LMFDB - Embedded newform 7623.2.a.bc.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(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ 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)

$f(q)$	$=$	$q+1.30278 q^{2} -0.302776 q^{4} +3.60555 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$	$q+1.30278 q^{2} -0.302776 q^{4} +3.60555 q^{5} -1.00000 q^{7} -3.00000 q^{8} +4.69722 q^{10} -0.605551 q^{13} -1.30278 q^{14} -3.30278 q^{16} -6.30278 q^{17} -3.00000 q^{19} -1.09167 q^{20} +6.30278 q^{23} +8.00000 q^{25} -0.788897 q^{26} +0.302776 q^{28} -8.30278 q^{29} +1.00000 q^{31} +1.69722 q^{32} -8.21110 q^{34} -3.60555 q^{35} +9.21110 q^{37} -3.90833 q^{38} -10.8167 q^{40} -7.00000 q^{41} +5.30278 q^{43} +8.21110 q^{46} -8.90833 q^{47} +1.00000 q^{49} +10.4222 q^{50} +0.183346 q^{52} +2.09167 q^{53} +3.00000 q^{56} -10.8167 q^{58} -10.3028 q^{59} -0.697224 q^{61} +1.30278 q^{62} +8.81665 q^{64} -2.18335 q^{65} -9.51388 q^{67} +1.90833 q^{68} -4.69722 q^{70} +0.697224 q^{71} -5.00000 q^{73} +12.0000 q^{74} +0.908327 q^{76} -4.69722 q^{79} -11.9083 q^{80} -9.11943 q^{82} -3.00000 q^{83} -22.7250 q^{85} +6.90833 q^{86} -17.7250 q^{89} +0.605551 q^{91} -1.90833 q^{92} -11.6056 q^{94} -10.8167 q^{95} +3.60555 q^{97} +1.30278 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8} + 13 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 9 q^{23} + 16 q^{25} - 16 q^{26} - 3 q^{28} - 13 q^{29} + 2 q^{31} + 7 q^{32} - 2 q^{34} + 4 q^{37} + 3 q^{38} - 14 q^{41} + 7 q^{43} + 2 q^{46} - 7 q^{47} + 2 q^{49} - 8 q^{50} + 22 q^{52} + 15 q^{53} + 6 q^{56} - 17 q^{59} - 5 q^{61} - q^{62} - 4 q^{64} - 26 q^{65} - q^{67} - 7 q^{68} - 13 q^{70} + 5 q^{71} - 10 q^{73} + 24 q^{74} - 9 q^{76} - 13 q^{79} - 13 q^{80} + 7 q^{82} - 6 q^{83} - 13 q^{85} + 3 q^{86} - 3 q^{89} - 6 q^{91} + 7 q^{92} - 16 q^{94} - q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8 + 13 * q^10 + 6 * q^13 + q^14 - 3 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 9 * q^23 + 16 * q^25 - 16 * q^26 - 3 * q^28 - 13 * q^29 + 2 * q^31 + 7 * q^32 - 2 * q^34 + 4 * q^37 + 3 * q^38 - 14 * q^41 + 7 * q^43 + 2 * q^46 - 7 * q^47 + 2 * q^49 - 8 * q^50 + 22 * q^52 + 15 * q^53 + 6 * q^56 - 17 * q^59 - 5 * q^61 - q^62 - 4 * q^64 - 26 * q^65 - q^67 - 7 * q^68 - 13 * q^70 + 5 * q^71 - 10 * q^73 + 24 * q^74 - 9 * q^76 - 13 * q^79 - 13 * q^80 + 7 * q^82 - 6 * q^83 - 13 * q^85 + 3 * q^86 - 3 * q^89 - 6 * q^91 + 7 * q^92 - 16 * q^94 - 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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	3.60555	1.61245	0.806226	−	0.591608i	$-0.201507\pi$
$5$	3.60555	1.61245	0.806226	+	0.591608i	$0.201507\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	4.69722	1.48539
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.605551	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551	−0.167950	−0.0839749	+	0.996468i	$0.526762\pi$
$14$	−1.30278	−0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−6.30278	−1.52865	−0.764324	−	0.644833i	$-0.776927\pi$
$17$	−6.30278	−1.52865	−0.764324	+	0.644833i	$0.776927\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−1.09167	−0.244106
$21$	0	0
$22$	0	0
$23$	6.30278	1.31422	0.657110	−	0.753795i	$-0.271779\pi$
$23$	6.30278	1.31422	0.657110	+	0.753795i	$0.271779\pi$
$24$	0	0
$25$	8.00000	1.60000
$26$	−0.788897	−0.154716
$27$	0	0
$28$	0.302776	0.0572192
$29$	−8.30278	−1.54179	−0.770893	−	0.636964i	$-0.780190\pi$
$29$	−8.30278	−1.54179	−0.770893	+	0.636964i	$0.780190\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.69722	0.300030
$33$	0	0
$34$	−8.21110	−1.40819
$35$	−3.60555	−0.609449
$36$	0	0
$37$	9.21110	1.51430	0.757148	−	0.653243i	$-0.226592\pi$
$37$	9.21110	1.51430	0.757148	+	0.653243i	$0.226592\pi$
$38$	−3.90833	−0.634014
$39$	0	0
$40$	−10.8167	−1.71026
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	5.30278	0.808666	0.404333	−	0.914612i	$-0.367504\pi$
$43$	5.30278	0.808666	0.404333	+	0.914612i	$0.367504\pi$
$44$	0	0
$45$	0	0
$46$	8.21110	1.21066
$47$	−8.90833	−1.29941	−0.649707	−	0.760185i	$-0.725108\pi$
$47$	−8.90833	−1.29941	−0.649707	+	0.760185i	$0.725108\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.4222	1.47392
$51$	0	0
$52$	0.183346	0.0254255
$53$	2.09167	0.287313	0.143657	−	0.989628i	$-0.454114\pi$
$53$	2.09167	0.287313	0.143657	+	0.989628i	$0.454114\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	−10.8167	−1.42030
$59$	−10.3028	−1.34131	−0.670654	−	0.741771i	$-0.733986\pi$
$59$	−10.3028	−1.34131	−0.670654	+	0.741771i	$0.733986\pi$
$60$	0	0
$61$	−0.697224	−0.0892704	−0.0446352	−	0.999003i	$-0.514213\pi$
$61$	−0.697224	−0.0892704	−0.0446352	+	0.999003i	$0.514213\pi$
$62$	1.30278	0.165453
$63$	0	0
$64$	8.81665	1.10208
$65$	−2.18335	−0.270811
$66$	0	0
$67$	−9.51388	−1.16231	−0.581153	−	0.813795i	$-0.697398\pi$
$67$	−9.51388	−1.16231	−0.581153	+	0.813795i	$0.697398\pi$
$68$	1.90833	0.231419
$69$	0	0
$70$	−4.69722	−0.561426
$71$	0.697224	0.0827453	0.0413727	−	0.999144i	$-0.486827\pi$
$71$	0.697224	0.0827453	0.0413727	+	0.999144i	$0.486827\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	0.908327	0.104192
$77$	0	0
$78$	0	0
$79$	−4.69722	−0.528479	−0.264240	−	0.964457i	$-0.585121\pi$
$79$	−4.69722	−0.528479	−0.264240	+	0.964457i	$0.585121\pi$
$80$	−11.9083	−1.33139
$81$	0	0
$82$	−9.11943	−1.00707
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	−22.7250	−2.46487
$86$	6.90833	0.744944
$87$	0	0
$88$	0	0
$89$	−17.7250	−1.87884	−0.939422	−	0.342762i	$-0.888637\pi$
$89$	−17.7250	−1.87884	−0.939422	+	0.342762i	$0.888637\pi$
$90$	0	0
$91$	0.605551	0.0634790
$92$	−1.90833	−0.198957
$93$	0	0
$94$	−11.6056	−1.19702
$95$	−10.8167	−1.10977
$96$	0	0
$97$	3.60555	0.366088	0.183044	−	0.983105i	$-0.441405\pi$
$97$	3.60555	0.366088	0.183044	+	0.983105i	$0.441405\pi$
$98$	1.30278	0.131600
$99$	0	0
$100$	−2.42221	−0.242221
$101$	1.69722	0.168880	0.0844401	−	0.996429i	$-0.473090\pi$
$101$	1.69722	0.168880	0.0844401	+	0.996429i	$0.473090\pi$
$102$	0	0
$103$	−10.6972	−1.05403	−0.527014	−	0.849856i	$-0.676689\pi$
$103$	−10.6972	−1.05403	−0.527014	+	0.849856i	$0.676689\pi$
$104$	1.81665	0.178138
$105$	0	0
$106$	2.72498	0.264674
$107$	−19.6056	−1.89534	−0.947670	−	0.319251i	$-0.896569\pi$
$107$	−19.6056	−1.89534	−0.947670	+	0.319251i	$0.896569\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	3.30278	0.312083
$113$	14.3028	1.34549	0.672746	−	0.739874i	$-0.265115\pi$
$113$	14.3028	1.34549	0.672746	+	0.739874i	$0.265115\pi$
$114$	0	0
$115$	22.7250	2.11912
$116$	2.51388	0.233408
$117$	0	0
$118$	−13.4222	−1.23561
$119$	6.30278	0.577774
$120$	0	0
$121$	0	0
$122$	−0.908327	−0.0822361
$123$	0	0
$124$	−0.302776	−0.0271901
$125$	10.8167	0.967471
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	8.09167	0.715210
$129$	0	0
$130$	−2.84441	−0.249471
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	−12.3944	−1.07072
$135$	0	0
$136$	18.9083	1.62138
$137$	−15.1194	−1.29174	−0.645870	−	0.763447i	$-0.723505\pi$
$137$	−15.1194	−1.29174	−0.645870	+	0.763447i	$0.723505\pi$
$138$	0	0
$139$	14.3944	1.22092	0.610461	−	0.792047i	$-0.290984\pi$
$139$	14.3944	1.22092	0.610461	+	0.792047i	$0.290984\pi$
$140$	1.09167	0.0922632
$141$	0	0
$142$	0.908327	0.0762251
$143$	0	0
$144$	0	0
$145$	−29.9361	−2.48606
$146$	−6.51388	−0.539092
$147$	0	0
$148$	−2.78890	−0.229246
$149$	−5.90833	−0.484029	−0.242015	−	0.970273i	$-0.577808\pi$
$149$	−5.90833	−0.484029	−0.242015	+	0.970273i	$0.577808\pi$
$150$	0	0
$151$	14.2111	1.15648	0.578242	−	0.815866i	$-0.303739\pi$
$151$	14.2111	1.15648	0.578242	+	0.815866i	$0.303739\pi$
$152$	9.00000	0.729996
$153$	0	0
$154$	0	0
$155$	3.60555	0.289605
$156$	0	0
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	−6.11943	−0.486836
$159$	0	0
$160$	6.11943	0.483783
$161$	−6.30278	−0.496728
$162$	0	0
$163$	−18.8167	−1.47383	−0.736917	−	0.675983i	$-0.763719\pi$
$163$	−18.8167	−1.47383	−0.736917	+	0.675983i	$0.763719\pi$
$164$	2.11943	0.165500
$165$	0	0
$166$	−3.90833	−0.303345
$167$	−9.21110	−0.712777	−0.356388	−	0.934338i	$-0.615992\pi$
$167$	−9.21110	−0.712777	−0.356388	+	0.934338i	$0.615992\pi$
$168$	0	0
$169$	−12.6333	−0.971793
$170$	−29.6056	−2.27064
$171$	0	0
$172$	−1.60555	−0.122422
$173$	−2.30278	−0.175077	−0.0875384	−	0.996161i	$-0.527900\pi$
$173$	−2.30278	−0.175077	−0.0875384	+	0.996161i	$0.527900\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	0	0
$177$	0	0
$178$	−23.0917	−1.73079
$179$	13.6056	1.01693	0.508463	−	0.861084i	$-0.330214\pi$
$179$	13.6056	1.01693	0.508463	+	0.861084i	$0.330214\pi$
$180$	0	0
$181$	−10.7889	−0.801932	−0.400966	−	0.916093i	$-0.631326\pi$
$181$	−10.7889	−0.801932	−0.400966	+	0.916093i	$0.631326\pi$
$182$	0.788897	0.0584770
$183$	0	0
$184$	−18.9083	−1.39394
$185$	33.2111	2.44173
$186$	0	0
$187$	0	0
$188$	2.69722	0.196715
$189$	0	0
$190$	−14.0917	−1.02232
$191$	5.18335	0.375054	0.187527	−	0.982259i	$-0.439953\pi$
$191$	5.18335	0.375054	0.187527	+	0.982259i	$0.439953\pi$
$192$	0	0
$193$	23.1194	1.66417	0.832086	−	0.554646i	$-0.187146\pi$
$193$	23.1194	1.66417	0.832086	+	0.554646i	$0.187146\pi$
$194$	4.69722	0.337241
$195$	0	0
$196$	−0.302776	−0.0216268
$197$	3.39445	0.241844	0.120922	−	0.992662i	$-0.461415\pi$
$197$	3.39445	0.241844	0.120922	+	0.992662i	$0.461415\pi$
$198$	0	0
$199$	−9.42221	−0.667922	−0.333961	−	0.942587i	$-0.608385\pi$
$199$	−9.42221	−0.667922	−0.333961	+	0.942587i	$0.608385\pi$
$200$	−24.0000	−1.69706
$201$	0	0
$202$	2.21110	0.155573
$203$	8.30278	0.582741
$204$	0	0
$205$	−25.2389	−1.76276
$206$	−13.9361	−0.970973
$207$	0	0
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	−2.51388	−0.173063	−0.0865313	−	0.996249i	$-0.527578\pi$
$211$	−2.51388	−0.173063	−0.0865313	+	0.996249i	$0.527578\pi$
$212$	−0.633308	−0.0434957
$213$	0	0
$214$	−25.5416	−1.74599
$215$	19.1194	1.30393
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	−10.4222	−0.705881
$219$	0	0
$220$	0	0
$221$	3.81665	0.256736
$222$	0	0
$223$	3.09167	0.207034	0.103517	−	0.994628i	$-0.466990\pi$
$223$	3.09167	0.207034	0.103517	+	0.994628i	$0.466990\pi$
$224$	−1.69722	−0.113401
$225$	0	0
$226$	18.6333	1.23947
$227$	11.5139	0.764203	0.382101	−	0.924120i	$-0.375200\pi$
$227$	11.5139	0.764203	0.382101	+	0.924120i	$0.375200\pi$
$228$	0	0
$229$	4.60555	0.304343	0.152172	−	0.988354i	$-0.451373\pi$
$229$	4.60555	0.304343	0.152172	+	0.988354i	$0.451373\pi$
$230$	29.6056	1.95213
$231$	0	0
$232$	24.9083	1.63531
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−32.1194	−2.09524
$236$	3.11943	0.203058
$237$	0	0
$238$	8.21110	0.532247
$239$	13.3305	0.862280	0.431140	−	0.902285i	$-0.358111\pi$
$239$	13.3305	0.862280	0.431140	+	0.902285i	$0.358111\pi$
$240$	0	0
$241$	18.5139	1.19258	0.596292	−	0.802768i	$-0.296640\pi$
$241$	18.5139	1.19258	0.596292	+	0.802768i	$0.296640\pi$
$242$	0	0
$243$	0	0
$244$	0.211103	0.0135145
$245$	3.60555	0.230350
$246$	0	0
$247$	1.81665	0.115591
$248$	−3.00000	−0.190500
$249$	0	0
$250$	14.0917	0.891236
$251$	−11.6056	−0.732536	−0.366268	−	0.930509i	$-0.619365\pi$
$251$	−11.6056	−0.732536	−0.366268	+	0.930509i	$0.619365\pi$
$252$	0	0
$253$	0	0
$254$	2.60555	0.163487
$255$	0	0
$256$	−7.09167	−0.443230
$257$	−24.5139	−1.52913	−0.764567	−	0.644544i	$-0.777047\pi$
$257$	−24.5139	−1.52913	−0.764567	+	0.644544i	$0.777047\pi$
$258$	0	0
$259$	−9.21110	−0.572350
$260$	0.661064	0.0409975
$261$	0	0
$262$	7.81665	0.482914
$263$	30.2389	1.86461	0.932304	−	0.361676i	$-0.117795\pi$
$263$	30.2389	1.86461	0.932304	+	0.361676i	$0.117795\pi$
$264$	0	0
$265$	7.54163	0.463279
$266$	3.90833	0.239635
$267$	0	0
$268$	2.88057	0.175959
$269$	9.11943	0.556021	0.278011	−	0.960578i	$-0.410325\pi$
$269$	9.11943	0.556021	0.278011	+	0.960578i	$0.410325\pi$
$270$	0	0
$271$	−10.4861	−0.636987	−0.318493	−	0.947925i	$-0.603177\pi$
$271$	−10.4861	−0.636987	−0.318493	+	0.947925i	$0.603177\pi$
$272$	20.8167	1.26220
$273$	0	0
$274$	−19.6972	−1.18995
$275$	0	0
$276$	0	0
$277$	−16.9361	−1.01759	−0.508795	−	0.860888i	$-0.669909\pi$
$277$	−16.9361	−1.01759	−0.508795	+	0.860888i	$0.669909\pi$
$278$	18.7527	1.12471
$279$	0	0
$280$	10.8167	0.646419
$281$	−13.6056	−0.811639	−0.405820	−	0.913953i	$-0.633014\pi$
$281$	−13.6056	−0.811639	−0.405820	+	0.913953i	$0.633014\pi$
$282$	0	0
$283$	11.6056	0.689878	0.344939	−	0.938625i	$-0.387900\pi$
$283$	11.6056	0.689878	0.344939	+	0.938625i	$0.387900\pi$
$284$	−0.211103	−0.0125266
$285$	0	0
$286$	0	0
$287$	7.00000	0.413197
$288$	0	0
$289$	22.7250	1.33676
$290$	−39.0000	−2.29016
$291$	0	0
$292$	1.51388	0.0885930
$293$	16.8167	0.982439	0.491220	−	0.871036i	$-0.336551\pi$
$293$	16.8167	0.982439	0.491220	+	0.871036i	$0.336551\pi$
$294$	0	0
$295$	−37.1472	−2.16279
$296$	−27.6333	−1.60615
$297$	0	0
$298$	−7.69722	−0.445888
$299$	−3.81665	−0.220723
$300$	0	0
$301$	−5.30278	−0.305647
$302$	18.5139	1.06535
$303$	0	0
$304$	9.90833	0.568282
$305$	−2.51388	−0.143944
$306$	0	0
$307$	26.6333	1.52004	0.760022	−	0.649898i	$-0.225188\pi$
$307$	26.6333	1.52004	0.760022	+	0.649898i	$0.225188\pi$
$308$	0	0
$309$	0	0
$310$	4.69722	0.266784
$311$	−13.6056	−0.771500	−0.385750	−	0.922603i	$-0.626057\pi$
$311$	−13.6056	−0.771500	−0.385750	+	0.922603i	$0.626057\pi$
$312$	0	0
$313$	11.6972	0.661166	0.330583	−	0.943777i	$-0.392755\pi$
$313$	11.6972	0.661166	0.330583	+	0.943777i	$0.392755\pi$
$314$	−9.39445	−0.530159
$315$	0	0
$316$	1.42221	0.0800053
$317$	−13.6972	−0.769313	−0.384656	−	0.923060i	$-0.625680\pi$
$317$	−13.6972	−0.769313	−0.384656	+	0.923060i	$0.625680\pi$
$318$	0	0
$319$	0	0
$320$	31.7889	1.77705
$321$	0	0
$322$	−8.21110	−0.457587
$323$	18.9083	1.05209
$324$	0	0
$325$	−4.84441	−0.268720
$326$	−24.5139	−1.35770
$327$	0	0
$328$	21.0000	1.15953
$329$	8.90833	0.491132
$330$	0	0
$331$	2.18335	0.120008	0.0600038	−	0.998198i	$-0.480889\pi$
$331$	2.18335	0.120008	0.0600038	+	0.998198i	$0.480889\pi$
$332$	0.908327	0.0498509
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−34.3028	−1.87416
$336$	0	0
$337$	26.2111	1.42781	0.713905	−	0.700243i	$-0.246925\pi$
$337$	26.2111	1.42781	0.713905	+	0.700243i	$0.246925\pi$
$338$	−16.4584	−0.895217
$339$	0	0
$340$	6.88057	0.373151
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−15.9083	−0.857720
$345$	0	0
$346$	−3.00000	−0.161281
$347$	−2.39445	−0.128541	−0.0642704	−	0.997933i	$-0.520472\pi$
$347$	−2.39445	−0.128541	−0.0642704	+	0.997933i	$0.520472\pi$
$348$	0	0
$349$	−8.30278	−0.444437	−0.222219	−	0.974997i	$-0.571330\pi$
$349$	−8.30278	−0.444437	−0.222219	+	0.974997i	$0.571330\pi$
$350$	−10.4222	−0.557090
$351$	0	0
$352$	0	0
$353$	−15.9083	−0.846715	−0.423357	−	0.905963i	$-0.639149\pi$
$353$	−15.9083	−0.846715	−0.423357	+	0.905963i	$0.639149\pi$
$354$	0	0
$355$	2.51388	0.133423
$356$	5.36669	0.284434
$357$	0	0
$358$	17.7250	0.936794
$359$	−12.9361	−0.682740	−0.341370	−	0.939929i	$-0.610891\pi$
$359$	−12.9361	−0.682740	−0.341370	+	0.939929i	$0.610891\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−14.0555	−0.738741
$363$	0	0
$364$	−0.183346	−0.00960995
$365$	−18.0278	−0.943616
$366$	0	0
$367$	25.6333	1.33805	0.669024	−	0.743241i	$-0.266712\pi$
$367$	25.6333	1.33805	0.669024	+	0.743241i	$0.266712\pi$
$368$	−20.8167	−1.08514
$369$	0	0
$370$	43.2666	2.24932
$371$	−2.09167	−0.108594
$372$	0	0
$373$	5.11943	0.265074	0.132537	−	0.991178i	$-0.457688\pi$
$373$	5.11943	0.265074	0.132537	+	0.991178i	$0.457688\pi$
$374$	0	0
$375$	0	0
$376$	26.7250	1.37824
$377$	5.02776	0.258943
$378$	0	0
$379$	5.81665	0.298781	0.149391	−	0.988778i	$-0.452269\pi$
$379$	5.81665	0.298781	0.149391	+	0.988778i	$0.452269\pi$
$380$	3.27502	0.168005
$381$	0	0
$382$	6.75274	0.345500
$383$	4.63331	0.236751	0.118375	−	0.992969i	$-0.462231\pi$
$383$	4.63331	0.236751	0.118375	+	0.992969i	$0.462231\pi$
$384$	0	0
$385$	0	0
$386$	30.1194	1.53304
$387$	0	0
$388$	−1.09167	−0.0554213
$389$	−13.8167	−0.700532	−0.350266	−	0.936650i	$-0.613909\pi$
$389$	−13.8167	−0.700532	−0.350266	+	0.936650i	$0.613909\pi$
$390$	0	0
$391$	−39.7250	−2.00898
$392$	−3.00000	−0.151523
$393$	0	0
$394$	4.42221	0.222787
$395$	−16.9361	−0.852147
$396$	0	0
$397$	15.7889	0.792422	0.396211	−	0.918159i	$-0.370325\pi$
$397$	15.7889	0.792422	0.396211	+	0.918159i	$0.370325\pi$
$398$	−12.2750	−0.615291
$399$	0	0
$400$	−26.4222	−1.32111
$401$	6.78890	0.339021	0.169511	−	0.985528i	$-0.445781\pi$
$401$	6.78890	0.339021	0.169511	+	0.985528i	$0.445781\pi$
$402$	0	0
$403$	−0.605551	−0.0301647
$404$	−0.513878	−0.0255664
$405$	0	0
$406$	10.8167	0.536822
$407$	0	0
$408$	0	0
$409$	13.3944	0.662313	0.331156	−	0.943576i	$-0.392561\pi$
$409$	13.3944	0.662313	0.331156	+	0.943576i	$0.392561\pi$
$410$	−32.8806	−1.62386
$411$	0	0
$412$	3.23886	0.159567
$413$	10.3028	0.506966
$414$	0	0
$415$	−10.8167	−0.530969
$416$	−1.02776	−0.0503899
$417$	0	0
$418$	0	0
$419$	9.57779	0.467906	0.233953	−	0.972248i	$-0.424834\pi$
$419$	9.57779	0.467906	0.233953	+	0.972248i	$0.424834\pi$
$420$	0	0
$421$	−0.183346	−0.00893575	−0.00446787	−	0.999990i	$-0.501422\pi$
$421$	−0.183346	−0.00893575	−0.00446787	+	0.999990i	$0.501422\pi$
$422$	−3.27502	−0.159425
$423$	0	0
$424$	−6.27502	−0.304742
$425$	−50.4222	−2.44584
$426$	0	0
$427$	0.697224	0.0337411
$428$	5.93608	0.286931
$429$	0	0
$430$	24.9083	1.20119
$431$	−22.1194	−1.06546	−0.532728	−	0.846287i	$-0.678833\pi$
$431$	−22.1194	−1.06546	−0.532728	+	0.846287i	$0.678833\pi$
$432$	0	0
$433$	−32.1472	−1.54490	−0.772448	−	0.635079i	$-0.780968\pi$
$433$	−32.1472	−1.54490	−0.772448	+	0.635079i	$0.780968\pi$
$434$	−1.30278	−0.0625352
$435$	0	0
$436$	2.42221	0.116003
$437$	−18.9083	−0.904508
$438$	0	0
$439$	−4.72498	−0.225511	−0.112756	−	0.993623i	$-0.535968\pi$
$439$	−4.72498	−0.225511	−0.112756	+	0.993623i	$0.535968\pi$
$440$	0	0
$441$	0	0
$442$	4.97224	0.236506
$443$	19.9361	0.947192	0.473596	−	0.880742i	$-0.342956\pi$
$443$	19.9361	0.947192	0.473596	+	0.880742i	$0.342956\pi$
$444$	0	0
$445$	−63.9083	−3.02955
$446$	4.02776	0.190720
$447$	0	0
$448$	−8.81665	−0.416548
$449$	−0.669468	−0.0315941	−0.0157971	−	0.999875i	$-0.505029\pi$
$449$	−0.669468	−0.0315941	−0.0157971	+	0.999875i	$0.505029\pi$
$450$	0	0
$451$	0	0
$452$	−4.33053	−0.203691
$453$	0	0
$454$	15.0000	0.703985
$455$	2.18335	0.102357
$456$	0	0
$457$	−4.11943	−0.192699	−0.0963494	−	0.995348i	$-0.530717\pi$
$457$	−4.11943	−0.192699	−0.0963494	+	0.995348i	$0.530717\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	−6.88057	−0.320808
$461$	−18.9083	−0.880649	−0.440324	−	0.897839i	$-0.645137\pi$
$461$	−18.9083	−0.880649	−0.440324	+	0.897839i	$0.645137\pi$
$462$	0	0
$463$	−9.18335	−0.426786	−0.213393	−	0.976966i	$-0.568452\pi$
$463$	−9.18335	−0.426786	−0.213393	+	0.976966i	$0.568452\pi$
$464$	27.4222	1.27304
$465$	0	0
$466$	−23.4500	−1.08630
$467$	−40.5416	−1.87604	−0.938022	−	0.346577i	$-0.887344\pi$
$467$	−40.5416	−1.87604	−0.938022	+	0.346577i	$0.887344\pi$
$468$	0	0
$469$	9.51388	0.439310
$470$	−41.8444	−1.93014
$471$	0	0
$472$	30.9083	1.42267
$473$	0	0
$474$	0	0
$475$	−24.0000	−1.10120
$476$	−1.90833	−0.0874680
$477$	0	0
$478$	17.3667	0.794334
$479$	−14.6333	−0.668613	−0.334306	−	0.942464i	$-0.608502\pi$
$479$	−14.6333	−0.668613	−0.334306	+	0.942464i	$0.608502\pi$
$480$	0	0
$481$	−5.57779	−0.254326
$482$	24.1194	1.09861
$483$	0	0
$484$	0	0
$485$	13.0000	0.590300
$486$	0	0
$487$	18.8167	0.852664	0.426332	−	0.904567i	$-0.359806\pi$
$487$	18.8167	0.852664	0.426332	+	0.904567i	$0.359806\pi$
$488$	2.09167	0.0946856
$489$	0	0
$490$	4.69722	0.212199
$491$	16.3028	0.735734	0.367867	−	0.929878i	$-0.380088\pi$
$491$	16.3028	0.735734	0.367867	+	0.929878i	$0.380088\pi$
$492$	0	0
$493$	52.3305	2.35685
$494$	2.36669	0.106483
$495$	0	0
$496$	−3.30278	−0.148299
$497$	−0.697224	−0.0312748
$498$	0	0
$499$	−7.90833	−0.354025	−0.177013	−	0.984209i	$-0.556643\pi$
$499$	−7.90833	−0.354025	−0.177013	+	0.984209i	$0.556643\pi$
$500$	−3.27502	−0.146463
$501$	0	0
$502$	−15.1194	−0.674813
$503$	−34.4222	−1.53481	−0.767405	−	0.641163i	$-0.778452\pi$
$503$	−34.4222	−1.53481	−0.767405	+	0.641163i	$0.778452\pi$
$504$	0	0
$505$	6.11943	0.272311
$506$	0	0
$507$	0	0
$508$	−0.605551	−0.0268670
$509$	−18.6972	−0.828740	−0.414370	−	0.910109i	$-0.635998\pi$
$509$	−18.6972	−0.828740	−0.414370	+	0.910109i	$0.635998\pi$
$510$	0	0
$511$	5.00000	0.221187
$512$	−25.4222	−1.12351
$513$	0	0
$514$	−31.9361	−1.40864
$515$	−38.5694	−1.69957
$516$	0	0
$517$	0	0
$518$	−12.0000	−0.527250
$519$	0	0
$520$	6.55004	0.287238
$521$	21.4222	0.938524	0.469262	−	0.883059i	$-0.344520\pi$
$521$	21.4222	0.938524	0.469262	+	0.883059i	$0.344520\pi$
$522$	0	0
$523$	9.54163	0.417227	0.208613	−	0.977998i	$-0.433105\pi$
$523$	9.54163	0.417227	0.208613	+	0.977998i	$0.433105\pi$
$524$	−1.81665	−0.0793609
$525$	0	0
$526$	39.3944	1.71768
$527$	−6.30278	−0.274553
$528$	0	0
$529$	16.7250	0.727173
$530$	9.82506	0.426773
$531$	0	0
$532$	−0.908327	−0.0393810
$533$	4.23886	0.183605
$534$	0	0
$535$	−70.6888	−3.05614
$536$	28.5416	1.23281
$537$	0	0
$538$	11.8806	0.512208
$539$	0	0
$540$	0	0
$541$	−39.6056	−1.70278	−0.851388	−	0.524537i	$-0.824239\pi$
$541$	−39.6056	−1.70278	−0.851388	+	0.524537i	$0.824239\pi$
$542$	−13.6611	−0.586793
$543$	0	0
$544$	−10.6972	−0.458640
$545$	−28.8444	−1.23556
$546$	0	0
$547$	16.9361	0.724135	0.362067	−	0.932152i	$-0.382071\pi$
$547$	16.9361	0.724135	0.362067	+	0.932152i	$0.382071\pi$
$548$	4.57779	0.195554
$549$	0	0
$550$	0	0
$551$	24.9083	1.06113
$552$	0	0
$553$	4.69722	0.199746
$554$	−22.0639	−0.937406
$555$	0	0
$556$	−4.35829	−0.184833
$557$	43.2389	1.83209	0.916045	−	0.401076i	$-0.131364\pi$
$557$	43.2389	1.83209	0.916045	+	0.401076i	$0.131364\pi$
$558$	0	0
$559$	−3.21110	−0.135815
$560$	11.9083	0.503219
$561$	0	0
$562$	−17.7250	−0.747683
$563$	−3.30278	−0.139195	−0.0695977	−	0.997575i	$-0.522172\pi$
$563$	−3.30278	−0.139195	−0.0695977	+	0.997575i	$0.522172\pi$
$564$	0	0
$565$	51.5694	2.16954
$566$	15.1194	0.635517
$567$	0	0
$568$	−2.09167	−0.0877647
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−31.8444	−1.33265	−0.666324	−	0.745663i	$-0.732133\pi$
$571$	−31.8444	−1.33265	−0.666324	+	0.745663i	$0.732133\pi$
$572$	0	0
$573$	0	0
$574$	9.11943	0.380638
$575$	50.4222	2.10275
$576$	0	0
$577$	−12.2111	−0.508355	−0.254177	−	0.967158i	$-0.581805\pi$
$577$	−12.2111	−0.508355	−0.254177	+	0.967158i	$0.581805\pi$
$578$	29.6056	1.23143
$579$	0	0
$580$	9.06392	0.376359
$581$	3.00000	0.124461
$582$	0	0
$583$	0	0
$584$	15.0000	0.620704
$585$	0	0
$586$	21.9083	0.905025
$587$	6.39445	0.263927	0.131964	−	0.991255i	$-0.457872\pi$
$587$	6.39445	0.263927	0.131964	+	0.991255i	$0.457872\pi$
$588$	0	0
$589$	−3.00000	−0.123613
$590$	−48.3944	−1.99237
$591$	0	0
$592$	−30.4222	−1.25034
$593$	−15.3944	−0.632174	−0.316087	−	0.948730i	$-0.602369\pi$
$593$	−15.3944	−0.632174	−0.316087	+	0.948730i	$0.602369\pi$
$594$	0	0
$595$	22.7250	0.931633
$596$	1.78890	0.0732761
$597$	0	0
$598$	−4.97224	−0.203330
$599$	29.2111	1.19353	0.596767	−	0.802415i	$-0.296452\pi$
$599$	29.2111	1.19353	0.596767	+	0.802415i	$0.296452\pi$
$600$	0	0
$601$	15.8167	0.645175	0.322587	−	0.946540i	$-0.395447\pi$
$601$	15.8167	0.645175	0.322587	+	0.946540i	$0.395447\pi$
$602$	−6.90833	−0.281562
$603$	0	0
$604$	−4.30278	−0.175077
$605$	0	0
$606$	0	0
$607$	33.5416	1.36141	0.680706	−	0.732556i	$-0.261673\pi$
$607$	33.5416	1.36141	0.680706	+	0.732556i	$0.261673\pi$
$608$	−5.09167	−0.206495
$609$	0	0
$610$	−3.27502	−0.132602
$611$	5.39445	0.218236
$612$	0	0
$613$	39.0555	1.57744	0.788719	−	0.614754i	$-0.210745\pi$
$613$	39.0555	1.57744	0.788719	+	0.614754i	$0.210745\pi$
$614$	34.6972	1.40027
$615$	0	0
$616$	0	0
$617$	21.6333	0.870924	0.435462	−	0.900207i	$-0.356585\pi$
$617$	21.6333	0.870924	0.435462	+	0.900207i	$0.356585\pi$
$618$	0	0
$619$	18.1833	0.730850	0.365425	−	0.930841i	$-0.380924\pi$
$619$	18.1833	0.730850	0.365425	+	0.930841i	$0.380924\pi$
$620$	−1.09167	−0.0438426
$621$	0	0
$622$	−17.7250	−0.710707
$623$	17.7250	0.710136
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	15.2389	0.609067
$627$	0	0
$628$	2.18335	0.0871250
$629$	−58.0555	−2.31482
$630$	0	0
$631$	41.0555	1.63439	0.817197	−	0.576358i	$-0.195527\pi$
$631$	41.0555	1.63439	0.817197	+	0.576358i	$0.195527\pi$
$632$	14.0917	0.560537
$633$	0	0
$634$	−17.8444	−0.708692
$635$	7.21110	0.286164
$636$	0	0
$637$	−0.605551	−0.0239928
$638$	0	0
$639$	0	0
$640$	29.1749	1.15324
$641$	7.18335	0.283725	0.141863	−	0.989886i	$-0.454691\pi$
$641$	7.18335	0.283725	0.141863	+	0.989886i	$0.454691\pi$
$642$	0	0
$643$	49.2389	1.94179	0.970896	−	0.239503i	$-0.0769846\pi$
$643$	49.2389	1.94179	0.970896	+	0.239503i	$0.0769846\pi$
$644$	1.90833	0.0751986
$645$	0	0
$646$	24.6333	0.969185
$647$	−3.39445	−0.133450	−0.0667248	−	0.997771i	$-0.521255\pi$
$647$	−3.39445	−0.133450	−0.0667248	+	0.997771i	$0.521255\pi$
$648$	0	0
$649$	0	0
$650$	−6.31118	−0.247545
$651$	0	0
$652$	5.69722	0.223121
$653$	−14.8167	−0.579820	−0.289910	−	0.957054i	$-0.593625\pi$
$653$	−14.8167	−0.579820	−0.289910	+	0.957054i	$0.593625\pi$
$654$	0	0
$655$	21.6333	0.845283
$656$	23.1194	0.902662
$657$	0	0
$658$	11.6056	0.452431
$659$	38.6333	1.50494	0.752470	−	0.658627i	$-0.228862\pi$
$659$	38.6333	1.50494	0.752470	+	0.658627i	$0.228862\pi$
$660$	0	0
$661$	−14.6972	−0.571656	−0.285828	−	0.958281i	$-0.592269\pi$
$661$	−14.6972	−0.571656	−0.285828	+	0.958281i	$0.592269\pi$
$662$	2.84441	0.110551
$663$	0	0
$664$	9.00000	0.349268
$665$	10.8167	0.419452
$666$	0	0
$667$	−52.3305	−2.02625
$668$	2.78890	0.107906
$669$	0	0
$670$	−44.6888	−1.72648
$671$	0	0
$672$	0	0
$673$	20.4222	0.787218	0.393609	−	0.919278i	$-0.371226\pi$
$673$	20.4222	0.787218	0.393609	+	0.919278i	$0.371226\pi$
$674$	34.1472	1.31530
$675$	0	0
$676$	3.82506	0.147118
$677$	−6.69722	−0.257395	−0.128698	−	0.991684i	$-0.541080\pi$
$677$	−6.69722	−0.257395	−0.128698	+	0.991684i	$0.541080\pi$
$678$	0	0
$679$	−3.60555	−0.138368
$680$	68.1749	2.61439
$681$	0	0
$682$	0	0
$683$	−19.3028	−0.738600	−0.369300	−	0.929310i	$-0.620403\pi$
$683$	−19.3028	−0.738600	−0.369300	+	0.929310i	$0.620403\pi$
$684$	0	0
$685$	−54.5139	−2.08287
$686$	−1.30278	−0.0497402
$687$	0	0
$688$	−17.5139	−0.667710
$689$	−1.26662	−0.0482542
$690$	0	0
$691$	19.0278	0.723850	0.361925	−	0.932207i	$-0.382120\pi$
$691$	19.0278	0.723850	0.361925	+	0.932207i	$0.382120\pi$
$692$	0.697224	0.0265045
$693$	0	0
$694$	−3.11943	−0.118412
$695$	51.8999	1.96868
$696$	0	0
$697$	44.1194	1.67114
$698$	−10.8167	−0.409416
$699$	0	0
$700$	2.42221	0.0915507
$701$	5.88057	0.222106	0.111053	−	0.993814i	$-0.464578\pi$
$701$	5.88057	0.222106	0.111053	+	0.993814i	$0.464578\pi$
$702$	0	0
$703$	−27.6333	−1.04221
$704$	0	0
$705$	0	0
$706$	−20.7250	−0.779995
$707$	−1.69722	−0.0638307
$708$	0	0
$709$	−14.3305	−0.538194	−0.269097	−	0.963113i	$-0.586725\pi$
$709$	−14.3305	−0.538194	−0.269097	+	0.963113i	$0.586725\pi$
$710$	3.27502	0.122909
$711$	0	0
$712$	53.1749	1.99282
$713$	6.30278	0.236041
$714$	0	0
$715$	0	0
$716$	−4.11943	−0.153950
$717$	0	0
$718$	−16.8528	−0.628941
$719$	29.2389	1.09043	0.545213	−	0.838298i	$-0.316449\pi$
$719$	29.2389	1.09043	0.545213	+	0.838298i	$0.316449\pi$
$720$	0	0
$721$	10.6972	0.398385
$722$	−13.0278	−0.484843
$723$	0	0
$724$	3.26662	0.121403
$725$	−66.4222	−2.46686
$726$	0	0
$727$	−1.11943	−0.0415173	−0.0207587	−	0.999785i	$-0.506608\pi$
$727$	−1.11943	−0.0415173	−0.0207587	+	0.999785i	$0.506608\pi$
$728$	−1.81665	−0.0673297
$729$	0	0
$730$	−23.4861	−0.869260
$731$	−33.4222	−1.23616
$732$	0	0
$733$	−21.2111	−0.783450	−0.391725	−	0.920082i	$-0.628122\pi$
$733$	−21.2111	−0.783450	−0.391725	+	0.920082i	$0.628122\pi$
$734$	33.3944	1.23261
$735$	0	0
$736$	10.6972	0.394305
$737$	0	0
$738$	0	0
$739$	−17.8806	−0.657747	−0.328874	−	0.944374i	$-0.606669\pi$
$739$	−17.8806	−0.657747	−0.328874	+	0.944374i	$0.606669\pi$
$740$	−10.0555	−0.369648
$741$	0	0
$742$	−2.72498	−0.100037
$743$	1.93608	0.0710280	0.0355140	−	0.999369i	$-0.488693\pi$
$743$	1.93608	0.0710280	0.0355140	+	0.999369i	$0.488693\pi$
$744$	0	0
$745$	−21.3028	−0.780473
$746$	6.66947	0.244187
$747$	0	0
$748$	0	0
$749$	19.6056	0.716371
$750$	0	0
$751$	−35.6333	−1.30028	−0.650139	−	0.759815i	$-0.725289\pi$
$751$	−35.6333	−1.30028	−0.650139	+	0.759815i	$0.725289\pi$
$752$	29.4222	1.07292
$753$	0	0
$754$	6.55004	0.238538
$755$	51.2389	1.86477
$756$	0	0
$757$	−2.18335	−0.0793551	−0.0396775	−	0.999213i	$-0.512633\pi$
$757$	−2.18335	−0.0793551	−0.0396775	+	0.999213i	$0.512633\pi$
$758$	7.57779	0.275238
$759$	0	0
$760$	32.4500	1.17708
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	−1.56939	−0.0567786
$765$	0	0
$766$	6.03616	0.218095
$767$	6.23886	0.225272
$768$	0	0
$769$	0.330532	0.0119193	0.00595964	−	0.999982i	$-0.498103\pi$
$769$	0.330532	0.0119193	0.00595964	+	0.999982i	$0.498103\pi$
$770$	0	0
$771$	0	0
$772$	−7.00000	−0.251936
$773$	12.6333	0.454388	0.227194	−	0.973849i	$-0.427045\pi$
$773$	12.6333	0.454388	0.227194	+	0.973849i	$0.427045\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−10.8167	−0.388295
$777$	0	0
$778$	−18.0000	−0.645331
$779$	21.0000	0.752403
$780$	0	0
$781$	0	0
$782$	−51.7527	−1.85067
$783$	0	0
$784$	−3.30278	−0.117956
$785$	−26.0000	−0.927980
$786$	0	0
$787$	−40.5139	−1.44416	−0.722082	−	0.691808i	$-0.756815\pi$
$787$	−40.5139	−1.44416	−0.722082	+	0.691808i	$0.756815\pi$
$788$	−1.02776	−0.0366123
$789$	0	0
$790$	−22.0639	−0.784999
$791$	−14.3028	−0.508548
$792$	0	0
$793$	0.422205	0.0149929
$794$	20.5694	0.729980
$795$	0	0
$796$	2.85281	0.101115
$797$	−14.6333	−0.518338	−0.259169	−	0.965832i	$-0.583449\pi$
$797$	−14.6333	−0.518338	−0.259169	+	0.965832i	$0.583449\pi$
$798$	0	0
$799$	56.1472	1.98634
$800$	13.5778	0.480048
$801$	0	0
$802$	8.84441	0.312307
$803$	0	0
$804$	0	0
$805$	−22.7250	−0.800950
$806$	−0.788897	−0.0277877
$807$	0	0
$808$	−5.09167	−0.179124
$809$	−18.9083	−0.664781	−0.332391	−	0.943142i	$-0.607855\pi$
$809$	−18.9083	−0.664781	−0.332391	+	0.943142i	$0.607855\pi$
$810$	0	0
$811$	16.8444	0.591487	0.295744	−	0.955267i	$-0.404433\pi$
$811$	16.8444	0.591487	0.295744	+	0.955267i	$0.404433\pi$
$812$	−2.51388	−0.0882198
$813$	0	0
$814$	0	0
$815$	−67.8444	−2.37649
$816$	0	0
$817$	−15.9083	−0.556562
$818$	17.4500	0.610124
$819$	0	0
$820$	7.64171	0.266860
$821$	−1.42221	−0.0496353	−0.0248177	−	0.999692i	$-0.507901\pi$
$821$	−1.42221	−0.0496353	−0.0248177	+	0.999692i	$0.507901\pi$
$822$	0	0
$823$	20.7889	0.724655	0.362328	−	0.932051i	$-0.381982\pi$
$823$	20.7889	0.724655	0.362328	+	0.932051i	$0.381982\pi$
$824$	32.0917	1.11797
$825$	0	0
$826$	13.4222	0.467018
$827$	35.8167	1.24547	0.622734	−	0.782434i	$-0.286022\pi$
$827$	35.8167	1.24547	0.622734	+	0.782434i	$0.286022\pi$
$828$	0	0
$829$	45.7250	1.58809	0.794047	−	0.607856i	$-0.207970\pi$
$829$	45.7250	1.58809	0.794047	+	0.607856i	$0.207970\pi$
$830$	−14.0917	−0.489129
$831$	0	0
$832$	−5.33894	−0.185094
$833$	−6.30278	−0.218378
$834$	0	0
$835$	−33.2111	−1.14932
$836$	0	0
$837$	0	0
$838$	12.4777	0.431036
$839$	−27.7889	−0.959379	−0.479690	−	0.877438i	$-0.659251\pi$
$839$	−27.7889	−0.959379	−0.479690	+	0.877438i	$0.659251\pi$
$840$	0	0
$841$	39.9361	1.37711
$842$	−0.238859	−0.00823162
$843$	0	0
$844$	0.761141	0.0261996
$845$	−45.5500	−1.56697
$846$	0	0
$847$	0	0
$848$	−6.90833	−0.237233
$849$	0	0
$850$	−65.6888	−2.25311
$851$	58.0555	1.99012
$852$	0	0
$853$	−13.2750	−0.454528	−0.227264	−	0.973833i	$-0.572978\pi$
$853$	−13.2750	−0.454528	−0.227264	+	0.973833i	$0.572978\pi$
$854$	0.908327	0.0310823
$855$	0	0
$856$	58.8167	2.01031
$857$	40.3583	1.37861	0.689306	−	0.724470i	$-0.257915\pi$
$857$	40.3583	1.37861	0.689306	+	0.724470i	$0.257915\pi$
$858$	0	0
$859$	−21.6972	−0.740300	−0.370150	−	0.928972i	$-0.620694\pi$
$859$	−21.6972	−0.740300	−0.370150	+	0.928972i	$0.620694\pi$
$860$	−5.78890	−0.197400
$861$	0	0
$862$	−28.8167	−0.981499
$863$	−2.57779	−0.0877492	−0.0438746	−	0.999037i	$-0.513970\pi$
$863$	−2.57779	−0.0877492	−0.0438746	+	0.999037i	$0.513970\pi$
$864$	0	0
$865$	−8.30278	−0.282303
$866$	−41.8806	−1.42316
$867$	0	0
$868$	0.302776	0.0102769
$869$	0	0
$870$	0	0
$871$	5.76114	0.195209
$872$	24.0000	0.812743
$873$	0	0
$874$	−24.6333	−0.833234
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−25.0555	−0.846065	−0.423032	−	0.906115i	$-0.639034\pi$
$877$	−25.0555	−0.846065	−0.423032	+	0.906115i	$0.639034\pi$
$878$	−6.15559	−0.207741
$879$	0	0
$880$	0	0
$881$	5.33053	0.179590	0.0897951	−	0.995960i	$-0.471379\pi$
$881$	5.33053	0.179590	0.0897951	+	0.995960i	$0.471379\pi$
$882$	0	0
$883$	24.7250	0.832062	0.416031	−	0.909350i	$-0.363421\pi$
$883$	24.7250	0.832062	0.416031	+	0.909350i	$0.363421\pi$
$884$	−1.15559	−0.0388667
$885$	0	0
$886$	25.9722	0.872555
$887$	−18.8806	−0.633948	−0.316974	−	0.948434i	$-0.602667\pi$
$887$	−18.8806	−0.633948	−0.316974	+	0.948434i	$0.602667\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	−83.2582	−2.79082
$891$	0	0
$892$	−0.936083	−0.0313424
$893$	26.7250	0.894317
$894$	0	0
$895$	49.0555	1.63974
$896$	−8.09167	−0.270324
$897$	0	0
$898$	−0.872167	−0.0291046
$899$	−8.30278	−0.276913
$900$	0	0
$901$	−13.1833	−0.439201
$902$	0	0
$903$	0	0
$904$	−42.9083	−1.42711
$905$	−38.8999	−1.29308
$906$	0	0
$907$	7.60555	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$907$	7.60555	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$908$	−3.48612	−0.115691
$909$	0	0
$910$	2.84441	0.0942913
$911$	−14.4500	−0.478749	−0.239374	−	0.970927i	$-0.576942\pi$
$911$	−14.4500	−0.478749	−0.239374	+	0.970927i	$0.576942\pi$
$912$	0	0
$913$	0	0
$914$	−5.36669	−0.177514
$915$	0	0
$916$	−1.39445	−0.0460739
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−45.4777	−1.50017	−0.750086	−	0.661341i	$-0.769988\pi$
$919$	−45.4777	−1.50017	−0.750086	+	0.661341i	$0.769988\pi$
$920$	−68.1749	−2.24766
$921$	0	0
$922$	−24.6333	−0.811255
$923$	−0.422205	−0.0138971
$924$	0	0
$925$	73.6888	2.42287
$926$	−11.9638	−0.393156
$927$	0	0
$928$	−14.0917	−0.462582
$929$	−15.6333	−0.512912	−0.256456	−	0.966556i	$-0.582555\pi$
$929$	−15.6333	−0.512912	−0.256456	+	0.966556i	$0.582555\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	5.44996	0.178519
$933$	0	0
$934$	−52.8167	−1.72821
$935$	0	0
$936$	0	0
$937$	24.3944	0.796932	0.398466	−	0.917183i	$-0.369543\pi$
$937$	24.3944	0.796932	0.398466	+	0.917183i	$0.369543\pi$
$938$	12.3944	0.404693
$939$	0	0
$940$	9.72498	0.317194
$941$	−8.57779	−0.279628	−0.139814	−	0.990178i	$-0.544650\pi$
$941$	−8.57779	−0.279628	−0.139814	+	0.990178i	$0.544650\pi$
$942$	0	0
$943$	−44.1194	−1.43673
$944$	34.0278	1.10751
$945$	0	0
$946$	0	0
$947$	32.6611	1.06134	0.530671	−	0.847578i	$-0.321940\pi$
$947$	32.6611	1.06134	0.530671	+	0.847578i	$0.321940\pi$
$948$	0	0
$949$	3.02776	0.0982851
$950$	−31.2666	−1.01442
$951$	0	0
$952$	−18.9083	−0.612822
$953$	−30.0278	−0.972694	−0.486347	−	0.873766i	$-0.661671\pi$
$953$	−30.0278	−0.972694	−0.486347	+	0.873766i	$0.661671\pi$
$954$	0	0
$955$	18.6888	0.604756
$956$	−4.03616	−0.130539
$957$	0	0
$958$	−19.0639	−0.615927
$959$	15.1194	0.488232
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−7.26662	−0.234285
$963$	0	0
$964$	−5.60555	−0.180543
$965$	83.3583	2.68340
$966$	0	0
$967$	22.4861	0.723105	0.361552	−	0.932352i	$-0.382247\pi$
$967$	22.4861	0.723105	0.361552	+	0.932352i	$0.382247\pi$
$968$	0	0
$969$	0	0
$970$	16.9361	0.543785
$971$	49.6333	1.59281	0.796404	−	0.604765i	$-0.206733\pi$
$971$	49.6333	1.59281	0.796404	+	0.604765i	$0.206733\pi$
$972$	0	0
$973$	−14.3944	−0.461465
$974$	24.5139	0.785475
$975$	0	0
$976$	2.30278	0.0737101
$977$	−3.97224	−0.127083	−0.0635417	−	0.997979i	$-0.520240\pi$
$977$	−3.97224	−0.127083	−0.0635417	+	0.997979i	$0.520240\pi$
$978$	0	0
$979$	0	0
$980$	−1.09167	−0.0348722
$981$	0	0
$982$	21.2389	0.677759
$983$	41.7250	1.33082	0.665410	−	0.746478i	$-0.268257\pi$
$983$	41.7250	1.33082	0.665410	+	0.746478i	$0.268257\pi$
$984$	0	0
$985$	12.2389	0.389962
$986$	68.1749	2.17113
$987$	0	0
$988$	−0.550039	−0.0174991
$989$	33.4222	1.06276
$990$	0	0
$991$	−11.2750	−0.358163	−0.179081	−	0.983834i	$-0.557313\pi$
$991$	−11.2750	−0.358163	−0.179081	+	0.983834i	$0.557313\pi$
$992$	1.69722	0.0538869
$993$	0	0
$994$	−0.908327	−0.0288104
$995$	−33.9722	−1.07699
$996$	0	0
$997$	49.3028	1.56143	0.780717	−	0.624884i	$-0.214854\pi$
$997$	49.3028	1.56143	0.780717	+	0.624884i	$0.214854\pi$
$998$	−10.3028	−0.326129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bc.1.2		2
3.2	odd	2		847.2.a.g.1.1	yes	2
11.10	odd	2		7623.2.a.bs.1.1		2
21.20	even	2		5929.2.a.p.1.1		2
33.2	even	10		847.2.f.r.323.1		8
33.5	odd	10		847.2.f.o.729.2		8
33.8	even	10		847.2.f.r.372.2		8
33.14	odd	10		847.2.f.o.372.1		8
33.17	even	10		847.2.f.r.729.1		8
33.20	odd	10		847.2.f.o.323.2		8
33.26	odd	10		847.2.f.o.148.1		8
33.29	even	10		847.2.f.r.148.2		8
33.32	even	2		847.2.a.e.1.2	✓	2
231.230	odd	2		5929.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.e.1.2	✓	2	33.32	even	2
847.2.a.g.1.1	yes	2	3.2	odd	2
847.2.f.o.148.1		8	33.26	odd	10
847.2.f.o.323.2		8	33.20	odd	10
847.2.f.o.372.1		8	33.14	odd	10
847.2.f.o.729.2		8	33.5	odd	10
847.2.f.r.148.2		8	33.29	even	10
847.2.f.r.323.1		8	33.2	even	10
847.2.f.r.372.2		8	33.8	even	10
847.2.f.r.729.1		8	33.17	even	10
5929.2.a.k.1.2		2	231.230	odd	2
5929.2.a.p.1.1		2	21.20	even	2
7623.2.a.bc.1.2		2	1.1	even	1	trivial
7623.2.a.bs.1.1		2	11.10	odd	2

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)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -0.302776 q^{4} +3.60555 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.30278 q^{2} -0.302776 q^{4} +3.60555 q^{5} -1.00000 q^{7} -3.00000 q^{8} +4.69722 q^{10} -0.605551 q^{13} -1.30278 q^{14} -3.30278 q^{16} -6.30278 q^{17} -3.00000 q^{19} -1.09167 q^{20} +6.30278 q^{23} +8.00000 q^{25} -0.788897 q^{26} +0.302776 q^{28} -8.30278 q^{29} +1.00000 q^{31} +1.69722 q^{32} -8.21110 q^{34} -3.60555 q^{35} +9.21110 q^{37} -3.90833 q^{38} -10.8167 q^{40} -7.00000 q^{41} +5.30278 q^{43} +8.21110 q^{46} -8.90833 q^{47} +1.00000 q^{49} +10.4222 q^{50} +0.183346 q^{52} +2.09167 q^{53} +3.00000 q^{56} -10.8167 q^{58} -10.3028 q^{59} -0.697224 q^{61} +1.30278 q^{62} +8.81665 q^{64} -2.18335 q^{65} -9.51388 q^{67} +1.90833 q^{68} -4.69722 q^{70} +0.697224 q^{71} -5.00000 q^{73} +12.0000 q^{74} +0.908327 q^{76} -4.69722 q^{79} -11.9083 q^{80} -9.11943 q^{82} -3.00000 q^{83} -22.7250 q^{85} +6.90833 q^{86} -17.7250 q^{89} +0.605551 q^{91} -1.90833 q^{92} -11.6056 q^{94} -10.8167 q^{95} +3.60555 q^{97} +1.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} - 2 q^{7} - 6 q^{8} + 13 q^{10} + 6 q^{13} + q^{14} - 3 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 9 q^{23} + 16 q^{25} - 16 q^{26} - 3 q^{28} - 13 q^{29} + 2 q^{31} + 7 q^{32} - 2 q^{34} + 4 q^{37} + 3 q^{38} - 14 q^{41} + 7 q^{43} + 2 q^{46} - 7 q^{47} + 2 q^{49} - 8 q^{50} + 22 q^{52} + 15 q^{53} + 6 q^{56} - 17 q^{59} - 5 q^{61} - q^{62} - 4 q^{64} - 26 q^{65} - q^{67} - 7 q^{68} - 13 q^{70} + 5 q^{71} - 10 q^{73} + 24 q^{74} - 9 q^{76} - 13 q^{79} - 13 q^{80} + 7 q^{82} - 6 q^{83} - 13 q^{85} + 3 q^{86} - 3 q^{89} - 6 q^{91} + 7 q^{92} - 16 q^{94} - q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^7 - 6 * q^8 + 13 * q^10 + 6 * q^13 + q^14 - 3 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 9 * q^23 + 16 * q^25 - 16 * q^26 - 3 * q^28 - 13 * q^29 + 2 * q^31 + 7 * q^32 - 2 * q^34 + 4 * q^37 + 3 * q^38 - 14 * q^41 + 7 * q^43 + 2 * q^46 - 7 * q^47 + 2 * q^49 - 8 * q^50 + 22 * q^52 + 15 * q^53 + 6 * q^56 - 17 * q^59 - 5 * q^61 - q^62 - 4 * q^64 - 26 * q^65 - q^67 - 7 * q^68 - 13 * q^70 + 5 * q^71 - 10 * q^73 + 24 * q^74 - 9 * q^76 - 13 * q^79 - 13 * q^80 + 7 * q^82 - 6 * q^83 - 13 * q^85 + 3 * q^86 - 3 * q^89 - 6 * q^91 + 7 * q^92 - 16 * q^94 - q^98

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