LMFDB - Embedded newform 7623.2.a.cv.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6988960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 $ x^8 - 9x^6 + 22x^4 - 11*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.80692$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80692 q^{2} +1.26498 q^{4} -2.14896 q^{5} +1.00000 q^{7} -1.32813 q^{8} +O(q^{10})$	$q+1.80692 q^{2} +1.26498 q^{4} -2.14896 q^{5} +1.00000 q^{7} -1.32813 q^{8} -3.88301 q^{10} +3.66481 q^{13} +1.80692 q^{14} -4.92979 q^{16} +1.89145 q^{17} -4.44661 q^{19} -2.71839 q^{20} +5.20938 q^{23} -0.381966 q^{25} +6.62204 q^{26} +1.26498 q^{28} -3.76049 q^{29} +2.21054 q^{31} -6.25149 q^{32} +3.41770 q^{34} -2.14896 q^{35} -8.31175 q^{37} -8.03469 q^{38} +2.85410 q^{40} +0.0621247 q^{41} +8.99443 q^{43} +9.41297 q^{46} -12.3547 q^{47} +1.00000 q^{49} -0.690184 q^{50} +4.63590 q^{52} +7.01020 q^{53} -1.32813 q^{56} -6.79493 q^{58} +12.0268 q^{59} -5.13486 q^{61} +3.99428 q^{62} -1.43640 q^{64} -7.87553 q^{65} -9.38744 q^{67} +2.39264 q^{68} -3.88301 q^{70} +4.35232 q^{71} -7.65242 q^{73} -15.0187 q^{74} -5.62486 q^{76} -16.1679 q^{79} +10.5939 q^{80} +0.112255 q^{82} -7.24748 q^{83} -4.06464 q^{85} +16.2523 q^{86} +6.68300 q^{89} +3.66481 q^{91} +6.58975 q^{92} -22.3240 q^{94} +9.55559 q^{95} -13.1033 q^{97} +1.80692 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} + 8 q^{7}+O(q^{10}) $ 8 * q + 2 * q^4 + 8 * q^7	$ 8 q + 2 q^{4} + 8 q^{7} - 14 q^{10} - 2 q^{16} + 6 q^{19} - 12 q^{25} + 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} - 4 q^{40} - 16 q^{43} + 42 q^{46} + 8 q^{49} - 2 q^{52} - 30 q^{58} - 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} - 14 q^{73} + 34 q^{76} - 22 q^{79} + 36 q^{82} + 18 q^{85} - 32 q^{94} - 48 q^{97}+O(q^{100}) $ 8 * q + 2 * q^4 + 8 * q^7 - 14 * q^10 - 2 * q^16 + 6 * q^19 - 12 * q^25 + 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 - 4 * q^40 - 16 * q^43 + 42 * q^46 + 8 * q^49 - 2 * q^52 - 30 * q^58 - 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 - 14 * q^73 + 34 * q^76 - 22 * q^79 + 36 * q^82 + 18 * q^85 - 32 * q^94 - 48 * 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 $

$2$	1.80692	1.27769	0.638844	−	0.769336i	$-0.279413\pi$
$2$	1.80692	1.27769	0.638844	+	0.769336i	$0.279413\pi$
$3$	0	0
$3$	0	0
$4$	1.26498	0.632489
$5$	−2.14896	−0.961045	−0.480522	−	0.876982i	$-0.659553\pi$
$5$	−2.14896	−0.961045	−0.480522	+	0.876982i	$0.659553\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.32813	−0.469565
$9$	0	0
$10$	−3.88301	−1.22792
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.66481	1.01644	0.508218	−	0.861229i	$-0.330305\pi$
$13$	3.66481	1.01644	0.508218	+	0.861229i	$0.330305\pi$
$14$	1.80692	0.482921
$15$	0	0
$16$	−4.92979	−1.23245
$17$	1.89145	0.458743	0.229371	−	0.973339i	$-0.426333\pi$
$17$	1.89145	0.458743	0.229371	+	0.973339i	$0.426333\pi$
$18$	0	0
$19$	−4.44661	−1.02012	−0.510061	−	0.860138i	$-0.670377\pi$
$19$	−4.44661	−1.02012	−0.510061	+	0.860138i	$0.670377\pi$
$20$	−2.71839	−0.607850
$21$	0	0
$22$	0	0
$23$	5.20938	1.08623	0.543116	−	0.839658i	$-0.317244\pi$
$23$	5.20938	1.08623	0.543116	+	0.839658i	$0.317244\pi$
$24$	0	0
$25$	−0.381966	−0.0763932
$26$	6.62204	1.29869
$27$	0	0
$28$	1.26498	0.239058
$29$	−3.76049	−0.698306	−0.349153	−	0.937066i	$-0.613531\pi$
$29$	−3.76049	−0.698306	−0.349153	+	0.937066i	$0.613531\pi$
$30$	0	0
$31$	2.21054	0.397025	0.198512	−	0.980098i	$-0.436389\pi$
$31$	2.21054	0.397025	0.198512	+	0.980098i	$0.436389\pi$
$32$	−6.25149	−1.10512
$33$	0	0
$34$	3.41770	0.586131
$35$	−2.14896	−0.363241
$36$	0	0
$37$	−8.31175	−1.36644	−0.683222	−	0.730211i	$-0.739422\pi$
$37$	−8.31175	−1.36644	−0.683222	+	0.730211i	$0.739422\pi$
$38$	−8.03469	−1.30340
$39$	0	0
$40$	2.85410	0.451273
$41$	0.0621247	0.00970224	0.00485112	−	0.999988i	$-0.498456\pi$
$41$	0.0621247	0.00970224	0.00485112	+	0.999988i	$0.498456\pi$
$42$	0	0
$43$	8.99443	1.37164	0.685819	−	0.727772i	$-0.259444\pi$
$43$	8.99443	1.37164	0.685819	+	0.727772i	$0.259444\pi$
$44$	0	0
$45$	0	0
$46$	9.41297	1.38787
$47$	−12.3547	−1.80212	−0.901061	−	0.433693i	$-0.857210\pi$
$47$	−12.3547	−1.80212	−0.901061	+	0.433693i	$0.857210\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.690184	−0.0976067
$51$	0	0
$52$	4.63590	0.642884
$53$	7.01020	0.962925	0.481462	−	0.876467i	$-0.340106\pi$
$53$	7.01020	0.962925	0.481462	+	0.876467i	$0.340106\pi$
$54$	0	0
$55$	0	0
$56$	−1.32813	−0.177479
$57$	0	0
$58$	−6.79493	−0.892218
$59$	12.0268	1.56576	0.782879	−	0.622174i	$-0.213751\pi$
$59$	12.0268	1.56576	0.782879	+	0.622174i	$0.213751\pi$
$60$	0	0
$61$	−5.13486	−0.657451	−0.328725	−	0.944426i	$-0.606619\pi$
$61$	−5.13486	−0.657451	−0.328725	+	0.944426i	$0.606619\pi$
$62$	3.99428	0.507274
$63$	0	0
$64$	−1.43640	−0.179550
$65$	−7.87553	−0.976840
$66$	0	0
$67$	−9.38744	−1.14686	−0.573429	−	0.819255i	$-0.694387\pi$
$67$	−9.38744	−1.14686	−0.573429	+	0.819255i	$0.694387\pi$
$68$	2.39264	0.290150
$69$	0	0
$70$	−3.88301	−0.464109
$71$	4.35232	0.516525	0.258262	−	0.966075i	$-0.416850\pi$
$71$	4.35232	0.516525	0.258262	+	0.966075i	$0.416850\pi$
$72$	0	0
$73$	−7.65242	−0.895647	−0.447824	−	0.894122i	$-0.647801\pi$
$73$	−7.65242	−0.895647	−0.447824	+	0.894122i	$0.647801\pi$
$74$	−15.0187	−1.74589
$75$	0	0
$76$	−5.62486	−0.645216
$77$	0	0
$78$	0	0
$79$	−16.1679	−1.81904	−0.909518	−	0.415665i	$-0.863549\pi$
$79$	−16.1679	−1.81904	−0.909518	+	0.415665i	$0.863549\pi$
$80$	10.5939	1.18444
$81$	0	0
$82$	0.112255	0.0123964
$83$	−7.24748	−0.795514	−0.397757	−	0.917491i	$-0.630211\pi$
$83$	−7.24748	−0.795514	−0.397757	+	0.917491i	$0.630211\pi$
$84$	0	0
$85$	−4.06464	−0.440872
$86$	16.2523	1.75253
$87$	0	0
$88$	0	0
$89$	6.68300	0.708397	0.354198	−	0.935170i	$-0.384754\pi$
$89$	6.68300	0.708397	0.354198	+	0.935170i	$0.384754\pi$
$90$	0	0
$91$	3.66481	0.384176
$92$	6.58975	0.687029
$93$	0	0
$94$	−22.3240	−2.30255
$95$	9.55559	0.980383
$96$	0	0
$97$	−13.1033	−1.33044	−0.665219	−	0.746648i	$-0.731662\pi$
$97$	−13.1033	−1.33044	−0.665219	+	0.746648i	$0.731662\pi$
$98$	1.80692	0.182527
$99$	0	0
$100$	−0.483178	−0.0483178
$101$	1.24739	0.124120	0.0620598	−	0.998072i	$-0.480233\pi$
$101$	1.24739	0.124120	0.0620598	+	0.998072i	$0.480233\pi$
$102$	0	0
$103$	−14.9689	−1.47493	−0.737465	−	0.675385i	$-0.763977\pi$
$103$	−14.9689	−1.47493	−0.737465	+	0.675385i	$0.763977\pi$
$104$	−4.86735	−0.477283
$105$	0	0
$106$	12.6669	1.23032
$107$	−10.2458	−0.990496	−0.495248	−	0.868752i	$-0.664923\pi$
$107$	−10.2458	−0.990496	−0.495248	+	0.868752i	$0.664923\pi$
$108$	0	0
$109$	−3.39172	−0.324868	−0.162434	−	0.986719i	$-0.551934\pi$
$109$	−3.39172	−0.324868	−0.162434	+	0.986719i	$0.551934\pi$
$110$	0	0
$111$	0	0
$112$	−4.92979	−0.465821
$113$	−8.21990	−0.773264	−0.386632	−	0.922234i	$-0.626362\pi$
$113$	−8.21990	−0.773264	−0.386632	+	0.922234i	$0.626362\pi$
$114$	0	0
$115$	−11.1948	−1.04392
$116$	−4.75694	−0.441671
$117$	0	0
$118$	21.7315	2.00055
$119$	1.89145	0.173389
$120$	0	0
$121$	0	0
$122$	−9.27830	−0.840018
$123$	0	0
$124$	2.79628	0.251114
$125$	11.5656	1.03446
$126$	0	0
$127$	3.25941	0.289226	0.144613	−	0.989488i	$-0.453806\pi$
$127$	3.25941	0.289226	0.144613	+	0.989488i	$0.453806\pi$
$128$	9.90751	0.875709
$129$	0	0
$130$	−14.2305	−1.24810
$131$	0.349304	0.0305189	0.0152594	−	0.999884i	$-0.495143\pi$
$131$	0.349304	0.0305189	0.0152594	+	0.999884i	$0.495143\pi$
$132$	0	0
$133$	−4.44661	−0.385570
$134$	−16.9624	−1.46533
$135$	0	0
$136$	−2.51209	−0.215410
$137$	0.776324	0.0663258	0.0331629	−	0.999450i	$-0.489442\pi$
$137$	0.776324	0.0663258	0.0331629	+	0.999450i	$0.489442\pi$
$138$	0	0
$139$	−8.76940	−0.743811	−0.371906	−	0.928271i	$-0.621295\pi$
$139$	−8.76940	−0.743811	−0.371906	+	0.928271i	$0.621295\pi$
$140$	−2.71839	−0.229746
$141$	0	0
$142$	7.86431	0.659958
$143$	0	0
$144$	0	0
$145$	8.08116	0.671104
$146$	−13.8273	−1.14436
$147$	0	0
$148$	−10.5142	−0.864260
$149$	−2.71490	−0.222413	−0.111206	−	0.993797i	$-0.535471\pi$
$149$	−2.71490	−0.222413	−0.111206	+	0.993797i	$0.535471\pi$
$150$	0	0
$151$	3.67720	0.299247	0.149623	−	0.988743i	$-0.452194\pi$
$151$	3.67720	0.299247	0.149623	+	0.988743i	$0.452194\pi$
$152$	5.90568	0.479014
$153$	0	0
$154$	0	0
$155$	−4.75037	−0.381559
$156$	0	0
$157$	21.0688	1.68148	0.840738	−	0.541443i	$-0.182122\pi$
$157$	21.0688	1.68148	0.840738	+	0.541443i	$0.182122\pi$
$158$	−29.2143	−2.32416
$159$	0	0
$160$	13.4342	1.06207
$161$	5.20938	0.410557
$162$	0	0
$163$	−16.9490	−1.32755	−0.663774	−	0.747933i	$-0.731046\pi$
$163$	−16.9490	−1.32755	−0.663774	+	0.747933i	$0.731046\pi$
$164$	0.0785863	0.00613656
$165$	0	0
$166$	−13.0956	−1.01642
$167$	−15.7297	−1.21720	−0.608599	−	0.793478i	$-0.708268\pi$
$167$	−15.7297	−1.21720	−0.608599	+	0.793478i	$0.708268\pi$
$168$	0	0
$169$	0.430832	0.0331409
$170$	−7.34450	−0.563298
$171$	0	0
$172$	11.3777	0.867545
$173$	9.79142	0.744428	0.372214	−	0.928147i	$-0.378599\pi$
$173$	9.79142	0.744428	0.372214	+	0.928147i	$0.378599\pi$
$174$	0	0
$175$	−0.381966	−0.0288739
$176$	0	0
$177$	0	0
$178$	12.0757	0.905111
$179$	−13.4597	−1.00603	−0.503013	−	0.864279i	$-0.667775\pi$
$179$	−13.4597	−1.00603	−0.503013	+	0.864279i	$0.667775\pi$
$180$	0	0
$181$	−15.7516	−1.17081	−0.585404	−	0.810742i	$-0.699064\pi$
$181$	−15.7516	−1.17081	−0.585404	+	0.810742i	$0.699064\pi$
$182$	6.62204	0.490858
$183$	0	0
$184$	−6.91874	−0.510057
$185$	17.8616	1.31321
$186$	0	0
$187$	0	0
$188$	−15.6284	−1.13982
$189$	0	0
$190$	17.2662	1.25262
$191$	9.10153	0.658563	0.329282	−	0.944232i	$-0.393193\pi$
$191$	9.10153	0.658563	0.329282	+	0.944232i	$0.393193\pi$
$192$	0	0
$193$	−14.3475	−1.03275	−0.516377	−	0.856361i	$-0.672720\pi$
$193$	−14.3475	−1.03275	−0.516377	+	0.856361i	$0.672720\pi$
$194$	−23.6767	−1.69989
$195$	0	0
$196$	1.26498	0.0903555
$197$	−20.6333	−1.47006	−0.735029	−	0.678035i	$-0.762832\pi$
$197$	−20.6333	−1.47006	−0.735029	+	0.678035i	$0.762832\pi$
$198$	0	0
$199$	19.5733	1.38751	0.693755	−	0.720211i	$-0.255955\pi$
$199$	19.5733	1.38751	0.693755	+	0.720211i	$0.255955\pi$
$200$	0.507301	0.0358716
$201$	0	0
$202$	2.25393	0.158586
$203$	−3.76049	−0.263935
$204$	0	0
$205$	−0.133503	−0.00932429
$206$	−27.0477	−1.88450
$207$	0	0
$208$	−18.0667	−1.25270
$209$	0	0
$210$	0	0
$211$	−10.1880	−0.701371	−0.350685	−	0.936493i	$-0.614051\pi$
$211$	−10.1880	−0.701371	−0.350685	+	0.936493i	$0.614051\pi$
$212$	8.86774	0.609039
$213$	0	0
$214$	−18.5133	−1.26555
$215$	−19.3287	−1.31821
$216$	0	0
$217$	2.21054	0.150061
$218$	−6.12858	−0.415080
$219$	0	0
$220$	0	0
$221$	6.93179	0.466283
$222$	0	0
$223$	−24.5903	−1.64669	−0.823345	−	0.567542i	$-0.807895\pi$
$223$	−24.5903	−1.64669	−0.823345	+	0.567542i	$0.807895\pi$
$224$	−6.25149	−0.417695
$225$	0	0
$226$	−14.8527	−0.987990
$227$	23.2865	1.54558	0.772789	−	0.634663i	$-0.218861\pi$
$227$	23.2865	1.54558	0.772789	+	0.634663i	$0.218861\pi$
$228$	0	0
$229$	−6.48746	−0.428703	−0.214352	−	0.976757i	$-0.568764\pi$
$229$	−6.48746	−0.428703	−0.214352	+	0.976757i	$0.568764\pi$
$230$	−20.2281	−1.33380
$231$	0	0
$232$	4.99443	0.327900
$233$	−23.0166	−1.50787	−0.753935	−	0.656949i	$-0.771846\pi$
$233$	−23.0166	−1.50787	−0.753935	+	0.656949i	$0.771846\pi$
$234$	0	0
$235$	26.5498	1.73192
$236$	15.2136	0.990324
$237$	0	0
$238$	3.41770	0.221537
$239$	9.08553	0.587694	0.293847	−	0.955852i	$-0.405064\pi$
$239$	9.08553	0.587694	0.293847	+	0.955852i	$0.405064\pi$
$240$	0	0
$241$	−6.60908	−0.425728	−0.212864	−	0.977082i	$-0.568279\pi$
$241$	−6.60908	−0.425728	−0.212864	+	0.977082i	$0.568279\pi$
$242$	0	0
$243$	0	0
$244$	−6.49548	−0.415830
$245$	−2.14896	−0.137292
$246$	0	0
$247$	−16.2960	−1.03689
$248$	−2.93589	−0.186429
$249$	0	0
$250$	20.8982	1.32172
$251$	−3.59669	−0.227021	−0.113511	−	0.993537i	$-0.536210\pi$
$251$	−3.59669	−0.227021	−0.113511	+	0.993537i	$0.536210\pi$
$252$	0	0
$253$	0	0
$254$	5.88950	0.369540
$255$	0	0
$256$	20.7749	1.29843
$257$	8.69374	0.542301	0.271150	−	0.962537i	$-0.412596\pi$
$257$	8.69374	0.542301	0.271150	+	0.962537i	$0.412596\pi$
$258$	0	0
$259$	−8.31175	−0.516467
$260$	−9.96237	−0.617840
$261$	0	0
$262$	0.631167	0.0389936
$263$	−23.6423	−1.45785	−0.728925	−	0.684594i	$-0.759980\pi$
$263$	−23.6423	−1.45785	−0.728925	+	0.684594i	$0.759980\pi$
$264$	0	0
$265$	−15.0646	−0.925414
$266$	−8.03469	−0.492638
$267$	0	0
$268$	−11.8749	−0.725375
$269$	−11.9533	−0.728808	−0.364404	−	0.931241i	$-0.618727\pi$
$269$	−11.9533	−0.728808	−0.364404	+	0.931241i	$0.618727\pi$
$270$	0	0
$271$	28.7131	1.74420	0.872099	−	0.489330i	$-0.162758\pi$
$271$	28.7131	1.74420	0.872099	+	0.489330i	$0.162758\pi$
$272$	−9.32442	−0.565376
$273$	0	0
$274$	1.40276	0.0847438
$275$	0	0
$276$	0	0
$277$	−15.6358	−0.939464	−0.469732	−	0.882809i	$-0.655650\pi$
$277$	−15.6358	−0.939464	−0.469732	+	0.882809i	$0.655650\pi$
$278$	−15.8457	−0.950359
$279$	0	0
$280$	2.85410	0.170565
$281$	16.2703	0.970607	0.485304	−	0.874346i	$-0.338709\pi$
$281$	16.2703	0.970607	0.485304	+	0.874346i	$0.338709\pi$
$282$	0	0
$283$	2.18428	0.129842	0.0649209	−	0.997890i	$-0.479321\pi$
$283$	2.18428	0.129842	0.0649209	+	0.997890i	$0.479321\pi$
$284$	5.50558	0.326696
$285$	0	0
$286$	0	0
$287$	0.0621247	0.00366710
$288$	0	0
$289$	−13.4224	−0.789555
$290$	14.6020	0.857462
$291$	0	0
$292$	−9.68013	−0.566487
$293$	12.3305	0.720353	0.360176	−	0.932884i	$-0.382717\pi$
$293$	12.3305	0.720353	0.360176	+	0.932884i	$0.382717\pi$
$294$	0	0
$295$	−25.8451	−1.50476
$296$	11.0391	0.641634
$297$	0	0
$298$	−4.90561	−0.284174
$299$	19.0914	1.10408
$300$	0	0
$301$	8.99443	0.518430
$302$	6.64443	0.382344
$303$	0	0
$304$	21.9208	1.25725
$305$	11.0346	0.631840
$306$	0	0
$307$	1.73840	0.0992160	0.0496080	−	0.998769i	$-0.484203\pi$
$307$	1.73840	0.0992160	0.0496080	+	0.998769i	$0.484203\pi$
$308$	0	0
$309$	0	0
$310$	−8.58355	−0.487513
$311$	−11.2208	−0.636273	−0.318137	−	0.948045i	$-0.603057\pi$
$311$	−11.2208	−0.636273	−0.318137	+	0.948045i	$0.603057\pi$
$312$	0	0
$313$	29.5442	1.66994	0.834970	−	0.550296i	$-0.185485\pi$
$313$	29.5442	1.66994	0.834970	+	0.550296i	$0.185485\pi$
$314$	38.0698	2.14840
$315$	0	0
$316$	−20.4521	−1.15052
$317$	−8.53297	−0.479259	−0.239630	−	0.970864i	$-0.577026\pi$
$317$	−8.53297	−0.479259	−0.239630	+	0.970864i	$0.577026\pi$
$318$	0	0
$319$	0	0
$320$	3.08677	0.172556
$321$	0	0
$322$	9.41297	0.524564
$323$	−8.41052	−0.467974
$324$	0	0
$325$	−1.39983	−0.0776488
$326$	−30.6256	−1.69619
$327$	0	0
$328$	−0.0825097	−0.00455583
$329$	−12.3547	−0.681138
$330$	0	0
$331$	3.22164	0.177078	0.0885388	−	0.996073i	$-0.471780\pi$
$331$	3.22164	0.177078	0.0885388	+	0.996073i	$0.471780\pi$
$332$	−9.16789	−0.503153
$333$	0	0
$334$	−28.4223	−1.55520
$335$	20.1732	1.10218
$336$	0	0
$337$	3.52393	0.191961	0.0959803	−	0.995383i	$-0.469401\pi$
$337$	3.52393	0.191961	0.0959803	+	0.995383i	$0.469401\pi$
$338$	0.778481	0.0423438
$339$	0	0
$340$	−5.14168	−0.278847
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−11.9458	−0.644073
$345$	0	0
$346$	17.6924	0.951148
$347$	−4.90171	−0.263138	−0.131569	−	0.991307i	$-0.542001\pi$
$347$	−4.90171	−0.263138	−0.131569	+	0.991307i	$0.542001\pi$
$348$	0	0
$349$	11.5598	0.618780	0.309390	−	0.950935i	$-0.399875\pi$
$349$	11.5598	0.618780	0.309390	+	0.950935i	$0.399875\pi$
$350$	−0.690184	−0.0368919
$351$	0	0
$352$	0	0
$353$	29.5076	1.57053	0.785264	−	0.619161i	$-0.212527\pi$
$353$	29.5076	1.57053	0.785264	+	0.619161i	$0.212527\pi$
$354$	0	0
$355$	−9.35296	−0.496404
$356$	8.45385	0.448053
$357$	0	0
$358$	−24.3207	−1.28539
$359$	7.57351	0.399715	0.199857	−	0.979825i	$-0.435952\pi$
$359$	7.57351	0.399715	0.199857	+	0.979825i	$0.435952\pi$
$360$	0	0
$361$	0.772330	0.0406490
$362$	−28.4620	−1.49593
$363$	0	0
$364$	4.63590	0.242987
$365$	16.4447	0.860757
$366$	0	0
$367$	−9.52922	−0.497421	−0.248711	−	0.968578i	$-0.580007\pi$
$367$	−9.52922	−0.497421	−0.248711	+	0.968578i	$0.580007\pi$
$368$	−25.6812	−1.33872
$369$	0	0
$370$	32.2746	1.67788
$371$	7.01020	0.363951
$372$	0	0
$373$	−17.3921	−0.900527	−0.450264	−	0.892896i	$-0.648670\pi$
$373$	−17.3921	−0.900527	−0.450264	+	0.892896i	$0.648670\pi$
$374$	0	0
$375$	0	0
$376$	16.4087	0.846213
$377$	−13.7815	−0.709783
$378$	0	0
$379$	18.3456	0.942349	0.471175	−	0.882040i	$-0.343830\pi$
$379$	18.3456	0.942349	0.471175	+	0.882040i	$0.343830\pi$
$380$	12.0876	0.620081
$381$	0	0
$382$	16.4458	0.841439
$383$	−2.83652	−0.144940	−0.0724698	−	0.997371i	$-0.523088\pi$
$383$	−2.83652	−0.144940	−0.0724698	+	0.997371i	$0.523088\pi$
$384$	0	0
$385$	0	0
$386$	−25.9248	−1.31954
$387$	0	0
$388$	−16.5754	−0.841487
$389$	−20.6813	−1.04858	−0.524291	−	0.851539i	$-0.675670\pi$
$389$	−20.6813	−1.04858	−0.524291	+	0.851539i	$0.675670\pi$
$390$	0	0
$391$	9.85327	0.498301
$392$	−1.32813	−0.0670807
$393$	0	0
$394$	−37.2827	−1.87828
$395$	34.7443	1.74817
$396$	0	0
$397$	26.6868	1.33937	0.669685	−	0.742645i	$-0.266429\pi$
$397$	26.6868	1.33937	0.669685	+	0.742645i	$0.266429\pi$
$398$	35.3674	1.77281
$399$	0	0
$400$	1.88301	0.0941506
$401$	−12.6012	−0.629272	−0.314636	−	0.949212i	$-0.601883\pi$
$401$	−12.6012	−0.629272	−0.314636	+	0.949212i	$0.601883\pi$
$402$	0	0
$403$	8.10121	0.403550
$404$	1.57792	0.0785043
$405$	0	0
$406$	−6.79493	−0.337227
$407$	0	0
$408$	0	0
$409$	0.0123946	0.000612872 0	0.000306436	−	1.00000i	$-0.499902\pi$
$409$	0.0123946	0.000612872 0	0.000306436		1.00000i	$0.499902\pi$
$410$	−0.241231	−0.0119135
$411$	0	0
$412$	−18.9353	−0.932876
$413$	12.0268	0.591801
$414$	0	0
$415$	15.5745	0.764524
$416$	−22.9105	−1.12328
$417$	0	0
$418$	0	0
$419$	−32.9560	−1.61001	−0.805003	−	0.593271i	$-0.797836\pi$
$419$	−32.9560	−1.61001	−0.805003	+	0.593271i	$0.797836\pi$
$420$	0	0
$421$	−27.3708	−1.33397	−0.666986	−	0.745070i	$-0.732416\pi$
$421$	−27.3708	−1.33397	−0.666986	+	0.745070i	$0.732416\pi$
$422$	−18.4089	−0.896134
$423$	0	0
$424$	−9.31046	−0.452156
$425$	−0.722468	−0.0350448
$426$	0	0
$427$	−5.13486	−0.248493
$428$	−12.9607	−0.626478
$429$	0	0
$430$	−34.9255	−1.68426
$431$	−24.7125	−1.19036	−0.595180	−	0.803593i	$-0.702919\pi$
$431$	−24.7125	−1.19036	−0.595180	+	0.803593i	$0.702919\pi$
$432$	0	0
$433$	−27.4299	−1.31820	−0.659099	−	0.752056i	$-0.729062\pi$
$433$	−27.4299	−1.31820	−0.659099	+	0.752056i	$0.729062\pi$
$434$	3.99428	0.191732
$435$	0	0
$436$	−4.29044	−0.205475
$437$	−23.1641	−1.10809
$438$	0	0
$439$	−3.36929	−0.160807	−0.0804037	−	0.996762i	$-0.525621\pi$
$439$	−3.36929	−0.160807	−0.0804037	+	0.996762i	$0.525621\pi$
$440$	0	0
$441$	0	0
$442$	12.5252	0.595764
$443$	7.97012	0.378672	0.189336	−	0.981912i	$-0.439366\pi$
$443$	7.97012	0.378672	0.189336	+	0.981912i	$0.439366\pi$
$444$	0	0
$445$	−14.3615	−0.680801
$446$	−44.4329	−2.10396
$447$	0	0
$448$	−1.43640	−0.0678636
$449$	5.55142	0.261988	0.130994	−	0.991383i	$-0.458183\pi$
$449$	5.55142	0.261988	0.130994	+	0.991383i	$0.458183\pi$
$450$	0	0
$451$	0	0
$452$	−10.3980	−0.489080
$453$	0	0
$454$	42.0769	1.97477
$455$	−7.87553	−0.369211
$456$	0	0
$457$	16.7915	0.785471	0.392735	−	0.919651i	$-0.371529\pi$
$457$	16.7915	0.785471	0.392735	+	0.919651i	$0.371529\pi$
$458$	−11.7223	−0.547749
$459$	0	0
$460$	−14.1611	−0.660266
$461$	17.3351	0.807376	0.403688	−	0.914897i	$-0.367728\pi$
$461$	17.3351	0.807376	0.403688	+	0.914897i	$0.367728\pi$
$462$	0	0
$463$	24.9175	1.15801	0.579006	−	0.815323i	$-0.303441\pi$
$463$	24.9175	1.15801	0.579006	+	0.815323i	$0.303441\pi$
$464$	18.5384	0.860625
$465$	0	0
$466$	−41.5893	−1.92659
$467$	−22.4093	−1.03698	−0.518490	−	0.855083i	$-0.673506\pi$
$467$	−22.4093	−1.03698	−0.518490	+	0.855083i	$0.673506\pi$
$468$	0	0
$469$	−9.38744	−0.433472
$470$	47.9735	2.21285
$471$	0	0
$472$	−15.9732	−0.735225
$473$	0	0
$474$	0	0
$475$	1.69845	0.0779304
$476$	2.39264	0.109666
$477$	0	0
$478$	16.4169	0.750890
$479$	3.20429	0.146408	0.0732039	−	0.997317i	$-0.476678\pi$
$479$	3.20429	0.146408	0.0732039	+	0.997317i	$0.476678\pi$
$480$	0	0
$481$	−30.4610	−1.38890
$482$	−11.9421	−0.543948
$483$	0	0
$484$	0	0
$485$	28.1585	1.27861
$486$	0	0
$487$	8.39844	0.380570	0.190285	−	0.981729i	$-0.439059\pi$
$487$	8.39844	0.380570	0.190285	+	0.981729i	$0.439059\pi$
$488$	6.81976	0.308716
$489$	0	0
$490$	−3.88301	−0.175417
$491$	34.7888	1.57000	0.784998	−	0.619498i	$-0.212664\pi$
$491$	34.7888	1.57000	0.784998	+	0.619498i	$0.212664\pi$
$492$	0	0
$493$	−7.11277	−0.320343
$494$	−29.4456	−1.32482
$495$	0	0
$496$	−10.8975	−0.489312
$497$	4.35232	0.195228
$498$	0	0
$499$	15.1305	0.677333	0.338667	−	0.940906i	$-0.390024\pi$
$499$	15.1305	0.677333	0.338667	+	0.940906i	$0.390024\pi$
$500$	14.6303	0.654285
$501$	0	0
$502$	−6.49895	−0.290063
$503$	36.7358	1.63797	0.818985	−	0.573815i	$-0.194537\pi$
$503$	36.7358	1.63797	0.818985	+	0.573815i	$0.194537\pi$
$504$	0	0
$505$	−2.68059	−0.119285
$506$	0	0
$507$	0	0
$508$	4.12308	0.182932
$509$	31.2822	1.38656	0.693279	−	0.720669i	$-0.256165\pi$
$509$	31.2822	1.38656	0.693279	+	0.720669i	$0.256165\pi$
$510$	0	0
$511$	−7.65242	−0.338523
$512$	17.7237	0.783285
$513$	0	0
$514$	15.7089	0.692892
$515$	32.1676	1.41747
$516$	0	0
$517$	0	0
$518$	−15.0187	−0.659884
$519$	0	0
$520$	10.4597	0.458690
$521$	41.5773	1.82153	0.910767	−	0.412921i	$-0.135491\pi$
$521$	41.5773	1.82153	0.910767	+	0.412921i	$0.135491\pi$
$522$	0	0
$523$	17.4520	0.763122	0.381561	−	0.924344i	$-0.375387\pi$
$523$	17.4520	0.763122	0.381561	+	0.924344i	$0.375387\pi$
$524$	0.441862	0.0193028
$525$	0	0
$526$	−42.7199	−1.86268
$527$	4.18112	0.182132
$528$	0	0
$529$	4.13768	0.179899
$530$	−27.2207	−1.18239
$531$	0	0
$532$	−5.62486	−0.243869
$533$	0.227675	0.00986170
$534$	0	0
$535$	22.0178	0.951911
$536$	12.4677	0.538525
$537$	0	0
$538$	−21.5988	−0.931189
$539$	0	0
$540$	0	0
$541$	−27.3372	−1.17532	−0.587659	−	0.809109i	$-0.699950\pi$
$541$	−27.3372	−1.17532	−0.587659	+	0.809109i	$0.699950\pi$
$542$	51.8825	2.22854
$543$	0	0
$544$	−11.8244	−0.506965
$545$	7.28867	0.312212
$546$	0	0
$547$	31.1721	1.33282	0.666411	−	0.745585i	$-0.267830\pi$
$547$	31.1721	1.33282	0.666411	+	0.745585i	$0.267830\pi$
$548$	0.982032	0.0419503
$549$	0	0
$550$	0	0
$551$	16.7214	0.712358
$552$	0	0
$553$	−16.1679	−0.687531
$554$	−28.2527	−1.20034
$555$	0	0
$556$	−11.0931	−0.470452
$557$	4.18066	0.177140	0.0885700	−	0.996070i	$-0.471770\pi$
$557$	4.18066	0.177140	0.0885700	+	0.996070i	$0.471770\pi$
$558$	0	0
$559$	32.9629	1.39418
$560$	10.5939	0.447675
$561$	0	0
$562$	29.3993	1.24013
$563$	24.7878	1.04468	0.522340	−	0.852737i	$-0.325059\pi$
$563$	24.7878	1.04468	0.522340	+	0.852737i	$0.325059\pi$
$564$	0	0
$565$	17.6643	0.743141
$566$	3.94682	0.165897
$567$	0	0
$568$	−5.78045	−0.242542
$569$	−35.0889	−1.47100	−0.735502	−	0.677522i	$-0.763054\pi$
$569$	−35.0889	−1.47100	−0.735502	+	0.677522i	$0.763054\pi$
$570$	0	0
$571$	−6.83359	−0.285977	−0.142988	−	0.989724i	$-0.545671\pi$
$571$	−6.83359	−0.285977	−0.142988	+	0.989724i	$0.545671\pi$
$572$	0	0
$573$	0	0
$574$	0.112255	0.00468542
$575$	−1.98981	−0.0829807
$576$	0	0
$577$	−45.3214	−1.88675	−0.943377	−	0.331721i	$-0.892371\pi$
$577$	−45.3214	−1.88675	−0.943377	+	0.331721i	$0.892371\pi$
$578$	−24.2533	−1.00881
$579$	0	0
$580$	10.2225	0.424465
$581$	−7.24748	−0.300676
$582$	0	0
$583$	0	0
$584$	10.1634	0.420565
$585$	0	0
$586$	22.2802	0.920387
$587$	−44.8902	−1.85282	−0.926409	−	0.376518i	$-0.877121\pi$
$587$	−44.8902	−1.85282	−0.926409	+	0.376518i	$0.877121\pi$
$588$	0	0
$589$	−9.82941	−0.405014
$590$	−46.7002	−1.92262
$591$	0	0
$592$	40.9752	1.68407
$593$	38.8436	1.59512	0.797558	−	0.603242i	$-0.206125\pi$
$593$	38.8436	1.59512	0.797558	+	0.603242i	$0.206125\pi$
$594$	0	0
$595$	−4.06464	−0.166634
$596$	−3.43428	−0.140674
$597$	0	0
$598$	34.4967	1.41068
$599$	17.2289	0.703954	0.351977	−	0.936009i	$-0.385510\pi$
$599$	17.2289	0.703954	0.351977	+	0.936009i	$0.385510\pi$
$600$	0	0
$601$	29.0667	1.18565	0.592827	−	0.805330i	$-0.298012\pi$
$601$	29.0667	1.18565	0.592827	+	0.805330i	$0.298012\pi$
$602$	16.2523	0.662393
$603$	0	0
$604$	4.65158	0.189270
$605$	0	0
$606$	0	0
$607$	24.0011	0.974174	0.487087	−	0.873353i	$-0.338059\pi$
$607$	24.0011	0.974174	0.487087	+	0.873353i	$0.338059\pi$
$608$	27.7979	1.12736
$609$	0	0
$610$	19.9387	0.807294
$611$	−45.2777	−1.83174
$612$	0	0
$613$	47.4567	1.91676	0.958379	−	0.285498i	$-0.0921589\pi$
$613$	47.4567	1.91676	0.958379	+	0.285498i	$0.0921589\pi$
$614$	3.14117	0.126767
$615$	0	0
$616$	0	0
$617$	14.0679	0.566354	0.283177	−	0.959068i	$-0.408612\pi$
$617$	14.0679	0.566354	0.283177	+	0.959068i	$0.408612\pi$
$618$	0	0
$619$	35.1265	1.41185	0.705927	−	0.708284i	$-0.250530\pi$
$619$	35.1265	1.41185	0.705927	+	0.708284i	$0.250530\pi$
$620$	−6.00910	−0.241331
$621$	0	0
$622$	−20.2751	−0.812959
$623$	6.68300	0.267749
$624$	0	0
$625$	−22.9443	−0.917771
$626$	53.3842	2.13366
$627$	0	0
$628$	26.6516	1.06351
$629$	−15.7212	−0.626846
$630$	0	0
$631$	−35.2530	−1.40340	−0.701700	−	0.712473i	$-0.747575\pi$
$631$	−35.2530	−1.40340	−0.701700	+	0.712473i	$0.747575\pi$
$632$	21.4731	0.854156
$633$	0	0
$634$	−15.4184	−0.612344
$635$	−7.00434	−0.277959
$636$	0	0
$637$	3.66481	0.145205
$638$	0	0
$639$	0	0
$640$	−21.2909	−0.841595
$641$	−36.1668	−1.42850	−0.714252	−	0.699889i	$-0.753233\pi$
$641$	−36.1668	−1.42850	−0.714252	+	0.699889i	$0.753233\pi$
$642$	0	0
$643$	−2.33953	−0.0922620	−0.0461310	−	0.998935i	$-0.514689\pi$
$643$	−2.33953	−0.0922620	−0.0461310	+	0.998935i	$0.514689\pi$
$644$	6.58975	0.259673
$645$	0	0
$646$	−15.1972	−0.597925
$647$	−28.6902	−1.12793	−0.563965	−	0.825799i	$-0.690725\pi$
$647$	−28.6902	−1.12793	−0.563965	+	0.825799i	$0.690725\pi$
$648$	0	0
$649$	0	0
$650$	−2.52939	−0.0992109
$651$	0	0
$652$	−21.4401	−0.839659
$653$	35.8827	1.40420	0.702099	−	0.712079i	$-0.252246\pi$
$653$	35.8827	1.40420	0.702099	+	0.712079i	$0.252246\pi$
$654$	0	0
$655$	−0.750641	−0.0293300
$656$	−0.306261	−0.0119575
$657$	0	0
$658$	−22.3240	−0.870282
$659$	48.1529	1.87577	0.937885	−	0.346945i	$-0.112781\pi$
$659$	48.1529	1.87577	0.937885	+	0.346945i	$0.112781\pi$
$660$	0	0
$661$	−26.9186	−1.04701	−0.523506	−	0.852022i	$-0.675376\pi$
$661$	−26.9186	−1.04701	−0.523506	+	0.852022i	$0.675376\pi$
$662$	5.82127	0.226250
$663$	0	0
$664$	9.62560	0.373546
$665$	9.55559	0.370550
$666$	0	0
$667$	−19.5899	−0.758523
$668$	−19.8977	−0.769864
$669$	0	0
$670$	36.4515	1.40825
$671$	0	0
$672$	0	0
$673$	6.53324	0.251838	0.125919	−	0.992041i	$-0.459812\pi$
$673$	6.53324	0.251838	0.125919	+	0.992041i	$0.459812\pi$
$674$	6.36747	0.245266
$675$	0	0
$676$	0.544993	0.0209613
$677$	41.9065	1.61060	0.805299	−	0.592869i	$-0.202005\pi$
$677$	41.9065	1.61060	0.805299	+	0.592869i	$0.202005\pi$
$678$	0	0
$679$	−13.1033	−0.502859
$680$	5.39838	0.207018
$681$	0	0
$682$	0	0
$683$	−17.5429	−0.671259	−0.335630	−	0.941994i	$-0.608949\pi$
$683$	−17.5429	−0.671259	−0.335630	+	0.941994i	$0.608949\pi$
$684$	0	0
$685$	−1.66829	−0.0637421
$686$	1.80692	0.0689887
$687$	0	0
$688$	−44.3406	−1.69047
$689$	25.6910	0.978751
$690$	0	0
$691$	−36.6341	−1.39363	−0.696813	−	0.717253i	$-0.745399\pi$
$691$	−36.6341	−1.39363	−0.696813	+	0.717253i	$0.745399\pi$
$692$	12.3859	0.470842
$693$	0	0
$694$	−8.85703	−0.336208
$695$	18.8451	0.714836
$696$	0	0
$697$	0.117505	0.00445083
$698$	20.8876	0.790608
$699$	0	0
$700$	−0.483178	−0.0182624
$701$	16.5248	0.624132	0.312066	−	0.950060i	$-0.398979\pi$
$701$	16.5248	0.624132	0.312066	+	0.950060i	$0.398979\pi$
$702$	0	0
$703$	36.9591	1.39394
$704$	0	0
$705$	0	0
$706$	53.3179	2.00665
$707$	1.24739	0.0469128
$708$	0	0
$709$	−17.8668	−0.671002	−0.335501	−	0.942040i	$-0.608906\pi$
$709$	−17.8668	−0.671002	−0.335501	+	0.942040i	$0.608906\pi$
$710$	−16.9001	−0.634249
$711$	0	0
$712$	−8.87590	−0.332639
$713$	11.5156	0.431261
$714$	0	0
$715$	0	0
$716$	−17.0262	−0.636300
$717$	0	0
$718$	13.6848	0.510711
$719$	−19.7323	−0.735891	−0.367945	−	0.929847i	$-0.619939\pi$
$719$	−19.7323	−0.735891	−0.367945	+	0.929847i	$0.619939\pi$
$720$	0	0
$721$	−14.9689	−0.557471
$722$	1.39554	0.0519367
$723$	0	0
$724$	−19.9254	−0.740522
$725$	1.43638	0.0533459
$726$	0	0
$727$	−28.6895	−1.06403	−0.532017	−	0.846733i	$-0.678566\pi$
$727$	−28.6895	−1.06403	−0.532017	+	0.846733i	$0.678566\pi$
$728$	−4.86735	−0.180396
$729$	0	0
$730$	29.7144	1.09978
$731$	17.0125	0.629229
$732$	0	0
$733$	5.36355	0.198107	0.0990535	−	0.995082i	$-0.468418\pi$
$733$	5.36355	0.198107	0.0990535	+	0.995082i	$0.468418\pi$
$734$	−17.2186	−0.635549
$735$	0	0
$736$	−32.5664	−1.20041
$737$	0	0
$738$	0	0
$739$	−45.1733	−1.66173	−0.830864	−	0.556476i	$-0.812153\pi$
$739$	−45.1733	−1.66173	−0.830864	+	0.556476i	$0.812153\pi$
$740$	22.5946	0.830593
$741$	0	0
$742$	12.6669	0.465017
$743$	19.8716	0.729020	0.364510	−	0.931199i	$-0.381236\pi$
$743$	19.8716	0.729020	0.364510	+	0.931199i	$0.381236\pi$
$744$	0	0
$745$	5.83421	0.213749
$746$	−31.4262	−1.15059
$747$	0	0
$748$	0	0
$749$	−10.2458	−0.374372
$750$	0	0
$751$	3.72318	0.135861	0.0679304	−	0.997690i	$-0.478360\pi$
$751$	3.72318	0.135861	0.0679304	+	0.997690i	$0.478360\pi$
$752$	60.9061	2.22102
$753$	0	0
$754$	−24.9021	−0.906882
$755$	−7.90217	−0.287589
$756$	0	0
$757$	37.2656	1.35444	0.677220	−	0.735780i	$-0.263184\pi$
$757$	37.2656	1.35444	0.677220	+	0.735780i	$0.263184\pi$
$758$	33.1491	1.20403
$759$	0	0
$760$	−12.6911	−0.460354
$761$	21.3908	0.775416	0.387708	−	0.921782i	$-0.373267\pi$
$761$	21.3908	0.775416	0.387708	+	0.921782i	$0.373267\pi$
$762$	0	0
$763$	−3.39172	−0.122788
$764$	11.5132	0.416534
$765$	0	0
$766$	−5.12539	−0.185188
$767$	44.0760	1.59149
$768$	0	0
$769$	20.2160	0.729006	0.364503	−	0.931202i	$-0.381239\pi$
$769$	20.2160	0.729006	0.364503	+	0.931202i	$0.381239\pi$
$770$	0	0
$771$	0	0
$772$	−18.1492	−0.653206
$773$	39.5108	1.42111	0.710553	−	0.703644i	$-0.248445\pi$
$773$	39.5108	1.42111	0.710553	+	0.703644i	$0.248445\pi$
$774$	0	0
$775$	−0.844351	−0.0303300
$776$	17.4029	0.624728
$777$	0	0
$778$	−37.3695	−1.33976
$779$	−0.276244	−0.00989747
$780$	0	0
$781$	0	0
$782$	17.8041	0.636674
$783$	0	0
$784$	−4.92979	−0.176064
$785$	−45.2761	−1.61597
$786$	0	0
$787$	34.2457	1.22073	0.610364	−	0.792121i	$-0.291023\pi$
$787$	34.2457	1.22073	0.610364	+	0.792121i	$0.291023\pi$
$788$	−26.1006	−0.929795
$789$	0	0
$790$	62.7803	2.23362
$791$	−8.21990	−0.292266
$792$	0	0
$793$	−18.8183	−0.668256
$794$	48.2210	1.71130
$795$	0	0
$796$	24.7597	0.877585
$797$	−16.1138	−0.570779	−0.285390	−	0.958412i	$-0.592123\pi$
$797$	−16.1138	−0.570779	−0.285390	+	0.958412i	$0.592123\pi$
$798$	0	0
$799$	−23.3683	−0.826710
$800$	2.38786	0.0844235
$801$	0	0
$802$	−22.7694	−0.804014
$803$	0	0
$804$	0	0
$805$	−11.1948	−0.394564
$806$	14.6383	0.515611
$807$	0	0
$808$	−1.65669	−0.0582823
$809$	−7.67619	−0.269880	−0.134940	−	0.990854i	$-0.543084\pi$
$809$	−7.67619	−0.269880	−0.134940	+	0.990854i	$0.543084\pi$
$810$	0	0
$811$	1.03802	0.0364498	0.0182249	−	0.999834i	$-0.494199\pi$
$811$	1.03802	0.0364498	0.0182249	+	0.999834i	$0.494199\pi$
$812$	−4.75694	−0.166936
$813$	0	0
$814$	0	0
$815$	36.4228	1.27583
$816$	0	0
$817$	−39.9947	−1.39924
$818$	0.0223961	0.000783060 0
$819$	0	0
$820$	−0.168879	−0.00589750
$821$	7.44620	0.259874	0.129937	−	0.991522i	$-0.458522\pi$
$821$	7.44620	0.259874	0.129937	+	0.991522i	$0.458522\pi$
$822$	0	0
$823$	2.94956	0.102815	0.0514076	−	0.998678i	$-0.483629\pi$
$823$	2.94956	0.102815	0.0514076	+	0.998678i	$0.483629\pi$
$824$	19.8807	0.692576
$825$	0	0
$826$	21.7315	0.756137
$827$	−42.6293	−1.48237	−0.741183	−	0.671303i	$-0.765735\pi$
$827$	−42.6293	−1.48237	−0.741183	+	0.671303i	$0.765735\pi$
$828$	0	0
$829$	6.02334	0.209199	0.104600	−	0.994514i	$-0.466644\pi$
$829$	6.02334	0.209199	0.104600	+	0.994514i	$0.466644\pi$
$830$	28.1420	0.976824
$831$	0	0
$832$	−5.26414	−0.182501
$833$	1.89145	0.0655347
$834$	0	0
$835$	33.8025	1.16978
$836$	0	0
$837$	0	0
$838$	−59.5490	−2.05709
$839$	−2.90271	−0.100213	−0.0501064	−	0.998744i	$-0.515956\pi$
$839$	−2.90271	−0.100213	−0.0501064	+	0.998744i	$0.515956\pi$
$840$	0	0
$841$	−14.8587	−0.512368
$842$	−49.4570	−1.70440
$843$	0	0
$844$	−12.8876	−0.443609
$845$	−0.925842	−0.0318499
$846$	0	0
$847$	0	0
$848$	−34.5588	−1.18675
$849$	0	0
$850$	−1.30545	−0.0447764
$851$	−43.2991	−1.48427
$852$	0	0
$853$	1.92232	0.0658191	0.0329095	−	0.999458i	$-0.489523\pi$
$853$	1.92232	0.0658191	0.0329095	+	0.999458i	$0.489523\pi$
$854$	−9.27830	−0.317497
$855$	0	0
$856$	13.6077	0.465103
$857$	55.9198	1.91019	0.955093	−	0.296308i	$-0.0957554\pi$
$857$	55.9198	1.91019	0.955093	+	0.296308i	$0.0957554\pi$
$858$	0	0
$859$	0.723243	0.0246767	0.0123384	−	0.999924i	$-0.496072\pi$
$859$	0.723243	0.0246767	0.0123384	+	0.999924i	$0.496072\pi$
$860$	−24.4503	−0.833750
$861$	0	0
$862$	−44.6536	−1.52091
$863$	−41.6422	−1.41752	−0.708759	−	0.705451i	$-0.750744\pi$
$863$	−41.6422	−1.41752	−0.708759	+	0.705451i	$0.750744\pi$
$864$	0	0
$865$	−21.0414	−0.715429
$866$	−49.5638	−1.68425
$867$	0	0
$868$	2.79628	0.0949121
$869$	0	0
$870$	0	0
$871$	−34.4032	−1.16571
$872$	4.50464	0.152546
$873$	0	0
$874$	−41.8558	−1.41579
$875$	11.5656	0.390990
$876$	0	0
$877$	−40.1509	−1.35580	−0.677900	−	0.735154i	$-0.737110\pi$
$877$	−40.1509	−1.35580	−0.677900	+	0.735154i	$0.737110\pi$
$878$	−6.08805	−0.205462
$879$	0	0
$880$	0	0
$881$	−3.28404	−0.110642	−0.0553210	−	0.998469i	$-0.517618\pi$
$881$	−3.28404	−0.110642	−0.0553210	+	0.998469i	$0.517618\pi$
$882$	0	0
$883$	−4.76427	−0.160331	−0.0801653	−	0.996782i	$-0.525545\pi$
$883$	−4.76427	−0.160331	−0.0801653	+	0.996782i	$0.525545\pi$
$884$	8.76855	0.294918
$885$	0	0
$886$	14.4014	0.483825
$887$	8.82252	0.296231	0.148116	−	0.988970i	$-0.452679\pi$
$887$	8.82252	0.296231	0.148116	+	0.988970i	$0.452679\pi$
$888$	0	0
$889$	3.25941	0.109317
$890$	−25.9502	−0.869852
$891$	0	0
$892$	−31.1062	−1.04151
$893$	54.9366	1.83838
$894$	0	0
$895$	28.9244	0.966835
$896$	9.90751	0.330987
$897$	0	0
$898$	10.0310	0.334739
$899$	−8.31273	−0.277245
$900$	0	0
$901$	13.2594	0.441735
$902$	0	0
$903$	0	0
$904$	10.9171	0.363098
$905$	33.8496	1.12520
$906$	0	0
$907$	−18.1246	−0.601816	−0.300908	−	0.953653i	$-0.597290\pi$
$907$	−18.1246	−0.601816	−0.300908	+	0.953653i	$0.597290\pi$
$908$	29.4569	0.977560
$909$	0	0
$910$	−14.2305	−0.471736
$911$	−40.4176	−1.33910	−0.669548	−	0.742769i	$-0.733512\pi$
$911$	−40.4176	−1.33910	−0.669548	+	0.742769i	$0.733512\pi$
$912$	0	0
$913$	0	0
$914$	30.3409	1.00359
$915$	0	0
$916$	−8.20648	−0.271150
$917$	0.349304	0.0115350
$918$	0	0
$919$	−29.0293	−0.957587	−0.478794	−	0.877928i	$-0.658926\pi$
$919$	−29.0293	−0.957587	−0.478794	+	0.877928i	$0.658926\pi$
$920$	14.8681	0.490187
$921$	0	0
$922$	31.3232	1.03158
$923$	15.9504	0.525014
$924$	0	0
$925$	3.17481	0.104387
$926$	45.0240	1.47958
$927$	0	0
$928$	23.5087	0.771711
$929$	−21.6963	−0.711832	−0.355916	−	0.934518i	$-0.615831\pi$
$929$	−21.6963	−0.711832	−0.355916	+	0.934518i	$0.615831\pi$
$930$	0	0
$931$	−4.44661	−0.145732
$932$	−29.1155	−0.953710
$933$	0	0
$934$	−40.4920	−1.32494
$935$	0	0
$936$	0	0
$937$	49.4997	1.61709	0.808543	−	0.588438i	$-0.200257\pi$
$937$	49.4997	1.61709	0.808543	+	0.588438i	$0.200257\pi$
$938$	−16.9624	−0.553842
$939$	0	0
$940$	33.5849	1.09542
$941$	−59.9530	−1.95441	−0.977206	−	0.212293i	$-0.931907\pi$
$941$	−59.9530	−1.95441	−0.977206	+	0.212293i	$0.931907\pi$
$942$	0	0
$943$	0.323631	0.0105389
$944$	−59.2896	−1.92971
$945$	0	0
$946$	0	0
$947$	59.3880	1.92985	0.964925	−	0.262526i	$-0.0845554\pi$
$947$	59.3880	1.92985	0.964925	+	0.262526i	$0.0845554\pi$
$948$	0	0
$949$	−28.0446	−0.910368
$950$	3.06898	0.0995708
$951$	0	0
$952$	−2.51209	−0.0814172
$953$	−56.6435	−1.83486	−0.917432	−	0.397893i	$-0.869742\pi$
$953$	−56.6435	−1.83486	−0.917432	+	0.397893i	$0.869742\pi$
$954$	0	0
$955$	−19.5588	−0.632909
$956$	11.4930	0.371710
$957$	0	0
$958$	5.78992	0.187064
$959$	0.776324	0.0250688
$960$	0	0
$961$	−26.1135	−0.842371
$962$	−55.0407	−1.77458
$963$	0	0
$964$	−8.36034	−0.269268
$965$	30.8322	0.992523
$966$	0	0
$967$	−8.51193	−0.273725	−0.136863	−	0.990590i	$-0.543702\pi$
$967$	−8.51193	−0.273725	−0.136863	+	0.990590i	$0.543702\pi$
$968$	0	0
$969$	0	0
$970$	50.8803	1.63367
$971$	8.61967	0.276618	0.138309	−	0.990389i	$-0.455833\pi$
$971$	8.61967	0.276618	0.138309	+	0.990389i	$0.455833\pi$
$972$	0	0
$973$	−8.76940	−0.281134
$974$	15.1754	0.486250
$975$	0	0
$976$	25.3137	0.810273
$977$	−34.3030	−1.09745	−0.548726	−	0.836002i	$-0.684887\pi$
$977$	−34.3030	−1.09745	−0.548726	+	0.836002i	$0.684887\pi$
$978$	0	0
$979$	0	0
$980$	−2.71839	−0.0868357
$981$	0	0
$982$	62.8607	2.00597
$983$	55.0178	1.75479	0.877397	−	0.479765i	$-0.159278\pi$
$983$	55.0178	1.75479	0.877397	+	0.479765i	$0.159278\pi$
$984$	0	0
$985$	44.3401	1.41279
$986$	−12.8522	−0.409299
$987$	0	0
$988$	−20.6140	−0.655820
$989$	46.8554	1.48992
$990$	0	0
$991$	42.7838	1.35907	0.679536	−	0.733642i	$-0.262181\pi$
$991$	42.7838	1.35907	0.679536	+	0.733642i	$0.262181\pi$
$992$	−13.8192	−0.438759
$993$	0	0
$994$	7.86431	0.249441
$995$	−42.0622	−1.33346
$996$	0	0
$997$	−43.5608	−1.37958	−0.689792	−	0.724007i	$-0.742298\pi$
$997$	−43.5608	−1.37958	−0.689792	+	0.724007i	$0.742298\pi$
$998$	27.3396	0.865421
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cv.1.7		8
3.2	odd	2	inner	7623.2.a.cv.1.2		8
11.2	odd	10		693.2.m.h.631.4	yes	16
11.6	odd	10		693.2.m.h.190.4	yes	16
11.10	odd	2		7623.2.a.cu.1.2		8
33.2	even	10		693.2.m.h.631.1	yes	16
33.17	even	10		693.2.m.h.190.1	✓	16
33.32	even	2		7623.2.a.cu.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.h.190.1	✓	16	33.17	even	10
693.2.m.h.190.4	yes	16	11.6	odd	10
693.2.m.h.631.1	yes	16	33.2	even	10
693.2.m.h.631.4	yes	16	11.2	odd	10
7623.2.a.cu.1.2		8	11.10	odd	2
7623.2.a.cu.1.7		8	33.32	even	2
7623.2.a.cv.1.2		8	3.2	odd	2	inner
7623.2.a.cv.1.7		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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80692 q^{2} +1.26498 q^{4} -2.14896 q^{5} +1.00000 q^{7} -1.32813 q^{8} +O(q^{10})\)	\(q+1.80692 q^{2} +1.26498 q^{4} -2.14896 q^{5} +1.00000 q^{7} -1.32813 q^{8} -3.88301 q^{10} +3.66481 q^{13} +1.80692 q^{14} -4.92979 q^{16} +1.89145 q^{17} -4.44661 q^{19} -2.71839 q^{20} +5.20938 q^{23} -0.381966 q^{25} +6.62204 q^{26} +1.26498 q^{28} -3.76049 q^{29} +2.21054 q^{31} -6.25149 q^{32} +3.41770 q^{34} -2.14896 q^{35} -8.31175 q^{37} -8.03469 q^{38} +2.85410 q^{40} +0.0621247 q^{41} +8.99443 q^{43} +9.41297 q^{46} -12.3547 q^{47} +1.00000 q^{49} -0.690184 q^{50} +4.63590 q^{52} +7.01020 q^{53} -1.32813 q^{56} -6.79493 q^{58} +12.0268 q^{59} -5.13486 q^{61} +3.99428 q^{62} -1.43640 q^{64} -7.87553 q^{65} -9.38744 q^{67} +2.39264 q^{68} -3.88301 q^{70} +4.35232 q^{71} -7.65242 q^{73} -15.0187 q^{74} -5.62486 q^{76} -16.1679 q^{79} +10.5939 q^{80} +0.112255 q^{82} -7.24748 q^{83} -4.06464 q^{85} +16.2523 q^{86} +6.68300 q^{89} +3.66481 q^{91} +6.58975 q^{92} -22.3240 q^{94} +9.55559 q^{95} -13.1033 q^{97} +1.80692 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) 8 * q + 2 * q^4 + 8 * q^7	\( 8 q + 2 q^{4} + 8 q^{7} - 14 q^{10} - 2 q^{16} + 6 q^{19} - 12 q^{25} + 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} - 4 q^{40} - 16 q^{43} + 42 q^{46} + 8 q^{49} - 2 q^{52} - 30 q^{58} - 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} - 14 q^{73} + 34 q^{76} - 22 q^{79} + 36 q^{82} + 18 q^{85} - 32 q^{94} - 48 q^{97}+O(q^{100}) \) 8 * q + 2 * q^4 + 8 * q^7 - 14 * q^10 - 2 * q^16 + 6 * q^19 - 12 * q^25 + 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 - 4 * q^40 - 16 * q^43 + 42 * q^46 + 8 * q^49 - 2 * q^52 - 30 * q^58 - 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 - 14 * q^73 + 34 * q^76 - 22 * q^79 + 36 * q^82 + 18 * q^85 - 32 * q^94 - 48 * q^97

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