LMFDB - Embedded newform 1148.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,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, 2, names="a")

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

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
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:	$9.16682615204$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-2.53209 q^{3} +3.75877 q^{5} +1.00000 q^{7} +3.41147 q^{9} +O(q^{10})$	$q-2.53209 q^{3} +3.75877 q^{5} +1.00000 q^{7} +3.41147 q^{9} -6.45336 q^{11} -3.57398 q^{13} -9.51754 q^{15} +1.34730 q^{17} -5.71688 q^{19} -2.53209 q^{21} +1.83750 q^{23} +9.12836 q^{25} -1.04189 q^{27} -7.06418 q^{29} +3.51754 q^{31} +16.3405 q^{33} +3.75877 q^{35} -2.05644 q^{37} +9.04963 q^{39} -1.00000 q^{41} -7.47565 q^{43} +12.8229 q^{45} -8.31315 q^{47} +1.00000 q^{49} -3.41147 q^{51} +3.63041 q^{53} -24.2567 q^{55} +14.4757 q^{57} +5.51754 q^{59} -13.0642 q^{61} +3.41147 q^{63} -13.4338 q^{65} +1.51754 q^{67} -4.65270 q^{69} -1.14796 q^{71} -11.5175 q^{73} -23.1138 q^{75} -6.45336 q^{77} +15.2763 q^{79} -7.59627 q^{81} -12.1284 q^{83} +5.06418 q^{85} +17.8871 q^{87} -11.0915 q^{89} -3.57398 q^{91} -8.90673 q^{93} -21.4884 q^{95} -8.04963 q^{97} -22.0155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^7	$ 3 q - 3 q^{3} + 3 q^{7} - 6 q^{11} - 3 q^{13} - 6 q^{15} + 3 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} + 9 q^{25} - 12 q^{29} - 12 q^{31} + 6 q^{33} - 21 q^{37} - 3 q^{41} - 3 q^{43} + 18 q^{45} - 3 q^{47} + 3 q^{49} + 18 q^{53} - 36 q^{55} + 24 q^{57} - 6 q^{59} - 30 q^{61} - 24 q^{65} - 18 q^{67} - 15 q^{69} + 12 q^{71} - 12 q^{73} - 33 q^{75} - 6 q^{77} + 12 q^{79} - 9 q^{81} - 18 q^{83} + 6 q^{85} + 24 q^{87} - 3 q^{89} - 3 q^{91} - 6 q^{95} + 3 q^{97} - 18 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 - 6 * q^11 - 3 * q^13 - 6 * q^15 + 3 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 + 9 * q^25 - 12 * q^29 - 12 * q^31 + 6 * q^33 - 21 * q^37 - 3 * q^41 - 3 * q^43 + 18 * q^45 - 3 * q^47 + 3 * q^49 + 18 * q^53 - 36 * q^55 + 24 * q^57 - 6 * q^59 - 30 * q^61 - 24 * q^65 - 18 * q^67 - 15 * q^69 + 12 * q^71 - 12 * q^73 - 33 * q^75 - 6 * q^77 + 12 * q^79 - 9 * q^81 - 18 * q^83 + 6 * q^85 + 24 * q^87 - 3 * q^89 - 3 * q^91 - 6 * q^95 + 3 * q^97 - 18 * 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
$3$	−2.53209	−1.46190	−0.730951	−	0.682430i	$-0.760923\pi$
$3$	−2.53209	−1.46190	−0.730951	+	0.682430i	$0.760923\pi$
$4$	0	0
$5$	3.75877	1.68097	0.840487	−	0.541832i	$-0.182269\pi$
$5$	3.75877	1.68097	0.840487	+	0.541832i	$0.182269\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.41147	1.13716
$10$	0	0
$11$	−6.45336	−1.94576	−0.972881	−	0.231306i	$-0.925700\pi$
$11$	−6.45336	−1.94576	−0.972881	+	0.231306i	$0.925700\pi$
$12$	0	0
$13$	−3.57398	−0.991243	−0.495622	−	0.868539i	$-0.665060\pi$
$13$	−3.57398	−0.991243	−0.495622	+	0.868539i	$0.665060\pi$
$14$	0	0
$15$	−9.51754	−2.45742
$16$	0	0
$17$	1.34730	0.326767	0.163384	−	0.986563i	$-0.447759\pi$
$17$	1.34730	0.326767	0.163384	+	0.986563i	$0.447759\pi$
$18$	0	0
$19$	−5.71688	−1.31154	−0.655771	−	0.754960i	$-0.727656\pi$
$19$	−5.71688	−1.31154	−0.655771	+	0.754960i	$0.727656\pi$
$20$	0	0
$21$	−2.53209	−0.552547
$22$	0	0
$23$	1.83750	0.383144	0.191572	−	0.981479i	$-0.438641\pi$
$23$	1.83750	0.383144	0.191572	+	0.981479i	$0.438641\pi$
$24$	0	0
$25$	9.12836	1.82567
$26$	0	0
$27$	−1.04189	−0.200512
$28$	0	0
$29$	−7.06418	−1.31178	−0.655892	−	0.754854i	$-0.727708\pi$
$29$	−7.06418	−1.31178	−0.655892	+	0.754854i	$0.727708\pi$
$30$	0	0
$31$	3.51754	0.631769	0.315885	−	0.948798i	$-0.397699\pi$
$31$	3.51754	0.631769	0.315885	+	0.948798i	$0.397699\pi$
$32$	0	0
$33$	16.3405	2.84451
$34$	0	0
$35$	3.75877	0.635348
$36$	0	0
$37$	−2.05644	−0.338076	−0.169038	−	0.985610i	$-0.554066\pi$
$37$	−2.05644	−0.338076	−0.169038	+	0.985610i	$0.554066\pi$
$38$	0	0
$39$	9.04963	1.44910
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−7.47565	−1.14003	−0.570013	−	0.821636i	$-0.693062\pi$
$43$	−7.47565	−1.14003	−0.570013	+	0.821636i	$0.693062\pi$
$44$	0	0
$45$	12.8229	1.91153
$46$	0	0
$47$	−8.31315	−1.21260	−0.606299	−	0.795237i	$-0.707346\pi$
$47$	−8.31315	−1.21260	−0.606299	+	0.795237i	$0.707346\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.41147	−0.477702
$52$	0	0
$53$	3.63041	0.498676	0.249338	−	0.968417i	$-0.419787\pi$
$53$	3.63041	0.498676	0.249338	+	0.968417i	$0.419787\pi$
$54$	0	0
$55$	−24.2567	−3.27077
$56$	0	0
$57$	14.4757	1.91735
$58$	0	0
$59$	5.51754	0.718323	0.359161	−	0.933275i	$-0.383063\pi$
$59$	5.51754	0.718323	0.359161	+	0.933275i	$0.383063\pi$
$60$	0	0
$61$	−13.0642	−1.67270	−0.836348	−	0.548198i	$-0.815314\pi$
$61$	−13.0642	−1.67270	−0.836348	+	0.548198i	$0.815314\pi$
$62$	0	0
$63$	3.41147	0.429805
$64$	0	0
$65$	−13.4338	−1.66625
$66$	0	0
$67$	1.51754	0.185397	0.0926986	−	0.995694i	$-0.470451\pi$
$67$	1.51754	0.185397	0.0926986	+	0.995694i	$0.470451\pi$
$68$	0	0
$69$	−4.65270	−0.560120
$70$	0	0
$71$	−1.14796	−0.136237	−0.0681187	−	0.997677i	$-0.521700\pi$
$71$	−1.14796	−0.136237	−0.0681187	+	0.997677i	$0.521700\pi$
$72$	0	0
$73$	−11.5175	−1.34803	−0.674013	−	0.738719i	$-0.735431\pi$
$73$	−11.5175	−1.34803	−0.674013	+	0.738719i	$0.735431\pi$
$74$	0	0
$75$	−23.1138	−2.66895
$76$	0	0
$77$	−6.45336	−0.735429
$78$	0	0
$79$	15.2763	1.71872	0.859360	−	0.511372i	$-0.170862\pi$
$79$	15.2763	1.71872	0.859360	+	0.511372i	$0.170862\pi$
$80$	0	0
$81$	−7.59627	−0.844030
$82$	0	0
$83$	−12.1284	−1.33126	−0.665630	−	0.746282i	$-0.731837\pi$
$83$	−12.1284	−1.33126	−0.665630	+	0.746282i	$0.731837\pi$
$84$	0	0
$85$	5.06418	0.549287
$86$	0	0
$87$	17.8871	1.91770
$88$	0	0
$89$	−11.0915	−1.17570	−0.587849	−	0.808970i	$-0.700025\pi$
$89$	−11.0915	−1.17570	−0.587849	+	0.808970i	$0.700025\pi$
$90$	0	0
$91$	−3.57398	−0.374655
$92$	0	0
$93$	−8.90673	−0.923585
$94$	0	0
$95$	−21.4884	−2.20467
$96$	0	0
$97$	−8.04963	−0.817316	−0.408658	−	0.912688i	$-0.634003\pi$
$97$	−8.04963	−0.817316	−0.408658	+	0.912688i	$0.634003\pi$
$98$	0	0
$99$	−22.0155	−2.21264
$100$	0	0
$101$	9.59627	0.954864	0.477432	−	0.878669i	$-0.341568\pi$
$101$	9.59627	0.954864	0.477432	+	0.878669i	$0.341568\pi$
$102$	0	0
$103$	14.0155	1.38099	0.690493	−	0.723339i	$-0.257394\pi$
$103$	14.0155	1.38099	0.690493	+	0.723339i	$0.257394\pi$
$104$	0	0
$105$	−9.51754	−0.928817
$106$	0	0
$107$	−3.21894	−0.311187	−0.155593	−	0.987821i	$-0.549729\pi$
$107$	−3.21894	−0.311187	−0.155593	+	0.987821i	$0.549729\pi$
$108$	0	0
$109$	8.12836	0.778555	0.389278	−	0.921120i	$-0.372725\pi$
$109$	8.12836	0.778555	0.389278	+	0.921120i	$0.372725\pi$
$110$	0	0
$111$	5.20708	0.494234
$112$	0	0
$113$	0.487511	0.0458612	0.0229306	−	0.999737i	$-0.492700\pi$
$113$	0.487511	0.0458612	0.0229306	+	0.999737i	$0.492700\pi$
$114$	0	0
$115$	6.90673	0.644056
$116$	0	0
$117$	−12.1925	−1.12720
$118$	0	0
$119$	1.34730	0.123506
$120$	0	0
$121$	30.6459	2.78599
$122$	0	0
$123$	2.53209	0.228311
$124$	0	0
$125$	15.5175	1.38793
$126$	0	0
$127$	−16.5544	−1.46896	−0.734482	−	0.678628i	$-0.762575\pi$
$127$	−16.5544	−1.46896	−0.734482	+	0.678628i	$0.762575\pi$
$128$	0	0
$129$	18.9290	1.66661
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	0	0
$133$	−5.71688	−0.495716
$134$	0	0
$135$	−3.91622	−0.337055
$136$	0	0
$137$	−8.58172	−0.733186	−0.366593	−	0.930381i	$-0.619476\pi$
$137$	−8.58172	−0.733186	−0.366593	+	0.930381i	$0.619476\pi$
$138$	0	0
$139$	−4.28581	−0.363517	−0.181759	−	0.983343i	$-0.558179\pi$
$139$	−4.28581	−0.363517	−0.181759	+	0.983343i	$0.558179\pi$
$140$	0	0
$141$	21.0496	1.77270
$142$	0	0
$143$	23.0642	1.92872
$144$	0	0
$145$	−26.5526	−2.20508
$146$	0	0
$147$	−2.53209	−0.208843
$148$	0	0
$149$	20.5817	1.68612	0.843060	−	0.537819i	$-0.180752\pi$
$149$	20.5817	1.68612	0.843060	+	0.537819i	$0.180752\pi$
$150$	0	0
$151$	−20.3851	−1.65891	−0.829457	−	0.558571i	$-0.811350\pi$
$151$	−20.3851	−1.65891	−0.829457	+	0.558571i	$0.811350\pi$
$152$	0	0
$153$	4.59627	0.371586
$154$	0	0
$155$	13.2216	1.06199
$156$	0	0
$157$	−12.0419	−0.961047	−0.480524	−	0.876982i	$-0.659553\pi$
$157$	−12.0419	−0.961047	−0.480524	+	0.876982i	$0.659553\pi$
$158$	0	0
$159$	−9.19253	−0.729015
$160$	0	0
$161$	1.83750	0.144815
$162$	0	0
$163$	15.8084	1.23821	0.619105	−	0.785308i	$-0.287496\pi$
$163$	15.8084	1.23821	0.619105	+	0.785308i	$0.287496\pi$
$164$	0	0
$165$	61.4201	4.78155
$166$	0	0
$167$	−3.50475	−0.271206	−0.135603	−	0.990763i	$-0.543297\pi$
$167$	−3.50475	−0.271206	−0.135603	+	0.990763i	$0.543297\pi$
$168$	0	0
$169$	−0.226682	−0.0174370
$170$	0	0
$171$	−19.5030	−1.49143
$172$	0	0
$173$	−0.980400	−0.0745384	−0.0372692	−	0.999305i	$-0.511866\pi$
$173$	−0.980400	−0.0745384	−0.0372692	+	0.999305i	$0.511866\pi$
$174$	0	0
$175$	9.12836	0.690039
$176$	0	0
$177$	−13.9709	−1.05012
$178$	0	0
$179$	19.2371	1.43785	0.718925	−	0.695088i	$-0.244635\pi$
$179$	19.2371	1.43785	0.718925	+	0.695088i	$0.244635\pi$
$180$	0	0
$181$	−13.4388	−0.998899	−0.499450	−	0.866343i	$-0.666464\pi$
$181$	−13.4388	−0.998899	−0.499450	+	0.866343i	$0.666464\pi$
$182$	0	0
$183$	33.0797	2.44532
$184$	0	0
$185$	−7.72967	−0.568297
$186$	0	0
$187$	−8.69459	−0.635812
$188$	0	0
$189$	−1.04189	−0.0757863
$190$	0	0
$191$	−8.82295	−0.638406	−0.319203	−	0.947686i	$-0.603415\pi$
$191$	−8.82295	−0.638406	−0.319203	+	0.947686i	$0.603415\pi$
$192$	0	0
$193$	−16.2276	−1.16809	−0.584045	−	0.811722i	$-0.698531\pi$
$193$	−16.2276	−1.16809	−0.584045	+	0.811722i	$0.698531\pi$
$194$	0	0
$195$	34.0155	2.43590
$196$	0	0
$197$	13.8844	0.989225	0.494613	−	0.869114i	$-0.335310\pi$
$197$	13.8844	0.989225	0.494613	+	0.869114i	$0.335310\pi$
$198$	0	0
$199$	21.9590	1.55664	0.778318	−	0.627871i	$-0.216073\pi$
$199$	21.9590	1.55664	0.778318	+	0.627871i	$0.216073\pi$
$200$	0	0
$201$	−3.84255	−0.271032
$202$	0	0
$203$	−7.06418	−0.495808
$204$	0	0
$205$	−3.75877	−0.262524
$206$	0	0
$207$	6.26857	0.435696
$208$	0	0
$209$	36.8931	2.55195
$210$	0	0
$211$	16.1284	1.11032	0.555161	−	0.831743i	$-0.312657\pi$
$211$	16.1284	1.11032	0.555161	+	0.831743i	$0.312657\pi$
$212$	0	0
$213$	2.90673	0.199166
$214$	0	0
$215$	−28.0993	−1.91635
$216$	0	0
$217$	3.51754	0.238786
$218$	0	0
$219$	29.1634	1.97068
$220$	0	0
$221$	−4.81521	−0.323906
$222$	0	0
$223$	0.822948	0.0551087	0.0275543	−	0.999620i	$-0.491228\pi$
$223$	0.822948	0.0551087	0.0275543	+	0.999620i	$0.491228\pi$
$224$	0	0
$225$	31.1411	2.07608
$226$	0	0
$227$	−14.3851	−0.954770	−0.477385	−	0.878694i	$-0.658415\pi$
$227$	−14.3851	−0.954770	−0.477385	+	0.878694i	$0.658415\pi$
$228$	0	0
$229$	11.8922	0.785857	0.392929	−	0.919569i	$-0.371462\pi$
$229$	11.8922	0.785857	0.392929	+	0.919569i	$0.371462\pi$
$230$	0	0
$231$	16.3405	1.07513
$232$	0	0
$233$	3.17705	0.208136	0.104068	−	0.994570i	$-0.466814\pi$
$233$	3.17705	0.208136	0.104068	+	0.994570i	$0.466814\pi$
$234$	0	0
$235$	−31.2472	−2.03834
$236$	0	0
$237$	−38.6810	−2.51260
$238$	0	0
$239$	−0.537141	−0.0347448	−0.0173724	−	0.999849i	$-0.505530\pi$
$239$	−0.537141	−0.0347448	−0.0173724	+	0.999849i	$0.505530\pi$
$240$	0	0
$241$	−14.8384	−0.955827	−0.477914	−	0.878407i	$-0.658607\pi$
$241$	−14.8384	−0.955827	−0.477914	+	0.878407i	$0.658607\pi$
$242$	0	0
$243$	22.3601	1.43440
$244$	0	0
$245$	3.75877	0.240139
$246$	0	0
$247$	20.4320	1.30006
$248$	0	0
$249$	30.7101	1.94617
$250$	0	0
$251$	−8.98040	−0.566838	−0.283419	−	0.958996i	$-0.591469\pi$
$251$	−8.98040	−0.566838	−0.283419	+	0.958996i	$0.591469\pi$
$252$	0	0
$253$	−11.8580	−0.745508
$254$	0	0
$255$	−12.8229	−0.803004
$256$	0	0
$257$	−5.43882	−0.339264	−0.169632	−	0.985507i	$-0.554258\pi$
$257$	−5.43882	−0.339264	−0.169632	+	0.985507i	$0.554258\pi$
$258$	0	0
$259$	−2.05644	−0.127781
$260$	0	0
$261$	−24.0993	−1.49171
$262$	0	0
$263$	17.2918	1.06626	0.533129	−	0.846034i	$-0.321016\pi$
$263$	17.2918	1.06626	0.533129	+	0.846034i	$0.321016\pi$
$264$	0	0
$265$	13.6459	0.838261
$266$	0	0
$267$	28.0847	1.71876
$268$	0	0
$269$	15.7196	0.958439	0.479220	−	0.877695i	$-0.340920\pi$
$269$	15.7196	0.958439	0.479220	+	0.877695i	$0.340920\pi$
$270$	0	0
$271$	17.8033	1.08148	0.540738	−	0.841191i	$-0.318145\pi$
$271$	17.8033	1.08148	0.540738	+	0.841191i	$0.318145\pi$
$272$	0	0
$273$	9.04963	0.547709
$274$	0	0
$275$	−58.9086	−3.55232
$276$	0	0
$277$	20.9118	1.25647	0.628234	−	0.778025i	$-0.283778\pi$
$277$	20.9118	1.25647	0.628234	+	0.778025i	$0.283778\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	14.3250	0.854558	0.427279	−	0.904120i	$-0.359472\pi$
$281$	14.3250	0.854558	0.427279	+	0.904120i	$0.359472\pi$
$282$	0	0
$283$	−23.4884	−1.39624	−0.698122	−	0.715979i	$-0.745981\pi$
$283$	−23.4884	−1.39624	−0.698122	+	0.715979i	$0.745981\pi$
$284$	0	0
$285$	54.4107	3.22301
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−15.1848	−0.893223
$290$	0	0
$291$	20.3824	1.19484
$292$	0	0
$293$	28.0702	1.63988	0.819938	−	0.572452i	$-0.194008\pi$
$293$	28.0702	1.63988	0.819938	+	0.572452i	$0.194008\pi$
$294$	0	0
$295$	20.7392	1.20748
$296$	0	0
$297$	6.72369	0.390148
$298$	0	0
$299$	−6.56717	−0.379789
$300$	0	0
$301$	−7.47565	−0.430889
$302$	0	0
$303$	−24.2986	−1.39592
$304$	0	0
$305$	−49.1052	−2.81176
$306$	0	0
$307$	−26.5526	−1.51544	−0.757719	−	0.652581i	$-0.773686\pi$
$307$	−26.5526	−1.51544	−0.757719	+	0.652581i	$0.773686\pi$
$308$	0	0
$309$	−35.4884	−2.01887
$310$	0	0
$311$	31.4466	1.78317	0.891585	−	0.452853i	$-0.149594\pi$
$311$	31.4466	1.78317	0.891585	+	0.452853i	$0.149594\pi$
$312$	0	0
$313$	−14.2540	−0.805685	−0.402843	−	0.915269i	$-0.631978\pi$
$313$	−14.2540	−0.805685	−0.402843	+	0.915269i	$0.631978\pi$
$314$	0	0
$315$	12.8229	0.722491
$316$	0	0
$317$	18.9222	1.06278	0.531389	−	0.847128i	$-0.321670\pi$
$317$	18.9222	1.06278	0.531389	+	0.847128i	$0.321670\pi$
$318$	0	0
$319$	45.5877	2.55242
$320$	0	0
$321$	8.15064	0.454925
$322$	0	0
$323$	−7.70233	−0.428569
$324$	0	0
$325$	−32.6245	−1.80968
$326$	0	0
$327$	−20.5817	−1.13817
$328$	0	0
$329$	−8.31315	−0.458319
$330$	0	0
$331$	22.0993	1.21469	0.607343	−	0.794440i	$-0.292236\pi$
$331$	22.0993	1.21469	0.607343	+	0.794440i	$0.292236\pi$
$332$	0	0
$333$	−7.01548	−0.384446
$334$	0	0
$335$	5.70409	0.311648
$336$	0	0
$337$	−18.3800	−1.00122	−0.500612	−	0.865672i	$-0.666892\pi$
$337$	−18.3800	−1.00122	−0.500612	+	0.865672i	$0.666892\pi$
$338$	0	0
$339$	−1.23442	−0.0670446
$340$	0	0
$341$	−22.7000	−1.22927
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−17.4884	−0.941546
$346$	0	0
$347$	−24.4979	−1.31512	−0.657559	−	0.753403i	$-0.728411\pi$
$347$	−24.4979	−1.31512	−0.657559	+	0.753403i	$0.728411\pi$
$348$	0	0
$349$	−29.6851	−1.58901	−0.794503	−	0.607260i	$-0.792269\pi$
$349$	−29.6851	−1.58901	−0.794503	+	0.607260i	$0.792269\pi$
$350$	0	0
$351$	3.72369	0.198756
$352$	0	0
$353$	−13.4338	−0.715007	−0.357504	−	0.933912i	$-0.616372\pi$
$353$	−13.4338	−0.715007	−0.357504	+	0.933912i	$0.616372\pi$
$354$	0	0
$355$	−4.31490	−0.229011
$356$	0	0
$357$	−3.41147	−0.180554
$358$	0	0
$359$	4.07192	0.214908	0.107454	−	0.994210i	$-0.465730\pi$
$359$	4.07192	0.214908	0.107454	+	0.994210i	$0.465730\pi$
$360$	0	0
$361$	13.6827	0.720144
$362$	0	0
$363$	−77.5981	−4.07285
$364$	0	0
$365$	−43.2918	−2.26600
$366$	0	0
$367$	−18.5526	−0.968439	−0.484220	−	0.874947i	$-0.660896\pi$
$367$	−18.5526	−0.968439	−0.484220	+	0.874947i	$0.660896\pi$
$368$	0	0
$369$	−3.41147	−0.177594
$370$	0	0
$371$	3.63041	0.188482
$372$	0	0
$373$	−13.2344	−0.685252	−0.342626	−	0.939472i	$-0.611316\pi$
$373$	−13.2344	−0.685252	−0.342626	+	0.939472i	$0.611316\pi$
$374$	0	0
$375$	−39.2918	−2.02902
$376$	0	0
$377$	25.2472	1.30030
$378$	0	0
$379$	11.0591	0.568069	0.284035	−	0.958814i	$-0.408327\pi$
$379$	11.0591	0.568069	0.284035	+	0.958814i	$0.408327\pi$
$380$	0	0
$381$	41.9172	2.14748
$382$	0	0
$383$	23.2030	1.18562	0.592808	−	0.805344i	$-0.298019\pi$
$383$	23.2030	1.18562	0.592808	+	0.805344i	$0.298019\pi$
$384$	0	0
$385$	−24.2567	−1.23624
$386$	0	0
$387$	−25.5030	−1.29639
$388$	0	0
$389$	6.42602	0.325812	0.162906	−	0.986642i	$-0.447913\pi$
$389$	6.42602	0.325812	0.162906	+	0.986642i	$0.447913\pi$
$390$	0	0
$391$	2.47565	0.125199
$392$	0	0
$393$	35.4492	1.78818
$394$	0	0
$395$	57.4201	2.88912
$396$	0	0
$397$	−17.2608	−0.866296	−0.433148	−	0.901323i	$-0.642597\pi$
$397$	−17.2608	−0.866296	−0.433148	+	0.901323i	$0.642597\pi$
$398$	0	0
$399$	14.4757	0.724689
$400$	0	0
$401$	15.2781	0.762950	0.381475	−	0.924379i	$-0.375416\pi$
$401$	15.2781	0.762950	0.381475	+	0.924379i	$0.375416\pi$
$402$	0	0
$403$	−12.5716	−0.626237
$404$	0	0
$405$	−28.5526	−1.41879
$406$	0	0
$407$	13.2709	0.657816
$408$	0	0
$409$	11.8425	0.585576	0.292788	−	0.956177i	$-0.405417\pi$
$409$	11.8425	0.585576	0.292788	+	0.956177i	$0.405417\pi$
$410$	0	0
$411$	21.7297	1.07185
$412$	0	0
$413$	5.51754	0.271500
$414$	0	0
$415$	−45.5877	−2.23781
$416$	0	0
$417$	10.8520	0.531427
$418$	0	0
$419$	−4.69459	−0.229346	−0.114673	−	0.993403i	$-0.536582\pi$
$419$	−4.69459	−0.229346	−0.114673	+	0.993403i	$0.536582\pi$
$420$	0	0
$421$	13.8135	0.673226	0.336613	−	0.941643i	$-0.390719\pi$
$421$	13.8135	0.673226	0.336613	+	0.941643i	$0.390719\pi$
$422$	0	0
$423$	−28.3601	−1.37891
$424$	0	0
$425$	12.2986	0.596570
$426$	0	0
$427$	−13.0642	−0.632220
$428$	0	0
$429$	−58.4005	−2.81961
$430$	0	0
$431$	20.4243	0.983802	0.491901	−	0.870651i	$-0.336302\pi$
$431$	20.4243	0.983802	0.491901	+	0.870651i	$0.336302\pi$
$432$	0	0
$433$	21.5175	1.03407	0.517034	−	0.855965i	$-0.327036\pi$
$433$	21.5175	1.03407	0.517034	+	0.855965i	$0.327036\pi$
$434$	0	0
$435$	67.2336	3.22360
$436$	0	0
$437$	−10.5047	−0.502510
$438$	0	0
$439$	−3.34966	−0.159871	−0.0799353	−	0.996800i	$-0.525471\pi$
$439$	−3.34966	−0.159871	−0.0799353	+	0.996800i	$0.525471\pi$
$440$	0	0
$441$	3.41147	0.162451
$442$	0	0
$443$	3.16519	0.150383	0.0751914	−	0.997169i	$-0.476043\pi$
$443$	3.16519	0.150383	0.0751914	+	0.997169i	$0.476043\pi$
$444$	0	0
$445$	−41.6905	−1.97632
$446$	0	0
$447$	−52.1147	−2.46494
$448$	0	0
$449$	14.7050	0.693973	0.346986	−	0.937870i	$-0.387205\pi$
$449$	14.7050	0.693973	0.346986	+	0.937870i	$0.387205\pi$
$450$	0	0
$451$	6.45336	0.303877
$452$	0	0
$453$	51.6168	2.42517
$454$	0	0
$455$	−13.4338	−0.629785
$456$	0	0
$457$	11.9608	0.559503	0.279751	−	0.960073i	$-0.409748\pi$
$457$	11.9608	0.559503	0.279751	+	0.960073i	$0.409748\pi$
$458$	0	0
$459$	−1.40373	−0.0655207
$460$	0	0
$461$	18.7101	0.871415	0.435708	−	0.900088i	$-0.356498\pi$
$461$	18.7101	0.871415	0.435708	+	0.900088i	$0.356498\pi$
$462$	0	0
$463$	−10.6209	−0.493596	−0.246798	−	0.969067i	$-0.579378\pi$
$463$	−10.6209	−0.493596	−0.246798	+	0.969067i	$0.579378\pi$
$464$	0	0
$465$	−33.4783	−1.55252
$466$	0	0
$467$	3.51754	0.162772	0.0813862	−	0.996683i	$-0.474065\pi$
$467$	3.51754	0.162772	0.0813862	+	0.996683i	$0.474065\pi$
$468$	0	0
$469$	1.51754	0.0700735
$470$	0	0
$471$	30.4911	1.40496
$472$	0	0
$473$	48.2431	2.21822
$474$	0	0
$475$	−52.1857	−2.39445
$476$	0	0
$477$	12.3851	0.567073
$478$	0	0
$479$	−35.0455	−1.60127	−0.800635	−	0.599152i	$-0.795504\pi$
$479$	−35.0455	−1.60127	−0.800635	+	0.599152i	$0.795504\pi$
$480$	0	0
$481$	7.34966	0.335116
$482$	0	0
$483$	−4.65270	−0.211705
$484$	0	0
$485$	−30.2567	−1.37389
$486$	0	0
$487$	24.7229	1.12030	0.560150	−	0.828391i	$-0.310743\pi$
$487$	24.7229	1.12030	0.560150	+	0.828391i	$0.310743\pi$
$488$	0	0
$489$	−40.0283	−1.81014
$490$	0	0
$491$	12.1334	0.547573	0.273787	−	0.961790i	$-0.411724\pi$
$491$	12.1334	0.547573	0.273787	+	0.961790i	$0.411724\pi$
$492$	0	0
$493$	−9.51754	−0.428648
$494$	0	0
$495$	−82.7511	−3.71939
$496$	0	0
$497$	−1.14796	−0.0514929
$498$	0	0
$499$	15.7142	0.703464	0.351732	−	0.936101i	$-0.385593\pi$
$499$	15.7142	0.703464	0.351732	+	0.936101i	$0.385593\pi$
$500$	0	0
$501$	8.87433	0.396476
$502$	0	0
$503$	42.2481	1.88375	0.941876	−	0.335961i	$-0.109061\pi$
$503$	42.2481	1.88375	0.941876	+	0.335961i	$0.109061\pi$
$504$	0	0
$505$	36.0702	1.60510
$506$	0	0
$507$	0.573978	0.0254913
$508$	0	0
$509$	42.2158	1.87118	0.935590	−	0.353088i	$-0.114868\pi$
$509$	42.2158	1.87118	0.935590	+	0.353088i	$0.114868\pi$
$510$	0	0
$511$	−11.5175	−0.509506
$512$	0	0
$513$	5.95636	0.262980
$514$	0	0
$515$	52.6810	2.32140
$516$	0	0
$517$	53.6478	2.35943
$518$	0	0
$519$	2.48246	0.108968
$520$	0	0
$521$	−10.2739	−0.450110	−0.225055	−	0.974346i	$-0.572256\pi$
$521$	−10.2739	−0.450110	−0.225055	+	0.974346i	$0.572256\pi$
$522$	0	0
$523$	−25.1581	−1.10009	−0.550043	−	0.835137i	$-0.685389\pi$
$523$	−25.1581	−1.10009	−0.550043	+	0.835137i	$0.685389\pi$
$524$	0	0
$525$	−23.1138	−1.00877
$526$	0	0
$527$	4.73917	0.206441
$528$	0	0
$529$	−19.6236	−0.853200
$530$	0	0
$531$	18.8229	0.816846
$532$	0	0
$533$	3.57398	0.154806
$534$	0	0
$535$	−12.0993	−0.523097
$536$	0	0
$537$	−48.7101	−2.10199
$538$	0	0
$539$	−6.45336	−0.277966
$540$	0	0
$541$	−13.5493	−0.582531	−0.291265	−	0.956642i	$-0.594076\pi$
$541$	−13.5493	−0.582531	−0.291265	+	0.956642i	$0.594076\pi$
$542$	0	0
$543$	34.0283	1.46029
$544$	0	0
$545$	30.5526	1.30873
$546$	0	0
$547$	16.9067	0.722879	0.361440	−	0.932395i	$-0.382285\pi$
$547$	16.9067	0.722879	0.361440	+	0.932395i	$0.382285\pi$
$548$	0	0
$549$	−44.5681	−1.90212
$550$	0	0
$551$	40.3851	1.72046
$552$	0	0
$553$	15.2763	0.649615
$554$	0	0
$555$	19.5722	0.830795
$556$	0	0
$557$	22.2276	0.941814	0.470907	−	0.882183i	$-0.343927\pi$
$557$	22.2276	0.941814	0.470907	+	0.882183i	$0.343927\pi$
$558$	0	0
$559$	26.7178	1.13004
$560$	0	0
$561$	22.0155	0.929494
$562$	0	0
$563$	17.0523	0.718670	0.359335	−	0.933209i	$-0.383004\pi$
$563$	17.0523	0.718670	0.359335	+	0.933209i	$0.383004\pi$
$564$	0	0
$565$	1.83244	0.0770915
$566$	0	0
$567$	−7.59627	−0.319013
$568$	0	0
$569$	−23.0223	−0.965145	−0.482572	−	0.875856i	$-0.660297\pi$
$569$	−23.0223	−0.965145	−0.482572	+	0.875856i	$0.660297\pi$
$570$	0	0
$571$	20.4397	0.855377	0.427688	−	0.903926i	$-0.359328\pi$
$571$	20.4397	0.855377	0.427688	+	0.903926i	$0.359328\pi$
$572$	0	0
$573$	22.3405	0.933287
$574$	0	0
$575$	16.7733	0.699496
$576$	0	0
$577$	30.5921	1.27357	0.636784	−	0.771042i	$-0.280264\pi$
$577$	30.5921	1.27357	0.636784	+	0.771042i	$0.280264\pi$
$578$	0	0
$579$	41.0898	1.70763
$580$	0	0
$581$	−12.1284	−0.503169
$582$	0	0
$583$	−23.4284	−0.970305
$584$	0	0
$585$	−45.8289	−1.89479
$586$	0	0
$587$	−12.1857	−0.502959	−0.251479	−	0.967863i	$-0.580917\pi$
$587$	−12.1857	−0.502959	−0.251479	+	0.967863i	$0.580917\pi$
$588$	0	0
$589$	−20.1094	−0.828592
$590$	0	0
$591$	−35.1566	−1.44615
$592$	0	0
$593$	10.9240	0.448593	0.224297	−	0.974521i	$-0.427992\pi$
$593$	10.9240	0.448593	0.224297	+	0.974521i	$0.427992\pi$
$594$	0	0
$595$	5.06418	0.207611
$596$	0	0
$597$	−55.6023	−2.27565
$598$	0	0
$599$	−19.9044	−0.813270	−0.406635	−	0.913591i	$-0.633298\pi$
$599$	−19.9044	−0.813270	−0.406635	+	0.913591i	$0.633298\pi$
$600$	0	0
$601$	−19.0823	−0.778385	−0.389193	−	0.921156i	$-0.627246\pi$
$601$	−19.0823	−0.778385	−0.389193	+	0.921156i	$0.627246\pi$
$602$	0	0
$603$	5.17705	0.210826
$604$	0	0
$605$	115.191	4.68318
$606$	0	0
$607$	−23.8716	−0.968920	−0.484460	−	0.874813i	$-0.660984\pi$
$607$	−23.8716	−0.968920	−0.484460	+	0.874813i	$0.660984\pi$
$608$	0	0
$609$	17.8871	0.724823
$610$	0	0
$611$	29.7110	1.20198
$612$	0	0
$613$	−36.4192	−1.47096	−0.735479	−	0.677547i	$-0.763043\pi$
$613$	−36.4192	−1.47096	−0.735479	+	0.677547i	$0.763043\pi$
$614$	0	0
$615$	9.51754	0.383784
$616$	0	0
$617$	4.72638	0.190277	0.0951384	−	0.995464i	$-0.469671\pi$
$617$	4.72638	0.190277	0.0951384	+	0.995464i	$0.469671\pi$
$618$	0	0
$619$	−13.3345	−0.535959	−0.267979	−	0.963425i	$-0.586356\pi$
$619$	−13.3345	−0.535959	−0.267979	+	0.963425i	$0.586356\pi$
$620$	0	0
$621$	−1.91447	−0.0768249
$622$	0	0
$623$	−11.0915	−0.444372
$624$	0	0
$625$	12.6851	0.507404
$626$	0	0
$627$	−93.4166	−3.73070
$628$	0	0
$629$	−2.77063	−0.110472
$630$	0	0
$631$	−43.4766	−1.73078	−0.865388	−	0.501103i	$-0.832928\pi$
$631$	−43.4766	−1.73078	−0.865388	+	0.501103i	$0.832928\pi$
$632$	0	0
$633$	−40.8384	−1.62318
$634$	0	0
$635$	−62.2241	−2.46929
$636$	0	0
$637$	−3.57398	−0.141606
$638$	0	0
$639$	−3.91622	−0.154923
$640$	0	0
$641$	18.1129	0.715416	0.357708	−	0.933834i	$-0.383558\pi$
$641$	18.1129	0.715416	0.357708	+	0.933834i	$0.383558\pi$
$642$	0	0
$643$	−38.9205	−1.53487	−0.767436	−	0.641125i	$-0.778468\pi$
$643$	−38.9205	−1.53487	−0.767436	+	0.641125i	$0.778468\pi$
$644$	0	0
$645$	71.1498	2.80152
$646$	0	0
$647$	−5.31078	−0.208788	−0.104394	−	0.994536i	$-0.533290\pi$
$647$	−5.31078	−0.208788	−0.104394	+	0.994536i	$0.533290\pi$
$648$	0	0
$649$	−35.6067	−1.39769
$650$	0	0
$651$	−8.90673	−0.349082
$652$	0	0
$653$	−23.3756	−0.914757	−0.457378	−	0.889272i	$-0.651211\pi$
$653$	−23.3756	−0.914757	−0.457378	+	0.889272i	$0.651211\pi$
$654$	0	0
$655$	−52.6228	−2.05614
$656$	0	0
$657$	−39.2918	−1.53292
$658$	0	0
$659$	−34.9222	−1.36038	−0.680188	−	0.733038i	$-0.738102\pi$
$659$	−34.9222	−1.36038	−0.680188	+	0.733038i	$0.738102\pi$
$660$	0	0
$661$	−20.7885	−0.808578	−0.404289	−	0.914631i	$-0.632481\pi$
$661$	−20.7885	−0.808578	−0.404289	+	0.914631i	$0.632481\pi$
$662$	0	0
$663$	12.1925	0.473519
$664$	0	0
$665$	−21.4884	−0.833286
$666$	0	0
$667$	−12.9804	−0.502603
$668$	0	0
$669$	−2.08378	−0.0805635
$670$	0	0
$671$	84.3079	3.25467
$672$	0	0
$673$	−13.1480	−0.506816	−0.253408	−	0.967359i	$-0.581552\pi$
$673$	−13.1480	−0.506816	−0.253408	+	0.967359i	$0.581552\pi$
$674$	0	0
$675$	−9.51073	−0.366068
$676$	0	0
$677$	21.3655	0.821142	0.410571	−	0.911829i	$-0.365329\pi$
$677$	21.3655	0.821142	0.410571	+	0.911829i	$0.365329\pi$
$678$	0	0
$679$	−8.04963	−0.308916
$680$	0	0
$681$	36.4243	1.39578
$682$	0	0
$683$	−5.80335	−0.222059	−0.111029	−	0.993817i	$-0.535415\pi$
$683$	−5.80335	−0.222059	−0.111029	+	0.993817i	$0.535415\pi$
$684$	0	0
$685$	−32.2567	−1.23247
$686$	0	0
$687$	−30.1121	−1.14885
$688$	0	0
$689$	−12.9750	−0.494309
$690$	0	0
$691$	−17.0760	−0.649603	−0.324802	−	0.945782i	$-0.605298\pi$
$691$	−17.0760	−0.649603	−0.324802	+	0.945782i	$0.605298\pi$
$692$	0	0
$693$	−22.0155	−0.836299
$694$	0	0
$695$	−16.1094	−0.611063
$696$	0	0
$697$	−1.34730	−0.0510325
$698$	0	0
$699$	−8.04458	−0.304274
$700$	0	0
$701$	−49.0333	−1.85196	−0.925982	−	0.377569i	$-0.876760\pi$
$701$	−49.0333	−1.85196	−0.925982	+	0.377569i	$0.876760\pi$
$702$	0	0
$703$	11.7564	0.443401
$704$	0	0
$705$	79.1207	2.97986
$706$	0	0
$707$	9.59627	0.360905
$708$	0	0
$709$	0.152075	0.00571128	0.00285564	−	0.999996i	$-0.499091\pi$
$709$	0.152075	0.00571128	0.00285564	+	0.999996i	$0.499091\pi$
$710$	0	0
$711$	52.1147	1.95446
$712$	0	0
$713$	6.46347	0.242059
$714$	0	0
$715$	86.6930	3.24213
$716$	0	0
$717$	1.36009	0.0507935
$718$	0	0
$719$	7.94181	0.296179	0.148090	−	0.988974i	$-0.452688\pi$
$719$	7.94181	0.296179	0.148090	+	0.988974i	$0.452688\pi$
$720$	0	0
$721$	14.0155	0.521964
$722$	0	0
$723$	37.5722	1.39733
$724$	0	0
$725$	−64.4843	−2.39489
$726$	0	0
$727$	−2.36690	−0.0877833	−0.0438917	−	0.999036i	$-0.513976\pi$
$727$	−2.36690	−0.0877833	−0.0438917	+	0.999036i	$0.513976\pi$
$728$	0	0
$729$	−33.8289	−1.25292
$730$	0	0
$731$	−10.0719	−0.372523
$732$	0	0
$733$	16.6364	0.614479	0.307240	−	0.951632i	$-0.400595\pi$
$733$	16.6364	0.614479	0.307240	+	0.951632i	$0.400595\pi$
$734$	0	0
$735$	−9.51754	−0.351060
$736$	0	0
$737$	−9.79324	−0.360739
$738$	0	0
$739$	−38.3678	−1.41138	−0.705692	−	0.708519i	$-0.749364\pi$
$739$	−38.3678	−1.41138	−0.705692	+	0.708519i	$0.749364\pi$
$740$	0	0
$741$	−51.7357	−1.90056
$742$	0	0
$743$	17.9135	0.657184	0.328592	−	0.944472i	$-0.393426\pi$
$743$	17.9135	0.657184	0.328592	+	0.944472i	$0.393426\pi$
$744$	0	0
$745$	77.3620	2.83432
$746$	0	0
$747$	−41.3756	−1.51385
$748$	0	0
$749$	−3.21894	−0.117618
$750$	0	0
$751$	−8.98040	−0.327699	−0.163850	−	0.986485i	$-0.552391\pi$
$751$	−8.98040	−0.327699	−0.163850	+	0.986485i	$0.552391\pi$
$752$	0	0
$753$	22.7392	0.828661
$754$	0	0
$755$	−76.6228	−2.78859
$756$	0	0
$757$	−41.0060	−1.49039	−0.745194	−	0.666848i	$-0.767643\pi$
$757$	−41.0060	−1.49039	−0.745194	+	0.666848i	$0.767643\pi$
$758$	0	0
$759$	30.0256	1.08986
$760$	0	0
$761$	22.4534	0.813934	0.406967	−	0.913443i	$-0.366586\pi$
$761$	22.4534	0.813934	0.406967	+	0.913443i	$0.366586\pi$
$762$	0	0
$763$	8.12836	0.294266
$764$	0	0
$765$	17.2763	0.624626
$766$	0	0
$767$	−19.7196	−0.712032
$768$	0	0
$769$	16.4979	0.594931	0.297465	−	0.954733i	$-0.403859\pi$
$769$	16.4979	0.594931	0.297465	+	0.954733i	$0.403859\pi$
$770$	0	0
$771$	13.7716	0.495971
$772$	0	0
$773$	−12.2676	−0.441236	−0.220618	−	0.975360i	$-0.570807\pi$
$773$	−12.2676	−0.441236	−0.220618	+	0.975360i	$0.570807\pi$
$774$	0	0
$775$	32.1094	1.15340
$776$	0	0
$777$	5.20708	0.186803
$778$	0	0
$779$	5.71688	0.204829
$780$	0	0
$781$	7.40818	0.265085
$782$	0	0
$783$	7.36009	0.263028
$784$	0	0
$785$	−45.2627	−1.61549
$786$	0	0
$787$	23.7897	0.848012	0.424006	−	0.905659i	$-0.360623\pi$
$787$	23.7897	0.848012	0.424006	+	0.905659i	$0.360623\pi$
$788$	0	0
$789$	−43.7844	−1.55876
$790$	0	0
$791$	0.487511	0.0173339
$792$	0	0
$793$	46.6911	1.65805
$794$	0	0
$795$	−34.5526	−1.22546
$796$	0	0
$797$	−29.2763	−1.03702	−0.518510	−	0.855072i	$-0.673513\pi$
$797$	−29.2763	−1.03702	−0.518510	+	0.855072i	$0.673513\pi$
$798$	0	0
$799$	−11.2003	−0.396237
$800$	0	0
$801$	−37.8384	−1.33696
$802$	0	0
$803$	74.3269	2.62294
$804$	0	0
$805$	6.90673	0.243430
$806$	0	0
$807$	−39.8033	−1.40114
$808$	0	0
$809$	−29.2627	−1.02882	−0.514411	−	0.857544i	$-0.671989\pi$
$809$	−29.2627	−1.02882	−0.514411	+	0.857544i	$0.671989\pi$
$810$	0	0
$811$	22.0291	0.773546	0.386773	−	0.922175i	$-0.373590\pi$
$811$	22.0291	0.773546	0.386773	+	0.922175i	$0.373590\pi$
$812$	0	0
$813$	−45.0797	−1.58101
$814$	0	0
$815$	59.4201	2.08140
$816$	0	0
$817$	42.7374	1.49519
$818$	0	0
$819$	−12.1925	−0.426042
$820$	0	0
$821$	39.6195	1.38273	0.691365	−	0.722506i	$-0.257010\pi$
$821$	39.6195	1.38273	0.691365	+	0.722506i	$0.257010\pi$
$822$	0	0
$823$	−42.1729	−1.47006	−0.735028	−	0.678037i	$-0.762831\pi$
$823$	−42.1729	−1.47006	−0.735028	+	0.678037i	$0.762831\pi$
$824$	0	0
$825$	149.162	5.19315
$826$	0	0
$827$	43.2080	1.50249	0.751245	−	0.660024i	$-0.229454\pi$
$827$	43.2080	1.50249	0.751245	+	0.660024i	$0.229454\pi$
$828$	0	0
$829$	30.5134	1.05977	0.529887	−	0.848068i	$-0.322234\pi$
$829$	30.5134	1.05977	0.529887	+	0.848068i	$0.322234\pi$
$830$	0	0
$831$	−52.9505	−1.83683
$832$	0	0
$833$	1.34730	0.0466811
$834$	0	0
$835$	−13.1735	−0.455889
$836$	0	0
$837$	−3.66489	−0.126677
$838$	0	0
$839$	4.59215	0.158539	0.0792693	−	0.996853i	$-0.474741\pi$
$839$	4.59215	0.158539	0.0792693	+	0.996853i	$0.474741\pi$
$840$	0	0
$841$	20.9026	0.720780
$842$	0	0
$843$	−36.2722	−1.24928
$844$	0	0
$845$	−0.852044	−0.0293112
$846$	0	0
$847$	30.6459	1.05301
$848$	0	0
$849$	59.4748	2.04117
$850$	0	0
$851$	−3.77870	−0.129532
$852$	0	0
$853$	−7.39929	−0.253347	−0.126673	−	0.991944i	$-0.540430\pi$
$853$	−7.39929	−0.253347	−0.126673	+	0.991944i	$0.540430\pi$
$854$	0	0
$855$	−73.3073	−2.50706
$856$	0	0
$857$	−35.5039	−1.21279	−0.606396	−	0.795163i	$-0.707385\pi$
$857$	−35.5039	−1.21279	−0.606396	+	0.795163i	$0.707385\pi$
$858$	0	0
$859$	53.7107	1.83258	0.916292	−	0.400510i	$-0.131167\pi$
$859$	53.7107	1.83258	0.916292	+	0.400510i	$0.131167\pi$
$860$	0	0
$861$	2.53209	0.0862934
$862$	0	0
$863$	26.7784	0.911546	0.455773	−	0.890096i	$-0.349363\pi$
$863$	26.7784	0.911546	0.455773	+	0.890096i	$0.349363\pi$
$864$	0	0
$865$	−3.68510	−0.125297
$866$	0	0
$867$	38.4492	1.30580
$868$	0	0
$869$	−98.5836	−3.34422
$870$	0	0
$871$	−5.42366	−0.183774
$872$	0	0
$873$	−27.4611	−0.929418
$874$	0	0
$875$	15.5175	0.524589
$876$	0	0
$877$	−1.97596	−0.0667233	−0.0333617	−	0.999443i	$-0.510621\pi$
$877$	−1.97596	−0.0667233	−0.0333617	+	0.999443i	$0.510621\pi$
$878$	0	0
$879$	−71.0761	−2.39734
$880$	0	0
$881$	−45.0452	−1.51761	−0.758805	−	0.651317i	$-0.774217\pi$
$881$	−45.0452	−1.51761	−0.758805	+	0.651317i	$0.774217\pi$
$882$	0	0
$883$	−33.6851	−1.13359	−0.566797	−	0.823858i	$-0.691818\pi$
$883$	−33.6851	−1.13359	−0.566797	+	0.823858i	$0.691818\pi$
$884$	0	0
$885$	−52.5134	−1.76522
$886$	0	0
$887$	9.16519	0.307737	0.153869	−	0.988091i	$-0.450827\pi$
$887$	9.16519	0.307737	0.153869	+	0.988091i	$0.450827\pi$
$888$	0	0
$889$	−16.5544	−0.555216
$890$	0	0
$891$	49.0215	1.64228
$892$	0	0
$893$	47.5253	1.59037
$894$	0	0
$895$	72.3079	2.41699
$896$	0	0
$897$	16.6287	0.555215
$898$	0	0
$899$	−24.8485	−0.828745
$900$	0	0
$901$	4.89124	0.162951
$902$	0	0
$903$	18.9290	0.629918
$904$	0	0
$905$	−50.5134	−1.67912
$906$	0	0
$907$	−56.7184	−1.88330	−0.941652	−	0.336588i	$-0.890727\pi$
$907$	−56.7184	−1.88330	−0.941652	+	0.336588i	$0.890727\pi$
$908$	0	0
$909$	32.7374	1.08583
$910$	0	0
$911$	−3.89992	−0.129210	−0.0646050	−	0.997911i	$-0.520579\pi$
$911$	−3.89992	−0.129210	−0.0646050	+	0.997911i	$0.520579\pi$
$912$	0	0
$913$	78.2687	2.59032
$914$	0	0
$915$	124.339	4.11052
$916$	0	0
$917$	−14.0000	−0.462321
$918$	0	0
$919$	8.83305	0.291376	0.145688	−	0.989331i	$-0.453461\pi$
$919$	8.83305	0.291376	0.145688	+	0.989331i	$0.453461\pi$
$920$	0	0
$921$	67.2336	2.21542
$922$	0	0
$923$	4.10277	0.135044
$924$	0	0
$925$	−18.7719	−0.617216
$926$	0	0
$927$	47.8135	1.57040
$928$	0	0
$929$	−37.6373	−1.23484	−0.617420	−	0.786633i	$-0.711822\pi$
$929$	−37.6373	−1.23484	−0.617420	+	0.786633i	$0.711822\pi$
$930$	0	0
$931$	−5.71688	−0.187363
$932$	0	0
$933$	−79.6255	−2.60682
$934$	0	0
$935$	−32.6810	−1.06878
$936$	0	0
$937$	−31.6833	−1.03505	−0.517525	−	0.855668i	$-0.673146\pi$
$937$	−31.6833	−1.03505	−0.517525	+	0.855668i	$0.673146\pi$
$938$	0	0
$939$	36.0925	1.17783
$940$	0	0
$941$	30.0209	0.978652	0.489326	−	0.872101i	$-0.337243\pi$
$941$	30.0209	0.978652	0.489326	+	0.872101i	$0.337243\pi$
$942$	0	0
$943$	−1.83750	−0.0598371
$944$	0	0
$945$	−3.91622	−0.127395
$946$	0	0
$947$	−34.9118	−1.13448	−0.567240	−	0.823552i	$-0.691989\pi$
$947$	−34.9118	−1.13448	−0.567240	+	0.823552i	$0.691989\pi$
$948$	0	0
$949$	41.1634	1.33622
$950$	0	0
$951$	−47.9127	−1.55368
$952$	0	0
$953$	14.0874	0.456336	0.228168	−	0.973622i	$-0.426727\pi$
$953$	14.0874	0.456336	0.228168	+	0.973622i	$0.426727\pi$
$954$	0	0
$955$	−33.1634	−1.07314
$956$	0	0
$957$	−115.432	−3.73139
$958$	0	0
$959$	−8.58172	−0.277118
$960$	0	0
$961$	−18.6269	−0.600868
$962$	0	0
$963$	−10.9813	−0.353869
$964$	0	0
$965$	−60.9959	−1.96353
$966$	0	0
$967$	−4.76827	−0.153337	−0.0766685	−	0.997057i	$-0.524428\pi$
$967$	−4.76827	−0.153337	−0.0766685	+	0.997057i	$0.524428\pi$
$968$	0	0
$969$	19.5030	0.626526
$970$	0	0
$971$	−21.8316	−0.700610	−0.350305	−	0.936636i	$-0.613922\pi$
$971$	−21.8316	−0.700610	−0.350305	+	0.936636i	$0.613922\pi$
$972$	0	0
$973$	−4.28581	−0.137397
$974$	0	0
$975$	82.6082	2.64558
$976$	0	0
$977$	8.89311	0.284516	0.142258	−	0.989830i	$-0.454564\pi$
$977$	8.89311	0.284516	0.142258	+	0.989830i	$0.454564\pi$
$978$	0	0
$979$	71.5776	2.28763
$980$	0	0
$981$	27.7297	0.885340
$982$	0	0
$983$	−49.0215	−1.56354	−0.781771	−	0.623566i	$-0.785683\pi$
$983$	−49.0215	−1.56354	−0.781771	+	0.623566i	$0.785683\pi$
$984$	0	0
$985$	52.1884	1.66286
$986$	0	0
$987$	21.0496	0.670017
$988$	0	0
$989$	−13.7365	−0.436795
$990$	0	0
$991$	−8.63640	−0.274344	−0.137172	−	0.990547i	$-0.543801\pi$
$991$	−8.63640	−0.274344	−0.137172	+	0.990547i	$0.543801\pi$
$992$	0	0
$993$	−55.9573	−1.77575
$994$	0	0
$995$	82.5390	2.61666
$996$	0	0
$997$	−40.0806	−1.26937	−0.634683	−	0.772773i	$-0.718869\pi$
$997$	−40.0806	−1.26937	−0.634683	+	0.772773i	$0.718869\pi$
$998$	0	0
$999$	2.14258	0.0677882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.b.1.1	✓	3
4.3	odd	2		4592.2.a.v.1.3		3
7.6	odd	2		8036.2.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.b.1.1	✓	3	1.1	even	1	trivial
4592.2.a.v.1.3		3	4.3	odd	2
8036.2.a.h.1.3		3	7.6	odd	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53209 q^{3} +3.75877 q^{5} +1.00000 q^{7} +3.41147 q^{9} +O(q^{10})\)	\(q-2.53209 q^{3} +3.75877 q^{5} +1.00000 q^{7} +3.41147 q^{9} -6.45336 q^{11} -3.57398 q^{13} -9.51754 q^{15} +1.34730 q^{17} -5.71688 q^{19} -2.53209 q^{21} +1.83750 q^{23} +9.12836 q^{25} -1.04189 q^{27} -7.06418 q^{29} +3.51754 q^{31} +16.3405 q^{33} +3.75877 q^{35} -2.05644 q^{37} +9.04963 q^{39} -1.00000 q^{41} -7.47565 q^{43} +12.8229 q^{45} -8.31315 q^{47} +1.00000 q^{49} -3.41147 q^{51} +3.63041 q^{53} -24.2567 q^{55} +14.4757 q^{57} +5.51754 q^{59} -13.0642 q^{61} +3.41147 q^{63} -13.4338 q^{65} +1.51754 q^{67} -4.65270 q^{69} -1.14796 q^{71} -11.5175 q^{73} -23.1138 q^{75} -6.45336 q^{77} +15.2763 q^{79} -7.59627 q^{81} -12.1284 q^{83} +5.06418 q^{85} +17.8871 q^{87} -11.0915 q^{89} -3.57398 q^{91} -8.90673 q^{93} -21.4884 q^{95} -8.04963 q^{97} -22.0155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7	\( 3 q - 3 q^{3} + 3 q^{7} - 6 q^{11} - 3 q^{13} - 6 q^{15} + 3 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} + 9 q^{25} - 12 q^{29} - 12 q^{31} + 6 q^{33} - 21 q^{37} - 3 q^{41} - 3 q^{43} + 18 q^{45} - 3 q^{47} + 3 q^{49} + 18 q^{53} - 36 q^{55} + 24 q^{57} - 6 q^{59} - 30 q^{61} - 24 q^{65} - 18 q^{67} - 15 q^{69} + 12 q^{71} - 12 q^{73} - 33 q^{75} - 6 q^{77} + 12 q^{79} - 9 q^{81} - 18 q^{83} + 6 q^{85} + 24 q^{87} - 3 q^{89} - 3 q^{91} - 6 q^{95} + 3 q^{97} - 18 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 - 6 * q^11 - 3 * q^13 - 6 * q^15 + 3 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 + 9 * q^25 - 12 * q^29 - 12 * q^31 + 6 * q^33 - 21 * q^37 - 3 * q^41 - 3 * q^43 + 18 * q^45 - 3 * q^47 + 3 * q^49 + 18 * q^53 - 36 * q^55 + 24 * q^57 - 6 * q^59 - 30 * q^61 - 24 * q^65 - 18 * q^67 - 15 * q^69 + 12 * q^71 - 12 * q^73 - 33 * q^75 - 6 * q^77 + 12 * q^79 - 9 * q^81 - 18 * q^83 + 6 * q^85 + 24 * q^87 - 3 * q^89 - 3 * q^91 - 6 * q^95 + 3 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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