LMFDB - Embedded newform 9075.2.a.dw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 $ x^8 - x^7 - 11x^6 + 9x^5 + 38x^4 - 25x^3 - 41x^2 + 20x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.09438$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09438 q^{2} +1.00000 q^{3} +2.38641 q^{4} -2.09438 q^{6} +2.59420 q^{7} -0.809299 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.09438 q^{2} +1.00000 q^{3} +2.38641 q^{4} -2.09438 q^{6} +2.59420 q^{7} -0.809299 q^{8} +1.00000 q^{9} +2.38641 q^{12} +6.70399 q^{13} -5.43324 q^{14} -3.07785 q^{16} -5.98152 q^{17} -2.09438 q^{18} -2.93650 q^{19} +2.59420 q^{21} +1.71725 q^{23} -0.809299 q^{24} -14.0407 q^{26} +1.00000 q^{27} +6.19084 q^{28} -1.41526 q^{29} +3.54665 q^{31} +8.06478 q^{32} +12.5276 q^{34} +2.38641 q^{36} -3.24168 q^{37} +6.15014 q^{38} +6.70399 q^{39} +9.91526 q^{41} -5.43324 q^{42} +0.378316 q^{43} -3.59657 q^{46} -5.21954 q^{47} -3.07785 q^{48} -0.270111 q^{49} -5.98152 q^{51} +15.9985 q^{52} -2.34136 q^{53} -2.09438 q^{54} -2.09948 q^{56} -2.93650 q^{57} +2.96408 q^{58} -11.8968 q^{59} +12.1112 q^{61} -7.42802 q^{62} +2.59420 q^{63} -10.7350 q^{64} -8.51092 q^{67} -14.2744 q^{68} +1.71725 q^{69} -6.97366 q^{71} -0.809299 q^{72} +2.53943 q^{73} +6.78930 q^{74} -7.00771 q^{76} -14.0407 q^{78} +9.29741 q^{79} +1.00000 q^{81} -20.7663 q^{82} -8.94133 q^{83} +6.19084 q^{84} -0.792336 q^{86} -1.41526 q^{87} +9.83200 q^{89} +17.3915 q^{91} +4.09807 q^{92} +3.54665 q^{93} +10.9317 q^{94} +8.06478 q^{96} +17.0745 q^{97} +0.565714 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 7 q^{4} + q^{6} + 8 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 7 * q^4 + q^6 + 8 * q^7 + 3 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 7 q^{4} + q^{6} + 8 q^{7} + 3 q^{8} + 8 q^{9} + 7 q^{12} + 7 q^{13} - 8 q^{14} - 7 q^{16} - q^{17} + q^{18} + 6 q^{19} + 8 q^{21} + 7 q^{23} + 3 q^{24} - q^{26} + 8 q^{27} + 11 q^{28} + 28 q^{29} + 10 q^{31} + 12 q^{32} - 8 q^{34} + 7 q^{36} + 14 q^{37} + 19 q^{38} + 7 q^{39} + 14 q^{41} - 8 q^{42} + 11 q^{43} + 5 q^{47} - 7 q^{48} + 2 q^{49} - q^{51} + q^{52} - 5 q^{53} + q^{54} - 4 q^{56} + 6 q^{57} - 3 q^{58} - 2 q^{59} + 36 q^{61} - 17 q^{62} + 8 q^{63} - 23 q^{64} + 6 q^{67} - 3 q^{68} + 7 q^{69} - 17 q^{71} + 3 q^{72} + 32 q^{73} + 47 q^{74} + 5 q^{76} - q^{78} + 41 q^{79} + 8 q^{81} - 6 q^{83} + 11 q^{84} - 9 q^{86} + 28 q^{87} + 21 q^{89} - 9 q^{91} + 69 q^{92} + 10 q^{93} + 51 q^{94} + 12 q^{96} - 4 q^{97} - 27 q^{98}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 7 * q^4 + q^6 + 8 * q^7 + 3 * q^8 + 8 * q^9 + 7 * q^12 + 7 * q^13 - 8 * q^14 - 7 * q^16 - q^17 + q^18 + 6 * q^19 + 8 * q^21 + 7 * q^23 + 3 * q^24 - q^26 + 8 * q^27 + 11 * q^28 + 28 * q^29 + 10 * q^31 + 12 * q^32 - 8 * q^34 + 7 * q^36 + 14 * q^37 + 19 * q^38 + 7 * q^39 + 14 * q^41 - 8 * q^42 + 11 * q^43 + 5 * q^47 - 7 * q^48 + 2 * q^49 - q^51 + q^52 - 5 * q^53 + q^54 - 4 * q^56 + 6 * q^57 - 3 * q^58 - 2 * q^59 + 36 * q^61 - 17 * q^62 + 8 * q^63 - 23 * q^64 + 6 * q^67 - 3 * q^68 + 7 * q^69 - 17 * q^71 + 3 * q^72 + 32 * q^73 + 47 * q^74 + 5 * q^76 - q^78 + 41 * q^79 + 8 * q^81 - 6 * q^83 + 11 * q^84 - 9 * q^86 + 28 * q^87 + 21 * q^89 - 9 * q^91 + 69 * q^92 + 10 * q^93 + 51 * q^94 + 12 * q^96 - 4 * q^97 - 27 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.09438	−1.48095	−0.740474	−	0.672085i	$-0.765399\pi$
$2$	−2.09438	−1.48095	−0.740474	+	0.672085i	$0.765399\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.38641	1.19321
$5$	0	0
$5$	0	0
$6$	−2.09438	−0.855026
$7$	2.59420	0.980517	0.490258	−	0.871577i	$-0.336902\pi$
$7$	2.59420	0.980517	0.490258	+	0.871577i	$0.336902\pi$
$8$	−0.809299	−0.286130
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	2.38641	0.688899
$13$	6.70399	1.85935	0.929676	−	0.368379i	$-0.120087\pi$
$13$	6.70399	1.85935	0.929676	+	0.368379i	$0.120087\pi$
$14$	−5.43324	−1.45209
$15$	0	0
$16$	−3.07785	−0.769463
$17$	−5.98152	−1.45073	−0.725366	−	0.688364i	$-0.758329\pi$
$17$	−5.98152	−1.45073	−0.725366	+	0.688364i	$0.758329\pi$
$18$	−2.09438	−0.493649
$19$	−2.93650	−0.673679	−0.336840	−	0.941562i	$-0.609358\pi$
$19$	−2.93650	−0.673679	−0.336840	+	0.941562i	$0.609358\pi$
$20$	0	0
$21$	2.59420	0.566101
$22$	0	0
$23$	1.71725	0.358072	0.179036	−	0.983843i	$-0.442702\pi$
$23$	1.71725	0.358072	0.179036	+	0.983843i	$0.442702\pi$
$24$	−0.809299	−0.165197
$25$	0	0
$26$	−14.0407	−2.75360
$27$	1.00000	0.192450
$28$	6.19084	1.16996
$29$	−1.41526	−0.262807	−0.131403	−	0.991329i	$-0.541948\pi$
$29$	−1.41526	−0.262807	−0.131403	+	0.991329i	$0.541948\pi$
$30$	0	0
$31$	3.54665	0.636997	0.318499	−	0.947923i	$-0.396821\pi$
$31$	3.54665	0.636997	0.318499	+	0.947923i	$0.396821\pi$
$32$	8.06478	1.42567
$33$	0	0
$34$	12.5276	2.14846
$35$	0	0
$36$	2.38641	0.397736
$37$	−3.24168	−0.532929	−0.266464	−	0.963845i	$-0.585855\pi$
$37$	−3.24168	−0.532929	−0.266464	+	0.963845i	$0.585855\pi$
$38$	6.15014	0.997684
$39$	6.70399	1.07350
$40$	0	0
$41$	9.91526	1.54850	0.774251	−	0.632878i	$-0.218127\pi$
$41$	9.91526	1.54850	0.774251	+	0.632878i	$0.218127\pi$
$42$	−5.43324	−0.838367
$43$	0.378316	0.0576926	0.0288463	−	0.999584i	$-0.490817\pi$
$43$	0.378316	0.0576926	0.0288463	+	0.999584i	$0.490817\pi$
$44$	0	0
$45$	0	0
$46$	−3.59657	−0.530286
$47$	−5.21954	−0.761348	−0.380674	−	0.924709i	$-0.624308\pi$
$47$	−5.21954	−0.761348	−0.380674	+	0.924709i	$0.624308\pi$
$48$	−3.07785	−0.444250
$49$	−0.270111	−0.0385873
$50$	0	0
$51$	−5.98152	−0.837580
$52$	15.9985	2.21859
$53$	−2.34136	−0.321610	−0.160805	−	0.986986i	$-0.551409\pi$
$53$	−2.34136	−0.321610	−0.160805	+	0.986986i	$0.551409\pi$
$54$	−2.09438	−0.285009
$55$	0	0
$56$	−2.09948	−0.280555
$57$	−2.93650	−0.388949
$58$	2.96408	0.389203
$59$	−11.8968	−1.54884	−0.774418	−	0.632674i	$-0.781957\pi$
$59$	−11.8968	−1.54884	−0.774418	+	0.632674i	$0.781957\pi$
$60$	0	0
$61$	12.1112	1.55068	0.775342	−	0.631542i	$-0.217578\pi$
$61$	12.1112	1.55068	0.775342	+	0.631542i	$0.217578\pi$
$62$	−7.42802	−0.943360
$63$	2.59420	0.326839
$64$	−10.7350	−1.34187
$65$	0	0
$66$	0	0
$67$	−8.51092	−1.03977	−0.519887	−	0.854235i	$-0.674026\pi$
$67$	−8.51092	−1.03977	−0.519887	+	0.854235i	$0.674026\pi$
$68$	−14.2744	−1.73102
$69$	1.71725	0.206733
$70$	0	0
$71$	−6.97366	−0.827621	−0.413810	−	0.910363i	$-0.635802\pi$
$71$	−6.97366	−0.827621	−0.413810	+	0.910363i	$0.635802\pi$
$72$	−0.809299	−0.0953768
$73$	2.53943	0.297217	0.148609	−	0.988896i	$-0.452521\pi$
$73$	2.53943	0.297217	0.148609	+	0.988896i	$0.452521\pi$
$74$	6.78930	0.789240
$75$	0	0
$76$	−7.00771	−0.803839
$77$	0	0
$78$	−14.0407	−1.58979
$79$	9.29741	1.04604	0.523020	−	0.852320i	$-0.324805\pi$
$79$	9.29741	1.04604	0.523020	+	0.852320i	$0.324805\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−20.7663	−2.29325
$83$	−8.94133	−0.981438	−0.490719	−	0.871318i	$-0.663266\pi$
$83$	−8.94133	−0.981438	−0.490719	+	0.871318i	$0.663266\pi$
$84$	6.19084	0.675477
$85$	0	0
$86$	−0.792336	−0.0854397
$87$	−1.41526	−0.151732
$88$	0	0
$89$	9.83200	1.04219	0.521095	−	0.853499i	$-0.325524\pi$
$89$	9.83200	1.04219	0.521095	+	0.853499i	$0.325524\pi$
$90$	0	0
$91$	17.3915	1.82312
$92$	4.09807	0.427254
$93$	3.54665	0.367770
$94$	10.9317	1.12752
$95$	0	0
$96$	8.06478	0.823108
$97$	17.0745	1.73366	0.866828	−	0.498607i	$-0.166155\pi$
$97$	17.0745	1.73366	0.866828	+	0.498607i	$0.166155\pi$
$98$	0.565714	0.0571458
$99$	0	0
$100$	0	0
$101$	8.39172	0.835008	0.417504	−	0.908675i	$-0.362905\pi$
$101$	8.39172	0.835008	0.417504	+	0.908675i	$0.362905\pi$
$102$	12.5276	1.24041
$103$	9.04392	0.891124	0.445562	−	0.895251i	$-0.353004\pi$
$103$	9.04392	0.891124	0.445562	+	0.895251i	$0.353004\pi$
$104$	−5.42553	−0.532017
$105$	0	0
$106$	4.90369	0.476288
$107$	3.83774	0.371008	0.185504	−	0.982644i	$-0.440608\pi$
$107$	3.83774	0.371008	0.185504	+	0.982644i	$0.440608\pi$
$108$	2.38641	0.229633
$109$	16.9858	1.62694	0.813472	−	0.581604i	$-0.197575\pi$
$109$	16.9858	1.62694	0.813472	+	0.581604i	$0.197575\pi$
$110$	0	0
$111$	−3.24168	−0.307687
$112$	−7.98458	−0.754472
$113$	8.16771	0.768353	0.384177	−	0.923260i	$-0.374485\pi$
$113$	8.16771	0.768353	0.384177	+	0.923260i	$0.374485\pi$
$114$	6.15014	0.576013
$115$	0	0
$116$	−3.37739	−0.313583
$117$	6.70399	0.619784
$118$	24.9165	2.29375
$119$	−15.5173	−1.42247
$120$	0	0
$121$	0	0
$122$	−25.3655	−2.29648
$123$	9.91526	0.894028
$124$	8.46378	0.760070
$125$	0	0
$126$	−5.43324	−0.484031
$127$	−18.9674	−1.68309	−0.841543	−	0.540190i	$-0.818353\pi$
$127$	−18.9674	−1.68309	−0.841543	+	0.540190i	$0.818353\pi$
$128$	6.35355	0.561580
$129$	0.378316	0.0333088
$130$	0	0
$131$	18.1082	1.58212	0.791058	−	0.611741i	$-0.209530\pi$
$131$	18.1082	1.58212	0.791058	+	0.611741i	$0.209530\pi$
$132$	0	0
$133$	−7.61788	−0.660554
$134$	17.8251	1.53985
$135$	0	0
$136$	4.84083	0.415098
$137$	−0.886110	−0.0757055	−0.0378528	−	0.999283i	$-0.512052\pi$
$137$	−0.886110	−0.0757055	−0.0378528	+	0.999283i	$0.512052\pi$
$138$	−3.59657	−0.306161
$139$	11.2359	0.953014	0.476507	−	0.879171i	$-0.341903\pi$
$139$	11.2359	0.953014	0.476507	+	0.879171i	$0.341903\pi$
$140$	0	0
$141$	−5.21954	−0.439565
$142$	14.6055	1.22566
$143$	0	0
$144$	−3.07785	−0.256488
$145$	0	0
$146$	−5.31852	−0.440164
$147$	−0.270111	−0.0222784
$148$	−7.73599	−0.635894
$149$	4.08354	0.334536	0.167268	−	0.985911i	$-0.446505\pi$
$149$	4.08354	0.334536	0.167268	+	0.985911i	$0.446505\pi$
$150$	0	0
$151$	19.2681	1.56802	0.784010	−	0.620749i	$-0.213171\pi$
$151$	19.2681	1.56802	0.784010	+	0.620749i	$0.213171\pi$
$152$	2.37651	0.192760
$153$	−5.98152	−0.483577
$154$	0	0
$155$	0	0
$156$	15.9985	1.28090
$157$	−15.1694	−1.21065	−0.605325	−	0.795978i	$-0.706957\pi$
$157$	−15.1694	−1.21065	−0.605325	+	0.795978i	$0.706957\pi$
$158$	−19.4723	−1.54913
$159$	−2.34136	−0.185682
$160$	0	0
$161$	4.45490	0.351095
$162$	−2.09438	−0.164550
$163$	7.87487	0.616807	0.308404	−	0.951256i	$-0.400205\pi$
$163$	7.87487	0.616807	0.308404	+	0.951256i	$0.400205\pi$
$164$	23.6619	1.84769
$165$	0	0
$166$	18.7265	1.45346
$167$	3.75789	0.290794	0.145397	−	0.989373i	$-0.453554\pi$
$167$	3.75789	0.290794	0.145397	+	0.989373i	$0.453554\pi$
$168$	−2.09948	−0.161979
$169$	31.9434	2.45719
$170$	0	0
$171$	−2.93650	−0.224560
$172$	0.902818	0.0688392
$173$	26.0277	1.97885	0.989425	−	0.145048i	$-0.0463337\pi$
$173$	26.0277	1.97885	0.989425	+	0.145048i	$0.0463337\pi$
$174$	2.96408	0.224707
$175$	0	0
$176$	0	0
$177$	−11.8968	−0.894221
$178$	−20.5919	−1.54343
$179$	−5.82809	−0.435612	−0.217806	−	0.975992i	$-0.569890\pi$
$179$	−5.82809	−0.435612	−0.217806	+	0.975992i	$0.569890\pi$
$180$	0	0
$181$	−0.00869156	−0.000646039 0	−0.000323019	−	1.00000i	$-0.500103\pi$
$181$	−0.00869156	−0.000646039 0	−0.000323019		1.00000i	$0.500103\pi$
$182$	−36.4244	−2.69995
$183$	12.1112	0.895288
$184$	−1.38977	−0.102455
$185$	0	0
$186$	−7.42802	−0.544649
$187$	0	0
$188$	−12.4560	−0.908446
$189$	2.59420	0.188700
$190$	0	0
$191$	−12.3714	−0.895162	−0.447581	−	0.894244i	$-0.647714\pi$
$191$	−12.3714	−0.895162	−0.447581	+	0.894244i	$0.647714\pi$
$192$	−10.7350	−0.774731
$193$	−3.42605	−0.246613	−0.123306	−	0.992369i	$-0.539350\pi$
$193$	−3.42605	−0.246613	−0.123306	+	0.992369i	$0.539350\pi$
$194$	−35.7605	−2.56746
$195$	0	0
$196$	−0.644597	−0.0460426
$197$	−2.12435	−0.151354	−0.0756770	−	0.997132i	$-0.524112\pi$
$197$	−2.12435	−0.151354	−0.0756770	+	0.997132i	$0.524112\pi$
$198$	0	0
$199$	0.644319	0.0456745	0.0228373	−	0.999739i	$-0.492730\pi$
$199$	0.644319	0.0456745	0.0228373	+	0.999739i	$0.492730\pi$
$200$	0	0
$201$	−8.51092	−0.600314
$202$	−17.5754	−1.23660
$203$	−3.67147	−0.257686
$204$	−14.2744	−0.999407
$205$	0	0
$206$	−18.9414	−1.31971
$207$	1.71725	0.119357
$208$	−20.6339	−1.43070
$209$	0	0
$210$	0	0
$211$	−22.1552	−1.52523	−0.762615	−	0.646853i	$-0.776085\pi$
$211$	−22.1552	−1.52523	−0.762615	+	0.646853i	$0.776085\pi$
$212$	−5.58745	−0.383748
$213$	−6.97366	−0.477827
$214$	−8.03767	−0.549444
$215$	0	0
$216$	−0.809299	−0.0550658
$217$	9.20073	0.624586
$218$	−35.5747	−2.40942
$219$	2.53943	0.171599
$220$	0	0
$221$	−40.1000	−2.69742
$222$	6.78930	0.455668
$223$	12.6051	0.844098	0.422049	−	0.906573i	$-0.361311\pi$
$223$	12.6051	0.844098	0.422049	+	0.906573i	$0.361311\pi$
$224$	20.9217	1.39789
$225$	0	0
$226$	−17.1063	−1.13789
$227$	29.3038	1.94496	0.972481	−	0.232984i	$-0.0748490\pi$
$227$	29.3038	1.94496	0.972481	+	0.232984i	$0.0748490\pi$
$228$	−7.00771	−0.464097
$229$	21.1921	1.40042	0.700208	−	0.713939i	$-0.253091\pi$
$229$	21.1921	1.40042	0.700208	+	0.713939i	$0.253091\pi$
$230$	0	0
$231$	0	0
$232$	1.14537	0.0751970
$233$	−24.8127	−1.62553	−0.812767	−	0.582589i	$-0.802040\pi$
$233$	−24.8127	−1.62553	−0.812767	+	0.582589i	$0.802040\pi$
$234$	−14.0407	−0.917868
$235$	0	0
$236$	−28.3908	−1.84808
$237$	9.29741	0.603931
$238$	32.4990	2.10660
$239$	2.18978	0.141645	0.0708224	−	0.997489i	$-0.477438\pi$
$239$	2.18978	0.141645	0.0708224	+	0.997489i	$0.477438\pi$
$240$	0	0
$241$	−7.20534	−0.464137	−0.232068	−	0.972699i	$-0.574549\pi$
$241$	−7.20534	−0.464137	−0.232068	+	0.972699i	$0.574549\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	28.9024	1.85029
$245$	0	0
$246$	−20.7663	−1.32401
$247$	−19.6863	−1.25261
$248$	−2.87030	−0.182264
$249$	−8.94133	−0.566634
$250$	0	0
$251$	−3.68279	−0.232455	−0.116228	−	0.993223i	$-0.537080\pi$
$251$	−3.68279	−0.232455	−0.116228	+	0.993223i	$0.537080\pi$
$252$	6.19084	0.389987
$253$	0	0
$254$	39.7249	2.49256
$255$	0	0
$256$	8.16325	0.510203
$257$	13.6015	0.848439	0.424220	−	0.905559i	$-0.360548\pi$
$257$	13.6015	0.848439	0.424220	+	0.905559i	$0.360548\pi$
$258$	−0.792336	−0.0493287
$259$	−8.40957	−0.522545
$260$	0	0
$261$	−1.41526	−0.0876023
$262$	−37.9253	−2.34303
$263$	2.96621	0.182904	0.0914522	−	0.995809i	$-0.470849\pi$
$263$	2.96621	0.182904	0.0914522	+	0.995809i	$0.470849\pi$
$264$	0	0
$265$	0	0
$266$	15.9547	0.978246
$267$	9.83200	0.601709
$268$	−20.3106	−1.24067
$269$	16.6684	1.01629	0.508146	−	0.861271i	$-0.330331\pi$
$269$	16.6684	1.01629	0.508146	+	0.861271i	$0.330331\pi$
$270$	0	0
$271$	1.89139	0.114894	0.0574470	−	0.998349i	$-0.481704\pi$
$271$	1.89139	0.114894	0.0574470	+	0.998349i	$0.481704\pi$
$272$	18.4102	1.11628
$273$	17.3915	1.05258
$274$	1.85585	0.112116
$275$	0	0
$276$	4.09807	0.246675
$277$	16.9593	1.01899	0.509494	−	0.860474i	$-0.329833\pi$
$277$	16.9593	1.01899	0.509494	+	0.860474i	$0.329833\pi$
$278$	−23.5321	−1.41136
$279$	3.54665	0.212332
$280$	0	0
$281$	−7.38363	−0.440470	−0.220235	−	0.975447i	$-0.570682\pi$
$281$	−7.38363	−0.440470	−0.220235	+	0.975447i	$0.570682\pi$
$282$	10.9317	0.650972
$283$	−27.9661	−1.66241	−0.831207	−	0.555964i	$-0.812349\pi$
$283$	−27.9661	−1.66241	−0.831207	+	0.555964i	$0.812349\pi$
$284$	−16.6420	−0.987524
$285$	0	0
$286$	0	0
$287$	25.7222	1.51833
$288$	8.06478	0.475222
$289$	18.7786	1.10462
$290$	0	0
$291$	17.0745	1.00093
$292$	6.06013	0.354642
$293$	−34.1829	−1.99699	−0.998493	−	0.0548806i	$-0.982522\pi$
$293$	−34.1829	−1.99699	−0.998493	+	0.0548806i	$0.982522\pi$
$294$	0.565714	0.0329931
$295$	0	0
$296$	2.62349	0.152487
$297$	0	0
$298$	−8.55246	−0.495431
$299$	11.5124	0.665781
$300$	0	0
$301$	0.981428	0.0565685
$302$	−40.3548	−2.32216
$303$	8.39172	0.482092
$304$	9.03812	0.518372
$305$	0	0
$306$	12.5276	0.716153
$307$	7.31822	0.417673	0.208836	−	0.977951i	$-0.433032\pi$
$307$	7.31822	0.417673	0.208836	+	0.977951i	$0.433032\pi$
$308$	0	0
$309$	9.04392	0.514491
$310$	0	0
$311$	−1.54636	−0.0876861	−0.0438431	−	0.999038i	$-0.513960\pi$
$311$	−1.54636	−0.0876861	−0.0438431	+	0.999038i	$0.513960\pi$
$312$	−5.42553	−0.307160
$313$	16.3281	0.922921	0.461461	−	0.887161i	$-0.347326\pi$
$313$	16.3281	0.922921	0.461461	+	0.887161i	$0.347326\pi$
$314$	31.7705	1.79291
$315$	0	0
$316$	22.1875	1.24814
$317$	−3.61928	−0.203279	−0.101639	−	0.994821i	$-0.532409\pi$
$317$	−3.61928	−0.203279	−0.101639	+	0.994821i	$0.532409\pi$
$318$	4.90369	0.274985
$319$	0	0
$320$	0	0
$321$	3.83774	0.214202
$322$	−9.33024	−0.519954
$323$	17.5647	0.977328
$324$	2.38641	0.132579
$325$	0	0
$326$	−16.4929	−0.913459
$327$	16.9858	0.939316
$328$	−8.02440	−0.443073
$329$	−13.5405	−0.746514
$330$	0	0
$331$	7.65089	0.420531	0.210265	−	0.977644i	$-0.432567\pi$
$331$	7.65089	0.420531	0.210265	+	0.977644i	$0.432567\pi$
$332$	−21.3377	−1.17106
$333$	−3.24168	−0.177643
$334$	−7.87044	−0.430651
$335$	0	0
$336$	−7.98458	−0.435594
$337$	−28.2921	−1.54117	−0.770583	−	0.637339i	$-0.780035\pi$
$337$	−28.2921	−1.54117	−0.770583	+	0.637339i	$0.780035\pi$
$338$	−66.9016	−3.63897
$339$	8.16771	0.443609
$340$	0	0
$341$	0	0
$342$	6.15014	0.332561
$343$	−18.8601	−1.01835
$344$	−0.306170	−0.0165076
$345$	0	0
$346$	−54.5118	−2.93057
$347$	1.62216	0.0870821	0.0435410	−	0.999052i	$-0.486136\pi$
$347$	1.62216	0.0870821	0.0435410	+	0.999052i	$0.486136\pi$
$348$	−3.37739	−0.181047
$349$	15.4063	0.824681	0.412340	−	0.911030i	$-0.364711\pi$
$349$	15.4063	0.824681	0.412340	+	0.911030i	$0.364711\pi$
$350$	0	0
$351$	6.70399	0.357832
$352$	0	0
$353$	−11.2275	−0.597580	−0.298790	−	0.954319i	$-0.596583\pi$
$353$	−11.2275	−0.597580	−0.298790	+	0.954319i	$0.596583\pi$
$354$	24.9165	1.32430
$355$	0	0
$356$	23.4632	1.24355
$357$	−15.5173	−0.821261
$358$	12.2062	0.645118
$359$	−4.05957	−0.214256	−0.107128	−	0.994245i	$-0.534165\pi$
$359$	−4.05957	−0.214256	−0.107128	+	0.994245i	$0.534165\pi$
$360$	0	0
$361$	−10.3770	−0.546156
$362$	0.0182034	0.000956750 0
$363$	0	0
$364$	41.5033	2.17537
$365$	0	0
$366$	−25.3655	−1.32587
$367$	−17.4266	−0.909664	−0.454832	−	0.890577i	$-0.650301\pi$
$367$	−17.4266	−0.909664	−0.454832	+	0.890577i	$0.650301\pi$
$368$	−5.28545	−0.275523
$369$	9.91526	0.516168
$370$	0	0
$371$	−6.07396	−0.315344
$372$	8.46378	0.438826
$373$	−3.18117	−0.164715	−0.0823573	−	0.996603i	$-0.526245\pi$
$373$	−3.18117	−0.164715	−0.0823573	+	0.996603i	$0.526245\pi$
$374$	0	0
$375$	0	0
$376$	4.22417	0.217845
$377$	−9.48787	−0.488650
$378$	−5.43324	−0.279456
$379$	13.2440	0.680299	0.340149	−	0.940371i	$-0.389522\pi$
$379$	13.2440	0.680299	0.340149	+	0.940371i	$0.389522\pi$
$380$	0	0
$381$	−18.9674	−0.971731
$382$	25.9103	1.32569
$383$	25.9231	1.32461	0.662303	−	0.749236i	$-0.269579\pi$
$383$	25.9231	1.32461	0.662303	+	0.749236i	$0.269579\pi$
$384$	6.35355	0.324228
$385$	0	0
$386$	7.17544	0.365220
$387$	0.378316	0.0192309
$388$	40.7469	2.06861
$389$	23.0440	1.16838	0.584188	−	0.811619i	$-0.301413\pi$
$389$	23.0440	1.16838	0.584188	+	0.811619i	$0.301413\pi$
$390$	0	0
$391$	−10.2718	−0.519466
$392$	0.218600	0.0110410
$393$	18.1082	0.913436
$394$	4.44920	0.224147
$395$	0	0
$396$	0	0
$397$	−23.7564	−1.19230	−0.596150	−	0.802873i	$-0.703304\pi$
$397$	−23.7564	−1.19230	−0.596150	+	0.802873i	$0.703304\pi$
$398$	−1.34945	−0.0676416
$399$	−7.61788	−0.381371
$400$	0	0
$401$	−10.2187	−0.510297	−0.255148	−	0.966902i	$-0.582124\pi$
$401$	−10.2187	−0.510297	−0.255148	+	0.966902i	$0.582124\pi$
$402$	17.8251	0.889034
$403$	23.7767	1.18440
$404$	20.0261	0.996337
$405$	0	0
$406$	7.68944	0.381620
$407$	0	0
$408$	4.84083	0.239657
$409$	−6.16945	−0.305060	−0.152530	−	0.988299i	$-0.548742\pi$
$409$	−6.16945	−0.305060	−0.152530	+	0.988299i	$0.548742\pi$
$410$	0	0
$411$	−0.886110	−0.0437086
$412$	21.5826	1.06330
$413$	−30.8628	−1.51866
$414$	−3.59657	−0.176762
$415$	0	0
$416$	54.0662	2.65081
$417$	11.2359	0.550223
$418$	0	0
$419$	20.6901	1.01078	0.505389	−	0.862892i	$-0.331349\pi$
$419$	20.6901	1.01078	0.505389	+	0.862892i	$0.331349\pi$
$420$	0	0
$421$	−10.9400	−0.533185	−0.266593	−	0.963809i	$-0.585898\pi$
$421$	−10.9400	−0.533185	−0.266593	+	0.963809i	$0.585898\pi$
$422$	46.4014	2.25879
$423$	−5.21954	−0.253783
$424$	1.89486	0.0920224
$425$	0	0
$426$	14.6055	0.707637
$427$	31.4190	1.52047
$428$	9.15843	0.442690
$429$	0	0
$430$	0	0
$431$	−13.9898	−0.673863	−0.336932	−	0.941529i	$-0.609389\pi$
$431$	−13.9898	−0.673863	−0.336932	+	0.941529i	$0.609389\pi$
$432$	−3.07785	−0.148083
$433$	31.8360	1.52994	0.764971	−	0.644065i	$-0.222753\pi$
$433$	31.8360	1.52994	0.764971	+	0.644065i	$0.222753\pi$
$434$	−19.2698	−0.924980
$435$	0	0
$436$	40.5351	1.94128
$437$	−5.04271	−0.241225
$438$	−5.31852	−0.254129
$439$	−33.8677	−1.61642	−0.808208	−	0.588897i	$-0.799562\pi$
$439$	−33.8677	−1.61642	−0.808208	+	0.588897i	$0.799562\pi$
$440$	0	0
$441$	−0.270111	−0.0128624
$442$	83.9846	3.99474
$443$	−37.6628	−1.78941	−0.894707	−	0.446653i	$-0.852616\pi$
$443$	−37.6628	−1.78941	−0.894707	+	0.446653i	$0.852616\pi$
$444$	−7.73599	−0.367134
$445$	0	0
$446$	−26.3998	−1.25007
$447$	4.08354	0.193145
$448$	−27.8487	−1.31573
$449$	22.1180	1.04381	0.521906	−	0.853003i	$-0.325221\pi$
$449$	22.1180	1.04381	0.521906	+	0.853003i	$0.325221\pi$
$450$	0	0
$451$	0	0
$452$	19.4915	0.916805
$453$	19.2681	0.905296
$454$	−61.3732	−2.88039
$455$	0	0
$456$	2.37651	0.111290
$457$	11.2328	0.525450	0.262725	−	0.964871i	$-0.415379\pi$
$457$	11.2328	0.525450	0.262725	+	0.964871i	$0.415379\pi$
$458$	−44.3843	−2.07394
$459$	−5.98152	−0.279193
$460$	0	0
$461$	15.0161	0.699368	0.349684	−	0.936868i	$-0.386289\pi$
$461$	15.0161	0.699368	0.349684	+	0.936868i	$0.386289\pi$
$462$	0	0
$463$	−14.2902	−0.664121	−0.332061	−	0.943258i	$-0.607744\pi$
$463$	−14.2902	−0.664121	−0.332061	+	0.943258i	$0.607744\pi$
$464$	4.35596	0.202220
$465$	0	0
$466$	51.9671	2.40733
$467$	−3.53872	−0.163753	−0.0818763	−	0.996643i	$-0.526091\pi$
$467$	−3.53872	−0.163753	−0.0818763	+	0.996643i	$0.526091\pi$
$468$	15.9985	0.739531
$469$	−22.0790	−1.01952
$470$	0	0
$471$	−15.1694	−0.698969
$472$	9.62810	0.443169
$473$	0	0
$474$	−19.4723	−0.894391
$475$	0	0
$476$	−37.0307	−1.69730
$477$	−2.34136	−0.107203
$478$	−4.58622	−0.209769
$479$	−36.7651	−1.67984	−0.839921	−	0.542709i	$-0.817399\pi$
$479$	−36.7651	−1.67984	−0.839921	+	0.542709i	$0.817399\pi$
$480$	0	0
$481$	−21.7322	−0.990902
$482$	15.0907	0.687362
$483$	4.45490	0.202705
$484$	0	0
$485$	0	0
$486$	−2.09438	−0.0950029
$487$	11.1992	0.507486	0.253743	−	0.967272i	$-0.418338\pi$
$487$	11.1992	0.507486	0.253743	+	0.967272i	$0.418338\pi$
$488$	−9.80160	−0.443698
$489$	7.87487	0.356114
$490$	0	0
$491$	37.0328	1.67127	0.835634	−	0.549287i	$-0.185101\pi$
$491$	37.0328	1.67127	0.835634	+	0.549287i	$0.185101\pi$
$492$	23.6619	1.06676
$493$	8.46539	0.381262
$494$	41.2304	1.85505
$495$	0	0
$496$	−10.9161	−0.490146
$497$	−18.0911	−0.811496
$498$	18.7265	0.839155
$499$	31.9261	1.42921	0.714603	−	0.699530i	$-0.246607\pi$
$499$	31.9261	1.42921	0.714603	+	0.699530i	$0.246607\pi$
$500$	0	0
$501$	3.75789	0.167890
$502$	7.71314	0.344254
$503$	18.7406	0.835604	0.417802	−	0.908538i	$-0.362801\pi$
$503$	18.7406	0.835604	0.417802	+	0.908538i	$0.362801\pi$
$504$	−2.09948	−0.0935185
$505$	0	0
$506$	0	0
$507$	31.9434	1.41866
$508$	−45.2641	−2.00827
$509$	−9.77357	−0.433206	−0.216603	−	0.976260i	$-0.569498\pi$
$509$	−9.77357	−0.433206	−0.216603	+	0.976260i	$0.569498\pi$
$510$	0	0
$511$	6.58779	0.291427
$512$	−29.8040	−1.31716
$513$	−2.93650	−0.129650
$514$	−28.4867	−1.25649
$515$	0	0
$516$	0.902818	0.0397444
$517$	0	0
$518$	17.6128	0.773863
$519$	26.0277	1.14249
$520$	0	0
$521$	2.64263	0.115776	0.0578878	−	0.998323i	$-0.481563\pi$
$521$	2.64263	0.115776	0.0578878	+	0.998323i	$0.481563\pi$
$522$	2.96408	0.129734
$523$	19.5304	0.854007	0.427003	−	0.904250i	$-0.359569\pi$
$523$	19.5304	0.854007	0.427003	+	0.904250i	$0.359569\pi$
$524$	43.2136	1.88779
$525$	0	0
$526$	−6.21237	−0.270872
$527$	−21.2144	−0.924112
$528$	0	0
$529$	−20.0510	−0.871785
$530$	0	0
$531$	−11.8968	−0.516279
$532$	−18.1794	−0.788178
$533$	66.4717	2.87921
$534$	−20.5919	−0.891099
$535$	0	0
$536$	6.88787	0.297511
$537$	−5.82809	−0.251501
$538$	−34.9099	−1.50507
$539$	0	0
$540$	0	0
$541$	−19.1916	−0.825111	−0.412556	−	0.910932i	$-0.635364\pi$
$541$	−19.1916	−0.825111	−0.412556	+	0.910932i	$0.635364\pi$
$542$	−3.96129	−0.170152
$543$	−0.00869156	−0.000372991 0
$544$	−48.2396	−2.06826
$545$	0	0
$546$	−36.4244	−1.55882
$547$	−12.6403	−0.540458	−0.270229	−	0.962796i	$-0.587099\pi$
$547$	−12.6403	−0.540458	−0.270229	+	0.962796i	$0.587099\pi$
$548$	−2.11463	−0.0903324
$549$	12.1112	0.516895
$550$	0	0
$551$	4.15591	0.177048
$552$	−1.38977	−0.0591525
$553$	24.1194	1.02566
$554$	−35.5192	−1.50907
$555$	0	0
$556$	26.8134	1.13714
$557$	−40.2581	−1.70579	−0.852895	−	0.522082i	$-0.825156\pi$
$557$	−40.2581	−1.70579	−0.852895	+	0.522082i	$0.825156\pi$
$558$	−7.42802	−0.314453
$559$	2.53622	0.107271
$560$	0	0
$561$	0	0
$562$	15.4641	0.652314
$563$	4.38310	0.184725	0.0923627	−	0.995725i	$-0.470558\pi$
$563$	4.38310	0.184725	0.0923627	+	0.995725i	$0.470558\pi$
$564$	−12.4560	−0.524492
$565$	0	0
$566$	58.5716	2.46195
$567$	2.59420	0.108946
$568$	5.64377	0.236807
$569$	8.71900	0.365519	0.182760	−	0.983158i	$-0.441497\pi$
$569$	8.71900	0.365519	0.182760	+	0.983158i	$0.441497\pi$
$570$	0	0
$571$	19.2314	0.804807	0.402404	−	0.915462i	$-0.368175\pi$
$571$	19.2314	0.804807	0.402404	+	0.915462i	$0.368175\pi$
$572$	0	0
$573$	−12.3714	−0.516822
$574$	−53.8720	−2.24857
$575$	0	0
$576$	−10.7350	−0.447291
$577$	9.10116	0.378886	0.189443	−	0.981892i	$-0.439332\pi$
$577$	9.10116	0.378886	0.189443	+	0.981892i	$0.439332\pi$
$578$	−39.3294	−1.63589
$579$	−3.42605	−0.142382
$580$	0	0
$581$	−23.1956	−0.962317
$582$	−35.7605	−1.48232
$583$	0	0
$584$	−2.05515	−0.0850429
$585$	0	0
$586$	71.5919	2.95743
$587$	−1.03002	−0.0425134	−0.0212567	−	0.999774i	$-0.506767\pi$
$587$	−1.03002	−0.0425134	−0.0212567	+	0.999774i	$0.506767\pi$
$588$	−0.644597	−0.0265827
$589$	−10.4147	−0.429132
$590$	0	0
$591$	−2.12435	−0.0873843
$592$	9.97741	0.410069
$593$	−13.9515	−0.572921	−0.286461	−	0.958092i	$-0.592479\pi$
$593$	−13.9515	−0.572921	−0.286461	+	0.958092i	$0.592479\pi$
$594$	0	0
$595$	0	0
$596$	9.74501	0.399171
$597$	0.644319	0.0263702
$598$	−24.1114	−0.985987
$599$	−33.2785	−1.35972	−0.679861	−	0.733341i	$-0.737960\pi$
$599$	−33.2785	−1.35972	−0.679861	+	0.733341i	$0.737960\pi$
$600$	0	0
$601$	43.6371	1.78000	0.889998	−	0.455964i	$-0.150706\pi$
$601$	43.6371	1.78000	0.889998	+	0.455964i	$0.150706\pi$
$602$	−2.05548	−0.0837751
$603$	−8.51092	−0.346591
$604$	45.9818	1.87097
$605$	0	0
$606$	−17.5754	−0.713953
$607$	37.3995	1.51800	0.759000	−	0.651091i	$-0.225688\pi$
$607$	37.3995	1.51800	0.759000	+	0.651091i	$0.225688\pi$
$608$	−23.6822	−0.960442
$609$	−3.67147	−0.148775
$610$	0	0
$611$	−34.9917	−1.41561
$612$	−14.2744	−0.577008
$613$	14.4497	0.583618	0.291809	−	0.956477i	$-0.405743\pi$
$613$	14.4497	0.583618	0.291809	+	0.956477i	$0.405743\pi$
$614$	−15.3271	−0.618552
$615$	0	0
$616$	0	0
$617$	1.12365	0.0452365	0.0226182	−	0.999744i	$-0.492800\pi$
$617$	1.12365	0.0452365	0.0226182	+	0.999744i	$0.492800\pi$
$618$	−18.9414	−0.761934
$619$	9.19131	0.369430	0.184715	−	0.982792i	$-0.440864\pi$
$619$	9.19131	0.369430	0.184715	+	0.982792i	$0.440864\pi$
$620$	0	0
$621$	1.71725	0.0689109
$622$	3.23867	0.129859
$623$	25.5062	1.02188
$624$	−20.6339	−0.826017
$625$	0	0
$626$	−34.1973	−1.36680
$627$	0	0
$628$	−36.2005	−1.44456
$629$	19.3902	0.773136
$630$	0	0
$631$	32.9997	1.31370	0.656848	−	0.754023i	$-0.271889\pi$
$631$	32.9997	1.31370	0.656848	+	0.754023i	$0.271889\pi$
$632$	−7.52438	−0.299304
$633$	−22.1552	−0.880592
$634$	7.58014	0.301046
$635$	0	0
$636$	−5.58745	−0.221557
$637$	−1.81082	−0.0717473
$638$	0	0
$639$	−6.97366	−0.275874
$640$	0	0
$641$	−43.2560	−1.70851	−0.854255	−	0.519854i	$-0.825987\pi$
$641$	−43.2560	−1.70851	−0.854255	+	0.519854i	$0.825987\pi$
$642$	−8.03767	−0.317221
$643$	10.6965	0.421827	0.210914	−	0.977505i	$-0.432356\pi$
$643$	10.6965	0.421827	0.210914	+	0.977505i	$0.432356\pi$
$644$	10.6312	0.418929
$645$	0	0
$646$	−36.7872	−1.44737
$647$	−11.1508	−0.438384	−0.219192	−	0.975682i	$-0.570342\pi$
$647$	−11.1508	−0.438384	−0.219192	+	0.975682i	$0.570342\pi$
$648$	−0.809299	−0.0317923
$649$	0	0
$650$	0	0
$651$	9.20073	0.360605
$652$	18.7927	0.735979
$653$	−25.3233	−0.990977	−0.495488	−	0.868615i	$-0.665011\pi$
$653$	−25.3233	−0.990977	−0.495488	+	0.868615i	$0.665011\pi$
$654$	−35.5747	−1.39108
$655$	0	0
$656$	−30.5177	−1.19152
$657$	2.53943	0.0990725
$658$	28.3590	1.10555
$659$	−14.0141	−0.545910	−0.272955	−	0.962027i	$-0.588001\pi$
$659$	−14.0141	−0.545910	−0.272955	+	0.962027i	$0.588001\pi$
$660$	0	0
$661$	40.8495	1.58886	0.794431	−	0.607354i	$-0.207769\pi$
$661$	40.8495	1.58886	0.794431	+	0.607354i	$0.207769\pi$
$662$	−16.0238	−0.622784
$663$	−40.1000	−1.55736
$664$	7.23621	0.280819
$665$	0	0
$666$	6.78930	0.263080
$667$	−2.43035	−0.0941037
$668$	8.96789	0.346978
$669$	12.6051	0.487340
$670$	0	0
$671$	0	0
$672$	20.9217	0.807071
$673$	32.7738	1.26334	0.631668	−	0.775239i	$-0.282370\pi$
$673$	32.7738	1.26334	0.631668	+	0.775239i	$0.282370\pi$
$674$	59.2542	2.28239
$675$	0	0
$676$	76.2303	2.93193
$677$	27.6819	1.06390	0.531950	−	0.846776i	$-0.321459\pi$
$677$	27.6819	1.06390	0.531950	+	0.846776i	$0.321459\pi$
$678$	−17.1063	−0.656962
$679$	44.2948	1.69988
$680$	0	0
$681$	29.3038	1.12292
$682$	0	0
$683$	−26.4160	−1.01078	−0.505391	−	0.862891i	$-0.668652\pi$
$683$	−26.4160	−1.01078	−0.505391	+	0.862891i	$0.668652\pi$
$684$	−7.00771	−0.267946
$685$	0	0
$686$	39.5003	1.50813
$687$	21.1921	0.808530
$688$	−1.16440	−0.0443923
$689$	−15.6964	−0.597986
$690$	0	0
$691$	−28.6032	−1.08812	−0.544059	−	0.839047i	$-0.683113\pi$
$691$	−28.6032	−1.08812	−0.544059	+	0.839047i	$0.683113\pi$
$692$	62.1129	2.36118
$693$	0	0
$694$	−3.39741	−0.128964
$695$	0	0
$696$	1.14537	0.0434150
$697$	−59.3083	−2.24646
$698$	−32.2666	−1.22131
$699$	−24.8127	−0.938502
$700$	0	0
$701$	23.9603	0.904968	0.452484	−	0.891773i	$-0.350538\pi$
$701$	23.9603	0.904968	0.452484	+	0.891773i	$0.350538\pi$
$702$	−14.0407	−0.529931
$703$	9.51919	0.359023
$704$	0	0
$705$	0	0
$706$	23.5146	0.884985
$707$	21.7698	0.818739
$708$	−28.3908	−1.06699
$709$	−33.0392	−1.24081	−0.620406	−	0.784281i	$-0.713032\pi$
$709$	−33.0392	−1.24081	−0.620406	+	0.784281i	$0.713032\pi$
$710$	0	0
$711$	9.29741	0.348680
$712$	−7.95702	−0.298202
$713$	6.09049	0.228091
$714$	32.4990	1.21625
$715$	0	0
$716$	−13.9082	−0.519775
$717$	2.18978	0.0817787
$718$	8.50227	0.317302
$719$	−7.74741	−0.288930	−0.144465	−	0.989510i	$-0.546146\pi$
$719$	−7.74741	−0.288930	−0.144465	+	0.989510i	$0.546146\pi$
$720$	0	0
$721$	23.4618	0.873762
$722$	21.7333	0.808829
$723$	−7.20534	−0.267969
$724$	−0.0207417	−0.000770858 0
$725$	0	0
$726$	0	0
$727$	−7.18494	−0.266475	−0.133237	−	0.991084i	$-0.542537\pi$
$727$	−7.18494	−0.266475	−0.133237	+	0.991084i	$0.542537\pi$
$728$	−14.0749	−0.521651
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.26290	−0.0836965
$732$	28.9024	1.06826
$733$	8.49537	0.313783	0.156892	−	0.987616i	$-0.449853\pi$
$733$	8.49537	0.313783	0.156892	+	0.987616i	$0.449853\pi$
$734$	36.4980	1.34716
$735$	0	0
$736$	13.8493	0.510490
$737$	0	0
$738$	−20.7663	−0.764417
$739$	33.5021	1.23239	0.616197	−	0.787592i	$-0.288673\pi$
$739$	33.5021	1.23239	0.616197	+	0.787592i	$0.288673\pi$
$740$	0	0
$741$	−19.6863	−0.723193
$742$	12.7212	0.467008
$743$	−7.60661	−0.279060	−0.139530	−	0.990218i	$-0.544559\pi$
$743$	−7.60661	−0.279060	−0.139530	+	0.990218i	$0.544559\pi$
$744$	−2.87030	−0.105230
$745$	0	0
$746$	6.66257	0.243934
$747$	−8.94133	−0.327146
$748$	0	0
$749$	9.95587	0.363780
$750$	0	0
$751$	29.5324	1.07765	0.538826	−	0.842417i	$-0.318868\pi$
$751$	29.5324	1.07765	0.538826	+	0.842417i	$0.318868\pi$
$752$	16.0650	0.585830
$753$	−3.68279	−0.134208
$754$	19.8712	0.723666
$755$	0	0
$756$	6.19084	0.225159
$757$	−40.1472	−1.45918	−0.729588	−	0.683887i	$-0.760288\pi$
$757$	−40.1472	−1.45918	−0.729588	+	0.683887i	$0.760288\pi$
$758$	−27.7379	−1.00749
$759$	0	0
$760$	0	0
$761$	2.41740	0.0876306	0.0438153	−	0.999040i	$-0.486049\pi$
$761$	2.41740	0.0876306	0.0438153	+	0.999040i	$0.486049\pi$
$762$	39.7249	1.43908
$763$	44.0646	1.59525
$764$	−29.5232	−1.06811
$765$	0	0
$766$	−54.2927	−1.96167
$767$	−79.7563	−2.87983
$768$	8.16325	0.294566
$769$	18.1464	0.654375	0.327187	−	0.944959i	$-0.393899\pi$
$769$	18.1464	0.654375	0.327187	+	0.944959i	$0.393899\pi$
$770$	0	0
$771$	13.6015	0.489847
$772$	−8.17598	−0.294260
$773$	−19.0082	−0.683679	−0.341839	−	0.939758i	$-0.611050\pi$
$773$	−19.0082	−0.683679	−0.341839	+	0.939758i	$0.611050\pi$
$774$	−0.792336	−0.0284799
$775$	0	0
$776$	−13.8184	−0.496052
$777$	−8.40957	−0.301692
$778$	−48.2628	−1.73030
$779$	−29.1161	−1.04319
$780$	0	0
$781$	0	0
$782$	21.5130	0.769302
$783$	−1.41526	−0.0505772
$784$	0.831362	0.0296915
$785$	0	0
$786$	−37.9253	−1.35275
$787$	18.3482	0.654043	0.327021	−	0.945017i	$-0.393955\pi$
$787$	18.3482	0.654043	0.327021	+	0.945017i	$0.393955\pi$
$788$	−5.06959	−0.180597
$789$	2.96621	0.105600
$790$	0	0
$791$	21.1887	0.753383
$792$	0	0
$793$	81.1935	2.88327
$794$	49.7549	1.76574
$795$	0	0
$796$	1.53761	0.0544992
$797$	42.8402	1.51748	0.758739	−	0.651395i	$-0.225816\pi$
$797$	42.8402	1.51748	0.758739	+	0.651395i	$0.225816\pi$
$798$	15.9547	0.564791
$799$	31.2208	1.10451
$800$	0	0
$801$	9.83200	0.347397
$802$	21.4018	0.755723
$803$	0	0
$804$	−20.3106	−0.716299
$805$	0	0
$806$	−49.7974	−1.75404
$807$	16.6684	0.586756
$808$	−6.79141	−0.238921
$809$	10.8625	0.381906	0.190953	−	0.981599i	$-0.438842\pi$
$809$	10.8625	0.381906	0.190953	+	0.981599i	$0.438842\pi$
$810$	0	0
$811$	30.1690	1.05938	0.529689	−	0.848192i	$-0.322309\pi$
$811$	30.1690	1.05938	0.529689	+	0.848192i	$0.322309\pi$
$812$	−8.76164	−0.307473
$813$	1.89139	0.0663341
$814$	0	0
$815$	0	0
$816$	18.4102	0.644487
$817$	−1.11092	−0.0388663
$818$	12.9212	0.451778
$819$	17.3915	0.607708
$820$	0	0
$821$	−8.73026	−0.304688	−0.152344	−	0.988328i	$-0.548682\pi$
$821$	−8.73026	−0.304688	−0.152344	+	0.988328i	$0.548682\pi$
$822$	1.85585	0.0647302
$823$	−13.3377	−0.464924	−0.232462	−	0.972605i	$-0.574678\pi$
$823$	−13.3377	−0.464924	−0.232462	+	0.972605i	$0.574678\pi$
$824$	−7.31923	−0.254978
$825$	0	0
$826$	64.6384	2.24906
$827$	−36.8109	−1.28004	−0.640020	−	0.768358i	$-0.721074\pi$
$827$	−36.8109	−1.28004	−0.640020	+	0.768358i	$0.721074\pi$
$828$	4.09807	0.142418
$829$	−22.1512	−0.769343	−0.384671	−	0.923054i	$-0.625685\pi$
$829$	−22.1512	−0.769343	−0.384671	+	0.923054i	$0.625685\pi$
$830$	0	0
$831$	16.9593	0.588313
$832$	−71.9672	−2.49501
$833$	1.61567	0.0559798
$834$	−23.5321	−0.814851
$835$	0	0
$836$	0	0
$837$	3.54665	0.122590
$838$	−43.3329	−1.49691
$839$	−24.2728	−0.837990	−0.418995	−	0.907989i	$-0.637618\pi$
$839$	−24.2728	−0.837990	−0.418995	+	0.907989i	$0.637618\pi$
$840$	0	0
$841$	−26.9970	−0.930933
$842$	22.9126	0.789620
$843$	−7.38363	−0.254306
$844$	−52.8716	−1.81992
$845$	0	0
$846$	10.9317	0.375839
$847$	0	0
$848$	7.20636	0.247467
$849$	−27.9661	−0.959795
$850$	0	0
$851$	−5.56678	−0.190827
$852$	−16.6420	−0.570147
$853$	12.3766	0.423766	0.211883	−	0.977295i	$-0.432040\pi$
$853$	12.3766	0.423766	0.211883	+	0.977295i	$0.432040\pi$
$854$	−65.8032	−2.25174
$855$	0	0
$856$	−3.10588	−0.106157
$857$	38.5293	1.31614	0.658068	−	0.752959i	$-0.271374\pi$
$857$	38.5293	1.31614	0.658068	+	0.752959i	$0.271374\pi$
$858$	0	0
$859$	−24.2702	−0.828088	−0.414044	−	0.910257i	$-0.635884\pi$
$859$	−24.2702	−0.828088	−0.414044	+	0.910257i	$0.635884\pi$
$860$	0	0
$861$	25.7222	0.876610
$862$	29.2998	0.997957
$863$	6.91737	0.235470	0.117735	−	0.993045i	$-0.462437\pi$
$863$	6.91737	0.235470	0.117735	+	0.993045i	$0.462437\pi$
$864$	8.06478	0.274369
$865$	0	0
$866$	−66.6767	−2.26577
$867$	18.7786	0.637754
$868$	21.9568	0.745261
$869$	0	0
$870$	0	0
$871$	−57.0571	−1.93331
$872$	−13.7466	−0.465518
$873$	17.0745	0.577885
$874$	10.5613	0.357242
$875$	0	0
$876$	6.06013	0.204753
$877$	5.09266	0.171967	0.0859835	−	0.996297i	$-0.472597\pi$
$877$	5.09266	0.171967	0.0859835	+	0.996297i	$0.472597\pi$
$878$	70.9317	2.39383
$879$	−34.1829	−1.15296
$880$	0	0
$881$	12.2472	0.412619	0.206309	−	0.978487i	$-0.433855\pi$
$881$	12.2472	0.412619	0.206309	+	0.978487i	$0.433855\pi$
$882$	0.565714	0.0190486
$883$	20.2777	0.682399	0.341199	−	0.939991i	$-0.389167\pi$
$883$	20.2777	0.682399	0.341199	+	0.939991i	$0.389167\pi$
$884$	−95.6953	−3.21858
$885$	0	0
$886$	78.8801	2.65003
$887$	5.90131	0.198147	0.0990734	−	0.995080i	$-0.468412\pi$
$887$	5.90131	0.198147	0.0990734	+	0.995080i	$0.468412\pi$
$888$	2.62349	0.0880384
$889$	−49.2053	−1.65029
$890$	0	0
$891$	0	0
$892$	30.0809	1.00718
$893$	15.3272	0.512905
$894$	−8.55246	−0.286037
$895$	0	0
$896$	16.4824	0.550638
$897$	11.5124	0.384389
$898$	−46.3234	−1.54583
$899$	−5.01943	−0.167407
$900$	0	0
$901$	14.0049	0.466570
$902$	0	0
$903$	0.981428	0.0326599
$904$	−6.61011	−0.219849
$905$	0	0
$906$	−40.3548	−1.34070
$907$	−11.5289	−0.382812	−0.191406	−	0.981511i	$-0.561305\pi$
$907$	−11.5289	−0.382812	−0.191406	+	0.981511i	$0.561305\pi$
$908$	69.9310	2.32074
$909$	8.39172	0.278336
$910$	0	0
$911$	−7.12406	−0.236031	−0.118015	−	0.993012i	$-0.537653\pi$
$911$	−7.12406	−0.236031	−0.118015	+	0.993012i	$0.537653\pi$
$912$	9.03812	0.299282
$913$	0	0
$914$	−23.5258	−0.778165
$915$	0	0
$916$	50.5732	1.67099
$917$	46.9762	1.55129
$918$	12.5276	0.413471
$919$	−48.1125	−1.58708	−0.793542	−	0.608516i	$-0.791765\pi$
$919$	−48.1125	−1.58708	−0.793542	+	0.608516i	$0.791765\pi$
$920$	0	0
$921$	7.31822	0.241144
$922$	−31.4493	−1.03573
$923$	−46.7513	−1.53884
$924$	0	0
$925$	0	0
$926$	29.9290	0.983529
$927$	9.04392	0.297041
$928$	−11.4137	−0.374675
$929$	8.23535	0.270193	0.135097	−	0.990832i	$-0.456866\pi$
$929$	8.23535	0.270193	0.135097	+	0.990832i	$0.456866\pi$
$930$	0	0
$931$	0.793181	0.0259955
$932$	−59.2134	−1.93960
$933$	−1.54636	−0.0506256
$934$	7.41142	0.242509
$935$	0	0
$936$	−5.42553	−0.177339
$937$	5.40569	0.176596	0.0882982	−	0.996094i	$-0.471857\pi$
$937$	5.40569	0.176596	0.0882982	+	0.996094i	$0.471857\pi$
$938$	46.2418	1.50985
$939$	16.3281	0.532849
$940$	0	0
$941$	36.0086	1.17385	0.586924	−	0.809642i	$-0.300339\pi$
$941$	36.0086	1.17385	0.586924	+	0.809642i	$0.300339\pi$
$942$	31.7705	1.03514
$943$	17.0270	0.554475
$944$	36.6167	1.19177
$945$	0	0
$946$	0	0
$947$	7.57154	0.246042	0.123021	−	0.992404i	$-0.460742\pi$
$947$	7.57154	0.246042	0.123021	+	0.992404i	$0.460742\pi$
$948$	22.1875	0.720615
$949$	17.0243	0.552632
$950$	0	0
$951$	−3.61928	−0.117363
$952$	12.5581	0.407011
$953$	25.1644	0.815156	0.407578	−	0.913170i	$-0.366373\pi$
$953$	25.1644	0.815156	0.407578	+	0.913170i	$0.366373\pi$
$954$	4.90369	0.158763
$955$	0	0
$956$	5.22571	0.169012
$957$	0	0
$958$	77.0001	2.48776
$959$	−2.29875	−0.0742305
$960$	0	0
$961$	−18.4213	−0.594235
$962$	45.5154	1.46747
$963$	3.83774	0.123669
$964$	−17.1949	−0.553811
$965$	0	0
$966$	−9.33024	−0.300195
$967$	−4.24511	−0.136514	−0.0682568	−	0.997668i	$-0.521744\pi$
$967$	−4.24511	−0.136514	−0.0682568	+	0.997668i	$0.521744\pi$
$968$	0	0
$969$	17.5647	0.564260
$970$	0	0
$971$	5.87200	0.188441	0.0942207	−	0.995551i	$-0.469964\pi$
$971$	5.87200	0.188441	0.0942207	+	0.995551i	$0.469964\pi$
$972$	2.38641	0.0765443
$973$	29.1481	0.934446
$974$	−23.4554	−0.751560
$975$	0	0
$976$	−37.2766	−1.19319
$977$	53.3337	1.70630	0.853149	−	0.521667i	$-0.174690\pi$
$977$	53.3337	1.70630	0.853149	+	0.521667i	$0.174690\pi$
$978$	−16.4929	−0.527386
$979$	0	0
$980$	0	0
$981$	16.9858	0.542315
$982$	−77.5607	−2.47506
$983$	21.7446	0.693546	0.346773	−	0.937949i	$-0.387277\pi$
$983$	21.7446	0.693546	0.346773	+	0.937949i	$0.387277\pi$
$984$	−8.02440	−0.255809
$985$	0	0
$986$	−17.7297	−0.564629
$987$	−13.5405	−0.431000
$988$	−46.9796	−1.49462
$989$	0.649663	0.0206581
$990$	0	0
$991$	−2.64276	−0.0839500	−0.0419750	−	0.999119i	$-0.513365\pi$
$991$	−2.64276	−0.0839500	−0.0419750	+	0.999119i	$0.513365\pi$
$992$	28.6030	0.908145
$993$	7.65089	0.242794
$994$	37.8896	1.20178
$995$	0	0
$996$	−21.3377	−0.676112
$997$	−42.2745	−1.33885	−0.669424	−	0.742881i	$-0.733459\pi$
$997$	−42.2745	−1.33885	−0.669424	+	0.742881i	$0.733459\pi$
$998$	−66.8652	−2.11658
$999$	−3.24168	−0.102562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dw.1.1		8
5.4	even	2		9075.2.a.dt.1.8		8
11.7	odd	10		825.2.n.m.676.4	yes	16
11.8	odd	10		825.2.n.m.526.4	✓	16
11.10	odd	2		9075.2.a.du.1.8		8
55.7	even	20		825.2.bx.j.49.2		32
55.8	even	20		825.2.bx.j.724.2		32
55.18	even	20		825.2.bx.j.49.7		32
55.19	odd	10		825.2.n.n.526.1	yes	16
55.29	odd	10		825.2.n.n.676.1	yes	16
55.52	even	20		825.2.bx.j.724.7		32
55.54	odd	2		9075.2.a.dv.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.m.526.4	✓	16	11.8	odd	10
825.2.n.m.676.4	yes	16	11.7	odd	10
825.2.n.n.526.1	yes	16	55.19	odd	10
825.2.n.n.676.1	yes	16	55.29	odd	10
825.2.bx.j.49.2		32	55.7	even	20
825.2.bx.j.49.7		32	55.18	even	20
825.2.bx.j.724.2		32	55.8	even	20
825.2.bx.j.724.7		32	55.52	even	20
9075.2.a.dt.1.8		8	5.4	even	2
9075.2.a.du.1.8		8	11.10	odd	2
9075.2.a.dv.1.1		8	55.54	odd	2
9075.2.a.dw.1.1		8	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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09438 q^{2} +1.00000 q^{3} +2.38641 q^{4} -2.09438 q^{6} +2.59420 q^{7} -0.809299 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.09438 q^{2} +1.00000 q^{3} +2.38641 q^{4} -2.09438 q^{6} +2.59420 q^{7} -0.809299 q^{8} +1.00000 q^{9} +2.38641 q^{12} +6.70399 q^{13} -5.43324 q^{14} -3.07785 q^{16} -5.98152 q^{17} -2.09438 q^{18} -2.93650 q^{19} +2.59420 q^{21} +1.71725 q^{23} -0.809299 q^{24} -14.0407 q^{26} +1.00000 q^{27} +6.19084 q^{28} -1.41526 q^{29} +3.54665 q^{31} +8.06478 q^{32} +12.5276 q^{34} +2.38641 q^{36} -3.24168 q^{37} +6.15014 q^{38} +6.70399 q^{39} +9.91526 q^{41} -5.43324 q^{42} +0.378316 q^{43} -3.59657 q^{46} -5.21954 q^{47} -3.07785 q^{48} -0.270111 q^{49} -5.98152 q^{51} +15.9985 q^{52} -2.34136 q^{53} -2.09438 q^{54} -2.09948 q^{56} -2.93650 q^{57} +2.96408 q^{58} -11.8968 q^{59} +12.1112 q^{61} -7.42802 q^{62} +2.59420 q^{63} -10.7350 q^{64} -8.51092 q^{67} -14.2744 q^{68} +1.71725 q^{69} -6.97366 q^{71} -0.809299 q^{72} +2.53943 q^{73} +6.78930 q^{74} -7.00771 q^{76} -14.0407 q^{78} +9.29741 q^{79} +1.00000 q^{81} -20.7663 q^{82} -8.94133 q^{83} +6.19084 q^{84} -0.792336 q^{86} -1.41526 q^{87} +9.83200 q^{89} +17.3915 q^{91} +4.09807 q^{92} +3.54665 q^{93} +10.9317 q^{94} +8.06478 q^{96} +17.0745 q^{97} +0.565714 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 7 q^{4} + q^{6} + 8 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 7 * q^4 + q^6 + 8 * q^7 + 3 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 7 q^{4} + q^{6} + 8 q^{7} + 3 q^{8} + 8 q^{9} + 7 q^{12} + 7 q^{13} - 8 q^{14} - 7 q^{16} - q^{17} + q^{18} + 6 q^{19} + 8 q^{21} + 7 q^{23} + 3 q^{24} - q^{26} + 8 q^{27} + 11 q^{28} + 28 q^{29} + 10 q^{31} + 12 q^{32} - 8 q^{34} + 7 q^{36} + 14 q^{37} + 19 q^{38} + 7 q^{39} + 14 q^{41} - 8 q^{42} + 11 q^{43} + 5 q^{47} - 7 q^{48} + 2 q^{49} - q^{51} + q^{52} - 5 q^{53} + q^{54} - 4 q^{56} + 6 q^{57} - 3 q^{58} - 2 q^{59} + 36 q^{61} - 17 q^{62} + 8 q^{63} - 23 q^{64} + 6 q^{67} - 3 q^{68} + 7 q^{69} - 17 q^{71} + 3 q^{72} + 32 q^{73} + 47 q^{74} + 5 q^{76} - q^{78} + 41 q^{79} + 8 q^{81} - 6 q^{83} + 11 q^{84} - 9 q^{86} + 28 q^{87} + 21 q^{89} - 9 q^{91} + 69 q^{92} + 10 q^{93} + 51 q^{94} + 12 q^{96} - 4 q^{97} - 27 q^{98}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 7 * q^4 + q^6 + 8 * q^7 + 3 * q^8 + 8 * q^9 + 7 * q^12 + 7 * q^13 - 8 * q^14 - 7 * q^16 - q^17 + q^18 + 6 * q^19 + 8 * q^21 + 7 * q^23 + 3 * q^24 - q^26 + 8 * q^27 + 11 * q^28 + 28 * q^29 + 10 * q^31 + 12 * q^32 - 8 * q^34 + 7 * q^36 + 14 * q^37 + 19 * q^38 + 7 * q^39 + 14 * q^41 - 8 * q^42 + 11 * q^43 + 5 * q^47 - 7 * q^48 + 2 * q^49 - q^51 + q^52 - 5 * q^53 + q^54 - 4 * q^56 + 6 * q^57 - 3 * q^58 - 2 * q^59 + 36 * q^61 - 17 * q^62 + 8 * q^63 - 23 * q^64 + 6 * q^67 - 3 * q^68 + 7 * q^69 - 17 * q^71 + 3 * q^72 + 32 * q^73 + 47 * q^74 + 5 * q^76 - q^78 + 41 * q^79 + 8 * q^81 - 6 * q^83 + 11 * q^84 - 9 * q^86 + 28 * q^87 + 21 * q^89 - 9 * q^91 + 69 * q^92 + 10 * q^93 + 51 * q^94 + 12 * q^96 - 4 * q^97 - 27 * q^98

Introduction

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