LMFDB - Embedded newform 9075.2.a.dq.1.3

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:	$6$
Coefficient field:	6.6.437199552.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 49x^{2} - 48 $ x^6 - 13x^4 + 49x^2 - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.23396$ 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)

$f(q)$	$=$	$q-1.23396 q^{2} +1.00000 q^{3} -0.477352 q^{4} -1.23396 q^{6} -3.79281 q^{7} +3.05694 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.23396 q^{2} +1.00000 q^{3} -0.477352 q^{4} -1.23396 q^{6} -3.79281 q^{7} +3.05694 q^{8} +1.00000 q^{9} -0.477352 q^{12} -2.46791 q^{13} +4.68016 q^{14} -2.81743 q^{16} +6.52105 q^{17} -1.23396 q^{18} +8.25310 q^{19} -3.79281 q^{21} +2.34008 q^{23} +3.05694 q^{24} +3.04530 q^{26} +1.00000 q^{27} +1.81050 q^{28} +2.46791 q^{29} +5.27455 q^{31} -2.63730 q^{32} -8.04668 q^{34} -0.477352 q^{36} -3.34008 q^{37} -10.1840 q^{38} -2.46791 q^{39} -3.46410 q^{41} +4.68016 q^{42} -12.0459 q^{43} -2.88755 q^{46} +9.56933 q^{47} -2.81743 q^{48} +7.38537 q^{49} +6.52105 q^{51} +1.17806 q^{52} +6.61463 q^{53} -1.23396 q^{54} -11.5944 q^{56} +8.25310 q^{57} -3.04530 q^{58} +10.1840 q^{59} -6.66788 q^{61} -6.50856 q^{62} -3.79281 q^{63} +8.88918 q^{64} -3.34008 q^{67} -3.11284 q^{68} +2.34008 q^{69} +7.22925 q^{71} +3.05694 q^{72} -9.72482 q^{73} +4.12151 q^{74} -3.93963 q^{76} +3.04530 q^{78} -9.65644 q^{79} +1.00000 q^{81} +4.27455 q^{82} -11.0497 q^{83} +1.81050 q^{84} +14.8641 q^{86} +2.46791 q^{87} +10.2745 q^{89} +9.36031 q^{91} -1.11704 q^{92} +5.27455 q^{93} -11.8081 q^{94} -2.63730 q^{96} -13.2495 q^{97} -9.11323 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9} + 14 q^{12} + 8 q^{14} + 10 q^{16} + 4 q^{23} + 52 q^{26} + 6 q^{27} + 18 q^{31} + 26 q^{34} + 14 q^{36} - 10 q^{37} + 20 q^{38} + 8 q^{42} + 10 q^{48} + 68 q^{49} + 16 q^{53} - 76 q^{56} - 52 q^{58} - 20 q^{59} + 16 q^{64} - 10 q^{67} + 4 q^{69} - 4 q^{71} + 52 q^{78} + 6 q^{81} + 12 q^{82} - 12 q^{86} + 48 q^{89} + 16 q^{91} - 30 q^{92} + 18 q^{93} - 2 q^{97}+O(q^{100}) $ 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9 + 14 * q^12 + 8 * q^14 + 10 * q^16 + 4 * q^23 + 52 * q^26 + 6 * q^27 + 18 * q^31 + 26 * q^34 + 14 * q^36 - 10 * q^37 + 20 * q^38 + 8 * q^42 + 10 * q^48 + 68 * q^49 + 16 * q^53 - 76 * q^56 - 52 * q^58 - 20 * q^59 + 16 * q^64 - 10 * q^67 + 4 * q^69 - 4 * q^71 + 52 * q^78 + 6 * q^81 + 12 * q^82 - 12 * q^86 + 48 * q^89 + 16 * q^91 - 30 * q^92 + 18 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.23396	−0.872539	−0.436269	−	0.899816i	$-0.643701\pi$
$2$	−1.23396	−0.872539	−0.436269	+	0.899816i	$0.643701\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.477352	−0.238676
$5$	0	0
$5$	0	0
$6$	−1.23396	−0.503760
$7$	−3.79281	−1.43355	−0.716773	−	0.697307i	$-0.754382\pi$
$7$	−3.79281	−1.43355	−0.716773	+	0.697307i	$0.754382\pi$
$8$	3.05694	1.08079
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.477352	−0.137800
$13$	−2.46791	−0.684476	−0.342238	−	0.939613i	$-0.611185\pi$
$13$	−2.46791	−0.684476	−0.342238	+	0.939613i	$0.611185\pi$
$14$	4.68016	1.25082
$15$	0	0
$16$	−2.81743	−0.704358
$17$	6.52105	1.58159	0.790793	−	0.612084i	$-0.209668\pi$
$17$	6.52105	1.58159	0.790793	+	0.612084i	$0.209668\pi$
$18$	−1.23396	−0.290846
$19$	8.25310	1.89339	0.946695	−	0.322131i	$-0.104399\pi$
$19$	8.25310	1.89339	0.946695	+	0.322131i	$0.104399\pi$
$20$	0	0
$21$	−3.79281	−0.827658
$22$	0	0
$23$	2.34008	0.487940	0.243970	−	0.969783i	$-0.421550\pi$
$23$	2.34008	0.487940	0.243970	+	0.969783i	$0.421550\pi$
$24$	3.05694	0.623996
$25$	0	0
$26$	3.04530	0.597232
$27$	1.00000	0.192450
$28$	1.81050	0.342153
$29$	2.46791	0.458280	0.229140	−	0.973394i	$-0.426409\pi$
$29$	2.46791	0.458280	0.229140	+	0.973394i	$0.426409\pi$
$30$	0	0
$31$	5.27455	0.947337	0.473669	−	0.880703i	$-0.342929\pi$
$31$	5.27455	0.947337	0.473669	+	0.880703i	$0.342929\pi$
$32$	−2.63730	−0.466214
$33$	0	0
$34$	−8.04668	−1.37999
$35$	0	0
$36$	−0.477352	−0.0795587
$37$	−3.34008	−0.549105	−0.274553	−	0.961572i	$-0.588530\pi$
$37$	−3.34008	−0.549105	−0.274553	+	0.961572i	$0.588530\pi$
$38$	−10.1840	−1.65206
$39$	−2.46791	−0.395182
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	4.68016	0.722164
$43$	−12.0459	−1.83698	−0.918491	−	0.395441i	$-0.870592\pi$
$43$	−12.0459	−1.83698	−0.918491	+	0.395441i	$0.870592\pi$
$44$	0	0
$45$	0	0
$46$	−2.88755	−0.425746
$47$	9.56933	1.39583	0.697915	−	0.716180i	$-0.254111\pi$
$47$	9.56933	1.39583	0.697915	+	0.716180i	$0.254111\pi$
$48$	−2.81743	−0.406661
$49$	7.38537	1.05505
$50$	0	0
$51$	6.52105	0.913129
$52$	1.17806	0.163368
$53$	6.61463	0.908589	0.454294	−	0.890852i	$-0.349891\pi$
$53$	6.61463	0.908589	0.454294	+	0.890852i	$0.349891\pi$
$54$	−1.23396	−0.167920
$55$	0	0
$56$	−11.5944	−1.54937
$57$	8.25310	1.09315
$58$	−3.04530	−0.399867
$59$	10.1840	1.32584	0.662919	−	0.748691i	$-0.269317\pi$
$59$	10.1840	1.32584	0.662919	+	0.748691i	$0.269317\pi$
$60$	0	0
$61$	−6.66788	−0.853734	−0.426867	−	0.904314i	$-0.640383\pi$
$61$	−6.66788	−0.853734	−0.426867	+	0.904314i	$0.640383\pi$
$62$	−6.50856	−0.826588
$63$	−3.79281	−0.477849
$64$	8.88918	1.11115
$65$	0	0
$66$	0	0
$67$	−3.34008	−0.408055	−0.204028	−	0.978965i	$-0.565403\pi$
$67$	−3.34008	−0.408055	−0.204028	+	0.978965i	$0.565403\pi$
$68$	−3.11284	−0.377487
$69$	2.34008	0.281712
$70$	0	0
$71$	7.22925	0.857955	0.428977	−	0.903315i	$-0.358874\pi$
$71$	7.22925	0.857955	0.428977	+	0.903315i	$0.358874\pi$
$72$	3.05694	0.360264
$73$	−9.72482	−1.13820	−0.569102	−	0.822267i	$-0.692709\pi$
$73$	−9.72482	−1.13820	−0.569102	+	0.822267i	$0.692709\pi$
$74$	4.12151	0.479116
$75$	0	0
$76$	−3.93963	−0.451907
$77$	0	0
$78$	3.04530	0.344812
$79$	−9.65644	−1.08643	−0.543217	−	0.839592i	$-0.682794\pi$
$79$	−9.65644	−1.08643	−0.543217	+	0.839592i	$0.682794\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.27455	0.472045
$83$	−11.0497	−1.21286	−0.606432	−	0.795136i	$-0.707400\pi$
$83$	−11.0497	−1.21286	−0.606432	+	0.795136i	$0.707400\pi$
$84$	1.81050	0.197542
$85$	0	0
$86$	14.8641	1.60284
$87$	2.46791	0.264588
$88$	0	0
$89$	10.2745	1.08910	0.544550	−	0.838728i	$-0.316700\pi$
$89$	10.2745	1.08910	0.544550	+	0.838728i	$0.316700\pi$
$90$	0	0
$91$	9.36031	0.981227
$92$	−1.11704	−0.116460
$93$	5.27455	0.546945
$94$	−11.8081	−1.21792
$95$	0	0
$96$	−2.63730	−0.269169
$97$	−13.2495	−1.34528	−0.672641	−	0.739969i	$-0.734840\pi$
$97$	−13.2495	−1.34528	−0.672641	+	0.739969i	$0.734840\pi$
$98$	−9.11323	−0.920575
$99$	0	0
$100$	0	0
$101$	1.99238	0.198249	0.0991246	−	0.995075i	$-0.468396\pi$
$101$	1.99238	0.198249	0.0991246	+	0.995075i	$0.468396\pi$
$102$	−8.04668	−0.796740
$103$	1.61463	0.159094	0.0795470	−	0.996831i	$-0.474653\pi$
$103$	1.61463	0.159094	0.0795470	+	0.996831i	$0.474653\pi$
$104$	−7.54427	−0.739776
$105$	0	0
$106$	−8.16216	−0.792779
$107$	−0.589032	−0.0569439	−0.0284719	−	0.999595i	$-0.509064\pi$
$107$	−0.589032	−0.0569439	−0.0284719	+	0.999595i	$0.509064\pi$
$108$	−0.477352	−0.0459333
$109$	4.78899	0.458703	0.229351	−	0.973344i	$-0.426340\pi$
$109$	4.78899	0.458703	0.229351	+	0.973344i	$0.426340\pi$
$110$	0	0
$111$	−3.34008	−0.317026
$112$	10.6860	1.00973
$113$	−18.5240	−1.74259	−0.871297	−	0.490755i	$-0.836721\pi$
$113$	−18.5240	−1.74259	−0.871297	+	0.490755i	$0.836721\pi$
$114$	−10.1840	−0.953815
$115$	0	0
$116$	−1.17806	−0.109380
$117$	−2.46791	−0.228159
$118$	−12.5666	−1.15685
$119$	−24.7331	−2.26728
$120$	0	0
$121$	0	0
$122$	8.22787	0.744916
$123$	−3.46410	−0.312348
$124$	−2.51782	−0.226107
$125$	0	0
$126$	4.68016	0.416941
$127$	12.1927	1.08193	0.540965	−	0.841045i	$-0.318059\pi$
$127$	12.1927	1.08193	0.540965	+	0.841045i	$0.318059\pi$
$128$	−5.69425	−0.503305
$129$	−12.0459	−1.06058
$130$	0	0
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	−31.3024	−2.71426
$134$	4.12151	0.356044
$135$	0	0
$136$	19.9345	1.70937
$137$	−7.93447	−0.677888	−0.338944	−	0.940807i	$-0.610070\pi$
$137$	−7.93447	−0.677888	−0.338944	+	0.940807i	$0.610070\pi$
$138$	−2.88755	−0.245805
$139$	−6.18226	−0.524373	−0.262186	−	0.965017i	$-0.584444\pi$
$139$	−6.18226	−0.524373	−0.262186	+	0.965017i	$0.584444\pi$
$140$	0	0
$141$	9.56933	0.805883
$142$	−8.92058	−0.748599
$143$	0	0
$144$	−2.81743	−0.234786
$145$	0	0
$146$	12.0000	0.993127
$147$	7.38537	0.609135
$148$	1.59439	0.131058
$149$	1.81050	0.148322	0.0741612	−	0.997246i	$-0.476372\pi$
$149$	1.81050	0.148322	0.0741612	+	0.997246i	$0.476372\pi$
$150$	0	0
$151$	−20.0387	−1.63072	−0.815362	−	0.578952i	$-0.803462\pi$
$151$	−20.0387	−1.63072	−0.815362	+	0.578952i	$0.803462\pi$
$152$	25.2293	2.04636
$153$	6.52105	0.527195
$154$	0	0
$155$	0	0
$156$	1.17806	0.0943206
$157$	14.5693	1.16276	0.581380	−	0.813632i	$-0.302513\pi$
$157$	14.5693	1.16276	0.581380	+	0.813632i	$0.302513\pi$
$158$	11.9156	0.947956
$159$	6.61463	0.524574
$160$	0	0
$161$	−8.87546	−0.699484
$162$	−1.23396	−0.0969487
$163$	5.88918	0.461276	0.230638	−	0.973040i	$-0.425919\pi$
$163$	5.88918	0.461276	0.230638	+	0.973040i	$0.425919\pi$
$164$	1.65360	0.129124
$165$	0	0
$166$	13.6349	1.05827
$167$	22.8454	1.76783	0.883914	−	0.467650i	$-0.154899\pi$
$167$	22.8454	1.76783	0.883914	+	0.467650i	$0.154899\pi$
$168$	−11.5944	−0.894527
$169$	−6.90941	−0.531493
$170$	0	0
$171$	8.25310	0.631130
$172$	5.75014	0.438444
$173$	12.5214	0.951987	0.475994	−	0.879449i	$-0.342089\pi$
$173$	12.5214	0.951987	0.475994	+	0.879449i	$0.342089\pi$
$174$	−3.04530	−0.230863
$175$	0	0
$176$	0	0
$177$	10.1840	0.765473
$178$	−12.6783	−0.950282
$179$	−3.04530	−0.227616	−0.113808	−	0.993503i	$-0.536305\pi$
$179$	−3.04530	−0.227616	−0.113808	+	0.993503i	$0.536305\pi$
$180$	0	0
$181$	4.38537	0.325962	0.162981	−	0.986629i	$-0.447889\pi$
$181$	4.38537	0.325962	0.162981	+	0.986629i	$0.447889\pi$
$182$	−11.5502	−0.856159
$183$	−6.66788	−0.492904
$184$	7.15349	0.527362
$185$	0	0
$186$	−6.50856	−0.477231
$187$	0	0
$188$	−4.56794	−0.333151
$189$	−3.79281	−0.275886
$190$	0	0
$191$	23.4132	1.69412	0.847060	−	0.531497i	$-0.178370\pi$
$191$	23.4132	1.69412	0.847060	+	0.531497i	$0.178370\pi$
$192$	8.88918	0.641521
$193$	23.2424	1.67303	0.836514	−	0.547946i	$-0.184590\pi$
$193$	23.2424	1.67303	0.836514	+	0.547946i	$0.184590\pi$
$194$	16.3493	1.17381
$195$	0	0
$196$	−3.52543	−0.251816
$197$	−1.99238	−0.141951	−0.0709756	−	0.997478i	$-0.522611\pi$
$197$	−1.99238	−0.141951	−0.0709756	+	0.997478i	$0.522611\pi$
$198$	0	0
$199$	8.72545	0.618531	0.309265	−	0.950976i	$-0.399917\pi$
$199$	8.72545	0.618531	0.309265	+	0.950976i	$0.399917\pi$
$200$	0	0
$201$	−3.34008	−0.235591
$202$	−2.45851	−0.172980
$203$	−9.36031	−0.656965
$204$	−3.11284	−0.217942
$205$	0	0
$206$	−1.99238	−0.138816
$207$	2.34008	0.162647
$208$	6.95317	0.482116
$209$	0	0
$210$	0	0
$211$	−16.0640	−1.10589	−0.552945	−	0.833218i	$-0.686496\pi$
$211$	−16.0640	−1.10589	−0.552945	+	0.833218i	$0.686496\pi$
$212$	−3.15751	−0.216859
$213$	7.22925	0.495340
$214$	0.726839	0.0496857
$215$	0	0
$216$	3.05694	0.207999
$217$	−20.0053	−1.35805
$218$	−5.90941	−0.400236
$219$	−9.72482	−0.657142
$220$	0	0
$221$	−16.0934	−1.08256
$222$	4.12151	0.276618
$223$	11.0655	0.741003	0.370501	−	0.928832i	$-0.379186\pi$
$223$	11.0655	0.741003	0.370501	+	0.928832i	$0.379186\pi$
$224$	10.0028	0.668339
$225$	0	0
$226$	22.8578	1.52048
$227$	15.3965	1.02190	0.510951	−	0.859610i	$-0.329293\pi$
$227$	15.3965	1.02190	0.510951	+	0.859610i	$0.329293\pi$
$228$	−3.93963	−0.260909
$229$	4.70522	0.310930	0.155465	−	0.987841i	$-0.450312\pi$
$229$	4.70522	0.310930	0.155465	+	0.987841i	$0.450312\pi$
$230$	0	0
$231$	0	0
$232$	7.54427	0.495306
$233$	14.9210	0.977505	0.488753	−	0.872422i	$-0.337452\pi$
$233$	14.9210	0.977505	0.488753	+	0.872422i	$0.337452\pi$
$234$	3.04530	0.199077
$235$	0	0
$236$	−4.86134	−0.316446
$237$	−9.65644	−0.627253
$238$	30.5195	1.97829
$239$	11.3885	0.736660	0.368330	−	0.929695i	$-0.379930\pi$
$239$	11.3885	0.736660	0.368330	+	0.929695i	$0.379930\pi$
$240$	0	0
$241$	−12.1143	−0.780349	−0.390175	−	0.920741i	$-0.627585\pi$
$241$	−12.1143	−0.780349	−0.390175	+	0.920741i	$0.627585\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	3.18293	0.203766
$245$	0	0
$246$	4.27455	0.272535
$247$	−20.3679	−1.29598
$248$	16.1240	1.02388
$249$	−11.0497	−0.700247
$250$	0	0
$251$	−17.4132	−1.09911	−0.549556	−	0.835457i	$-0.685203\pi$
$251$	−17.4132	−1.09911	−0.549556	+	0.835457i	$0.685203\pi$
$252$	1.81050	0.114051
$253$	0	0
$254$	−15.0453	−0.944026
$255$	0	0
$256$	−10.7519	−0.671994
$257$	18.4334	1.14985	0.574923	−	0.818207i	$-0.305032\pi$
$257$	18.4334	1.14985	0.574923	+	0.818207i	$0.305032\pi$
$258$	14.8641	0.925399
$259$	12.6683	0.787168
$260$	0	0
$261$	2.46791	0.152760
$262$	4.27455	0.264083
$263$	−2.06075	−0.127072	−0.0635358	−	0.997980i	$-0.520238\pi$
$263$	−2.06075	−0.127072	−0.0635358	+	0.997980i	$0.520238\pi$
$264$	0	0
$265$	0	0
$266$	38.6258	2.36830
$267$	10.2745	0.628792
$268$	1.59439	0.0973931
$269$	3.45090	0.210405	0.105203	−	0.994451i	$-0.466451\pi$
$269$	3.45090	0.210405	0.105203	+	0.994451i	$0.466451\pi$
$270$	0	0
$271$	−21.2167	−1.28882	−0.644412	−	0.764678i	$-0.722898\pi$
$271$	−21.2167	−1.28882	−0.644412	+	0.764678i	$0.722898\pi$
$272$	−18.3726	−1.11400
$273$	9.36031	0.566512
$274$	9.79079	0.591483
$275$	0	0
$276$	−1.11704	−0.0672380
$277$	4.13159	0.248243	0.124122	−	0.992267i	$-0.460389\pi$
$277$	4.13159	0.248243	0.124122	+	0.992267i	$0.460389\pi$
$278$	7.62864	0.457536
$279$	5.27455	0.315779
$280$	0	0
$281$	23.0956	1.37777	0.688884	−	0.724871i	$-0.258101\pi$
$281$	23.0956	1.37777	0.688884	+	0.724871i	$0.258101\pi$
$282$	−11.8081	−0.703164
$283$	24.7593	1.47179	0.735893	−	0.677097i	$-0.236762\pi$
$283$	24.7593	1.47179	0.735893	+	0.677097i	$0.236762\pi$
$284$	−3.45090	−0.204773
$285$	0	0
$286$	0	0
$287$	13.1387	0.775551
$288$	−2.63730	−0.155405
$289$	25.5240	1.50141
$290$	0	0
$291$	−13.2495	−0.776699
$292$	4.64217	0.271662
$293$	−7.33536	−0.428536	−0.214268	−	0.976775i	$-0.568737\pi$
$293$	−7.33536	−0.428536	−0.214268	+	0.976775i	$0.568737\pi$
$294$	−9.11323	−0.531494
$295$	0	0
$296$	−10.2104	−0.593469
$297$	0	0
$298$	−2.23408	−0.129417
$299$	−5.77511	−0.333983
$300$	0	0
$301$	45.6878	2.63340
$302$	24.7268	1.42287
$303$	1.99238	0.114459
$304$	−23.2525	−1.33362
$305$	0	0
$306$	−8.04668	−0.459998
$307$	−22.7669	−1.29938	−0.649688	−	0.760201i	$-0.725101\pi$
$307$	−22.7669	−1.29938	−0.649688	+	0.760201i	$0.725101\pi$
$308$	0	0
$309$	1.61463	0.0918529
$310$	0	0
$311$	−25.5443	−1.44848	−0.724241	−	0.689547i	$-0.757810\pi$
$311$	−25.5443	−1.44848	−0.724241	+	0.689547i	$0.757810\pi$
$312$	−7.54427	−0.427110
$313$	34.7735	1.96552	0.982758	−	0.184897i	$-0.0591952\pi$
$313$	34.7735	1.96552	0.982758	+	0.184897i	$0.0591952\pi$
$314$	−17.9779	−1.01455
$315$	0	0
$316$	4.60953	0.259306
$317$	5.38537	0.302473	0.151236	−	0.988498i	$-0.451675\pi$
$317$	5.38537	0.302473	0.151236	+	0.988498i	$0.451675\pi$
$318$	−8.16216	−0.457711
$319$	0	0
$320$	0	0
$321$	−0.589032	−0.0328766
$322$	10.9519	0.610327
$323$	53.8188	2.99456
$324$	−0.477352	−0.0265196
$325$	0	0
$326$	−7.26699	−0.402481
$327$	4.78899	0.264832
$328$	−10.5896	−0.584711
$329$	−36.2946	−2.00099
$330$	0	0
$331$	19.8641	1.09183	0.545915	−	0.837840i	$-0.316182\pi$
$331$	19.8641	1.09183	0.545915	+	0.837840i	$0.316182\pi$
$332$	5.27461	0.289482
$333$	−3.34008	−0.183035
$334$	−28.1902	−1.54250
$335$	0	0
$336$	10.6860	0.582967
$337$	−10.9029	−0.593918	−0.296959	−	0.954890i	$-0.595972\pi$
$337$	−10.9029	−0.593918	−0.296959	+	0.954890i	$0.595972\pi$
$338$	8.52591	0.463748
$339$	−18.5240	−1.00609
$340$	0	0
$341$	0	0
$342$	−10.1840	−0.550685
$343$	−1.46164	−0.0789214
$344$	−36.8236	−1.98540
$345$	0	0
$346$	−15.4509	−0.830646
$347$	5.00420	0.268640	0.134320	−	0.990938i	$-0.457115\pi$
$347$	5.00420	0.268640	0.134320	+	0.990938i	$0.457115\pi$
$348$	−1.17806	−0.0631508
$349$	26.7749	1.43323	0.716614	−	0.697470i	$-0.245691\pi$
$349$	26.7749	1.43323	0.716614	+	0.697470i	$0.245691\pi$
$350$	0	0
$351$	−2.46791	−0.131727
$352$	0	0
$353$	−17.2042	−0.915687	−0.457843	−	0.889033i	$-0.651378\pi$
$353$	−17.2042	−0.915687	−0.457843	+	0.889033i	$0.651378\pi$
$354$	−12.5666	−0.667905
$355$	0	0
$356$	−4.90458	−0.259942
$357$	−24.7331	−1.30901
$358$	3.75776	0.198604
$359$	0.657408	0.0346966	0.0173483	−	0.999850i	$-0.494478\pi$
$359$	0.657408	0.0346966	0.0173483	+	0.999850i	$0.494478\pi$
$360$	0	0
$361$	49.1136	2.58493
$362$	−5.41136	−0.284415
$363$	0	0
$364$	−4.46817	−0.234196
$365$	0	0
$366$	8.22787	0.430077
$367$	33.5038	1.74888	0.874442	−	0.485130i	$-0.161228\pi$
$367$	33.5038	1.74888	0.874442	+	0.485130i	$0.161228\pi$
$368$	−6.59301	−0.343684
$369$	−3.46410	−0.180334
$370$	0	0
$371$	−25.0880	−1.30250
$372$	−2.51782	−0.130543
$373$	7.73244	0.400371	0.200185	−	0.979758i	$-0.435846\pi$
$373$	7.73244	0.400371	0.200185	+	0.979758i	$0.435846\pi$
$374$	0	0
$375$	0	0
$376$	29.2529	1.50860
$377$	−6.09059	−0.313681
$378$	4.68016	0.240721
$379$	−0.340078	−0.0174686	−0.00873431	−	0.999962i	$-0.502780\pi$
$379$	−0.340078	−0.0174686	−0.00873431	+	0.999962i	$0.502780\pi$
$380$	0	0
$381$	12.1927	0.624653
$382$	−28.8909	−1.47819
$383$	−31.0481	−1.58648	−0.793241	−	0.608908i	$-0.791608\pi$
$383$	−31.0481	−1.58648	−0.793241	+	0.608908i	$0.791608\pi$
$384$	−5.69425	−0.290583
$385$	0	0
$386$	−28.6802	−1.45978
$387$	−12.0459	−0.612328
$388$	6.32467	0.321087
$389$	−24.3150	−1.23282	−0.616410	−	0.787425i	$-0.711414\pi$
$389$	−24.3150	−1.23282	−0.616410	+	0.787425i	$0.711414\pi$
$390$	0	0
$391$	15.2598	0.771719
$392$	22.5767	1.14029
$393$	−3.46410	−0.174741
$394$	2.45851	0.123858
$395$	0	0
$396$	0	0
$397$	−10.2948	−0.516680	−0.258340	−	0.966054i	$-0.583176\pi$
$397$	−10.2948	−0.516680	−0.258340	+	0.966054i	$0.583176\pi$
$398$	−10.7668	−0.539692
$399$	−31.3024	−1.56708
$400$	0	0
$401$	−19.2293	−0.960263	−0.480132	−	0.877196i	$-0.659411\pi$
$401$	−19.2293	−0.960263	−0.480132	+	0.877196i	$0.659411\pi$
$402$	4.12151	0.205562
$403$	−13.0171	−0.648429
$404$	−0.951067	−0.0473173
$405$	0	0
$406$	11.5502	0.573227
$407$	0	0
$408$	19.9345	0.986903
$409$	6.53112	0.322943	0.161472	−	0.986877i	$-0.448376\pi$
$409$	6.53112	0.322943	0.161472	+	0.986877i	$0.448376\pi$
$410$	0	0
$411$	−7.93447	−0.391379
$412$	−0.770746	−0.0379719
$413$	−38.6258	−1.90065
$414$	−2.88755	−0.141915
$415$	0	0
$416$	6.50863	0.319112
$417$	−6.18226	−0.302747
$418$	0	0
$419$	27.6878	1.35264	0.676318	−	0.736610i	$-0.263575\pi$
$419$	27.6878	1.35264	0.676318	+	0.736610i	$0.263575\pi$
$420$	0	0
$421$	−11.1637	−0.544087	−0.272043	−	0.962285i	$-0.587699\pi$
$421$	−11.1637	−0.544087	−0.272043	+	0.962285i	$0.587699\pi$
$422$	19.8223	0.964932
$423$	9.56933	0.465277
$424$	20.2205	0.981996
$425$	0	0
$426$	−8.92058	−0.432204
$427$	25.2900	1.22387
$428$	0.281176	0.0135911
$429$	0	0
$430$	0	0
$431$	13.3607	0.643563	0.321782	−	0.946814i	$-0.395718\pi$
$431$	13.3607	0.643563	0.321782	+	0.946814i	$0.395718\pi$
$432$	−2.81743	−0.135554
$433$	−19.2495	−0.925071	−0.462536	−	0.886601i	$-0.653060\pi$
$433$	−19.2495	−0.925071	−0.462536	+	0.886601i	$0.653060\pi$
$434$	24.6857	1.18495
$435$	0	0
$436$	−2.28604	−0.109481
$437$	19.3129	0.923861
$438$	12.0000	0.573382
$439$	21.6572	1.03364	0.516821	−	0.856093i	$-0.327115\pi$
$439$	21.6572	1.03364	0.516821	+	0.856093i	$0.327115\pi$
$440$	0	0
$441$	7.38537	0.351684
$442$	19.8585	0.944573
$443$	−1.13866	−0.0540995	−0.0270498	−	0.999634i	$-0.508611\pi$
$443$	−1.13866	−0.0540995	−0.0270498	+	0.999634i	$0.508611\pi$
$444$	1.59439	0.0756666
$445$	0	0
$446$	−13.6544	−0.646553
$447$	1.81050	0.0856339
$448$	−33.7149	−1.59288
$449$	−0.0905906	−0.00427524	−0.00213762	−	0.999998i	$-0.500680\pi$
$449$	−0.0905906	−0.00427524	−0.00213762	+	0.999998i	$0.500680\pi$
$450$	0	0
$451$	0	0
$452$	8.84249	0.415916
$453$	−20.0387	−0.941499
$454$	−18.9986	−0.891649
$455$	0	0
$456$	25.2293	1.18147
$457$	−4.12151	−0.192796	−0.0963980	−	0.995343i	$-0.530732\pi$
$457$	−4.12151	−0.192796	−0.0963980	+	0.995343i	$0.530732\pi$
$458$	−5.80603	−0.271298
$459$	6.52105	0.304376
$460$	0	0
$461$	−6.74633	−0.314208	−0.157104	−	0.987582i	$-0.550216\pi$
$461$	−6.74633	−0.314208	−0.157104	+	0.987582i	$0.550216\pi$
$462$	0	0
$463$	20.4990	0.952668	0.476334	−	0.879264i	$-0.341965\pi$
$463$	20.4990	0.952668	0.476334	+	0.879264i	$0.341965\pi$
$464$	−6.95317	−0.322793
$465$	0	0
$466$	−18.4118	−0.852911
$467$	4.79859	0.222052	0.111026	−	0.993817i	$-0.464586\pi$
$467$	4.79859	0.222052	0.111026	+	0.993817i	$0.464586\pi$
$468$	1.17806	0.0544560
$469$	12.6683	0.584966
$470$	0	0
$471$	14.5693	0.671319
$472$	31.1318	1.43296
$473$	0	0
$474$	11.9156	0.547303
$475$	0	0
$476$	11.8064	0.541145
$477$	6.61463	0.302863
$478$	−14.0529	−0.642765
$479$	−9.39612	−0.429319	−0.214660	−	0.976689i	$-0.568864\pi$
$479$	−9.39612	−0.429319	−0.214660	+	0.976689i	$0.568864\pi$
$480$	0	0
$481$	8.24302	0.375849
$482$	14.9485	0.680885
$483$	−8.87546	−0.403847
$484$	0	0
$485$	0	0
$486$	−1.23396	−0.0559734
$487$	24.0934	1.09177	0.545887	−	0.837859i	$-0.316193\pi$
$487$	24.0934	1.09177	0.545887	+	0.837859i	$0.316193\pi$
$488$	−20.3833	−0.922710
$489$	5.88918	0.266318
$490$	0	0
$491$	−6.63454	−0.299413	−0.149706	−	0.988730i	$-0.547833\pi$
$491$	−6.63454	−0.299413	−0.149706	+	0.988730i	$0.547833\pi$
$492$	1.65360	0.0745499
$493$	16.0934	0.724809
$494$	25.1331	1.13079
$495$	0	0
$496$	−14.8607	−0.667264
$497$	−27.4192	−1.22992
$498$	13.6349	0.610993
$499$	22.1637	0.992185	0.496092	−	0.868270i	$-0.334768\pi$
$499$	22.1637	0.992185	0.496092	+	0.868270i	$0.334768\pi$
$500$	0	0
$501$	22.8454	1.02066
$502$	21.4871	0.959018
$503$	−37.8597	−1.68808	−0.844040	−	0.536281i	$-0.819829\pi$
$503$	−37.8597	−1.68808	−0.844040	+	0.536281i	$0.819829\pi$
$504$	−11.5944	−0.516455
$505$	0	0
$506$	0	0
$507$	−6.90941	−0.306858
$508$	−5.82023	−0.258231
$509$	−0.405606	−0.0179782	−0.00898909	−	0.999960i	$-0.502861\pi$
$509$	−0.405606	−0.0179782	−0.00898909	+	0.999960i	$0.502861\pi$
$510$	0	0
$511$	36.8843	1.63167
$512$	24.6559	1.08965
$513$	8.25310	0.364383
$514$	−22.7461	−1.00329
$515$	0	0
$516$	5.75014	0.253136
$517$	0	0
$518$	−15.6321	−0.686834
$519$	12.5214	0.549630
$520$	0	0
$521$	22.9952	1.00744	0.503718	−	0.863868i	$-0.331965\pi$
$521$	22.9952	1.00744	0.503718	+	0.863868i	$0.331965\pi$
$522$	−3.04530	−0.133289
$523$	1.14302	0.0499807	0.0249904	−	0.999688i	$-0.492044\pi$
$523$	1.14302	0.0499807	0.0249904	+	0.999688i	$0.492044\pi$
$524$	1.65360	0.0722377
$525$	0	0
$526$	2.54288	0.110875
$527$	34.3956	1.49829
$528$	0	0
$529$	−17.5240	−0.761915
$530$	0	0
$531$	10.1840	0.441946
$532$	14.9423	0.647830
$533$	8.54910	0.370303
$534$	−12.6783	−0.548646
$535$	0	0
$536$	−10.2104	−0.441023
$537$	−3.04530	−0.131414
$538$	−4.25826	−0.183587
$539$	0	0
$540$	0	0
$541$	8.92058	0.383526	0.191763	−	0.981441i	$-0.438580\pi$
$541$	8.92058	0.383526	0.191763	+	0.981441i	$0.438580\pi$
$542$	26.1805	1.12455
$543$	4.38537	0.188194
$544$	−17.1980	−0.737357
$545$	0	0
$546$	−11.5502	−0.494303
$547$	−8.73871	−0.373640	−0.186820	−	0.982394i	$-0.559818\pi$
$547$	−8.73871	−0.373640	−0.186820	+	0.982394i	$0.559818\pi$
$548$	3.78754	0.161796
$549$	−6.66788	−0.284578
$550$	0	0
$551$	20.3679	0.867702
$552$	7.15349	0.304473
$553$	36.6250	1.55745
$554$	−5.09820	−0.216602
$555$	0	0
$556$	2.95112	0.125155
$557$	−16.6197	−0.704199	−0.352099	−	0.935963i	$-0.614532\pi$
$557$	−16.6197	−0.704199	−0.352099	+	0.935963i	$0.614532\pi$
$558$	−6.50856	−0.275529
$559$	29.7282	1.25737
$560$	0	0
$561$	0	0
$562$	−28.4990	−1.20216
$563$	41.7060	1.75770	0.878849	−	0.477101i	$-0.158312\pi$
$563$	41.7060	1.75770	0.878849	+	0.477101i	$0.158312\pi$
$564$	−4.56794	−0.192345
$565$	0	0
$566$	−30.5519	−1.28419
$567$	−3.79281	−0.159283
$568$	22.0994	0.927271
$569$	−7.76749	−0.325630	−0.162815	−	0.986657i	$-0.552057\pi$
$569$	−7.76749	−0.325630	−0.162815	+	0.986657i	$0.552057\pi$
$570$	0	0
$571$	−26.9568	−1.12811	−0.564053	−	0.825738i	$-0.690759\pi$
$571$	−26.9568	−1.12811	−0.564053	+	0.825738i	$0.690759\pi$
$572$	0	0
$573$	23.4132	0.978101
$574$	−16.2125	−0.676698
$575$	0	0
$576$	8.88918	0.370382
$577$	−26.9345	−1.12130	−0.560648	−	0.828054i	$-0.689448\pi$
$577$	−26.9345	−1.12130	−0.560648	+	0.828054i	$0.689448\pi$
$578$	−31.4955	−1.31004
$579$	23.2424	0.965923
$580$	0	0
$581$	41.9094	1.73870
$582$	16.3493	0.677700
$583$	0	0
$584$	−29.7282	−1.23016
$585$	0	0
$586$	9.05151	0.373915
$587$	13.4383	0.554657	0.277328	−	0.960775i	$-0.410551\pi$
$587$	13.4383	0.554657	0.277328	+	0.960775i	$0.410551\pi$
$588$	−3.52543	−0.145386
$589$	43.5314	1.79368
$590$	0	0
$591$	−1.99238	−0.0819555
$592$	9.41044	0.386767
$593$	23.4344	0.962335	0.481168	−	0.876629i	$-0.340213\pi$
$593$	23.4344	0.962335	0.481168	+	0.876629i	$0.340213\pi$
$594$	0	0
$595$	0	0
$596$	−0.864249	−0.0354010
$597$	8.72545	0.357109
$598$	7.12623	0.291413
$599$	−40.0028	−1.63447	−0.817235	−	0.576305i	$-0.804494\pi$
$599$	−40.0028	−1.63447	−0.817235	+	0.576305i	$0.804494\pi$
$600$	0	0
$601$	15.3181	0.624836	0.312418	−	0.949945i	$-0.398861\pi$
$601$	15.3181	0.624836	0.312418	+	0.949945i	$0.398861\pi$
$602$	−56.3767	−2.29774
$603$	−3.34008	−0.136018
$604$	9.56551	0.389215
$605$	0	0
$606$	−2.45851	−0.0998701
$607$	34.0086	1.38037	0.690183	−	0.723635i	$-0.257530\pi$
$607$	34.0086	1.38037	0.690183	+	0.723635i	$0.257530\pi$
$608$	−21.7659	−0.882724
$609$	−9.36031	−0.379299
$610$	0	0
$611$	−23.6163	−0.955412
$612$	−3.11284	−0.125829
$613$	40.0874	1.61912	0.809558	−	0.587040i	$-0.199707\pi$
$613$	40.0874	1.61912	0.809558	+	0.587040i	$0.199707\pi$
$614$	28.0934	1.13376
$615$	0	0
$616$	0	0
$617$	−3.36031	−0.135281	−0.0676405	−	0.997710i	$-0.521547\pi$
$617$	−3.36031	−0.135281	−0.0676405	+	0.997710i	$0.521547\pi$
$618$	−1.99238	−0.0801452
$619$	28.5491	1.14749	0.573743	−	0.819036i	$-0.305491\pi$
$619$	28.5491	1.14749	0.573743	+	0.819036i	$0.305491\pi$
$620$	0	0
$621$	2.34008	0.0939041
$622$	31.5205	1.26386
$623$	−38.9694	−1.56127
$624$	6.95317	0.278350
$625$	0	0
$626$	−42.9090	−1.71499
$627$	0	0
$628$	−6.95470	−0.277523
$629$	−21.7808	−0.868457
$630$	0	0
$631$	−18.7986	−0.748360	−0.374180	−	0.927356i	$-0.622076\pi$
$631$	−18.7986	−0.748360	−0.374180	+	0.927356i	$0.622076\pi$
$632$	−29.5192	−1.17421
$633$	−16.0640	−0.638486
$634$	−6.64531	−0.263919
$635$	0	0
$636$	−3.15751	−0.125203
$637$	−18.2265	−0.722158
$638$	0	0
$639$	7.22925	0.285985
$640$	0	0
$641$	45.1010	1.78138	0.890691	−	0.454610i	$-0.150221\pi$
$641$	45.1010	1.78138	0.890691	+	0.454610i	$0.150221\pi$
$642$	0.726839	0.0286861
$643$	−30.8439	−1.21636	−0.608182	−	0.793798i	$-0.708101\pi$
$643$	−30.8439	−1.21636	−0.608182	+	0.793798i	$0.708101\pi$
$644$	4.23672	0.166950
$645$	0	0
$646$	−66.4101	−2.61287
$647$	−31.3477	−1.23240	−0.616202	−	0.787588i	$-0.711330\pi$
$647$	−31.3477	−1.23240	−0.616202	+	0.787588i	$0.711330\pi$
$648$	3.05694	0.120088
$649$	0	0
$650$	0	0
$651$	−20.0053	−0.784071
$652$	−2.81121	−0.110096
$653$	−32.4585	−1.27020	−0.635100	−	0.772430i	$-0.719041\pi$
$653$	−32.4585	−1.27020	−0.635100	+	0.772430i	$0.719041\pi$
$654$	−5.90941	−0.231076
$655$	0	0
$656$	9.75986	0.381059
$657$	−9.72482	−0.379401
$658$	44.7860	1.74594
$659$	21.7808	0.848459	0.424230	−	0.905555i	$-0.360545\pi$
$659$	21.7808	0.848459	0.424230	+	0.905555i	$0.360545\pi$
$660$	0	0
$661$	3.83627	0.149214	0.0746069	−	0.997213i	$-0.476230\pi$
$661$	3.83627	0.149214	0.0746069	+	0.997213i	$0.476230\pi$
$662$	−24.5114	−0.952664
$663$	−16.0934	−0.625015
$664$	−33.7784	−1.31085
$665$	0	0
$666$	4.12151	0.159705
$667$	5.77511	0.223613
$668$	−10.9053	−0.421938
$669$	11.0655	0.427818
$670$	0	0
$671$	0	0
$672$	10.0028	0.385866
$673$	−24.8711	−0.958709	−0.479355	−	0.877621i	$-0.659129\pi$
$673$	−24.8711	−0.958709	−0.479355	+	0.877621i	$0.659129\pi$
$674$	13.4537	0.518216
$675$	0	0
$676$	3.29822	0.126855
$677$	14.2202	0.546525	0.273262	−	0.961940i	$-0.411897\pi$
$677$	14.2202	0.546525	0.273262	+	0.961940i	$0.411897\pi$
$678$	22.8578	0.877850
$679$	50.2527	1.92852
$680$	0	0
$681$	15.3965	0.589995
$682$	0	0
$683$	9.68776	0.370692	0.185346	−	0.982673i	$-0.440659\pi$
$683$	9.68776	0.370692	0.185346	+	0.982673i	$0.440659\pi$
$684$	−3.93963	−0.150636
$685$	0	0
$686$	1.80361	0.0688620
$687$	4.70522	0.179515
$688$	33.9385	1.29389
$689$	−16.3243	−0.621907
$690$	0	0
$691$	−0.594394	−0.0226118	−0.0113059	−	0.999936i	$-0.503599\pi$
$691$	−0.594394	−0.0226118	−0.0113059	+	0.999936i	$0.503599\pi$
$692$	−5.97714	−0.227217
$693$	0	0
$694$	−6.17496	−0.234398
$695$	0	0
$696$	7.54427	0.285965
$697$	−22.5896	−0.855641
$698$	−33.0391	−1.25055
$699$	14.9210	0.564363
$700$	0	0
$701$	18.1598	0.685886	0.342943	−	0.939356i	$-0.388576\pi$
$701$	18.1598	0.685886	0.342943	+	0.939356i	$0.388576\pi$
$702$	3.04530	0.114937
$703$	−27.5660	−1.03967
$704$	0	0
$705$	0	0
$706$	21.2292	0.798972
$707$	−7.55671	−0.284199
$708$	−4.86134	−0.182700
$709$	−1.80341	−0.0677287	−0.0338643	−	0.999426i	$-0.510781\pi$
$709$	−1.80341	−0.0677287	−0.0338643	+	0.999426i	$0.510781\pi$
$710$	0	0
$711$	−9.65644	−0.362145
$712$	31.4087	1.17709
$713$	12.3429	0.462244
$714$	30.5195	1.14216
$715$	0	0
$716$	1.45368	0.0543265
$717$	11.3885	0.425311
$718$	−0.811212	−0.0302742
$719$	−37.0481	−1.38166	−0.690830	−	0.723017i	$-0.742755\pi$
$719$	−37.0481	−1.38166	−0.690830	+	0.723017i	$0.742755\pi$
$720$	0	0
$721$	−6.12397	−0.228068
$722$	−60.6040	−2.25545
$723$	−12.1143	−0.450535
$724$	−2.09337	−0.0777994
$725$	0	0
$726$	0	0
$727$	−23.3198	−0.864885	−0.432443	−	0.901661i	$-0.642348\pi$
$727$	−23.3198	−0.864885	−0.432443	+	0.901661i	$0.642348\pi$
$728$	28.6139	1.06050
$729$	1.00000	0.0370370
$730$	0	0
$731$	−78.5519	−2.90535
$732$	3.18293	0.117644
$733$	15.9855	0.590439	0.295220	−	0.955429i	$-0.404607\pi$
$733$	15.9855	0.590439	0.295220	+	0.955429i	$0.404607\pi$
$734$	−41.3422	−1.52597
$735$	0	0
$736$	−6.17149	−0.227484
$737$	0	0
$738$	4.27455	0.157348
$739$	−32.9339	−1.21149	−0.605747	−	0.795657i	$-0.707126\pi$
$739$	−32.9339	−1.21149	−0.605747	+	0.795657i	$0.707126\pi$
$740$	0	0
$741$	−20.3679	−0.748234
$742$	30.9575	1.13648
$743$	−23.2758	−0.853905	−0.426953	−	0.904274i	$-0.640413\pi$
$743$	−23.2758	−0.853905	−0.426953	+	0.904274i	$0.640413\pi$
$744$	16.1240	0.591135
$745$	0	0
$746$	−9.54149	−0.349339
$747$	−11.0497	−0.404288
$748$	0	0
$749$	2.23408	0.0816316
$750$	0	0
$751$	−9.18396	−0.335127	−0.167564	−	0.985861i	$-0.553590\pi$
$751$	−9.18396	−0.335127	−0.167564	+	0.985861i	$0.553590\pi$
$752$	−26.9609	−0.983164
$753$	−17.4132	−0.634573
$754$	7.51552	0.273699
$755$	0	0
$756$	1.81050	0.0658474
$757$	47.2523	1.71741	0.858706	−	0.512468i	$-0.171269\pi$
$757$	47.2523	1.71741	0.858706	+	0.512468i	$0.171269\pi$
$758$	0.419641	0.0152420
$759$	0	0
$760$	0	0
$761$	49.9941	1.81229	0.906143	−	0.422972i	$-0.139013\pi$
$761$	49.9941	1.81229	0.906143	+	0.422972i	$0.139013\pi$
$762$	−15.0453	−0.545034
$763$	−18.1637	−0.657571
$764$	−11.1764	−0.404346
$765$	0	0
$766$	38.3120	1.38427
$767$	−25.1331	−0.907504
$768$	−10.7519	−0.387976
$769$	−4.53874	−0.163671	−0.0818357	−	0.996646i	$-0.526078\pi$
$769$	−4.53874	−0.163671	−0.0818357	+	0.996646i	$0.526078\pi$
$770$	0	0
$771$	18.4334	0.663864
$772$	−11.0948	−0.399312
$773$	31.3352	1.12705	0.563525	−	0.826099i	$-0.309445\pi$
$773$	31.3352	1.12705	0.563525	+	0.826099i	$0.309445\pi$
$774$	14.8641	0.534280
$775$	0	0
$776$	−40.5029	−1.45397
$777$	12.6683	0.454471
$778$	30.0037	1.07568
$779$	−28.5896	−1.02433
$780$	0	0
$781$	0	0
$782$	−18.8299	−0.673355
$783$	2.46791	0.0881960
$784$	−20.8078	−0.743135
$785$	0	0
$786$	4.27455	0.152468
$787$	5.59323	0.199377	0.0996886	−	0.995019i	$-0.468215\pi$
$787$	5.59323	0.199377	0.0996886	+	0.995019i	$0.468215\pi$
$788$	0.951067	0.0338803
$789$	−2.06075	−0.0733648
$790$	0	0
$791$	70.2581	2.49809
$792$	0	0
$793$	16.4557	0.584360
$794$	12.7033	0.450824
$795$	0	0
$796$	−4.16511	−0.147629
$797$	32.4585	1.14974	0.574870	−	0.818245i	$-0.305053\pi$
$797$	32.4585	1.14974	0.574870	+	0.818245i	$0.305053\pi$
$798$	38.6258	1.36734
$799$	62.4020	2.20763
$800$	0	0
$801$	10.2745	0.363033
$802$	23.7281	0.837867
$803$	0	0
$804$	1.59439	0.0562299
$805$	0	0
$806$	16.0626	0.565780
$807$	3.45090	0.121477
$808$	6.09059	0.214266
$809$	−41.9128	−1.47358	−0.736788	−	0.676124i	$-0.763658\pi$
$809$	−41.9128	−1.47358	−0.736788	+	0.676124i	$0.763658\pi$
$810$	0	0
$811$	−28.4285	−0.998260	−0.499130	−	0.866527i	$-0.666347\pi$
$811$	−28.4285	−0.998260	−0.499130	+	0.866527i	$0.666347\pi$
$812$	4.46817	0.156802
$813$	−21.2167	−0.744103
$814$	0	0
$815$	0	0
$816$	−18.3726	−0.643169
$817$	−99.4160	−3.47813
$818$	−8.05912	−0.281781
$819$	9.36031	0.327076
$820$	0	0
$821$	−53.6401	−1.87205	−0.936026	−	0.351931i	$-0.885525\pi$
$821$	−53.6401	−1.87205	−0.936026	+	0.351931i	$0.885525\pi$
$822$	9.79079	0.341493
$823$	11.1561	0.388878	0.194439	−	0.980915i	$-0.437711\pi$
$823$	11.1561	0.388878	0.194439	+	0.980915i	$0.437711\pi$
$824$	4.93582	0.171948
$825$	0	0
$826$	47.6625	1.65839
$827$	27.8496	0.968424	0.484212	−	0.874951i	$-0.339106\pi$
$827$	27.8496	0.968424	0.484212	+	0.874951i	$0.339106\pi$
$828$	−1.11704	−0.0388199
$829$	39.4084	1.36871	0.684355	−	0.729149i	$-0.260084\pi$
$829$	39.4084	1.36871	0.684355	+	0.729149i	$0.260084\pi$
$830$	0	0
$831$	4.13159	0.143323
$832$	−21.9377	−0.760553
$833$	48.1604	1.66866
$834$	7.62864	0.264158
$835$	0	0
$836$	0	0
$837$	5.27455	0.182315
$838$	−34.1655	−1.18023
$839$	35.8188	1.23660	0.618301	−	0.785941i	$-0.287821\pi$
$839$	35.8188	1.23660	0.618301	+	0.785941i	$0.287821\pi$
$840$	0	0
$841$	−22.9094	−0.789980
$842$	13.7755	0.474737
$843$	23.0956	0.795455
$844$	7.66818	0.263950
$845$	0	0
$846$	−11.8081	−0.405972
$847$	0	0
$848$	−18.6362	−0.639971
$849$	24.7593	0.849737
$850$	0	0
$851$	−7.81604	−0.267930
$852$	−3.45090	−0.118226
$853$	27.4091	0.938469	0.469234	−	0.883074i	$-0.344530\pi$
$853$	27.4091	0.938469	0.469234	+	0.883074i	$0.344530\pi$
$854$	−31.2067	−1.06787
$855$	0	0
$856$	−1.80064	−0.0615445
$857$	−9.82824	−0.335726	−0.167863	−	0.985810i	$-0.553687\pi$
$857$	−9.82824	−0.335726	−0.167863	+	0.985810i	$0.553687\pi$
$858$	0	0
$859$	−33.9825	−1.15947	−0.579735	−	0.814805i	$-0.696844\pi$
$859$	−33.9825	−1.15947	−0.579735	+	0.814805i	$0.696844\pi$
$860$	0	0
$861$	13.1387	0.447764
$862$	−16.4865	−0.561534
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−2.63730	−0.0897229
$865$	0	0
$866$	23.7530	0.807161
$867$	25.5240	0.866842
$868$	9.54960	0.324134
$869$	0	0
$870$	0	0
$871$	8.24302	0.279304
$872$	14.6397	0.495762
$873$	−13.2495	−0.448427
$874$	−23.8313	−0.806104
$875$	0	0
$876$	4.64217	0.156844
$877$	42.8941	1.44843	0.724216	−	0.689574i	$-0.242202\pi$
$877$	42.8941	1.44843	0.724216	+	0.689574i	$0.242202\pi$
$878$	−26.7241	−0.901893
$879$	−7.33536	−0.247416
$880$	0	0
$881$	−17.7282	−0.597279	−0.298640	−	0.954366i	$-0.596533\pi$
$881$	−17.7282	−0.597279	−0.298640	+	0.954366i	$0.596533\pi$
$882$	−9.11323	−0.306858
$883$	0.201414	0.00677813	0.00338907	−	0.999994i	$-0.498921\pi$
$883$	0.201414	0.00677813	0.00338907	+	0.999994i	$0.498921\pi$
$884$	7.68221	0.258381
$885$	0	0
$886$	1.40506	0.0472039
$887$	43.0409	1.44517	0.722587	−	0.691280i	$-0.242953\pi$
$887$	43.0409	1.44517	0.722587	+	0.691280i	$0.242953\pi$
$888$	−10.2104	−0.342640
$889$	−46.2447	−1.55100
$890$	0	0
$891$	0	0
$892$	−5.28216	−0.176860
$893$	78.9766	2.64285
$894$	−2.23408	−0.0747189
$895$	0	0
$896$	21.5972	0.721511
$897$	−5.77511	−0.192825
$898$	0.111785	0.00373031
$899$	13.0171	0.434145
$900$	0	0
$901$	43.1343	1.43701
$902$	0	0
$903$	45.6878	1.52039
$904$	−56.6269	−1.88338
$905$	0	0
$906$	24.7268	0.821494
$907$	0.204192	0.00678008	0.00339004	−	0.999994i	$-0.498921\pi$
$907$	0.204192	0.00678008	0.00339004	+	0.999994i	$0.498921\pi$
$908$	−7.34956	−0.243904
$909$	1.99238	0.0660830
$910$	0	0
$911$	−32.9547	−1.09184	−0.545919	−	0.837838i	$-0.683819\pi$
$911$	−32.9547	−1.09184	−0.545919	+	0.837838i	$0.683819\pi$
$912$	−23.2525	−0.769968
$913$	0	0
$914$	5.08576	0.168222
$915$	0	0
$916$	−2.24605	−0.0742115
$917$	13.1387	0.433877
$918$	−8.04668	−0.265580
$919$	−50.8435	−1.67717	−0.838586	−	0.544770i	$-0.816617\pi$
$919$	−50.8435	−1.67717	−0.838586	+	0.544770i	$0.816617\pi$
$920$	0	0
$921$	−22.7669	−0.750195
$922$	8.32467	0.274159
$923$	−17.8412	−0.587249
$924$	0	0
$925$	0	0
$926$	−25.2948	−0.831240
$927$	1.61463	0.0530313
$928$	−6.50863	−0.213656
$929$	48.5394	1.59253	0.796264	−	0.604950i	$-0.206807\pi$
$929$	48.5394	1.59253	0.796264	+	0.604950i	$0.206807\pi$
$930$	0	0
$931$	60.9522	1.99763
$932$	−7.12256	−0.233307
$933$	−25.5443	−0.836282
$934$	−5.92124	−0.193749
$935$	0	0
$936$	−7.54427	−0.246592
$937$	−10.3573	−0.338357	−0.169178	−	0.985585i	$-0.554111\pi$
$937$	−10.3573	−0.338357	−0.169178	+	0.985585i	$0.554111\pi$
$938$	−15.6321	−0.510406
$939$	34.7735	1.13479
$940$	0	0
$941$	48.9528	1.59582	0.797908	−	0.602779i	$-0.205940\pi$
$941$	48.9528	1.59582	0.797908	+	0.602779i	$0.205940\pi$
$942$	−17.9779	−0.585752
$943$	−8.10627	−0.263976
$944$	−28.6926	−0.933864
$945$	0	0
$946$	0	0
$947$	20.5213	0.666851	0.333426	−	0.942776i	$-0.391795\pi$
$947$	20.5213	0.666851	0.333426	+	0.942776i	$0.391795\pi$
$948$	4.60953	0.149710
$949$	24.0000	0.779073
$950$	0	0
$951$	5.38537	0.174633
$952$	−75.6076	−2.45046
$953$	−11.2066	−0.363018	−0.181509	−	0.983389i	$-0.558098\pi$
$953$	−11.2066	−0.363018	−0.181509	+	0.983389i	$0.558098\pi$
$954$	−8.16216	−0.264260
$955$	0	0
$956$	−5.43632	−0.175823
$957$	0	0
$958$	11.5944	0.374598
$959$	30.0939	0.971783
$960$	0	0
$961$	−3.17913	−0.102553
$962$	−10.1715	−0.327943
$963$	−0.589032	−0.0189813
$964$	5.78278	0.186251
$965$	0	0
$966$	10.9519	0.352372
$967$	−5.76022	−0.185236	−0.0926181	−	0.995702i	$-0.529524\pi$
$967$	−5.76022	−0.185236	−0.0926181	+	0.995702i	$0.529524\pi$
$968$	0	0
$969$	53.8188	1.72891
$970$	0	0
$971$	14.9547	0.479919	0.239960	−	0.970783i	$-0.422866\pi$
$971$	14.9547	0.479919	0.239960	+	0.970783i	$0.422866\pi$
$972$	−0.477352	−0.0153111
$973$	23.4481	0.751712
$974$	−29.7302	−0.952616
$975$	0	0
$976$	18.7863	0.601334
$977$	−14.7457	−0.471756	−0.235878	−	0.971783i	$-0.575797\pi$
$977$	−14.7457	−0.471756	−0.235878	+	0.971783i	$0.575797\pi$
$978$	−7.26699	−0.232373
$979$	0	0
$980$	0	0
$981$	4.78899	0.152901
$982$	8.18674	0.261249
$983$	−34.8892	−1.11279	−0.556396	−	0.830917i	$-0.687816\pi$
$983$	−34.8892	−1.11279	−0.556396	+	0.830917i	$0.687816\pi$
$984$	−10.5896	−0.337583
$985$	0	0
$986$	−19.8585	−0.632424
$987$	−36.2946	−1.15527
$988$	9.72267	0.309319
$989$	−28.1883	−0.896337
$990$	0	0
$991$	−15.5289	−0.493291	−0.246645	−	0.969106i	$-0.579328\pi$
$991$	−15.5289	−0.493291	−0.246645	+	0.969106i	$0.579328\pi$
$992$	−13.9106	−0.441662
$993$	19.8641	0.630369
$994$	33.8340	1.07315
$995$	0	0
$996$	5.27461	0.167132
$997$	3.63590	0.115150	0.0575750	−	0.998341i	$-0.481663\pi$
$997$	3.63590	0.115150	0.0575750	+	0.998341i	$0.481663\pi$
$998$	−27.3491	−0.865720
$999$	−3.34008	−0.105675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.dq.1.3		6
5.4	even	2		1815.2.a.y.1.4	yes	6
11.10	odd	2	inner	9075.2.a.dq.1.4		6
15.14	odd	2		5445.2.a.bz.1.3		6
55.54	odd	2		1815.2.a.y.1.3	✓	6
165.164	even	2		5445.2.a.bz.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.y.1.3	✓	6	55.54	odd	2
1815.2.a.y.1.4	yes	6	5.4	even	2
5445.2.a.bz.1.3		6	15.14	odd	2
5445.2.a.bz.1.4		6	165.164	even	2
9075.2.a.dq.1.3		6	1.1	even	1	trivial
9075.2.a.dq.1.4		6	11.10	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.23396 q^{2} +1.00000 q^{3} -0.477352 q^{4} -1.23396 q^{6} -3.79281 q^{7} +3.05694 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.23396 q^{2} +1.00000 q^{3} -0.477352 q^{4} -1.23396 q^{6} -3.79281 q^{7} +3.05694 q^{8} +1.00000 q^{9} -0.477352 q^{12} -2.46791 q^{13} +4.68016 q^{14} -2.81743 q^{16} +6.52105 q^{17} -1.23396 q^{18} +8.25310 q^{19} -3.79281 q^{21} +2.34008 q^{23} +3.05694 q^{24} +3.04530 q^{26} +1.00000 q^{27} +1.81050 q^{28} +2.46791 q^{29} +5.27455 q^{31} -2.63730 q^{32} -8.04668 q^{34} -0.477352 q^{36} -3.34008 q^{37} -10.1840 q^{38} -2.46791 q^{39} -3.46410 q^{41} +4.68016 q^{42} -12.0459 q^{43} -2.88755 q^{46} +9.56933 q^{47} -2.81743 q^{48} +7.38537 q^{49} +6.52105 q^{51} +1.17806 q^{52} +6.61463 q^{53} -1.23396 q^{54} -11.5944 q^{56} +8.25310 q^{57} -3.04530 q^{58} +10.1840 q^{59} -6.66788 q^{61} -6.50856 q^{62} -3.79281 q^{63} +8.88918 q^{64} -3.34008 q^{67} -3.11284 q^{68} +2.34008 q^{69} +7.22925 q^{71} +3.05694 q^{72} -9.72482 q^{73} +4.12151 q^{74} -3.93963 q^{76} +3.04530 q^{78} -9.65644 q^{79} +1.00000 q^{81} +4.27455 q^{82} -11.0497 q^{83} +1.81050 q^{84} +14.8641 q^{86} +2.46791 q^{87} +10.2745 q^{89} +9.36031 q^{91} -1.11704 q^{92} +5.27455 q^{93} -11.8081 q^{94} -2.63730 q^{96} -13.2495 q^{97} -9.11323 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} + 14 q^{4} + 6 q^{9} + 14 q^{12} + 8 q^{14} + 10 q^{16} + 4 q^{23} + 52 q^{26} + 6 q^{27} + 18 q^{31} + 26 q^{34} + 14 q^{36} - 10 q^{37} + 20 q^{38} + 8 q^{42} + 10 q^{48} + 68 q^{49} + 16 q^{53} - 76 q^{56} - 52 q^{58} - 20 q^{59} + 16 q^{64} - 10 q^{67} + 4 q^{69} - 4 q^{71} + 52 q^{78} + 6 q^{81} + 12 q^{82} - 12 q^{86} + 48 q^{89} + 16 q^{91} - 30 q^{92} + 18 q^{93} - 2 q^{97}+O(q^{100}) \) 6 * q + 6 * q^3 + 14 * q^4 + 6 * q^9 + 14 * q^12 + 8 * q^14 + 10 * q^16 + 4 * q^23 + 52 * q^26 + 6 * q^27 + 18 * q^31 + 26 * q^34 + 14 * q^36 - 10 * q^37 + 20 * q^38 + 8 * q^42 + 10 * q^48 + 68 * q^49 + 16 * q^53 - 76 * q^56 - 52 * q^58 - 20 * q^59 + 16 * q^64 - 10 * q^67 + 4 * q^69 - 4 * q^71 + 52 * q^78 + 6 * q^81 + 12 * q^82 - 12 * q^86 + 48 * q^89 + 16 * q^91 - 30 * q^92 + 18 * q^93 - 2 * q^97

Introduction

L-functions

Modular forms

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Database

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