LMFDB - Embedded newform 7440.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.56155 q^{7} +1.00000 q^{9} +1.56155 q^{11} +2.00000 q^{13} +1.00000 q^{15} +5.12311 q^{17} -4.68466 q^{19} +1.56155 q^{21} -5.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.12311 q^{29} -1.00000 q^{31} +1.56155 q^{33} +1.56155 q^{35} +5.12311 q^{37} +2.00000 q^{39} -1.12311 q^{41} +7.80776 q^{43} +1.00000 q^{45} +3.12311 q^{47} -4.56155 q^{49} +5.12311 q^{51} +11.5616 q^{53} +1.56155 q^{55} -4.68466 q^{57} +4.87689 q^{59} +6.00000 q^{61} +1.56155 q^{63} +2.00000 q^{65} -9.36932 q^{67} -5.56155 q^{69} -4.68466 q^{71} +9.80776 q^{73} +1.00000 q^{75} +2.43845 q^{77} +16.6847 q^{79} +1.00000 q^{81} +2.24621 q^{83} +5.12311 q^{85} -1.12311 q^{87} -1.31534 q^{89} +3.12311 q^{91} -1.00000 q^{93} -4.68466 q^{95} -6.00000 q^{97} +1.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 + 3 * q^19 - q^21 - 7 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^31 - q^33 - q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 - 5 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 + 2 * q^51 + 19 * q^53 - q^55 + 3 * q^57 + 18 * q^59 + 12 * q^61 - q^63 + 4 * q^65 + 6 * q^67 - 7 * q^69 + 3 * q^71 - q^73 + 2 * q^75 + 9 * q^77 + 21 * q^79 + 2 * q^81 - 12 * q^83 + 2 * q^85 + 6 * q^87 - 15 * q^89 - 2 * q^91 - 2 * q^93 + 3 * q^95 - 12 * q^97 - q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	−4.68466	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$19$	−4.68466	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$20$	0	0
$21$	1.56155	0.340759
$22$	0	0
$23$	−5.56155	−1.15966	−0.579832	−	0.814736i	$-0.696882\pi$
$23$	−5.56155	−1.15966	−0.579832	+	0.814736i	$0.696882\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	1.56155	0.271831
$34$	0	0
$35$	1.56155	0.263951
$36$	0	0
$37$	5.12311	0.842233	0.421117	−	0.907006i	$-0.361638\pi$
$37$	5.12311	0.842233	0.421117	+	0.907006i	$0.361638\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	7.80776	1.19067	0.595336	−	0.803477i	$-0.297019\pi$
$43$	7.80776	1.19067	0.595336	+	0.803477i	$0.297019\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	3.12311	0.455552	0.227776	−	0.973714i	$-0.426855\pi$
$47$	3.12311	0.455552	0.227776	+	0.973714i	$0.426855\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	5.12311	0.717378
$52$	0	0
$53$	11.5616	1.58810	0.794051	−	0.607852i	$-0.207968\pi$
$53$	11.5616	1.58810	0.794051	+	0.607852i	$0.207968\pi$
$54$	0	0
$55$	1.56155	0.210560
$56$	0	0
$57$	−4.68466	−0.620498
$58$	0	0
$59$	4.87689	0.634918	0.317459	−	0.948272i	$-0.397170\pi$
$59$	4.87689	0.634918	0.317459	+	0.948272i	$0.397170\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	1.56155	0.196737
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−9.36932	−1.14464	−0.572322	−	0.820029i	$-0.693957\pi$
$67$	−9.36932	−1.14464	−0.572322	+	0.820029i	$0.693957\pi$
$68$	0	0
$69$	−5.56155	−0.669532
$70$	0	0
$71$	−4.68466	−0.555967	−0.277983	−	0.960586i	$-0.589666\pi$
$71$	−4.68466	−0.555967	−0.277983	+	0.960586i	$0.589666\pi$
$72$	0	0
$73$	9.80776	1.14791	0.573956	−	0.818886i	$-0.305408\pi$
$73$	9.80776	1.14791	0.573956	+	0.818886i	$0.305408\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.43845	0.277887
$78$	0	0
$79$	16.6847	1.87717	0.938585	−	0.345047i	$-0.112137\pi$
$79$	16.6847	1.87717	0.938585	+	0.345047i	$0.112137\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	5.12311	0.555679
$86$	0	0
$87$	−1.12311	−0.120410
$88$	0	0
$89$	−1.31534	−0.139426	−0.0697130	−	0.997567i	$-0.522208\pi$
$89$	−1.31534	−0.139426	−0.0697130	+	0.997567i	$0.522208\pi$
$90$	0	0
$91$	3.12311	0.327390
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−4.68466	−0.480636
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.56155	0.156942
$100$	0	0
$101$	3.56155	0.354388	0.177194	−	0.984176i	$-0.443298\pi$
$101$	3.56155	0.354388	0.177194	+	0.984176i	$0.443298\pi$
$102$	0	0
$103$	18.2462	1.79785	0.898926	−	0.438100i	$-0.144348\pi$
$103$	18.2462	1.79785	0.898926	+	0.438100i	$0.144348\pi$
$104$	0	0
$105$	1.56155	0.152392
$106$	0	0
$107$	−1.56155	−0.150961	−0.0754805	−	0.997147i	$-0.524049\pi$
$107$	−1.56155	−0.150961	−0.0754805	+	0.997147i	$0.524049\pi$
$108$	0	0
$109$	−19.3693	−1.85524	−0.927622	−	0.373520i	$-0.878151\pi$
$109$	−19.3693	−1.85524	−0.927622	+	0.373520i	$0.878151\pi$
$110$	0	0
$111$	5.12311	0.486264
$112$	0	0
$113$	10.6847	1.00513	0.502564	−	0.864540i	$-0.332390\pi$
$113$	10.6847	1.00513	0.502564	+	0.864540i	$0.332390\pi$
$114$	0	0
$115$	−5.56155	−0.518617
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	−1.12311	−0.101267
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	7.80776	0.687435
$130$	0	0
$131$	−9.36932	−0.818601	−0.409301	−	0.912400i	$-0.634227\pi$
$131$	−9.36932	−0.818601	−0.409301	+	0.912400i	$0.634227\pi$
$132$	0	0
$133$	−7.31534	−0.634321
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	3.75379	0.320708	0.160354	−	0.987060i	$-0.448736\pi$
$137$	3.75379	0.320708	0.160354	+	0.987060i	$0.448736\pi$
$138$	0	0
$139$	8.87689	0.752928	0.376464	−	0.926431i	$-0.377140\pi$
$139$	8.87689	0.752928	0.376464	+	0.926431i	$0.377140\pi$
$140$	0	0
$141$	3.12311	0.263013
$142$	0	0
$143$	3.12311	0.261167
$144$	0	0
$145$	−1.12311	−0.0932688
$146$	0	0
$147$	−4.56155	−0.376231
$148$	0	0
$149$	−20.0540	−1.64289	−0.821443	−	0.570291i	$-0.806830\pi$
$149$	−20.0540	−1.64289	−0.821443	+	0.570291i	$0.806830\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	5.12311	0.414179
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−20.0540	−1.60048	−0.800241	−	0.599679i	$-0.795295\pi$
$157$	−20.0540	−1.60048	−0.800241	+	0.599679i	$0.795295\pi$
$158$	0	0
$159$	11.5616	0.916891
$160$	0	0
$161$	−8.68466	−0.684447
$162$	0	0
$163$	−9.36932	−0.733862	−0.366931	−	0.930248i	$-0.619591\pi$
$163$	−9.36932	−0.733862	−0.366931	+	0.930248i	$0.619591\pi$
$164$	0	0
$165$	1.56155	0.121567
$166$	0	0
$167$	2.43845	0.188693	0.0943464	−	0.995539i	$-0.469924\pi$
$167$	2.43845	0.188693	0.0943464	+	0.995539i	$0.469924\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.68466	−0.358245
$172$	0	0
$173$	−8.24621	−0.626948	−0.313474	−	0.949597i	$-0.601493\pi$
$173$	−8.24621	−0.626948	−0.313474	+	0.949597i	$0.601493\pi$
$174$	0	0
$175$	1.56155	0.118042
$176$	0	0
$177$	4.87689	0.366570
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	24.0540	1.78792	0.893959	−	0.448149i	$-0.147917\pi$
$181$	24.0540	1.78792	0.893959	+	0.448149i	$0.147917\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	5.12311	0.376658
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	1.56155	0.113586
$190$	0	0
$191$	−2.24621	−0.162530	−0.0812651	−	0.996693i	$-0.525896\pi$
$191$	−2.24621	−0.162530	−0.0812651	+	0.996693i	$0.525896\pi$
$192$	0	0
$193$	−18.4924	−1.33111	−0.665557	−	0.746347i	$-0.731806\pi$
$193$	−18.4924	−1.33111	−0.665557	+	0.746347i	$0.731806\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−3.80776	−0.269925	−0.134963	−	0.990851i	$-0.543091\pi$
$199$	−3.80776	−0.269925	−0.134963	+	0.990851i	$0.543091\pi$
$200$	0	0
$201$	−9.36932	−0.660861
$202$	0	0
$203$	−1.75379	−0.123092
$204$	0	0
$205$	−1.12311	−0.0784411
$206$	0	0
$207$	−5.56155	−0.386555
$208$	0	0
$209$	−7.31534	−0.506013
$210$	0	0
$211$	11.3153	0.778980	0.389490	−	0.921031i	$-0.372651\pi$
$211$	11.3153	0.778980	0.389490	+	0.921031i	$0.372651\pi$
$212$	0	0
$213$	−4.68466	−0.320988
$214$	0	0
$215$	7.80776	0.532485
$216$	0	0
$217$	−1.56155	−0.106005
$218$	0	0
$219$	9.80776	0.662747
$220$	0	0
$221$	10.2462	0.689235
$222$	0	0
$223$	12.8769	0.862301	0.431150	−	0.902280i	$-0.358108\pi$
$223$	12.8769	0.862301	0.431150	+	0.902280i	$0.358108\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	4.68466	0.310932	0.155466	−	0.987841i	$-0.450312\pi$
$227$	4.68466	0.310932	0.155466	+	0.987841i	$0.450312\pi$
$228$	0	0
$229$	−12.4384	−0.821956	−0.410978	−	0.911645i	$-0.634813\pi$
$229$	−12.4384	−0.821956	−0.410978	+	0.911645i	$0.634813\pi$
$230$	0	0
$231$	2.43845	0.160438
$232$	0	0
$233$	20.0540	1.31378	0.656890	−	0.753987i	$-0.271872\pi$
$233$	20.0540	1.31378	0.656890	+	0.753987i	$0.271872\pi$
$234$	0	0
$235$	3.12311	0.203729
$236$	0	0
$237$	16.6847	1.08379
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.56155	−0.291427
$246$	0	0
$247$	−9.36932	−0.596155
$248$	0	0
$249$	2.24621	0.142348
$250$	0	0
$251$	16.4924	1.04099	0.520496	−	0.853864i	$-0.325747\pi$
$251$	16.4924	1.04099	0.520496	+	0.853864i	$0.325747\pi$
$252$	0	0
$253$	−8.68466	−0.546000
$254$	0	0
$255$	5.12311	0.320821
$256$	0	0
$257$	10.6847	0.666491	0.333245	−	0.942840i	$-0.391856\pi$
$257$	10.6847	0.666491	0.333245	+	0.942840i	$0.391856\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−1.12311	−0.0695185
$262$	0	0
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	0	0
$265$	11.5616	0.710221
$266$	0	0
$267$	−1.31534	−0.0804976
$268$	0	0
$269$	16.2462	0.990549	0.495274	−	0.868737i	$-0.335067\pi$
$269$	16.2462	0.990549	0.495274	+	0.868737i	$0.335067\pi$
$270$	0	0
$271$	10.0540	0.610736	0.305368	−	0.952234i	$-0.401221\pi$
$271$	10.0540	0.610736	0.305368	+	0.952234i	$0.401221\pi$
$272$	0	0
$273$	3.12311	0.189019
$274$	0	0
$275$	1.56155	0.0941652
$276$	0	0
$277$	19.3693	1.16379	0.581895	−	0.813264i	$-0.302312\pi$
$277$	19.3693	1.16379	0.581895	+	0.813264i	$0.302312\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	13.1231	0.782859	0.391429	−	0.920208i	$-0.371981\pi$
$281$	13.1231	0.782859	0.391429	+	0.920208i	$0.371981\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	−4.68466	−0.277495
$286$	0	0
$287$	−1.75379	−0.103523
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−6.49242	−0.379291	−0.189646	−	0.981853i	$-0.560734\pi$
$293$	−6.49242	−0.379291	−0.189646	+	0.981853i	$0.560734\pi$
$294$	0	0
$295$	4.87689	0.283944
$296$	0	0
$297$	1.56155	0.0906105
$298$	0	0
$299$	−11.1231	−0.643266
$300$	0	0
$301$	12.1922	0.702749
$302$	0	0
$303$	3.56155	0.204606
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	18.2462	1.03799
$310$	0	0
$311$	18.2462	1.03465	0.517324	−	0.855790i	$-0.326928\pi$
$311$	18.2462	1.03465	0.517324	+	0.855790i	$0.326928\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	1.56155	0.0879835
$316$	0	0
$317$	25.1231	1.41105	0.705527	−	0.708683i	$-0.250710\pi$
$317$	25.1231	1.41105	0.705527	+	0.708683i	$0.250710\pi$
$318$	0	0
$319$	−1.75379	−0.0981933
$320$	0	0
$321$	−1.56155	−0.0871574
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−19.3693	−1.07113
$328$	0	0
$329$	4.87689	0.268872
$330$	0	0
$331$	−10.2462	−0.563183	−0.281591	−	0.959534i	$-0.590862\pi$
$331$	−10.2462	−0.563183	−0.281591	+	0.959534i	$0.590862\pi$
$332$	0	0
$333$	5.12311	0.280744
$334$	0	0
$335$	−9.36932	−0.511900
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	10.6847	0.580311
$340$	0	0
$341$	−1.56155	−0.0845628
$342$	0	0
$343$	−18.0540	−0.974823
$344$	0	0
$345$	−5.56155	−0.299424
$346$	0	0
$347$	2.24621	0.120583	0.0602915	−	0.998181i	$-0.480797\pi$
$347$	2.24621	0.120583	0.0602915	+	0.998181i	$0.480797\pi$
$348$	0	0
$349$	−19.3693	−1.03682	−0.518408	−	0.855133i	$-0.673475\pi$
$349$	−19.3693	−1.03682	−0.518408	+	0.855133i	$0.673475\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	16.2462	0.864699	0.432349	−	0.901706i	$-0.357685\pi$
$353$	16.2462	0.864699	0.432349	+	0.901706i	$0.357685\pi$
$354$	0	0
$355$	−4.68466	−0.248636
$356$	0	0
$357$	8.00000	0.423405
$358$	0	0
$359$	19.3153	1.01942	0.509712	−	0.860345i	$-0.329752\pi$
$359$	19.3153	1.01942	0.509712	+	0.860345i	$0.329752\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	0	0
$363$	−8.56155	−0.449365
$364$	0	0
$365$	9.80776	0.513362
$366$	0	0
$367$	9.36932	0.489074	0.244537	−	0.969640i	$-0.421364\pi$
$367$	9.36932	0.489074	0.244537	+	0.969640i	$0.421364\pi$
$368$	0	0
$369$	−1.12311	−0.0584665
$370$	0	0
$371$	18.0540	0.937316
$372$	0	0
$373$	6.68466	0.346118	0.173059	−	0.984911i	$-0.444635\pi$
$373$	6.68466	0.346118	0.173059	+	0.984911i	$0.444635\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−2.24621	−0.115686
$378$	0	0
$379$	−30.0540	−1.54377	−0.771885	−	0.635763i	$-0.780686\pi$
$379$	−30.0540	−1.54377	−0.771885	+	0.635763i	$0.780686\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	0	0
$385$	2.43845	0.124275
$386$	0	0
$387$	7.80776	0.396891
$388$	0	0
$389$	24.2462	1.22933	0.614666	−	0.788788i	$-0.289291\pi$
$389$	24.2462	1.22933	0.614666	+	0.788788i	$0.289291\pi$
$390$	0	0
$391$	−28.4924	−1.44092
$392$	0	0
$393$	−9.36932	−0.472620
$394$	0	0
$395$	16.6847	0.839496
$396$	0	0
$397$	−28.0540	−1.40799	−0.703994	−	0.710206i	$-0.748602\pi$
$397$	−28.0540	−1.40799	−0.703994	+	0.710206i	$0.748602\pi$
$398$	0	0
$399$	−7.31534	−0.366225
$400$	0	0
$401$	−1.31534	−0.0656850	−0.0328425	−	0.999461i	$-0.510456\pi$
$401$	−1.31534	−0.0656850	−0.0328425	+	0.999461i	$0.510456\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−12.6307	−0.624547	−0.312274	−	0.949992i	$-0.601091\pi$
$409$	−12.6307	−0.624547	−0.312274	+	0.949992i	$0.601091\pi$
$410$	0	0
$411$	3.75379	0.185161
$412$	0	0
$413$	7.61553	0.374736
$414$	0	0
$415$	2.24621	0.110262
$416$	0	0
$417$	8.87689	0.434703
$418$	0	0
$419$	−22.2462	−1.08680	−0.543399	−	0.839474i	$-0.682863\pi$
$419$	−22.2462	−1.08680	−0.543399	+	0.839474i	$0.682863\pi$
$420$	0	0
$421$	18.4924	0.901266	0.450633	−	0.892709i	$-0.351198\pi$
$421$	18.4924	0.901266	0.450633	+	0.892709i	$0.351198\pi$
$422$	0	0
$423$	3.12311	0.151851
$424$	0	0
$425$	5.12311	0.248507
$426$	0	0
$427$	9.36932	0.453413
$428$	0	0
$429$	3.12311	0.150785
$430$	0	0
$431$	13.7538	0.662497	0.331248	−	0.943544i	$-0.392530\pi$
$431$	13.7538	0.662497	0.331248	+	0.943544i	$0.392530\pi$
$432$	0	0
$433$	−18.3002	−0.879451	−0.439725	−	0.898132i	$-0.644924\pi$
$433$	−18.3002	−0.879451	−0.439725	+	0.898132i	$0.644924\pi$
$434$	0	0
$435$	−1.12311	−0.0538488
$436$	0	0
$437$	26.0540	1.24633
$438$	0	0
$439$	31.6155	1.50893	0.754463	−	0.656342i	$-0.227897\pi$
$439$	31.6155	1.50893	0.754463	+	0.656342i	$0.227897\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	14.0540	0.667725	0.333862	−	0.942622i	$-0.391648\pi$
$443$	14.0540	0.667725	0.333862	+	0.942622i	$0.391648\pi$
$444$	0	0
$445$	−1.31534	−0.0623532
$446$	0	0
$447$	−20.0540	−0.948520
$448$	0	0
$449$	−22.4924	−1.06148	−0.530742	−	0.847534i	$-0.678087\pi$
$449$	−22.4924	−1.06148	−0.530742	+	0.847534i	$0.678087\pi$
$450$	0	0
$451$	−1.75379	−0.0825827
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.12311	0.146413
$456$	0	0
$457$	24.7386	1.15722	0.578612	−	0.815603i	$-0.303594\pi$
$457$	24.7386	1.15722	0.578612	+	0.815603i	$0.303594\pi$
$458$	0	0
$459$	5.12311	0.239126
$460$	0	0
$461$	−15.3693	−0.715820	−0.357910	−	0.933756i	$-0.616511\pi$
$461$	−15.3693	−0.715820	−0.357910	+	0.933756i	$0.616511\pi$
$462$	0	0
$463$	18.7386	0.870858	0.435429	−	0.900223i	$-0.356597\pi$
$463$	18.7386	0.870858	0.435429	+	0.900223i	$0.356597\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	26.2462	1.21453	0.607265	−	0.794499i	$-0.292267\pi$
$467$	26.2462	1.21453	0.607265	+	0.794499i	$0.292267\pi$
$468$	0	0
$469$	−14.6307	−0.675582
$470$	0	0
$471$	−20.0540	−0.924038
$472$	0	0
$473$	12.1922	0.560600
$474$	0	0
$475$	−4.68466	−0.214947
$476$	0	0
$477$	11.5616	0.529367
$478$	0	0
$479$	−6.43845	−0.294180	−0.147090	−	0.989123i	$-0.546991\pi$
$479$	−6.43845	−0.294180	−0.147090	+	0.989123i	$0.546991\pi$
$480$	0	0
$481$	10.2462	0.467187
$482$	0	0
$483$	−8.68466	−0.395166
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	−9.36932	−0.423695
$490$	0	0
$491$	−2.93087	−0.132268	−0.0661341	−	0.997811i	$-0.521067\pi$
$491$	−2.93087	−0.132268	−0.0661341	+	0.997811i	$0.521067\pi$
$492$	0	0
$493$	−5.75379	−0.259138
$494$	0	0
$495$	1.56155	0.0701866
$496$	0	0
$497$	−7.31534	−0.328138
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	2.43845	0.108942
$502$	0	0
$503$	−9.75379	−0.434900	−0.217450	−	0.976071i	$-0.569774\pi$
$503$	−9.75379	−0.434900	−0.217450	+	0.976071i	$0.569774\pi$
$504$	0	0
$505$	3.56155	0.158487
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	−21.6155	−0.958091	−0.479046	−	0.877790i	$-0.659017\pi$
$509$	−21.6155	−0.958091	−0.479046	+	0.877790i	$0.659017\pi$
$510$	0	0
$511$	15.3153	0.677511
$512$	0	0
$513$	−4.68466	−0.206833
$514$	0	0
$515$	18.2462	0.804024
$516$	0	0
$517$	4.87689	0.214486
$518$	0	0
$519$	−8.24621	−0.361968
$520$	0	0
$521$	−15.7538	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$521$	−15.7538	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$522$	0	0
$523$	−39.4233	−1.72386	−0.861930	−	0.507027i	$-0.830744\pi$
$523$	−39.4233	−1.72386	−0.861930	+	0.507027i	$0.830744\pi$
$524$	0	0
$525$	1.56155	0.0681518
$526$	0	0
$527$	−5.12311	−0.223166
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	4.87689	0.211639
$532$	0	0
$533$	−2.24621	−0.0972942
$534$	0	0
$535$	−1.56155	−0.0675118
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−7.12311	−0.306814
$540$	0	0
$541$	36.2462	1.55835	0.779173	−	0.626809i	$-0.215639\pi$
$541$	36.2462	1.55835	0.779173	+	0.626809i	$0.215639\pi$
$542$	0	0
$543$	24.0540	1.03225
$544$	0	0
$545$	−19.3693	−0.829690
$546$	0	0
$547$	−35.1231	−1.50176	−0.750878	−	0.660441i	$-0.770369\pi$
$547$	−35.1231	−1.50176	−0.750878	+	0.660441i	$0.770369\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	5.26137	0.224142
$552$	0	0
$553$	26.0540	1.10793
$554$	0	0
$555$	5.12311	0.217464
$556$	0	0
$557$	−6.19224	−0.262373	−0.131187	−	0.991358i	$-0.541879\pi$
$557$	−6.19224	−0.262373	−0.131187	+	0.991358i	$0.541879\pi$
$558$	0	0
$559$	15.6155	0.660466
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	16.4924	0.695073	0.347536	−	0.937667i	$-0.387018\pi$
$563$	16.4924	0.695073	0.347536	+	0.937667i	$0.387018\pi$
$564$	0	0
$565$	10.6847	0.449507
$566$	0	0
$567$	1.56155	0.0655791
$568$	0	0
$569$	10.1922	0.427281	0.213640	−	0.976912i	$-0.431468\pi$
$569$	10.1922	0.427281	0.213640	+	0.976912i	$0.431468\pi$
$570$	0	0
$571$	−32.8769	−1.37586	−0.687928	−	0.725779i	$-0.741479\pi$
$571$	−32.8769	−1.37586	−0.687928	+	0.725779i	$0.741479\pi$
$572$	0	0
$573$	−2.24621	−0.0938368
$574$	0	0
$575$	−5.56155	−0.231933
$576$	0	0
$577$	6.87689	0.286289	0.143144	−	0.989702i	$-0.454279\pi$
$577$	6.87689	0.286289	0.143144	+	0.989702i	$0.454279\pi$
$578$	0	0
$579$	−18.4924	−0.768519
$580$	0	0
$581$	3.50758	0.145519
$582$	0	0
$583$	18.0540	0.747719
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−32.4924	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$587$	−32.4924	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$588$	0	0
$589$	4.68466	0.193028
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	−3.80776	−0.155841
$598$	0	0
$599$	23.4233	0.957050	0.478525	−	0.878074i	$-0.341172\pi$
$599$	23.4233	0.957050	0.478525	+	0.878074i	$0.341172\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−9.36932	−0.381548
$604$	0	0
$605$	−8.56155	−0.348077
$606$	0	0
$607$	30.0540	1.21985	0.609927	−	0.792458i	$-0.291199\pi$
$607$	30.0540	1.21985	0.609927	+	0.792458i	$0.291199\pi$
$608$	0	0
$609$	−1.75379	−0.0710671
$610$	0	0
$611$	6.24621	0.252695
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−1.12311	−0.0452880
$616$	0	0
$617$	−19.5616	−0.787518	−0.393759	−	0.919214i	$-0.628826\pi$
$617$	−19.5616	−0.787518	−0.393759	+	0.919214i	$0.628826\pi$
$618$	0	0
$619$	13.3693	0.537358	0.268679	−	0.963230i	$-0.413413\pi$
$619$	13.3693	0.537358	0.268679	+	0.963230i	$0.413413\pi$
$620$	0	0
$621$	−5.56155	−0.223177
$622$	0	0
$623$	−2.05398	−0.0822908
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.31534	−0.292147
$628$	0	0
$629$	26.2462	1.04650
$630$	0	0
$631$	24.3002	0.967375	0.483688	−	0.875241i	$-0.339297\pi$
$631$	24.3002	0.967375	0.483688	+	0.875241i	$0.339297\pi$
$632$	0	0
$633$	11.3153	0.449744
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.12311	−0.361471
$638$	0	0
$639$	−4.68466	−0.185322
$640$	0	0
$641$	−38.4924	−1.52036	−0.760180	−	0.649713i	$-0.774889\pi$
$641$	−38.4924	−1.52036	−0.760180	+	0.649713i	$0.774889\pi$
$642$	0	0
$643$	−30.0540	−1.18521	−0.592607	−	0.805492i	$-0.701901\pi$
$643$	−30.0540	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$644$	0	0
$645$	7.80776	0.307430
$646$	0	0
$647$	17.0691	0.671057	0.335528	−	0.942030i	$-0.391085\pi$
$647$	17.0691	0.671057	0.335528	+	0.942030i	$0.391085\pi$
$648$	0	0
$649$	7.61553	0.298936
$650$	0	0
$651$	−1.56155	−0.0612021
$652$	0	0
$653$	23.7538	0.929558	0.464779	−	0.885427i	$-0.346134\pi$
$653$	23.7538	0.929558	0.464779	+	0.885427i	$0.346134\pi$
$654$	0	0
$655$	−9.36932	−0.366090
$656$	0	0
$657$	9.80776	0.382637
$658$	0	0
$659$	−18.7386	−0.729954	−0.364977	−	0.931017i	$-0.618923\pi$
$659$	−18.7386	−0.729954	−0.364977	+	0.931017i	$0.618923\pi$
$660$	0	0
$661$	−6.49242	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$661$	−6.49242	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$662$	0	0
$663$	10.2462	0.397930
$664$	0	0
$665$	−7.31534	−0.283677
$666$	0	0
$667$	6.24621	0.241854
$668$	0	0
$669$	12.8769	0.497849
$670$	0	0
$671$	9.36932	0.361698
$672$	0	0
$673$	28.2462	1.08881	0.544406	−	0.838822i	$-0.316755\pi$
$673$	28.2462	1.08881	0.544406	+	0.838822i	$0.316755\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−29.4233	−1.13083	−0.565414	−	0.824807i	$-0.691284\pi$
$677$	−29.4233	−1.13083	−0.565414	+	0.824807i	$0.691284\pi$
$678$	0	0
$679$	−9.36932	−0.359561
$680$	0	0
$681$	4.68466	0.179517
$682$	0	0
$683$	−19.3153	−0.739081	−0.369541	−	0.929215i	$-0.620485\pi$
$683$	−19.3153	−0.739081	−0.369541	+	0.929215i	$0.620485\pi$
$684$	0	0
$685$	3.75379	0.143425
$686$	0	0
$687$	−12.4384	−0.474556
$688$	0	0
$689$	23.1231	0.880920
$690$	0	0
$691$	22.0540	0.838973	0.419486	−	0.907762i	$-0.362210\pi$
$691$	22.0540	0.838973	0.419486	+	0.907762i	$0.362210\pi$
$692$	0	0
$693$	2.43845	0.0926289
$694$	0	0
$695$	8.87689	0.336720
$696$	0	0
$697$	−5.75379	−0.217940
$698$	0	0
$699$	20.0540	0.758511
$700$	0	0
$701$	25.8078	0.974746	0.487373	−	0.873194i	$-0.337955\pi$
$701$	25.8078	0.974746	0.487373	+	0.873194i	$0.337955\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	3.12311	0.117623
$706$	0	0
$707$	5.56155	0.209164
$708$	0	0
$709$	−20.0540	−0.753143	−0.376571	−	0.926388i	$-0.622897\pi$
$709$	−20.0540	−0.753143	−0.376571	+	0.926388i	$0.622897\pi$
$710$	0	0
$711$	16.6847	0.625724
$712$	0	0
$713$	5.56155	0.208282
$714$	0	0
$715$	3.12311	0.116798
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	43.1231	1.60822	0.804110	−	0.594480i	$-0.202642\pi$
$719$	43.1231	1.60822	0.804110	+	0.594480i	$0.202642\pi$
$720$	0	0
$721$	28.4924	1.06111
$722$	0	0
$723$	−4.24621	−0.157918
$724$	0	0
$725$	−1.12311	−0.0417111
$726$	0	0
$727$	14.0540	0.521233	0.260617	−	0.965442i	$-0.416074\pi$
$727$	14.0540	0.521233	0.260617	+	0.965442i	$0.416074\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	26.4924	0.978520	0.489260	−	0.872138i	$-0.337267\pi$
$733$	26.4924	0.978520	0.489260	+	0.872138i	$0.337267\pi$
$734$	0	0
$735$	−4.56155	−0.168255
$736$	0	0
$737$	−14.6307	−0.538928
$738$	0	0
$739$	12.9848	0.477655	0.238828	−	0.971062i	$-0.423237\pi$
$739$	12.9848	0.477655	0.238828	+	0.971062i	$0.423237\pi$
$740$	0	0
$741$	−9.36932	−0.344190
$742$	0	0
$743$	−11.4233	−0.419080	−0.209540	−	0.977800i	$-0.567197\pi$
$743$	−11.4233	−0.419080	−0.209540	+	0.977800i	$0.567197\pi$
$744$	0	0
$745$	−20.0540	−0.734721
$746$	0	0
$747$	2.24621	0.0821846
$748$	0	0
$749$	−2.43845	−0.0890989
$750$	0	0
$751$	−36.8769	−1.34566	−0.672828	−	0.739798i	$-0.734921\pi$
$751$	−36.8769	−1.34566	−0.672828	+	0.739798i	$0.734921\pi$
$752$	0	0
$753$	16.4924	0.601017
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	−8.68466	−0.315233
$760$	0	0
$761$	41.4233	1.50159	0.750797	−	0.660533i	$-0.229670\pi$
$761$	41.4233	1.50159	0.750797	+	0.660533i	$0.229670\pi$
$762$	0	0
$763$	−30.2462	−1.09499
$764$	0	0
$765$	5.12311	0.185226
$766$	0	0
$767$	9.75379	0.352189
$768$	0	0
$769$	−16.4384	−0.592786	−0.296393	−	0.955066i	$-0.595784\pi$
$769$	−16.4384	−0.592786	−0.296393	+	0.955066i	$0.595784\pi$
$770$	0	0
$771$	10.6847	0.384799
$772$	0	0
$773$	−15.5616	−0.559710	−0.279855	−	0.960042i	$-0.590286\pi$
$773$	−15.5616	−0.559710	−0.279855	+	0.960042i	$0.590286\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	5.26137	0.188508
$780$	0	0
$781$	−7.31534	−0.261764
$782$	0	0
$783$	−1.12311	−0.0401365
$784$	0	0
$785$	−20.0540	−0.715757
$786$	0	0
$787$	−10.9309	−0.389643	−0.194822	−	0.980839i	$-0.562413\pi$
$787$	−10.9309	−0.389643	−0.194822	+	0.980839i	$0.562413\pi$
$788$	0	0
$789$	−12.4924	−0.444742
$790$	0	0
$791$	16.6847	0.593238
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	11.5616	0.410046
$796$	0	0
$797$	38.9848	1.38091	0.690457	−	0.723373i	$-0.257409\pi$
$797$	38.9848	1.38091	0.690457	+	0.723373i	$0.257409\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−1.31534	−0.0464753
$802$	0	0
$803$	15.3153	0.540467
$804$	0	0
$805$	−8.68466	−0.306094
$806$	0	0
$807$	16.2462	0.571894
$808$	0	0
$809$	−25.3153	−0.890040	−0.445020	−	0.895521i	$-0.646803\pi$
$809$	−25.3153	−0.890040	−0.445020	+	0.895521i	$0.646803\pi$
$810$	0	0
$811$	−53.6695	−1.88459	−0.942296	−	0.334782i	$-0.891337\pi$
$811$	−53.6695	−1.88459	−0.942296	+	0.334782i	$0.891337\pi$
$812$	0	0
$813$	10.0540	0.352608
$814$	0	0
$815$	−9.36932	−0.328193
$816$	0	0
$817$	−36.5767	−1.27966
$818$	0	0
$819$	3.12311	0.109130
$820$	0	0
$821$	−21.6155	−0.754387	−0.377194	−	0.926134i	$-0.623111\pi$
$821$	−21.6155	−0.754387	−0.377194	+	0.926134i	$0.623111\pi$
$822$	0	0
$823$	−15.6155	−0.544323	−0.272162	−	0.962252i	$-0.587739\pi$
$823$	−15.6155	−0.544323	−0.272162	+	0.962252i	$0.587739\pi$
$824$	0	0
$825$	1.56155	0.0543663
$826$	0	0
$827$	−18.2462	−0.634483	−0.317241	−	0.948345i	$-0.602757\pi$
$827$	−18.2462	−0.634483	−0.317241	+	0.948345i	$0.602757\pi$
$828$	0	0
$829$	−37.4233	−1.29976	−0.649882	−	0.760035i	$-0.725182\pi$
$829$	−37.4233	−1.29976	−0.649882	+	0.760035i	$0.725182\pi$
$830$	0	0
$831$	19.3693	0.671914
$832$	0	0
$833$	−23.3693	−0.809699
$834$	0	0
$835$	2.43845	0.0843859
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−34.9309	−1.20595	−0.602974	−	0.797761i	$-0.706018\pi$
$839$	−34.9309	−1.20595	−0.602974	+	0.797761i	$0.706018\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	0	0
$843$	13.1231	0.451984
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−13.3693	−0.459375
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−28.4924	−0.976708
$852$	0	0
$853$	−38.7926	−1.32823	−0.664117	−	0.747629i	$-0.731192\pi$
$853$	−38.7926	−1.32823	−0.664117	+	0.747629i	$0.731192\pi$
$854$	0	0
$855$	−4.68466	−0.160212
$856$	0	0
$857$	−53.2311	−1.81834	−0.909169	−	0.416427i	$-0.863282\pi$
$857$	−53.2311	−1.81834	−0.909169	+	0.416427i	$0.863282\pi$
$858$	0	0
$859$	22.7386	0.775832	0.387916	−	0.921695i	$-0.373195\pi$
$859$	22.7386	0.775832	0.387916	+	0.921695i	$0.373195\pi$
$860$	0	0
$861$	−1.75379	−0.0597690
$862$	0	0
$863$	30.5464	1.03981	0.519906	−	0.854224i	$-0.325967\pi$
$863$	30.5464	1.03981	0.519906	+	0.854224i	$0.325967\pi$
$864$	0	0
$865$	−8.24621	−0.280380
$866$	0	0
$867$	9.24621	0.314018
$868$	0	0
$869$	26.0540	0.883821
$870$	0	0
$871$	−18.7386	−0.634934
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	1.56155	0.0527901
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	−6.49242	−0.218984
$880$	0	0
$881$	42.4924	1.43161	0.715803	−	0.698302i	$-0.246061\pi$
$881$	42.4924	1.43161	0.715803	+	0.698302i	$0.246061\pi$
$882$	0	0
$883$	31.4233	1.05748	0.528739	−	0.848784i	$-0.322665\pi$
$883$	31.4233	1.05748	0.528739	+	0.848784i	$0.322665\pi$
$884$	0	0
$885$	4.87689	0.163935
$886$	0	0
$887$	57.3693	1.92627	0.963137	−	0.269013i	$-0.0866974\pi$
$887$	57.3693	1.92627	0.963137	+	0.269013i	$0.0866974\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.56155	0.0523140
$892$	0	0
$893$	−14.6307	−0.489597
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−11.1231	−0.371390
$898$	0	0
$899$	1.12311	0.0374577
$900$	0	0
$901$	59.2311	1.97327
$902$	0	0
$903$	12.1922	0.405732
$904$	0	0
$905$	24.0540	0.799581
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	0	0
$909$	3.56155	0.118129
$910$	0	0
$911$	44.4924	1.47410	0.737050	−	0.675838i	$-0.236218\pi$
$911$	44.4924	1.47410	0.737050	+	0.675838i	$0.236218\pi$
$912$	0	0
$913$	3.50758	0.116084
$914$	0	0
$915$	6.00000	0.198354
$916$	0	0
$917$	−14.6307	−0.483148
$918$	0	0
$919$	−39.6155	−1.30680	−0.653398	−	0.757015i	$-0.726657\pi$
$919$	−39.6155	−1.30680	−0.653398	+	0.757015i	$0.726657\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.36932	−0.308395
$924$	0	0
$925$	5.12311	0.168447
$926$	0	0
$927$	18.2462	0.599284
$928$	0	0
$929$	11.5616	0.379322	0.189661	−	0.981850i	$-0.439261\pi$
$929$	11.5616	0.379322	0.189661	+	0.981850i	$0.439261\pi$
$930$	0	0
$931$	21.3693	0.700351
$932$	0	0
$933$	18.2462	0.597354
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	16.6307	0.543301	0.271650	−	0.962396i	$-0.412431\pi$
$937$	16.6307	0.543301	0.271650	+	0.962396i	$0.412431\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	27.3693	0.892214	0.446107	−	0.894980i	$-0.352810\pi$
$941$	27.3693	0.892214	0.446107	+	0.894980i	$0.352810\pi$
$942$	0	0
$943$	6.24621	0.203405
$944$	0	0
$945$	1.56155	0.0507973
$946$	0	0
$947$	0.876894	0.0284952	0.0142476	−	0.999898i	$-0.495465\pi$
$947$	0.876894	0.0284952	0.0142476	+	0.999898i	$0.495465\pi$
$948$	0	0
$949$	19.6155	0.636747
$950$	0	0
$951$	25.1231	0.814673
$952$	0	0
$953$	−58.1080	−1.88230	−0.941151	−	0.337988i	$-0.890254\pi$
$953$	−58.1080	−1.88230	−0.941151	+	0.337988i	$0.890254\pi$
$954$	0	0
$955$	−2.24621	−0.0726857
$956$	0	0
$957$	−1.75379	−0.0566919
$958$	0	0
$959$	5.86174	0.189285
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−1.56155	−0.0503203
$964$	0	0
$965$	−18.4924	−0.595292
$966$	0	0
$967$	25.7538	0.828186	0.414093	−	0.910235i	$-0.364099\pi$
$967$	25.7538	0.828186	0.414093	+	0.910235i	$0.364099\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	28.8769	0.926704	0.463352	−	0.886174i	$-0.346647\pi$
$971$	28.8769	0.926704	0.463352	+	0.886174i	$0.346647\pi$
$972$	0	0
$973$	13.8617	0.444387
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−22.9848	−0.735350	−0.367675	−	0.929954i	$-0.619846\pi$
$977$	−22.9848	−0.735350	−0.367675	+	0.929954i	$0.619846\pi$
$978$	0	0
$979$	−2.05398	−0.0656453
$980$	0	0
$981$	−19.3693	−0.618415
$982$	0	0
$983$	24.9848	0.796893	0.398446	−	0.917192i	$-0.369549\pi$
$983$	24.9848	0.796893	0.398446	+	0.917192i	$0.369549\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	4.87689	0.155233
$988$	0	0
$989$	−43.4233	−1.38078
$990$	0	0
$991$	−23.3153	−0.740636	−0.370318	−	0.928905i	$-0.620751\pi$
$991$	−23.3153	−0.740636	−0.370318	+	0.928905i	$0.620751\pi$
$992$	0	0
$993$	−10.2462	−0.325154
$994$	0	0
$995$	−3.80776	−0.120714
$996$	0	0
$997$	−62.4924	−1.97915	−0.989577	−	0.144002i	$-0.954003\pi$
$997$	−62.4924	−1.97915	−0.989577	+	0.144002i	$0.954003\pi$
$998$	0	0
$999$	5.12311	0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bl.1.2		2
4.3	odd	2		930.2.a.p.1.1	✓	2
12.11	even	2		2790.2.a.be.1.1		2
20.3	even	4		4650.2.d.bd.3349.2		4
20.7	even	4		4650.2.d.bd.3349.3		4
20.19	odd	2		4650.2.a.ce.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.1	✓	2	4.3	odd	2
2790.2.a.be.1.1		2	12.11	even	2
4650.2.a.ce.1.2		2	20.19	odd	2
4650.2.d.bd.3349.2		4	20.3	even	4
4650.2.d.bd.3349.3		4	20.7	even	4
7440.2.a.bl.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.56155 q^{7} +1.00000 q^{9} +1.56155 q^{11} +2.00000 q^{13} +1.00000 q^{15} +5.12311 q^{17} -4.68466 q^{19} +1.56155 q^{21} -5.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.12311 q^{29} -1.00000 q^{31} +1.56155 q^{33} +1.56155 q^{35} +5.12311 q^{37} +2.00000 q^{39} -1.12311 q^{41} +7.80776 q^{43} +1.00000 q^{45} +3.12311 q^{47} -4.56155 q^{49} +5.12311 q^{51} +11.5616 q^{53} +1.56155 q^{55} -4.68466 q^{57} +4.87689 q^{59} +6.00000 q^{61} +1.56155 q^{63} +2.00000 q^{65} -9.36932 q^{67} -5.56155 q^{69} -4.68466 q^{71} +9.80776 q^{73} +1.00000 q^{75} +2.43845 q^{77} +16.6847 q^{79} +1.00000 q^{81} +2.24621 q^{83} +5.12311 q^{85} -1.12311 q^{87} -1.31534 q^{89} +3.12311 q^{91} -1.00000 q^{93} -4.68466 q^{95} -6.00000 q^{97} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 + 3 * q^19 - q^21 - 7 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^31 - q^33 - q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 - 5 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 + 2 * q^51 + 19 * q^53 - q^55 + 3 * q^57 + 18 * q^59 + 12 * q^61 - q^63 + 4 * q^65 + 6 * q^67 - 7 * q^69 + 3 * q^71 - q^73 + 2 * q^75 + 9 * q^77 + 21 * q^79 + 2 * q^81 - 12 * q^83 + 2 * q^85 + 6 * q^87 - 15 * q^89 - 2 * q^91 - 2 * q^93 + 3 * q^95 - 12 * q^97 - q^99

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