LMFDB - Embedded newform 1148.4.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 $ x^15 - 247x^13 - 6x^12 + 23870x^11 + 940x^10 - 1147074x^9 - 8966x^8 + 28798759x^7 - 3139244x^6 - 363586916x^5 + 114207044x^4 + 2060613491x^3 - 892545712x^2 - 4098298668*x + 1720288256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$8.23201$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.23201 q^{3} -8.08041 q^{5} -7.00000 q^{7} +40.7659 q^{9} +O(q^{10})$	$q+8.23201 q^{3} -8.08041 q^{5} -7.00000 q^{7} +40.7659 q^{9} +14.7813 q^{11} -43.1520 q^{13} -66.5180 q^{15} +44.6224 q^{17} +2.31676 q^{19} -57.6240 q^{21} -169.449 q^{23} -59.7069 q^{25} +113.321 q^{27} -19.5342 q^{29} -173.348 q^{31} +121.680 q^{33} +56.5629 q^{35} -80.6605 q^{37} -355.228 q^{39} -41.0000 q^{41} +313.552 q^{43} -329.406 q^{45} -441.334 q^{47} +49.0000 q^{49} +367.332 q^{51} +294.832 q^{53} -119.439 q^{55} +19.0716 q^{57} -41.5144 q^{59} -202.787 q^{61} -285.362 q^{63} +348.686 q^{65} +548.144 q^{67} -1394.90 q^{69} -803.122 q^{71} +602.852 q^{73} -491.508 q^{75} -103.469 q^{77} -998.028 q^{79} -167.819 q^{81} +401.262 q^{83} -360.567 q^{85} -160.806 q^{87} -1515.08 q^{89} +302.064 q^{91} -1427.00 q^{93} -18.7204 q^{95} +329.903 q^{97} +602.576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * 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
$3$	8.23201	1.58425	0.792125	−	0.610359i	$-0.208975\pi$
$3$	8.23201	1.58425	0.792125	+	0.610359i	$0.208975\pi$
$4$	0	0
$5$	−8.08041	−0.722734	−0.361367	−	0.932424i	$-0.617690\pi$
$5$	−8.08041	−0.722734	−0.361367	+	0.932424i	$0.617690\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	40.7659	1.50985
$10$	0	0
$11$	14.7813	0.405159	0.202579	−	0.979266i	$-0.435068\pi$
$11$	14.7813	0.405159	0.202579	+	0.979266i	$0.435068\pi$
$12$	0	0
$13$	−43.1520	−0.920632	−0.460316	−	0.887755i	$-0.652264\pi$
$13$	−43.1520	−0.920632	−0.460316	+	0.887755i	$0.652264\pi$
$14$	0	0
$15$	−66.5180	−1.14499
$16$	0	0
$17$	44.6224	0.636619	0.318309	−	0.947987i	$-0.396885\pi$
$17$	44.6224	0.636619	0.318309	+	0.947987i	$0.396885\pi$
$18$	0	0
$19$	2.31676	0.0279737	0.0139869	−	0.999902i	$-0.495548\pi$
$19$	2.31676	0.0279737	0.0139869	+	0.999902i	$0.495548\pi$
$20$	0	0
$21$	−57.6240	−0.598790
$22$	0	0
$23$	−169.449	−1.53619	−0.768097	−	0.640334i	$-0.778796\pi$
$23$	−169.449	−1.53619	−0.768097	+	0.640334i	$0.778796\pi$
$24$	0	0
$25$	−59.7069	−0.477655
$26$	0	0
$27$	113.321	0.807729
$28$	0	0
$29$	−19.5342	−0.125083	−0.0625416	−	0.998042i	$-0.519921\pi$
$29$	−19.5342	−0.125083	−0.0625416	+	0.998042i	$0.519921\pi$
$30$	0	0
$31$	−173.348	−1.00433	−0.502166	−	0.864771i	$-0.667463\pi$
$31$	−173.348	−1.00433	−0.502166	+	0.864771i	$0.667463\pi$
$32$	0	0
$33$	121.680	0.641873
$34$	0	0
$35$	56.5629	0.273168
$36$	0	0
$37$	−80.6605	−0.358392	−0.179196	−	0.983813i	$-0.557350\pi$
$37$	−80.6605	−0.358392	−0.179196	+	0.983813i	$0.557350\pi$
$38$	0	0
$39$	−355.228	−1.45851
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	313.552	1.11200	0.556002	−	0.831181i	$-0.312335\pi$
$43$	313.552	1.11200	0.556002	+	0.831181i	$0.312335\pi$
$44$	0	0
$45$	−329.406	−1.09122
$46$	0	0
$47$	−441.334	−1.36968	−0.684842	−	0.728692i	$-0.740129\pi$
$47$	−441.334	−1.36968	−0.684842	+	0.728692i	$0.740129\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	367.332	1.00856
$52$	0	0
$53$	294.832	0.764119	0.382060	−	0.924138i	$-0.375215\pi$
$53$	294.832	0.764119	0.382060	+	0.924138i	$0.375215\pi$
$54$	0	0
$55$	−119.439	−0.292822
$56$	0	0
$57$	19.0716	0.0443174
$58$	0	0
$59$	−41.5144	−0.0916052	−0.0458026	−	0.998951i	$-0.514585\pi$
$59$	−41.5144	−0.0916052	−0.0458026	+	0.998951i	$0.514585\pi$
$60$	0	0
$61$	−202.787	−0.425644	−0.212822	−	0.977091i	$-0.568265\pi$
$61$	−202.787	−0.425644	−0.212822	+	0.977091i	$0.568265\pi$
$62$	0	0
$63$	−285.362	−0.570670
$64$	0	0
$65$	348.686	0.665372
$66$	0	0
$67$	548.144	0.999498	0.499749	−	0.866170i	$-0.333425\pi$
$67$	548.144	0.999498	0.499749	+	0.866170i	$0.333425\pi$
$68$	0	0
$69$	−1394.90	−2.43372
$70$	0	0
$71$	−803.122	−1.34244	−0.671219	−	0.741259i	$-0.734229\pi$
$71$	−803.122	−1.34244	−0.671219	+	0.741259i	$0.734229\pi$
$72$	0	0
$73$	602.852	0.966555	0.483278	−	0.875467i	$-0.339446\pi$
$73$	602.852	0.966555	0.483278	+	0.875467i	$0.339446\pi$
$74$	0	0
$75$	−491.508	−0.756726
$76$	0	0
$77$	−103.469	−0.153136
$78$	0	0
$79$	−998.028	−1.42135	−0.710677	−	0.703519i	$-0.751611\pi$
$79$	−998.028	−1.42135	−0.710677	+	0.703519i	$0.751611\pi$
$80$	0	0
$81$	−167.819	−0.230204
$82$	0	0
$83$	401.262	0.530653	0.265327	−	0.964159i	$-0.414520\pi$
$83$	401.262	0.530653	0.265327	+	0.964159i	$0.414520\pi$
$84$	0	0
$85$	−360.567	−0.460106
$86$	0	0
$87$	−160.806	−0.198163
$88$	0	0
$89$	−1515.08	−1.80447	−0.902237	−	0.431241i	$-0.858076\pi$
$89$	−1515.08	−1.80447	−0.902237	+	0.431241i	$0.858076\pi$
$90$	0	0
$91$	302.064	0.347966
$92$	0	0
$93$	−1427.00	−1.59111
$94$	0	0
$95$	−18.7204	−0.0202176
$96$	0	0
$97$	329.903	0.345326	0.172663	−	0.984981i	$-0.444763\pi$
$97$	329.903	0.345326	0.172663	+	0.984981i	$0.444763\pi$
$98$	0	0
$99$	602.576	0.611728
$100$	0	0
$101$	1350.63	1.33062	0.665311	−	0.746566i	$-0.268299\pi$
$101$	1350.63	1.33062	0.665311	+	0.746566i	$0.268299\pi$
$102$	0	0
$103$	−773.686	−0.740131	−0.370066	−	0.929006i	$-0.620665\pi$
$103$	−773.686	−0.740131	−0.370066	+	0.929006i	$0.620665\pi$
$104$	0	0
$105$	465.626	0.432766
$106$	0	0
$107$	−1092.17	−0.986770	−0.493385	−	0.869811i	$-0.664241\pi$
$107$	−1092.17	−0.986770	−0.493385	+	0.869811i	$0.664241\pi$
$108$	0	0
$109$	−1343.86	−1.18090	−0.590451	−	0.807074i	$-0.701050\pi$
$109$	−1343.86	−1.18090	−0.590451	+	0.807074i	$0.701050\pi$
$110$	0	0
$111$	−663.998	−0.567783
$112$	0	0
$113$	−1863.19	−1.55110	−0.775551	−	0.631285i	$-0.782528\pi$
$113$	−1863.19	−1.55110	−0.775551	+	0.631285i	$0.782528\pi$
$114$	0	0
$115$	1369.21	1.11026
$116$	0	0
$117$	−1759.13	−1.39002
$118$	0	0
$119$	−312.357	−0.240619
$120$	0	0
$121$	−1112.51	−0.835847
$122$	0	0
$123$	−337.512	−0.247418
$124$	0	0
$125$	1492.51	1.06795
$126$	0	0
$127$	1114.39	0.778630	0.389315	−	0.921105i	$-0.372712\pi$
$127$	1114.39	0.778630	0.389315	+	0.921105i	$0.372712\pi$
$128$	0	0
$129$	2581.16	1.76169
$130$	0	0
$131$	−1652.81	−1.10234	−0.551172	−	0.834392i	$-0.685819\pi$
$131$	−1652.81	−1.10234	−0.551172	+	0.834392i	$0.685819\pi$
$132$	0	0
$133$	−16.2173	−0.0105731
$134$	0	0
$135$	−915.683	−0.583774
$136$	0	0
$137$	−2809.90	−1.75231	−0.876153	−	0.482033i	$-0.839899\pi$
$137$	−2809.90	−1.75231	−0.876153	+	0.482033i	$0.839899\pi$
$138$	0	0
$139$	−95.3415	−0.0581781	−0.0290891	−	0.999577i	$-0.509261\pi$
$139$	−95.3415	−0.0581781	−0.0290891	+	0.999577i	$0.509261\pi$
$140$	0	0
$141$	−3633.06	−2.16992
$142$	0	0
$143$	−637.845	−0.373002
$144$	0	0
$145$	157.844	0.0904018
$146$	0	0
$147$	403.368	0.226321
$148$	0	0
$149$	2296.55	1.26269	0.631345	−	0.775502i	$-0.282503\pi$
$149$	2296.55	1.26269	0.631345	+	0.775502i	$0.282503\pi$
$150$	0	0
$151$	−1051.30	−0.566581	−0.283291	−	0.959034i	$-0.591426\pi$
$151$	−1051.30	−0.566581	−0.283291	+	0.959034i	$0.591426\pi$
$152$	0	0
$153$	1819.07	0.961199
$154$	0	0
$155$	1400.73	0.725865
$156$	0	0
$157$	1228.34	0.624410	0.312205	−	0.950015i	$-0.398932\pi$
$157$	1228.34	0.624410	0.312205	+	0.950015i	$0.398932\pi$
$158$	0	0
$159$	2427.06	1.21056
$160$	0	0
$161$	1186.14	0.580627
$162$	0	0
$163$	3178.58	1.52740	0.763699	−	0.645572i	$-0.223381\pi$
$163$	3178.58	1.52740	0.763699	+	0.645572i	$0.223381\pi$
$164$	0	0
$165$	−983.226	−0.463903
$166$	0	0
$167$	3200.37	1.48295	0.741474	−	0.670982i	$-0.234127\pi$
$167$	3200.37	1.48295	0.741474	+	0.670982i	$0.234127\pi$
$168$	0	0
$169$	−334.903	−0.152436
$170$	0	0
$171$	94.4449	0.0422361
$172$	0	0
$173$	−964.858	−0.424028	−0.212014	−	0.977267i	$-0.568002\pi$
$173$	−964.858	−0.424028	−0.212014	+	0.977267i	$0.568002\pi$
$174$	0	0
$175$	417.948	0.180537
$176$	0	0
$177$	−341.747	−0.145126
$178$	0	0
$179$	1567.47	0.654515	0.327258	−	0.944935i	$-0.393875\pi$
$179$	1567.47	0.654515	0.327258	+	0.944935i	$0.393875\pi$
$180$	0	0
$181$	−3840.23	−1.57703	−0.788513	−	0.615018i	$-0.789149\pi$
$181$	−3840.23	−1.57703	−0.788513	+	0.615018i	$0.789149\pi$
$182$	0	0
$183$	−1669.35	−0.674326
$184$	0	0
$185$	651.771	0.259022
$186$	0	0
$187$	659.579	0.257932
$188$	0	0
$189$	−793.249	−0.305293
$190$	0	0
$191$	2568.85	0.973171	0.486586	−	0.873633i	$-0.338242\pi$
$191$	2568.85	0.973171	0.486586	+	0.873633i	$0.338242\pi$
$192$	0	0
$193$	2864.67	1.06841	0.534206	−	0.845355i	$-0.320611\pi$
$193$	2864.67	1.06841	0.534206	+	0.845355i	$0.320611\pi$
$194$	0	0
$195$	2870.39	1.05412
$196$	0	0
$197$	3989.91	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$197$	3989.91	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$198$	0	0
$199$	4475.87	1.59440	0.797200	−	0.603715i	$-0.206313\pi$
$199$	4475.87	1.59440	0.797200	+	0.603715i	$0.206313\pi$
$200$	0	0
$201$	4512.32	1.58346
$202$	0	0
$203$	136.739	0.0472770
$204$	0	0
$205$	331.297	0.112872
$206$	0	0
$207$	−6907.73	−2.31942
$208$	0	0
$209$	34.2448	0.0113338
$210$	0	0
$211$	−2065.66	−0.673962	−0.336981	−	0.941511i	$-0.609406\pi$
$211$	−2065.66	−0.673962	−0.336981	+	0.941511i	$0.609406\pi$
$212$	0	0
$213$	−6611.31	−2.12676
$214$	0	0
$215$	−2533.63	−0.803684
$216$	0	0
$217$	1213.44	0.379602
$218$	0	0
$219$	4962.69	1.53127
$220$	0	0
$221$	−1925.55	−0.586092
$222$	0	0
$223$	1609.01	0.483172	0.241586	−	0.970379i	$-0.422332\pi$
$223$	1609.01	0.483172	0.241586	+	0.970379i	$0.422332\pi$
$224$	0	0
$225$	−2434.01	−0.721188
$226$	0	0
$227$	−2071.57	−0.605705	−0.302852	−	0.953038i	$-0.597939\pi$
$227$	−2071.57	−0.605705	−0.302852	+	0.953038i	$0.597939\pi$
$228$	0	0
$229$	−3371.11	−0.972793	−0.486396	−	0.873738i	$-0.661689\pi$
$229$	−3371.11	−0.972793	−0.486396	+	0.873738i	$0.661689\pi$
$230$	0	0
$231$	−851.761	−0.242605
$232$	0	0
$233$	−2789.20	−0.784233	−0.392117	−	0.919916i	$-0.628257\pi$
$233$	−2789.20	−0.784233	−0.392117	+	0.919916i	$0.628257\pi$
$234$	0	0
$235$	3566.16	0.989917
$236$	0	0
$237$	−8215.77	−2.25178
$238$	0	0
$239$	995.615	0.269460	0.134730	−	0.990882i	$-0.456983\pi$
$239$	995.615	0.269460	0.134730	+	0.990882i	$0.456983\pi$
$240$	0	0
$241$	541.771	0.144807	0.0724036	−	0.997375i	$-0.476933\pi$
$241$	541.771	0.144807	0.0724036	+	0.997375i	$0.476933\pi$
$242$	0	0
$243$	−4441.16	−1.17243
$244$	0	0
$245$	−395.940	−0.103248
$246$	0	0
$247$	−99.9729	−0.0257535
$248$	0	0
$249$	3303.19	0.840688
$250$	0	0
$251$	−681.430	−0.171361	−0.0856803	−	0.996323i	$-0.527306\pi$
$251$	−681.430	−0.171361	−0.0856803	+	0.996323i	$0.527306\pi$
$252$	0	0
$253$	−2504.68	−0.622402
$254$	0	0
$255$	−2968.19	−0.728923
$256$	0	0
$257$	−4182.47	−1.01516	−0.507579	−	0.861605i	$-0.669459\pi$
$257$	−4182.47	−1.01516	−0.507579	+	0.861605i	$0.669459\pi$
$258$	0	0
$259$	564.624	0.135460
$260$	0	0
$261$	−796.330	−0.188857
$262$	0	0
$263$	3655.22	0.856999	0.428499	−	0.903542i	$-0.359042\pi$
$263$	3655.22	0.856999	0.428499	+	0.903542i	$0.359042\pi$
$264$	0	0
$265$	−2382.37	−0.552255
$266$	0	0
$267$	−12472.2	−2.85874
$268$	0	0
$269$	5522.88	1.25181	0.625903	−	0.779901i	$-0.284731\pi$
$269$	5522.88	1.25181	0.625903	+	0.779901i	$0.284731\pi$
$270$	0	0
$271$	−3877.43	−0.869140	−0.434570	−	0.900638i	$-0.643100\pi$
$271$	−3877.43	−0.869140	−0.434570	+	0.900638i	$0.643100\pi$
$272$	0	0
$273$	2486.59	0.551266
$274$	0	0
$275$	−882.549	−0.193526
$276$	0	0
$277$	−5925.83	−1.28537	−0.642687	−	0.766129i	$-0.722181\pi$
$277$	−5925.83	−1.28537	−0.642687	+	0.766129i	$0.722181\pi$
$278$	0	0
$279$	−7066.71	−1.51639
$280$	0	0
$281$	−94.0667	−0.0199699	−0.00998496	−	0.999950i	$-0.503178\pi$
$281$	−94.0667	−0.0199699	−0.00998496	+	0.999950i	$0.503178\pi$
$282$	0	0
$283$	−3274.34	−0.687772	−0.343886	−	0.939011i	$-0.611743\pi$
$283$	−3274.34	−0.687772	−0.343886	+	0.939011i	$0.611743\pi$
$284$	0	0
$285$	−154.106	−0.0320297
$286$	0	0
$287$	287.000	0.0590281
$288$	0	0
$289$	−2921.84	−0.594716
$290$	0	0
$291$	2715.77	0.547082
$292$	0	0
$293$	535.263	0.106725	0.0533624	−	0.998575i	$-0.483006\pi$
$293$	535.263	0.106725	0.0533624	+	0.998575i	$0.483006\pi$
$294$	0	0
$295$	335.453	0.0662062
$296$	0	0
$297$	1675.04	0.327258
$298$	0	0
$299$	7312.05	1.41427
$300$	0	0
$301$	−2194.86	−0.420298
$302$	0	0
$303$	11118.4	2.10804
$304$	0	0
$305$	1638.61	0.307627
$306$	0	0
$307$	4100.48	0.762301	0.381151	−	0.924513i	$-0.375528\pi$
$307$	4100.48	0.762301	0.381151	+	0.924513i	$0.375528\pi$
$308$	0	0
$309$	−6368.99	−1.17255
$310$	0	0
$311$	1834.95	0.334568	0.167284	−	0.985909i	$-0.446500\pi$
$311$	1834.95	0.334568	0.167284	+	0.985909i	$0.446500\pi$
$312$	0	0
$313$	5458.62	0.985749	0.492874	−	0.870100i	$-0.335946\pi$
$313$	5458.62	0.985749	0.492874	+	0.870100i	$0.335946\pi$
$314$	0	0
$315$	2305.84	0.412442
$316$	0	0
$317$	−788.457	−0.139698	−0.0698488	−	0.997558i	$-0.522252\pi$
$317$	−788.457	−0.139698	−0.0698488	+	0.997558i	$0.522252\pi$
$318$	0	0
$319$	−288.742	−0.0506785
$320$	0	0
$321$	−8990.78	−1.56329
$322$	0	0
$323$	103.379	0.0178086
$324$	0	0
$325$	2576.47	0.439745
$326$	0	0
$327$	−11062.7	−1.87084
$328$	0	0
$329$	3089.34	0.517692
$330$	0	0
$331$	9065.72	1.50543	0.752715	−	0.658347i	$-0.228744\pi$
$331$	9065.72	1.50543	0.752715	+	0.658347i	$0.228744\pi$
$332$	0	0
$333$	−3288.20	−0.541118
$334$	0	0
$335$	−4429.23	−0.722372
$336$	0	0
$337$	699.136	0.113010	0.0565050	−	0.998402i	$-0.482004\pi$
$337$	699.136	0.113010	0.0565050	+	0.998402i	$0.482004\pi$
$338$	0	0
$339$	−15337.8	−2.45733
$340$	0	0
$341$	−2562.32	−0.406913
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	11271.4	1.75893
$346$	0	0
$347$	8512.88	1.31699	0.658495	−	0.752585i	$-0.271193\pi$
$347$	8512.88	1.31699	0.658495	+	0.752585i	$0.271193\pi$
$348$	0	0
$349$	1472.03	0.225776	0.112888	−	0.993608i	$-0.463990\pi$
$349$	1472.03	0.225776	0.112888	+	0.993608i	$0.463990\pi$
$350$	0	0
$351$	−4890.04	−0.743622
$352$	0	0
$353$	6896.02	1.03977	0.519884	−	0.854237i	$-0.325975\pi$
$353$	6896.02	1.03977	0.519884	+	0.854237i	$0.325975\pi$
$354$	0	0
$355$	6489.56	0.970226
$356$	0	0
$357$	−2571.32	−0.381201
$358$	0	0
$359$	1228.64	0.180627	0.0903135	−	0.995913i	$-0.471213\pi$
$359$	1228.64	0.180627	0.0903135	+	0.995913i	$0.471213\pi$
$360$	0	0
$361$	−6853.63	−0.999217
$362$	0	0
$363$	−9158.20	−1.32419
$364$	0	0
$365$	−4871.30	−0.698563
$366$	0	0
$367$	−5321.76	−0.756931	−0.378466	−	0.925615i	$-0.623548\pi$
$367$	−5321.76	−0.756931	−0.378466	+	0.925615i	$0.623548\pi$
$368$	0	0
$369$	−1671.40	−0.235799
$370$	0	0
$371$	−2063.83	−0.288810
$372$	0	0
$373$	−3410.67	−0.473452	−0.236726	−	0.971576i	$-0.576074\pi$
$373$	−3410.67	−0.473452	−0.236726	+	0.971576i	$0.576074\pi$
$374$	0	0
$375$	12286.3	1.69190
$376$	0	0
$377$	842.940	0.115156
$378$	0	0
$379$	−13136.3	−1.78039	−0.890195	−	0.455579i	$-0.849432\pi$
$379$	−13136.3	−1.78039	−0.890195	+	0.455579i	$0.849432\pi$
$380$	0	0
$381$	9173.66	1.23355
$382$	0	0
$383$	−2467.92	−0.329255	−0.164628	−	0.986356i	$-0.552642\pi$
$383$	−2467.92	−0.329255	−0.164628	+	0.986356i	$0.552642\pi$
$384$	0	0
$385$	836.076	0.110676
$386$	0	0
$387$	12782.2	1.67896
$388$	0	0
$389$	8457.57	1.10235	0.551177	−	0.834388i	$-0.314179\pi$
$389$	8457.57	1.10235	0.551177	+	0.834388i	$0.314179\pi$
$390$	0	0
$391$	−7561.20	−0.977970
$392$	0	0
$393$	−13606.0	−1.74639
$394$	0	0
$395$	8064.48	1.02726
$396$	0	0
$397$	2070.44	0.261744	0.130872	−	0.991399i	$-0.458222\pi$
$397$	2070.44	0.261744	0.130872	+	0.991399i	$0.458222\pi$
$398$	0	0
$399$	−133.501	−0.0167504
$400$	0	0
$401$	2517.31	0.313487	0.156744	−	0.987639i	$-0.449900\pi$
$401$	2517.31	0.313487	0.156744	+	0.987639i	$0.449900\pi$
$402$	0	0
$403$	7480.33	0.924620
$404$	0	0
$405$	1356.04	0.166376
$406$	0	0
$407$	−1192.27	−0.145206
$408$	0	0
$409$	9319.41	1.12669	0.563344	−	0.826222i	$-0.309515\pi$
$409$	9319.41	1.12669	0.563344	+	0.826222i	$0.309515\pi$
$410$	0	0
$411$	−23131.1	−2.77609
$412$	0	0
$413$	290.601	0.0346235
$414$	0	0
$415$	−3242.36	−0.383521
$416$	0	0
$417$	−784.852	−0.0921687
$418$	0	0
$419$	8155.16	0.950849	0.475424	−	0.879757i	$-0.342294\pi$
$419$	8155.16	0.950849	0.475424	+	0.879757i	$0.342294\pi$
$420$	0	0
$421$	2555.67	0.295857	0.147929	−	0.988998i	$-0.452739\pi$
$421$	2555.67	0.295857	0.147929	+	0.988998i	$0.452739\pi$
$422$	0	0
$423$	−17991.4	−2.06802
$424$	0	0
$425$	−2664.27	−0.304084
$426$	0	0
$427$	1419.51	0.160878
$428$	0	0
$429$	−5250.75	−0.590929
$430$	0	0
$431$	6396.35	0.714853	0.357426	−	0.933941i	$-0.383654\pi$
$431$	6396.35	0.714853	0.357426	+	0.933941i	$0.383654\pi$
$432$	0	0
$433$	−4571.84	−0.507410	−0.253705	−	0.967282i	$-0.581649\pi$
$433$	−4571.84	−0.507410	−0.253705	+	0.967282i	$0.581649\pi$
$434$	0	0
$435$	1299.38	0.143219
$436$	0	0
$437$	−392.571	−0.0429731
$438$	0	0
$439$	4585.84	0.498565	0.249282	−	0.968431i	$-0.419805\pi$
$439$	4585.84	0.498565	0.249282	+	0.968431i	$0.419805\pi$
$440$	0	0
$441$	1997.53	0.215693
$442$	0	0
$443$	8728.71	0.936148	0.468074	−	0.883689i	$-0.344948\pi$
$443$	8728.71	0.936148	0.468074	+	0.883689i	$0.344948\pi$
$444$	0	0
$445$	12242.5	1.30415
$446$	0	0
$447$	18905.2	2.00042
$448$	0	0
$449$	17921.6	1.88368	0.941842	−	0.336055i	$-0.109093\pi$
$449$	17921.6	1.88368	0.941842	+	0.336055i	$0.109093\pi$
$450$	0	0
$451$	−606.035	−0.0632751
$452$	0	0
$453$	−8654.33	−0.897606
$454$	0	0
$455$	−2440.80	−0.251487
$456$	0	0
$457$	7757.48	0.794047	0.397024	−	0.917808i	$-0.370043\pi$
$457$	7757.48	0.794047	0.397024	+	0.917808i	$0.370043\pi$
$458$	0	0
$459$	5056.67	0.514216
$460$	0	0
$461$	10081.7	1.01856	0.509278	−	0.860602i	$-0.329913\pi$
$461$	10081.7	1.01856	0.509278	+	0.860602i	$0.329913\pi$
$462$	0	0
$463$	−17061.3	−1.71254	−0.856271	−	0.516526i	$-0.827225\pi$
$463$	−17061.3	−1.71254	−0.856271	+	0.516526i	$0.827225\pi$
$464$	0	0
$465$	11530.8	1.14995
$466$	0	0
$467$	−8316.25	−0.824048	−0.412024	−	0.911173i	$-0.635178\pi$
$467$	−8316.25	−0.824048	−0.412024	+	0.911173i	$0.635178\pi$
$468$	0	0
$469$	−3837.01	−0.377775
$470$	0	0
$471$	10111.7	0.989222
$472$	0	0
$473$	4634.72	0.450538
$474$	0	0
$475$	−138.327	−0.0133618
$476$	0	0
$477$	12019.1	1.15371
$478$	0	0
$479$	3202.94	0.305525	0.152762	−	0.988263i	$-0.451183\pi$
$479$	3202.94	0.305525	0.152762	+	0.988263i	$0.451183\pi$
$480$	0	0
$481$	3480.67	0.329947
$482$	0	0
$483$	9764.31	0.919858
$484$	0	0
$485$	−2665.75	−0.249579
$486$	0	0
$487$	−4613.00	−0.429230	−0.214615	−	0.976699i	$-0.568850\pi$
$487$	−4613.00	−0.429230	−0.214615	+	0.976699i	$0.568850\pi$
$488$	0	0
$489$	26166.1	2.41978
$490$	0	0
$491$	11752.2	1.08018	0.540090	−	0.841608i	$-0.318390\pi$
$491$	11752.2	1.08018	0.540090	+	0.841608i	$0.318390\pi$
$492$	0	0
$493$	−871.663	−0.0796303
$494$	0	0
$495$	−4869.06	−0.442117
$496$	0	0
$497$	5621.86	0.507394
$498$	0	0
$499$	−10176.5	−0.912952	−0.456476	−	0.889736i	$-0.650888\pi$
$499$	−10176.5	−0.912952	−0.456476	+	0.889736i	$0.650888\pi$
$500$	0	0
$501$	26345.5	2.34936
$502$	0	0
$503$	−12065.4	−1.06953	−0.534763	−	0.845002i	$-0.679599\pi$
$503$	−12065.4	−1.06953	−0.534763	+	0.845002i	$0.679599\pi$
$504$	0	0
$505$	−10913.7	−0.961686
$506$	0	0
$507$	−2756.92	−0.241497
$508$	0	0
$509$	3747.91	0.326372	0.163186	−	0.986595i	$-0.447823\pi$
$509$	3747.91	0.326372	0.163186	+	0.986595i	$0.447823\pi$
$510$	0	0
$511$	−4219.97	−0.365324
$512$	0	0
$513$	262.538	0.0225952
$514$	0	0
$515$	6251.70	0.534918
$516$	0	0
$517$	−6523.51	−0.554939
$518$	0	0
$519$	−7942.72	−0.671766
$520$	0	0
$521$	14988.2	1.26036	0.630179	−	0.776450i	$-0.282982\pi$
$521$	14988.2	1.26036	0.630179	+	0.776450i	$0.282982\pi$
$522$	0	0
$523$	−7481.72	−0.625531	−0.312766	−	0.949830i	$-0.601255\pi$
$523$	−7481.72	−0.625531	−0.312766	+	0.949830i	$0.601255\pi$
$524$	0	0
$525$	3440.55	0.286015
$526$	0	0
$527$	−7735.22	−0.639376
$528$	0	0
$529$	16545.8	1.35989
$530$	0	0
$531$	−1692.37	−0.138310
$532$	0	0
$533$	1769.23	0.143779
$534$	0	0
$535$	8825.21	0.713172
$536$	0	0
$537$	12903.4	1.03692
$538$	0	0
$539$	724.286	0.0578798
$540$	0	0
$541$	7269.87	0.577737	0.288869	−	0.957369i	$-0.406721\pi$
$541$	7269.87	0.577737	0.288869	+	0.957369i	$0.406721\pi$
$542$	0	0
$543$	−31612.8	−2.49841
$544$	0	0
$545$	10858.9	0.853478
$546$	0	0
$547$	−7430.09	−0.580782	−0.290391	−	0.956908i	$-0.593785\pi$
$547$	−7430.09	−0.580782	−0.290391	+	0.956908i	$0.593785\pi$
$548$	0	0
$549$	−8266.82	−0.642658
$550$	0	0
$551$	−45.2560	−0.00349904
$552$	0	0
$553$	6986.20	0.537221
$554$	0	0
$555$	5365.38	0.410356
$556$	0	0
$557$	10165.0	0.773258	0.386629	−	0.922235i	$-0.373639\pi$
$557$	10165.0	0.773258	0.386629	+	0.922235i	$0.373639\pi$
$558$	0	0
$559$	−13530.4	−1.02375
$560$	0	0
$561$	5429.66	0.408628
$562$	0	0
$563$	−15572.7	−1.16574	−0.582868	−	0.812567i	$-0.698070\pi$
$563$	−15572.7	−1.16574	−0.582868	+	0.812567i	$0.698070\pi$
$564$	0	0
$565$	15055.4	1.12103
$566$	0	0
$567$	1174.73	0.0870089
$568$	0	0
$569$	−10750.7	−0.792076	−0.396038	−	0.918234i	$-0.629615\pi$
$569$	−10750.7	−0.792076	−0.396038	+	0.918234i	$0.629615\pi$
$570$	0	0
$571$	−7553.64	−0.553608	−0.276804	−	0.960926i	$-0.589275\pi$
$571$	−7553.64	−0.553608	−0.276804	+	0.960926i	$0.589275\pi$
$572$	0	0
$573$	21146.8	1.54175
$574$	0	0
$575$	10117.2	0.733771
$576$	0	0
$577$	2397.55	0.172983	0.0864915	−	0.996253i	$-0.472434\pi$
$577$	2397.55	0.172983	0.0864915	+	0.996253i	$0.472434\pi$
$578$	0	0
$579$	23582.0	1.69263
$580$	0	0
$581$	−2808.83	−0.200568
$582$	0	0
$583$	4358.02	0.309589
$584$	0	0
$585$	14214.5	1.00461
$586$	0	0
$587$	−16577.9	−1.16566	−0.582832	−	0.812593i	$-0.698055\pi$
$587$	−16577.9	−1.16566	−0.582832	+	0.812593i	$0.698055\pi$
$588$	0	0
$589$	−401.606	−0.0280949
$590$	0	0
$591$	32844.9	2.28606
$592$	0	0
$593$	120.242	0.00832675	0.00416338	−	0.999991i	$-0.498675\pi$
$593$	120.242	0.00832675	0.00416338	+	0.999991i	$0.498675\pi$
$594$	0	0
$595$	2523.97	0.173904
$596$	0	0
$597$	36845.4	2.52593
$598$	0	0
$599$	−578.087	−0.0394324	−0.0197162	−	0.999806i	$-0.506276\pi$
$599$	−578.087	−0.0394324	−0.0197162	+	0.999806i	$0.506276\pi$
$600$	0	0
$601$	26005.7	1.76505	0.882526	−	0.470263i	$-0.155841\pi$
$601$	26005.7	1.76505	0.882526	+	0.470263i	$0.155841\pi$
$602$	0	0
$603$	22345.6	1.50909
$604$	0	0
$605$	8989.55	0.604095
$606$	0	0
$607$	15126.1	1.01145	0.505723	−	0.862696i	$-0.331226\pi$
$607$	15126.1	1.01145	0.505723	+	0.862696i	$0.331226\pi$
$608$	0	0
$609$	1125.64	0.0748986
$610$	0	0
$611$	19044.4	1.26098
$612$	0	0
$613$	−13983.0	−0.921321	−0.460661	−	0.887576i	$-0.652387\pi$
$613$	−13983.0	−0.921321	−0.460661	+	0.887576i	$0.652387\pi$
$614$	0	0
$615$	2727.24	0.178818
$616$	0	0
$617$	−3094.51	−0.201913	−0.100957	−	0.994891i	$-0.532190\pi$
$617$	−3094.51	−0.201913	−0.100957	+	0.994891i	$0.532190\pi$
$618$	0	0
$619$	−244.451	−0.0158729	−0.00793645	−	0.999969i	$-0.502526\pi$
$619$	−244.451	−0.0158729	−0.00793645	+	0.999969i	$0.502526\pi$
$620$	0	0
$621$	−19202.1	−1.24083
$622$	0	0
$623$	10605.6	0.682027
$624$	0	0
$625$	−4596.72	−0.294190
$626$	0	0
$627$	281.904	0.0179556
$628$	0	0
$629$	−3599.27	−0.228159
$630$	0	0
$631$	9329.07	0.588565	0.294282	−	0.955719i	$-0.404919\pi$
$631$	9329.07	0.588565	0.294282	+	0.955719i	$0.404919\pi$
$632$	0	0
$633$	−17004.5	−1.06773
$634$	0	0
$635$	−9004.73	−0.562743
$636$	0	0
$637$	−2114.45	−0.131519
$638$	0	0
$639$	−32740.0	−2.02688
$640$	0	0
$641$	−7885.18	−0.485875	−0.242937	−	0.970042i	$-0.578111\pi$
$641$	−7885.18	−0.485875	−0.242937	+	0.970042i	$0.578111\pi$
$642$	0	0
$643$	−7906.06	−0.484890	−0.242445	−	0.970165i	$-0.577949\pi$
$643$	−7906.06	−0.484890	−0.242445	+	0.970165i	$0.577949\pi$
$644$	0	0
$645$	−20856.8	−1.27324
$646$	0	0
$647$	−10056.7	−0.611079	−0.305539	−	0.952179i	$-0.598837\pi$
$647$	−10056.7	−0.611079	−0.305539	+	0.952179i	$0.598837\pi$
$648$	0	0
$649$	−613.638	−0.0371147
$650$	0	0
$651$	9989.03	0.601384
$652$	0	0
$653$	−15207.3	−0.911341	−0.455671	−	0.890148i	$-0.650601\pi$
$653$	−15207.3	−0.911341	−0.455671	+	0.890148i	$0.650601\pi$
$654$	0	0
$655$	13355.4	0.796701
$656$	0	0
$657$	24575.8	1.45935
$658$	0	0
$659$	−13239.8	−0.782623	−0.391312	−	0.920258i	$-0.627979\pi$
$659$	−13239.8	−0.782623	−0.391312	+	0.920258i	$0.627979\pi$
$660$	0	0
$661$	−1051.55	−0.0618765	−0.0309383	−	0.999521i	$-0.509850\pi$
$661$	−1051.55	−0.0618765	−0.0309383	+	0.999521i	$0.509850\pi$
$662$	0	0
$663$	−15851.1	−0.928516
$664$	0	0
$665$	131.043	0.00764152
$666$	0	0
$667$	3310.04	0.192152
$668$	0	0
$669$	13245.4	0.765465
$670$	0	0
$671$	−2997.47	−0.172453
$672$	0	0
$673$	−14752.2	−0.844955	−0.422478	−	0.906373i	$-0.638839\pi$
$673$	−14752.2	−0.844955	−0.422478	+	0.906373i	$0.638839\pi$
$674$	0	0
$675$	−6766.07	−0.385816
$676$	0	0
$677$	20705.2	1.17543	0.587716	−	0.809068i	$-0.300027\pi$
$677$	20705.2	1.17543	0.587716	+	0.809068i	$0.300027\pi$
$678$	0	0
$679$	−2309.32	−0.130521
$680$	0	0
$681$	−17053.2	−0.959588
$682$	0	0
$683$	−453.942	−0.0254313	−0.0127157	−	0.999919i	$-0.504048\pi$
$683$	−453.942	−0.0254313	−0.0127157	+	0.999919i	$0.504048\pi$
$684$	0	0
$685$	22705.2	1.26645
$686$	0	0
$687$	−27751.0	−1.54115
$688$	0	0
$689$	−12722.6	−0.703473
$690$	0	0
$691$	13540.2	0.745431	0.372716	−	0.927946i	$-0.378427\pi$
$691$	13540.2	0.745431	0.372716	+	0.927946i	$0.378427\pi$
$692$	0	0
$693$	−4218.03	−0.231212
$694$	0	0
$695$	770.399	0.0420473
$696$	0	0
$697$	−1829.52	−0.0994232
$698$	0	0
$699$	−22960.7	−1.24242
$700$	0	0
$701$	−2998.90	−0.161579	−0.0807896	−	0.996731i	$-0.525744\pi$
$701$	−2998.90	−0.161579	−0.0807896	+	0.996731i	$0.525744\pi$
$702$	0	0
$703$	−186.871	−0.0100256
$704$	0	0
$705$	29356.6	1.56828
$706$	0	0
$707$	−9454.42	−0.502928
$708$	0	0
$709$	29655.2	1.57084	0.785418	−	0.618965i	$-0.212448\pi$
$709$	29655.2	1.57084	0.785418	+	0.618965i	$0.212448\pi$
$710$	0	0
$711$	−40685.6	−2.14603
$712$	0	0
$713$	29373.6	1.54285
$714$	0	0
$715$	5154.05	0.269581
$716$	0	0
$717$	8195.91	0.426892
$718$	0	0
$719$	−31462.1	−1.63190	−0.815951	−	0.578122i	$-0.803786\pi$
$719$	−31462.1	−1.63190	−0.815951	+	0.578122i	$0.803786\pi$
$720$	0	0
$721$	5415.80	0.279743
$722$	0	0
$723$	4459.86	0.229411
$724$	0	0
$725$	1166.33	0.0597466
$726$	0	0
$727$	17537.1	0.894658	0.447329	−	0.894369i	$-0.352375\pi$
$727$	17537.1	0.894658	0.447329	+	0.894369i	$0.352375\pi$
$728$	0	0
$729$	−32028.5	−1.62722
$730$	0	0
$731$	13991.4	0.707923
$732$	0	0
$733$	−23753.9	−1.19696	−0.598478	−	0.801139i	$-0.704228\pi$
$733$	−23753.9	−1.19696	−0.598478	+	0.801139i	$0.704228\pi$
$734$	0	0
$735$	−3259.38	−0.163570
$736$	0	0
$737$	8102.30	0.404955
$738$	0	0
$739$	−3659.06	−0.182139	−0.0910696	−	0.995845i	$-0.529029\pi$
$739$	−3659.06	−0.182139	−0.0910696	+	0.995845i	$0.529029\pi$
$740$	0	0
$741$	−822.977	−0.0408000
$742$	0	0
$743$	−19194.6	−0.947752	−0.473876	−	0.880592i	$-0.657146\pi$
$743$	−19194.6	−0.947752	−0.473876	+	0.880592i	$0.657146\pi$
$744$	0	0
$745$	−18557.1	−0.912589
$746$	0	0
$747$	16357.8	0.801207
$748$	0	0
$749$	7645.21	0.372964
$750$	0	0
$751$	1653.17	0.0803264	0.0401632	−	0.999193i	$-0.487212\pi$
$751$	1653.17	0.0803264	0.0401632	+	0.999193i	$0.487212\pi$
$752$	0	0
$753$	−5609.54	−0.271478
$754$	0	0
$755$	8494.95	0.409487
$756$	0	0
$757$	33776.9	1.62172	0.810860	−	0.585240i	$-0.199000\pi$
$757$	33776.9	1.62172	0.810860	+	0.585240i	$0.199000\pi$
$758$	0	0
$759$	−20618.5	−0.986041
$760$	0	0
$761$	3578.93	0.170481	0.0852405	−	0.996360i	$-0.472834\pi$
$761$	3578.93	0.170481	0.0852405	+	0.996360i	$0.472834\pi$
$762$	0	0
$763$	9407.01	0.446339
$764$	0	0
$765$	−14698.9	−0.694691
$766$	0	0
$767$	1791.43	0.0843347
$768$	0	0
$769$	−31106.9	−1.45871	−0.729353	−	0.684138i	$-0.760179\pi$
$769$	−31106.9	−1.45871	−0.729353	+	0.684138i	$0.760179\pi$
$770$	0	0
$771$	−34430.1	−1.60826
$772$	0	0
$773$	−13016.1	−0.605635	−0.302817	−	0.953049i	$-0.597927\pi$
$773$	−13016.1	−0.605635	−0.302817	+	0.953049i	$0.597927\pi$
$774$	0	0
$775$	10350.1	0.479724
$776$	0	0
$777$	4647.99	0.214602
$778$	0	0
$779$	−94.9871	−0.00436876
$780$	0	0
$781$	−11871.2	−0.543900
$782$	0	0
$783$	−2213.64	−0.101033
$784$	0	0
$785$	−9925.51	−0.451282
$786$	0	0
$787$	−31389.7	−1.42176	−0.710878	−	0.703315i	$-0.751702\pi$
$787$	−31389.7	−1.42176	−0.710878	+	0.703315i	$0.751702\pi$
$788$	0	0
$789$	30089.8	1.35770
$790$	0	0
$791$	13042.4	0.586261
$792$	0	0
$793$	8750.69	0.391861
$794$	0	0
$795$	−19611.7	−0.874910
$796$	0	0
$797$	−12608.3	−0.560362	−0.280181	−	0.959947i	$-0.590394\pi$
$797$	−12608.3	−0.560362	−0.280181	+	0.959947i	$0.590394\pi$
$798$	0	0
$799$	−19693.4	−0.871967
$800$	0	0
$801$	−61763.7	−2.72448
$802$	0	0
$803$	8910.97	0.391608
$804$	0	0
$805$	−9584.50	−0.419639
$806$	0	0
$807$	45464.4	1.98317
$808$	0	0
$809$	−22105.5	−0.960676	−0.480338	−	0.877084i	$-0.659486\pi$
$809$	−22105.5	−0.960676	−0.480338	+	0.877084i	$0.659486\pi$
$810$	0	0
$811$	25313.2	1.09601	0.548007	−	0.836474i	$-0.315387\pi$
$811$	25313.2	1.09601	0.548007	+	0.836474i	$0.315387\pi$
$812$	0	0
$813$	−31919.0	−1.37694
$814$	0	0
$815$	−25684.3	−1.10390
$816$	0	0
$817$	726.424	0.0311069
$818$	0	0
$819$	12313.9	0.525377
$820$	0	0
$821$	−31094.0	−1.32179	−0.660895	−	0.750478i	$-0.729823\pi$
$821$	−31094.0	−1.32179	−0.660895	+	0.750478i	$0.729823\pi$
$822$	0	0
$823$	−28450.4	−1.20501	−0.602503	−	0.798117i	$-0.705830\pi$
$823$	−28450.4	−1.20501	−0.602503	+	0.798117i	$0.705830\pi$
$824$	0	0
$825$	−7265.15	−0.306594
$826$	0	0
$827$	30236.2	1.27136	0.635680	−	0.771953i	$-0.280720\pi$
$827$	30236.2	1.27136	0.635680	+	0.771953i	$0.280720\pi$
$828$	0	0
$829$	−9378.74	−0.392928	−0.196464	−	0.980511i	$-0.562946\pi$
$829$	−9378.74	−0.392928	−0.196464	+	0.980511i	$0.562946\pi$
$830$	0	0
$831$	−48781.5	−2.03636
$832$	0	0
$833$	2186.50	0.0909456
$834$	0	0
$835$	−25860.3	−1.07178
$836$	0	0
$837$	−19644.1	−0.811228
$838$	0	0
$839$	10449.0	0.429962	0.214981	−	0.976618i	$-0.431031\pi$
$839$	10449.0	0.429962	0.214981	+	0.976618i	$0.431031\pi$
$840$	0	0
$841$	−24007.4	−0.984354
$842$	0	0
$843$	−774.357	−0.0316374
$844$	0	0
$845$	2706.15	0.110171
$846$	0	0
$847$	7787.58	0.315920
$848$	0	0
$849$	−26954.4	−1.08960
$850$	0	0
$851$	13667.8	0.550560
$852$	0	0
$853$	18568.8	0.745351	0.372675	−	0.927962i	$-0.378440\pi$
$853$	18568.8	0.745351	0.372675	+	0.927962i	$0.378440\pi$
$854$	0	0
$855$	−763.154	−0.0305255
$856$	0	0
$857$	43591.8	1.73754	0.868768	−	0.495219i	$-0.164912\pi$
$857$	43591.8	1.73754	0.868768	+	0.495219i	$0.164912\pi$
$858$	0	0
$859$	−3393.15	−0.134776	−0.0673882	−	0.997727i	$-0.521467\pi$
$859$	−3393.15	−0.134776	−0.0673882	+	0.997727i	$0.521467\pi$
$860$	0	0
$861$	2362.59	0.0935153
$862$	0	0
$863$	9454.75	0.372936	0.186468	−	0.982461i	$-0.440296\pi$
$863$	9454.75	0.372936	0.186468	+	0.982461i	$0.440296\pi$
$864$	0	0
$865$	7796.45	0.306459
$866$	0	0
$867$	−24052.6	−0.942180
$868$	0	0
$869$	−14752.2	−0.575874
$870$	0	0
$871$	−23653.5	−0.920170
$872$	0	0
$873$	13448.8	0.521390
$874$	0	0
$875$	−10447.6	−0.403648
$876$	0	0
$877$	−12291.0	−0.473249	−0.236624	−	0.971601i	$-0.576041\pi$
$877$	−12291.0	−0.473249	−0.236624	+	0.971601i	$0.576041\pi$
$878$	0	0
$879$	4406.29	0.169079
$880$	0	0
$881$	−6957.14	−0.266052	−0.133026	−	0.991113i	$-0.542469\pi$
$881$	−6957.14	−0.266052	−0.133026	+	0.991113i	$0.542469\pi$
$882$	0	0
$883$	3006.57	0.114586	0.0572928	−	0.998357i	$-0.481753\pi$
$883$	3006.57	0.114586	0.0572928	+	0.998357i	$0.481753\pi$
$884$	0	0
$885$	2761.45	0.104887
$886$	0	0
$887$	1556.12	0.0589057	0.0294528	−	0.999566i	$-0.490624\pi$
$887$	1556.12	0.0589057	0.0294528	+	0.999566i	$0.490624\pi$
$888$	0	0
$889$	−7800.73	−0.294295
$890$	0	0
$891$	−2480.58	−0.0932690
$892$	0	0
$893$	−1022.46	−0.0383152
$894$	0	0
$895$	−12665.8	−0.473041
$896$	0	0
$897$	60192.8	2.24056
$898$	0	0
$899$	3386.22	0.125625
$900$	0	0
$901$	13156.1	0.486453
$902$	0	0
$903$	−18068.1	−0.665858
$904$	0	0
$905$	31030.6	1.13977
$906$	0	0
$907$	−35669.9	−1.30584	−0.652921	−	0.757426i	$-0.726457\pi$
$907$	−35669.9	−1.30584	−0.652921	+	0.757426i	$0.726457\pi$
$908$	0	0
$909$	55059.8	2.00904
$910$	0	0
$911$	15920.2	0.578991	0.289495	−	0.957179i	$-0.406513\pi$
$911$	15920.2	0.578991	0.289495	+	0.957179i	$0.406513\pi$
$912$	0	0
$913$	5931.19	0.214999
$914$	0	0
$915$	13489.0	0.487359
$916$	0	0
$917$	11569.7	0.416647
$918$	0	0
$919$	−47207.7	−1.69449	−0.847246	−	0.531201i	$-0.821741\pi$
$919$	−47207.7	−1.69449	−0.847246	+	0.531201i	$0.821741\pi$
$920$	0	0
$921$	33755.1	1.20768
$922$	0	0
$923$	34656.4	1.23589
$924$	0	0
$925$	4815.99	0.171188
$926$	0	0
$927$	−31540.0	−1.11749
$928$	0	0
$929$	−13372.3	−0.472262	−0.236131	−	0.971721i	$-0.575879\pi$
$929$	−13372.3	−0.472262	−0.236131	+	0.971721i	$0.575879\pi$
$930$	0	0
$931$	113.521	0.00399625
$932$	0	0
$933$	15105.3	0.530039
$934$	0	0
$935$	−5329.67	−0.186416
$936$	0	0
$937$	−16657.8	−0.580776	−0.290388	−	0.956909i	$-0.593784\pi$
$937$	−16657.8	−0.580776	−0.290388	+	0.956909i	$0.593784\pi$
$938$	0	0
$939$	44935.4	1.56167
$940$	0	0
$941$	−1975.10	−0.0684236	−0.0342118	−	0.999415i	$-0.510892\pi$
$941$	−1975.10	−0.0684236	−0.0342118	+	0.999415i	$0.510892\pi$
$942$	0	0
$943$	6947.39	0.239913
$944$	0	0
$945$	6409.78	0.220646
$946$	0	0
$947$	26600.6	0.912780	0.456390	−	0.889780i	$-0.349142\pi$
$947$	26600.6	0.912780	0.456390	+	0.889780i	$0.349142\pi$
$948$	0	0
$949$	−26014.3	−0.889842
$950$	0	0
$951$	−6490.58	−0.221316
$952$	0	0
$953$	−44098.3	−1.49893	−0.749467	−	0.662042i	$-0.769690\pi$
$953$	−44098.3	−1.49893	−0.749467	+	0.662042i	$0.769690\pi$
$954$	0	0
$955$	−20757.4	−0.703344
$956$	0	0
$957$	−2376.92	−0.0802874
$958$	0	0
$959$	19669.3	0.662310
$960$	0	0
$961$	258.635	0.00868165
$962$	0	0
$963$	−44523.5	−1.48987
$964$	0	0
$965$	−23147.7	−0.772177
$966$	0	0
$967$	−2460.05	−0.0818096	−0.0409048	−	0.999163i	$-0.513024\pi$
$967$	−2460.05	−0.0818096	−0.0409048	+	0.999163i	$0.513024\pi$
$968$	0	0
$969$	851.020	0.0282133
$970$	0	0
$971$	46539.4	1.53813	0.769063	−	0.639174i	$-0.220723\pi$
$971$	46539.4	1.53813	0.769063	+	0.639174i	$0.220723\pi$
$972$	0	0
$973$	667.390	0.0219893
$974$	0	0
$975$	21209.6	0.696666
$976$	0	0
$977$	7331.13	0.240065	0.120033	−	0.992770i	$-0.461700\pi$
$977$	7331.13	0.240065	0.120033	+	0.992770i	$0.461700\pi$
$978$	0	0
$979$	−22394.9	−0.731098
$980$	0	0
$981$	−54783.7	−1.78298
$982$	0	0
$983$	22809.1	0.740079	0.370039	−	0.929016i	$-0.379344\pi$
$983$	22809.1	0.740079	0.370039	+	0.929016i	$0.379344\pi$
$984$	0	0
$985$	−32240.1	−1.04290
$986$	0	0
$987$	25431.4	0.820154
$988$	0	0
$989$	−53130.9	−1.70825
$990$	0	0
$991$	−10510.9	−0.336921	−0.168460	−	0.985708i	$-0.553880\pi$
$991$	−10510.9	−0.336921	−0.168460	+	0.985708i	$0.553880\pi$
$992$	0	0
$993$	74629.1	2.38498
$994$	0	0
$995$	−36166.8	−1.15233
$996$	0	0
$997$	9171.22	0.291329	0.145665	−	0.989334i	$-0.453468\pi$
$997$	9171.22	0.291329	0.145665	+	0.989334i	$0.453468\pi$
$998$	0	0
$999$	−9140.56	−0.289484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.b.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.b.1.14	✓	15	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+8.23201 q^{3} -8.08041 q^{5} -7.00000 q^{7} +40.7659 q^{9} +O(q^{10})\)	\(q+8.23201 q^{3} -8.08041 q^{5} -7.00000 q^{7} +40.7659 q^{9} +14.7813 q^{11} -43.1520 q^{13} -66.5180 q^{15} +44.6224 q^{17} +2.31676 q^{19} -57.6240 q^{21} -169.449 q^{23} -59.7069 q^{25} +113.321 q^{27} -19.5342 q^{29} -173.348 q^{31} +121.680 q^{33} +56.5629 q^{35} -80.6605 q^{37} -355.228 q^{39} -41.0000 q^{41} +313.552 q^{43} -329.406 q^{45} -441.334 q^{47} +49.0000 q^{49} +367.332 q^{51} +294.832 q^{53} -119.439 q^{55} +19.0716 q^{57} -41.5144 q^{59} -202.787 q^{61} -285.362 q^{63} +348.686 q^{65} +548.144 q^{67} -1394.90 q^{69} -803.122 q^{71} +602.852 q^{73} -491.508 q^{75} -103.469 q^{77} -998.028 q^{79} -167.819 q^{81} +401.262 q^{83} -360.567 q^{85} -160.806 q^{87} -1515.08 q^{89} +302.064 q^{91} -1427.00 q^{93} -18.7204 q^{95} +329.903 q^{97} +602.576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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