0148	−1.29479
$863$	53.9732	1.83727	0.918634	−	0.395110i	$-0.129294\pi$
$863$	53.9732	1.83727	0.918634	+	0.395110i	$0.129294\pi$
$864$	0	0
$865$	−9.70383	−0.329940
$866$	−36.4405	−1.23830
$867$	0	0
$868$	21.1490	0.717843
$869$	−17.5754	−0.596205
$870$	0	0
$871$	23.9681	0.812127
$872$	7.02860	0.238019
$873$	0	0
$874$	−19.1436	−0.647542
$875$	−34.9280	−1.18078
$876$	0	0
$877$	−36.8411	−1.24404	−0.622018	−	0.783003i	$-0.713687\pi$
$877$	−36.8411	−1.24404	−0.622018	+	0.783003i	$0.713687\pi$
$878$	21.3279	0.719782
$879$	0	0
$880$	1.73102	0.0583527
$881$	−9.98724	−0.336479	−0.168239	−	0.985746i	$-0.553808\pi$
$881$	−9.98724	−0.336479	−0.168239	+	0.985746i	$0.553808\pi$
$882$	0	0
$883$	14.4722	0.487029	0.243514	−	0.969897i	$-0.421700\pi$
$883$	14.4722	0.487029	0.243514	+	0.969897i	$0.421700\pi$
$884$	20.1183	0.676651
$885$	0	0
$886$	8.11476	0.272621
$887$	51.9741	1.74512	0.872559	−	0.488508i	$-0.162459\pi$
$887$	51.9741	1.74512	0.872559	+	0.488508i	$0.162459\pi$
$888$	0	0
$889$	8.10847	0.271949
$890$	−14.3329	−0.480440
$891$	0	0
$892$	−10.7527	−0.360026
$893$	−8.16126	−0.273106
$894$	0	0
$895$	20.0928	0.671627
$896$	−4.04317	−0.135073
$897$	0	0
$898$	−11.0770	−0.369645
$899$	−11.7847	−0.393041
$900$	0	0
$901$	−37.8374	−1.26055
$902$	4.42536	0.147348
$903$	0	0
$904$	12.7285	0.423345
$905$	9.27110	0.308182
$906$	0	0
$907$	36.9710	1.22760	0.613801	−	0.789461i	$-0.289640\pi$
$907$	36.9710	1.22760	0.613801	+	0.789461i	$0.289640\pi$
$908$	1.16889	0.0387910
$909$	0	0
$910$	10.1839	0.337592
$911$	5.80919	0.192467	0.0962335	−	0.995359i	$-0.469320\pi$
$911$	5.80919	0.192467	0.0962335	+	0.995359i	$0.469320\pi$
$912$	0	0
$913$	−9.64605	−0.319238
$914$	18.4871	0.611498
$915$	0	0
$916$	17.5125	0.578628
$917$	22.6698	0.748621
$918$	0	0
$919$	−44.1079	−1.45499	−0.727493	−	0.686115i	$-0.759315\pi$
$919$	−44.1079	−1.45499	−0.727493	+	0.686115i	$0.759315\pi$
$920$	−3.36414	−0.110913
$921$	0	0
$922$	13.6124	0.448299
$923$	−27.4791	−0.904485
$924$	0	0
$925$	4.09850	0.134758
$926$	−7.87343	−0.258737
$927$	0	0
$928$	2.25294	0.0739563
$929$	−14.0254	−0.460159	−0.230080	−	0.973172i	$-0.573899\pi$
$929$	−14.0254	−0.460159	−0.230080	+	0.973172i	$0.573899\pi$
$930$	0	0
$931$	−50.5025	−1.65515
$932$	13.6563	0.447328
$933$	0	0
$934$	7.42635	0.242997
$935$	13.1276	0.429317
$936$	0	0
$937$	−26.5435	−0.867138	−0.433569	−	0.901120i	$-0.642746\pi$
$937$	−26.5435	−0.867138	−0.433569	+	0.901120i	$0.642746\pi$
$938$	−36.5297	−1.19274
$939$	0	0
$940$	−1.43419	−0.0467782
$941$	15.4165	0.502562	0.251281	−	0.967914i	$-0.419148\pi$
$941$	15.4165	0.502562	0.251281	+	0.967914i	$0.419148\pi$
$942$	0	0
$943$	−8.60045	−0.280069
$944$	−12.2070	−0.397305
$945$	0	0
$946$	−17.8580	−0.580615
$947$	−12.5592	−0.408119	−0.204060	−	0.978958i	$-0.565414\pi$
$947$	−12.5592	−0.408119	−0.204060	+	0.978958i	$0.565414\pi$
$948$	0	0
$949$	7.77693	0.252450
$950$	22.1441	0.718448
$951$	0	0
$952$	−30.6622	−0.993768
$953$	41.9349	1.35841	0.679203	−	0.733951i	$-0.262326\pi$
$953$	41.9349	1.35841	0.679203	+	0.733951i	$0.262326\pi$
$954$	0	0
$955$	−7.61617	−0.246453
$956$	−20.0715	−0.649160
$957$	0	0
$958$	−24.5363	−0.792732
$959$	24.7137	0.798049
$960$	0	0
$961$	−3.63877	−0.117380
$962$	−2.65283	−0.0855306
$963$	0	0
$964$	3.04219	0.0979822
$965$	−11.6743	−0.375808
$966$	0	0
$967$	7.33982	0.236033	0.118016	−	0.993012i	$-0.462346\pi$
$967$	7.33982	0.236033	0.118016	+	0.993012i	$0.462346\pi$
$968$	−7.67617	−0.246722
$969$	0	0
$970$	−8.85912	−0.284449
$971$	9.43859	0.302899	0.151449	−	0.988465i	$-0.451606\pi$
$971$	9.43859	0.302899	0.151449	+	0.988465i	$0.451606\pi$
$972$	0	0
$973$	−37.9762	−1.21746
$974$	22.7588	0.729239
$975$	0	0
$976$	1.61261	0.0516182
$977$	43.2930	1.38507	0.692533	−	0.721386i	$-0.256495\pi$
$977$	43.2930	1.38507	0.692533	+	0.721386i	$0.256495\pi$
$978$	0	0
$979$	−27.5214	−0.879587
$980$	−8.87491	−0.283499
$981$	0	0
$982$	34.1327	1.08922
$983$	46.3782	1.47923	0.739617	−	0.673028i	$-0.235007\pi$
$983$	46.3782	1.47923	0.739617	+	0.673028i	$0.235007\pi$
$984$	0	0
$985$	−22.2118	−0.707727
$986$	17.0856	0.544118
$987$	0	0
$988$	−14.3331	−0.455998
$989$	34.7061	1.10359
$990$	0	0
$991$	56.1550	1.78382	0.891911	−	0.452211i	$-0.149365\pi$
$991$	56.1550	1.78382	0.891911	+	0.452211i	$0.149365\pi$
$992$	−5.23080	−0.166078
$993$	0	0
$994$	41.8808	1.32838
$995$	−8.54783	−0.270984
$996$	0	0
$997$	39.1868	1.24106	0.620529	−	0.784184i	$-0.286918\pi$
$997$	39.1868	1.24106	0.620529	+	0.784184i	$0.286918\pi$
$998$	40.5029	1.28210
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5994.2.a.bb.1.4		10
3.2	odd	2		5994.2.a.ba.1.7		10
9.2	odd	6		1998.2.e.e.1333.4		20
9.4	even	3		666.2.e.e.223.9	✓	20
9.5	odd	6		1998.2.e.e.667.4		20
9.7	even	3		666.2.e.e.445.9	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
666.2.e.e.223.9	✓	20	9.4	even	3
666.2.e.e.445.9	yes	20	9.7	even	3
1998.2.e.e.667.4		20	9.5	odd	6
1998.2.e.e.1333.4		20	9.2	odd	6
5994.2.a.ba.1.7		10	3.2	odd	2
5994.2.a.bb.1.4		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5994 = 2 \cdot 3^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5994.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.949474 q^{5} -4.04317 q^{7} +1.00000 q^{8} -0.949474 q^{10} -1.82314 q^{11} +2.65283 q^{13} -4.04317 q^{14} +1.00000 q^{16} +7.58372 q^{17} -5.40297 q^{19} -0.949474 q^{20} -1.82314 q^{22} +3.54317 q^{23} -4.09850 q^{25} +2.65283 q^{26} -4.04317 q^{28} +2.25294 q^{29} -5.23080 q^{31} +1.00000 q^{32} +7.58372 q^{34} +3.83888 q^{35} -1.00000 q^{37} -5.40297 q^{38} -0.949474 q^{40} -2.42733 q^{41} +9.79523 q^{43} -1.82314 q^{44} +3.54317 q^{46} +1.51051 q^{47} +9.34719 q^{49} -4.09850 q^{50} +2.65283 q^{52} -4.98930 q^{53} +1.73102 q^{55} -4.04317 q^{56} +2.25294 q^{58} -12.2070 q^{59} +1.61261 q^{61} -5.23080 q^{62} +1.00000 q^{64} -2.51879 q^{65} +9.03491 q^{67} +7.58372 q^{68} +3.83888 q^{70} -10.3584 q^{71} +2.93156 q^{73} -1.00000 q^{74} -5.40297 q^{76} +7.37124 q^{77} +9.64020 q^{79} -0.949474 q^{80} -2.42733 q^{82} +5.29091 q^{83} -7.20054 q^{85} +9.79523 q^{86} -1.82314 q^{88} +15.0956 q^{89} -10.7258 q^{91} +3.54317 q^{92} +1.51051 q^{94} +5.12997 q^{95} +9.33056 q^{97} +9.34719 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} + q^{5} + q^{7} + 10 q^{8} + q^{10} + 3 q^{11} + 12 q^{13} + q^{14} + 10 q^{16} + 12 q^{17} + 24 q^{19} + q^{20} + 3 q^{22} + 3 q^{23} + 21 q^{25} + 12 q^{26} + q^{28} + 4 q^{29}+ \cdots + 35 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 + q^5 + q^7 + 10 * q^8 + q^10 + 3 * q^11 + 12 * q^13 + q^14 + 10 * q^16 + 12 * q^17 + 24 * q^19 + q^20 + 3 * q^22 + 3 * q^23 + 21 * q^25 + 12 * q^26 + q^28 + 4 * q^29 + 11 * q^31 + 10 * q^32 + 12 * q^34 - 7 * q^35 - 10 * q^37 + 24 * q^38 + q^40 - 3 * q^41 + 6 * q^43 + 3 * q^44 + 3 * q^46 + 8 * q^47 + 35 * q^49 + 21 * q^50 + 12 * q^52 - 10 * q^53 + 2 * q^55 + q^56 + 4 * q^58 + 10 * q^59 + 2 * q^61 + 11 * q^62 + 10 * q^64 + 26 * q^65 + 7 * q^67 + 12 * q^68 - 7 * q^70 + 15 * q^71 + 3 * q^73 - 10 * q^74 + 24 * q^76 + 9 * q^77 + 16 * q^79 + q^80 - 3 * q^82 - 23 * q^83 + 15 * q^85 + 6 * q^86 + 3 * q^88 + 21 * q^89 + 59 * q^91 + 3 * q^92 + 8 * q^94 - 6 * q^95 + 24 * q^97 + 35 * q^98

Introduction

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