LMFDB - Embedded newform 6975.2.a.bz.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-1,0,7,0,0,-6,-3,0,0,-9,0,-4,0,0,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.136751504.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 7x^{3} + 20x^{2} - 8x - 12 $ x^6 - x^5 - 9x^4 + 7x^3 + 20x^2 - 8x - 12
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2325)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.759793$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.759793 q^{2} -1.42271 q^{4} -4.29226 q^{7} -2.60056 q^{8} +1.35462 q^{11} -0.931910 q^{13} -3.26123 q^{14} +0.869544 q^{16} +3.18251 q^{17} -1.13077 q^{19} +1.02923 q^{22} +4.97154 q^{23} -0.708059 q^{26} +6.10666 q^{28} +4.66324 q^{29} +1.00000 q^{31} +5.86178 q^{32} +2.41805 q^{34} -10.9138 q^{37} -0.859154 q^{38} +11.4282 q^{41} +5.71497 q^{43} -1.92724 q^{44} +3.77734 q^{46} +0.503231 q^{47} +11.4235 q^{49} +1.32584 q^{52} +2.52183 q^{53} +11.1623 q^{56} +3.54310 q^{58} -10.5500 q^{59} +3.74164 q^{61} +0.759793 q^{62} +2.71466 q^{64} -10.7877 q^{67} -4.52780 q^{68} -4.27758 q^{71} -10.0678 q^{73} -8.29226 q^{74} +1.60877 q^{76} -5.81440 q^{77} +2.28187 q^{79} +8.68303 q^{82} -8.83536 q^{83} +4.34220 q^{86} -3.52278 q^{88} -1.21577 q^{89} +4.00000 q^{91} -7.07307 q^{92} +0.382351 q^{94} -9.32497 q^{97} +8.67949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 7 q^{4} - 6 q^{7} - 3 q^{8} - 9 q^{11} - 4 q^{13} + q^{16} - 2 q^{17} + 17 q^{19} + 2 q^{22} - q^{23} + 4 q^{26} - 14 q^{28} - 10 q^{29} + 6 q^{31} + 3 q^{32} + 23 q^{34} - 8 q^{37} - 26 q^{38}+ \cdots - 61 q^{98}+O(q^{100}) $ 6 * q - q^2 + 7 * q^4 - 6 * q^7 - 3 * q^8 - 9 * q^11 - 4 * q^13 + q^16 - 2 * q^17 + 17 * q^19 + 2 * q^22 - q^23 + 4 * q^26 - 14 * q^28 - 10 * q^29 + 6 * q^31 + 3 * q^32 + 23 * q^34 - 8 * q^37 - 26 * q^38 - 6 * q^41 - q^43 - 34 * q^44 - 10 * q^46 - 7 * q^47 + 18 * q^49 - 12 * q^52 + q^53 + 36 * q^56 + 7 * q^58 - 22 * q^59 + 14 * q^61 - q^62 + 9 * q^64 - 7 * q^67 - 37 * q^68 - 5 * q^71 - 4 * q^73 - 30 * q^74 + 18 * q^76 - 4 * q^77 + 19 * q^79 - 16 * q^82 - 19 * q^83 + 5 * q^86 + 46 * q^88 - 14 * q^89 + 24 * q^91 + 8 * q^92 + 35 * q^94 - 34 * q^97 - 61 * 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 $

$2$	0.759793	0.537255	0.268627	−	0.963244i	$-0.413430\pi$
$2$	0.759793	0.537255	0.268627	+	0.963244i	$0.413430\pi$
$3$	0	0
$3$	0	0
$4$	−1.42271	−0.711357
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.29226	−1.62232	−0.811161	−	0.584823i	$-0.801164\pi$
$7$	−4.29226	−1.62232	−0.811161	+	0.584823i	$0.801164\pi$
$8$	−2.60056	−0.919435
$9$	0	0
$10$	0	0
$11$	1.35462	0.408435	0.204217	−	0.978926i	$-0.434535\pi$
$11$	1.35462	0.408435	0.204217	+	0.978926i	$0.434535\pi$
$12$	0	0
$13$	−0.931910	−0.258465	−0.129233	−	0.991614i	$-0.541251\pi$
$13$	−0.931910	−0.258465	−0.129233	+	0.991614i	$0.541251\pi$
$14$	−3.26123	−0.871600
$15$	0	0
$16$	0.869544	0.217386
$17$	3.18251	0.771871	0.385936	−	0.922526i	$-0.373879\pi$
$17$	3.18251	0.771871	0.385936	+	0.922526i	$0.373879\pi$
$18$	0	0
$19$	−1.13077	−0.259417	−0.129709	−	0.991552i	$-0.541404\pi$
$19$	−1.13077	−0.259417	−0.129709	+	0.991552i	$0.541404\pi$
$20$	0	0
$21$	0	0
$22$	1.02923	0.219434
$23$	4.97154	1.03664	0.518318	−	0.855188i	$-0.326558\pi$
$23$	4.97154	1.03664	0.518318	+	0.855188i	$0.326558\pi$
$24$	0	0
$25$	0	0
$26$	−0.708059	−0.138862
$27$	0	0
$28$	6.10666	1.15405
$29$	4.66324	0.865942	0.432971	−	0.901408i	$-0.357465\pi$
$29$	4.66324	0.865942	0.432971	+	0.901408i	$0.357465\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.86178	1.03623
$33$	0	0
$34$	2.41805	0.414692
$35$	0	0
$36$	0	0
$37$	−10.9138	−1.79422	−0.897112	−	0.441804i	$-0.854339\pi$
$37$	−10.9138	−1.79422	−0.897112	+	0.441804i	$0.854339\pi$
$38$	−0.859154	−0.139373
$39$	0	0
$40$	0	0
$41$	11.4282	1.78478	0.892389	−	0.451268i	$-0.149028\pi$
$41$	11.4282	1.78478	0.892389	+	0.451268i	$0.149028\pi$
$42$	0	0
$43$	5.71497	0.871525	0.435763	−	0.900062i	$-0.356479\pi$
$43$	5.71497	0.871525	0.435763	+	0.900062i	$0.356479\pi$
$44$	−1.92724	−0.290543
$45$	0	0
$46$	3.77734	0.556938
$47$	0.503231	0.0734038	0.0367019	−	0.999326i	$-0.488315\pi$
$47$	0.503231	0.0734038	0.0367019	+	0.999326i	$0.488315\pi$
$48$	0	0
$49$	11.4235	1.63193
$50$	0	0
$51$	0	0
$52$	1.32584	0.183861
$53$	2.52183	0.346400	0.173200	−	0.984887i	$-0.444589\pi$
$53$	2.52183	0.346400	0.173200	+	0.984887i	$0.444589\pi$
$54$	0	0
$55$	0	0
$56$	11.1623	1.49162
$57$	0	0
$58$	3.54310	0.465231
$59$	−10.5500	−1.37349	−0.686747	−	0.726896i	$-0.740962\pi$
$59$	−10.5500	−1.37349	−0.686747	+	0.726896i	$0.740962\pi$
$60$	0	0
$61$	3.74164	0.479068	0.239534	−	0.970888i	$-0.423005\pi$
$61$	3.74164	0.479068	0.239534	+	0.970888i	$0.423005\pi$
$62$	0.759793	0.0964938
$63$	0	0
$64$	2.71466	0.339332
$65$	0	0
$66$	0	0
$67$	−10.7877	−1.31793	−0.658965	−	0.752173i	$-0.729006\pi$
$67$	−10.7877	−1.31793	−0.658965	+	0.752173i	$0.729006\pi$
$68$	−4.52780	−0.549076
$69$	0	0
$70$	0	0
$71$	−4.27758	−0.507656	−0.253828	−	0.967249i	$-0.581690\pi$
$71$	−4.27758	−0.507656	−0.253828	+	0.967249i	$0.581690\pi$
$72$	0	0
$73$	−10.0678	−1.17835	−0.589174	−	0.808007i	$-0.700547\pi$
$73$	−10.0678	−1.17835	−0.589174	+	0.808007i	$0.700547\pi$
$74$	−8.29226	−0.963955
$75$	0	0
$76$	1.60877	0.184538
$77$	−5.81440	−0.662612
$78$	0	0
$79$	2.28187	0.256730	0.128365	−	0.991727i	$-0.459027\pi$
$79$	2.28187	0.256730	0.128365	+	0.991727i	$0.459027\pi$
$80$	0	0
$81$	0	0
$82$	8.68303	0.958880
$83$	−8.83536	−0.969806	−0.484903	−	0.874568i	$-0.661145\pi$
$83$	−8.83536	−0.969806	−0.484903	+	0.874568i	$0.661145\pi$
$84$	0	0
$85$	0	0
$86$	4.34220	0.468231
$87$	0	0
$88$	−3.52278	−0.375529
$89$	−1.21577	−0.128872	−0.0644358	−	0.997922i	$-0.520525\pi$
$89$	−1.21577	−0.128872	−0.0644358	+	0.997922i	$0.520525\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	−7.07307	−0.737419
$93$	0	0
$94$	0.382351	0.0394365
$95$	0	0
$96$	0	0
$97$	−9.32497	−0.946807	−0.473404	−	0.880846i	$-0.656975\pi$
$97$	−9.32497	−0.946807	−0.473404	+	0.880846i	$0.656975\pi$
$98$	8.67949	0.876760
$99$	0	0
$100$	0	0
$101$	−11.4627	−1.14058	−0.570288	−	0.821444i	$-0.693169\pi$
$101$	−11.4627	−1.14058	−0.570288	+	0.821444i	$0.693169\pi$
$102$	0	0
$103$	6.26727	0.617533	0.308766	−	0.951138i	$-0.400084\pi$
$103$	6.26727	0.617533	0.308766	+	0.951138i	$0.400084\pi$
$104$	2.42348	0.237642
$105$	0	0
$106$	1.91607	0.186105
$107$	5.72381	0.553342	0.276671	−	0.960965i	$-0.410769\pi$
$107$	5.72381	0.553342	0.276671	+	0.960965i	$0.410769\pi$
$108$	0	0
$109$	−3.77840	−0.361905	−0.180952	−	0.983492i	$-0.557918\pi$
$109$	−3.77840	−0.361905	−0.180952	+	0.983492i	$0.557918\pi$
$110$	0	0
$111$	0	0
$112$	−3.73231	−0.352670
$113$	−10.2428	−0.963565	−0.481783	−	0.876291i	$-0.660010\pi$
$113$	−10.2428	−0.963565	−0.481783	+	0.876291i	$0.660010\pi$
$114$	0	0
$115$	0	0
$116$	−6.63446	−0.615994
$117$	0	0
$118$	−8.01583	−0.737917
$119$	−13.6601	−1.25222
$120$	0	0
$121$	−9.16499	−0.833181
$122$	2.84287	0.257382
$123$	0	0
$124$	−1.42271	−0.127764
$125$	0	0
$126$	0	0
$127$	20.4646	1.81594	0.907971	−	0.419034i	$-0.137631\pi$
$127$	20.4646	1.81594	0.907971	+	0.419034i	$0.137631\pi$
$128$	−9.66099	−0.853919
$129$	0	0
$130$	0	0
$131$	5.92205	0.517412	0.258706	−	0.965956i	$-0.416704\pi$
$131$	5.92205	0.517412	0.258706	+	0.965956i	$0.416704\pi$
$132$	0	0
$133$	4.85357	0.420858
$134$	−8.19644	−0.708065
$135$	0	0
$136$	−8.27629	−0.709686
$137$	−20.0932	−1.71668	−0.858338	−	0.513084i	$-0.828503\pi$
$137$	−20.0932	−1.71668	−0.858338	+	0.513084i	$0.828503\pi$
$138$	0	0
$139$	6.51070	0.552231	0.276115	−	0.961125i	$-0.410953\pi$
$139$	6.51070	0.552231	0.276115	+	0.961125i	$0.410953\pi$
$140$	0	0
$141$	0	0
$142$	−3.25008	−0.272741
$143$	−1.26239	−0.105566
$144$	0	0
$145$	0	0
$146$	−7.64945	−0.633073
$147$	0	0
$148$	15.5273	1.27633
$149$	0.346029	0.0283478	0.0141739	−	0.999900i	$-0.495488\pi$
$149$	0.346029	0.0283478	0.0141739	+	0.999900i	$0.495488\pi$
$150$	0	0
$151$	15.7889	1.28488	0.642442	−	0.766334i	$-0.277921\pi$
$151$	15.7889	1.28488	0.642442	+	0.766334i	$0.277921\pi$
$152$	2.94064	0.238517
$153$	0	0
$154$	−4.41774	−0.355992
$155$	0	0
$156$	0	0
$157$	−19.8274	−1.58240	−0.791198	−	0.611561i	$-0.790542\pi$
$157$	−19.8274	−1.58240	−0.791198	+	0.611561i	$0.790542\pi$
$158$	1.73375	0.137930
$159$	0	0
$160$	0	0
$161$	−21.3391	−1.68176
$162$	0	0
$163$	9.76559	0.764900	0.382450	−	0.923976i	$-0.375080\pi$
$163$	9.76559	0.764900	0.382450	+	0.923976i	$0.375080\pi$
$164$	−16.2590	−1.26961
$165$	0	0
$166$	−6.71304	−0.521033
$167$	3.65200	0.282600	0.141300	−	0.989967i	$-0.454872\pi$
$167$	3.65200	0.282600	0.141300	+	0.989967i	$0.454872\pi$
$168$	0	0
$169$	−12.1315	−0.933196
$170$	0	0
$171$	0	0
$172$	−8.13077	−0.619966
$173$	19.9940	1.52011	0.760056	−	0.649857i	$-0.225171\pi$
$173$	19.9940	1.52011	0.760056	+	0.649857i	$0.225171\pi$
$174$	0	0
$175$	0	0
$176$	1.17791	0.0887880
$177$	0	0
$178$	−0.923736	−0.0692370
$179$	2.08420	0.155781	0.0778904	−	0.996962i	$-0.475182\pi$
$179$	2.08420	0.155781	0.0778904	+	0.996962i	$0.475182\pi$
$180$	0	0
$181$	1.69194	0.125761	0.0628806	−	0.998021i	$-0.479971\pi$
$181$	1.69194	0.125761	0.0628806	+	0.998021i	$0.479971\pi$
$182$	3.03917	0.225278
$183$	0	0
$184$	−12.9288	−0.953120
$185$	0	0
$186$	0	0
$187$	4.31110	0.315259
$188$	−0.715954	−0.0522163
$189$	0	0
$190$	0	0
$191$	−21.1386	−1.52954	−0.764769	−	0.644305i	$-0.777147\pi$
$191$	−21.1386	−1.52954	−0.764769	+	0.644305i	$0.777147\pi$
$192$	0	0
$193$	−1.75322	−0.126200	−0.0630999	−	0.998007i	$-0.520099\pi$
$193$	−1.75322	−0.126200	−0.0630999	+	0.998007i	$0.520099\pi$
$194$	−7.08505	−0.508677
$195$	0	0
$196$	−16.2524	−1.16088
$197$	−12.8682	−0.916825	−0.458412	−	0.888740i	$-0.651582\pi$
$197$	−12.8682	−0.916825	−0.458412	+	0.888740i	$0.651582\pi$
$198$	0	0
$199$	−15.9399	−1.12995	−0.564975	−	0.825108i	$-0.691114\pi$
$199$	−15.9399	−1.12995	−0.564975	+	0.825108i	$0.691114\pi$
$200$	0	0
$201$	0	0
$202$	−8.70925	−0.612781
$203$	−20.0158	−1.40484
$204$	0	0
$205$	0	0
$206$	4.76183	0.331772
$207$	0	0
$208$	−0.810337	−0.0561868
$209$	−1.53177	−0.105955
$210$	0	0
$211$	11.6144	0.799567	0.399783	−	0.916610i	$-0.369085\pi$
$211$	11.6144	0.799567	0.399783	+	0.916610i	$0.369085\pi$
$212$	−3.58785	−0.246414
$213$	0	0
$214$	4.34891	0.297285
$215$	0	0
$216$	0	0
$217$	−4.29226	−0.291378
$218$	−2.87080	−0.194435
$219$	0	0
$220$	0	0
$221$	−2.96581	−0.199502
$222$	0	0
$223$	10.8903	0.729270	0.364635	−	0.931151i	$-0.381194\pi$
$223$	10.8903	0.729270	0.364635	+	0.931151i	$0.381194\pi$
$224$	−25.1603	−1.68109
$225$	0	0
$226$	−7.78244	−0.517680
$227$	−20.7528	−1.37741	−0.688707	−	0.725039i	$-0.741822\pi$
$227$	−20.7528	−1.37741	−0.688707	+	0.725039i	$0.741822\pi$
$228$	0	0
$229$	−15.4796	−1.02292	−0.511462	−	0.859306i	$-0.670896\pi$
$229$	−15.4796	−1.02292	−0.511462	+	0.859306i	$0.670896\pi$
$230$	0	0
$231$	0	0
$232$	−12.1270	−0.796177
$233$	−2.20143	−0.144220	−0.0721102	−	0.997397i	$-0.522973\pi$
$233$	−2.20143	−0.144220	−0.0721102	+	0.997397i	$0.522973\pi$
$234$	0	0
$235$	0	0
$236$	15.0097	0.977045
$237$	0	0
$238$	−10.3789	−0.672763
$239$	−14.8091	−0.957924	−0.478962	−	0.877836i	$-0.658987\pi$
$239$	−14.8091	−0.957924	−0.478962	+	0.877836i	$0.658987\pi$
$240$	0	0
$241$	24.0442	1.54882	0.774411	−	0.632683i	$-0.218046\pi$
$241$	24.0442	1.54882	0.774411	+	0.632683i	$0.218046\pi$
$242$	−6.96350	−0.447631
$243$	0	0
$244$	−5.32329	−0.340789
$245$	0	0
$246$	0	0
$247$	1.05378	0.0670504
$248$	−2.60056	−0.165135
$249$	0	0
$250$	0	0
$251$	4.23395	0.267245	0.133622	−	0.991032i	$-0.457339\pi$
$251$	4.23395	0.267245	0.133622	+	0.991032i	$0.457339\pi$
$252$	0	0
$253$	6.73456	0.423398
$254$	15.5489	0.975623
$255$	0	0
$256$	−12.7697	−0.798104
$257$	−24.3613	−1.51962	−0.759808	−	0.650147i	$-0.774707\pi$
$257$	−24.3613	−1.51962	−0.759808	+	0.650147i	$0.774707\pi$
$258$	0	0
$259$	46.8450	2.91081
$260$	0	0
$261$	0	0
$262$	4.49953	0.277982
$263$	8.58093	0.529123	0.264561	−	0.964369i	$-0.414773\pi$
$263$	8.58093	0.529123	0.264561	+	0.964369i	$0.414773\pi$
$264$	0	0
$265$	0	0
$266$	3.68771	0.226108
$267$	0	0
$268$	15.3479	0.937520
$269$	13.9684	0.851671	0.425835	−	0.904801i	$-0.359980\pi$
$269$	13.9684	0.851671	0.425835	+	0.904801i	$0.359980\pi$
$270$	0	0
$271$	−17.8507	−1.08435	−0.542176	−	0.840265i	$-0.682400\pi$
$271$	−17.8507	−1.08435	−0.542176	+	0.840265i	$0.682400\pi$
$272$	2.76733	0.167794
$273$	0	0
$274$	−15.2667	−0.922293
$275$	0	0
$276$	0	0
$277$	−21.2156	−1.27472	−0.637360	−	0.770566i	$-0.719974\pi$
$277$	−21.2156	−1.27472	−0.637360	+	0.770566i	$0.719974\pi$
$278$	4.94679	0.296689
$279$	0	0
$280$	0	0
$281$	12.3013	0.733834	0.366917	−	0.930254i	$-0.380413\pi$
$281$	12.3013	0.733834	0.366917	+	0.930254i	$0.380413\pi$
$282$	0	0
$283$	−7.30671	−0.434339	−0.217169	−	0.976134i	$-0.569682\pi$
$283$	−7.30671	−0.434339	−0.217169	+	0.976134i	$0.569682\pi$
$284$	6.08578	0.361125
$285$	0	0
$286$	−0.959154	−0.0567160
$287$	−49.0526	−2.89548
$288$	0	0
$289$	−6.87165	−0.404214
$290$	0	0
$291$	0	0
$292$	14.3236	0.838226
$293$	2.40359	0.140419	0.0702097	−	0.997532i	$-0.477633\pi$
$293$	2.40359	0.140419	0.0702097	+	0.997532i	$0.477633\pi$
$294$	0	0
$295$	0	0
$296$	28.3820	1.64967
$297$	0	0
$298$	0.262911	0.0152300
$299$	−4.63302	−0.267935
$300$	0	0
$301$	−24.5301	−1.41389
$302$	11.9963	0.690310
$303$	0	0
$304$	−0.983258	−0.0563937
$305$	0	0
$306$	0	0
$307$	27.1451	1.54925	0.774626	−	0.632419i	$-0.217938\pi$
$307$	27.1451	1.54925	0.774626	+	0.632419i	$0.217938\pi$
$308$	8.27223	0.471354
$309$	0	0
$310$	0	0
$311$	28.3041	1.60498	0.802489	−	0.596667i	$-0.203509\pi$
$311$	28.3041	1.60498	0.802489	+	0.596667i	$0.203509\pi$
$312$	0	0
$313$	18.1802	1.02761	0.513803	−	0.857908i	$-0.328236\pi$
$313$	18.1802	1.02761	0.513803	+	0.857908i	$0.328236\pi$
$314$	−15.0647	−0.850150
$315$	0	0
$316$	−3.24645	−0.182627
$317$	−9.28110	−0.521278	−0.260639	−	0.965436i	$-0.583933\pi$
$317$	−9.28110	−0.521278	−0.260639	+	0.965436i	$0.583933\pi$
$318$	0	0
$319$	6.31694	0.353681
$320$	0	0
$321$	0	0
$322$	−16.2133	−0.903533
$323$	−3.59869	−0.200237
$324$	0	0
$325$	0	0
$326$	7.41983	0.410946
$327$	0	0
$328$	−29.7195	−1.64099
$329$	−2.16000	−0.119084
$330$	0	0
$331$	−6.54506	−0.359749	−0.179874	−	0.983690i	$-0.557569\pi$
$331$	−6.54506	−0.359749	−0.179874	+	0.983690i	$0.557569\pi$
$332$	12.5702	0.689879
$333$	0	0
$334$	2.77477	0.151828
$335$	0	0
$336$	0	0
$337$	1.68787	0.0919442	0.0459721	−	0.998943i	$-0.485361\pi$
$337$	1.68787	0.0919442	0.0459721	+	0.998943i	$0.485361\pi$
$338$	−9.21746	−0.501364
$339$	0	0
$340$	0	0
$341$	1.35462	0.0733570
$342$	0	0
$343$	−18.9867	−1.02519
$344$	−14.8621	−0.801311
$345$	0	0
$346$	15.1913	0.816688
$347$	−8.71061	−0.467610	−0.233805	−	0.972283i	$-0.575118\pi$
$347$	−8.71061	−0.467610	−0.233805	+	0.972283i	$0.575118\pi$
$348$	0	0
$349$	18.8459	1.00880	0.504399	−	0.863471i	$-0.331714\pi$
$349$	18.8459	1.00880	0.504399	+	0.863471i	$0.331714\pi$
$350$	0	0
$351$	0	0
$352$	7.94052	0.423231
$353$	−27.5743	−1.46763	−0.733815	−	0.679349i	$-0.762262\pi$
$353$	−27.5743	−1.46763	−0.733815	+	0.679349i	$0.762262\pi$
$354$	0	0
$355$	0	0
$356$	1.72970	0.0916738
$357$	0	0
$358$	1.58356	0.0836940
$359$	16.4160	0.866401	0.433200	−	0.901298i	$-0.357384\pi$
$359$	16.4160	0.866401	0.433200	+	0.901298i	$0.357384\pi$
$360$	0	0
$361$	−17.7214	−0.932703
$362$	1.28553	0.0675659
$363$	0	0
$364$	−5.69086	−0.298282
$365$	0	0
$366$	0	0
$367$	−11.3882	−0.594461	−0.297231	−	0.954806i	$-0.596063\pi$
$367$	−11.3882	−0.594461	−0.297231	+	0.954806i	$0.596063\pi$
$368$	4.32297	0.225350
$369$	0	0
$370$	0	0
$371$	−10.8244	−0.561973
$372$	0	0
$373$	−27.6598	−1.43217	−0.716086	−	0.698012i	$-0.754068\pi$
$373$	−27.6598	−1.43217	−0.716086	+	0.698012i	$0.754068\pi$
$374$	3.27555	0.169374
$375$	0	0
$376$	−1.30868	−0.0674900
$377$	−4.34572	−0.223816
$378$	0	0
$379$	−21.5637	−1.10765	−0.553827	−	0.832632i	$-0.686833\pi$
$379$	−21.5637	−1.10765	−0.553827	+	0.832632i	$0.686833\pi$
$380$	0	0
$381$	0	0
$382$	−16.0610	−0.821752
$383$	−35.2566	−1.80153	−0.900764	−	0.434309i	$-0.856993\pi$
$383$	−35.2566	−1.80153	−0.900764	+	0.434309i	$0.856993\pi$
$384$	0	0
$385$	0	0
$386$	−1.33209	−0.0678015
$387$	0	0
$388$	13.2668	0.673518
$389$	12.3703	0.627200	0.313600	−	0.949555i	$-0.398465\pi$
$389$	12.3703	0.627200	0.313600	+	0.949555i	$0.398465\pi$
$390$	0	0
$391$	15.8219	0.800150
$392$	−29.7074	−1.50045
$393$	0	0
$394$	−9.77721	−0.492569
$395$	0	0
$396$	0	0
$397$	25.8597	1.29786	0.648931	−	0.760847i	$-0.275216\pi$
$397$	25.8597	1.29786	0.648931	+	0.760847i	$0.275216\pi$
$398$	−12.1110	−0.607071
$399$	0	0
$400$	0	0
$401$	−25.7209	−1.28444	−0.642220	−	0.766520i	$-0.721987\pi$
$401$	−25.7209	−1.28444	−0.642220	+	0.766520i	$0.721987\pi$
$402$	0	0
$403$	−0.931910	−0.0464218
$404$	16.3081	0.811358
$405$	0	0
$406$	−15.2079	−0.754755
$407$	−14.7842	−0.732823
$408$	0	0
$409$	−18.1825	−0.899069	−0.449534	−	0.893263i	$-0.648410\pi$
$409$	−18.1825	−0.899069	−0.449534	+	0.893263i	$0.648410\pi$
$410$	0	0
$411$	0	0
$412$	−8.91654	−0.439286
$413$	45.2834	2.22825
$414$	0	0
$415$	0	0
$416$	−5.46266	−0.267829
$417$	0	0
$418$	−1.16383	−0.0569248
$419$	14.3499	0.701038	0.350519	−	0.936556i	$-0.386005\pi$
$419$	14.3499	0.701038	0.350519	+	0.936556i	$0.386005\pi$
$420$	0	0
$421$	23.9088	1.16525	0.582623	−	0.812743i	$-0.302026\pi$
$421$	23.9088	1.16525	0.582623	+	0.812743i	$0.302026\pi$
$422$	8.82452	0.429571
$423$	0	0
$424$	−6.55817	−0.318493
$425$	0	0
$426$	0	0
$427$	−16.0601	−0.777203
$428$	−8.14335	−0.393623
$429$	0	0
$430$	0	0
$431$	25.8546	1.24537	0.622686	−	0.782472i	$-0.286041\pi$
$431$	25.8546	1.24537	0.622686	+	0.782472i	$0.286041\pi$
$432$	0	0
$433$	−17.4970	−0.840853	−0.420427	−	0.907327i	$-0.638120\pi$
$433$	−17.4970	−0.840853	−0.420427	+	0.907327i	$0.638120\pi$
$434$	−3.26123	−0.156544
$435$	0	0
$436$	5.37558	0.257443
$437$	−5.62168	−0.268921
$438$	0	0
$439$	−11.3551	−0.541948	−0.270974	−	0.962587i	$-0.587346\pi$
$439$	−11.3551	−0.541948	−0.270974	+	0.962587i	$0.587346\pi$
$440$	0	0
$441$	0	0
$442$	−2.25340	−0.107183
$443$	17.0042	0.807892	0.403946	−	0.914783i	$-0.367638\pi$
$443$	17.0042	0.807892	0.403946	+	0.914783i	$0.367638\pi$
$444$	0	0
$445$	0	0
$446$	8.27440	0.391804
$447$	0	0
$448$	−11.6520	−0.550505
$449$	29.0151	1.36931	0.684654	−	0.728868i	$-0.259953\pi$
$449$	29.0151	1.36931	0.684654	+	0.728868i	$0.259953\pi$
$450$	0	0
$451$	15.4809	0.728965
$452$	14.5726	0.685439
$453$	0	0
$454$	−15.7679	−0.740023
$455$	0	0
$456$	0	0
$457$	7.88481	0.368836	0.184418	−	0.982848i	$-0.440960\pi$
$457$	7.88481	0.368836	0.184418	+	0.982848i	$0.440960\pi$
$458$	−11.7613	−0.549571
$459$	0	0
$460$	0	0
$461$	−20.8317	−0.970231	−0.485116	−	0.874450i	$-0.661222\pi$
$461$	−20.8317	−0.970231	−0.485116	+	0.874450i	$0.661222\pi$
$462$	0	0
$463$	−33.0145	−1.53431	−0.767157	−	0.641460i	$-0.778329\pi$
$463$	−33.0145	−1.53431	−0.767157	+	0.641460i	$0.778329\pi$
$464$	4.05489	0.188244
$465$	0	0
$466$	−1.67263	−0.0774831
$467$	−9.98212	−0.461917	−0.230959	−	0.972964i	$-0.574186\pi$
$467$	−9.98212	−0.461917	−0.230959	+	0.972964i	$0.574186\pi$
$468$	0	0
$469$	46.3037	2.13811
$470$	0	0
$471$	0	0
$472$	27.4359	1.26284
$473$	7.74164	0.355961
$474$	0	0
$475$	0	0
$476$	19.4345	0.890778
$477$	0	0
$478$	−11.2519	−0.514649
$479$	16.5755	0.757355	0.378678	−	0.925529i	$-0.376379\pi$
$479$	16.5755	0.757355	0.378678	+	0.925529i	$0.376379\pi$
$480$	0	0
$481$	10.1707	0.463745
$482$	18.2686	0.832112
$483$	0	0
$484$	13.0392	0.592689
$485$	0	0
$486$	0	0
$487$	−31.3786	−1.42190	−0.710951	−	0.703242i	$-0.751735\pi$
$487$	−31.3786	−1.42190	−0.710951	+	0.703242i	$0.751735\pi$
$488$	−9.73035	−0.440472
$489$	0	0
$490$	0	0
$491$	−6.07019	−0.273944	−0.136972	−	0.990575i	$-0.543737\pi$
$491$	−6.07019	−0.273944	−0.136972	+	0.990575i	$0.543737\pi$
$492$	0	0
$493$	14.8408	0.668396
$494$	0.800654	0.0360231
$495$	0	0
$496$	0.869544	0.0390437
$497$	18.3605	0.823581
$498$	0	0
$499$	−2.36790	−0.106002	−0.0530008	−	0.998594i	$-0.516879\pi$
$499$	−2.36790	−0.106002	−0.0530008	+	0.998594i	$0.516879\pi$
$500$	0	0
$501$	0	0
$502$	3.21693	0.143579
$503$	−30.2479	−1.34869	−0.674343	−	0.738418i	$-0.735573\pi$
$503$	−30.2479	−1.34869	−0.674343	+	0.738418i	$0.735573\pi$
$504$	0	0
$505$	0	0
$506$	5.11688	0.227473
$507$	0	0
$508$	−29.1153	−1.29178
$509$	−14.4792	−0.641780	−0.320890	−	0.947116i	$-0.603982\pi$
$509$	−14.4792	−0.641780	−0.320890	+	0.947116i	$0.603982\pi$
$510$	0	0
$511$	43.2136	1.91166
$512$	9.61968	0.425134
$513$	0	0
$514$	−18.5096	−0.816421
$515$	0	0
$516$	0	0
$517$	0.681689	0.0299806
$518$	35.5925	1.56385
$519$	0	0
$520$	0	0
$521$	31.3640	1.37408	0.687041	−	0.726619i	$-0.258909\pi$
$521$	31.3640	1.37408	0.687041	+	0.726619i	$0.258909\pi$
$522$	0	0
$523$	−30.1333	−1.31764	−0.658819	−	0.752302i	$-0.728944\pi$
$523$	−30.1333	−1.31764	−0.658819	+	0.752302i	$0.728944\pi$
$524$	−8.42538	−0.368065
$525$	0	0
$526$	6.51973	0.284274
$527$	3.18251	0.138632
$528$	0	0
$529$	1.71616	0.0746158
$530$	0	0
$531$	0	0
$532$	−6.90525	−0.299380
$533$	−10.6500	−0.461303
$534$	0	0
$535$	0	0
$536$	28.0541	1.21175
$537$	0	0
$538$	10.6131	0.457564
$539$	15.4745	0.666535
$540$	0	0
$541$	28.4347	1.22250	0.611251	−	0.791437i	$-0.290667\pi$
$541$	28.4347	1.22250	0.611251	+	0.791437i	$0.290667\pi$
$542$	−13.5628	−0.582574
$543$	0	0
$544$	18.6552	0.799834
$545$	0	0
$546$	0	0
$547$	30.0885	1.28649	0.643247	−	0.765659i	$-0.277587\pi$
$547$	30.0885	1.28649	0.643247	+	0.765659i	$0.277587\pi$
$548$	28.5869	1.22117
$549$	0	0
$550$	0	0
$551$	−5.27307	−0.224640
$552$	0	0
$553$	−9.79437	−0.416499
$554$	−16.1194	−0.684850
$555$	0	0
$556$	−9.26287	−0.392833
$557$	−2.97797	−0.126181	−0.0630903	−	0.998008i	$-0.520096\pi$
$557$	−2.97797	−0.126181	−0.0630903	+	0.998008i	$0.520096\pi$
$558$	0	0
$559$	−5.32584	−0.225259
$560$	0	0
$561$	0	0
$562$	9.34645	0.394256
$563$	−13.5587	−0.571430	−0.285715	−	0.958315i	$-0.592231\pi$
$563$	−13.5587	−0.571430	−0.285715	+	0.958315i	$0.592231\pi$
$564$	0	0
$565$	0	0
$566$	−5.55159	−0.233351
$567$	0	0
$568$	11.1241	0.466757
$569$	−0.614510	−0.0257616	−0.0128808	−	0.999917i	$-0.504100\pi$
$569$	−0.614510	−0.0257616	−0.0128808	+	0.999917i	$0.504100\pi$
$570$	0	0
$571$	1.23557	0.0517068	0.0258534	−	0.999666i	$-0.491770\pi$
$571$	1.23557	0.0517068	0.0258534	+	0.999666i	$0.491770\pi$
$572$	1.79602	0.0750953
$573$	0	0
$574$	−37.2698	−1.55561
$575$	0	0
$576$	0	0
$577$	−4.04626	−0.168448	−0.0842240	−	0.996447i	$-0.526841\pi$
$577$	−4.04626	−0.168448	−0.0842240	+	0.996447i	$0.526841\pi$
$578$	−5.22103	−0.217166
$579$	0	0
$580$	0	0
$581$	37.9236	1.57334
$582$	0	0
$583$	3.41614	0.141482
$584$	26.1819	1.08341
$585$	0	0
$586$	1.82623	0.0754411
$587$	19.5517	0.806985	0.403493	−	0.914983i	$-0.367796\pi$
$587$	19.5517	0.806985	0.403493	+	0.914983i	$0.367796\pi$
$588$	0	0
$589$	−1.13077	−0.0465927
$590$	0	0
$591$	0	0
$592$	−9.49006	−0.390039
$593$	4.97964	0.204489	0.102245	−	0.994759i	$-0.467398\pi$
$593$	4.97964	0.204489	0.102245	+	0.994759i	$0.467398\pi$
$594$	0	0
$595$	0	0
$596$	−0.492301	−0.0201654
$597$	0	0
$598$	−3.52014	−0.143949
$599$	45.0854	1.84214	0.921069	−	0.389399i	$-0.127317\pi$
$599$	45.0854	1.84214	0.921069	+	0.389399i	$0.127317\pi$
$600$	0	0
$601$	−35.4753	−1.44707	−0.723534	−	0.690289i	$-0.757483\pi$
$601$	−35.4753	−1.44707	−0.723534	+	0.690289i	$0.757483\pi$
$602$	−18.6378	−0.759621
$603$	0	0
$604$	−22.4631	−0.914011
$605$	0	0
$606$	0	0
$607$	−29.1844	−1.18456	−0.592280	−	0.805732i	$-0.701772\pi$
$607$	−29.1844	−1.18456	−0.592280	+	0.805732i	$0.701772\pi$
$608$	−6.62835	−0.268815
$609$	0	0
$610$	0	0
$611$	−0.468966	−0.0189723
$612$	0	0
$613$	27.6353	1.11618	0.558089	−	0.829781i	$-0.311535\pi$
$613$	27.6353	1.11618	0.558089	+	0.829781i	$0.311535\pi$
$614$	20.6247	0.832344
$615$	0	0
$616$	15.1207	0.609229
$617$	−31.6280	−1.27330	−0.636648	−	0.771154i	$-0.719680\pi$
$617$	−31.6280	−1.27330	−0.636648	+	0.771154i	$0.719680\pi$
$618$	0	0
$619$	12.1802	0.489562	0.244781	−	0.969578i	$-0.421284\pi$
$619$	12.1802	0.489562	0.244781	+	0.969578i	$0.421284\pi$
$620$	0	0
$621$	0	0
$622$	21.5052	0.862282
$623$	5.21841	0.209071
$624$	0	0
$625$	0	0
$626$	13.8132	0.552086
$627$	0	0
$628$	28.2087	1.12565
$629$	−34.7334	−1.38491
$630$	0	0
$631$	3.68350	0.146638	0.0733190	−	0.997309i	$-0.476641\pi$
$631$	3.68350	0.146638	0.0733190	+	0.997309i	$0.476641\pi$
$632$	−5.93412	−0.236047
$633$	0	0
$634$	−7.05172	−0.280059
$635$	0	0
$636$	0	0
$637$	−10.6457	−0.421797
$638$	4.79957	0.190017
$639$	0	0
$640$	0	0
$641$	−36.4209	−1.43854	−0.719269	−	0.694731i	$-0.755523\pi$
$641$	−36.4209	−1.43854	−0.719269	+	0.694731i	$0.755523\pi$
$642$	0	0
$643$	−11.2298	−0.442858	−0.221429	−	0.975176i	$-0.571072\pi$
$643$	−11.2298	−0.442858	−0.221429	+	0.975176i	$0.571072\pi$
$644$	30.3595	1.19633
$645$	0	0
$646$	−2.73426	−0.107578
$647$	32.2242	1.26686	0.633432	−	0.773798i	$-0.281645\pi$
$647$	32.2242	1.26686	0.633432	+	0.773798i	$0.281645\pi$
$648$	0	0
$649$	−14.2913	−0.560983
$650$	0	0
$651$	0	0
$652$	−13.8936	−0.544117
$653$	39.4220	1.54270	0.771351	−	0.636410i	$-0.219581\pi$
$653$	39.4220	1.54270	0.771351	+	0.636410i	$0.219581\pi$
$654$	0	0
$655$	0	0
$656$	9.93728	0.387986
$657$	0	0
$658$	−1.64115	−0.0639787
$659$	−36.0597	−1.40469	−0.702343	−	0.711839i	$-0.747863\pi$
$659$	−36.0597	−1.40469	−0.702343	+	0.711839i	$0.747863\pi$
$660$	0	0
$661$	−12.8543	−0.499974	−0.249987	−	0.968249i	$-0.580426\pi$
$661$	−12.8543	−0.499974	−0.249987	+	0.968249i	$0.580426\pi$
$662$	−4.97289	−0.193277
$663$	0	0
$664$	22.9768	0.891674
$665$	0	0
$666$	0	0
$667$	23.1835	0.897667
$668$	−5.19576	−0.201030
$669$	0	0
$670$	0	0
$671$	5.06852	0.195668
$672$	0	0
$673$	−36.1228	−1.39243	−0.696216	−	0.717832i	$-0.745134\pi$
$673$	−36.1228	−1.39243	−0.696216	+	0.717832i	$0.745134\pi$
$674$	1.28243	0.0493975
$675$	0	0
$676$	17.2597	0.663835
$677$	−10.6701	−0.410086	−0.205043	−	0.978753i	$-0.565733\pi$
$677$	−10.6701	−0.410086	−0.205043	+	0.978753i	$0.565733\pi$
$678$	0	0
$679$	40.0252	1.53603
$680$	0	0
$681$	0	0
$682$	1.02923	0.0394114
$683$	−35.2542	−1.34897	−0.674483	−	0.738290i	$-0.735633\pi$
$683$	−35.2542	−1.34897	−0.674483	+	0.738290i	$0.735633\pi$
$684$	0	0
$685$	0	0
$686$	−14.4260	−0.550787
$687$	0	0
$688$	4.96942	0.189457
$689$	−2.35012	−0.0895325
$690$	0	0
$691$	−9.34244	−0.355403	−0.177702	−	0.984084i	$-0.556866\pi$
$691$	−9.34244	−0.355403	−0.177702	+	0.984084i	$0.556866\pi$
$692$	−28.4457	−1.08134
$693$	0	0
$694$	−6.61826	−0.251226
$695$	0	0
$696$	0	0
$697$	36.3702	1.37762
$698$	14.3190	0.541982
$699$	0	0
$700$	0	0
$701$	−10.2061	−0.385479	−0.192739	−	0.981250i	$-0.561737\pi$
$701$	−10.2061	−0.385479	−0.192739	+	0.981250i	$0.561737\pi$
$702$	0	0
$703$	12.3411	0.465452
$704$	3.67734	0.138595
$705$	0	0
$706$	−20.9507	−0.788491
$707$	49.2007	1.85038
$708$	0	0
$709$	−25.1752	−0.945473	−0.472737	−	0.881204i	$-0.656734\pi$
$709$	−25.1752	−0.945473	−0.472737	+	0.881204i	$0.656734\pi$
$710$	0	0
$711$	0	0
$712$	3.16169	0.118489
$713$	4.97154	0.186185
$714$	0	0
$715$	0	0
$716$	−2.96523	−0.110816
$717$	0	0
$718$	12.4727	0.465478
$719$	0.997764	0.0372103	0.0186052	−	0.999827i	$-0.494077\pi$
$719$	0.997764	0.0372103	0.0186052	+	0.999827i	$0.494077\pi$
$720$	0	0
$721$	−26.9007	−1.00184
$722$	−13.4646	−0.501099
$723$	0	0
$724$	−2.40715	−0.0894612
$725$	0	0
$726$	0	0
$727$	17.8993	0.663847	0.331923	−	0.943306i	$-0.392302\pi$
$727$	17.8993	0.663847	0.331923	+	0.943306i	$0.392302\pi$
$728$	−10.4022	−0.385532
$729$	0	0
$730$	0	0
$731$	18.1879	0.672705
$732$	0	0
$733$	−36.4270	−1.34546	−0.672731	−	0.739887i	$-0.734879\pi$
$733$	−36.4270	−1.34546	−0.672731	+	0.739887i	$0.734879\pi$
$734$	−8.65271	−0.319377
$735$	0	0
$736$	29.1421	1.07419
$737$	−14.6133	−0.538289
$738$	0	0
$739$	−13.7225	−0.504789	−0.252394	−	0.967624i	$-0.581218\pi$
$739$	−13.7225	−0.504789	−0.252394	+	0.967624i	$0.581218\pi$
$740$	0	0
$741$	0	0
$742$	−8.22428	−0.301923
$743$	−20.5884	−0.755317	−0.377658	−	0.925945i	$-0.623271\pi$
$743$	−20.5884	−0.755317	−0.377658	+	0.925945i	$0.623271\pi$
$744$	0	0
$745$	0	0
$746$	−21.0157	−0.769441
$747$	0	0
$748$	−6.13347	−0.224262
$749$	−24.5681	−0.897698
$750$	0	0
$751$	−4.23331	−0.154476	−0.0772379	−	0.997013i	$-0.524610\pi$
$751$	−4.23331	−0.154476	−0.0772379	+	0.997013i	$0.524610\pi$
$752$	0.437582	0.0159570
$753$	0	0
$754$	−3.30185	−0.120246
$755$	0	0
$756$	0	0
$757$	43.2109	1.57053	0.785263	−	0.619162i	$-0.212528\pi$
$757$	43.2109	1.57053	0.785263	+	0.619162i	$0.212528\pi$
$758$	−16.3840	−0.595093
$759$	0	0
$760$	0	0
$761$	−52.1836	−1.89165	−0.945827	−	0.324670i	$-0.894747\pi$
$761$	−52.1836	−1.89165	−0.945827	+	0.324670i	$0.894747\pi$
$762$	0	0
$763$	16.2179	0.587126
$764$	30.0742	1.08805
$765$	0	0
$766$	−26.7877	−0.967880
$767$	9.83166	0.355001
$768$	0	0
$769$	−12.6632	−0.456645	−0.228323	−	0.973586i	$-0.573324\pi$
$769$	−12.6632	−0.456645	−0.228323	+	0.973586i	$0.573324\pi$
$770$	0	0
$771$	0	0
$772$	2.49434	0.0897732
$773$	29.7529	1.07014	0.535069	−	0.844809i	$-0.320286\pi$
$773$	29.7529	1.07014	0.535069	+	0.844809i	$0.320286\pi$
$774$	0	0
$775$	0	0
$776$	24.2501	0.870528
$777$	0	0
$778$	9.39888	0.336966
$779$	−12.9226	−0.463002
$780$	0	0
$781$	−5.79452	−0.207344
$782$	12.0214	0.429885
$783$	0	0
$784$	9.93323	0.354758
$785$	0	0
$786$	0	0
$787$	8.35273	0.297743	0.148871	−	0.988857i	$-0.452436\pi$
$787$	8.35273	0.297743	0.148871	+	0.988857i	$0.452436\pi$
$788$	18.3078	0.652190
$789$	0	0
$790$	0	0
$791$	43.9649	1.56321
$792$	0	0
$793$	−3.48688	−0.123823
$794$	19.6480	0.697283
$795$	0	0
$796$	22.6779	0.803798
$797$	−35.2382	−1.24820	−0.624101	−	0.781344i	$-0.714534\pi$
$797$	−35.2382	−1.24820	−0.624101	+	0.781344i	$0.714534\pi$
$798$	0	0
$799$	1.60154	0.0566583
$800$	0	0
$801$	0	0
$802$	−19.5426	−0.690072
$803$	−13.6381	−0.481278
$804$	0	0
$805$	0	0
$806$	−0.708059	−0.0249403
$807$	0	0
$808$	29.8093	1.04869
$809$	−29.4886	−1.03676	−0.518382	−	0.855149i	$-0.673465\pi$
$809$	−29.4886	−1.03676	−0.518382	+	0.855149i	$0.673465\pi$
$810$	0	0
$811$	45.4610	1.59635	0.798176	−	0.602425i	$-0.205799\pi$
$811$	45.4610	1.59635	0.798176	+	0.602425i	$0.205799\pi$
$812$	28.4768	0.999340
$813$	0	0
$814$	−11.2329	−0.393713
$815$	0	0
$816$	0	0
$817$	−6.46234	−0.226089
$818$	−13.8150	−0.483029
$819$	0	0
$820$	0	0
$821$	40.3261	1.40739	0.703695	−	0.710503i	$-0.251532\pi$
$821$	40.3261	1.40739	0.703695	+	0.710503i	$0.251532\pi$
$822$	0	0
$823$	39.9929	1.39406	0.697032	−	0.717040i	$-0.254503\pi$
$823$	39.9929	1.39406	0.697032	+	0.717040i	$0.254503\pi$
$824$	−16.2984	−0.567781
$825$	0	0
$826$	34.4060	1.19714
$827$	5.43802	0.189098	0.0945492	−	0.995520i	$-0.469859\pi$
$827$	5.43802	0.189098	0.0945492	+	0.995520i	$0.469859\pi$
$828$	0	0
$829$	−2.77873	−0.0965094	−0.0482547	−	0.998835i	$-0.515366\pi$
$829$	−2.77873	−0.0965094	−0.0482547	+	0.998835i	$0.515366\pi$
$830$	0	0
$831$	0	0
$832$	−2.52982	−0.0877056
$833$	36.3553	1.25964
$834$	0	0
$835$	0	0
$836$	2.17928	0.0753718
$837$	0	0
$838$	10.9029	0.376636
$839$	−49.1509	−1.69688	−0.848439	−	0.529293i	$-0.822457\pi$
$839$	−49.1509	−1.69688	−0.848439	+	0.529293i	$0.822457\pi$
$840$	0	0
$841$	−7.25420	−0.250145
$842$	18.1658	0.626034
$843$	0	0
$844$	−16.5239	−0.568777
$845$	0	0
$846$	0	0
$847$	39.3385	1.35169
$848$	2.19285	0.0753026
$849$	0	0
$850$	0	0
$851$	−54.2585	−1.85996
$852$	0	0
$853$	−20.7849	−0.711660	−0.355830	−	0.934551i	$-0.615802\pi$
$853$	−20.7849	−0.711660	−0.355830	+	0.934551i	$0.615802\pi$
$854$	−12.2024	−0.417556
$855$	0	0
$856$	−14.8851	−0.508762
$857$	−23.3327	−0.797031	−0.398516	−	0.917162i	$-0.630475\pi$
$857$	−23.3327	−0.797031	−0.398516	+	0.917162i	$0.630475\pi$
$858$	0	0
$859$	29.8998	1.02017	0.510084	−	0.860124i	$-0.329614\pi$
$859$	29.8998	1.02017	0.510084	+	0.860124i	$0.329614\pi$
$860$	0	0
$861$	0	0
$862$	19.6441	0.669082
$863$	−51.4297	−1.75069	−0.875344	−	0.483501i	$-0.839365\pi$
$863$	−51.4297	−1.75069	−0.875344	+	0.483501i	$0.839365\pi$
$864$	0	0
$865$	0	0
$866$	−13.2941	−0.451752
$867$	0	0
$868$	6.10666	0.207273
$869$	3.09107	0.104858
$870$	0	0
$871$	10.0532	0.340640
$872$	9.82593	0.332748
$873$	0	0
$874$	−4.27131	−0.144479
$875$	0	0
$876$	0	0
$877$	−30.3170	−1.02373	−0.511866	−	0.859065i	$-0.671046\pi$
$877$	−30.3170	−1.02373	−0.511866	+	0.859065i	$0.671046\pi$
$878$	−8.62751	−0.291164
$879$	0	0
$880$	0	0
$881$	−49.6125	−1.67149	−0.835743	−	0.549120i	$-0.814963\pi$
$881$	−49.6125	−1.67149	−0.835743	+	0.549120i	$0.814963\pi$
$882$	0	0
$883$	−48.6385	−1.63682	−0.818408	−	0.574638i	$-0.805143\pi$
$883$	−48.6385	−1.63682	−0.818408	+	0.574638i	$0.805143\pi$
$884$	4.21950	0.141917
$885$	0	0
$886$	12.9197	0.434044
$887$	10.9891	0.368978	0.184489	−	0.982835i	$-0.440937\pi$
$887$	10.9891	0.368978	0.184489	+	0.982835i	$0.440937\pi$
$888$	0	0
$889$	−87.8394	−2.94604
$890$	0	0
$891$	0	0
$892$	−15.4938	−0.518771
$893$	−0.569040	−0.0190422
$894$	0	0
$895$	0	0
$896$	41.4675	1.38533
$897$	0	0
$898$	22.0455	0.735668
$899$	4.66324	0.155528
$900$	0	0
$901$	8.02575	0.267377
$902$	11.7622	0.391640
$903$	0	0
$904$	26.6371	0.885935
$905$	0	0
$906$	0	0
$907$	−22.6235	−0.751201	−0.375600	−	0.926782i	$-0.622563\pi$
$907$	−22.6235	−0.751201	−0.375600	+	0.926782i	$0.622563\pi$
$908$	29.5254	0.979834
$909$	0	0
$910$	0	0
$911$	−12.5226	−0.414894	−0.207447	−	0.978246i	$-0.566515\pi$
$911$	−12.5226	−0.414894	−0.207447	+	0.978246i	$0.566515\pi$
$912$	0	0
$913$	−11.9686	−0.396103
$914$	5.99082	0.198159
$915$	0	0
$916$	22.0231	0.727664
$917$	−25.4190	−0.839408
$918$	0	0
$919$	4.33540	0.143012	0.0715059	−	0.997440i	$-0.477220\pi$
$919$	4.33540	0.143012	0.0715059	+	0.997440i	$0.477220\pi$
$920$	0	0
$921$	0	0
$922$	−15.8278	−0.521261
$923$	3.98633	0.131211
$924$	0	0
$925$	0	0
$926$	−25.0842	−0.824317
$927$	0	0
$928$	27.3349	0.897312
$929$	−13.4205	−0.440312	−0.220156	−	0.975465i	$-0.570657\pi$
$929$	−13.4205	−0.440312	−0.220156	+	0.975465i	$0.570657\pi$
$930$	0	0
$931$	−12.9174	−0.423350
$932$	3.13200	0.102592
$933$	0	0
$934$	−7.58434	−0.248167
$935$	0	0
$936$	0	0
$937$	29.3362	0.958372	0.479186	−	0.877713i	$-0.340932\pi$
$937$	29.3362	0.958372	0.479186	+	0.877713i	$0.340932\pi$
$938$	35.1813	1.14871
$939$	0	0
$940$	0	0
$941$	9.99467	0.325817	0.162908	−	0.986641i	$-0.447912\pi$
$941$	9.99467	0.325817	0.162908	+	0.986641i	$0.447912\pi$
$942$	0	0
$943$	56.8155	1.85017
$944$	−9.17370	−0.298579
$945$	0	0
$946$	5.88205	0.191242
$947$	58.8338	1.91184	0.955921	−	0.293624i	$-0.0948613\pi$
$947$	58.8338	1.91184	0.955921	+	0.293624i	$0.0948613\pi$
$948$	0	0
$949$	9.38229	0.304562
$950$	0	0
$951$	0	0
$952$	35.5240	1.15134
$953$	−53.4153	−1.73029	−0.865146	−	0.501520i	$-0.832774\pi$
$953$	−53.4153	−1.73029	−0.865146	+	0.501520i	$0.832774\pi$
$954$	0	0
$955$	0	0
$956$	21.0692	0.681426
$957$	0	0
$958$	12.5940	0.406893
$959$	86.2451	2.78500
$960$	0	0
$961$	1.00000	0.0322581
$962$	7.72764	0.249149
$963$	0	0
$964$	−34.2080	−1.10177
$965$	0	0
$966$	0	0
$967$	−42.0597	−1.35255	−0.676275	−	0.736649i	$-0.736407\pi$
$967$	−42.0597	−1.35255	−0.676275	+	0.736649i	$0.736407\pi$
$968$	23.8341	0.766056
$969$	0	0
$970$	0	0
$971$	28.2778	0.907479	0.453740	−	0.891134i	$-0.350090\pi$
$971$	28.2778	0.907479	0.453740	+	0.891134i	$0.350090\pi$
$972$	0	0
$973$	−27.9456	−0.895896
$974$	−23.8413	−0.763923
$975$	0	0
$976$	3.25352	0.104143
$977$	28.3622	0.907386	0.453693	−	0.891158i	$-0.350106\pi$
$977$	28.3622	0.907386	0.453693	+	0.891158i	$0.350106\pi$
$978$	0	0
$979$	−1.64692	−0.0526357
$980$	0	0
$981$	0	0
$982$	−4.61209	−0.147178
$983$	38.4019	1.22483	0.612416	−	0.790536i	$-0.290198\pi$
$983$	38.4019	1.22483	0.612416	+	0.790536i	$0.290198\pi$
$984$	0	0
$985$	0	0
$986$	11.2759	0.359099
$987$	0	0
$988$	−1.49923	−0.0476968
$989$	28.4122	0.903455
$990$	0	0
$991$	−29.5490	−0.938655	−0.469327	−	0.883024i	$-0.655504\pi$
$991$	−29.5490	−0.938655	−0.469327	+	0.883024i	$0.655504\pi$
$992$	5.86178	0.186112
$993$	0	0
$994$	13.9502	0.442473
$995$	0	0
$996$	0	0
$997$	31.7648	1.00600	0.503001	−	0.864286i	$-0.332229\pi$
$997$	31.7648	1.00600	0.503001	+	0.864286i	$0.332229\pi$
$998$	−1.79911	−0.0569499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bz.1.4		6
3.2	odd	2		2325.2.a.ba.1.3	yes	6
5.4	even	2		6975.2.a.cd.1.3		6
15.2	even	4		2325.2.c.q.1024.6		12
15.8	even	4		2325.2.c.q.1024.7		12
15.14	odd	2		2325.2.a.z.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.z.1.4	✓	6	15.14	odd	2
2325.2.a.ba.1.3	yes	6	3.2	odd	2
2325.2.c.q.1024.6		12	15.2	even	4
2325.2.c.q.1024.7		12	15.8	even	4
6975.2.a.bz.1.4		6	1.1	even	1	trivial
6975.2.a.cd.1.3		6	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+0.759793 q^{2} -1.42271 q^{4} -4.29226 q^{7} -2.60056 q^{8} +1.35462 q^{11} -0.931910 q^{13} -3.26123 q^{14} +0.869544 q^{16} +3.18251 q^{17} -1.13077 q^{19} +1.02923 q^{22} +4.97154 q^{23} -0.708059 q^{26} +6.10666 q^{28} +4.66324 q^{29} +1.00000 q^{31} +5.86178 q^{32} +2.41805 q^{34} -10.9138 q^{37} -0.859154 q^{38} +11.4282 q^{41} +5.71497 q^{43} -1.92724 q^{44} +3.77734 q^{46} +0.503231 q^{47} +11.4235 q^{49} +1.32584 q^{52} +2.52183 q^{53} +11.1623 q^{56} +3.54310 q^{58} -10.5500 q^{59} +3.74164 q^{61} +0.759793 q^{62} +2.71466 q^{64} -10.7877 q^{67} -4.52780 q^{68} -4.27758 q^{71} -10.0678 q^{73} -8.29226 q^{74} +1.60877 q^{76} -5.81440 q^{77} +2.28187 q^{79} +8.68303 q^{82} -8.83536 q^{83} +4.34220 q^{86} -3.52278 q^{88} -1.21577 q^{89} +4.00000 q^{91} -7.07307 q^{92} +0.382351 q^{94} -9.32497 q^{97} +8.67949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 7 q^{4} - 6 q^{7} - 3 q^{8} - 9 q^{11} - 4 q^{13} + q^{16} - 2 q^{17} + 17 q^{19} + 2 q^{22} - q^{23} + 4 q^{26} - 14 q^{28} - 10 q^{29} + 6 q^{31} + 3 q^{32} + 23 q^{34} - 8 q^{37} - 26 q^{38}+ \cdots - 61 q^{98}+O(q^{100}) \) 6 * q - q^2 + 7 * q^4 - 6 * q^7 - 3 * q^8 - 9 * q^11 - 4 * q^13 + q^16 - 2 * q^17 + 17 * q^19 + 2 * q^22 - q^23 + 4 * q^26 - 14 * q^28 - 10 * q^29 + 6 * q^31 + 3 * q^32 + 23 * q^34 - 8 * q^37 - 26 * q^38 - 6 * q^41 - q^43 - 34 * q^44 - 10 * q^46 - 7 * q^47 + 18 * q^49 - 12 * q^52 + q^53 + 36 * q^56 + 7 * q^58 - 22 * q^59 + 14 * q^61 - q^62 + 9 * q^64 - 7 * q^67 - 37 * q^68 - 5 * q^71 - 4 * q^73 - 30 * q^74 + 18 * q^76 - 4 * q^77 + 19 * q^79 - 16 * q^82 - 19 * q^83 + 5 * q^86 + 46 * q^88 - 14 * q^89 + 24 * q^91 + 8 * q^92 + 35 * q^94 - 34 * q^97 - 61 * q^98

Introduction

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