LMFDB - Embedded newform 7488.2.a.ct.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7488,2,Mod(1,7488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.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:	$59.7919810335$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{5} -1.23607 q^{7} +O(q^{10})$	$q-1.23607 q^{5} -1.23607 q^{7} +2.00000 q^{11} +1.00000 q^{13} -4.47214 q^{17} -5.23607 q^{19} +2.47214 q^{23} -3.47214 q^{25} +4.47214 q^{29} +7.70820 q^{31} +1.52786 q^{35} -0.472136 q^{37} -1.23607 q^{41} -6.47214 q^{43} -6.94427 q^{47} -5.47214 q^{49} +8.47214 q^{53} -2.47214 q^{55} +8.47214 q^{59} +12.4721 q^{61} -1.23607 q^{65} -6.76393 q^{67} -4.47214 q^{71} +0.472136 q^{73} -2.47214 q^{77} -8.94427 q^{79} +7.52786 q^{83} +5.52786 q^{85} +10.1803 q^{89} -1.23607 q^{91} +6.47214 q^{95} -4.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{13} - 6 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{31} + 12 q^{35} + 8 q^{37} + 2 q^{41} - 4 q^{43} + 4 q^{47} - 2 q^{49} + 8 q^{53} + 4 q^{55} + 8 q^{59} + 16 q^{61} + 2 q^{65} - 18 q^{67} - 8 q^{73} + 4 q^{77} + 24 q^{83} + 20 q^{85} - 2 q^{89} + 2 q^{91} + 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 + 2 * q^13 - 6 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^31 + 12 * q^35 + 8 * q^37 + 2 * q^41 - 4 * q^43 + 4 * q^47 - 2 * q^49 + 8 * q^53 + 4 * q^55 + 8 * q^59 + 16 * q^61 + 2 * q^65 - 18 * q^67 - 8 * q^73 + 4 * q^77 + 24 * q^83 + 20 * q^85 - 2 * q^89 + 2 * q^91 + 4 * q^95

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
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−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
$19$	−5.23607	−1.20124	−0.600618	−	0.799536i	$-0.705079\pi$
$19$	−5.23607	−1.20124	−0.600618	+	0.799536i	$0.705079\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	7.70820	1.38443	0.692217	−	0.721689i	$-0.256634\pi$
$31$	7.70820	1.38443	0.692217	+	0.721689i	$0.256634\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.52786	0.258256
$36$	0	0
$37$	−0.472136	−0.0776187	−0.0388093	−	0.999247i	$-0.512356\pi$
$37$	−0.472136	−0.0776187	−0.0388093	+	0.999247i	$0.512356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.23607	−0.193041	−0.0965207	−	0.995331i	$-0.530771\pi$
$41$	−1.23607	−0.193041	−0.0965207	+	0.995331i	$0.530771\pi$
$42$	0	0
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.94427	−1.01293	−0.506463	−	0.862262i	$-0.669047\pi$
$47$	−6.94427	−1.01293	−0.506463	+	0.862262i	$0.669047\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.47214	1.10298	0.551489	−	0.834182i	$-0.314060\pi$
$59$	8.47214	1.10298	0.551489	+	0.834182i	$0.314060\pi$
$60$	0	0
$61$	12.4721	1.59689	0.798447	−	0.602066i	$-0.205655\pi$
$61$	12.4721	1.59689	0.798447	+	0.602066i	$0.205655\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−6.76393	−0.826346	−0.413173	−	0.910653i	$-0.635579\pi$
$67$	−6.76393	−0.826346	−0.413173	+	0.910653i	$0.635579\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	0.472136	0.0552593	0.0276297	−	0.999618i	$-0.491204\pi$
$73$	0.472136	0.0552593	0.0276297	+	0.999618i	$0.491204\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.47214	−0.281726
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.52786	0.826290	0.413145	−	0.910665i	$-0.364430\pi$
$83$	7.52786	0.826290	0.413145	+	0.910665i	$0.364430\pi$
$84$	0	0
$85$	5.52786	0.599581
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.1803	1.07911	0.539557	−	0.841949i	$-0.318592\pi$
$89$	10.1803	1.07911	0.539557	+	0.841949i	$0.318592\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.47214	0.664027
$96$	0	0
$97$	−4.47214	−0.454077	−0.227038	−	0.973886i	$-0.572904\pi$
$97$	−4.47214	−0.454077	−0.227038	+	0.973886i	$0.572904\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.94427	−0.292966	−0.146483	−	0.989213i	$-0.546795\pi$
$101$	−2.94427	−0.292966	−0.146483	+	0.989213i	$0.546795\pi$
$102$	0	0
$103$	−1.52786	−0.150545	−0.0752725	−	0.997163i	$-0.523983\pi$
$103$	−1.52786	−0.150545	−0.0752725	+	0.997163i	$0.523983\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4721	1.39907	0.699537	−	0.714596i	$-0.253390\pi$
$107$	14.4721	1.39907	0.699537	+	0.714596i	$0.253390\pi$
$108$	0	0
$109$	1.05573	0.101120	0.0505602	−	0.998721i	$-0.483899\pi$
$109$	1.05573	0.101120	0.0505602	+	0.998721i	$0.483899\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−3.05573	−0.284948
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.52786	0.506738
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	6.47214	0.574309	0.287155	−	0.957884i	$-0.407291\pi$
$127$	6.47214	0.574309	0.287155	+	0.957884i	$0.407291\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.4164	−0.997456	−0.498728	−	0.866758i	$-0.666199\pi$
$131$	−11.4164	−0.997456	−0.498728	+	0.866758i	$0.666199\pi$
$132$	0	0
$133$	6.47214	0.561205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.76393	0.577882	0.288941	−	0.957347i	$-0.406697\pi$
$137$	6.76393	0.577882	0.288941	+	0.957347i	$0.406697\pi$
$138$	0	0
$139$	13.8885	1.17801	0.589005	−	0.808129i	$-0.299520\pi$
$139$	13.8885	1.17801	0.589005	+	0.808129i	$0.299520\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−5.52786	−0.459064
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.29180	0.679290	0.339645	−	0.940554i	$-0.389693\pi$
$149$	8.29180	0.679290	0.339645	+	0.940554i	$0.389693\pi$
$150$	0	0
$151$	−18.7639	−1.52699	−0.763494	−	0.645815i	$-0.776518\pi$
$151$	−18.7639	−1.52699	−0.763494	+	0.645815i	$0.776518\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.52786	−0.765296
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.05573	−0.240825
$162$	0	0
$163$	−2.76393	−0.216488	−0.108244	−	0.994124i	$-0.534523\pi$
$163$	−2.76393	−0.216488	−0.108244	+	0.994124i	$0.534523\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.8885	1.22949	0.614746	−	0.788725i	$-0.289258\pi$
$167$	15.8885	1.22949	0.614746	+	0.788725i	$0.289258\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	4.29180	0.324429
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.41641	0.554328	0.277164	−	0.960823i	$-0.410605\pi$
$179$	7.41641	0.554328	0.277164	+	0.960823i	$0.410605\pi$
$180$	0	0
$181$	−9.41641	−0.699916	−0.349958	−	0.936765i	$-0.613804\pi$
$181$	−9.41641	−0.699916	−0.349958	+	0.936765i	$0.613804\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.583592	0.0429065
$186$	0	0
$187$	−8.94427	−0.654070
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.52786	0.110552	0.0552762	−	0.998471i	$-0.482396\pi$
$191$	1.52786	0.110552	0.0552762	+	0.998471i	$0.482396\pi$
$192$	0	0
$193$	19.8885	1.43161	0.715804	−	0.698301i	$-0.246060\pi$
$193$	19.8885	1.43161	0.715804	+	0.698301i	$0.246060\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.7639	1.62186	0.810932	−	0.585141i	$-0.198961\pi$
$197$	22.7639	1.62186	0.810932	+	0.585141i	$0.198961\pi$
$198$	0	0
$199$	−9.52786	−0.675412	−0.337706	−	0.941252i	$-0.609651\pi$
$199$	−9.52786	−0.675412	−0.337706	+	0.941252i	$0.609651\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.52786	−0.387980
$204$	0	0
$205$	1.52786	0.106711
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.4721	−0.724373
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−9.52786	−0.646794
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.47214	−0.300828
$222$	0	0
$223$	4.65248	0.311553	0.155776	−	0.987792i	$-0.450212\pi$
$223$	4.65248	0.311553	0.155776	+	0.987792i	$0.450212\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.9443	0.726397	0.363198	−	0.931712i	$-0.381685\pi$
$227$	10.9443	0.726397	0.363198	+	0.931712i	$0.381685\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.4164	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$233$	17.4164	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$234$	0	0
$235$	8.58359	0.559932
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.4721	−0.806755	−0.403378	−	0.915034i	$-0.632164\pi$
$239$	−12.4721	−0.806755	−0.403378	+	0.915034i	$0.632164\pi$
$240$	0	0
$241$	−22.3607	−1.44038	−0.720189	−	0.693778i	$-0.755945\pi$
$241$	−22.3607	−1.44038	−0.720189	+	0.693778i	$0.755945\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.76393	0.432132
$246$	0	0
$247$	−5.23607	−0.333163
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.9443	−1.82695	−0.913473	−	0.406899i	$-0.866610\pi$
$251$	−28.9443	−1.82695	−0.913473	+	0.406899i	$0.866610\pi$
$252$	0	0
$253$	4.94427	0.310844
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.4721	0.777990	0.388995	−	0.921240i	$-0.372822\pi$
$257$	12.4721	0.777990	0.388995	+	0.921240i	$0.372822\pi$
$258$	0	0
$259$	0.583592	0.0362627
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.94427	0.551527	0.275764	−	0.961225i	$-0.411069\pi$
$263$	8.94427	0.551527	0.275764	+	0.961225i	$0.411069\pi$
$264$	0	0
$265$	−10.4721	−0.643298
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.4721	1.24821	0.624104	−	0.781341i	$-0.285464\pi$
$269$	20.4721	1.24821	0.624104	+	0.781341i	$0.285464\pi$
$270$	0	0
$271$	−19.1246	−1.16174	−0.580869	−	0.813997i	$-0.697287\pi$
$271$	−19.1246	−1.16174	−0.580869	+	0.813997i	$0.697287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.94427	−0.418755
$276$	0	0
$277$	14.9443	0.897914	0.448957	−	0.893553i	$-0.351796\pi$
$277$	14.9443	0.897914	0.448957	+	0.893553i	$0.351796\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.70820	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$281$	3.70820	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$282$	0	0
$283$	22.4721	1.33583	0.667915	−	0.744238i	$-0.267187\pi$
$283$	22.4721	1.33583	0.667915	+	0.744238i	$0.267187\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.52786	0.0901870
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.1803	0.594742	0.297371	−	0.954762i	$-0.403890\pi$
$293$	10.1803	0.594742	0.297371	+	0.954762i	$0.403890\pi$
$294$	0	0
$295$	−10.4721	−0.609711
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.47214	0.142967
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.4164	−0.882741
$306$	0	0
$307$	6.76393	0.386038	0.193019	−	0.981195i	$-0.438172\pi$
$307$	6.76393	0.386038	0.193019	+	0.981195i	$0.438172\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.583592	0.0330925	0.0165462	−	0.999863i	$-0.494733\pi$
$311$	0.583592	0.0330925	0.0165462	+	0.999863i	$0.494733\pi$
$312$	0	0
$313$	31.8885	1.80245	0.901224	−	0.433355i	$-0.142670\pi$
$313$	31.8885	1.80245	0.901224	+	0.433355i	$0.142670\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.23607	−0.518749	−0.259375	−	0.965777i	$-0.583516\pi$
$317$	−9.23607	−0.518749	−0.259375	+	0.965777i	$0.583516\pi$
$318$	0	0
$319$	8.94427	0.500783
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.4164	1.30292
$324$	0	0
$325$	−3.47214	−0.192599
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.58359	0.473229
$330$	0	0
$331$	14.1803	0.779422	0.389711	−	0.920937i	$-0.372575\pi$
$331$	14.1803	0.779422	0.389711	+	0.920937i	$0.372575\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.36068	0.456793
$336$	0	0
$337$	−6.94427	−0.378279	−0.189139	−	0.981950i	$-0.560570\pi$
$337$	−6.94427	−0.378279	−0.189139	+	0.981950i	$0.560570\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.4164	0.834845
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	14.3607	0.768710	0.384355	−	0.923185i	$-0.374424\pi$
$349$	14.3607	0.768710	0.384355	+	0.923185i	$0.374424\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.5967	1.14948	0.574739	−	0.818336i	$-0.305103\pi$
$353$	21.5967	1.14948	0.574739	+	0.818336i	$0.305103\pi$
$354$	0	0
$355$	5.52786	0.293389
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.8328	1.94396	0.971981	−	0.235060i	$-0.0755287\pi$
$359$	36.8328	1.94396	0.971981	+	0.235060i	$0.0755287\pi$
$360$	0	0
$361$	8.41641	0.442969
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.583592	−0.0305466
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.4721	−0.543686
$372$	0	0
$373$	17.0557	0.883112	0.441556	−	0.897234i	$-0.354427\pi$
$373$	17.0557	0.883112	0.441556	+	0.897234i	$0.354427\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.47214	0.230327
$378$	0	0
$379$	25.5967	1.31482	0.657408	−	0.753535i	$-0.271653\pi$
$379$	25.5967	1.31482	0.657408	+	0.753535i	$0.271653\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	3.05573	0.155734
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.8885	1.21120	0.605599	−	0.795770i	$-0.292934\pi$
$389$	23.8885	1.21120	0.605599	+	0.795770i	$0.292934\pi$
$390$	0	0
$391$	−11.0557	−0.559112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.0557	0.556274
$396$	0	0
$397$	7.52786	0.377813	0.188906	−	0.981995i	$-0.439506\pi$
$397$	7.52786	0.377813	0.188906	+	0.981995i	$0.439506\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.1803	0.907883	0.453941	−	0.891032i	$-0.350018\pi$
$401$	18.1803	0.907883	0.453941	+	0.891032i	$0.350018\pi$
$402$	0	0
$403$	7.70820	0.383973
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.944272	−0.0468058
$408$	0	0
$409$	34.3607	1.69903	0.849513	−	0.527567i	$-0.176896\pi$
$409$	34.3607	1.69903	0.849513	+	0.527567i	$0.176896\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.4721	−0.515300
$414$	0	0
$415$	−9.30495	−0.456762
$416$	0	0
$417$	0	0
$418$	0	0
$419$	38.4721	1.87949	0.939743	−	0.341881i	$-0.111064\pi$
$419$	38.4721	1.87949	0.939743	+	0.341881i	$0.111064\pi$
$420$	0	0
$421$	26.9443	1.31318	0.656592	−	0.754246i	$-0.271997\pi$
$421$	26.9443	1.31318	0.656592	+	0.754246i	$0.271997\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.5279	0.753212
$426$	0	0
$427$	−15.4164	−0.746052
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	−8.47214	−0.407145	−0.203572	−	0.979060i	$-0.565255\pi$
$433$	−8.47214	−0.407145	−0.203572	+	0.979060i	$0.565255\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.9443	−0.619208
$438$	0	0
$439$	22.4721	1.07254	0.536268	−	0.844048i	$-0.319834\pi$
$439$	22.4721	1.07254	0.536268	+	0.844048i	$0.319834\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.4721	−1.44777	−0.723887	−	0.689918i	$-0.757647\pi$
$443$	−30.4721	−1.44777	−0.723887	+	0.689918i	$0.757647\pi$
$444$	0	0
$445$	−12.5836	−0.596519
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.5410	−1.44132	−0.720660	−	0.693289i	$-0.756161\pi$
$449$	−30.5410	−1.44132	−0.720660	+	0.693289i	$0.756161\pi$
$450$	0	0
$451$	−2.47214	−0.116408
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.52786	0.0716274
$456$	0	0
$457$	10.3607	0.484652	0.242326	−	0.970195i	$-0.422090\pi$
$457$	10.3607	0.484652	0.242326	+	0.970195i	$0.422090\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.65248	0.402986	0.201493	−	0.979490i	$-0.435421\pi$
$461$	8.65248	0.402986	0.201493	+	0.979490i	$0.435421\pi$
$462$	0	0
$463$	−4.29180	−0.199457	−0.0997283	−	0.995015i	$-0.531797\pi$
$463$	−4.29180	−0.199457	−0.0997283	+	0.995015i	$0.531797\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.8885	−1.56817	−0.784087	−	0.620650i	$-0.786869\pi$
$467$	−33.8885	−1.56817	−0.784087	+	0.620650i	$0.786869\pi$
$468$	0	0
$469$	8.36068	0.386060
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12.9443	−0.595178
$474$	0	0
$475$	18.1803	0.834171
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−39.3050	−1.79589	−0.897945	−	0.440109i	$-0.854940\pi$
$479$	−39.3050	−1.79589	−0.897945	+	0.440109i	$0.854940\pi$
$480$	0	0
$481$	−0.472136	−0.0215275
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.52786	0.251007
$486$	0	0
$487$	39.7082	1.79935	0.899675	−	0.436560i	$-0.143803\pi$
$487$	39.7082	1.79935	0.899675	+	0.436560i	$0.143803\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−28.9443	−1.30624	−0.653118	−	0.757256i	$-0.726540\pi$
$491$	−28.9443	−1.30624	−0.653118	+	0.757256i	$0.726540\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.52786	0.247959
$498$	0	0
$499$	−17.8197	−0.797718	−0.398859	−	0.917012i	$-0.630594\pi$
$499$	−17.8197	−0.797718	−0.398859	+	0.917012i	$0.630594\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.7771	−1.41687	−0.708435	−	0.705776i	$-0.750599\pi$
$503$	−31.7771	−1.41687	−0.708435	+	0.705776i	$0.750599\pi$
$504$	0	0
$505$	3.63932	0.161948
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.18034	−0.273939	−0.136969	−	0.990575i	$-0.543736\pi$
$509$	−6.18034	−0.273939	−0.136969	+	0.990575i	$0.543736\pi$
$510$	0	0
$511$	−0.583592	−0.0258166
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.88854	0.0832192
$516$	0	0
$517$	−13.8885	−0.610817
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.05573	−0.0462523	−0.0231261	−	0.999733i	$-0.507362\pi$
$521$	−1.05573	−0.0462523	−0.0231261	+	0.999733i	$0.507362\pi$
$522$	0	0
$523$	−40.9443	−1.79037	−0.895184	−	0.445697i	$-0.852956\pi$
$523$	−40.9443	−1.79037	−0.895184	+	0.445697i	$0.852956\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.4721	−1.50163
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.23607	−0.0535400
$534$	0	0
$535$	−17.8885	−0.773389
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.9443	−0.471403
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.30495	−0.0558980
$546$	0	0
$547$	16.9443	0.724485	0.362242	−	0.932084i	$-0.382011\pi$
$547$	16.9443	0.724485	0.362242	+	0.932084i	$0.382011\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23.4164	−0.997573
$552$	0	0
$553$	11.0557	0.470137
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.291796	0.0123638	0.00618190	−	0.999981i	$-0.498032\pi$
$557$	0.291796	0.0123638	0.00618190	+	0.999981i	$0.498032\pi$
$558$	0	0
$559$	−6.47214	−0.273742
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.4721	0.441348	0.220674	−	0.975348i	$-0.429174\pi$
$563$	10.4721	0.441348	0.220674	+	0.975348i	$0.429174\pi$
$564$	0	0
$565$	−2.47214	−0.104004
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.3607	0.602031	0.301016	−	0.953619i	$-0.402674\pi$
$569$	14.3607	0.602031	0.301016	+	0.953619i	$0.402674\pi$
$570$	0	0
$571$	25.5279	1.06831	0.534154	−	0.845387i	$-0.320630\pi$
$571$	25.5279	1.06831	0.534154	+	0.845387i	$0.320630\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.58359	−0.357961
$576$	0	0
$577$	−1.05573	−0.0439505	−0.0219753	−	0.999759i	$-0.506996\pi$
$577$	−1.05573	−0.0439505	−0.0219753	+	0.999759i	$0.506996\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.30495	−0.386034
$582$	0	0
$583$	16.9443	0.701760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.4164	1.37924	0.689621	−	0.724170i	$-0.257777\pi$
$587$	33.4164	1.37924	0.689621	+	0.724170i	$0.257777\pi$
$588$	0	0
$589$	−40.3607	−1.66303
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10.7639	−0.442022	−0.221011	−	0.975271i	$-0.570936\pi$
$593$	−10.7639	−0.442022	−0.221011	+	0.975271i	$0.570936\pi$
$594$	0	0
$595$	−6.83282	−0.280118
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.05573	0.288289	0.144145	−	0.989557i	$-0.453957\pi$
$599$	7.05573	0.288289	0.144145	+	0.989557i	$0.453957\pi$
$600$	0	0
$601$	12.1115	0.494037	0.247018	−	0.969011i	$-0.420549\pi$
$601$	12.1115	0.494037	0.247018	+	0.969011i	$0.420549\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.65248	0.351773
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.94427	−0.280935
$612$	0	0
$613$	14.3607	0.580022	0.290011	−	0.957023i	$-0.406341\pi$
$613$	14.3607	0.580022	0.290011	+	0.957023i	$0.406341\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.5967	1.83566	0.917828	−	0.396978i	$-0.129941\pi$
$617$	45.5967	1.83566	0.917828	+	0.396978i	$0.129941\pi$
$618$	0	0
$619$	−7.34752	−0.295322	−0.147661	−	0.989038i	$-0.547174\pi$
$619$	−7.34752	−0.295322	−0.147661	+	0.989038i	$0.547174\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.5836	−0.504151
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.11146	0.0841893
$630$	0	0
$631$	−12.6525	−0.503687	−0.251844	−	0.967768i	$-0.581037\pi$
$631$	−12.6525	−0.503687	−0.251844	+	0.967768i	$0.581037\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−5.47214	−0.216814
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.41641	0.213935	0.106968	−	0.994263i	$-0.465886\pi$
$641$	5.41641	0.213935	0.106968	+	0.994263i	$0.465886\pi$
$642$	0	0
$643$	11.1246	0.438712	0.219356	−	0.975645i	$-0.429604\pi$
$643$	11.1246	0.438712	0.219356	+	0.975645i	$0.429604\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	16.9443	0.665121
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.3607	−1.34464	−0.672319	−	0.740262i	$-0.734702\pi$
$653$	−34.3607	−1.34464	−0.672319	+	0.740262i	$0.734702\pi$
$654$	0	0
$655$	14.1115	0.551380
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.8328	−0.733622	−0.366811	−	0.930295i	$-0.619550\pi$
$659$	−18.8328	−0.733622	−0.366811	+	0.930295i	$0.619550\pi$
$660$	0	0
$661$	−44.2492	−1.72110	−0.860548	−	0.509370i	$-0.829879\pi$
$661$	−44.2492	−1.72110	−0.860548	+	0.509370i	$0.829879\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	11.0557	0.428080
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.9443	0.962963
$672$	0	0
$673$	45.4164	1.75067	0.875337	−	0.483513i	$-0.160640\pi$
$673$	45.4164	1.75067	0.875337	+	0.483513i	$0.160640\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.3607	0.398193	0.199097	−	0.979980i	$-0.436199\pi$
$677$	10.3607	0.398193	0.199097	+	0.979980i	$0.436199\pi$
$678$	0	0
$679$	5.52786	0.212140
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.0000	0.382639	0.191320	−	0.981528i	$-0.438723\pi$
$683$	10.0000	0.382639	0.191320	+	0.981528i	$0.438723\pi$
$684$	0	0
$685$	−8.36068	−0.319445
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.47214	0.322763
$690$	0	0
$691$	−21.5967	−0.821579	−0.410790	−	0.911730i	$-0.634747\pi$
$691$	−21.5967	−0.821579	−0.410790	+	0.911730i	$0.634747\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.1672	−0.651188
$696$	0	0
$697$	5.52786	0.209383
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	2.47214	0.0932384
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.63932	0.136871
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.0557	0.713643
$714$	0	0
$715$	−2.47214	−0.0924526
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.47214	0.0921951	0.0460976	−	0.998937i	$-0.485321\pi$
$719$	2.47214	0.0921951	0.0460976	+	0.998937i	$0.485321\pi$
$720$	0	0
$721$	1.88854	0.0703330
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.5279	−0.576690
$726$	0	0
$727$	1.52786	0.0566653	0.0283327	−	0.999599i	$-0.490980\pi$
$727$	1.52786	0.0566653	0.0283327	+	0.999599i	$0.490980\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.9443	1.07054
$732$	0	0
$733$	−27.8885	−1.03009	−0.515043	−	0.857164i	$-0.672224\pi$
$733$	−27.8885	−1.03009	−0.515043	+	0.857164i	$0.672224\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.5279	−0.498305
$738$	0	0
$739$	−46.1803	−1.69877	−0.849386	−	0.527773i	$-0.823027\pi$
$739$	−46.1803	−1.69877	−0.849386	+	0.527773i	$0.823027\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.4721	0.604304	0.302152	−	0.953260i	$-0.402295\pi$
$743$	16.4721	0.604304	0.302152	+	0.953260i	$0.402295\pi$
$744$	0	0
$745$	−10.2492	−0.375502
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.8885	−0.653633
$750$	0	0
$751$	32.3607	1.18086	0.590429	−	0.807090i	$-0.298959\pi$
$751$	32.3607	1.18086	0.590429	+	0.807090i	$0.298959\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23.1935	0.844098
$756$	0	0
$757$	−37.4164	−1.35992	−0.679961	−	0.733248i	$-0.738003\pi$
$757$	−37.4164	−1.35992	−0.679961	+	0.733248i	$0.738003\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.81966	0.355962	0.177981	−	0.984034i	$-0.443043\pi$
$761$	9.81966	0.355962	0.177981	+	0.984034i	$0.443043\pi$
$762$	0	0
$763$	−1.30495	−0.0472424
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.47214	0.305911
$768$	0	0
$769$	−52.8328	−1.90520	−0.952600	−	0.304226i	$-0.901602\pi$
$769$	−52.8328	−1.90520	−0.952600	+	0.304226i	$0.901602\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.29180	0.298235	0.149118	−	0.988819i	$-0.452357\pi$
$773$	8.29180	0.298235	0.149118	+	0.988819i	$0.452357\pi$
$774$	0	0
$775$	−26.7639	−0.961389
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.47214	0.231888
$780$	0	0
$781$	−8.94427	−0.320051
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.5836	−0.591894
$786$	0	0
$787$	−4.29180	−0.152986	−0.0764930	−	0.997070i	$-0.524372\pi$
$787$	−4.29180	−0.152986	−0.0764930	+	0.997070i	$0.524372\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.47214	−0.0878990
$792$	0	0
$793$	12.4721	0.442899
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	31.0557	1.09867
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.944272	0.0333226
$804$	0	0
$805$	3.77709	0.133125
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.9443	1.08794	0.543971	−	0.839104i	$-0.316920\pi$
$809$	30.9443	1.08794	0.543971	+	0.839104i	$0.316920\pi$
$810$	0	0
$811$	−26.1803	−0.919316	−0.459658	−	0.888096i	$-0.652028\pi$
$811$	−26.1803	−0.919316	−0.459658	+	0.888096i	$0.652028\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.41641	0.119672
$816$	0	0
$817$	33.8885	1.18561
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1246	1.08626	0.543128	−	0.839650i	$-0.317240\pi$
$821$	31.1246	1.08626	0.543128	+	0.839650i	$0.317240\pi$
$822$	0	0
$823$	−45.3050	−1.57923	−0.789616	−	0.613602i	$-0.789720\pi$
$823$	−45.3050	−1.57923	−0.789616	+	0.613602i	$0.789720\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33.7771	−1.17454	−0.587272	−	0.809389i	$-0.699798\pi$
$827$	−33.7771	−1.17454	−0.587272	+	0.809389i	$0.699798\pi$
$828$	0	0
$829$	−48.4721	−1.68351	−0.841753	−	0.539862i	$-0.818476\pi$
$829$	−48.4721	−1.68351	−0.841753	+	0.539862i	$0.818476\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.4721	0.847909
$834$	0	0
$835$	−19.6393	−0.679647
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.3607	1.04817	0.524084	−	0.851667i	$-0.324408\pi$
$839$	30.3607	1.04817	0.524084	+	0.851667i	$0.324408\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.23607	−0.0425220
$846$	0	0
$847$	8.65248	0.297303
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.16718	−0.0400106
$852$	0	0
$853$	20.4721	0.700953	0.350476	−	0.936572i	$-0.386020\pi$
$853$	20.4721	0.700953	0.350476	+	0.936572i	$0.386020\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.5836	0.361529	0.180764	−	0.983526i	$-0.442143\pi$
$857$	10.5836	0.361529	0.180764	+	0.983526i	$0.442143\pi$
$858$	0	0
$859$	15.0557	0.513695	0.256847	−	0.966452i	$-0.417316\pi$
$859$	15.0557	0.513695	0.256847	+	0.966452i	$0.417316\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.1115	−0.412279	−0.206139	−	0.978523i	$-0.566090\pi$
$863$	−12.1115	−0.412279	−0.206139	+	0.978523i	$0.566090\pi$
$864$	0	0
$865$	−2.47214	−0.0840551
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−17.8885	−0.606827
$870$	0	0
$871$	−6.76393	−0.229187
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.9443	−0.437596
$876$	0	0
$877$	35.3050	1.19216	0.596082	−	0.802924i	$-0.296723\pi$
$877$	35.3050	1.19216	0.596082	+	0.802924i	$0.296723\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−50.3607	−1.69669	−0.848347	−	0.529440i	$-0.822402\pi$
$881$	−50.3607	−1.69669	−0.848347	+	0.529440i	$0.822402\pi$
$882$	0	0
$883$	48.3607	1.62747	0.813733	−	0.581239i	$-0.197432\pi$
$883$	48.3607	1.62747	0.813733	+	0.581239i	$0.197432\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.3050	−1.52119	−0.760596	−	0.649226i	$-0.775093\pi$
$887$	−45.3050	−1.52119	−0.760596	+	0.649226i	$0.775093\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.3607	1.21676
$894$	0	0
$895$	−9.16718	−0.306425
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.4721	1.14971
$900$	0	0
$901$	−37.8885	−1.26225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.6393	0.386904
$906$	0	0
$907$	24.3607	0.808883	0.404442	−	0.914564i	$-0.367466\pi$
$907$	24.3607	0.808883	0.404442	+	0.914564i	$0.367466\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	54.4721	1.80474	0.902371	−	0.430960i	$-0.141825\pi$
$911$	54.4721	1.80474	0.902371	+	0.430960i	$0.141825\pi$
$912$	0	0
$913$	15.0557	0.498272
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.1115	0.466001
$918$	0	0
$919$	21.8885	0.722036	0.361018	−	0.932559i	$-0.382429\pi$
$919$	21.8885	0.722036	0.361018	+	0.932559i	$0.382429\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.47214	−0.147202
$924$	0	0
$925$	1.63932	0.0539005
$926$	0	0
$927$	0	0
$928$	0	0
$929$	58.5410	1.92067	0.960334	−	0.278851i	$-0.0899537\pi$
$929$	58.5410	1.92067	0.960334	+	0.278851i	$0.0899537\pi$
$930$	0	0
$931$	28.6525	0.939047
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.0557	0.361561
$936$	0	0
$937$	−31.3050	−1.02269	−0.511344	−	0.859376i	$-0.670852\pi$
$937$	−31.3050	−1.02269	−0.511344	+	0.859376i	$0.670852\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.7639	−0.872479	−0.436240	−	0.899831i	$-0.643690\pi$
$941$	−26.7639	−0.872479	−0.436240	+	0.899831i	$0.643690\pi$
$942$	0	0
$943$	−3.05573	−0.0995082
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.5836	0.473903	0.236952	−	0.971521i	$-0.423852\pi$
$947$	14.5836	0.473903	0.236952	+	0.971521i	$0.423852\pi$
$948$	0	0
$949$	0.472136	0.0153262
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−40.4721	−1.31102	−0.655511	−	0.755186i	$-0.727547\pi$
$953$	−40.4721	−1.31102	−0.655511	+	0.755186i	$0.727547\pi$
$954$	0	0
$955$	−1.88854	−0.0611118
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.36068	−0.269980
$960$	0	0
$961$	28.4164	0.916658
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−24.5836	−0.791374
$966$	0	0
$967$	10.7639	0.346145	0.173072	−	0.984909i	$-0.444631\pi$
$967$	10.7639	0.346145	0.173072	+	0.984909i	$0.444631\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.83282	0.0909094	0.0454547	−	0.998966i	$-0.485526\pi$
$971$	2.83282	0.0909094	0.0454547	+	0.998966i	$0.485526\pi$
$972$	0	0
$973$	−17.1672	−0.550355
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.291796	0.00933538	0.00466769	−	0.999989i	$-0.498514\pi$
$977$	0.291796	0.00933538	0.00466769	+	0.999989i	$0.498514\pi$
$978$	0	0
$979$	20.3607	0.650730
$980$	0	0
$981$	0	0
$982$	0	0
$983$	49.4164	1.57614	0.788069	−	0.615587i	$-0.211081\pi$
$983$	49.4164	1.57614	0.788069	+	0.615587i	$0.211081\pi$
$984$	0	0
$985$	−28.1378	−0.896544
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−23.0557	−0.732389	−0.366195	−	0.930538i	$-0.619340\pi$
$991$	−23.0557	−0.732389	−0.366195	+	0.930538i	$0.619340\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.7771	0.373359
$996$	0	0
$997$	24.1115	0.763617	0.381809	−	0.924241i	$-0.375301\pi$
$997$	24.1115	0.763617	0.381809	+	0.924241i	$0.375301\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.a.ct.1.1		2
3.2	odd	2		2496.2.a.be.1.2		2
4.3	odd	2		7488.2.a.cs.1.1		2
8.3	odd	2		3744.2.a.r.1.2		2
8.5	even	2		3744.2.a.s.1.2		2
12.11	even	2		2496.2.a.bh.1.2		2
24.5	odd	2		1248.2.a.n.1.1	yes	2
24.11	even	2		1248.2.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.l.1.1	✓	2	24.11	even	2
1248.2.a.n.1.1	yes	2	24.5	odd	2
2496.2.a.be.1.2		2	3.2	odd	2
2496.2.a.bh.1.2		2	12.11	even	2
3744.2.a.r.1.2		2	8.3	odd	2
3744.2.a.s.1.2		2	8.5	even	2
7488.2.a.cs.1.1		2	4.3	odd	2
7488.2.a.ct.1.1		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:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{5} -1.23607 q^{7} +O(q^{10})\)	\(q-1.23607 q^{5} -1.23607 q^{7} +2.00000 q^{11} +1.00000 q^{13} -4.47214 q^{17} -5.23607 q^{19} +2.47214 q^{23} -3.47214 q^{25} +4.47214 q^{29} +7.70820 q^{31} +1.52786 q^{35} -0.472136 q^{37} -1.23607 q^{41} -6.47214 q^{43} -6.94427 q^{47} -5.47214 q^{49} +8.47214 q^{53} -2.47214 q^{55} +8.47214 q^{59} +12.4721 q^{61} -1.23607 q^{65} -6.76393 q^{67} -4.47214 q^{71} +0.472136 q^{73} -2.47214 q^{77} -8.94427 q^{79} +7.52786 q^{83} +5.52786 q^{85} +10.1803 q^{89} -1.23607 q^{91} +6.47214 q^{95} -4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{13} - 6 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{31} + 12 q^{35} + 8 q^{37} + 2 q^{41} - 4 q^{43} + 4 q^{47} - 2 q^{49} + 8 q^{53} + 4 q^{55} + 8 q^{59} + 16 q^{61} + 2 q^{65} - 18 q^{67} - 8 q^{73} + 4 q^{77} + 24 q^{83} + 20 q^{85} - 2 q^{89} + 2 q^{91} + 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 + 2 * q^13 - 6 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^31 + 12 * q^35 + 8 * q^37 + 2 * q^41 - 4 * q^43 + 4 * q^47 - 2 * q^49 + 8 * q^53 + 4 * q^55 + 8 * q^59 + 16 * q^61 + 2 * q^65 - 18 * q^67 - 8 * q^73 + 4 * q^77 + 24 * q^83 + 20 * q^85 - 2 * q^89 + 2 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more