LMFDB - Embedded newform 7623.2.a.bq.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +6.85410 q^{10} -6.23607 q^{13} +1.61803 q^{14} -4.85410 q^{16} -4.47214 q^{17} -3.00000 q^{19} +2.61803 q^{20} -8.47214 q^{23} +12.9443 q^{25} -10.0902 q^{26} +0.618034 q^{28} -3.00000 q^{29} -3.38197 q^{32} -7.23607 q^{34} +4.23607 q^{35} +3.47214 q^{37} -4.85410 q^{38} -9.47214 q^{40} +1.52786 q^{41} -10.9443 q^{43} -13.7082 q^{46} +3.00000 q^{47} +1.00000 q^{49} +20.9443 q^{50} -3.85410 q^{52} -8.94427 q^{53} -2.23607 q^{56} -4.85410 q^{58} +1.47214 q^{59} -3.52786 q^{61} +4.23607 q^{64} -26.4164 q^{65} -8.70820 q^{67} -2.76393 q^{68} +6.85410 q^{70} -1.52786 q^{71} -12.2361 q^{73} +5.61803 q^{74} -1.85410 q^{76} +13.4164 q^{79} -20.5623 q^{80} +2.47214 q^{82} +6.00000 q^{83} -18.9443 q^{85} -17.7082 q^{86} +4.47214 q^{89} -6.23607 q^{91} -5.23607 q^{92} +4.85410 q^{94} -12.7082 q^{95} +2.00000 q^{97} +1.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 4 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 + 4 * q^5 + 2 * q^7	$ 2 q + q^{2} - q^{4} + 4 q^{5} + 2 q^{7} + 7 q^{10} - 8 q^{13} + q^{14} - 3 q^{16} - 6 q^{19} + 3 q^{20} - 8 q^{23} + 8 q^{25} - 9 q^{26} - q^{28} - 6 q^{29} - 9 q^{32} - 10 q^{34} + 4 q^{35} - 2 q^{37} - 3 q^{38} - 10 q^{40} + 12 q^{41} - 4 q^{43} - 14 q^{46} + 6 q^{47} + 2 q^{49} + 24 q^{50} - q^{52} - 3 q^{58} - 6 q^{59} - 16 q^{61} + 4 q^{64} - 26 q^{65} - 4 q^{67} - 10 q^{68} + 7 q^{70} - 12 q^{71} - 20 q^{73} + 9 q^{74} + 3 q^{76} - 21 q^{80} - 4 q^{82} + 12 q^{83} - 20 q^{85} - 22 q^{86} - 8 q^{91} - 6 q^{92} + 3 q^{94} - 12 q^{95} + 4 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 4 * q^5 + 2 * q^7 + 7 * q^10 - 8 * q^13 + q^14 - 3 * q^16 - 6 * q^19 + 3 * q^20 - 8 * q^23 + 8 * q^25 - 9 * q^26 - q^28 - 6 * q^29 - 9 * q^32 - 10 * q^34 + 4 * q^35 - 2 * q^37 - 3 * q^38 - 10 * q^40 + 12 * q^41 - 4 * q^43 - 14 * q^46 + 6 * q^47 + 2 * q^49 + 24 * q^50 - q^52 - 3 * q^58 - 6 * q^59 - 16 * q^61 + 4 * q^64 - 26 * q^65 - 4 * q^67 - 10 * q^68 + 7 * q^70 - 12 * q^71 - 20 * q^73 + 9 * q^74 + 3 * q^76 - 21 * q^80 - 4 * q^82 + 12 * q^83 - 20 * q^85 - 22 * q^86 - 8 * q^91 - 6 * q^92 + 3 * q^94 - 12 * q^95 + 4 * q^97 + 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	4.23607	1.89443	0.947214	−	0.320603i	$-0.103886\pi$
$5$	4.23607	1.89443	0.947214	+	0.320603i	$0.103886\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	6.85410	2.16746
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	1.61803	0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	2.61803	0.585410
$21$	0	0
$22$	0	0
$23$	−8.47214	−1.76656	−0.883281	−	0.468844i	$-0.844671\pi$
$23$	−8.47214	−1.76656	−0.883281	+	0.468844i	$0.844671\pi$
$24$	0	0
$25$	12.9443	2.58885
$26$	−10.0902	−1.97885
$27$	0	0
$28$	0.618034	0.116797
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	−7.23607	−1.24098
$35$	4.23607	0.716026
$36$	0	0
$37$	3.47214	0.570816	0.285408	−	0.958406i	$-0.407871\pi$
$37$	3.47214	0.570816	0.285408	+	0.958406i	$0.407871\pi$
$38$	−4.85410	−0.787439
$39$	0	0
$40$	−9.47214	−1.49768
$41$	1.52786	0.238612	0.119306	−	0.992858i	$-0.461933\pi$
$41$	1.52786	0.238612	0.119306	+	0.992858i	$0.461933\pi$
$42$	0	0
$43$	−10.9443	−1.66899	−0.834493	−	0.551019i	$-0.814239\pi$
$43$	−10.9443	−1.66899	−0.834493	+	0.551019i	$0.814239\pi$
$44$	0	0
$45$	0	0
$46$	−13.7082	−2.02116
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	20.9443	2.96197
$51$	0	0
$52$	−3.85410	−0.534468
$53$	−8.94427	−1.22859	−0.614295	−	0.789076i	$-0.710560\pi$
$53$	−8.94427	−1.22859	−0.614295	+	0.789076i	$0.710560\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−4.85410	−0.637375
$59$	1.47214	0.191656	0.0958279	−	0.995398i	$-0.469450\pi$
$59$	1.47214	0.191656	0.0958279	+	0.995398i	$0.469450\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	0	0
$64$	4.23607	0.529508
$65$	−26.4164	−3.27655
$66$	0	0
$67$	−8.70820	−1.06388	−0.531938	−	0.846783i	$-0.678536\pi$
$67$	−8.70820	−1.06388	−0.531938	+	0.846783i	$0.678536\pi$
$68$	−2.76393	−0.335176
$69$	0	0
$70$	6.85410	0.819222
$71$	−1.52786	−0.181324	−0.0906621	−	0.995882i	$-0.528898\pi$
$71$	−1.52786	−0.181324	−0.0906621	+	0.995882i	$0.528898\pi$
$72$	0	0
$73$	−12.2361	−1.43212	−0.716062	−	0.698037i	$-0.754057\pi$
$73$	−12.2361	−1.43212	−0.716062	+	0.698037i	$0.754057\pi$
$74$	5.61803	0.653083
$75$	0	0
$76$	−1.85410	−0.212680
$77$	0	0
$78$	0	0
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	−20.5623	−2.29894
$81$	0	0
$82$	2.47214	0.273002
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−18.9443	−2.05479
$86$	−17.7082	−1.90952
$87$	0	0
$88$	0	0
$89$	4.47214	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$90$	0	0
$91$	−6.23607	−0.653718
$92$	−5.23607	−0.545898
$93$	0	0
$94$	4.85410	0.500662
$95$	−12.7082	−1.30383
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.61803	0.163446
$99$	0	0
$100$	8.00000	0.800000
$101$	−16.9443	−1.68602	−0.843009	−	0.537899i	$-0.819218\pi$
$101$	−16.9443	−1.68602	−0.843009	+	0.537899i	$0.819218\pi$
$102$	0	0
$103$	6.47214	0.637719	0.318859	−	0.947802i	$-0.396700\pi$
$103$	6.47214	0.637719	0.318859	+	0.947802i	$0.396700\pi$
$104$	13.9443	1.36735
$105$	0	0
$106$	−14.4721	−1.40566
$107$	9.76393	0.943915	0.471957	−	0.881621i	$-0.343548\pi$
$107$	9.76393	0.943915	0.471957	+	0.881621i	$0.343548\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	−4.85410	−0.458670
$113$	6.47214	0.608847	0.304424	−	0.952537i	$-0.401536\pi$
$113$	6.47214	0.608847	0.304424	+	0.952537i	$0.401536\pi$
$114$	0	0
$115$	−35.8885	−3.34662
$116$	−1.85410	−0.172149
$117$	0	0
$118$	2.38197	0.219278
$119$	−4.47214	−0.409960
$120$	0	0
$121$	0	0
$122$	−5.70820	−0.516797
$123$	0	0
$124$	0	0
$125$	33.6525	3.00997
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−42.7426	−3.74878
$131$	2.47214	0.215992	0.107996	−	0.994151i	$-0.465557\pi$
$131$	2.47214	0.215992	0.107996	+	0.994151i	$0.465557\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	−14.0902	−1.21721
$135$	0	0
$136$	10.0000	0.857493
$137$	−2.94427	−0.251546	−0.125773	−	0.992059i	$-0.540141\pi$
$137$	−2.94427	−0.251546	−0.125773	+	0.992059i	$0.540141\pi$
$138$	0	0
$139$	0.944272	0.0800921	0.0400460	−	0.999198i	$-0.487250\pi$
$139$	0.944272	0.0800921	0.0400460	+	0.999198i	$0.487250\pi$
$140$	2.61803	0.221264
$141$	0	0
$142$	−2.47214	−0.207457
$143$	0	0
$144$	0	0
$145$	−12.7082	−1.05536
$146$	−19.7984	−1.63853
$147$	0	0
$148$	2.14590	0.176392
$149$	17.9443	1.47005	0.735026	−	0.678039i	$-0.237170\pi$
$149$	17.9443	1.47005	0.735026	+	0.678039i	$0.237170\pi$
$150$	0	0
$151$	−17.4164	−1.41733	−0.708664	−	0.705547i	$-0.750702\pi$
$151$	−17.4164	−1.41733	−0.708664	+	0.705547i	$0.750702\pi$
$152$	6.70820	0.544107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.4164	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$158$	21.7082	1.72701
$159$	0	0
$160$	−14.3262	−1.13259
$161$	−8.47214	−0.667698
$162$	0	0
$163$	7.29180	0.571138	0.285569	−	0.958358i	$-0.407818\pi$
$163$	7.29180	0.571138	0.285569	+	0.958358i	$0.407818\pi$
$164$	0.944272	0.0737352
$165$	0	0
$166$	9.70820	0.753503
$167$	18.9443	1.46595	0.732976	−	0.680255i	$-0.238131\pi$
$167$	18.9443	1.46595	0.732976	+	0.680255i	$0.238131\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	−30.6525	−2.35094
$171$	0	0
$172$	−6.76393	−0.515745
$173$	7.52786	0.572333	0.286166	−	0.958180i	$-0.407619\pi$
$173$	7.52786	0.572333	0.286166	+	0.958180i	$0.407619\pi$
$174$	0	0
$175$	12.9443	0.978495
$176$	0	0
$177$	0	0
$178$	7.23607	0.542366
$179$	16.4721	1.23119	0.615593	−	0.788065i	$-0.288917\pi$
$179$	16.4721	1.23119	0.615593	+	0.788065i	$0.288917\pi$
$180$	0	0
$181$	−3.41641	−0.253940	−0.126970	−	0.991907i	$-0.540525\pi$
$181$	−3.41641	−0.253940	−0.126970	+	0.991907i	$0.540525\pi$
$182$	−10.0902	−0.747933
$183$	0	0
$184$	18.9443	1.39659
$185$	14.7082	1.08137
$186$	0	0
$187$	0	0
$188$	1.85410	0.135224
$189$	0	0
$190$	−20.5623	−1.49175
$191$	−2.94427	−0.213040	−0.106520	−	0.994311i	$-0.533971\pi$
$191$	−2.94427	−0.213040	−0.106520	+	0.994311i	$0.533971\pi$
$192$	0	0
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	3.23607	0.232336
$195$	0	0
$196$	0.618034	0.0441453
$197$	−1.05573	−0.0752175	−0.0376088	−	0.999293i	$-0.511974\pi$
$197$	−1.05573	−0.0752175	−0.0376088	+	0.999293i	$0.511974\pi$
$198$	0	0
$199$	−12.4721	−0.884126	−0.442063	−	0.896984i	$-0.645753\pi$
$199$	−12.4721	−0.884126	−0.442063	+	0.896984i	$0.645753\pi$
$200$	−28.9443	−2.04667
$201$	0	0
$202$	−27.4164	−1.92901
$203$	−3.00000	−0.210559
$204$	0	0
$205$	6.47214	0.452034
$206$	10.4721	0.729628
$207$	0	0
$208$	30.2705	2.09888
$209$	0	0
$210$	0	0
$211$	1.41641	0.0975095	0.0487548	−	0.998811i	$-0.484475\pi$
$211$	1.41641	0.0975095	0.0487548	+	0.998811i	$0.484475\pi$
$212$	−5.52786	−0.379655
$213$	0	0
$214$	15.7984	1.07995
$215$	−46.3607	−3.16177
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	27.8885	1.87599
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	−3.38197	−0.225967
$225$	0	0
$226$	10.4721	0.696596
$227$	−15.5279	−1.03062	−0.515310	−	0.857004i	$-0.672323\pi$
$227$	−15.5279	−1.03062	−0.515310	+	0.857004i	$0.672323\pi$
$228$	0	0
$229$	−24.4721	−1.61716	−0.808582	−	0.588383i	$-0.799765\pi$
$229$	−24.4721	−1.61716	−0.808582	+	0.588383i	$0.799765\pi$
$230$	−58.0689	−3.82895
$231$	0	0
$232$	6.70820	0.440415
$233$	19.8885	1.30294	0.651471	−	0.758674i	$-0.274152\pi$
$233$	19.8885	1.30294	0.651471	+	0.758674i	$0.274152\pi$
$234$	0	0
$235$	12.7082	0.828992
$236$	0.909830	0.0592249
$237$	0	0
$238$	−7.23607	−0.469045
$239$	12.7082	0.822025	0.411013	−	0.911630i	$-0.365175\pi$
$239$	12.7082	0.822025	0.411013	+	0.911630i	$0.365175\pi$
$240$	0	0
$241$	−12.7082	−0.818607	−0.409304	−	0.912398i	$-0.634228\pi$
$241$	−12.7082	−0.818607	−0.409304	+	0.912398i	$0.634228\pi$
$242$	0	0
$243$	0	0
$244$	−2.18034	−0.139582
$245$	4.23607	0.270632
$246$	0	0
$247$	18.7082	1.19037
$248$	0	0
$249$	0	0
$250$	54.4508	3.44377
$251$	16.4164	1.03619	0.518097	−	0.855322i	$-0.326641\pi$
$251$	16.4164	1.03619	0.518097	+	0.855322i	$0.326641\pi$
$252$	0	0
$253$	0	0
$254$	−3.23607	−0.203049
$255$	0	0
$256$	13.5623	0.847644
$257$	−0.347524	−0.0216780	−0.0108390	−	0.999941i	$-0.503450\pi$
$257$	−0.347524	−0.0216780	−0.0108390	+	0.999941i	$0.503450\pi$
$258$	0	0
$259$	3.47214	0.215748
$260$	−16.3262	−1.01251
$261$	0	0
$262$	4.00000	0.247121
$263$	0.236068	0.0145566	0.00727829	−	0.999974i	$-0.497683\pi$
$263$	0.236068	0.0145566	0.00727829	+	0.999974i	$0.497683\pi$
$264$	0	0
$265$	−37.8885	−2.32747
$266$	−4.85410	−0.297624
$267$	0	0
$268$	−5.38197	−0.328756
$269$	19.5279	1.19063	0.595317	−	0.803491i	$-0.297026\pi$
$269$	19.5279	1.19063	0.595317	+	0.803491i	$0.297026\pi$
$270$	0	0
$271$	3.47214	0.210917	0.105459	−	0.994424i	$-0.466369\pi$
$271$	3.47214	0.210917	0.105459	+	0.994424i	$0.466369\pi$
$272$	21.7082	1.31625
$273$	0	0
$274$	−4.76393	−0.287800
$275$	0	0
$276$	0	0
$277$	12.4721	0.749378	0.374689	−	0.927151i	$-0.377749\pi$
$277$	12.4721	0.749378	0.374689	+	0.927151i	$0.377749\pi$
$278$	1.52786	0.0916352
$279$	0	0
$280$	−9.47214	−0.566068
$281$	16.4164	0.979321	0.489660	−	0.871913i	$-0.337121\pi$
$281$	16.4164	0.979321	0.489660	+	0.871913i	$0.337121\pi$
$282$	0	0
$283$	18.8885	1.12281	0.561404	−	0.827542i	$-0.310262\pi$
$283$	18.8885	1.12281	0.561404	+	0.827542i	$0.310262\pi$
$284$	−0.944272	−0.0560322
$285$	0	0
$286$	0	0
$287$	1.52786	0.0901870
$288$	0	0
$289$	3.00000	0.176471
$290$	−20.5623	−1.20746
$291$	0	0
$292$	−7.56231	−0.442550
$293$	9.88854	0.577695	0.288847	−	0.957375i	$-0.406728\pi$
$293$	9.88854	0.577695	0.288847	+	0.957375i	$0.406728\pi$
$294$	0	0
$295$	6.23607	0.363078
$296$	−7.76393	−0.451269
$297$	0	0
$298$	29.0344	1.68192
$299$	52.8328	3.05540
$300$	0	0
$301$	−10.9443	−0.630817
$302$	−28.1803	−1.62160
$303$	0	0
$304$	14.5623	0.835206
$305$	−14.9443	−0.855707
$306$	0	0
$307$	−16.9443	−0.967061	−0.483530	−	0.875328i	$-0.660646\pi$
$307$	−16.9443	−0.967061	−0.483530	+	0.875328i	$0.660646\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.8885	1.01437	0.507183	−	0.861838i	$-0.330687\pi$
$311$	17.8885	1.01437	0.507183	+	0.861838i	$0.330687\pi$
$312$	0	0
$313$	−20.4721	−1.15715	−0.578577	−	0.815628i	$-0.696392\pi$
$313$	−20.4721	−1.15715	−0.578577	+	0.815628i	$0.696392\pi$
$314$	−21.7082	−1.22506
$315$	0	0
$316$	8.29180	0.466450
$317$	−30.4721	−1.71149	−0.855743	−	0.517401i	$-0.826899\pi$
$317$	−30.4721	−1.71149	−0.855743	+	0.517401i	$0.826899\pi$
$318$	0	0
$319$	0	0
$320$	17.9443	1.00312
$321$	0	0
$322$	−13.7082	−0.763928
$323$	13.4164	0.746509
$324$	0	0
$325$	−80.7214	−4.47762
$326$	11.7984	0.653451
$327$	0	0
$328$	−3.41641	−0.188640
$329$	3.00000	0.165395
$330$	0	0
$331$	24.0000	1.31916	0.659580	−	0.751635i	$-0.270734\pi$
$331$	24.0000	1.31916	0.659580	+	0.751635i	$0.270734\pi$
$332$	3.70820	0.203514
$333$	0	0
$334$	30.6525	1.67723
$335$	−36.8885	−2.01544
$336$	0	0
$337$	−24.8328	−1.35273	−0.676365	−	0.736567i	$-0.736446\pi$
$337$	−24.8328	−1.35273	−0.676365	+	0.736567i	$0.736446\pi$
$338$	41.8885	2.27844
$339$	0	0
$340$	−11.7082	−0.634967
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	24.4721	1.31945
$345$	0	0
$346$	12.1803	0.654819
$347$	−3.05573	−0.164040	−0.0820200	−	0.996631i	$-0.526137\pi$
$347$	−3.05573	−0.164040	−0.0820200	+	0.996631i	$0.526137\pi$
$348$	0	0
$349$	22.7082	1.21554	0.607771	−	0.794112i	$-0.292064\pi$
$349$	22.7082	1.21554	0.607771	+	0.794112i	$0.292064\pi$
$350$	20.9443	1.11952
$351$	0	0
$352$	0	0
$353$	−32.5967	−1.73495	−0.867475	−	0.497481i	$-0.834258\pi$
$353$	−32.5967	−1.73495	−0.867475	+	0.497481i	$0.834258\pi$
$354$	0	0
$355$	−6.47214	−0.343505
$356$	2.76393	0.146488
$357$	0	0
$358$	26.6525	1.40863
$359$	−19.0557	−1.00572	−0.502861	−	0.864367i	$-0.667719\pi$
$359$	−19.0557	−1.00572	−0.502861	+	0.864367i	$0.667719\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−5.52786	−0.290538
$363$	0	0
$364$	−3.85410	−0.202010
$365$	−51.8328	−2.71305
$366$	0	0
$367$	−22.8328	−1.19186	−0.595932	−	0.803035i	$-0.703217\pi$
$367$	−22.8328	−1.19186	−0.595932	+	0.803035i	$0.703217\pi$
$368$	41.1246	2.14377
$369$	0	0
$370$	23.7984	1.23722
$371$	−8.94427	−0.464363
$372$	0	0
$373$	−2.58359	−0.133773	−0.0668867	−	0.997761i	$-0.521307\pi$
$373$	−2.58359	−0.133773	−0.0668867	+	0.997761i	$0.521307\pi$
$374$	0	0
$375$	0	0
$376$	−6.70820	−0.345949
$377$	18.7082	0.963522
$378$	0	0
$379$	−4.23607	−0.217592	−0.108796	−	0.994064i	$-0.534700\pi$
$379$	−4.23607	−0.217592	−0.108796	+	0.994064i	$0.534700\pi$
$380$	−7.85410	−0.402907
$381$	0	0
$382$	−4.76393	−0.243744
$383$	−13.8885	−0.709671	−0.354836	−	0.934929i	$-0.615463\pi$
$383$	−13.8885	−0.709671	−0.354836	+	0.934929i	$0.615463\pi$
$384$	0	0
$385$	0	0
$386$	−19.4164	−0.988269
$387$	0	0
$388$	1.23607	0.0627518
$389$	−28.3607	−1.43794	−0.718972	−	0.695039i	$-0.755387\pi$
$389$	−28.3607	−1.43794	−0.718972	+	0.695039i	$0.755387\pi$
$390$	0	0
$391$	37.8885	1.91611
$392$	−2.23607	−0.112938
$393$	0	0
$394$	−1.70820	−0.0860581
$395$	56.8328	2.85957
$396$	0	0
$397$	12.4721	0.625959	0.312979	−	0.949760i	$-0.398673\pi$
$397$	12.4721	0.625959	0.312979	+	0.949760i	$0.398673\pi$
$398$	−20.1803	−1.01155
$399$	0	0
$400$	−62.8328	−3.14164
$401$	10.3607	0.517388	0.258694	−	0.965959i	$-0.416708\pi$
$401$	10.3607	0.517388	0.258694	+	0.965959i	$0.416708\pi$
$402$	0	0
$403$	0	0
$404$	−10.4721	−0.521008
$405$	0	0
$406$	−4.85410	−0.240905
$407$	0	0
$408$	0	0
$409$	−2.58359	−0.127750	−0.0638752	−	0.997958i	$-0.520346\pi$
$409$	−2.58359	−0.127750	−0.0638752	+	0.997958i	$0.520346\pi$
$410$	10.4721	0.517182
$411$	0	0
$412$	4.00000	0.197066
$413$	1.47214	0.0724391
$414$	0	0
$415$	25.4164	1.24764
$416$	21.0902	1.03403
$417$	0	0
$418$	0	0
$419$	−24.5279	−1.19826	−0.599132	−	0.800650i	$-0.704488\pi$
$419$	−24.5279	−1.19826	−0.599132	+	0.800650i	$0.704488\pi$
$420$	0	0
$421$	−16.4164	−0.800087	−0.400043	−	0.916496i	$-0.631005\pi$
$421$	−16.4164	−0.800087	−0.400043	+	0.916496i	$0.631005\pi$
$422$	2.29180	0.111563
$423$	0	0
$424$	20.0000	0.971286
$425$	−57.8885	−2.80801
$426$	0	0
$427$	−3.52786	−0.170725
$428$	6.03444	0.291686
$429$	0	0
$430$	−75.0132	−3.61746
$431$	−18.5967	−0.895774	−0.447887	−	0.894090i	$-0.647823\pi$
$431$	−18.5967	−0.895774	−0.447887	+	0.894090i	$0.647823\pi$
$432$	0	0
$433$	9.41641	0.452524	0.226262	−	0.974067i	$-0.427349\pi$
$433$	9.41641	0.452524	0.226262	+	0.974067i	$0.427349\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.4164	1.21583
$438$	0	0
$439$	9.36068	0.446761	0.223380	−	0.974731i	$-0.428291\pi$
$439$	9.36068	0.446761	0.223380	+	0.974731i	$0.428291\pi$
$440$	0	0
$441$	0	0
$442$	45.1246	2.14636
$443$	7.88854	0.374796	0.187398	−	0.982284i	$-0.439995\pi$
$443$	7.88854	0.374796	0.187398	+	0.982284i	$0.439995\pi$
$444$	0	0
$445$	18.9443	0.898045
$446$	9.70820	0.459697
$447$	0	0
$448$	4.23607	0.200135
$449$	−28.4721	−1.34368	−0.671842	−	0.740695i	$-0.734496\pi$
$449$	−28.4721	−1.34368	−0.671842	+	0.740695i	$0.734496\pi$
$450$	0	0
$451$	0	0
$452$	4.00000	0.188144
$453$	0	0
$454$	−25.1246	−1.17916
$455$	−26.4164	−1.23842
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−39.5967	−1.85023
$459$	0	0
$460$	−22.1803	−1.03416
$461$	26.9443	1.25492	0.627460	−	0.778649i	$-0.284095\pi$
$461$	26.9443	1.25492	0.627460	+	0.778649i	$0.284095\pi$
$462$	0	0
$463$	−32.1246	−1.49296	−0.746479	−	0.665409i	$-0.768257\pi$
$463$	−32.1246	−1.49296	−0.746479	+	0.665409i	$0.768257\pi$
$464$	14.5623	0.676038
$465$	0	0
$466$	32.1803	1.49073
$467$	−15.4721	−0.715965	−0.357983	−	0.933728i	$-0.616535\pi$
$467$	−15.4721	−0.715965	−0.357983	+	0.933728i	$0.616535\pi$
$468$	0	0
$469$	−8.70820	−0.402107
$470$	20.5623	0.948468
$471$	0	0
$472$	−3.29180	−0.151517
$473$	0	0
$474$	0	0
$475$	−38.8328	−1.78177
$476$	−2.76393	−0.126685
$477$	0	0
$478$	20.5623	0.940498
$479$	−28.4721	−1.30093	−0.650463	−	0.759538i	$-0.725425\pi$
$479$	−28.4721	−1.30093	−0.650463	+	0.759538i	$0.725425\pi$
$480$	0	0
$481$	−21.6525	−0.987268
$482$	−20.5623	−0.936587
$483$	0	0
$484$	0	0
$485$	8.47214	0.384700
$486$	0	0
$487$	13.8885	0.629350	0.314675	−	0.949199i	$-0.398104\pi$
$487$	13.8885	0.629350	0.314675	+	0.949199i	$0.398104\pi$
$488$	7.88854	0.357098
$489$	0	0
$490$	6.85410	0.309637
$491$	20.5967	0.929518	0.464759	−	0.885437i	$-0.346141\pi$
$491$	20.5967	0.929518	0.464759	+	0.885437i	$0.346141\pi$
$492$	0	0
$493$	13.4164	0.604245
$494$	30.2705	1.36193
$495$	0	0
$496$	0	0
$497$	−1.52786	−0.0685341
$498$	0	0
$499$	−23.2918	−1.04268	−0.521342	−	0.853348i	$-0.674568\pi$
$499$	−23.2918	−1.04268	−0.521342	+	0.853348i	$0.674568\pi$
$500$	20.7984	0.930132
$501$	0	0
$502$	26.5623	1.18553
$503$	−19.4164	−0.865735	−0.432867	−	0.901458i	$-0.642498\pi$
$503$	−19.4164	−0.865735	−0.432867	+	0.901458i	$0.642498\pi$
$504$	0	0
$505$	−71.7771	−3.19404
$506$	0	0
$507$	0	0
$508$	−1.23607	−0.0548416
$509$	−26.3607	−1.16842	−0.584208	−	0.811604i	$-0.698595\pi$
$509$	−26.3607	−1.16842	−0.584208	+	0.811604i	$0.698595\pi$
$510$	0	0
$511$	−12.2361	−0.541292
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−0.562306	−0.0248022
$515$	27.4164	1.20811
$516$	0	0
$517$	0	0
$518$	5.61803	0.246842
$519$	0	0
$520$	59.0689	2.59034
$521$	−14.2361	−0.623693	−0.311847	−	0.950132i	$-0.600948\pi$
$521$	−14.2361	−0.623693	−0.311847	+	0.950132i	$0.600948\pi$
$522$	0	0
$523$	3.00000	0.131181	0.0655904	−	0.997847i	$-0.479107\pi$
$523$	3.00000	0.131181	0.0655904	+	0.997847i	$0.479107\pi$
$524$	1.52786	0.0667451
$525$	0	0
$526$	0.381966	0.0166545
$527$	0	0
$528$	0	0
$529$	48.7771	2.12074
$530$	−61.3050	−2.66292
$531$	0	0
$532$	−1.85410	−0.0803855
$533$	−9.52786	−0.412698
$534$	0	0
$535$	41.3607	1.78818
$536$	19.4721	0.841068
$537$	0	0
$538$	31.5967	1.36223
$539$	0	0
$540$	0	0
$541$	−15.4164	−0.662803	−0.331402	−	0.943490i	$-0.607521\pi$
$541$	−15.4164	−0.662803	−0.331402	+	0.943490i	$0.607521\pi$
$542$	5.61803	0.241315
$543$	0	0
$544$	15.1246	0.648462
$545$	0	0
$546$	0	0
$547$	−36.8328	−1.57486	−0.787429	−	0.616406i	$-0.788588\pi$
$547$	−36.8328	−1.57486	−0.787429	+	0.616406i	$0.788588\pi$
$548$	−1.81966	−0.0777320
$549$	0	0
$550$	0	0
$551$	9.00000	0.383413
$552$	0	0
$553$	13.4164	0.570524
$554$	20.1803	0.857380
$555$	0	0
$556$	0.583592	0.0247498
$557$	−39.0000	−1.65248	−0.826242	−	0.563316i	$-0.809525\pi$
$557$	−39.0000	−1.65248	−0.826242	+	0.563316i	$0.809525\pi$
$558$	0	0
$559$	68.2492	2.88663
$560$	−20.5623	−0.868916
$561$	0	0
$562$	26.5623	1.12046
$563$	−2.47214	−0.104188	−0.0520941	−	0.998642i	$-0.516590\pi$
$563$	−2.47214	−0.104188	−0.0520941	+	0.998642i	$0.516590\pi$
$564$	0	0
$565$	27.4164	1.15342
$566$	30.5623	1.28463
$567$	0	0
$568$	3.41641	0.143349
$569$	−8.11146	−0.340050	−0.170025	−	0.985440i	$-0.554385\pi$
$569$	−8.11146	−0.340050	−0.170025	+	0.985440i	$0.554385\pi$
$570$	0	0
$571$	−22.9443	−0.960188	−0.480094	−	0.877217i	$-0.659397\pi$
$571$	−22.9443	−0.960188	−0.480094	+	0.877217i	$0.659397\pi$
$572$	0	0
$573$	0	0
$574$	2.47214	0.103185
$575$	−109.666	−4.57337
$576$	0	0
$577$	−24.0000	−0.999133	−0.499567	−	0.866276i	$-0.666507\pi$
$577$	−24.0000	−0.999133	−0.499567	+	0.866276i	$0.666507\pi$
$578$	4.85410	0.201904
$579$	0	0
$580$	−7.85410	−0.326124
$581$	6.00000	0.248922
$582$	0	0
$583$	0	0
$584$	27.3607	1.13219
$585$	0	0
$586$	16.0000	0.660954
$587$	0.639320	0.0263876	0.0131938	−	0.999913i	$-0.495800\pi$
$587$	0.639320	0.0263876	0.0131938	+	0.999913i	$0.495800\pi$
$588$	0	0
$589$	0	0
$590$	10.0902	0.415406
$591$	0	0
$592$	−16.8541	−0.692699
$593$	−29.7771	−1.22280	−0.611399	−	0.791322i	$-0.709393\pi$
$593$	−29.7771	−1.22280	−0.611399	+	0.791322i	$0.709393\pi$
$594$	0	0
$595$	−18.9443	−0.776639
$596$	11.0902	0.454271
$597$	0	0
$598$	85.4853	3.49575
$599$	−1.41641	−0.0578729	−0.0289364	−	0.999581i	$-0.509212\pi$
$599$	−1.41641	−0.0578729	−0.0289364	+	0.999581i	$0.509212\pi$
$600$	0	0
$601$	−9.18034	−0.374474	−0.187237	−	0.982315i	$-0.559953\pi$
$601$	−9.18034	−0.374474	−0.187237	+	0.982315i	$0.559953\pi$
$602$	−17.7082	−0.721733
$603$	0	0
$604$	−10.7639	−0.437978
$605$	0	0
$606$	0	0
$607$	43.2492	1.75543	0.877716	−	0.479181i	$-0.159066\pi$
$607$	43.2492	1.75543	0.877716	+	0.479181i	$0.159066\pi$
$608$	10.1459	0.411470
$609$	0	0
$610$	−24.1803	−0.979033
$611$	−18.7082	−0.756853
$612$	0	0
$613$	17.5279	0.707944	0.353972	−	0.935256i	$-0.384831\pi$
$613$	17.5279	0.707944	0.353972	+	0.935256i	$0.384831\pi$
$614$	−27.4164	−1.10644
$615$	0	0
$616$	0	0
$617$	29.8885	1.20327	0.601634	−	0.798772i	$-0.294517\pi$
$617$	29.8885	1.20327	0.601634	+	0.798772i	$0.294517\pi$
$618$	0	0
$619$	46.2492	1.85891	0.929457	−	0.368931i	$-0.120276\pi$
$619$	46.2492	1.85891	0.929457	+	0.368931i	$0.120276\pi$
$620$	0	0
$621$	0	0
$622$	28.9443	1.16056
$623$	4.47214	0.179172
$624$	0	0
$625$	77.8328	3.11331
$626$	−33.1246	−1.32393
$627$	0	0
$628$	−8.29180	−0.330879
$629$	−15.5279	−0.619136
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−30.0000	−1.19334
$633$	0	0
$634$	−49.3050	−1.95815
$635$	−8.47214	−0.336206
$636$	0	0
$637$	−6.23607	−0.247082
$638$	0	0
$639$	0	0
$640$	57.6869	2.28028
$641$	38.8328	1.53380	0.766902	−	0.641764i	$-0.221797\pi$
$641$	38.8328	1.53380	0.766902	+	0.641764i	$0.221797\pi$
$642$	0	0
$643$	−1.41641	−0.0558577	−0.0279288	−	0.999610i	$-0.508891\pi$
$643$	−1.41641	−0.0558577	−0.0279288	+	0.999610i	$0.508891\pi$
$644$	−5.23607	−0.206330
$645$	0	0
$646$	21.7082	0.854098
$647$	−4.88854	−0.192188	−0.0960942	−	0.995372i	$-0.530635\pi$
$647$	−4.88854	−0.192188	−0.0960942	+	0.995372i	$0.530635\pi$
$648$	0	0
$649$	0	0
$650$	−130.610	−5.12294
$651$	0	0
$652$	4.50658	0.176491
$653$	44.9443	1.75881	0.879403	−	0.476079i	$-0.157942\pi$
$653$	44.9443	1.75881	0.879403	+	0.476079i	$0.157942\pi$
$654$	0	0
$655$	10.4721	0.409180
$656$	−7.41641	−0.289562
$657$	0	0
$658$	4.85410	0.189233
$659$	25.7639	1.00362	0.501810	−	0.864978i	$-0.332668\pi$
$659$	25.7639	1.00362	0.501810	+	0.864978i	$0.332668\pi$
$660$	0	0
$661$	30.8328	1.19926	0.599629	−	0.800278i	$-0.295315\pi$
$661$	30.8328	1.19926	0.599629	+	0.800278i	$0.295315\pi$
$662$	38.8328	1.50928
$663$	0	0
$664$	−13.4164	−0.520658
$665$	−12.7082	−0.492803
$666$	0	0
$667$	25.4164	0.984127
$668$	11.7082	0.453004
$669$	0	0
$670$	−59.6869	−2.30591
$671$	0	0
$672$	0	0
$673$	−33.3050	−1.28381	−0.641906	−	0.766784i	$-0.721856\pi$
$673$	−33.3050	−1.28381	−0.641906	+	0.766784i	$0.721856\pi$
$674$	−40.1803	−1.54769
$675$	0	0
$676$	16.0000	0.615385
$677$	28.3607	1.08999	0.544995	−	0.838439i	$-0.316532\pi$
$677$	28.3607	1.08999	0.544995	+	0.838439i	$0.316532\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	42.3607	1.62446
$681$	0	0
$682$	0	0
$683$	9.52786	0.364574	0.182287	−	0.983245i	$-0.441650\pi$
$683$	9.52786	0.364574	0.182287	+	0.983245i	$0.441650\pi$
$684$	0	0
$685$	−12.4721	−0.476536
$686$	1.61803	0.0617768
$687$	0	0
$688$	53.1246	2.02536
$689$	55.7771	2.12494
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	4.65248	0.176861
$693$	0	0
$694$	−4.94427	−0.187682
$695$	4.00000	0.151729
$696$	0	0
$697$	−6.83282	−0.258811
$698$	36.7426	1.39073
$699$	0	0
$700$	8.00000	0.302372
$701$	6.94427	0.262282	0.131141	−	0.991364i	$-0.458136\pi$
$701$	6.94427	0.262282	0.131141	+	0.991364i	$0.458136\pi$
$702$	0	0
$703$	−10.4164	−0.392862
$704$	0	0
$705$	0	0
$706$	−52.7426	−1.98500
$707$	−16.9443	−0.637255
$708$	0	0
$709$	11.5836	0.435031	0.217515	−	0.976057i	$-0.430205\pi$
$709$	11.5836	0.435031	0.217515	+	0.976057i	$0.430205\pi$
$710$	−10.4721	−0.393012
$711$	0	0
$712$	−10.0000	−0.374766
$713$	0	0
$714$	0	0
$715$	0	0
$716$	10.1803	0.380457
$717$	0	0
$718$	−30.8328	−1.15067
$719$	24.7771	0.924029	0.462015	−	0.886872i	$-0.347127\pi$
$719$	24.7771	0.924029	0.462015	+	0.886872i	$0.347127\pi$
$720$	0	0
$721$	6.47214	0.241035
$722$	−16.1803	−0.602170
$723$	0	0
$724$	−2.11146	−0.0784717
$725$	−38.8328	−1.44221
$726$	0	0
$727$	44.8328	1.66276	0.831379	−	0.555706i	$-0.187552\pi$
$727$	44.8328	1.66276	0.831379	+	0.555706i	$0.187552\pi$
$728$	13.9443	0.516809
$729$	0	0
$730$	−83.8673	−3.10407
$731$	48.9443	1.81027
$732$	0	0
$733$	−24.4721	−0.903899	−0.451949	−	0.892044i	$-0.649271\pi$
$733$	−24.4721	−0.903899	−0.451949	+	0.892044i	$0.649271\pi$
$734$	−36.9443	−1.36364
$735$	0	0
$736$	28.6525	1.05614
$737$	0	0
$738$	0	0
$739$	−51.3050	−1.88728	−0.943642	−	0.330969i	$-0.892624\pi$
$739$	−51.3050	−1.88728	−0.943642	+	0.330969i	$0.892624\pi$
$740$	9.09017	0.334161
$741$	0	0
$742$	−14.4721	−0.531289
$743$	−15.7639	−0.578323	−0.289161	−	0.957280i	$-0.593376\pi$
$743$	−15.7639	−0.578323	−0.289161	+	0.957280i	$0.593376\pi$
$744$	0	0
$745$	76.0132	2.78491
$746$	−4.18034	−0.153053
$747$	0	0
$748$	0	0
$749$	9.76393	0.356766
$750$	0	0
$751$	−18.2361	−0.665444	−0.332722	−	0.943025i	$-0.607967\pi$
$751$	−18.2361	−0.665444	−0.332722	+	0.943025i	$0.607967\pi$
$752$	−14.5623	−0.531033
$753$	0	0
$754$	30.2705	1.10239
$755$	−73.7771	−2.68502
$756$	0	0
$757$	24.4164	0.887429	0.443715	−	0.896168i	$-0.353660\pi$
$757$	24.4164	0.887429	0.443715	+	0.896168i	$0.353660\pi$
$758$	−6.85410	−0.248952
$759$	0	0
$760$	28.4164	1.03077
$761$	6.36068	0.230574	0.115287	−	0.993332i	$-0.463221\pi$
$761$	6.36068	0.230574	0.115287	+	0.993332i	$0.463221\pi$
$762$	0	0
$763$	0	0
$764$	−1.81966	−0.0658330
$765$	0	0
$766$	−22.4721	−0.811951
$767$	−9.18034	−0.331483
$768$	0	0
$769$	14.1246	0.509347	0.254673	−	0.967027i	$-0.418032\pi$
$769$	14.1246	0.509347	0.254673	+	0.967027i	$0.418032\pi$
$770$	0	0
$771$	0	0
$772$	−7.41641	−0.266922
$773$	11.2918	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$773$	11.2918	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$774$	0	0
$775$	0	0
$776$	−4.47214	−0.160540
$777$	0	0
$778$	−45.8885	−1.64518
$779$	−4.58359	−0.164224
$780$	0	0
$781$	0	0
$782$	61.3050	2.19226
$783$	0	0
$784$	−4.85410	−0.173361
$785$	−56.8328	−2.02845
$786$	0	0
$787$	27.0000	0.962446	0.481223	−	0.876598i	$-0.340193\pi$
$787$	27.0000	0.962446	0.481223	+	0.876598i	$0.340193\pi$
$788$	−0.652476	−0.0232435
$789$	0	0
$790$	91.9574	3.27170
$791$	6.47214	0.230123
$792$	0	0
$793$	22.0000	0.781243
$794$	20.1803	0.716173
$795$	0	0
$796$	−7.70820	−0.273210
$797$	14.8197	0.524939	0.262470	−	0.964940i	$-0.415463\pi$
$797$	14.8197	0.524939	0.262470	+	0.964940i	$0.415463\pi$
$798$	0	0
$799$	−13.4164	−0.474638
$800$	−43.7771	−1.54775
$801$	0	0
$802$	16.7639	0.591955
$803$	0	0
$804$	0	0
$805$	−35.8885	−1.26490
$806$	0	0
$807$	0	0
$808$	37.8885	1.33291
$809$	−31.3607	−1.10258	−0.551291	−	0.834313i	$-0.685865\pi$
$809$	−31.3607	−1.10258	−0.551291	+	0.834313i	$0.685865\pi$
$810$	0	0
$811$	−55.8328	−1.96056	−0.980278	−	0.197625i	$-0.936677\pi$
$811$	−55.8328	−1.96056	−0.980278	+	0.197625i	$0.936677\pi$
$812$	−1.85410	−0.0650662
$813$	0	0
$814$	0	0
$815$	30.8885	1.08198
$816$	0	0
$817$	32.8328	1.14867
$818$	−4.18034	−0.146162
$819$	0	0
$820$	4.00000	0.139686
$821$	−17.9443	−0.626259	−0.313130	−	0.949710i	$-0.601377\pi$
$821$	−17.9443	−0.626259	−0.313130	+	0.949710i	$0.601377\pi$
$822$	0	0
$823$	−15.1803	−0.529153	−0.264577	−	0.964365i	$-0.585232\pi$
$823$	−15.1803	−0.529153	−0.264577	+	0.964365i	$0.585232\pi$
$824$	−14.4721	−0.504161
$825$	0	0
$826$	2.38197	0.0828792
$827$	−20.1246	−0.699801	−0.349901	−	0.936787i	$-0.613785\pi$
$827$	−20.1246	−0.699801	−0.349901	+	0.936787i	$0.613785\pi$
$828$	0	0
$829$	−38.8328	−1.34872	−0.674360	−	0.738403i	$-0.735580\pi$
$829$	−38.8328	−1.34872	−0.674360	+	0.738403i	$0.735580\pi$
$830$	41.1246	1.42746
$831$	0	0
$832$	−26.4164	−0.915824
$833$	−4.47214	−0.154950
$834$	0	0
$835$	80.2492	2.77714
$836$	0	0
$837$	0	0
$838$	−39.6869	−1.37096
$839$	6.05573	0.209067	0.104533	−	0.994521i	$-0.466665\pi$
$839$	6.05573	0.209067	0.104533	+	0.994521i	$0.466665\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−26.5623	−0.915398
$843$	0	0
$844$	0.875388	0.0301321
$845$	109.666	3.77261
$846$	0	0
$847$	0	0
$848$	43.4164	1.49093
$849$	0	0
$850$	−93.6656	−3.21270
$851$	−29.4164	−1.00838
$852$	0	0
$853$	−22.5836	−0.773247	−0.386624	−	0.922238i	$-0.626359\pi$
$853$	−22.5836	−0.773247	−0.386624	+	0.922238i	$0.626359\pi$
$854$	−5.70820	−0.195331
$855$	0	0
$856$	−21.8328	−0.746230
$857$	−28.4721	−0.972590	−0.486295	−	0.873795i	$-0.661652\pi$
$857$	−28.4721	−0.972590	−0.486295	+	0.873795i	$0.661652\pi$
$858$	0	0
$859$	37.7771	1.28894	0.644469	−	0.764631i	$-0.277079\pi$
$859$	37.7771	1.28894	0.644469	+	0.764631i	$0.277079\pi$
$860$	−28.6525	−0.977041
$861$	0	0
$862$	−30.0902	−1.02488
$863$	−7.63932	−0.260045	−0.130023	−	0.991511i	$-0.541505\pi$
$863$	−7.63932	−0.260045	−0.130023	+	0.991511i	$0.541505\pi$
$864$	0	0
$865$	31.8885	1.08424
$866$	15.2361	0.517743
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	54.3050	1.84005
$872$	0	0
$873$	0	0
$874$	41.1246	1.39106
$875$	33.6525	1.13766
$876$	0	0
$877$	2.83282	0.0956574	0.0478287	−	0.998856i	$-0.484770\pi$
$877$	2.83282	0.0956574	0.0478287	+	0.998856i	$0.484770\pi$
$878$	15.1459	0.511149
$879$	0	0
$880$	0	0
$881$	26.4853	0.892312	0.446156	−	0.894955i	$-0.352793\pi$
$881$	26.4853	0.892312	0.446156	+	0.894955i	$0.352793\pi$
$882$	0	0
$883$	40.7082	1.36994	0.684970	−	0.728571i	$-0.259815\pi$
$883$	40.7082	1.36994	0.684970	+	0.728571i	$0.259815\pi$
$884$	17.2361	0.579712
$885$	0	0
$886$	12.7639	0.428813
$887$	7.41641	0.249019	0.124509	−	0.992218i	$-0.460264\pi$
$887$	7.41641	0.249019	0.124509	+	0.992218i	$0.460264\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	30.6525	1.02747
$891$	0	0
$892$	3.70820	0.124160
$893$	−9.00000	−0.301174
$894$	0	0
$895$	69.7771	2.33239
$896$	13.6180	0.454947
$897$	0	0
$898$	−46.0689	−1.53734
$899$	0	0
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	0	0
$904$	−14.4721	−0.481336
$905$	−14.4721	−0.481070
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−9.59675	−0.318479
$909$	0	0
$910$	−42.7426	−1.41690
$911$	21.3050	0.705865	0.352932	−	0.935649i	$-0.385185\pi$
$911$	21.3050	0.705865	0.352932	+	0.935649i	$0.385185\pi$
$912$	0	0
$913$	0	0
$914$	−42.0689	−1.39151
$915$	0	0
$916$	−15.1246	−0.499731
$917$	2.47214	0.0816371
$918$	0	0
$919$	23.0557	0.760538	0.380269	−	0.924876i	$-0.375831\pi$
$919$	23.0557	0.760538	0.380269	+	0.924876i	$0.375831\pi$
$920$	80.2492	2.64574
$921$	0	0
$922$	43.5967	1.43578
$923$	9.52786	0.313613
$924$	0	0
$925$	44.9443	1.47776
$926$	−51.9787	−1.70813
$927$	0	0
$928$	10.1459	0.333055
$929$	29.0689	0.953719	0.476860	−	0.878979i	$-0.341775\pi$
$929$	29.0689	0.953719	0.476860	+	0.878979i	$0.341775\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	12.2918	0.402631
$933$	0	0
$934$	−25.0344	−0.819152
$935$	0	0
$936$	0	0
$937$	29.4164	0.960992	0.480496	−	0.876997i	$-0.340457\pi$
$937$	29.4164	0.960992	0.480496	+	0.876997i	$0.340457\pi$
$938$	−14.0902	−0.460060
$939$	0	0
$940$	7.85410	0.256173
$941$	−22.3607	−0.728937	−0.364469	−	0.931216i	$-0.618749\pi$
$941$	−22.3607	−0.728937	−0.364469	+	0.931216i	$0.618749\pi$
$942$	0	0
$943$	−12.9443	−0.421523
$944$	−7.14590	−0.232579
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	76.3050	2.47696
$950$	−62.8328	−2.03857
$951$	0	0
$952$	10.0000	0.324102
$953$	−11.3607	−0.368009	−0.184004	−	0.982925i	$-0.558906\pi$
$953$	−11.3607	−0.368009	−0.184004	+	0.982925i	$0.558906\pi$
$954$	0	0
$955$	−12.4721	−0.403589
$956$	7.85410	0.254020
$957$	0	0
$958$	−46.0689	−1.48842
$959$	−2.94427	−0.0950755
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−35.0344	−1.12956
$963$	0	0
$964$	−7.85410	−0.252964
$965$	−50.8328	−1.63637
$966$	0	0
$967$	29.5279	0.949552	0.474776	−	0.880107i	$-0.342529\pi$
$967$	29.5279	0.949552	0.474776	+	0.880107i	$0.342529\pi$
$968$	0	0
$969$	0	0
$970$	13.7082	0.440144
$971$	23.4721	0.753257	0.376628	−	0.926364i	$-0.377083\pi$
$971$	23.4721	0.753257	0.376628	+	0.926364i	$0.377083\pi$
$972$	0	0
$973$	0.944272	0.0302720
$974$	22.4721	0.720054
$975$	0	0
$976$	17.1246	0.548145
$977$	−57.7771	−1.84845	−0.924226	−	0.381845i	$-0.875289\pi$
$977$	−57.7771	−1.84845	−0.924226	+	0.381845i	$0.875289\pi$
$978$	0	0
$979$	0	0
$980$	2.61803	0.0836300
$981$	0	0
$982$	33.3262	1.06348
$983$	−41.8885	−1.33604	−0.668019	−	0.744145i	$-0.732857\pi$
$983$	−41.8885	−1.33604	−0.668019	+	0.744145i	$0.732857\pi$
$984$	0	0
$985$	−4.47214	−0.142494
$986$	21.7082	0.691330
$987$	0	0
$988$	11.5623	0.367846
$989$	92.7214	2.94837
$990$	0	0
$991$	−54.2361	−1.72287	−0.861433	−	0.507872i	$-0.830432\pi$
$991$	−54.2361	−1.72287	−0.861433	+	0.507872i	$0.830432\pi$
$992$	0	0
$993$	0	0
$994$	−2.47214	−0.0784114
$995$	−52.8328	−1.67491
$996$	0	0
$997$	10.5836	0.335186	0.167593	−	0.985856i	$-0.446401\pi$
$997$	10.5836	0.335186	0.167593	+	0.985856i	$0.446401\pi$
$998$	−37.6869	−1.19296
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bq.1.2		2
3.2	odd	2		2541.2.a.r.1.1	✓	2
11.10	odd	2		7623.2.a.bb.1.1		2
33.32	even	2		2541.2.a.ba.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.r.1.1	✓	2	3.2	odd	2
2541.2.a.ba.1.2	yes	2	33.32	even	2
7623.2.a.bb.1.1		2	11.10	odd	2
7623.2.a.bq.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +6.85410 q^{10} -6.23607 q^{13} +1.61803 q^{14} -4.85410 q^{16} -4.47214 q^{17} -3.00000 q^{19} +2.61803 q^{20} -8.47214 q^{23} +12.9443 q^{25} -10.0902 q^{26} +0.618034 q^{28} -3.00000 q^{29} -3.38197 q^{32} -7.23607 q^{34} +4.23607 q^{35} +3.47214 q^{37} -4.85410 q^{38} -9.47214 q^{40} +1.52786 q^{41} -10.9443 q^{43} -13.7082 q^{46} +3.00000 q^{47} +1.00000 q^{49} +20.9443 q^{50} -3.85410 q^{52} -8.94427 q^{53} -2.23607 q^{56} -4.85410 q^{58} +1.47214 q^{59} -3.52786 q^{61} +4.23607 q^{64} -26.4164 q^{65} -8.70820 q^{67} -2.76393 q^{68} +6.85410 q^{70} -1.52786 q^{71} -12.2361 q^{73} +5.61803 q^{74} -1.85410 q^{76} +13.4164 q^{79} -20.5623 q^{80} +2.47214 q^{82} +6.00000 q^{83} -18.9443 q^{85} -17.7082 q^{86} +4.47214 q^{89} -6.23607 q^{91} -5.23607 q^{92} +4.85410 q^{94} -12.7082 q^{95} +2.00000 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 4 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 + 4 * q^5 + 2 * q^7	\( 2 q + q^{2} - q^{4} + 4 q^{5} + 2 q^{7} + 7 q^{10} - 8 q^{13} + q^{14} - 3 q^{16} - 6 q^{19} + 3 q^{20} - 8 q^{23} + 8 q^{25} - 9 q^{26} - q^{28} - 6 q^{29} - 9 q^{32} - 10 q^{34} + 4 q^{35} - 2 q^{37} - 3 q^{38} - 10 q^{40} + 12 q^{41} - 4 q^{43} - 14 q^{46} + 6 q^{47} + 2 q^{49} + 24 q^{50} - q^{52} - 3 q^{58} - 6 q^{59} - 16 q^{61} + 4 q^{64} - 26 q^{65} - 4 q^{67} - 10 q^{68} + 7 q^{70} - 12 q^{71} - 20 q^{73} + 9 q^{74} + 3 q^{76} - 21 q^{80} - 4 q^{82} + 12 q^{83} - 20 q^{85} - 22 q^{86} - 8 q^{91} - 6 q^{92} + 3 q^{94} - 12 q^{95} + 4 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 4 * q^5 + 2 * q^7 + 7 * q^10 - 8 * q^13 + q^14 - 3 * q^16 - 6 * q^19 + 3 * q^20 - 8 * q^23 + 8 * q^25 - 9 * q^26 - q^28 - 6 * q^29 - 9 * q^32 - 10 * q^34 + 4 * q^35 - 2 * q^37 - 3 * q^38 - 10 * q^40 + 12 * q^41 - 4 * q^43 - 14 * q^46 + 6 * q^47 + 2 * q^49 + 24 * q^50 - q^52 - 3 * q^58 - 6 * q^59 - 16 * q^61 + 4 * q^64 - 26 * q^65 - 4 * q^67 - 10 * q^68 + 7 * q^70 - 12 * q^71 - 20 * q^73 + 9 * q^74 + 3 * q^76 - 21 * q^80 - 4 * q^82 + 12 * q^83 - 20 * q^85 - 22 * q^86 - 8 * q^91 - 6 * q^92 + 3 * q^94 - 12 * q^95 + 4 * q^97 + q^98

Introduction

L-functions

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Varieties

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