LMFDB - Embedded newform 2475.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +O(q^{10})$	$q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{11} -5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} -1.17157 q^{19} +0.414214 q^{22} -4.00000 q^{23} -2.34315 q^{26} -8.82843 q^{28} -0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} -0.343146 q^{37} -0.485281 q^{38} +0.828427 q^{41} +3.17157 q^{43} -1.82843 q^{44} -1.65685 q^{46} -4.00000 q^{47} +16.3137 q^{49} +10.3431 q^{52} -13.3137 q^{53} -7.65685 q^{56} -0.343146 q^{58} +4.00000 q^{59} -0.343146 q^{61} -4.17157 q^{64} -5.65685 q^{67} +12.4853 q^{68} -13.6569 q^{71} +11.3137 q^{73} -0.142136 q^{74} +2.14214 q^{76} +4.82843 q^{77} -8.48528 q^{79} +0.343146 q^{82} -10.0000 q^{83} +1.31371 q^{86} -1.58579 q^{88} +7.65685 q^{89} -27.3137 q^{91} +7.31371 q^{92} -1.65685 q^{94} -0.343146 q^{97} +6.75736 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 2 q^{11} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 8 q^{19} - 2 q^{22} - 8 q^{23} - 16 q^{26} - 12 q^{28} + 4 q^{29} + 6 q^{32} - 12 q^{37} + 16 q^{38} - 4 q^{41} + 12 q^{43} + 2 q^{44} + 8 q^{46} - 8 q^{47} + 10 q^{49} + 32 q^{52} - 4 q^{53} - 4 q^{56} - 12 q^{58} + 8 q^{59} - 12 q^{61} - 14 q^{64} + 8 q^{68} - 16 q^{71} + 28 q^{74} - 24 q^{76} + 4 q^{77} + 12 q^{82} - 20 q^{83} - 20 q^{86} - 6 q^{88} + 4 q^{89} - 32 q^{91} - 8 q^{92} + 8 q^{94} - 12 q^{97} + 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 2 * q^11 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 8 * q^19 - 2 * q^22 - 8 * q^23 - 16 * q^26 - 12 * q^28 + 4 * q^29 + 6 * q^32 - 12 * q^37 + 16 * q^38 - 4 * q^41 + 12 * q^43 + 2 * q^44 + 8 * q^46 - 8 * q^47 + 10 * q^49 + 32 * q^52 - 4 * q^53 - 4 * q^56 - 12 * q^58 + 8 * q^59 - 12 * q^61 - 14 * q^64 + 8 * q^68 - 16 * q^71 + 28 * q^74 - 24 * q^76 + 4 * q^77 + 12 * q^82 - 20 * q^83 - 20 * q^86 - 6 * q^88 + 4 * q^89 - 32 * q^91 - 8 * q^92 + 8 * q^94 - 12 * q^97 + 22 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.65685	−1.56893	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	−5.65685	−1.56893	−0.784465	+	0.620174i	$0.787062\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	−1.17157	−0.268777	−0.134389	−	0.990929i	$-0.542907\pi$
$19$	−1.17157	−0.268777	−0.134389	+	0.990929i	$0.542907\pi$
$20$	0	0
$21$	0	0
$22$	0.414214	0.0883106
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	−2.34315	−0.459529
$27$	0	0
$28$	−8.82843	−1.66842
$29$	−0.828427	−0.153835	−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427	−0.153835	−0.0769175	+	0.997037i	$0.524508\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	−2.82843	−0.485071
$35$	0	0
$36$	0	0
$37$	−0.343146	−0.0564128	−0.0282064	−	0.999602i	$-0.508980\pi$
$37$	−0.343146	−0.0564128	−0.0282064	+	0.999602i	$0.508980\pi$
$38$	−0.485281	−0.0787230
$39$	0	0
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	3.17157	0.483660	0.241830	−	0.970319i	$-0.422252\pi$
$43$	3.17157	0.483660	0.241830	+	0.970319i	$0.422252\pi$
$44$	−1.82843	−0.275646
$45$	0	0
$46$	−1.65685	−0.244290
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	0	0
$52$	10.3431	1.43434
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	0	0
$55$	0	0
$56$	−7.65685	−1.02319
$57$	0	0
$58$	−0.343146	−0.0450572
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−0.343146	−0.0439353	−0.0219677	−	0.999759i	$-0.506993\pi$
$61$	−0.343146	−0.0439353	−0.0219677	+	0.999759i	$0.506993\pi$
$62$	0	0
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	12.4853	1.51406
$69$	0	0
$70$	0	0
$71$	−13.6569	−1.62077	−0.810385	−	0.585897i	$-0.800742\pi$
$71$	−13.6569	−1.62077	−0.810385	+	0.585897i	$0.800742\pi$
$72$	0	0
$73$	11.3137	1.32417	0.662085	−	0.749429i	$-0.269672\pi$
$73$	11.3137	1.32417	0.662085	+	0.749429i	$0.269672\pi$
$74$	−0.142136	−0.0165229
$75$	0	0
$76$	2.14214	0.245720
$77$	4.82843	0.550250
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	0	0
$82$	0.343146	0.0378941
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	0	0
$86$	1.31371	0.141661
$87$	0	0
$88$	−1.58579	−0.169045
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	−27.3137	−2.86325
$92$	7.31371	0.762507
$93$	0	0
$94$	−1.65685	−0.170891
$95$	0	0
$96$	0	0
$97$	−0.343146	−0.0348412	−0.0174206	−	0.999848i	$-0.505545\pi$
$97$	−0.343146	−0.0348412	−0.0174206	+	0.999848i	$0.505545\pi$
$98$	6.75736	0.682596
$99$	0	0
$100$	0	0
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	0	0
$103$	−19.3137	−1.90304	−0.951518	−	0.307593i	$-0.900477\pi$
$103$	−19.3137	−1.90304	−0.951518	+	0.307593i	$0.900477\pi$
$104$	8.97056	0.879636
$105$	0	0
$106$	−5.51472	−0.535637
$107$	−5.31371	−0.513696	−0.256848	−	0.966452i	$-0.582684\pi$
$107$	−5.31371	−0.513696	−0.256848	+	0.966452i	$0.582684\pi$
$108$	0	0
$109$	−5.31371	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$109$	−5.31371	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$110$	0	0
$111$	0	0
$112$	14.4853	1.36873
$113$	14.9706	1.40831	0.704156	−	0.710045i	$-0.251326\pi$
$113$	14.9706	1.40831	0.704156	+	0.710045i	$0.251326\pi$
$114$	0	0
$115$	0	0
$116$	1.51472	0.140638
$117$	0	0
$118$	1.65685	0.152526
$119$	−32.9706	−3.02241
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.142136	−0.0128684
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.48528	−0.220533	−0.110267	−	0.993902i	$-0.535170\pi$
$127$	−2.48528	−0.220533	−0.110267	+	0.993902i	$0.535170\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	0	0
$131$	19.3137	1.68745	0.843723	−	0.536778i	$-0.180359\pi$
$131$	19.3137	1.68745	0.843723	+	0.536778i	$0.180359\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	−2.34315	−0.202417
$135$	0	0
$136$	10.8284	0.928530
$137$	9.31371	0.795724	0.397862	−	0.917445i	$-0.369752\pi$
$137$	9.31371	0.795724	0.397862	+	0.917445i	$0.369752\pi$
$138$	0	0
$139$	−16.4853	−1.39826	−0.699132	−	0.714993i	$-0.746430\pi$
$139$	−16.4853	−1.39826	−0.699132	+	0.714993i	$0.746430\pi$
$140$	0	0
$141$	0	0
$142$	−5.65685	−0.474713
$143$	−5.65685	−0.473050
$144$	0	0
$145$	0	0
$146$	4.68629	0.387840
$147$	0	0
$148$	0.627417	0.0515734
$149$	−18.4853	−1.51437	−0.757187	−	0.653199i	$-0.773427\pi$
$149$	−18.4853	−1.51437	−0.757187	+	0.653199i	$0.773427\pi$
$150$	0	0
$151$	−0.485281	−0.0394916	−0.0197458	−	0.999805i	$-0.506286\pi$
$151$	−0.485281	−0.0394916	−0.0197458	+	0.999805i	$0.506286\pi$
$152$	1.85786	0.150693
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−3.51472	−0.279616
$159$	0	0
$160$	0	0
$161$	−19.3137	−1.52213
$162$	0	0
$163$	−15.3137	−1.19946	−0.599731	−	0.800202i	$-0.704726\pi$
$163$	−15.3137	−1.19946	−0.599731	+	0.800202i	$0.704726\pi$
$164$	−1.51472	−0.118280
$165$	0	0
$166$	−4.14214	−0.321492
$167$	9.31371	0.720716	0.360358	−	0.932814i	$-0.382654\pi$
$167$	9.31371	0.720716	0.360358	+	0.932814i	$0.382654\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	0	0
$172$	−5.79899	−0.442169
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	3.17157	0.237719
$179$	6.34315	0.474109	0.237054	−	0.971496i	$-0.423818\pi$
$179$	6.34315	0.474109	0.237054	+	0.971496i	$0.423818\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−11.3137	−0.838628
$183$	0	0
$184$	6.34315	0.467623
$185$	0	0
$186$	0	0
$187$	−6.82843	−0.499344
$188$	7.31371	0.533407
$189$	0	0
$190$	0	0
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	0	0
$193$	−2.34315	−0.168663	−0.0843317	−	0.996438i	$-0.526876\pi$
$193$	−2.34315	−0.168663	−0.0843317	+	0.996438i	$0.526876\pi$
$194$	−0.142136	−0.0102047
$195$	0	0
$196$	−29.8284	−2.13060
$197$	8.48528	0.604551	0.302276	−	0.953221i	$-0.402254\pi$
$197$	8.48528	0.604551	0.302276	+	0.953221i	$0.402254\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	0	0
$202$	−2.00000	−0.140720
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−16.9706	−1.17670
$209$	−1.17157	−0.0810394
$210$	0	0
$211$	6.82843	0.470088	0.235044	−	0.971985i	$-0.424477\pi$
$211$	6.82843	0.470088	0.235044	+	0.971985i	$0.424477\pi$
$212$	24.3431	1.67189
$213$	0	0
$214$	−2.20101	−0.150458
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−2.20101	−0.149071
$219$	0	0
$220$	0	0
$221$	38.6274	2.59836
$222$	0	0
$223$	17.6569	1.18239	0.591195	−	0.806529i	$-0.298656\pi$
$223$	17.6569	1.18239	0.591195	+	0.806529i	$0.298656\pi$
$224$	21.3137	1.42408
$225$	0	0
$226$	6.20101	0.412485
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	1.31371	0.0862492
$233$	−13.1716	−0.862898	−0.431449	−	0.902137i	$-0.641998\pi$
$233$	−13.1716	−0.862898	−0.431449	+	0.902137i	$0.641998\pi$
$234$	0	0
$235$	0	0
$236$	−7.31371	−0.476082
$237$	0	0
$238$	−13.6569	−0.885242
$239$	−6.34315	−0.410304	−0.205152	−	0.978730i	$-0.565769\pi$
$239$	−6.34315	−0.410304	−0.205152	+	0.978730i	$0.565769\pi$
$240$	0	0
$241$	−23.6569	−1.52387	−0.761936	−	0.647652i	$-0.775751\pi$
$241$	−23.6569	−1.52387	−0.761936	+	0.647652i	$0.775751\pi$
$242$	0.414214	0.0266267
$243$	0	0
$244$	0.627417	0.0401663
$245$	0	0
$246$	0	0
$247$	6.62742	0.421692
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.9706	0.818695	0.409347	−	0.912379i	$-0.365756\pi$
$251$	12.9706	0.818695	0.409347	+	0.912379i	$0.365756\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−1.02944	−0.0645926
$255$	0	0
$256$	3.97056	0.248160
$257$	−27.6569	−1.72519	−0.862594	−	0.505898i	$-0.831161\pi$
$257$	−27.6569	−1.72519	−0.862594	+	0.505898i	$0.831161\pi$
$258$	0	0
$259$	−1.65685	−0.102952
$260$	0	0
$261$	0	0
$262$	8.00000	0.494242
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	−2.34315	−0.143667
$267$	0	0
$268$	10.3431	0.631808
$269$	24.6274	1.50156	0.750780	−	0.660552i	$-0.229678\pi$
$269$	24.6274	1.50156	0.750780	+	0.660552i	$0.229678\pi$
$270$	0	0
$271$	27.7990	1.68867	0.844334	−	0.535817i	$-0.179996\pi$
$271$	27.7990	1.68867	0.844334	+	0.535817i	$0.179996\pi$
$272$	−20.4853	−1.24210
$273$	0	0
$274$	3.85786	0.233062
$275$	0	0
$276$	0	0
$277$	13.6569	0.820561	0.410280	−	0.911959i	$-0.365431\pi$
$277$	13.6569	0.820561	0.410280	+	0.911959i	$0.365431\pi$
$278$	−6.82843	−0.409542
$279$	0	0
$280$	0	0
$281$	16.8284	1.00390	0.501950	−	0.864897i	$-0.332616\pi$
$281$	16.8284	1.00390	0.501950	+	0.864897i	$0.332616\pi$
$282$	0	0
$283$	3.17157	0.188530	0.0942652	−	0.995547i	$-0.469950\pi$
$283$	3.17157	0.188530	0.0942652	+	0.995547i	$0.469950\pi$
$284$	24.9706	1.48173
$285$	0	0
$286$	−2.34315	−0.138553
$287$	4.00000	0.236113
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	0	0
$292$	−20.6863	−1.21057
$293$	−1.17157	−0.0684440	−0.0342220	−	0.999414i	$-0.510895\pi$
$293$	−1.17157	−0.0684440	−0.0342220	+	0.999414i	$0.510895\pi$
$294$	0	0
$295$	0	0
$296$	0.544156	0.0316284
$297$	0	0
$298$	−7.65685	−0.443550
$299$	22.6274	1.30858
$300$	0	0
$301$	15.3137	0.882667
$302$	−0.201010	−0.0115668
$303$	0	0
$304$	−3.51472	−0.201583
$305$	0	0
$306$	0	0
$307$	8.82843	0.503865	0.251932	−	0.967745i	$-0.418934\pi$
$307$	8.82843	0.503865	0.251932	+	0.967745i	$0.418934\pi$
$308$	−8.82843	−0.503046
$309$	0	0
$310$	0	0
$311$	−19.3137	−1.09518	−0.547590	−	0.836747i	$-0.684455\pi$
$311$	−19.3137	−1.09518	−0.547590	+	0.836747i	$0.684455\pi$
$312$	0	0
$313$	−4.34315	−0.245489	−0.122745	−	0.992438i	$-0.539170\pi$
$313$	−4.34315	−0.245489	−0.122745	+	0.992438i	$0.539170\pi$
$314$	−7.45584	−0.420758
$315$	0	0
$316$	15.5147	0.872771
$317$	30.2843	1.70093	0.850467	−	0.526028i	$-0.176319\pi$
$317$	30.2843	1.70093	0.850467	+	0.526028i	$0.176319\pi$
$318$	0	0
$319$	−0.828427	−0.0463830
$320$	0	0
$321$	0	0
$322$	−8.00000	−0.445823
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	−6.34315	−0.351314
$327$	0	0
$328$	−1.31371	−0.0725374
$329$	−19.3137	−1.06480
$330$	0	0
$331$	17.6569	0.970508	0.485254	−	0.874373i	$-0.338727\pi$
$331$	17.6569	0.970508	0.485254	+	0.874373i	$0.338727\pi$
$332$	18.2843	1.00348
$333$	0	0
$334$	3.85786	0.211093
$335$	0	0
$336$	0	0
$337$	19.3137	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$337$	19.3137	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$338$	7.87006	0.428075
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	44.9706	2.42818
$344$	−5.02944	−0.271169
$345$	0	0
$346$	−1.17157	−0.0629841
$347$	−6.68629	−0.358939	−0.179469	−	0.983764i	$-0.557438\pi$
$347$	−6.68629	−0.358939	−0.179469	+	0.983764i	$0.557438\pi$
$348$	0	0
$349$	−22.9706	−1.22959	−0.614793	−	0.788688i	$-0.710760\pi$
$349$	−22.9706	−1.22959	−0.614793	+	0.788688i	$0.710760\pi$
$350$	0	0
$351$	0	0
$352$	4.41421	0.235278
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	2.62742	0.138863
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	−5.79899	−0.304788
$363$	0	0
$364$	49.9411	2.61763
$365$	0	0
$366$	0	0
$367$	1.65685	0.0864871	0.0432435	−	0.999065i	$-0.486231\pi$
$367$	1.65685	0.0864871	0.0432435	+	0.999065i	$0.486231\pi$
$368$	−12.0000	−0.625543
$369$	0	0
$370$	0	0
$371$	−64.2843	−3.33747
$372$	0	0
$373$	34.6274	1.79294	0.896470	−	0.443105i	$-0.146123\pi$
$373$	34.6274	1.79294	0.896470	+	0.443105i	$0.146123\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	6.34315	0.327123
$377$	4.68629	0.241356
$378$	0	0
$379$	−0.686292	−0.0352524	−0.0176262	−	0.999845i	$-0.505611\pi$
$379$	−0.686292	−0.0352524	−0.0176262	+	0.999845i	$0.505611\pi$
$380$	0	0
$381$	0	0
$382$	2.34315	0.119886
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	−0.970563	−0.0494003
$387$	0	0
$388$	0.627417	0.0318523
$389$	12.3431	0.625822	0.312911	−	0.949782i	$-0.398696\pi$
$389$	12.3431	0.625822	0.312911	+	0.949782i	$0.398696\pi$
$390$	0	0
$391$	27.3137	1.38131
$392$	−25.8701	−1.30664
$393$	0	0
$394$	3.51472	0.177069
$395$	0	0
$396$	0	0
$397$	−18.9706	−0.952105	−0.476053	−	0.879417i	$-0.657933\pi$
$397$	−18.9706	−0.952105	−0.476053	+	0.879417i	$0.657933\pi$
$398$	−4.28427	−0.214751
$399$	0	0
$400$	0	0
$401$	29.3137	1.46386	0.731928	−	0.681382i	$-0.238621\pi$
$401$	29.3137	1.46386	0.731928	+	0.681382i	$0.238621\pi$
$402$	0	0
$403$	0	0
$404$	8.82843	0.439231
$405$	0	0
$406$	−1.65685	−0.0822283
$407$	−0.343146	−0.0170091
$408$	0	0
$409$	−8.34315	−0.412542	−0.206271	−	0.978495i	$-0.566133\pi$
$409$	−8.34315	−0.412542	−0.206271	+	0.978495i	$0.566133\pi$
$410$	0	0
$411$	0	0
$412$	35.3137	1.73978
$413$	19.3137	0.950365
$414$	0	0
$415$	0	0
$416$	−24.9706	−1.22428
$417$	0	0
$418$	−0.485281	−0.0237359
$419$	3.02944	0.147998	0.0739988	−	0.997258i	$-0.476424\pi$
$419$	3.02944	0.147998	0.0739988	+	0.997258i	$0.476424\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	2.82843	0.137686
$423$	0	0
$424$	21.1127	1.02532
$425$	0	0
$426$	0	0
$427$	−1.65685	−0.0801808
$428$	9.71573	0.469627
$429$	0	0
$430$	0	0
$431$	−10.3431	−0.498212	−0.249106	−	0.968476i	$-0.580137\pi$
$431$	−10.3431	−0.498212	−0.249106	+	0.968476i	$0.580137\pi$
$432$	0	0
$433$	4.34315	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$433$	4.34315	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$434$	0	0
$435$	0	0
$436$	9.71573	0.465299
$437$	4.68629	0.224176
$438$	0	0
$439$	−3.51472	−0.167748	−0.0838742	−	0.996476i	$-0.526729\pi$
$439$	−3.51472	−0.167748	−0.0838742	+	0.996476i	$0.526729\pi$
$440$	0	0
$441$	0	0
$442$	16.0000	0.761042
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	0	0
$446$	7.31371	0.346314
$447$	0	0
$448$	−20.1421	−0.951626
$449$	2.97056	0.140190	0.0700948	−	0.997540i	$-0.477670\pi$
$449$	2.97056	0.140190	0.0700948	+	0.997540i	$0.477670\pi$
$450$	0	0
$451$	0.828427	0.0390091
$452$	−27.3726	−1.28750
$453$	0	0
$454$	−5.79899	−0.272160
$455$	0	0
$456$	0	0
$457$	0.686292	0.0321034	0.0160517	−	0.999871i	$-0.494890\pi$
$457$	0.686292	0.0321034	0.0160517	+	0.999871i	$0.494890\pi$
$458$	−0.828427	−0.0387099
$459$	0	0
$460$	0	0
$461$	28.1421	1.31071	0.655355	−	0.755321i	$-0.272519\pi$
$461$	28.1421	1.31071	0.655355	+	0.755321i	$0.272519\pi$
$462$	0	0
$463$	28.9706	1.34638	0.673188	−	0.739471i	$-0.264924\pi$
$463$	28.9706	1.34638	0.673188	+	0.739471i	$0.264924\pi$
$464$	−2.48528	−0.115376
$465$	0	0
$466$	−5.45584	−0.252737
$467$	22.6274	1.04707	0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274	1.04707	0.523536	+	0.852004i	$0.324613\pi$
$468$	0	0
$469$	−27.3137	−1.26123
$470$	0	0
$471$	0	0
$472$	−6.34315	−0.291967
$473$	3.17157	0.145829
$474$	0	0
$475$	0	0
$476$	60.2843	2.76313
$477$	0	0
$478$	−2.62742	−0.120175
$479$	−3.02944	−0.138419	−0.0692093	−	0.997602i	$-0.522048\pi$
$479$	−3.02944	−0.138419	−0.0692093	+	0.997602i	$0.522048\pi$
$480$	0	0
$481$	1.94113	0.0885077
$482$	−9.79899	−0.446332
$483$	0	0
$484$	−1.82843	−0.0831103
$485$	0	0
$486$	0	0
$487$	−20.9706	−0.950267	−0.475133	−	0.879914i	$-0.657600\pi$
$487$	−20.9706	−0.950267	−0.475133	+	0.879914i	$0.657600\pi$
$488$	0.544156	0.0246328
$489$	0	0
$490$	0	0
$491$	−25.6569	−1.15788	−0.578939	−	0.815371i	$-0.696533\pi$
$491$	−25.6569	−1.15788	−0.578939	+	0.815371i	$0.696533\pi$
$492$	0	0
$493$	5.65685	0.254772
$494$	2.74517	0.123511
$495$	0	0
$496$	0	0
$497$	−65.9411	−2.95786
$498$	0	0
$499$	−33.6569	−1.50669	−0.753344	−	0.657627i	$-0.771560\pi$
$499$	−33.6569	−1.50669	−0.753344	+	0.657627i	$0.771560\pi$
$500$	0	0
$501$	0	0
$502$	5.37258	0.239790
$503$	−5.31371	−0.236927	−0.118463	−	0.992958i	$-0.537797\pi$
$503$	−5.31371	−0.236927	−0.118463	+	0.992958i	$0.537797\pi$
$504$	0	0
$505$	0	0
$506$	−1.65685	−0.0736562
$507$	0	0
$508$	4.54416	0.201614
$509$	−41.3137	−1.83120	−0.915599	−	0.402093i	$-0.868283\pi$
$509$	−41.3137	−1.83120	−0.915599	+	0.402093i	$0.868283\pi$
$510$	0	0
$511$	54.6274	2.41657
$512$	22.7574	1.00574
$513$	0	0
$514$	−11.4558	−0.505296
$515$	0	0
$516$	0	0
$517$	−4.00000	−0.175920
$518$	−0.686292	−0.0301539
$519$	0	0
$520$	0	0
$521$	−12.6274	−0.553217	−0.276609	−	0.960983i	$-0.589211\pi$
$521$	−12.6274	−0.553217	−0.276609	+	0.960983i	$0.589211\pi$
$522$	0	0
$523$	26.4853	1.15812	0.579060	−	0.815285i	$-0.303420\pi$
$523$	26.4853	1.15812	0.579060	+	0.815285i	$0.303420\pi$
$524$	−35.3137	−1.54269
$525$	0	0
$526$	7.45584	0.325090
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	10.3431	0.448432
$533$	−4.68629	−0.202986
$534$	0	0
$535$	0	0
$536$	8.97056	0.387469
$537$	0	0
$538$	10.2010	0.439797
$539$	16.3137	0.702681
$540$	0	0
$541$	−5.31371	−0.228454	−0.114227	−	0.993455i	$-0.536439\pi$
$541$	−5.31371	−0.228454	−0.114227	+	0.993455i	$0.536439\pi$
$542$	11.5147	0.494600
$543$	0	0
$544$	−30.1421	−1.29233
$545$	0	0
$546$	0	0
$547$	−20.1421	−0.861216	−0.430608	−	0.902539i	$-0.641701\pi$
$547$	−20.1421	−0.861216	−0.430608	+	0.902539i	$0.641701\pi$
$548$	−17.0294	−0.727462
$549$	0	0
$550$	0	0
$551$	0.970563	0.0413474
$552$	0	0
$553$	−40.9706	−1.74225
$554$	5.65685	0.240337
$555$	0	0
$556$	30.1421	1.27831
$557$	10.8284	0.458815	0.229408	−	0.973330i	$-0.426321\pi$
$557$	10.8284	0.458815	0.229408	+	0.973330i	$0.426321\pi$
$558$	0	0
$559$	−17.9411	−0.758829
$560$	0	0
$561$	0	0
$562$	6.97056	0.294035
$563$	20.3431	0.857361	0.428681	−	0.903456i	$-0.358979\pi$
$563$	20.3431	0.857361	0.428681	+	0.903456i	$0.358979\pi$
$564$	0	0
$565$	0	0
$566$	1.31371	0.0552193
$567$	0	0
$568$	21.6569	0.908701
$569$	−15.4558	−0.647943	−0.323971	−	0.946067i	$-0.605018\pi$
$569$	−15.4558	−0.647943	−0.323971	+	0.946067i	$0.605018\pi$
$570$	0	0
$571$	0.485281	0.0203084	0.0101542	−	0.999948i	$-0.496768\pi$
$571$	0.485281	0.0203084	0.0101542	+	0.999948i	$0.496768\pi$
$572$	10.3431	0.432469
$573$	0	0
$574$	1.65685	0.0691558
$575$	0	0
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	12.2721	0.510451
$579$	0	0
$580$	0	0
$581$	−48.2843	−2.00317
$582$	0	0
$583$	−13.3137	−0.551397
$584$	−17.9411	−0.742409
$585$	0	0
$586$	−0.485281	−0.0200468
$587$	−30.6274	−1.26413	−0.632064	−	0.774916i	$-0.717792\pi$
$587$	−30.6274	−1.26413	−0.632064	+	0.774916i	$0.717792\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−1.02944	−0.0423096
$593$	−17.1716	−0.705152	−0.352576	−	0.935783i	$-0.614694\pi$
$593$	−17.1716	−0.705152	−0.352576	+	0.935783i	$0.614694\pi$
$594$	0	0
$595$	0	0
$596$	33.7990	1.38446
$597$	0	0
$598$	9.37258	0.383273
$599$	−4.68629	−0.191477	−0.0957383	−	0.995407i	$-0.530521\pi$
$599$	−4.68629	−0.191477	−0.0957383	+	0.995407i	$0.530521\pi$
$600$	0	0
$601$	17.3137	0.706241	0.353120	−	0.935578i	$-0.385121\pi$
$601$	17.3137	0.706241	0.353120	+	0.935578i	$0.385121\pi$
$602$	6.34315	0.258527
$603$	0	0
$604$	0.887302	0.0361038
$605$	0	0
$606$	0	0
$607$	−18.4853	−0.750294	−0.375147	−	0.926965i	$-0.622408\pi$
$607$	−18.4853	−0.750294	−0.375147	+	0.926965i	$0.622408\pi$
$608$	−5.17157	−0.209735
$609$	0	0
$610$	0	0
$611$	22.6274	0.915407
$612$	0	0
$613$	−21.9411	−0.886194	−0.443097	−	0.896474i	$-0.646120\pi$
$613$	−21.9411	−0.886194	−0.443097	+	0.896474i	$0.646120\pi$
$614$	3.65685	0.147579
$615$	0	0
$616$	−7.65685	−0.308503
$617$	11.6569	0.469287	0.234644	−	0.972081i	$-0.424608\pi$
$617$	11.6569	0.469287	0.234644	+	0.972081i	$0.424608\pi$
$618$	0	0
$619$	−25.6569	−1.03124	−0.515618	−	0.856819i	$-0.672438\pi$
$619$	−25.6569	−1.03124	−0.515618	+	0.856819i	$0.672438\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	36.9706	1.48119
$624$	0	0
$625$	0	0
$626$	−1.79899	−0.0719021
$627$	0	0
$628$	32.9117	1.31332
$629$	2.34315	0.0934273
$630$	0	0
$631$	34.3431	1.36718	0.683590	−	0.729867i	$-0.260418\pi$
$631$	34.3431	1.36718	0.683590	+	0.729867i	$0.260418\pi$
$632$	13.4558	0.535245
$633$	0	0
$634$	12.5442	0.498192
$635$	0	0
$636$	0	0
$637$	−92.2843	−3.65644
$638$	−0.343146	−0.0135853
$639$	0	0
$640$	0	0
$641$	26.9706	1.06527	0.532637	−	0.846344i	$-0.321201\pi$
$641$	26.9706	1.06527	0.532637	+	0.846344i	$0.321201\pi$
$642$	0	0
$643$	29.9411	1.18076	0.590381	−	0.807124i	$-0.298977\pi$
$643$	29.9411	1.18076	0.590381	+	0.807124i	$0.298977\pi$
$644$	35.3137	1.39156
$645$	0	0
$646$	3.31371	0.130376
$647$	27.3137	1.07381	0.536906	−	0.843642i	$-0.319593\pi$
$647$	27.3137	1.07381	0.536906	+	0.843642i	$0.319593\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	28.0000	1.09656
$653$	−26.9706	−1.05544	−0.527720	−	0.849418i	$-0.676953\pi$
$653$	−26.9706	−1.05544	−0.527720	+	0.849418i	$0.676953\pi$
$654$	0	0
$655$	0	0
$656$	2.48528	0.0970339
$657$	0	0
$658$	−8.00000	−0.311872
$659$	−7.31371	−0.284902	−0.142451	−	0.989802i	$-0.545498\pi$
$659$	−7.31371	−0.284902	−0.142451	+	0.989802i	$0.545498\pi$
$660$	0	0
$661$	−13.3137	−0.517843	−0.258922	−	0.965898i	$-0.583367\pi$
$661$	−13.3137	−0.517843	−0.258922	+	0.965898i	$0.583367\pi$
$662$	7.31371	0.284255
$663$	0	0
$664$	15.8579	0.615404
$665$	0	0
$666$	0	0
$667$	3.31371	0.128307
$668$	−17.0294	−0.658889
$669$	0	0
$670$	0	0
$671$	−0.343146	−0.0132470
$672$	0	0
$673$	−29.6569	−1.14319	−0.571594	−	0.820537i	$-0.693675\pi$
$673$	−29.6569	−1.14319	−0.571594	+	0.820537i	$0.693675\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−34.7401	−1.33616
$677$	−21.4558	−0.824615	−0.412308	−	0.911045i	$-0.635277\pi$
$677$	−21.4558	−0.824615	−0.412308	+	0.911045i	$0.635277\pi$
$678$	0	0
$679$	−1.65685	−0.0635842
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	0	0
$686$	18.6274	0.711198
$687$	0	0
$688$	9.51472	0.362745
$689$	75.3137	2.86922
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	5.17157	0.196594
$693$	0	0
$694$	−2.76955	−0.105131
$695$	0	0
$696$	0	0
$697$	−5.65685	−0.214269
$698$	−9.51472	−0.360137
$699$	0	0
$700$	0	0
$701$	−7.85786	−0.296787	−0.148394	−	0.988928i	$-0.547410\pi$
$701$	−7.85786	−0.296787	−0.148394	+	0.988928i	$0.547410\pi$
$702$	0	0
$703$	0.402020	0.0151625
$704$	−4.17157	−0.157222
$705$	0	0
$706$	10.7696	0.405317
$707$	−23.3137	−0.876802
$708$	0	0
$709$	29.3137	1.10090	0.550450	−	0.834868i	$-0.314456\pi$
$709$	29.3137	1.10090	0.550450	+	0.834868i	$0.314456\pi$
$710$	0	0
$711$	0	0
$712$	−12.1421	−0.455046
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−11.5980	−0.433437
$717$	0	0
$718$	−4.97056	−0.185500
$719$	−31.5980	−1.17841	−0.589203	−	0.807985i	$-0.700558\pi$
$719$	−31.5980	−1.17841	−0.589203	+	0.807985i	$0.700558\pi$
$720$	0	0
$721$	−93.2548	−3.47299
$722$	−7.30152	−0.271734
$723$	0	0
$724$	25.5980	0.951341
$725$	0	0
$726$	0	0
$727$	33.9411	1.25881	0.629403	−	0.777079i	$-0.283299\pi$
$727$	33.9411	1.25881	0.629403	+	0.777079i	$0.283299\pi$
$728$	43.3137	1.60531
$729$	0	0
$730$	0	0
$731$	−21.6569	−0.801008
$732$	0	0
$733$	17.6569	0.652171	0.326085	−	0.945340i	$-0.394270\pi$
$733$	17.6569	0.652171	0.326085	+	0.945340i	$0.394270\pi$
$734$	0.686292	0.0253315
$735$	0	0
$736$	−17.6569	−0.650840
$737$	−5.65685	−0.208373
$738$	0	0
$739$	47.1127	1.73307	0.866534	−	0.499118i	$-0.166342\pi$
$739$	47.1127	1.73307	0.866534	+	0.499118i	$0.166342\pi$
$740$	0	0
$741$	0	0
$742$	−26.6274	−0.977523
$743$	47.6569	1.74836	0.874180	−	0.485602i	$-0.161399\pi$
$743$	47.6569	1.74836	0.874180	+	0.485602i	$0.161399\pi$
$744$	0	0
$745$	0	0
$746$	14.3431	0.525140
$747$	0	0
$748$	12.4853	0.456507
$749$	−25.6569	−0.937481
$750$	0	0
$751$	−36.2843	−1.32403	−0.662016	−	0.749490i	$-0.730299\pi$
$751$	−36.2843	−1.32403	−0.662016	+	0.749490i	$0.730299\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	1.94113	0.0706916
$755$	0	0
$756$	0	0
$757$	−8.62742	−0.313569	−0.156784	−	0.987633i	$-0.550113\pi$
$757$	−8.62742	−0.313569	−0.156784	+	0.987633i	$0.550113\pi$
$758$	−0.284271	−0.0103252
$759$	0	0
$760$	0	0
$761$	23.1716	0.839969	0.419984	−	0.907531i	$-0.362036\pi$
$761$	23.1716	0.839969	0.419984	+	0.907531i	$0.362036\pi$
$762$	0	0
$763$	−25.6569	−0.928840
$764$	−10.3431	−0.374202
$765$	0	0
$766$	−3.31371	−0.119729
$767$	−22.6274	−0.817029
$768$	0	0
$769$	33.3137	1.20132	0.600662	−	0.799503i	$-0.294904\pi$
$769$	33.3137	1.20132	0.600662	+	0.799503i	$0.294904\pi$
$770$	0	0
$771$	0	0
$772$	4.28427	0.154194
$773$	−7.65685	−0.275398	−0.137699	−	0.990474i	$-0.543971\pi$
$773$	−7.65685	−0.275398	−0.137699	+	0.990474i	$0.543971\pi$
$774$	0	0
$775$	0	0
$776$	0.544156	0.0195341
$777$	0	0
$778$	5.11270	0.183299
$779$	−0.970563	−0.0347740
$780$	0	0
$781$	−13.6569	−0.488681
$782$	11.3137	0.404577
$783$	0	0
$784$	48.9411	1.74790
$785$	0	0
$786$	0	0
$787$	−8.14214	−0.290236	−0.145118	−	0.989414i	$-0.546356\pi$
$787$	−8.14214	−0.290236	−0.145118	+	0.989414i	$0.546356\pi$
$788$	−15.5147	−0.552689
$789$	0	0
$790$	0	0
$791$	72.2843	2.57013
$792$	0	0
$793$	1.94113	0.0689314
$794$	−7.85786	−0.278865
$795$	0	0
$796$	18.9117	0.670307
$797$	1.02944	0.0364645	0.0182323	−	0.999834i	$-0.494196\pi$
$797$	1.02944	0.0364645	0.0182323	+	0.999834i	$0.494196\pi$
$798$	0	0
$799$	27.3137	0.966290
$800$	0	0
$801$	0	0
$802$	12.1421	0.428754
$803$	11.3137	0.399252
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	7.65685	0.269367
$809$	56.4264	1.98385	0.991923	−	0.126838i	$-0.0404829\pi$
$809$	56.4264	1.98385	0.991923	+	0.126838i	$0.0404829\pi$
$810$	0	0
$811$	−16.4853	−0.578877	−0.289438	−	0.957197i	$-0.593468\pi$
$811$	−16.4853	−0.578877	−0.289438	+	0.957197i	$0.593468\pi$
$812$	7.31371	0.256661
$813$	0	0
$814$	−0.142136	−0.00498185
$815$	0	0
$816$	0	0
$817$	−3.71573	−0.129997
$818$	−3.45584	−0.120831
$819$	0	0
$820$	0	0
$821$	7.17157	0.250290	0.125145	−	0.992138i	$-0.460060\pi$
$821$	7.17157	0.250290	0.125145	+	0.992138i	$0.460060\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	30.6274	1.06696
$825$	0	0
$826$	8.00000	0.278356
$827$	18.6863	0.649786	0.324893	−	0.945751i	$-0.394672\pi$
$827$	18.6863	0.649786	0.324893	+	0.945751i	$0.394672\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	0	0
$832$	23.5980	0.818113
$833$	−111.397	−3.85968
$834$	0	0
$835$	0	0
$836$	2.14214	0.0740873
$837$	0	0
$838$	1.25483	0.0433475
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	−2.48528	−0.0856485
$843$	0	0
$844$	−12.4853	−0.429761
$845$	0	0
$846$	0	0
$847$	4.82843	0.165907
$848$	−39.9411	−1.37158
$849$	0	0
$850$	0	0
$851$	1.37258	0.0470515
$852$	0	0
$853$	31.3137	1.07216	0.536080	−	0.844167i	$-0.319904\pi$
$853$	31.3137	1.07216	0.536080	+	0.844167i	$0.319904\pi$
$854$	−0.686292	−0.0234844
$855$	0	0
$856$	8.42641	0.288009
$857$	−11.5147	−0.393335	−0.196668	−	0.980470i	$-0.563012\pi$
$857$	−11.5147	−0.393335	−0.196668	+	0.980470i	$0.563012\pi$
$858$	0	0
$859$	−19.0294	−0.649276	−0.324638	−	0.945838i	$-0.605242\pi$
$859$	−19.0294	−0.649276	−0.324638	+	0.945838i	$0.605242\pi$
$860$	0	0
$861$	0	0
$862$	−4.28427	−0.145923
$863$	−43.3137	−1.47442	−0.737208	−	0.675666i	$-0.763856\pi$
$863$	−43.3137	−1.47442	−0.737208	+	0.675666i	$0.763856\pi$
$864$	0	0
$865$	0	0
$866$	1.79899	0.0611322
$867$	0	0
$868$	0	0
$869$	−8.48528	−0.287843
$870$	0	0
$871$	32.0000	1.08428
$872$	8.42641	0.285354
$873$	0	0
$874$	1.94113	0.0656595
$875$	0	0
$876$	0	0
$877$	42.6274	1.43943	0.719713	−	0.694272i	$-0.244273\pi$
$877$	42.6274	1.43943	0.719713	+	0.694272i	$0.244273\pi$
$878$	−1.45584	−0.0491324
$879$	0	0
$880$	0	0
$881$	13.0294	0.438973	0.219486	−	0.975616i	$-0.429562\pi$
$881$	13.0294	0.438973	0.219486	+	0.975616i	$0.429562\pi$
$882$	0	0
$883$	50.6274	1.70375	0.851874	−	0.523747i	$-0.175466\pi$
$883$	50.6274	1.70375	0.851874	+	0.523747i	$0.175466\pi$
$884$	−70.6274	−2.37546
$885$	0	0
$886$	4.97056	0.166989
$887$	−4.34315	−0.145829	−0.0729143	−	0.997338i	$-0.523230\pi$
$887$	−4.34315	−0.145829	−0.0729143	+	0.997338i	$0.523230\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	−32.2843	−1.08096
$893$	4.68629	0.156821
$894$	0	0
$895$	0	0
$896$	−50.9706	−1.70281
$897$	0	0
$898$	1.23045	0.0410606
$899$	0	0
$900$	0	0
$901$	90.9117	3.02871
$902$	0.343146	0.0114255
$903$	0	0
$904$	−23.7401	−0.789584
$905$	0	0
$906$	0	0
$907$	−7.02944	−0.233409	−0.116704	−	0.993167i	$-0.537233\pi$
$907$	−7.02944	−0.233409	−0.116704	+	0.993167i	$0.537233\pi$
$908$	25.5980	0.849499
$909$	0	0
$910$	0	0
$911$	15.0294	0.497947	0.248974	−	0.968510i	$-0.419907\pi$
$911$	15.0294	0.497947	0.248974	+	0.968510i	$0.419907\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	0.284271	0.00940286
$915$	0	0
$916$	3.65685	0.120826
$917$	93.2548	3.07955
$918$	0	0
$919$	28.4853	0.939643	0.469821	−	0.882762i	$-0.344318\pi$
$919$	28.4853	0.939643	0.469821	+	0.882762i	$0.344318\pi$
$920$	0	0
$921$	0	0
$922$	11.6569	0.383898
$923$	77.2548	2.54287
$924$	0	0
$925$	0	0
$926$	12.0000	0.394344
$927$	0	0
$928$	−3.65685	−0.120042
$929$	−33.5980	−1.10231	−0.551157	−	0.834402i	$-0.685813\pi$
$929$	−33.5980	−1.10231	−0.551157	+	0.834402i	$0.685813\pi$
$930$	0	0
$931$	−19.1127	−0.626393
$932$	24.0833	0.788873
$933$	0	0
$934$	9.37258	0.306680
$935$	0	0
$936$	0	0
$937$	−44.9706	−1.46912	−0.734562	−	0.678541i	$-0.762612\pi$
$937$	−44.9706	−1.46912	−0.734562	+	0.678541i	$0.762612\pi$
$938$	−11.3137	−0.369406
$939$	0	0
$940$	0	0
$941$	−38.7696	−1.26385	−0.631926	−	0.775029i	$-0.717735\pi$
$941$	−38.7696	−1.26385	−0.631926	+	0.775029i	$0.717735\pi$
$942$	0	0
$943$	−3.31371	−0.107909
$944$	12.0000	0.390567
$945$	0	0
$946$	1.31371	0.0427123
$947$	−38.6274	−1.25522	−0.627611	−	0.778527i	$-0.715967\pi$
$947$	−38.6274	−1.25522	−0.627611	+	0.778527i	$0.715967\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	0	0
$951$	0	0
$952$	52.2843	1.69454
$953$	27.7990	0.900498	0.450249	−	0.892903i	$-0.351335\pi$
$953$	27.7990	0.900498	0.450249	+	0.892903i	$0.351335\pi$
$954$	0	0
$955$	0	0
$956$	11.5980	0.375105
$957$	0	0
$958$	−1.25483	−0.0405418
$959$	44.9706	1.45218
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0.804041	0.0259233
$963$	0	0
$964$	43.2548	1.39314
$965$	0	0
$966$	0	0
$967$	39.4558	1.26881	0.634407	−	0.772999i	$-0.281244\pi$
$967$	39.4558	1.26881	0.634407	+	0.772999i	$0.281244\pi$
$968$	−1.58579	−0.0509691
$969$	0	0
$970$	0	0
$971$	−10.6274	−0.341050	−0.170525	−	0.985353i	$-0.554546\pi$
$971$	−10.6274	−0.341050	−0.170525	+	0.985353i	$0.554546\pi$
$972$	0	0
$973$	−79.5980	−2.55179
$974$	−8.68629	−0.278327
$975$	0	0
$976$	−1.02944	−0.0329515
$977$	25.3137	0.809857	0.404929	−	0.914348i	$-0.367296\pi$
$977$	25.3137	0.809857	0.404929	+	0.914348i	$0.367296\pi$
$978$	0	0
$979$	7.65685	0.244714
$980$	0	0
$981$	0	0
$982$	−10.6274	−0.339135
$983$	14.6274	0.466542	0.233271	−	0.972412i	$-0.425057\pi$
$983$	14.6274	0.466542	0.233271	+	0.972412i	$0.425057\pi$
$984$	0	0
$985$	0	0
$986$	2.34315	0.0746210
$987$	0	0
$988$	−12.1177	−0.385517
$989$	−12.6863	−0.403401
$990$	0	0
$991$	14.6274	0.464655	0.232328	−	0.972638i	$-0.425366\pi$
$991$	14.6274	0.464655	0.232328	+	0.972638i	$0.425366\pi$
$992$	0	0
$993$	0	0
$994$	−27.3137	−0.866338
$995$	0	0
$996$	0	0
$997$	16.6863	0.528460	0.264230	−	0.964460i	$-0.414882\pi$
$997$	16.6863	0.528460	0.264230	+	0.964460i	$0.414882\pi$
$998$	−13.9411	−0.441299
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.m.1.2		2
3.2	odd	2		825.2.a.g.1.1		2
5.2	odd	4		2475.2.c.m.199.3		4
5.3	odd	4		2475.2.c.m.199.2		4
5.4	even	2		495.2.a.d.1.1		2
15.2	even	4		825.2.c.e.199.2		4
15.8	even	4		825.2.c.e.199.3		4
15.14	odd	2		165.2.a.a.1.2	✓	2
20.19	odd	2		7920.2.a.cg.1.2		2
33.32	even	2		9075.2.a.v.1.2		2
55.54	odd	2		5445.2.a.m.1.2		2
60.59	even	2		2640.2.a.bb.1.2		2
105.104	even	2		8085.2.a.ba.1.2		2
165.164	even	2		1815.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.a.1.2	✓	2	15.14	odd	2
495.2.a.d.1.1		2	5.4	even	2
825.2.a.g.1.1		2	3.2	odd	2
825.2.c.e.199.2		4	15.2	even	4
825.2.c.e.199.3		4	15.8	even	4
1815.2.a.k.1.1		2	165.164	even	2
2475.2.a.m.1.2		2	1.1	even	1	trivial
2475.2.c.m.199.2		4	5.3	odd	4
2475.2.c.m.199.3		4	5.2	odd	4
2640.2.a.bb.1.2		2	60.59	even	2
5445.2.a.m.1.2		2	55.54	odd	2
7920.2.a.cg.1.2		2	20.19	odd	2
8085.2.a.ba.1.2		2	105.104	even	2
9075.2.a.v.1.2		2	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{11} -5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} -1.17157 q^{19} +0.414214 q^{22} -4.00000 q^{23} -2.34315 q^{26} -8.82843 q^{28} -0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} -0.343146 q^{37} -0.485281 q^{38} +0.828427 q^{41} +3.17157 q^{43} -1.82843 q^{44} -1.65685 q^{46} -4.00000 q^{47} +16.3137 q^{49} +10.3431 q^{52} -13.3137 q^{53} -7.65685 q^{56} -0.343146 q^{58} +4.00000 q^{59} -0.343146 q^{61} -4.17157 q^{64} -5.65685 q^{67} +12.4853 q^{68} -13.6569 q^{71} +11.3137 q^{73} -0.142136 q^{74} +2.14214 q^{76} +4.82843 q^{77} -8.48528 q^{79} +0.343146 q^{82} -10.0000 q^{83} +1.31371 q^{86} -1.58579 q^{88} +7.65685 q^{89} -27.3137 q^{91} +7.31371 q^{92} -1.65685 q^{94} -0.343146 q^{97} +6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 2 q^{11} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 8 q^{19} - 2 q^{22} - 8 q^{23} - 16 q^{26} - 12 q^{28} + 4 q^{29} + 6 q^{32} - 12 q^{37} + 16 q^{38} - 4 q^{41} + 12 q^{43} + 2 q^{44} + 8 q^{46} - 8 q^{47} + 10 q^{49} + 32 q^{52} - 4 q^{53} - 4 q^{56} - 12 q^{58} + 8 q^{59} - 12 q^{61} - 14 q^{64} + 8 q^{68} - 16 q^{71} + 28 q^{74} - 24 q^{76} + 4 q^{77} + 12 q^{82} - 20 q^{83} - 20 q^{86} - 6 q^{88} + 4 q^{89} - 32 q^{91} - 8 q^{92} + 8 q^{94} - 12 q^{97} + 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 2 * q^11 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 8 * q^19 - 2 * q^22 - 8 * q^23 - 16 * q^26 - 12 * q^28 + 4 * q^29 + 6 * q^32 - 12 * q^37 + 16 * q^38 - 4 * q^41 + 12 * q^43 + 2 * q^44 + 8 * q^46 - 8 * q^47 + 10 * q^49 + 32 * q^52 - 4 * q^53 - 4 * q^56 - 12 * q^58 + 8 * q^59 - 12 * q^61 - 14 * q^64 + 8 * q^68 - 16 * q^71 + 28 * q^74 - 24 * q^76 + 4 * q^77 + 12 * q^82 - 20 * q^83 - 20 * q^86 - 6 * q^88 + 4 * q^89 - 32 * q^91 - 8 * q^92 + 8 * q^94 - 12 * q^97 + 22 * q^98

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