LMFDB - Embedded newform 6975.2.a.by.1.1

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 = [5,4,0,6,0,0,-2,9,0,0,2,0,-4,-2,0,4,19] 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:	$0$
Dimension:	$5$
Coefficient field:	5.5.144209.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - x^{2} + 6x - 1 $ x^5 - 6x^3 - x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 775)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.43848$ 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)

$f(q)$	$=$	$q-1.70816 q^{2} +0.917797 q^{4} -1.50771 q^{7} +1.84857 q^{8} -1.68949 q^{11} -4.04706 q^{13} +2.57540 q^{14} -4.99324 q^{16} -0.146636 q^{17} -3.59081 q^{19} +2.88591 q^{22} +4.77169 q^{23} +6.91300 q^{26} -1.38377 q^{28} -6.52727 q^{29} -1.00000 q^{31} +4.83209 q^{32} +0.250477 q^{34} -5.06769 q^{37} +6.13366 q^{38} -0.252300 q^{41} -0.0635347 q^{43} -1.55061 q^{44} -8.15079 q^{46} -0.392322 q^{47} -4.72682 q^{49} -3.71438 q^{52} +8.47346 q^{53} -2.78710 q^{56} +11.1496 q^{58} -7.08400 q^{59} -0.825505 q^{61} +1.70816 q^{62} +1.73251 q^{64} -13.4370 q^{67} -0.134582 q^{68} -14.1291 q^{71} +8.86670 q^{73} +8.65641 q^{74} -3.29563 q^{76} +2.54725 q^{77} +4.74952 q^{79} +0.430967 q^{82} -4.20538 q^{83} +0.108527 q^{86} -3.12314 q^{88} +7.58991 q^{89} +6.10177 q^{91} +4.37944 q^{92} +0.670146 q^{94} +2.57936 q^{97} +8.07415 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 6 q^{4} - 2 q^{7} + 9 q^{8} + 2 q^{11} - 4 q^{13} - 2 q^{14} + 4 q^{16} + 19 q^{17} + 8 q^{19} + 10 q^{22} + 12 q^{23} + 16 q^{26} - 6 q^{28} - 6 q^{29} - 5 q^{31} + 7 q^{32} + 31 q^{34}+ \cdots + 10 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 6 * q^4 - 2 * q^7 + 9 * q^8 + 2 * q^11 - 4 * q^13 - 2 * q^14 + 4 * q^16 + 19 * q^17 + 8 * q^19 + 10 * q^22 + 12 * q^23 + 16 * q^26 - 6 * q^28 - 6 * q^29 - 5 * q^31 + 7 * q^32 + 31 * q^34 - 16 * q^37 + 14 * q^38 + 2 * q^41 - q^43 + 4 * q^44 - 11 * q^46 + 8 * q^47 - 9 * q^49 + 26 * q^52 + 3 * q^53 + 9 * q^56 - 5 * q^58 + 4 * q^59 - 5 * q^61 - 4 * q^62 - 5 * q^64 - 13 * q^67 + 30 * q^68 - 6 * q^71 - 19 * q^73 - 4 * q^74 - 5 * q^76 + 23 * q^77 - 6 * q^79 + 27 * q^82 + 18 * q^83 + 7 * q^86 + q^88 + 31 * q^89 + 8 * q^91 - 16 * q^92 - 14 * q^94 + q^97 + 10 * 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 $

$2$	−1.70816	−1.20785	−0.603924	−	0.797042i	$-0.706397\pi$
$2$	−1.70816	−1.20785	−0.603924	+	0.797042i	$0.706397\pi$
$3$	0	0
$3$	0	0
$4$	0.917797	0.458898
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.50771	−0.569859	−0.284930	−	0.958548i	$-0.591970\pi$
$7$	−1.50771	−0.569859	−0.284930	+	0.958548i	$0.591970\pi$
$8$	1.84857	0.653569
$9$	0	0
$10$	0	0
$11$	−1.68949	−0.509400	−0.254700	−	0.967020i	$-0.581977\pi$
$11$	−1.68949	−0.509400	−0.254700	+	0.967020i	$0.581977\pi$
$12$	0	0
$13$	−4.04706	−1.12245	−0.561226	−	0.827663i	$-0.689670\pi$
$13$	−4.04706	−1.12245	−0.561226	+	0.827663i	$0.689670\pi$
$14$	2.57540	0.688304
$15$	0	0
$16$	−4.99324	−1.24831
$17$	−0.146636	−0.0355645	−0.0177822	−	0.999842i	$-0.505661\pi$
$17$	−0.146636	−0.0355645	−0.0177822	+	0.999842i	$0.505661\pi$
$18$	0	0
$19$	−3.59081	−0.823788	−0.411894	−	0.911232i	$-0.635133\pi$
$19$	−3.59081	−0.823788	−0.411894	+	0.911232i	$0.635133\pi$
$20$	0	0
$21$	0	0
$22$	2.88591	0.615278
$23$	4.77169	0.994966	0.497483	−	0.867474i	$-0.334258\pi$
$23$	4.77169	0.994966	0.497483	+	0.867474i	$0.334258\pi$
$24$	0	0
$25$	0	0
$26$	6.91300	1.35575
$27$	0	0
$28$	−1.38377	−0.261507
$29$	−6.52727	−1.21208	−0.606042	−	0.795433i	$-0.707244\pi$
$29$	−6.52727	−1.21208	−0.606042	+	0.795433i	$0.707244\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	4.83209	0.854202
$33$	0	0
$34$	0.250477	0.0429565
$35$	0	0
$36$	0	0
$37$	−5.06769	−0.833123	−0.416562	−	0.909107i	$-0.636765\pi$
$37$	−5.06769	−0.833123	−0.416562	+	0.909107i	$0.636765\pi$
$38$	6.13366	0.995011
$39$	0	0
$40$	0	0
$41$	−0.252300	−0.0394026	−0.0197013	−	0.999806i	$-0.506272\pi$
$41$	−0.252300	−0.0394026	−0.0197013	+	0.999806i	$0.506272\pi$
$42$	0	0
$43$	−0.0635347	−0.00968895	−0.00484448	−	0.999988i	$-0.501542\pi$
$43$	−0.0635347	−0.00968895	−0.00484448	+	0.999988i	$0.501542\pi$
$44$	−1.55061	−0.233763
$45$	0	0
$46$	−8.15079	−1.20177
$47$	−0.392322	−0.0572260	−0.0286130	−	0.999591i	$-0.509109\pi$
$47$	−0.392322	−0.0572260	−0.0286130	+	0.999591i	$0.509109\pi$
$48$	0	0
$49$	−4.72682	−0.675261
$50$	0	0
$51$	0	0
$52$	−3.71438	−0.515091
$53$	8.47346	1.16392	0.581959	−	0.813218i	$-0.302286\pi$
$53$	8.47346	1.16392	0.581959	+	0.813218i	$0.302286\pi$
$54$	0	0
$55$	0	0
$56$	−2.78710	−0.372442
$57$	0	0
$58$	11.1496	1.46401
$59$	−7.08400	−0.922258	−0.461129	−	0.887333i	$-0.652556\pi$
$59$	−7.08400	−0.922258	−0.461129	+	0.887333i	$0.652556\pi$
$60$	0	0
$61$	−0.825505	−0.105695	−0.0528475	−	0.998603i	$-0.516830\pi$
$61$	−0.825505	−0.105695	−0.0528475	+	0.998603i	$0.516830\pi$
$62$	1.70816	0.216936
$63$	0	0
$64$	1.73251	0.216564
$65$	0	0
$66$	0	0
$67$	−13.4370	−1.64159	−0.820797	−	0.571221i	$-0.806470\pi$
$67$	−13.4370	−1.64159	−0.820797	+	0.571221i	$0.806470\pi$
$68$	−0.134582	−0.0163205
$69$	0	0
$70$	0	0
$71$	−14.1291	−1.67681	−0.838407	−	0.545044i	$-0.816513\pi$
$71$	−14.1291	−1.67681	−0.838407	+	0.545044i	$0.816513\pi$
$72$	0	0
$73$	8.86670	1.03777	0.518885	−	0.854844i	$-0.326347\pi$
$73$	8.86670	1.03777	0.518885	+	0.854844i	$0.326347\pi$
$74$	8.65641	1.00629
$75$	0	0
$76$	−3.29563	−0.378035
$77$	2.54725	0.290286
$78$	0	0
$79$	4.74952	0.534363	0.267182	−	0.963646i	$-0.413908\pi$
$79$	4.74952	0.534363	0.267182	+	0.963646i	$0.413908\pi$
$80$	0	0
$81$	0	0
$82$	0.430967	0.0475924
$83$	−4.20538	−0.461600	−0.230800	−	0.973001i	$-0.574134\pi$
$83$	−4.20538	−0.461600	−0.230800	+	0.973001i	$0.574134\pi$
$84$	0	0
$85$	0	0
$86$	0.108527	0.0117028
$87$	0	0
$88$	−3.12314	−0.332928
$89$	7.58991	0.804529	0.402264	−	0.915524i	$-0.368223\pi$
$89$	7.58991	0.804529	0.402264	+	0.915524i	$0.368223\pi$
$90$	0	0
$91$	6.10177	0.639639
$92$	4.37944	0.456588
$93$	0	0
$94$	0.670146	0.0691203
$95$	0	0
$96$	0	0
$97$	2.57936	0.261895	0.130947	−	0.991389i	$-0.458198\pi$
$97$	2.57936	0.261895	0.130947	+	0.991389i	$0.458198\pi$
$98$	8.07415	0.815613
$99$	0	0
$100$	0	0
$101$	−5.01677	−0.499187	−0.249593	−	0.968351i	$-0.580297\pi$
$101$	−5.01677	−0.499187	−0.249593	+	0.968351i	$0.580297\pi$
$102$	0	0
$103$	−3.20953	−0.316245	−0.158122	−	0.987420i	$-0.550544\pi$
$103$	−3.20953	−0.316245	−0.158122	+	0.987420i	$0.550544\pi$
$104$	−7.48128	−0.733599
$105$	0	0
$106$	−14.4740	−1.40584
$107$	11.5370	1.11532	0.557662	−	0.830068i	$-0.311699\pi$
$107$	11.5370	1.11532	0.557662	+	0.830068i	$0.311699\pi$
$108$	0	0
$109$	11.7744	1.12778	0.563891	−	0.825849i	$-0.309304\pi$
$109$	11.7744	1.12778	0.563891	+	0.825849i	$0.309304\pi$
$110$	0	0
$111$	0	0
$112$	7.52834	0.711361
$113$	14.7365	1.38630	0.693149	−	0.720795i	$-0.256223\pi$
$113$	14.7365	1.38630	0.693149	+	0.720795i	$0.256223\pi$
$114$	0	0
$115$	0	0
$116$	−5.99071	−0.556223
$117$	0	0
$118$	12.1006	1.11395
$119$	0.221084	0.0202668
$120$	0	0
$121$	−8.14563	−0.740512
$122$	1.41009	0.127664
$123$	0	0
$124$	−0.917797	−0.0824206
$125$	0	0
$126$	0	0
$127$	1.86026	0.165071	0.0825356	−	0.996588i	$-0.473698\pi$
$127$	1.86026	0.165071	0.0825356	+	0.996588i	$0.473698\pi$
$128$	−12.6236	−1.11578
$129$	0	0
$130$	0	0
$131$	−5.96735	−0.521370	−0.260685	−	0.965424i	$-0.583948\pi$
$131$	−5.96735	−0.521370	−0.260685	+	0.965424i	$0.583948\pi$
$132$	0	0
$133$	5.41388	0.469443
$134$	22.9525	1.98280
$135$	0	0
$136$	−0.271067	−0.0232438
$137$	5.44434	0.465141	0.232571	−	0.972580i	$-0.425286\pi$
$137$	5.44434	0.465141	0.232571	+	0.972580i	$0.425286\pi$
$138$	0	0
$139$	−16.0083	−1.35780	−0.678901	−	0.734229i	$-0.737544\pi$
$139$	−16.0083	−1.35780	−0.678901	+	0.734229i	$0.737544\pi$
$140$	0	0
$141$	0	0
$142$	24.1347	2.02534
$143$	6.83745	0.571777
$144$	0	0
$145$	0	0
$146$	−15.1457	−1.25347
$147$	0	0
$148$	−4.65111	−0.382319
$149$	7.04333	0.577012	0.288506	−	0.957478i	$-0.406841\pi$
$149$	7.04333	0.577012	0.288506	+	0.957478i	$0.406841\pi$
$150$	0	0
$151$	4.90355	0.399045	0.199523	−	0.979893i	$-0.436061\pi$
$151$	4.90355	0.399045	0.199523	+	0.979893i	$0.436061\pi$
$152$	−6.63786	−0.538402
$153$	0	0
$154$	−4.35110	−0.350622
$155$	0	0
$156$	0	0
$157$	−1.45325	−0.115982	−0.0579912	−	0.998317i	$-0.518470\pi$
$157$	−1.45325	−0.115982	−0.0579912	+	0.998317i	$0.518470\pi$
$158$	−8.11293	−0.645430
$159$	0	0
$160$	0	0
$161$	−7.19430	−0.566991
$162$	0	0
$163$	24.2080	1.89612	0.948060	−	0.318092i	$-0.103042\pi$
$163$	24.2080	1.89612	0.948060	+	0.318092i	$0.103042\pi$
$164$	−0.231560	−0.0180818
$165$	0	0
$166$	7.18344	0.557543
$167$	−16.3393	−1.26437	−0.632187	−	0.774815i	$-0.717843\pi$
$167$	−16.3393	−1.26437	−0.632187	+	0.774815i	$0.717843\pi$
$168$	0	0
$169$	3.37867	0.259898
$170$	0	0
$171$	0	0
$172$	−0.0583120	−0.00444625
$173$	−4.14414	−0.315073	−0.157537	−	0.987513i	$-0.550355\pi$
$173$	−4.14414	−0.315073	−0.157537	+	0.987513i	$0.550355\pi$
$174$	0	0
$175$	0	0
$176$	8.43602	0.635889
$177$	0	0
$178$	−12.9647	−0.971749
$179$	23.3747	1.74711	0.873553	−	0.486728i	$-0.161810\pi$
$179$	23.3747	1.74711	0.873553	+	0.486728i	$0.161810\pi$
$180$	0	0
$181$	−24.8415	−1.84645	−0.923226	−	0.384256i	$-0.874458\pi$
$181$	−24.8415	−1.84645	−0.923226	+	0.384256i	$0.874458\pi$
$182$	−10.4228	−0.772587
$183$	0	0
$184$	8.82081	0.650279
$185$	0	0
$186$	0	0
$187$	0.247740	0.0181165
$188$	−0.360071	−0.0262609
$189$	0	0
$190$	0	0
$191$	−2.32373	−0.168139	−0.0840697	−	0.996460i	$-0.526792\pi$
$191$	−2.32373	−0.168139	−0.0840697	+	0.996460i	$0.526792\pi$
$192$	0	0
$193$	−4.74287	−0.341399	−0.170700	−	0.985323i	$-0.554603\pi$
$193$	−4.74287	−0.341399	−0.170700	+	0.985323i	$0.554603\pi$
$194$	−4.40596	−0.316329
$195$	0	0
$196$	−4.33826	−0.309876
$197$	14.6528	1.04397	0.521985	−	0.852955i	$-0.325192\pi$
$197$	14.6528	1.04397	0.521985	+	0.852955i	$0.325192\pi$
$198$	0	0
$199$	−10.0654	−0.713517	−0.356758	−	0.934197i	$-0.616118\pi$
$199$	−10.0654	−0.713517	−0.356758	+	0.934197i	$0.616118\pi$
$200$	0	0
$201$	0	0
$202$	8.56942	0.602942
$203$	9.84120	0.690717
$204$	0	0
$205$	0	0
$206$	5.48239	0.381976
$207$	0	0
$208$	20.2079	1.40117
$209$	6.06662	0.419637
$210$	0	0
$211$	11.3609	0.782117	0.391058	−	0.920366i	$-0.372109\pi$
$211$	11.3609	0.782117	0.391058	+	0.920366i	$0.372109\pi$
$212$	7.77691	0.534121
$213$	0	0
$214$	−19.7070	−1.34714
$215$	0	0
$216$	0	0
$217$	1.50771	0.102350
$218$	−20.1125	−1.36219
$219$	0	0
$220$	0	0
$221$	0.593445	0.0399194
$222$	0	0
$223$	10.9582	0.733813	0.366906	−	0.930258i	$-0.380417\pi$
$223$	10.9582	0.733813	0.366906	+	0.930258i	$0.380417\pi$
$224$	−7.28537	−0.486775
$225$	0	0
$226$	−25.1723	−1.67444
$227$	−23.2624	−1.54398	−0.771990	−	0.635634i	$-0.780739\pi$
$227$	−23.2624	−1.54398	−0.771990	+	0.635634i	$0.780739\pi$
$228$	0	0
$229$	−28.9501	−1.91307	−0.956537	−	0.291611i	$-0.905809\pi$
$229$	−28.9501	−1.91307	−0.956537	+	0.291611i	$0.905809\pi$
$230$	0	0
$231$	0	0
$232$	−12.0661	−0.792180
$233$	−17.7216	−1.16098	−0.580489	−	0.814268i	$-0.697139\pi$
$233$	−17.7216	−1.16098	−0.580489	+	0.814268i	$0.697139\pi$
$234$	0	0
$235$	0	0
$236$	−6.50167	−0.423223
$237$	0	0
$238$	−0.377646	−0.0244792
$239$	21.5484	1.39385	0.696926	−	0.717143i	$-0.254550\pi$
$239$	21.5484	1.39385	0.696926	+	0.717143i	$0.254550\pi$
$240$	0	0
$241$	−17.0153	−1.09605	−0.548026	−	0.836461i	$-0.684621\pi$
$241$	−17.0153	−1.09605	−0.548026	+	0.836461i	$0.684621\pi$
$242$	13.9140	0.894426
$243$	0	0
$244$	−0.757646	−0.0485033
$245$	0	0
$246$	0	0
$247$	14.5322	0.924662
$248$	−1.84857	−0.117384
$249$	0	0
$250$	0	0
$251$	7.52627	0.475054	0.237527	−	0.971381i	$-0.423663\pi$
$251$	7.52627	0.475054	0.237527	+	0.971381i	$0.423663\pi$
$252$	0	0
$253$	−8.06171	−0.506836
$254$	−3.17761	−0.199381
$255$	0	0
$256$	18.0980	1.13113
$257$	21.4877	1.34037	0.670184	−	0.742195i	$-0.266215\pi$
$257$	21.4877	1.34037	0.670184	+	0.742195i	$0.266215\pi$
$258$	0	0
$259$	7.64058	0.474763
$260$	0	0
$261$	0	0
$262$	10.1932	0.629736
$263$	7.79202	0.480477	0.240238	−	0.970714i	$-0.422774\pi$
$263$	7.79202	0.480477	0.240238	+	0.970714i	$0.422774\pi$
$264$	0	0
$265$	0	0
$266$	−9.24775	−0.567016
$267$	0	0
$268$	−12.3325	−0.753325
$269$	−27.1817	−1.65730	−0.828649	−	0.559768i	$-0.810890\pi$
$269$	−27.1817	−1.65730	−0.828649	+	0.559768i	$0.810890\pi$
$270$	0	0
$271$	6.46754	0.392875	0.196438	−	0.980516i	$-0.437063\pi$
$271$	6.46754	0.392875	0.196438	+	0.980516i	$0.437063\pi$
$272$	0.732190	0.0443955
$273$	0	0
$274$	−9.29978	−0.561820
$275$	0	0
$276$	0	0
$277$	−18.3195	−1.10071	−0.550357	−	0.834930i	$-0.685508\pi$
$277$	−18.3195	−1.10071	−0.550357	+	0.834930i	$0.685508\pi$
$278$	27.3446	1.64002
$279$	0	0
$280$	0	0
$281$	15.5765	0.929217	0.464609	−	0.885516i	$-0.346195\pi$
$281$	15.5765	0.929217	0.464609	+	0.885516i	$0.346195\pi$
$282$	0	0
$283$	29.7118	1.76619	0.883093	−	0.469199i	$-0.155457\pi$
$283$	29.7118	1.76619	0.883093	+	0.469199i	$0.155457\pi$
$284$	−12.9676	−0.769488
$285$	0	0
$286$	−11.6794	−0.690620
$287$	0.380394	0.0224539
$288$	0	0
$289$	−16.9785	−0.998735
$290$	0	0
$291$	0	0
$292$	8.13783	0.476231
$293$	5.90735	0.345111	0.172556	−	0.985000i	$-0.444798\pi$
$293$	5.90735	0.345111	0.172556	+	0.985000i	$0.444798\pi$
$294$	0	0
$295$	0	0
$296$	−9.36799	−0.544503
$297$	0	0
$298$	−12.0311	−0.696943
$299$	−19.3113	−1.11680
$300$	0	0
$301$	0.0957916	0.00552134
$302$	−8.37603	−0.481986
$303$	0	0
$304$	17.9298	1.02834
$305$	0	0
$306$	0	0
$307$	−24.5783	−1.40276	−0.701379	−	0.712788i	$-0.747432\pi$
$307$	−24.5783	−1.40276	−0.701379	+	0.712788i	$0.747432\pi$
$308$	2.33786	0.133212
$309$	0	0
$310$	0	0
$311$	−5.20092	−0.294917	−0.147459	−	0.989068i	$-0.547109\pi$
$311$	−5.20092	−0.294917	−0.147459	+	0.989068i	$0.547109\pi$
$312$	0	0
$313$	−25.0015	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$313$	−25.0015	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$314$	2.48239	0.140089
$315$	0	0
$316$	4.35910	0.245218
$317$	26.4784	1.48717	0.743587	−	0.668640i	$-0.233123\pi$
$317$	26.4784	1.48717	0.743587	+	0.668640i	$0.233123\pi$
$318$	0	0
$319$	11.0277	0.617435
$320$	0	0
$321$	0	0
$322$	12.2890	0.684839
$323$	0.526542	0.0292976
$324$	0	0
$325$	0	0
$326$	−41.3511	−2.29023
$327$	0	0
$328$	−0.466394	−0.0257523
$329$	0.591505	0.0326107
$330$	0	0
$331$	−9.66552	−0.531265	−0.265633	−	0.964074i	$-0.585581\pi$
$331$	−9.66552	−0.531265	−0.265633	+	0.964074i	$0.585581\pi$
$332$	−3.85968	−0.211828
$333$	0	0
$334$	27.9101	1.52717
$335$	0	0
$336$	0	0
$337$	−21.2134	−1.15557	−0.577783	−	0.816190i	$-0.696082\pi$
$337$	−21.2134	−1.15557	−0.577783	+	0.816190i	$0.696082\pi$
$338$	−5.77130	−0.313917
$339$	0	0
$340$	0	0
$341$	1.68949	0.0914909
$342$	0	0
$343$	17.6806	0.954662
$344$	−0.117448	−0.00633240
$345$	0	0
$346$	7.07884	0.380561
$347$	15.8255	0.849560	0.424780	−	0.905297i	$-0.360352\pi$
$347$	15.8255	0.849560	0.424780	+	0.905297i	$0.360352\pi$
$348$	0	0
$349$	34.6360	1.85402	0.927011	−	0.375034i	$-0.122369\pi$
$349$	34.6360	1.85402	0.927011	+	0.375034i	$0.122369\pi$
$350$	0	0
$351$	0	0
$352$	−8.16376	−0.435130
$353$	29.3051	1.55975	0.779875	−	0.625935i	$-0.215282\pi$
$353$	29.3051	1.55975	0.779875	+	0.625935i	$0.215282\pi$
$354$	0	0
$355$	0	0
$356$	6.96599	0.369197
$357$	0	0
$358$	−39.9276	−2.11024
$359$	34.9463	1.84439	0.922197	−	0.386721i	$-0.126392\pi$
$359$	34.9463	1.84439	0.922197	+	0.386721i	$0.126392\pi$
$360$	0	0
$361$	−6.10611	−0.321374
$362$	42.4331	2.23024
$363$	0	0
$364$	5.60019	0.293529
$365$	0	0
$366$	0	0
$367$	11.7486	0.613272	0.306636	−	0.951827i	$-0.400797\pi$
$367$	11.7486	0.613272	0.306636	+	0.951827i	$0.400797\pi$
$368$	−23.8262	−1.24203
$369$	0	0
$370$	0	0
$371$	−12.7755	−0.663270
$372$	0	0
$373$	22.2888	1.15407	0.577034	−	0.816720i	$-0.304210\pi$
$373$	22.2888	1.15407	0.577034	+	0.816720i	$0.304210\pi$
$374$	−0.423179	−0.0218820
$375$	0	0
$376$	−0.725234	−0.0374011
$377$	26.4162	1.36051
$378$	0	0
$379$	28.1163	1.44424	0.722118	−	0.691770i	$-0.243169\pi$
$379$	28.1163	1.44424	0.722118	+	0.691770i	$0.243169\pi$
$380$	0	0
$381$	0	0
$382$	3.96930	0.203087
$383$	−24.5287	−1.25336	−0.626679	−	0.779277i	$-0.715586\pi$
$383$	−24.5287	−1.25336	−0.626679	+	0.779277i	$0.715586\pi$
$384$	0	0
$385$	0	0
$386$	8.10156	0.412359
$387$	0	0
$388$	2.36733	0.120183
$389$	−29.6625	−1.50395	−0.751974	−	0.659193i	$-0.770898\pi$
$389$	−29.6625	−1.50395	−0.751974	+	0.659193i	$0.770898\pi$
$390$	0	0
$391$	−0.699703	−0.0353855
$392$	−8.73787	−0.441329
$393$	0	0
$394$	−25.0293	−1.26096
$395$	0	0
$396$	0	0
$397$	32.9987	1.65616	0.828078	−	0.560613i	$-0.189435\pi$
$397$	32.9987	1.65616	0.828078	+	0.560613i	$0.189435\pi$
$398$	17.1933	0.861820
$399$	0	0
$400$	0	0
$401$	18.3043	0.914072	0.457036	−	0.889448i	$-0.348911\pi$
$401$	18.3043	0.914072	0.457036	+	0.889448i	$0.348911\pi$
$402$	0	0
$403$	4.04706	0.201598
$404$	−4.60437	−0.229076
$405$	0	0
$406$	−16.8103	−0.834282
$407$	8.56180	0.424393
$408$	0	0
$409$	32.7463	1.61920	0.809601	−	0.586980i	$-0.199683\pi$
$409$	32.7463	1.61920	0.809601	+	0.586980i	$0.199683\pi$
$410$	0	0
$411$	0	0
$412$	−2.94570	−0.145124
$413$	10.6806	0.525557
$414$	0	0
$415$	0	0
$416$	−19.5558	−0.958800
$417$	0	0
$418$	−10.3627	−0.506858
$419$	9.46641	0.462465	0.231232	−	0.972899i	$-0.425724\pi$
$419$	9.46641	0.462465	0.231232	+	0.972899i	$0.425724\pi$
$420$	0	0
$421$	−26.2012	−1.27697	−0.638484	−	0.769635i	$-0.720438\pi$
$421$	−26.2012	−1.27697	−0.638484	+	0.769635i	$0.720438\pi$
$422$	−19.4062	−0.944679
$423$	0	0
$424$	15.6638	0.760701
$425$	0	0
$426$	0	0
$427$	1.24462	0.0602313
$428$	10.5886	0.511820
$429$	0	0
$430$	0	0
$431$	7.84987	0.378115	0.189058	−	0.981966i	$-0.439457\pi$
$431$	7.84987	0.378115	0.189058	+	0.981966i	$0.439457\pi$
$432$	0	0
$433$	18.7559	0.901349	0.450674	−	0.892688i	$-0.351184\pi$
$433$	18.7559	0.901349	0.450674	+	0.892688i	$0.351184\pi$
$434$	−2.57540	−0.123623
$435$	0	0
$436$	10.8065	0.517537
$437$	−17.1342	−0.819641
$438$	0	0
$439$	2.29975	0.109761	0.0548806	−	0.998493i	$-0.482522\pi$
$439$	2.29975	0.109761	0.0548806	+	0.998493i	$0.482522\pi$
$440$	0	0
$441$	0	0
$442$	−1.01370	−0.0482166
$443$	−16.1095	−0.765385	−0.382692	−	0.923876i	$-0.625003\pi$
$443$	−16.1095	−0.765385	−0.382692	+	0.923876i	$0.625003\pi$
$444$	0	0
$445$	0	0
$446$	−18.7182	−0.886335
$447$	0	0
$448$	−2.61212	−0.123411
$449$	−2.51235	−0.118565	−0.0592825	−	0.998241i	$-0.518881\pi$
$449$	−2.51235	−0.118565	−0.0592825	+	0.998241i	$0.518881\pi$
$450$	0	0
$451$	0.426257	0.0200717
$452$	13.5252	0.636170
$453$	0	0
$454$	39.7358	1.86490
$455$	0	0
$456$	0	0
$457$	−8.32036	−0.389210	−0.194605	−	0.980882i	$-0.562342\pi$
$457$	−8.32036	−0.389210	−0.194605	+	0.980882i	$0.562342\pi$
$458$	49.4512	2.31070
$459$	0	0
$460$	0	0
$461$	−12.9178	−0.601640	−0.300820	−	0.953681i	$-0.597260\pi$
$461$	−12.9178	−0.601640	−0.300820	+	0.953681i	$0.597260\pi$
$462$	0	0
$463$	−8.54868	−0.397291	−0.198645	−	0.980071i	$-0.563654\pi$
$463$	−8.54868	−0.397291	−0.198645	+	0.980071i	$0.563654\pi$
$464$	32.5923	1.51306
$465$	0	0
$466$	30.2712	1.40229
$467$	−27.6109	−1.27768	−0.638840	−	0.769339i	$-0.720586\pi$
$467$	−27.6109	−1.27768	−0.638840	+	0.769339i	$0.720586\pi$
$468$	0	0
$469$	20.2591	0.935477
$470$	0	0
$471$	0	0
$472$	−13.0953	−0.602759
$473$	0.107341	0.00493555
$474$	0	0
$475$	0	0
$476$	0.202910	0.00930038
$477$	0	0
$478$	−36.8081	−1.68356
$479$	19.0396	0.869943	0.434971	−	0.900444i	$-0.356758\pi$
$479$	19.0396	0.869943	0.434971	+	0.900444i	$0.356758\pi$
$480$	0	0
$481$	20.5092	0.935141
$482$	29.0648	1.32387
$483$	0	0
$484$	−7.47604	−0.339820
$485$	0	0
$486$	0	0
$487$	2.70816	0.122718	0.0613591	−	0.998116i	$-0.480457\pi$
$487$	2.70816	0.122718	0.0613591	+	0.998116i	$0.480457\pi$
$488$	−1.52601	−0.0690790
$489$	0	0
$490$	0	0
$491$	17.3623	0.783549	0.391774	−	0.920061i	$-0.371861\pi$
$491$	17.3623	0.783549	0.391774	+	0.920061i	$0.371861\pi$
$492$	0	0
$493$	0.957134	0.0431072
$494$	−24.8233	−1.11685
$495$	0	0
$496$	4.99324	0.224203
$497$	21.3025	0.955548
$498$	0	0
$499$	−5.49072	−0.245799	−0.122899	−	0.992419i	$-0.539219\pi$
$499$	−5.49072	−0.245799	−0.122899	+	0.992419i	$0.539219\pi$
$500$	0	0
$501$	0	0
$502$	−12.8560	−0.573793
$503$	0.0479977	0.00214011	0.00107006	−	0.999999i	$-0.499659\pi$
$503$	0.0479977	0.00214011	0.00107006	+	0.999999i	$0.499659\pi$
$504$	0	0
$505$	0	0
$506$	13.7707	0.612181
$507$	0	0
$508$	1.70734	0.0757509
$509$	27.5596	1.22156	0.610778	−	0.791802i	$-0.290857\pi$
$509$	27.5596	1.22156	0.610778	+	0.791802i	$0.290857\pi$
$510$	0	0
$511$	−13.3684	−0.591382
$512$	−5.66708	−0.250452
$513$	0	0
$514$	−36.7044	−1.61896
$515$	0	0
$516$	0	0
$517$	0.662822	0.0291509
$518$	−13.0513	−0.573442
$519$	0	0
$520$	0	0
$521$	25.3222	1.10939	0.554693	−	0.832055i	$-0.312836\pi$
$521$	25.3222	1.10939	0.554693	+	0.832055i	$0.312836\pi$
$522$	0	0
$523$	−4.21023	−0.184100	−0.0920502	−	0.995754i	$-0.529342\pi$
$523$	−4.21023	−0.184100	−0.0920502	+	0.995754i	$0.529342\pi$
$524$	−5.47681	−0.239256
$525$	0	0
$526$	−13.3100	−0.580343
$527$	0.146636	0.00638757
$528$	0	0
$529$	−0.230968	−0.0100421
$530$	0	0
$531$	0	0
$532$	4.96884	0.215427
$533$	1.02107	0.0442275
$534$	0	0
$535$	0	0
$536$	−24.8393	−1.07289
$537$	0	0
$538$	46.4306	2.00177
$539$	7.98591	0.343978
$540$	0	0
$541$	7.18074	0.308724	0.154362	−	0.988014i	$-0.450668\pi$
$541$	7.18074	0.308724	0.154362	+	0.988014i	$0.450668\pi$
$542$	−11.0476	−0.474534
$543$	0	0
$544$	−0.708560	−0.0303793
$545$	0	0
$546$	0	0
$547$	−11.2805	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$547$	−11.2805	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$548$	4.99680	0.213453
$549$	0	0
$550$	0	0
$551$	23.4382	0.998500
$552$	0	0
$553$	−7.16088	−0.304512
$554$	31.2926	1.32950
$555$	0	0
$556$	−14.6923	−0.623094
$557$	−4.29074	−0.181805	−0.0909023	−	0.995860i	$-0.528975\pi$
$557$	−4.29074	−0.181805	−0.0909023	+	0.995860i	$0.528975\pi$
$558$	0	0
$559$	0.257129	0.0108754
$560$	0	0
$561$	0	0
$562$	−26.6071	−1.12235
$563$	−4.71566	−0.198741	−0.0993707	−	0.995050i	$-0.531683\pi$
$563$	−4.71566	−0.198741	−0.0993707	+	0.995050i	$0.531683\pi$
$564$	0	0
$565$	0	0
$566$	−50.7525	−2.13328
$567$	0	0
$568$	−26.1186	−1.09591
$569$	−6.01377	−0.252110	−0.126055	−	0.992023i	$-0.540232\pi$
$569$	−6.01377	−0.252110	−0.126055	+	0.992023i	$0.540232\pi$
$570$	0	0
$571$	−2.62131	−0.109699	−0.0548493	−	0.998495i	$-0.517468\pi$
$571$	−2.62131	−0.109699	−0.0548493	+	0.998495i	$0.517468\pi$
$572$	6.27539	0.262387
$573$	0	0
$574$	−0.649772	−0.0271210
$575$	0	0
$576$	0	0
$577$	28.2940	1.17790	0.588948	−	0.808171i	$-0.299542\pi$
$577$	28.2940	1.17790	0.588948	+	0.808171i	$0.299542\pi$
$578$	29.0019	1.20632
$579$	0	0
$580$	0	0
$581$	6.34047	0.263047
$582$	0	0
$583$	−14.3158	−0.592900
$584$	16.3907	0.678254
$585$	0	0
$586$	−10.0907	−0.416842
$587$	3.89919	0.160937	0.0804684	−	0.996757i	$-0.474358\pi$
$587$	3.89919	0.160937	0.0804684	+	0.996757i	$0.474358\pi$
$588$	0	0
$589$	3.59081	0.147957
$590$	0	0
$591$	0	0
$592$	25.3042	1.04000
$593$	13.1943	0.541826	0.270913	−	0.962604i	$-0.412674\pi$
$593$	13.1943	0.541826	0.270913	+	0.962604i	$0.412674\pi$
$594$	0	0
$595$	0	0
$596$	6.46435	0.264790
$597$	0	0
$598$	32.9867	1.34893
$599$	24.8218	1.01419	0.507095	−	0.861890i	$-0.330719\pi$
$599$	24.8218	1.01419	0.507095	+	0.861890i	$0.330719\pi$
$600$	0	0
$601$	24.7518	1.00965	0.504824	−	0.863222i	$-0.331557\pi$
$601$	24.7518	1.00965	0.504824	+	0.863222i	$0.331557\pi$
$602$	−0.163627	−0.00666894
$603$	0	0
$604$	4.50047	0.183121
$605$	0	0
$606$	0	0
$607$	14.1035	0.572445	0.286222	−	0.958163i	$-0.407600\pi$
$607$	14.1035	0.572445	0.286222	+	0.958163i	$0.407600\pi$
$608$	−17.3511	−0.703681
$609$	0	0
$610$	0	0
$611$	1.58775	0.0642334
$612$	0	0
$613$	−40.2582	−1.62602	−0.813008	−	0.582253i	$-0.802171\pi$
$613$	−40.2582	−1.62602	−0.813008	+	0.582253i	$0.802171\pi$
$614$	41.9836	1.69432
$615$	0	0
$616$	4.70877	0.189722
$617$	−2.30815	−0.0929225	−0.0464612	−	0.998920i	$-0.514794\pi$
$617$	−2.30815	−0.0929225	−0.0464612	+	0.998920i	$0.514794\pi$
$618$	0	0
$619$	−13.7118	−0.551126	−0.275563	−	0.961283i	$-0.588864\pi$
$619$	−13.7118	−0.551126	−0.275563	+	0.961283i	$0.588864\pi$
$620$	0	0
$621$	0	0
$622$	8.88398	0.356215
$623$	−11.4433	−0.458468
$624$	0	0
$625$	0	0
$626$	42.7064	1.70689
$627$	0	0
$628$	−1.33379	−0.0532241
$629$	0.743107	0.0296296
$630$	0	0
$631$	39.9282	1.58952	0.794759	−	0.606925i	$-0.207597\pi$
$631$	39.9282	1.58952	0.794759	+	0.606925i	$0.207597\pi$
$632$	8.77983	0.349243
$633$	0	0
$634$	−45.2292	−1.79628
$635$	0	0
$636$	0	0
$637$	19.1297	0.757947
$638$	−18.8371	−0.745768
$639$	0	0
$640$	0	0
$641$	−26.8928	−1.06220	−0.531101	−	0.847309i	$-0.678221\pi$
$641$	−26.8928	−1.06220	−0.531101	+	0.847309i	$0.678221\pi$
$642$	0	0
$643$	2.01612	0.0795079	0.0397539	−	0.999210i	$-0.487343\pi$
$643$	2.01612	0.0795079	0.0397539	+	0.999210i	$0.487343\pi$
$644$	−6.60291	−0.260191
$645$	0	0
$646$	−0.899416	−0.0353871
$647$	−48.0278	−1.88817	−0.944084	−	0.329705i	$-0.893051\pi$
$647$	−48.0278	−1.88817	−0.944084	+	0.329705i	$0.893051\pi$
$648$	0	0
$649$	11.9683	0.469798
$650$	0	0
$651$	0	0
$652$	22.2181	0.870126
$653$	−13.8241	−0.540978	−0.270489	−	0.962723i	$-0.587185\pi$
$653$	−13.8241	−0.540978	−0.270489	+	0.962723i	$0.587185\pi$
$654$	0	0
$655$	0	0
$656$	1.25979	0.0491867
$657$	0	0
$658$	−1.01038	−0.0393888
$659$	−46.1112	−1.79624	−0.898118	−	0.439754i	$-0.855066\pi$
$659$	−46.1112	−1.79624	−0.898118	+	0.439754i	$0.855066\pi$
$660$	0	0
$661$	44.6278	1.73582	0.867910	−	0.496722i	$-0.165463\pi$
$661$	44.6278	1.73582	0.867910	+	0.496722i	$0.165463\pi$
$662$	16.5102	0.641688
$663$	0	0
$664$	−7.77394	−0.301688
$665$	0	0
$666$	0	0
$667$	−31.1461	−1.20598
$668$	−14.9962	−0.580220
$669$	0	0
$670$	0	0
$671$	1.39468	0.0538410
$672$	0	0
$673$	38.3586	1.47861	0.739307	−	0.673368i	$-0.235153\pi$
$673$	38.3586	1.47861	0.739307	+	0.673368i	$0.235153\pi$
$674$	36.2357	1.39575
$675$	0	0
$676$	3.10093	0.119267
$677$	−15.7801	−0.606479	−0.303239	−	0.952914i	$-0.598068\pi$
$677$	−15.7801	−0.606479	−0.303239	+	0.952914i	$0.598068\pi$
$678$	0	0
$679$	−3.88892	−0.149243
$680$	0	0
$681$	0	0
$682$	−2.88591	−0.110507
$683$	−12.3468	−0.472436	−0.236218	−	0.971700i	$-0.575908\pi$
$683$	−12.3468	−0.472436	−0.236218	+	0.971700i	$0.575908\pi$
$684$	0	0
$685$	0	0
$686$	−30.2012	−1.15309
$687$	0	0
$688$	0.317244	0.0120948
$689$	−34.2926	−1.30644
$690$	0	0
$691$	45.3865	1.72658	0.863291	−	0.504706i	$-0.168399\pi$
$691$	45.3865	1.72658	0.863291	+	0.504706i	$0.168399\pi$
$692$	−3.80348	−0.144587
$693$	0	0
$694$	−27.0325	−1.02614
$695$	0	0
$696$	0	0
$697$	0.0369963	0.00140133
$698$	−59.1637	−2.23938
$699$	0	0
$700$	0	0
$701$	39.8104	1.50362	0.751810	−	0.659380i	$-0.229181\pi$
$701$	39.8104	1.50362	0.751810	+	0.659380i	$0.229181\pi$
$702$	0	0
$703$	18.1971	0.686317
$704$	−2.92706	−0.110318
$705$	0	0
$706$	−50.0576	−1.88394
$707$	7.56380	0.284466
$708$	0	0
$709$	17.0671	0.640967	0.320484	−	0.947254i	$-0.396155\pi$
$709$	17.0671	0.640967	0.320484	+	0.947254i	$0.396155\pi$
$710$	0	0
$711$	0	0
$712$	14.0305	0.525815
$713$	−4.77169	−0.178701
$714$	0	0
$715$	0	0
$716$	21.4532	0.801745
$717$	0	0
$718$	−59.6937	−2.22775
$719$	−26.0121	−0.970089	−0.485045	−	0.874489i	$-0.661197\pi$
$719$	−26.0121	−0.970089	−0.485045	+	0.874489i	$0.661197\pi$
$720$	0	0
$721$	4.83903	0.180215
$722$	10.4302	0.388171
$723$	0	0
$724$	−22.7994	−0.847334
$725$	0	0
$726$	0	0
$727$	15.5941	0.578352	0.289176	−	0.957276i	$-0.406619\pi$
$727$	15.5941	0.578352	0.289176	+	0.957276i	$0.406619\pi$
$728$	11.2796	0.418048
$729$	0	0
$730$	0	0
$731$	0.00931649	0.000344583 0
$732$	0	0
$733$	−15.8360	−0.584916	−0.292458	−	0.956278i	$-0.594473\pi$
$733$	−15.8360	−0.584916	−0.292458	+	0.956278i	$0.594473\pi$
$734$	−20.0684	−0.740740
$735$	0	0
$736$	23.0573	0.849902
$737$	22.7017	0.836227
$738$	0	0
$739$	14.0347	0.516276	0.258138	−	0.966108i	$-0.416891\pi$
$739$	14.0347	0.516276	0.258138	+	0.966108i	$0.416891\pi$
$740$	0	0
$741$	0	0
$742$	21.8225	0.801130
$743$	50.6986	1.85995	0.929976	−	0.367621i	$-0.119828\pi$
$743$	50.6986	1.85995	0.929976	+	0.367621i	$0.119828\pi$
$744$	0	0
$745$	0	0
$746$	−38.0727	−1.39394
$747$	0	0
$748$	0.227375	0.00831365
$749$	−17.3944	−0.635577
$750$	0	0
$751$	−1.89436	−0.0691260	−0.0345630	−	0.999403i	$-0.511004\pi$
$751$	−1.89436	−0.0691260	−0.0345630	+	0.999403i	$0.511004\pi$
$752$	1.95896	0.0714358
$753$	0	0
$754$	−45.1231	−1.64329
$755$	0	0
$756$	0	0
$757$	31.4194	1.14196	0.570979	−	0.820965i	$-0.306564\pi$
$757$	31.4194	1.14196	0.570979	+	0.820965i	$0.306564\pi$
$758$	−48.0270	−1.74442
$759$	0	0
$760$	0	0
$761$	−22.0845	−0.800562	−0.400281	−	0.916392i	$-0.631087\pi$
$761$	−22.0845	−0.800562	−0.400281	+	0.916392i	$0.631087\pi$
$762$	0	0
$763$	−17.7523	−0.642677
$764$	−2.13271	−0.0771589
$765$	0	0
$766$	41.8988	1.51387
$767$	28.6694	1.03519
$768$	0	0
$769$	46.3915	1.67292	0.836459	−	0.548029i	$-0.184622\pi$
$769$	46.3915	1.67292	0.836459	+	0.548029i	$0.184622\pi$
$770$	0	0
$771$	0	0
$772$	−4.35299	−0.156668
$773$	31.6637	1.13886	0.569431	−	0.822039i	$-0.307163\pi$
$773$	31.6637	1.13886	0.569431	+	0.822039i	$0.307163\pi$
$774$	0	0
$775$	0	0
$776$	4.76814	0.171166
$777$	0	0
$778$	50.6681	1.81654
$779$	0.905960	0.0324594
$780$	0	0
$781$	23.8709	0.854169
$782$	1.19520	0.0427403
$783$	0	0
$784$	23.6022	0.842935
$785$	0	0
$786$	0	0
$787$	39.0199	1.39091	0.695455	−	0.718570i	$-0.255203\pi$
$787$	39.0199	1.39091	0.695455	+	0.718570i	$0.255203\pi$
$788$	13.4483	0.479076
$789$	0	0
$790$	0	0
$791$	−22.2184	−0.789994
$792$	0	0
$793$	3.34087	0.118638
$794$	−56.3669	−2.00038
$795$	0	0
$796$	−9.23799	−0.327432
$797$	−36.1603	−1.28087	−0.640433	−	0.768014i	$-0.721245\pi$
$797$	−36.1603	−1.28087	−0.640433	+	0.768014i	$0.721245\pi$
$798$	0	0
$799$	0.0575285	0.00203521
$800$	0	0
$801$	0	0
$802$	−31.2666	−1.10406
$803$	−14.9802	−0.528639
$804$	0	0
$805$	0	0
$806$	−6.91300	−0.243500
$807$	0	0
$808$	−9.27385	−0.326253
$809$	44.8762	1.57776	0.788881	−	0.614546i	$-0.210661\pi$
$809$	44.8762	1.57776	0.788881	+	0.614546i	$0.210661\pi$
$810$	0	0
$811$	−23.1881	−0.814243	−0.407121	−	0.913374i	$-0.633467\pi$
$811$	−23.1881	−0.814243	−0.407121	+	0.913374i	$0.633467\pi$
$812$	9.03223	0.316969
$813$	0	0
$814$	−14.6249	−0.512602
$815$	0	0
$816$	0	0
$817$	0.228141	0.00798164
$818$	−55.9359	−1.95575
$819$	0	0
$820$	0	0
$821$	45.2209	1.57822	0.789110	−	0.614252i	$-0.210542\pi$
$821$	45.2209	1.57822	0.789110	+	0.614252i	$0.210542\pi$
$822$	0	0
$823$	15.9408	0.555661	0.277830	−	0.960630i	$-0.410385\pi$
$823$	15.9408	0.555661	0.277830	+	0.960630i	$0.410385\pi$
$824$	−5.93305	−0.206688
$825$	0	0
$826$	−18.2441	−0.634794
$827$	−24.1361	−0.839296	−0.419648	−	0.907687i	$-0.637846\pi$
$827$	−24.1361	−0.839296	−0.419648	+	0.907687i	$0.637846\pi$
$828$	0	0
$829$	9.01909	0.313246	0.156623	−	0.987658i	$-0.449939\pi$
$829$	9.01909	0.313246	0.156623	+	0.987658i	$0.449939\pi$
$830$	0	0
$831$	0	0
$832$	−7.01159	−0.243083
$833$	0.693123	0.0240153
$834$	0	0
$835$	0	0
$836$	5.56793	0.192571
$837$	0	0
$838$	−16.1701	−0.558587
$839$	−37.8252	−1.30587	−0.652936	−	0.757413i	$-0.726463\pi$
$839$	−37.8252	−1.30587	−0.652936	+	0.757413i	$0.726463\pi$
$840$	0	0
$841$	13.6053	0.469148
$842$	44.7557	1.54238
$843$	0	0
$844$	10.4270	0.358912
$845$	0	0
$846$	0	0
$847$	12.2812	0.421987
$848$	−42.3100	−1.45293
$849$	0	0
$850$	0	0
$851$	−24.1815	−0.828930
$852$	0	0
$853$	26.8684	0.919955	0.459978	−	0.887931i	$-0.347857\pi$
$853$	26.8684	0.919955	0.459978	+	0.887931i	$0.347857\pi$
$854$	−2.12600	−0.0727503
$855$	0	0
$856$	21.3270	0.728940
$857$	55.0595	1.88080	0.940398	−	0.340077i	$-0.110453\pi$
$857$	55.0595	1.88080	0.940398	+	0.340077i	$0.110453\pi$
$858$	0	0
$859$	−34.7493	−1.18563	−0.592816	−	0.805338i	$-0.701984\pi$
$859$	−34.7493	−1.18563	−0.592816	+	0.805338i	$0.701984\pi$
$860$	0	0
$861$	0	0
$862$	−13.4088	−0.456706
$863$	23.6439	0.804848	0.402424	−	0.915453i	$-0.368168\pi$
$863$	23.6439	0.804848	0.402424	+	0.915453i	$0.368168\pi$
$864$	0	0
$865$	0	0
$866$	−32.0379	−1.08869
$867$	0	0
$868$	1.38377	0.0469681
$869$	−8.02426	−0.272204
$870$	0	0
$871$	54.3804	1.84261
$872$	21.7658	0.737083
$873$	0	0
$874$	29.2679	0.990002
$875$	0	0
$876$	0	0
$877$	17.2533	0.582604	0.291302	−	0.956631i	$-0.405912\pi$
$877$	17.2533	0.582604	0.291302	+	0.956631i	$0.405912\pi$
$878$	−3.92833	−0.132575
$879$	0	0
$880$	0	0
$881$	57.9997	1.95406	0.977031	−	0.213099i	$-0.0683557\pi$
$881$	57.9997	1.95406	0.977031	+	0.213099i	$0.0683557\pi$
$882$	0	0
$883$	51.3264	1.72727	0.863636	−	0.504116i	$-0.168182\pi$
$883$	51.3264	1.72727	0.863636	+	0.504116i	$0.168182\pi$
$884$	0.544662	0.0183190
$885$	0	0
$886$	27.5175	0.924469
$887$	10.4756	0.351736	0.175868	−	0.984414i	$-0.443727\pi$
$887$	10.4756	0.351736	0.175868	+	0.984414i	$0.443727\pi$
$888$	0	0
$889$	−2.80472	−0.0940673
$890$	0	0
$891$	0	0
$892$	10.0574	0.336745
$893$	1.40875	0.0471420
$894$	0	0
$895$	0	0
$896$	19.0327	0.635837
$897$	0	0
$898$	4.29148	0.143208
$899$	6.52727	0.217697
$900$	0	0
$901$	−1.24252	−0.0413942
$902$	−0.728114	−0.0242435
$903$	0	0
$904$	27.2416	0.906041
$905$	0	0
$906$	0	0
$907$	−17.6369	−0.585625	−0.292813	−	0.956170i	$-0.594591\pi$
$907$	−17.6369	−0.585625	−0.292813	+	0.956170i	$0.594591\pi$
$908$	−21.3502	−0.708530
$909$	0	0
$910$	0	0
$911$	23.6680	0.784156	0.392078	−	0.919932i	$-0.371756\pi$
$911$	23.6680	0.784156	0.392078	+	0.919932i	$0.371756\pi$
$912$	0	0
$913$	7.10494	0.235139
$914$	14.2125	0.470107
$915$	0	0
$916$	−26.5703	−0.877907
$917$	8.99700	0.297107
$918$	0	0
$919$	18.1214	0.597769	0.298884	−	0.954289i	$-0.403385\pi$
$919$	18.1214	0.597769	0.298884	+	0.954289i	$0.403385\pi$
$920$	0	0
$921$	0	0
$922$	22.0656	0.726690
$923$	57.1812	1.88214
$924$	0	0
$925$	0	0
$926$	14.6025	0.479867
$927$	0	0
$928$	−31.5404	−1.03536
$929$	18.6456	0.611741	0.305870	−	0.952073i	$-0.401053\pi$
$929$	18.6456	0.611741	0.305870	+	0.952073i	$0.401053\pi$
$930$	0	0
$931$	16.9731	0.556271
$932$	−16.2648	−0.532771
$933$	0	0
$934$	47.1638	1.54325
$935$	0	0
$936$	0	0
$937$	6.88877	0.225046	0.112523	−	0.993649i	$-0.464107\pi$
$937$	6.88877	0.225046	0.112523	+	0.993649i	$0.464107\pi$
$938$	−34.6056	−1.12991
$939$	0	0
$940$	0	0
$941$	12.6410	0.412085	0.206043	−	0.978543i	$-0.433941\pi$
$941$	12.6410	0.412085	0.206043	+	0.978543i	$0.433941\pi$
$942$	0	0
$943$	−1.20390	−0.0392043
$944$	35.3721	1.15126
$945$	0	0
$946$	−0.183355	−0.00596140
$947$	−25.5531	−0.830365	−0.415182	−	0.909738i	$-0.636282\pi$
$947$	−25.5531	−0.830365	−0.415182	+	0.909738i	$0.636282\pi$
$948$	0	0
$949$	−35.8841	−1.16485
$950$	0	0
$951$	0	0
$952$	0.408690	0.0132457
$953$	35.7180	1.15702	0.578510	−	0.815675i	$-0.303634\pi$
$953$	35.7180	1.15702	0.578510	+	0.815675i	$0.303634\pi$
$954$	0	0
$955$	0	0
$956$	19.7771	0.639637
$957$	0	0
$958$	−32.5227	−1.05076
$959$	−8.20846	−0.265065
$960$	0	0
$961$	1.00000	0.0322581
$962$	−35.0330	−1.12951
$963$	0	0
$964$	−15.6166	−0.502977
$965$	0	0
$966$	0	0
$967$	−28.8255	−0.926964	−0.463482	−	0.886106i	$-0.653400\pi$
$967$	−28.8255	−0.926964	−0.463482	+	0.886106i	$0.653400\pi$
$968$	−15.0578	−0.483976
$969$	0	0
$970$	0	0
$971$	22.0561	0.707815	0.353907	−	0.935280i	$-0.384853\pi$
$971$	22.0561	0.707815	0.353907	+	0.935280i	$0.384853\pi$
$972$	0	0
$973$	24.1357	0.773756
$974$	−4.62595	−0.148225
$975$	0	0
$976$	4.12195	0.131940
$977$	−1.18894	−0.0380375	−0.0190187	−	0.999819i	$-0.506054\pi$
$977$	−1.18894	−0.0380375	−0.0190187	+	0.999819i	$0.506054\pi$
$978$	0	0
$979$	−12.8231	−0.409827
$980$	0	0
$981$	0	0
$982$	−29.6575	−0.946408
$983$	−20.1277	−0.641973	−0.320987	−	0.947084i	$-0.604014\pi$
$983$	−20.1277	−0.641973	−0.320987	+	0.947084i	$0.604014\pi$
$984$	0	0
$985$	0	0
$986$	−1.63493	−0.0520669
$987$	0	0
$988$	13.3376	0.424326
$989$	−0.303168	−0.00964018
$990$	0	0
$991$	−9.37058	−0.297666	−0.148833	−	0.988862i	$-0.547552\pi$
$991$	−9.37058	−0.297666	−0.148833	+	0.988862i	$0.547552\pi$
$992$	−4.83209	−0.153419
$993$	0	0
$994$	−36.3880	−1.15416
$995$	0	0
$996$	0	0
$997$	2.96440	0.0938836	0.0469418	−	0.998898i	$-0.485052\pi$
$997$	2.96440	0.0938836	0.0469418	+	0.998898i	$0.485052\pi$
$998$	9.37901	0.296887
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.by.1.1		5
3.2	odd	2		775.2.a.h.1.5	✓	5
5.4	even	2		6975.2.a.bp.1.5		5
15.2	even	4		775.2.b.g.249.8		10
15.8	even	4		775.2.b.g.249.3		10
15.14	odd	2		775.2.a.k.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
775.2.a.h.1.5	✓	5	3.2	odd	2
775.2.a.k.1.1	yes	5	15.14	odd	2
775.2.b.g.249.3		10	15.8	even	4
775.2.b.g.249.8		10	15.2	even	4
6975.2.a.bp.1.5		5	5.4	even	2
6975.2.a.by.1.1		5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.70816 q^{2} +0.917797 q^{4} -1.50771 q^{7} +1.84857 q^{8} -1.68949 q^{11} -4.04706 q^{13} +2.57540 q^{14} -4.99324 q^{16} -0.146636 q^{17} -3.59081 q^{19} +2.88591 q^{22} +4.77169 q^{23} +6.91300 q^{26} -1.38377 q^{28} -6.52727 q^{29} -1.00000 q^{31} +4.83209 q^{32} +0.250477 q^{34} -5.06769 q^{37} +6.13366 q^{38} -0.252300 q^{41} -0.0635347 q^{43} -1.55061 q^{44} -8.15079 q^{46} -0.392322 q^{47} -4.72682 q^{49} -3.71438 q^{52} +8.47346 q^{53} -2.78710 q^{56} +11.1496 q^{58} -7.08400 q^{59} -0.825505 q^{61} +1.70816 q^{62} +1.73251 q^{64} -13.4370 q^{67} -0.134582 q^{68} -14.1291 q^{71} +8.86670 q^{73} +8.65641 q^{74} -3.29563 q^{76} +2.54725 q^{77} +4.74952 q^{79} +0.430967 q^{82} -4.20538 q^{83} +0.108527 q^{86} -3.12314 q^{88} +7.58991 q^{89} +6.10177 q^{91} +4.37944 q^{92} +0.670146 q^{94} +2.57936 q^{97} +8.07415 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 6 q^{4} - 2 q^{7} + 9 q^{8} + 2 q^{11} - 4 q^{13} - 2 q^{14} + 4 q^{16} + 19 q^{17} + 8 q^{19} + 10 q^{22} + 12 q^{23} + 16 q^{26} - 6 q^{28} - 6 q^{29} - 5 q^{31} + 7 q^{32} + 31 q^{34}+ \cdots + 10 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 6 * q^4 - 2 * q^7 + 9 * q^8 + 2 * q^11 - 4 * q^13 - 2 * q^14 + 4 * q^16 + 19 * q^17 + 8 * q^19 + 10 * q^22 + 12 * q^23 + 16 * q^26 - 6 * q^28 - 6 * q^29 - 5 * q^31 + 7 * q^32 + 31 * q^34 - 16 * q^37 + 14 * q^38 + 2 * q^41 - q^43 + 4 * q^44 - 11 * q^46 + 8 * q^47 - 9 * q^49 + 26 * q^52 + 3 * q^53 + 9 * q^56 - 5 * q^58 + 4 * q^59 - 5 * q^61 - 4 * q^62 - 5 * q^64 - 13 * q^67 + 30 * q^68 - 6 * q^71 - 19 * q^73 - 4 * q^74 - 5 * q^76 + 23 * q^77 - 6 * q^79 + 27 * q^82 + 18 * q^83 + 7 * q^86 + q^88 + 31 * q^89 + 8 * q^91 - 16 * q^92 - 14 * q^94 + q^97 + 10 * q^98

Introduction

L-functions

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