LMFDB - Embedded newform 7623.2.a.bh.1.1

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q-2.00000 q^{4} -3.60555 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-2.00000 q^{4} -3.60555 q^{5} +1.00000 q^{7} -2.00000 q^{13} +4.00000 q^{16} +3.60555 q^{17} -6.00000 q^{19} +7.21110 q^{20} -7.21110 q^{23} +8.00000 q^{25} -2.00000 q^{28} +7.21110 q^{29} -2.00000 q^{31} -3.60555 q^{35} -2.00000 q^{37} +7.21110 q^{41} +5.00000 q^{43} -3.60555 q^{47} +1.00000 q^{49} +4.00000 q^{52} +10.8167 q^{59} +14.0000 q^{61} -8.00000 q^{64} +7.21110 q^{65} -15.0000 q^{67} -7.21110 q^{68} +14.4222 q^{71} -4.00000 q^{73} +12.0000 q^{76} -14.4222 q^{80} +10.8167 q^{83} -13.0000 q^{85} -10.8167 q^{89} -2.00000 q^{91} +14.4222 q^{92} +21.6333 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 2 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 + 2 * q^7	$ 2 q - 4 q^{4} + 2 q^{7} - 4 q^{13} + 8 q^{16} - 12 q^{19} + 16 q^{25} - 4 q^{28} - 4 q^{31} - 4 q^{37} + 10 q^{43} + 2 q^{49} + 8 q^{52} + 28 q^{61} - 16 q^{64} - 30 q^{67} - 8 q^{73} + 24 q^{76} - 26 q^{85} - 4 q^{91} - 16 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 2 * q^7 - 4 * q^13 + 8 * q^16 - 12 * q^19 + 16 * q^25 - 4 * q^28 - 4 * q^31 - 4 * q^37 + 10 * q^43 + 2 * q^49 + 8 * q^52 + 28 * q^61 - 16 * q^64 - 30 * q^67 - 8 * q^73 + 24 * q^76 - 26 * q^85 - 4 * q^91 - 16 * 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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	−3.60555	−1.61245	−0.806226	−	0.591608i	$-0.798493\pi$
$5$	−3.60555	−1.61245	−0.806226	+	0.591608i	$0.798493\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	3.60555	0.874475	0.437237	−	0.899346i	$-0.355957\pi$
$17$	3.60555	0.874475	0.437237	+	0.899346i	$0.355957\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	7.21110	1.61245
$21$	0	0
$22$	0	0
$23$	−7.21110	−1.50362	−0.751809	−	0.659380i	$-0.770819\pi$
$23$	−7.21110	−1.50362	−0.751809	+	0.659380i	$0.770819\pi$
$24$	0	0
$25$	8.00000	1.60000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	7.21110	1.33907	0.669534	−	0.742781i	$-0.266494\pi$
$29$	7.21110	1.33907	0.669534	+	0.742781i	$0.266494\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.60555	−0.609449
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.21110	1.12619	0.563093	−	0.826394i	$-0.309611\pi$
$41$	7.21110	1.12619	0.563093	+	0.826394i	$0.309611\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.60555	−0.525924	−0.262962	−	0.964806i	$-0.584699\pi$
$47$	−3.60555	−0.525924	−0.262962	+	0.964806i	$0.584699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	4.00000	0.554700
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.8167	1.40821	0.704104	−	0.710097i	$-0.251349\pi$
$59$	10.8167	1.40821	0.704104	+	0.710097i	$0.251349\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	7.21110	0.894427
$66$	0	0
$67$	−15.0000	−1.83254	−0.916271	−	0.400559i	$-0.868816\pi$
$67$	−15.0000	−1.83254	−0.916271	+	0.400559i	$0.868816\pi$
$68$	−7.21110	−0.874475
$69$	0	0
$70$	0	0
$71$	14.4222	1.71160	0.855800	−	0.517306i	$-0.173065\pi$
$71$	14.4222	1.71160	0.855800	+	0.517306i	$0.173065\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−14.4222	−1.61245
$81$	0	0
$82$	0	0
$83$	10.8167	1.18728	0.593641	−	0.804730i	$-0.297690\pi$
$83$	10.8167	1.18728	0.593641	+	0.804730i	$0.297690\pi$
$84$	0	0
$85$	−13.0000	−1.41005
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.8167	−1.14656	−0.573282	−	0.819358i	$-0.694330\pi$
$89$	−10.8167	−1.14656	−0.573282	+	0.819358i	$0.694330\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	14.4222	1.50362
$93$	0	0
$94$	0	0
$95$	21.6333	2.21953
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	−16.0000	−1.60000
$101$	10.8167	1.07630	0.538149	−	0.842850i	$-0.319124\pi$
$101$	10.8167	1.07630	0.538149	+	0.842850i	$0.319124\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.21110	−0.697124	−0.348562	−	0.937286i	$-0.613330\pi$
$107$	−7.21110	−0.697124	−0.348562	+	0.937286i	$0.613330\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	7.21110	0.678363	0.339182	−	0.940721i	$-0.389850\pi$
$113$	7.21110	0.678363	0.339182	+	0.940721i	$0.389850\pi$
$114$	0	0
$115$	26.0000	2.42451
$116$	−14.4222	−1.33907
$117$	0	0
$118$	0	0
$119$	3.60555	0.330520
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	4.00000	0.359211
$125$	−10.8167	−0.967471
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.8167	0.945055	0.472528	−	0.881316i	$-0.343342\pi$
$131$	10.8167	0.945055	0.472528	+	0.881316i	$0.343342\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.6333	−1.84826	−0.924129	−	0.382080i	$-0.875208\pi$
$137$	−21.6333	−1.84826	−0.924129	+	0.382080i	$0.875208\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	7.21110	0.609449
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−26.0000	−2.15918
$146$	0	0
$147$	0	0
$148$	4.00000	0.328798
$149$	14.4222	1.18151	0.590757	−	0.806850i	$-0.298829\pi$
$149$	14.4222	1.18151	0.590757	+	0.806850i	$0.298829\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.21110	0.579210
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.21110	−0.568314
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−14.4222	−1.12619
$165$	0	0
$166$	0	0
$167$	−3.60555	−0.279006	−0.139503	−	0.990222i	$-0.544550\pi$
$167$	−3.60555	−0.279006	−0.139503	+	0.990222i	$0.544550\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−10.0000	−0.762493
$173$	−3.60555	−0.274125	−0.137062	−	0.990562i	$-0.543766\pi$
$173$	−3.60555	−0.274125	−0.137062	+	0.990562i	$0.543766\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.4222	1.07797	0.538983	−	0.842317i	$-0.318809\pi$
$179$	14.4222	1.07797	0.538983	+	0.842317i	$0.318809\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.21110	0.530171
$186$	0	0
$187$	0	0
$188$	7.21110	0.525924
$189$	0	0
$190$	0	0
$191$	−14.4222	−1.04355	−0.521777	−	0.853082i	$-0.674731\pi$
$191$	−14.4222	−1.04355	−0.521777	+	0.853082i	$0.674731\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	−21.6333	−1.54131	−0.770655	−	0.637253i	$-0.780071\pi$
$197$	−21.6333	−1.54131	−0.770655	+	0.637253i	$0.780071\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.21110	0.506120
$204$	0	0
$205$	−26.0000	−1.81592
$206$	0	0
$207$	0	0
$208$	−8.00000	−0.554700
$209$	0	0
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.0278	−1.22948
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.21110	−0.485071
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.2389	−1.67516	−0.837581	−	0.546313i	$-0.816031\pi$
$227$	−25.2389	−1.67516	−0.837581	+	0.546313i	$0.816031\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	13.0000	0.848026
$236$	−21.6333	−1.40821
$237$	0	0
$238$	0	0
$239$	7.21110	0.466447	0.233224	−	0.972423i	$-0.425073\pi$
$239$	7.21110	0.466447	0.233224	+	0.972423i	$0.425073\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	0	0
$243$	0	0
$244$	−28.0000	−1.79252
$245$	−3.60555	−0.230350
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−10.8167	−0.674724	−0.337362	−	0.941375i	$-0.609535\pi$
$257$	−10.8167	−0.674724	−0.337362	+	0.941375i	$0.609535\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	−14.4222	−0.894427
$261$	0	0
$262$	0	0
$263$	21.6333	1.33397	0.666983	−	0.745073i	$-0.267585\pi$
$263$	21.6333	1.33397	0.666983	+	0.745073i	$0.267585\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	30.0000	1.83254
$269$	21.6333	1.31901	0.659503	−	0.751702i	$-0.270767\pi$
$269$	21.6333	1.31901	0.659503	+	0.751702i	$0.270767\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	14.4222	0.874475
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−21.0000	−1.26177	−0.630884	−	0.775877i	$-0.717308\pi$
$277$	−21.0000	−1.26177	−0.630884	+	0.775877i	$0.717308\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.4222	0.860357	0.430178	−	0.902744i	$-0.358451\pi$
$281$	14.4222	0.860357	0.430178	+	0.902744i	$0.358451\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−28.8444	−1.71160
$285$	0	0
$286$	0	0
$287$	7.21110	0.425658
$288$	0	0
$289$	−4.00000	−0.235294
$290$	0	0
$291$	0	0
$292$	8.00000	0.468165
$293$	−32.4500	−1.89575	−0.947873	−	0.318647i	$-0.896772\pi$
$293$	−32.4500	−1.89575	−0.947873	+	0.318647i	$0.896772\pi$
$294$	0	0
$295$	−39.0000	−2.27067
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.4222	0.834058
$300$	0	0
$301$	5.00000	0.288195
$302$	0	0
$303$	0	0
$304$	−24.0000	−1.37649
$305$	−50.4777	−2.89035
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.0278	1.02226	0.511130	−	0.859503i	$-0.329227\pi$
$311$	18.0278	1.02226	0.511130	+	0.859503i	$0.329227\pi$
$312$	0	0
$313$	4.00000	0.226093	0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000	0.226093	0.113047	+	0.993590i	$0.463939\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.6333	−1.21505	−0.607524	−	0.794301i	$-0.707837\pi$
$317$	−21.6333	−1.21505	−0.607524	+	0.794301i	$0.707837\pi$
$318$	0	0
$319$	0	0
$320$	28.8444	1.61245
$321$	0	0
$322$	0	0
$323$	−21.6333	−1.20371
$324$	0	0
$325$	−16.0000	−0.887520
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.60555	−0.198780
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	−21.6333	−1.18728
$333$	0	0
$334$	0	0
$335$	54.0833	2.95488
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	26.0000	1.41005
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.8444	−1.54845	−0.774225	−	0.632911i	$-0.781860\pi$
$347$	−28.8444	−1.54845	−0.774225	+	0.632911i	$0.781860\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.6333	1.15142	0.575712	−	0.817652i	$-0.304725\pi$
$353$	21.6333	1.15142	0.575712	+	0.817652i	$0.304725\pi$
$354$	0	0
$355$	−52.0000	−2.75987
$356$	21.6333	1.14656
$357$	0	0
$358$	0	0
$359$	14.4222	0.761175	0.380587	−	0.924745i	$-0.375722\pi$
$359$	14.4222	0.761175	0.380587	+	0.924745i	$0.375722\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	4.00000	0.209657
$365$	14.4222	0.754893
$366$	0	0
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	−28.8444	−1.50362
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	25.0000	1.29445	0.647225	−	0.762299i	$-0.275929\pi$
$373$	25.0000	1.29445	0.647225	+	0.762299i	$0.275929\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.4222	−0.742781
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	−43.2666	−2.21953
$381$	0	0
$382$	0	0
$383$	−25.2389	−1.28965	−0.644823	−	0.764332i	$-0.723069\pi$
$383$	−25.2389	−1.28965	−0.644823	+	0.764332i	$0.723069\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	16.0000	0.812277
$389$	−21.6333	−1.09685	−0.548426	−	0.836199i	$-0.684773\pi$
$389$	−21.6333	−1.09685	−0.548426	+	0.836199i	$0.684773\pi$
$390$	0	0
$391$	−26.0000	−1.31488
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	0	0
$400$	32.0000	1.60000
$401$	−21.6333	−1.08032	−0.540158	−	0.841564i	$-0.681635\pi$
$401$	−21.6333	−1.08032	−0.540158	+	0.841564i	$0.681635\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	−21.6333	−1.07630
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	0	0
$412$	−12.0000	−0.591198
$413$	10.8167	0.532253
$414$	0	0
$415$	−39.0000	−1.91443
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.60555	0.176143	0.0880714	−	0.996114i	$-0.471930\pi$
$419$	3.60555	0.176143	0.0880714	+	0.996114i	$0.471930\pi$
$420$	0	0
$421$	−35.0000	−1.70580	−0.852898	−	0.522078i	$-0.825157\pi$
$421$	−35.0000	−1.70580	−0.852898	+	0.522078i	$0.825157\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	28.8444	1.39916
$426$	0	0
$427$	14.0000	0.677507
$428$	14.4222	0.697124
$429$	0	0
$430$	0	0
$431$	7.21110	0.347347	0.173673	−	0.984803i	$-0.444436\pi$
$431$	7.21110	0.347347	0.173673	+	0.984803i	$0.444436\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	43.2666	2.06972
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.6333	1.02783	0.513915	−	0.857841i	$-0.328195\pi$
$443$	21.6333	1.02783	0.513915	+	0.857841i	$0.328195\pi$
$444$	0	0
$445$	39.0000	1.84878
$446$	0	0
$447$	0	0
$448$	−8.00000	−0.377964
$449$	21.6333	1.02094	0.510469	−	0.859896i	$-0.329472\pi$
$449$	21.6333	1.02094	0.510469	+	0.859896i	$0.329472\pi$
$450$	0	0
$451$	0	0
$452$	−14.4222	−0.678363
$453$	0	0
$454$	0	0
$455$	7.21110	0.338062
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	0	0
$460$	−52.0000	−2.42451
$461$	−32.4500	−1.51135	−0.755673	−	0.654949i	$-0.772690\pi$
$461$	−32.4500	−1.51135	−0.755673	+	0.654949i	$0.772690\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	28.8444	1.33907
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−15.0000	−0.692636
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−48.0000	−2.20239
$476$	−7.21110	−0.330520
$477$	0	0
$478$	0	0
$479$	25.2389	1.15319	0.576596	−	0.817029i	$-0.304381\pi$
$479$	25.2389	1.15319	0.576596	+	0.817029i	$0.304381\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.8444	1.30976
$486$	0	0
$487$	5.00000	0.226572	0.113286	−	0.993562i	$-0.463862\pi$
$487$	5.00000	0.226572	0.113286	+	0.993562i	$0.463862\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−43.2666	−1.95260	−0.976298	−	0.216433i	$-0.930558\pi$
$491$	−43.2666	−1.95260	−0.976298	+	0.216433i	$0.930558\pi$
$492$	0	0
$493$	26.0000	1.17098
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	14.4222	0.646924
$498$	0	0
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	21.6333	0.967471
$501$	0	0
$502$	0	0
$503$	−10.8167	−0.482291	−0.241145	−	0.970489i	$-0.577523\pi$
$503$	−10.8167	−0.482291	−0.241145	+	0.970489i	$0.577523\pi$
$504$	0	0
$505$	−39.0000	−1.73548
$506$	0	0
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	39.6611	1.75795	0.878973	−	0.476872i	$-0.158229\pi$
$509$	39.6611	1.75795	0.878973	+	0.476872i	$0.158229\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−21.6333	−0.953277
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0278	−0.789810	−0.394905	−	0.918722i	$-0.629223\pi$
$521$	−18.0278	−0.789810	−0.394905	+	0.918722i	$0.629223\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−21.6333	−0.945055
$525$	0	0
$526$	0	0
$527$	−7.21110	−0.314121
$528$	0	0
$529$	29.0000	1.26087
$530$	0	0
$531$	0	0
$532$	12.0000	0.520266
$533$	−14.4222	−0.624695
$534$	0	0
$535$	26.0000	1.12408
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.2389	1.08111
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	43.2666	1.84826
$549$	0	0
$550$	0	0
$551$	−43.2666	−1.84322
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−32.0000	−1.35710
$557$	36.0555	1.52772	0.763861	−	0.645381i	$-0.223302\pi$
$557$	36.0555	1.52772	0.763861	+	0.645381i	$0.223302\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	−14.4222	−0.609449
$561$	0	0
$562$	0	0
$563$	−18.0278	−0.759779	−0.379890	−	0.925032i	$-0.624038\pi$
$563$	−18.0278	−0.759779	−0.379890	+	0.925032i	$0.624038\pi$
$564$	0	0
$565$	−26.0000	−1.09383
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−57.6888	−2.40579
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	0	0
$579$	0	0
$580$	52.0000	2.15918
$581$	10.8167	0.448750
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.4500	1.33935	0.669677	−	0.742653i	$-0.266433\pi$
$587$	32.4500	1.33935	0.669677	+	0.742653i	$0.266433\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−10.8167	−0.444187	−0.222093	−	0.975025i	$-0.571289\pi$
$593$	−10.8167	−0.444187	−0.222093	+	0.975025i	$0.571289\pi$
$594$	0	0
$595$	−13.0000	−0.532948
$596$	−28.8444	−1.18151
$597$	0	0
$598$	0	0
$599$	−43.2666	−1.76783	−0.883913	−	0.467651i	$-0.845100\pi$
$599$	−43.2666	−1.76783	−0.883913	+	0.467651i	$0.845100\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	18.0000	0.732410
$605$	0	0
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.21110	0.291730
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−14.4222	−0.579210
$621$	0	0
$622$	0	0
$623$	−10.8167	−0.433360
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	4.00000	0.159617
$629$	−7.21110	−0.287525
$630$	0	0
$631$	−21.0000	−0.835997	−0.417998	−	0.908448i	$-0.637268\pi$
$631$	−21.0000	−0.835997	−0.417998	+	0.908448i	$0.637268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.60555	−0.143082
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	14.4222	0.568314
$645$	0	0
$646$	0	0
$647$	32.4500	1.27574	0.637870	−	0.770144i	$-0.279816\pi$
$647$	32.4500	1.27574	0.637870	+	0.770144i	$0.279816\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	36.0555	1.41096	0.705481	−	0.708729i	$-0.250731\pi$
$653$	36.0555	1.41096	0.705481	+	0.708729i	$0.250731\pi$
$654$	0	0
$655$	−39.0000	−1.52386
$656$	28.8444	1.12619
$657$	0	0
$658$	0	0
$659$	28.8444	1.12362	0.561809	−	0.827267i	$-0.310105\pi$
$659$	28.8444	1.12362	0.561809	+	0.827267i	$0.310105\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21.6333	0.838904
$666$	0	0
$667$	−52.0000	−2.01345
$668$	7.21110	0.279006
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	0	0
$676$	18.0000	0.692308
$677$	−18.0278	−0.692863	−0.346431	−	0.938075i	$-0.612607\pi$
$677$	−18.0278	−0.692863	−0.346431	+	0.938075i	$0.612607\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−43.2666	−1.65555	−0.827776	−	0.561059i	$-0.810394\pi$
$683$	−43.2666	−1.65555	−0.827776	+	0.561059i	$0.810394\pi$
$684$	0	0
$685$	78.0000	2.98023
$686$	0	0
$687$	0	0
$688$	20.0000	0.762493
$689$	0	0
$690$	0	0
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	7.21110	0.274125
$693$	0	0
$694$	0	0
$695$	−57.6888	−2.18826
$696$	0	0
$697$	26.0000	0.984820
$698$	0	0
$699$	0	0
$700$	−16.0000	−0.604743
$701$	43.2666	1.63416	0.817079	−	0.576526i	$-0.195592\pi$
$701$	43.2666	1.63416	0.817079	+	0.576526i	$0.195592\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.8167	0.406802
$708$	0	0
$709$	−9.00000	−0.338002	−0.169001	−	0.985616i	$-0.554054\pi$
$709$	−9.00000	−0.338002	−0.169001	+	0.985616i	$0.554054\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.4222	0.540116
$714$	0	0
$715$	0	0
$716$	−28.8444	−1.07797
$717$	0	0
$718$	0	0
$719$	14.4222	0.537857	0.268929	−	0.963160i	$-0.413330\pi$
$719$	14.4222	0.537857	0.268929	+	0.963160i	$0.413330\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	0	0
$724$	−40.0000	−1.48659
$725$	57.6888	2.14251
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.0278	0.666781
$732$	0	0
$733$	28.0000	1.03420	0.517102	−	0.855924i	$-0.327011\pi$
$733$	28.0000	1.03420	0.517102	+	0.855924i	$0.327011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−14.4222	−0.530171
$741$	0	0
$742$	0	0
$743$	21.6333	0.793649	0.396825	−	0.917894i	$-0.370112\pi$
$743$	21.6333	0.793649	0.396825	+	0.917894i	$0.370112\pi$
$744$	0	0
$745$	−52.0000	−1.90513
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.21110	−0.263488
$750$	0	0
$751$	−41.0000	−1.49611	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−41.0000	−1.49611	−0.748056	+	0.663636i	$0.769012\pi$
$752$	−14.4222	−0.525924
$753$	0	0
$754$	0	0
$755$	32.4500	1.18098
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.8167	0.392103	0.196052	−	0.980594i	$-0.437188\pi$
$761$	10.8167	0.392103	0.196052	+	0.980594i	$0.437188\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	28.8444	1.04355
$765$	0	0
$766$	0	0
$767$	−21.6333	−0.781133
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	46.0000	1.65558
$773$	−10.8167	−0.389048	−0.194524	−	0.980898i	$-0.562316\pi$
$773$	−10.8167	−0.389048	−0.194524	+	0.980898i	$0.562316\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−43.2666	−1.55019
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	4.00000	0.142857
$785$	7.21110	0.257375
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	43.2666	1.54131
$789$	0	0
$790$	0	0
$791$	7.21110	0.256397
$792$	0	0
$793$	−28.0000	−0.994309
$794$	0	0
$795$	0	0
$796$	4.00000	0.141776
$797$	−25.2389	−0.894006	−0.447003	−	0.894532i	$-0.647509\pi$
$797$	−25.2389	−0.894006	−0.447003	+	0.894532i	$0.647509\pi$
$798$	0	0
$799$	−13.0000	−0.459907
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	26.0000	0.916380
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.6333	−0.760587	−0.380293	−	0.924866i	$-0.624177\pi$
$809$	−21.6333	−0.760587	−0.380293	+	0.924866i	$0.624177\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	−14.4222	−0.506120
$813$	0	0
$814$	0	0
$815$	−14.4222	−0.505188
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	0	0
$820$	52.0000	1.81592
$821$	43.2666	1.51002	0.755008	−	0.655716i	$-0.227633\pi$
$821$	43.2666	1.51002	0.755008	+	0.655716i	$0.227633\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.21110	−0.250755	−0.125377	−	0.992109i	$-0.540014\pi$
$827$	−7.21110	−0.250755	−0.125377	+	0.992109i	$0.540014\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	0	0
$832$	16.0000	0.554700
$833$	3.60555	0.124925
$834$	0	0
$835$	13.0000	0.449884
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.8167	−0.373432	−0.186716	−	0.982414i	$-0.559784\pi$
$839$	−10.8167	−0.373432	−0.186716	+	0.982414i	$0.559784\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	26.0000	0.894957
$845$	32.4500	1.11631
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.4222	0.494387
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.2389	0.862143	0.431071	−	0.902318i	$-0.358136\pi$
$857$	25.2389	0.862143	0.431071	+	0.902318i	$0.358136\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	36.0555	1.22948
$861$	0	0
$862$	0	0
$863$	−7.21110	−0.245469	−0.122734	−	0.992440i	$-0.539166\pi$
$863$	−7.21110	−0.245469	−0.122734	+	0.992440i	$0.539166\pi$
$864$	0	0
$865$	13.0000	0.442013
$866$	0	0
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	30.0000	1.01651
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.6333	0.728845	0.364422	−	0.931234i	$-0.381266\pi$
$881$	21.6333	0.728845	0.364422	+	0.931234i	$0.381266\pi$
$882$	0	0
$883$	−29.0000	−0.975928	−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000	−0.975928	−0.487964	+	0.872864i	$0.662260\pi$
$884$	14.4222	0.485071
$885$	0	0
$886$	0	0
$887$	−3.60555	−0.121063	−0.0605313	−	0.998166i	$-0.519279\pi$
$887$	−3.60555	−0.121063	−0.0605313	+	0.998166i	$0.519279\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	0	0
$891$	0	0
$892$	−20.0000	−0.669650
$893$	21.6333	0.723931
$894$	0	0
$895$	−52.0000	−1.73817
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.4222	−0.481007
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−72.1110	−2.39705
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	50.4777	1.67516
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	44.0000	1.45380
$917$	10.8167	0.357197
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−28.8444	−0.949425
$924$	0	0
$925$	−16.0000	−0.526077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.8167	−0.354883	−0.177441	−	0.984131i	$-0.556782\pi$
$929$	−10.8167	−0.354883	−0.177441	+	0.984131i	$0.556782\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	0	0
$940$	−26.0000	−0.848026
$941$	7.21110	0.235075	0.117538	−	0.993068i	$-0.462500\pi$
$941$	7.21110	0.235075	0.117538	+	0.993068i	$0.462500\pi$
$942$	0	0
$943$	−52.0000	−1.69335
$944$	43.2666	1.40821
$945$	0	0
$946$	0	0
$947$	−14.4222	−0.468659	−0.234329	−	0.972157i	$-0.575289\pi$
$947$	−14.4222	−0.468659	−0.234329	+	0.972157i	$0.575289\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.0555	1.16795	0.583976	−	0.811771i	$-0.301496\pi$
$953$	36.0555	1.16795	0.583976	+	0.811771i	$0.301496\pi$
$954$	0	0
$955$	52.0000	1.68268
$956$	−14.4222	−0.466447
$957$	0	0
$958$	0	0
$959$	−21.6333	−0.698576
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	−24.0000	−0.772988
$965$	82.9277	2.66954
$966$	0	0
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.60555	−0.115708	−0.0578538	−	0.998325i	$-0.518426\pi$
$971$	−3.60555	−0.115708	−0.0578538	+	0.998325i	$0.518426\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0	0
$976$	56.0000	1.79252
$977$	28.8444	0.922814	0.461407	−	0.887188i	$-0.347345\pi$
$977$	28.8444	0.922814	0.461407	+	0.887188i	$0.347345\pi$
$978$	0	0
$979$	0	0
$980$	7.21110	0.230350
$981$	0	0
$982$	0	0
$983$	43.2666	1.37999	0.689995	−	0.723814i	$-0.257613\pi$
$983$	43.2666	1.37999	0.689995	+	0.723814i	$0.257613\pi$
$984$	0	0
$985$	78.0000	2.48529
$986$	0	0
$987$	0	0
$988$	−24.0000	−0.763542
$989$	−36.0555	−1.14650
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.21110	0.228607
$996$	0	0
$997$	36.0000	1.14013	0.570066	−	0.821599i	$-0.306918\pi$
$997$	36.0000	1.14013	0.570066	+	0.821599i	$0.306918\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bh.1.1	yes	2
3.2	odd	2	inner	7623.2.a.bh.1.2	yes	2
11.10	odd	2		7623.2.a.bg.1.1	✓	2
33.32	even	2		7623.2.a.bg.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.bg.1.1	✓	2	11.10	odd	2
7623.2.a.bg.1.2	yes	2	33.32	even	2
7623.2.a.bh.1.1	yes	2	1.1	even	1	trivial
7623.2.a.bh.1.2	yes	2	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -3.60555 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} -3.60555 q^{5} +1.00000 q^{7} -2.00000 q^{13} +4.00000 q^{16} +3.60555 q^{17} -6.00000 q^{19} +7.21110 q^{20} -7.21110 q^{23} +8.00000 q^{25} -2.00000 q^{28} +7.21110 q^{29} -2.00000 q^{31} -3.60555 q^{35} -2.00000 q^{37} +7.21110 q^{41} +5.00000 q^{43} -3.60555 q^{47} +1.00000 q^{49} +4.00000 q^{52} +10.8167 q^{59} +14.0000 q^{61} -8.00000 q^{64} +7.21110 q^{65} -15.0000 q^{67} -7.21110 q^{68} +14.4222 q^{71} -4.00000 q^{73} +12.0000 q^{76} -14.4222 q^{80} +10.8167 q^{83} -13.0000 q^{85} -10.8167 q^{89} -2.00000 q^{91} +14.4222 q^{92} +21.6333 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 + 2 * q^7	\( 2 q - 4 q^{4} + 2 q^{7} - 4 q^{13} + 8 q^{16} - 12 q^{19} + 16 q^{25} - 4 q^{28} - 4 q^{31} - 4 q^{37} + 10 q^{43} + 2 q^{49} + 8 q^{52} + 28 q^{61} - 16 q^{64} - 30 q^{67} - 8 q^{73} + 24 q^{76} - 26 q^{85} - 4 q^{91} - 16 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 2 * q^7 - 4 * q^13 + 8 * q^16 - 12 * q^19 + 16 * q^25 - 4 * q^28 - 4 * q^31 - 4 * q^37 + 10 * q^43 + 2 * q^49 + 8 * q^52 + 28 * q^61 - 16 * q^64 - 30 * q^67 - 8 * q^73 + 24 * q^76 - 26 * q^85 - 4 * q^91 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more