LMFDB - Embedded newform 1232.4.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 94x^{4} + 161x^{3} + 533x^{2} - 384x - 468 $ x^6 - x^5 - 94x^4 + 161x^3 + 533x^2 - 384x - 468
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.17071$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.18220 q^{3} -10.8550 q^{5} -7.00000 q^{7} -9.50917 q^{9} +O(q^{10})$	$q-4.18220 q^{3} -10.8550 q^{5} -7.00000 q^{7} -9.50917 q^{9} +11.0000 q^{11} +43.5213 q^{13} +45.3976 q^{15} -10.4078 q^{17} -92.6056 q^{19} +29.2754 q^{21} +102.063 q^{23} -7.17004 q^{25} +152.689 q^{27} +49.3034 q^{29} +256.365 q^{31} -46.0042 q^{33} +75.9847 q^{35} +88.0910 q^{37} -182.015 q^{39} -409.140 q^{41} -12.4355 q^{43} +103.222 q^{45} +638.513 q^{47} +49.0000 q^{49} +43.5274 q^{51} +51.5874 q^{53} -119.404 q^{55} +387.296 q^{57} +355.930 q^{59} -472.606 q^{61} +66.5642 q^{63} -472.421 q^{65} +1067.21 q^{67} -426.847 q^{69} +8.37831 q^{71} +339.261 q^{73} +29.9866 q^{75} -77.0000 q^{77} -989.847 q^{79} -381.828 q^{81} -1393.36 q^{83} +112.976 q^{85} -206.197 q^{87} -1263.84 q^{89} -304.649 q^{91} -1072.17 q^{93} +1005.23 q^{95} +303.261 q^{97} -104.601 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{5} - 42 q^{7} + 28 q^{9}+O(q^{10}) $ 6 * q - 14 * q^5 - 42 * q^7 + 28 * q^9	$ 6 q - 14 q^{5} - 42 q^{7} + 28 q^{9} + 66 q^{11} - 70 q^{13} + 50 q^{15} - 102 q^{17} + 136 q^{19} + 146 q^{23} + 44 q^{25} + 30 q^{27} - 148 q^{29} - 308 q^{31} + 98 q^{35} - 6 q^{37} + 540 q^{39} - 90 q^{41} + 396 q^{43} - 812 q^{45} + 346 q^{47} + 294 q^{49} + 600 q^{51} - 984 q^{53} - 154 q^{55} - 1580 q^{57} + 44 q^{59} - 1414 q^{61} - 196 q^{63} - 1208 q^{65} + 926 q^{67} - 2550 q^{69} + 1478 q^{71} - 678 q^{73} + 1350 q^{75} - 462 q^{77} + 700 q^{79} - 966 q^{81} + 1192 q^{83} - 1596 q^{85} - 1900 q^{87} - 3314 q^{89} + 490 q^{91} - 2330 q^{93} - 668 q^{95} - 346 q^{97} + 308 q^{99}+O(q^{100}) $ 6 * q - 14 * q^5 - 42 * q^7 + 28 * q^9 + 66 * q^11 - 70 * q^13 + 50 * q^15 - 102 * q^17 + 136 * q^19 + 146 * q^23 + 44 * q^25 + 30 * q^27 - 148 * q^29 - 308 * q^31 + 98 * q^35 - 6 * q^37 + 540 * q^39 - 90 * q^41 + 396 * q^43 - 812 * q^45 + 346 * q^47 + 294 * q^49 + 600 * q^51 - 984 * q^53 - 154 * q^55 - 1580 * q^57 + 44 * q^59 - 1414 * q^61 - 196 * q^63 - 1208 * q^65 + 926 * q^67 - 2550 * q^69 + 1478 * q^71 - 678 * q^73 + 1350 * q^75 - 462 * q^77 + 700 * q^79 - 966 * q^81 + 1192 * q^83 - 1596 * q^85 - 1900 * q^87 - 3314 * q^89 + 490 * q^91 - 2330 * q^93 - 668 * q^95 - 346 * q^97 + 308 * 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
$3$	−4.18220	−0.804866	−0.402433	−	0.915450i	$-0.631835\pi$
$3$	−4.18220	−0.804866	−0.402433	+	0.915450i	$0.631835\pi$
$4$	0	0
$5$	−10.8550	−0.970896	−0.485448	−	0.874265i	$-0.661344\pi$
$5$	−10.8550	−0.970896	−0.485448	+	0.874265i	$0.661344\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.50917	−0.352191
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	43.5213	0.928510	0.464255	−	0.885702i	$-0.346322\pi$
$13$	43.5213	0.928510	0.464255	+	0.885702i	$0.346322\pi$
$14$	0	0
$15$	45.3976	0.781441
$16$	0	0
$17$	−10.4078	−0.148486	−0.0742428	−	0.997240i	$-0.523654\pi$
$17$	−10.4078	−0.148486	−0.0742428	+	0.997240i	$0.523654\pi$
$18$	0	0
$19$	−92.6056	−1.11817	−0.559084	−	0.829111i	$-0.688847\pi$
$19$	−92.6056	−1.11817	−0.559084	+	0.829111i	$0.688847\pi$
$20$	0	0
$21$	29.2754	0.304211
$22$	0	0
$23$	102.063	0.925284	0.462642	−	0.886545i	$-0.346901\pi$
$23$	102.063	0.925284	0.462642	+	0.886545i	$0.346901\pi$
$24$	0	0
$25$	−7.17004	−0.0573603
$26$	0	0
$27$	152.689	1.08833
$28$	0	0
$29$	49.3034	0.315704	0.157852	−	0.987463i	$-0.449543\pi$
$29$	49.3034	0.315704	0.157852	+	0.987463i	$0.449543\pi$
$30$	0	0
$31$	256.365	1.48531	0.742653	−	0.669676i	$-0.233567\pi$
$31$	256.365	1.48531	0.742653	+	0.669676i	$0.233567\pi$
$32$	0	0
$33$	−46.0042	−0.242676
$34$	0	0
$35$	75.9847	0.366964
$36$	0	0
$37$	88.0910	0.391407	0.195704	−	0.980663i	$-0.437301\pi$
$37$	88.0910	0.391407	0.195704	+	0.980663i	$0.437301\pi$
$38$	0	0
$39$	−182.015	−0.747325
$40$	0	0
$41$	−409.140	−1.55846	−0.779230	−	0.626738i	$-0.784390\pi$
$41$	−409.140	−1.55846	−0.779230	+	0.626738i	$0.784390\pi$
$42$	0	0
$43$	−12.4355	−0.0441022	−0.0220511	−	0.999757i	$-0.507020\pi$
$43$	−12.4355	−0.0441022	−0.0220511	+	0.999757i	$0.507020\pi$
$44$	0	0
$45$	103.222	0.341941
$46$	0	0
$47$	638.513	1.98163	0.990817	−	0.135213i	$-0.0431720\pi$
$47$	638.513	1.98163	0.990817	+	0.135213i	$0.0431720\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	43.5274	0.119511
$52$	0	0
$53$	51.5874	0.133699	0.0668497	−	0.997763i	$-0.478705\pi$
$53$	51.5874	0.133699	0.0668497	+	0.997763i	$0.478705\pi$
$54$	0	0
$55$	−119.404	−0.292736
$56$	0	0
$57$	387.296	0.899975
$58$	0	0
$59$	355.930	0.785391	0.392696	−	0.919668i	$-0.371543\pi$
$59$	355.930	0.785391	0.392696	+	0.919668i	$0.371543\pi$
$60$	0	0
$61$	−472.606	−0.991984	−0.495992	−	0.868327i	$-0.665195\pi$
$61$	−472.606	−0.991984	−0.495992	+	0.868327i	$0.665195\pi$
$62$	0	0
$63$	66.5642	0.133116
$64$	0	0
$65$	−472.421	−0.901487
$66$	0	0
$67$	1067.21	1.94598	0.972989	−	0.230852i	$-0.0741512\pi$
$67$	1067.21	1.94598	0.972989	+	0.230852i	$0.0741512\pi$
$68$	0	0
$69$	−426.847	−0.744729
$70$	0	0
$71$	8.37831	0.0140045	0.00700227	−	0.999975i	$-0.497771\pi$
$71$	8.37831	0.0140045	0.00700227	+	0.999975i	$0.497771\pi$
$72$	0	0
$73$	339.261	0.543938	0.271969	−	0.962306i	$-0.412325\pi$
$73$	339.261	0.543938	0.271969	+	0.962306i	$0.412325\pi$
$74$	0	0
$75$	29.9866	0.0461673
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−989.847	−1.40970	−0.704851	−	0.709355i	$-0.748986\pi$
$79$	−989.847	−1.40970	−0.704851	+	0.709355i	$0.748986\pi$
$80$	0	0
$81$	−381.828	−0.523770
$82$	0	0
$83$	−1393.36	−1.84266	−0.921331	−	0.388780i	$-0.872897\pi$
$83$	−1393.36	−1.84266	−0.921331	+	0.388780i	$0.872897\pi$
$84$	0	0
$85$	112.976	0.144164
$86$	0	0
$87$	−206.197	−0.254099
$88$	0	0
$89$	−1263.84	−1.50524	−0.752622	−	0.658453i	$-0.771211\pi$
$89$	−1263.84	−1.50524	−0.752622	+	0.658453i	$0.771211\pi$
$90$	0	0
$91$	−304.649	−0.350944
$92$	0	0
$93$	−1072.17	−1.19547
$94$	0	0
$95$	1005.23	1.08562
$96$	0	0
$97$	303.261	0.317438	0.158719	−	0.987324i	$-0.449264\pi$
$97$	303.261	0.317438	0.158719	+	0.987324i	$0.449264\pi$
$98$	0	0
$99$	−104.601	−0.106190
$100$	0	0
$101$	−411.384	−0.405290	−0.202645	−	0.979252i	$-0.564954\pi$
$101$	−411.384	−0.405290	−0.202645	+	0.979252i	$0.564954\pi$
$102$	0	0
$103$	−1338.20	−1.28016	−0.640081	−	0.768307i	$-0.721099\pi$
$103$	−1338.20	−1.28016	−0.640081	+	0.768307i	$0.721099\pi$
$104$	0	0
$105$	−317.783	−0.295357
$106$	0	0
$107$	183.446	0.165742	0.0828710	−	0.996560i	$-0.473591\pi$
$107$	183.446	0.165742	0.0828710	+	0.996560i	$0.473591\pi$
$108$	0	0
$109$	−214.447	−0.188443	−0.0942216	−	0.995551i	$-0.530036\pi$
$109$	−214.447	−0.188443	−0.0942216	+	0.995551i	$0.530036\pi$
$110$	0	0
$111$	−368.415	−0.315030
$112$	0	0
$113$	−189.831	−0.158034	−0.0790168	−	0.996873i	$-0.525178\pi$
$113$	−189.831	−0.158034	−0.0790168	+	0.996873i	$0.525178\pi$
$114$	0	0
$115$	−1107.89	−0.898355
$116$	0	0
$117$	−413.851	−0.327013
$118$	0	0
$119$	72.8544	0.0561223
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1711.11	1.25435
$124$	0	0
$125$	1434.70	1.02659
$126$	0	0
$127$	1207.65	0.843790	0.421895	−	0.906645i	$-0.361365\pi$
$127$	1207.65	0.843790	0.421895	+	0.906645i	$0.361365\pi$
$128$	0	0
$129$	52.0078	0.0354964
$130$	0	0
$131$	1591.72	1.06160	0.530800	−	0.847497i	$-0.321892\pi$
$131$	1591.72	1.06160	0.530800	+	0.847497i	$0.321892\pi$
$132$	0	0
$133$	648.239	0.422628
$134$	0	0
$135$	−1657.43	−1.05666
$136$	0	0
$137$	353.891	0.220693	0.110347	−	0.993893i	$-0.464804\pi$
$137$	353.891	0.220693	0.110347	+	0.993893i	$0.464804\pi$
$138$	0	0
$139$	−758.694	−0.462961	−0.231481	−	0.972840i	$-0.574357\pi$
$139$	−758.694	−0.462961	−0.231481	+	0.972840i	$0.574357\pi$
$140$	0	0
$141$	−2670.39	−1.59495
$142$	0	0
$143$	478.734	0.279956
$144$	0	0
$145$	−535.186	−0.306516
$146$	0	0
$147$	−204.928	−0.114981
$148$	0	0
$149$	−565.729	−0.311049	−0.155525	−	0.987832i	$-0.549707\pi$
$149$	−565.729	−0.311049	−0.155525	+	0.987832i	$0.549707\pi$
$150$	0	0
$151$	−2431.77	−1.31056	−0.655280	−	0.755386i	$-0.727449\pi$
$151$	−2431.77	−1.31056	−0.655280	+	0.755386i	$0.727449\pi$
$152$	0	0
$153$	98.9693	0.0522954
$154$	0	0
$155$	−2782.83	−1.44208
$156$	0	0
$157$	−2960.66	−1.50501	−0.752505	−	0.658586i	$-0.771155\pi$
$157$	−2960.66	−1.50501	−0.752505	+	0.658586i	$0.771155\pi$
$158$	0	0
$159$	−215.749	−0.107610
$160$	0	0
$161$	−714.439	−0.349725
$162$	0	0
$163$	3658.12	1.75783	0.878914	−	0.476980i	$-0.158269\pi$
$163$	3658.12	1.75783	0.878914	+	0.476980i	$0.158269\pi$
$164$	0	0
$165$	499.374	0.235613
$166$	0	0
$167$	−3729.64	−1.72820	−0.864098	−	0.503324i	$-0.832110\pi$
$167$	−3729.64	−1.72820	−0.864098	+	0.503324i	$0.832110\pi$
$168$	0	0
$169$	−302.900	−0.137870
$170$	0	0
$171$	880.602	0.393809
$172$	0	0
$173$	−1646.01	−0.723375	−0.361687	−	0.932299i	$-0.617799\pi$
$173$	−1646.01	−0.723375	−0.361687	+	0.932299i	$0.617799\pi$
$174$	0	0
$175$	50.1903	0.0216802
$176$	0	0
$177$	−1488.57	−0.632134
$178$	0	0
$179$	4127.69	1.72356	0.861782	−	0.507278i	$-0.169348\pi$
$179$	4127.69	1.72356	0.861782	+	0.507278i	$0.169348\pi$
$180$	0	0
$181$	260.155	0.106835	0.0534176	−	0.998572i	$-0.482989\pi$
$181$	260.155	0.106835	0.0534176	+	0.998572i	$0.482989\pi$
$182$	0	0
$183$	1976.53	0.798413
$184$	0	0
$185$	−956.224	−0.380016
$186$	0	0
$187$	−114.485	−0.0447701
$188$	0	0
$189$	−1068.82	−0.411351
$190$	0	0
$191$	−4065.22	−1.54005	−0.770023	−	0.638016i	$-0.779755\pi$
$191$	−4065.22	−1.54005	−0.770023	+	0.638016i	$0.779755\pi$
$192$	0	0
$193$	−5154.63	−1.92248	−0.961239	−	0.275715i	$-0.911085\pi$
$193$	−5154.63	−1.92248	−0.961239	+	0.275715i	$0.911085\pi$
$194$	0	0
$195$	1975.76	0.725576
$196$	0	0
$197$	234.392	0.0847702	0.0423851	−	0.999101i	$-0.486504\pi$
$197$	234.392	0.0847702	0.0423851	+	0.999101i	$0.486504\pi$
$198$	0	0
$199$	−1199.71	−0.427363	−0.213681	−	0.976903i	$-0.568545\pi$
$199$	−1199.71	−0.427363	−0.213681	+	0.976903i	$0.568545\pi$
$200$	0	0
$201$	−4463.29	−1.56625
$202$	0	0
$203$	−345.124	−0.119325
$204$	0	0
$205$	4441.19	1.51310
$206$	0	0
$207$	−970.531	−0.325877
$208$	0	0
$209$	−1018.66	−0.337140
$210$	0	0
$211$	−2796.16	−0.912301	−0.456150	−	0.889903i	$-0.650772\pi$
$211$	−2796.16	−0.912301	−0.456150	+	0.889903i	$0.650772\pi$
$212$	0	0
$213$	−35.0398	−0.0112718
$214$	0	0
$215$	134.987	0.0428187
$216$	0	0
$217$	−1794.55	−0.561393
$218$	0	0
$219$	−1418.86	−0.437797
$220$	0	0
$221$	−452.959	−0.137870
$222$	0	0
$223$	−3329.13	−0.999708	−0.499854	−	0.866110i	$-0.666613\pi$
$223$	−3329.13	−0.999708	−0.499854	+	0.866110i	$0.666613\pi$
$224$	0	0
$225$	68.1811	0.0202018
$226$	0	0
$227$	−306.792	−0.0897027	−0.0448513	−	0.998994i	$-0.514281\pi$
$227$	−306.792	−0.0897027	−0.0448513	+	0.998994i	$0.514281\pi$
$228$	0	0
$229$	6493.99	1.87395	0.936975	−	0.349395i	$-0.113613\pi$
$229$	6493.99	1.87395	0.936975	+	0.349395i	$0.113613\pi$
$230$	0	0
$231$	322.030	0.0917229
$232$	0	0
$233$	1469.14	0.413077	0.206538	−	0.978439i	$-0.433780\pi$
$233$	1469.14	0.413077	0.206538	+	0.978439i	$0.433780\pi$
$234$	0	0
$235$	−6931.03	−1.92396
$236$	0	0
$237$	4139.74	1.13462
$238$	0	0
$239$	−775.629	−0.209922	−0.104961	−	0.994476i	$-0.533472\pi$
$239$	−775.629	−0.209922	−0.104961	+	0.994476i	$0.533472\pi$
$240$	0	0
$241$	171.942	0.0459574	0.0229787	−	0.999736i	$-0.492685\pi$
$241$	171.942	0.0459574	0.0229787	+	0.999736i	$0.492685\pi$
$242$	0	0
$243$	−2525.71	−0.666768
$244$	0	0
$245$	−531.893	−0.138699
$246$	0	0
$247$	−4030.31	−1.03823
$248$	0	0
$249$	5827.31	1.48309
$250$	0	0
$251$	492.552	0.123863	0.0619315	−	0.998080i	$-0.480274\pi$
$251$	492.552	0.123863	0.0619315	+	0.998080i	$0.480274\pi$
$252$	0	0
$253$	1122.69	0.278984
$254$	0	0
$255$	−472.488	−0.116033
$256$	0	0
$257$	−2098.27	−0.509287	−0.254644	−	0.967035i	$-0.581958\pi$
$257$	−2098.27	−0.509287	−0.254644	+	0.967035i	$0.581958\pi$
$258$	0	0
$259$	−616.637	−0.147938
$260$	0	0
$261$	−468.835	−0.111188
$262$	0	0
$263$	4421.59	1.03668	0.518341	−	0.855174i	$-0.326550\pi$
$263$	4421.59	1.03668	0.518341	+	0.855174i	$0.326550\pi$
$264$	0	0
$265$	−559.979	−0.129808
$266$	0	0
$267$	5285.64	1.21152
$268$	0	0
$269$	6845.66	1.55163	0.775813	−	0.630963i	$-0.217340\pi$
$269$	6845.66	1.55163	0.775813	+	0.630963i	$0.217340\pi$
$270$	0	0
$271$	−4623.48	−1.03637	−0.518185	−	0.855269i	$-0.673392\pi$
$271$	−4623.48	−1.03637	−0.518185	+	0.855269i	$0.673392\pi$
$272$	0	0
$273$	1274.10	0.282462
$274$	0	0
$275$	−78.8704	−0.0172948
$276$	0	0
$277$	3984.89	0.864363	0.432182	−	0.901787i	$-0.357744\pi$
$277$	3984.89	0.864363	0.432182	+	0.901787i	$0.357744\pi$
$278$	0	0
$279$	−2437.82	−0.523112
$280$	0	0
$281$	6014.64	1.27688	0.638440	−	0.769672i	$-0.279580\pi$
$281$	6014.64	1.27688	0.638440	+	0.769672i	$0.279580\pi$
$282$	0	0
$283$	−1959.56	−0.411603	−0.205801	−	0.978594i	$-0.565980\pi$
$283$	−1959.56	−0.411603	−0.205801	+	0.978594i	$0.565980\pi$
$284$	0	0
$285$	−4204.07	−0.873782
$286$	0	0
$287$	2863.98	0.589043
$288$	0	0
$289$	−4804.68	−0.977952
$290$	0	0
$291$	−1268.30	−0.255495
$292$	0	0
$293$	−4540.43	−0.905307	−0.452653	−	0.891687i	$-0.649522\pi$
$293$	−4540.43	−0.905307	−0.452653	+	0.891687i	$0.649522\pi$
$294$	0	0
$295$	−3863.60	−0.762534
$296$	0	0
$297$	1679.58	0.328145
$298$	0	0
$299$	4441.90	0.859135
$300$	0	0
$301$	87.0485	0.0166691
$302$	0	0
$303$	1720.49	0.326204
$304$	0	0
$305$	5130.12	0.963113
$306$	0	0
$307$	−7600.82	−1.41303	−0.706517	−	0.707696i	$-0.749735\pi$
$307$	−7600.82	−1.41303	−0.706517	+	0.707696i	$0.749735\pi$
$308$	0	0
$309$	5596.62	1.03036
$310$	0	0
$311$	−6918.26	−1.26141	−0.630705	−	0.776023i	$-0.717234\pi$
$311$	−6918.26	−1.26141	−0.630705	+	0.776023i	$0.717234\pi$
$312$	0	0
$313$	−2805.75	−0.506679	−0.253339	−	0.967377i	$-0.581529\pi$
$313$	−2805.75	−0.506679	−0.253339	+	0.967377i	$0.581529\pi$
$314$	0	0
$315$	−722.551	−0.129242
$316$	0	0
$317$	−899.135	−0.159307	−0.0796537	−	0.996823i	$-0.525381\pi$
$317$	−899.135	−0.159307	−0.0796537	+	0.996823i	$0.525381\pi$
$318$	0	0
$319$	542.338	0.0951884
$320$	0	0
$321$	−767.209	−0.133400
$322$	0	0
$323$	963.818	0.166032
$324$	0	0
$325$	−312.049	−0.0532596
$326$	0	0
$327$	896.862	0.151671
$328$	0	0
$329$	−4469.59	−0.748987
$330$	0	0
$331$	−541.520	−0.0899233	−0.0449617	−	0.998989i	$-0.514317\pi$
$331$	−541.520	−0.0899233	−0.0449617	+	0.998989i	$0.514317\pi$
$332$	0	0
$333$	−837.672	−0.137850
$334$	0	0
$335$	−11584.5	−1.88934
$336$	0	0
$337$	−5659.24	−0.914773	−0.457387	−	0.889268i	$-0.651214\pi$
$337$	−5659.24	−0.914773	−0.457387	+	0.889268i	$0.651214\pi$
$338$	0	0
$339$	793.912	0.127196
$340$	0	0
$341$	2820.01	0.447837
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4633.40	0.723055
$346$	0	0
$347$	411.258	0.0636238	0.0318119	−	0.999494i	$-0.489872\pi$
$347$	411.258	0.0636238	0.0318119	+	0.999494i	$0.489872\pi$
$348$	0	0
$349$	1053.75	0.161622	0.0808109	−	0.996729i	$-0.474249\pi$
$349$	1053.75	0.161622	0.0808109	+	0.996729i	$0.474249\pi$
$350$	0	0
$351$	6645.21	1.01053
$352$	0	0
$353$	−2484.82	−0.374657	−0.187328	−	0.982297i	$-0.559983\pi$
$353$	−2484.82	−0.374657	−0.187328	+	0.982297i	$0.559983\pi$
$354$	0	0
$355$	−90.9462	−0.0135970
$356$	0	0
$357$	−304.692	−0.0451709
$358$	0	0
$359$	4198.60	0.617252	0.308626	−	0.951183i	$-0.400131\pi$
$359$	4198.60	0.617252	0.308626	+	0.951183i	$0.400131\pi$
$360$	0	0
$361$	1716.80	0.250299
$362$	0	0
$363$	−506.047	−0.0731696
$364$	0	0
$365$	−3682.66	−0.528107
$366$	0	0
$367$	8926.89	1.26970	0.634850	−	0.772636i	$-0.281062\pi$
$367$	8926.89	1.26970	0.634850	+	0.772636i	$0.281062\pi$
$368$	0	0
$369$	3890.58	0.548877
$370$	0	0
$371$	−361.112	−0.0505337
$372$	0	0
$373$	9513.13	1.32057	0.660283	−	0.751017i	$-0.270436\pi$
$373$	9513.13	1.32057	0.660283	+	0.751017i	$0.270436\pi$
$374$	0	0
$375$	−6000.21	−0.826265
$376$	0	0
$377$	2145.75	0.293134
$378$	0	0
$379$	1343.42	0.182076	0.0910381	−	0.995847i	$-0.470981\pi$
$379$	1343.42	0.182076	0.0910381	+	0.995847i	$0.470981\pi$
$380$	0	0
$381$	−5050.63	−0.679138
$382$	0	0
$383$	6200.71	0.827262	0.413631	−	0.910445i	$-0.364260\pi$
$383$	6200.71	0.827262	0.413631	+	0.910445i	$0.364260\pi$
$384$	0	0
$385$	835.831	0.110644
$386$	0	0
$387$	118.251	0.0155324
$388$	0	0
$389$	14047.7	1.83097	0.915483	−	0.402356i	$-0.131809\pi$
$389$	14047.7	1.83097	0.915483	+	0.402356i	$0.131809\pi$
$390$	0	0
$391$	−1062.24	−0.137391
$392$	0	0
$393$	−6656.91	−0.854444
$394$	0	0
$395$	10744.7	1.36868
$396$	0	0
$397$	−13776.7	−1.74165	−0.870823	−	0.491597i	$-0.836413\pi$
$397$	−13776.7	−1.74165	−0.870823	+	0.491597i	$0.836413\pi$
$398$	0	0
$399$	−2711.07	−0.340158
$400$	0	0
$401$	−3172.10	−0.395030	−0.197515	−	0.980300i	$-0.563287\pi$
$401$	−3172.10	−0.395030	−0.197515	+	0.980300i	$0.563287\pi$
$402$	0	0
$403$	11157.3	1.37912
$404$	0	0
$405$	4144.73	0.508526
$406$	0	0
$407$	969.001	0.118014
$408$	0	0
$409$	−4658.12	−0.563153	−0.281576	−	0.959539i	$-0.590857\pi$
$409$	−4658.12	−0.563153	−0.281576	+	0.959539i	$0.590857\pi$
$410$	0	0
$411$	−1480.04	−0.177628
$412$	0	0
$413$	−2491.51	−0.296850
$414$	0	0
$415$	15124.8	1.78903
$416$	0	0
$417$	3173.01	0.372621
$418$	0	0
$419$	−7856.05	−0.915974	−0.457987	−	0.888959i	$-0.651429\pi$
$419$	−7856.05	−0.915974	−0.457987	+	0.888959i	$0.651429\pi$
$420$	0	0
$421$	−6971.14	−0.807013	−0.403507	−	0.914977i	$-0.632209\pi$
$421$	−6971.14	−0.807013	−0.403507	+	0.914977i	$0.632209\pi$
$422$	0	0
$423$	−6071.73	−0.697914
$424$	0	0
$425$	74.6241	0.00851718
$426$	0	0
$427$	3308.24	0.374935
$428$	0	0
$429$	−2002.16	−0.225327
$430$	0	0
$431$	−5670.73	−0.633758	−0.316879	−	0.948466i	$-0.602635\pi$
$431$	−5670.73	−0.633758	−0.316879	+	0.948466i	$0.602635\pi$
$432$	0	0
$433$	11368.5	1.26174	0.630871	−	0.775888i	$-0.282698\pi$
$433$	11368.5	1.26174	0.630871	+	0.775888i	$0.282698\pi$
$434$	0	0
$435$	2238.26	0.246704
$436$	0	0
$437$	−9451.58	−1.03462
$438$	0	0
$439$	1509.40	0.164099	0.0820497	−	0.996628i	$-0.473853\pi$
$439$	1509.40	0.164099	0.0820497	+	0.996628i	$0.473853\pi$
$440$	0	0
$441$	−465.949	−0.0503131
$442$	0	0
$443$	12924.8	1.38617	0.693087	−	0.720854i	$-0.256250\pi$
$443$	12924.8	1.38617	0.693087	+	0.720854i	$0.256250\pi$
$444$	0	0
$445$	13718.9	1.46144
$446$	0	0
$447$	2366.00	0.250353
$448$	0	0
$449$	−740.654	−0.0778477	−0.0389239	−	0.999242i	$-0.512393\pi$
$449$	−740.654	−0.0778477	−0.0389239	+	0.999242i	$0.512393\pi$
$450$	0	0
$451$	−4500.54	−0.469894
$452$	0	0
$453$	10170.2	1.05482
$454$	0	0
$455$	3306.95	0.340730
$456$	0	0
$457$	11388.2	1.16568	0.582841	−	0.812586i	$-0.301941\pi$
$457$	11388.2	1.16568	0.582841	+	0.812586i	$0.301941\pi$
$458$	0	0
$459$	−1589.15	−0.161602
$460$	0	0
$461$	−9629.70	−0.972885	−0.486443	−	0.873713i	$-0.661706\pi$
$461$	−9629.70	−0.972885	−0.486443	+	0.873713i	$0.661706\pi$
$462$	0	0
$463$	3758.42	0.377254	0.188627	−	0.982049i	$-0.439596\pi$
$463$	3758.42	0.377254	0.188627	+	0.982049i	$0.439596\pi$
$464$	0	0
$465$	11638.4	1.16068
$466$	0	0
$467$	−7089.30	−0.702471	−0.351235	−	0.936287i	$-0.614238\pi$
$467$	−7089.30	−0.702471	−0.351235	+	0.936287i	$0.614238\pi$
$468$	0	0
$469$	−7470.47	−0.735511
$470$	0	0
$471$	12382.1	1.21133
$472$	0	0
$473$	−136.790	−0.0132973
$474$	0	0
$475$	663.986	0.0641385
$476$	0	0
$477$	−490.553	−0.0470878
$478$	0	0
$479$	1838.09	0.175333	0.0876663	−	0.996150i	$-0.472059\pi$
$479$	1838.09	0.175333	0.0876663	+	0.996150i	$0.472059\pi$
$480$	0	0
$481$	3833.83	0.363426
$482$	0	0
$483$	2987.93	0.281481
$484$	0	0
$485$	−3291.89	−0.308200
$486$	0	0
$487$	−6386.65	−0.594264	−0.297132	−	0.954836i	$-0.596030\pi$
$487$	−6386.65	−0.594264	−0.297132	+	0.954836i	$0.596030\pi$
$488$	0	0
$489$	−15299.0	−1.41482
$490$	0	0
$491$	12728.0	1.16987	0.584935	−	0.811080i	$-0.301120\pi$
$491$	12728.0	1.16987	0.584935	+	0.811080i	$0.301120\pi$
$492$	0	0
$493$	−513.139	−0.0468775
$494$	0	0
$495$	1135.44	0.103099
$496$	0	0
$497$	−58.6482	−0.00529322
$498$	0	0
$499$	2163.18	0.194062	0.0970311	−	0.995281i	$-0.469065\pi$
$499$	2163.18	0.194062	0.0970311	+	0.995281i	$0.469065\pi$
$500$	0	0
$501$	15598.1	1.39096
$502$	0	0
$503$	−2165.73	−0.191978	−0.0959891	−	0.995382i	$-0.530601\pi$
$503$	−2165.73	−0.191978	−0.0959891	+	0.995382i	$0.530601\pi$
$504$	0	0
$505$	4465.55	0.393494
$506$	0	0
$507$	1266.79	0.110967
$508$	0	0
$509$	−4981.96	−0.433834	−0.216917	−	0.976190i	$-0.569600\pi$
$509$	−4981.96	−0.433834	−0.216917	+	0.976190i	$0.569600\pi$
$510$	0	0
$511$	−2374.82	−0.205589
$512$	0	0
$513$	−14139.8	−1.21694
$514$	0	0
$515$	14526.1	1.24291
$516$	0	0
$517$	7023.65	0.597485
$518$	0	0
$519$	6883.95	0.582219
$520$	0	0
$521$	12676.7	1.06598	0.532990	−	0.846122i	$-0.321068\pi$
$521$	12676.7	1.06598	0.532990	+	0.846122i	$0.321068\pi$
$522$	0	0
$523$	−7790.27	−0.651329	−0.325664	−	0.945485i	$-0.605588\pi$
$523$	−7790.27	−0.651329	−0.325664	+	0.945485i	$0.605588\pi$
$524$	0	0
$525$	−209.906	−0.0174496
$526$	0	0
$527$	−2668.19	−0.220547
$528$	0	0
$529$	−1750.21	−0.143849
$530$	0	0
$531$	−3384.59	−0.276608
$532$	0	0
$533$	−17806.3	−1.44705
$534$	0	0
$535$	−1991.30	−0.160918
$536$	0	0
$537$	−17262.8	−1.38724
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−24538.6	−1.95009	−0.975045	−	0.222008i	$-0.928739\pi$
$541$	−24538.6	−1.95009	−0.975045	+	0.222008i	$0.928739\pi$
$542$	0	0
$543$	−1088.02	−0.0859879
$544$	0	0
$545$	2327.81	0.182959
$546$	0	0
$547$	4202.86	0.328522	0.164261	−	0.986417i	$-0.447476\pi$
$547$	4202.86	0.328522	0.164261	+	0.986417i	$0.447476\pi$
$548$	0	0
$549$	4494.09	0.349368
$550$	0	0
$551$	−4565.78	−0.353010
$552$	0	0
$553$	6928.93	0.532818
$554$	0	0
$555$	3999.12	0.305862
$556$	0	0
$557$	16954.9	1.28977	0.644886	−	0.764279i	$-0.276905\pi$
$557$	16954.9	1.28977	0.644886	+	0.764279i	$0.276905\pi$
$558$	0	0
$559$	−541.209	−0.0409493
$560$	0	0
$561$	478.802	0.0360339
$562$	0	0
$563$	19583.3	1.46596	0.732981	−	0.680249i	$-0.238128\pi$
$563$	19583.3	1.46596	0.732981	+	0.680249i	$0.238128\pi$
$564$	0	0
$565$	2060.61	0.153434
$566$	0	0
$567$	2672.80	0.197966
$568$	0	0
$569$	2823.38	0.208018	0.104009	−	0.994576i	$-0.466833\pi$
$569$	2823.38	0.208018	0.104009	+	0.994576i	$0.466833\pi$
$570$	0	0
$571$	18386.8	1.34757	0.673787	−	0.738925i	$-0.264666\pi$
$571$	18386.8	1.34757	0.673787	+	0.738925i	$0.264666\pi$
$572$	0	0
$573$	17001.6	1.23953
$574$	0	0
$575$	−731.794	−0.0530746
$576$	0	0
$577$	11211.7	0.808922	0.404461	−	0.914555i	$-0.367459\pi$
$577$	11211.7	0.808922	0.404461	+	0.914555i	$0.367459\pi$
$578$	0	0
$579$	21557.7	1.54734
$580$	0	0
$581$	9753.51	0.696461
$582$	0	0
$583$	567.461	0.0403119
$584$	0	0
$585$	4492.33	0.317496
$586$	0	0
$587$	9839.07	0.691826	0.345913	−	0.938267i	$-0.387569\pi$
$587$	9839.07	0.691826	0.345913	+	0.938267i	$0.387569\pi$
$588$	0	0
$589$	−23740.8	−1.66082
$590$	0	0
$591$	−980.274	−0.0682286
$592$	0	0
$593$	−16111.2	−1.11569	−0.557847	−	0.829943i	$-0.688373\pi$
$593$	−16111.2	−1.11569	−0.557847	+	0.829943i	$0.688373\pi$
$594$	0	0
$595$	−790.831	−0.0544889
$596$	0	0
$597$	5017.43	0.343969
$598$	0	0
$599$	17251.3	1.17674	0.588372	−	0.808591i	$-0.299769\pi$
$599$	17251.3	1.17674	0.588372	+	0.808591i	$0.299769\pi$
$600$	0	0
$601$	8142.82	0.552666	0.276333	−	0.961062i	$-0.410881\pi$
$601$	8142.82	0.552666	0.276333	+	0.961062i	$0.410881\pi$
$602$	0	0
$603$	−10148.3	−0.685357
$604$	0	0
$605$	−1313.45	−0.0882633
$606$	0	0
$607$	−8033.40	−0.537176	−0.268588	−	0.963255i	$-0.586557\pi$
$607$	−8033.40	−0.537176	−0.268588	+	0.963255i	$0.586557\pi$
$608$	0	0
$609$	1443.38	0.0960405
$610$	0	0
$611$	27788.9	1.83997
$612$	0	0
$613$	−15284.3	−1.00706	−0.503530	−	0.863978i	$-0.667966\pi$
$613$	−15284.3	−1.00706	−0.503530	+	0.863978i	$0.667966\pi$
$614$	0	0
$615$	−18574.0	−1.21785
$616$	0	0
$617$	−57.5979	−0.00375819	−0.00187910	−	0.999998i	$-0.500598\pi$
$617$	−57.5979	−0.00375819	−0.00187910	+	0.999998i	$0.500598\pi$
$618$	0	0
$619$	−18083.8	−1.17423	−0.587115	−	0.809504i	$-0.699736\pi$
$619$	−18083.8	−1.17423	−0.587115	+	0.809504i	$0.699736\pi$
$620$	0	0
$621$	15583.8	1.00702
$622$	0	0
$623$	8846.88	0.568929
$624$	0	0
$625$	−14677.3	−0.939349
$626$	0	0
$627$	4260.25	0.271353
$628$	0	0
$629$	−916.831	−0.0581184
$630$	0	0
$631$	−6874.21	−0.433690	−0.216845	−	0.976206i	$-0.569577\pi$
$631$	−6874.21	−0.433690	−0.216845	+	0.976206i	$0.569577\pi$
$632$	0	0
$633$	11694.1	0.734279
$634$	0	0
$635$	−13109.0	−0.819233
$636$	0	0
$637$	2132.54	0.132644
$638$	0	0
$639$	−79.6708	−0.00493228
$640$	0	0
$641$	−7024.72	−0.432854	−0.216427	−	0.976299i	$-0.569440\pi$
$641$	−7024.72	−0.432854	−0.216427	+	0.976299i	$0.569440\pi$
$642$	0	0
$643$	−9017.24	−0.553041	−0.276520	−	0.961008i	$-0.589181\pi$
$643$	−9017.24	−0.553041	−0.276520	+	0.961008i	$0.589181\pi$
$644$	0	0
$645$	−564.542	−0.0344633
$646$	0	0
$647$	−9652.79	−0.586538	−0.293269	−	0.956030i	$-0.594743\pi$
$647$	−9652.79	−0.586538	−0.293269	+	0.956030i	$0.594743\pi$
$648$	0	0
$649$	3915.23	0.236804
$650$	0	0
$651$	7505.19	0.451846
$652$	0	0
$653$	3981.31	0.238592	0.119296	−	0.992859i	$-0.461936\pi$
$653$	3981.31	0.238592	0.119296	+	0.992859i	$0.461936\pi$
$654$	0	0
$655$	−17278.1	−1.03070
$656$	0	0
$657$	−3226.09	−0.191570
$658$	0	0
$659$	−817.650	−0.0483325	−0.0241663	−	0.999708i	$-0.507693\pi$
$659$	−817.650	−0.0483325	−0.0241663	+	0.999708i	$0.507693\pi$
$660$	0	0
$661$	176.286	0.0103733	0.00518664	−	0.999987i	$-0.498349\pi$
$661$	176.286	0.0103733	0.00518664	+	0.999987i	$0.498349\pi$
$662$	0	0
$663$	1894.37	0.110967
$664$	0	0
$665$	−7036.61	−0.410328
$666$	0	0
$667$	5032.04	0.292116
$668$	0	0
$669$	13923.1	0.804630
$670$	0	0
$671$	−5198.67	−0.299094
$672$	0	0
$673$	−29274.7	−1.67676	−0.838378	−	0.545089i	$-0.816496\pi$
$673$	−29274.7	−1.67676	−0.838378	+	0.545089i	$0.816496\pi$
$674$	0	0
$675$	−1094.78	−0.0624271
$676$	0	0
$677$	−676.291	−0.0383929	−0.0191964	−	0.999816i	$-0.506111\pi$
$677$	−676.291	−0.0383929	−0.0191964	+	0.999816i	$0.506111\pi$
$678$	0	0
$679$	−2122.83	−0.119980
$680$	0	0
$681$	1283.07	0.0721986
$682$	0	0
$683$	−8653.32	−0.484788	−0.242394	−	0.970178i	$-0.577933\pi$
$683$	−8653.32	−0.484788	−0.242394	+	0.970178i	$0.577933\pi$
$684$	0	0
$685$	−3841.47	−0.214270
$686$	0	0
$687$	−27159.2	−1.50828
$688$	0	0
$689$	2245.15	0.124141
$690$	0	0
$691$	21543.5	1.18604	0.593021	−	0.805187i	$-0.297935\pi$
$691$	21543.5	1.18604	0.593021	+	0.805187i	$0.297935\pi$
$692$	0	0
$693$	732.206	0.0401359
$694$	0	0
$695$	8235.59	0.449487
$696$	0	0
$697$	4258.23	0.231409
$698$	0	0
$699$	−6144.26	−0.332471
$700$	0	0
$701$	−25010.3	−1.34754	−0.673771	−	0.738940i	$-0.735327\pi$
$701$	−25010.3	−1.34754	−0.673771	+	0.738940i	$0.735327\pi$
$702$	0	0
$703$	−8157.72	−0.437659
$704$	0	0
$705$	28987.0	1.54853
$706$	0	0
$707$	2879.69	0.153185
$708$	0	0
$709$	−20546.4	−1.08834	−0.544171	−	0.838974i	$-0.683156\pi$
$709$	−20546.4	−1.08834	−0.544171	+	0.838974i	$0.683156\pi$
$710$	0	0
$711$	9412.63	0.496485
$712$	0	0
$713$	26165.3	1.37433
$714$	0	0
$715$	−5196.63	−0.271808
$716$	0	0
$717$	3243.84	0.168959
$718$	0	0
$719$	20771.0	1.07737	0.538685	−	0.842508i	$-0.318921\pi$
$719$	20771.0	1.07737	0.538685	+	0.842508i	$0.318921\pi$
$720$	0	0
$721$	9367.40	0.483856
$722$	0	0
$723$	−719.095	−0.0369895
$724$	0	0
$725$	−353.508	−0.0181089
$726$	0	0
$727$	−11214.4	−0.572104	−0.286052	−	0.958214i	$-0.592343\pi$
$727$	−11214.4	−0.572104	−0.286052	+	0.958214i	$0.592343\pi$
$728$	0	0
$729$	20872.4	1.06043
$730$	0	0
$731$	129.426	0.00654854
$732$	0	0
$733$	−14242.9	−0.717701	−0.358850	−	0.933395i	$-0.616831\pi$
$733$	−14242.9	−0.717701	−0.358850	+	0.933395i	$0.616831\pi$
$734$	0	0
$735$	2224.48	0.111634
$736$	0	0
$737$	11739.3	0.586734
$738$	0	0
$739$	−1505.92	−0.0749612	−0.0374806	−	0.999297i	$-0.511933\pi$
$739$	−1505.92	−0.0749612	−0.0374806	+	0.999297i	$0.511933\pi$
$740$	0	0
$741$	16855.6	0.835635
$742$	0	0
$743$	−30820.0	−1.52177	−0.760886	−	0.648885i	$-0.775236\pi$
$743$	−30820.0	−1.52177	−0.760886	+	0.648885i	$0.775236\pi$
$744$	0	0
$745$	6140.97	0.301997
$746$	0	0
$747$	13249.7	0.648970
$748$	0	0
$749$	−1284.12	−0.0626446
$750$	0	0
$751$	11620.0	0.564606	0.282303	−	0.959325i	$-0.408902\pi$
$751$	11620.0	0.564606	0.282303	+	0.959325i	$0.408902\pi$
$752$	0	0
$753$	−2059.95	−0.0996930
$754$	0	0
$755$	26396.7	1.27242
$756$	0	0
$757$	−20054.0	−0.962847	−0.481424	−	0.876488i	$-0.659880\pi$
$757$	−20054.0	−0.962847	−0.481424	+	0.876488i	$0.659880\pi$
$758$	0	0
$759$	−4695.32	−0.224544
$760$	0	0
$761$	−39794.3	−1.89559	−0.947794	−	0.318882i	$-0.896693\pi$
$761$	−39794.3	−1.89559	−0.947794	+	0.318882i	$0.896693\pi$
$762$	0	0
$763$	1501.13	0.0712248
$764$	0	0
$765$	−1074.31	−0.0507734
$766$	0	0
$767$	15490.5	0.729243
$768$	0	0
$769$	34050.2	1.59672	0.798362	−	0.602178i	$-0.205700\pi$
$769$	34050.2	1.59672	0.798362	+	0.602178i	$0.205700\pi$
$770$	0	0
$771$	8775.41	0.409908
$772$	0	0
$773$	−26079.7	−1.21348	−0.606740	−	0.794900i	$-0.707523\pi$
$773$	−26079.7	−1.21348	−0.606740	+	0.794900i	$0.707523\pi$
$774$	0	0
$775$	−1838.15	−0.0851977
$776$	0	0
$777$	2578.90	0.119070
$778$	0	0
$779$	37888.6	1.74262
$780$	0	0
$781$	92.1614	0.00422253
$782$	0	0
$783$	7528.08	0.343591
$784$	0	0
$785$	32137.8	1.46121
$786$	0	0
$787$	−36384.7	−1.64800	−0.824000	−	0.566590i	$-0.808262\pi$
$787$	−36384.7	−1.64800	−0.824000	+	0.566590i	$0.808262\pi$
$788$	0	0
$789$	−18492.0	−0.834389
$790$	0	0
$791$	1328.82	0.0597311
$792$	0	0
$793$	−20568.4	−0.921066
$794$	0	0
$795$	2341.94	0.104478
$796$	0	0
$797$	−22177.0	−0.985633	−0.492817	−	0.870133i	$-0.664033\pi$
$797$	−22177.0	−0.985633	−0.492817	+	0.870133i	$0.664033\pi$
$798$	0	0
$799$	−6645.50	−0.294244
$800$	0	0
$801$	12018.1	0.530134
$802$	0	0
$803$	3731.87	0.164003
$804$	0	0
$805$	7755.20	0.339546
$806$	0	0
$807$	−28630.0	−1.24885
$808$	0	0
$809$	16993.4	0.738511	0.369256	−	0.929328i	$-0.379613\pi$
$809$	16993.4	0.738511	0.369256	+	0.929328i	$0.379613\pi$
$810$	0	0
$811$	−25640.2	−1.11017	−0.555086	−	0.831793i	$-0.687315\pi$
$811$	−25640.2	−1.11017	−0.555086	+	0.831793i	$0.687315\pi$
$812$	0	0
$813$	19336.3	0.834138
$814$	0	0
$815$	−39708.7	−1.70667
$816$	0	0
$817$	1151.60	0.0493137
$818$	0	0
$819$	2896.96	0.123599
$820$	0	0
$821$	5413.07	0.230106	0.115053	−	0.993359i	$-0.463296\pi$
$821$	5413.07	0.230106	0.115053	+	0.993359i	$0.463296\pi$
$822$	0	0
$823$	−33164.5	−1.40467	−0.702334	−	0.711848i	$-0.747859\pi$
$823$	−33164.5	−1.40467	−0.702334	+	0.711848i	$0.747859\pi$
$824$	0	0
$825$	329.852	0.0139200
$826$	0	0
$827$	36243.6	1.52396	0.761980	−	0.647601i	$-0.224228\pi$
$827$	36243.6	1.52396	0.761980	+	0.647601i	$0.224228\pi$
$828$	0	0
$829$	23772.6	0.995969	0.497984	−	0.867186i	$-0.334074\pi$
$829$	23772.6	0.995969	0.497984	+	0.867186i	$0.334074\pi$
$830$	0	0
$831$	−16665.6	−0.695696
$832$	0	0
$833$	−509.981	−0.0212122
$834$	0	0
$835$	40485.1	1.67790
$836$	0	0
$837$	39144.1	1.61651
$838$	0	0
$839$	−41771.7	−1.71885	−0.859427	−	0.511258i	$-0.829180\pi$
$839$	−41771.7	−1.71885	−0.859427	+	0.511258i	$0.829180\pi$
$840$	0	0
$841$	−21958.2	−0.900331
$842$	0	0
$843$	−25154.4	−1.02772
$844$	0	0
$845$	3287.96	0.133857
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	8195.26	0.331285
$850$	0	0
$851$	8990.80	0.362163
$852$	0	0
$853$	28458.9	1.14234	0.571169	−	0.820832i	$-0.306490\pi$
$853$	28458.9	1.14234	0.571169	+	0.820832i	$0.306490\pi$
$854$	0	0
$855$	−9558.90	−0.382348
$856$	0	0
$857$	17798.5	0.709434	0.354717	−	0.934974i	$-0.384577\pi$
$857$	17798.5	0.709434	0.354717	+	0.934974i	$0.384577\pi$
$858$	0	0
$859$	−27382.8	−1.08765	−0.543823	−	0.839200i	$-0.683024\pi$
$859$	−27382.8	−1.08765	−0.543823	+	0.839200i	$0.683024\pi$
$860$	0	0
$861$	−11977.7	−0.474100
$862$	0	0
$863$	43276.0	1.70699	0.853495	−	0.521100i	$-0.174478\pi$
$863$	43276.0	1.70699	0.853495	+	0.521100i	$0.174478\pi$
$864$	0	0
$865$	17867.4	0.702322
$866$	0	0
$867$	20094.1	0.787120
$868$	0	0
$869$	−10888.3	−0.425041
$870$	0	0
$871$	46446.3	1.80686
$872$	0	0
$873$	−2883.76	−0.111799
$874$	0	0
$875$	−10042.9	−0.388014
$876$	0	0
$877$	15792.3	0.608061	0.304031	−	0.952662i	$-0.401667\pi$
$877$	15792.3	0.608061	0.304031	+	0.952662i	$0.401667\pi$
$878$	0	0
$879$	18989.0	0.728650
$880$	0	0
$881$	−30284.3	−1.15812	−0.579059	−	0.815285i	$-0.696580\pi$
$881$	−30284.3	−1.15812	−0.579059	+	0.815285i	$0.696580\pi$
$882$	0	0
$883$	32514.7	1.23919	0.619596	−	0.784921i	$-0.287296\pi$
$883$	32514.7	1.23919	0.619596	+	0.784921i	$0.287296\pi$
$884$	0	0
$885$	16158.4	0.613737
$886$	0	0
$887$	114.991	0.00435290	0.00217645	−	0.999998i	$-0.499307\pi$
$887$	114.991	0.00435290	0.00217645	+	0.999998i	$0.499307\pi$
$888$	0	0
$889$	−8453.53	−0.318923
$890$	0	0
$891$	−4200.11	−0.157923
$892$	0	0
$893$	−59129.9	−2.21580
$894$	0	0
$895$	−44805.9	−1.67340
$896$	0	0
$897$	−18576.9	−0.691488
$898$	0	0
$899$	12639.7	0.468917
$900$	0	0
$901$	−536.910	−0.0198524
$902$	0	0
$903$	−364.055	−0.0134164
$904$	0	0
$905$	−2823.97	−0.103726
$906$	0	0
$907$	−41840.1	−1.53173	−0.765864	−	0.643003i	$-0.777688\pi$
$907$	−41840.1	−1.53173	−0.765864	+	0.643003i	$0.777688\pi$
$908$	0	0
$909$	3911.92	0.142740
$910$	0	0
$911$	−918.744	−0.0334131	−0.0167066	−	0.999860i	$-0.505318\pi$
$911$	−918.744	−0.0334131	−0.0167066	+	0.999860i	$0.505318\pi$
$912$	0	0
$913$	−15326.9	−0.555583
$914$	0	0
$915$	−21455.2	−0.775177
$916$	0	0
$917$	−11142.1	−0.401247
$918$	0	0
$919$	−36189.9	−1.29901	−0.649507	−	0.760356i	$-0.725025\pi$
$919$	−36189.9	−1.29901	−0.649507	+	0.760356i	$0.725025\pi$
$920$	0	0
$921$	31788.2	1.13730
$922$	0	0
$923$	364.635	0.0130034
$924$	0	0
$925$	−631.616	−0.0224513
$926$	0	0
$927$	12725.2	0.450862
$928$	0	0
$929$	−10738.3	−0.379237	−0.189618	−	0.981858i	$-0.560725\pi$
$929$	−10738.3	−0.379237	−0.189618	+	0.981858i	$0.560725\pi$
$930$	0	0
$931$	−4537.67	−0.159738
$932$	0	0
$933$	28933.6	1.01527
$934$	0	0
$935$	1242.73	0.0434671
$936$	0	0
$937$	34056.1	1.18737	0.593684	−	0.804698i	$-0.297673\pi$
$937$	34056.1	1.18737	0.593684	+	0.804698i	$0.297673\pi$
$938$	0	0
$939$	11734.2	0.407808
$940$	0	0
$941$	17151.4	0.594175	0.297088	−	0.954850i	$-0.403985\pi$
$941$	17151.4	0.594175	0.297088	+	0.954850i	$0.403985\pi$
$942$	0	0
$943$	−41757.9	−1.44202
$944$	0	0
$945$	11602.0	0.399379
$946$	0	0
$947$	33234.5	1.14042	0.570208	−	0.821500i	$-0.306862\pi$
$947$	33234.5	1.14042	0.570208	+	0.821500i	$0.306862\pi$
$948$	0	0
$949$	14765.0	0.505051
$950$	0	0
$951$	3760.37	0.128221
$952$	0	0
$953$	−55016.2	−1.87004	−0.935021	−	0.354592i	$-0.884620\pi$
$953$	−55016.2	−1.87004	−0.935021	+	0.354592i	$0.884620\pi$
$954$	0	0
$955$	44127.7	1.49522
$956$	0	0
$957$	−2268.17	−0.0766138
$958$	0	0
$959$	−2477.24	−0.0834141
$960$	0	0
$961$	35932.0	1.20614
$962$	0	0
$963$	−1744.42	−0.0583730
$964$	0	0
$965$	55953.3	1.86653
$966$	0	0
$967$	−5385.55	−0.179098	−0.0895489	−	0.995982i	$-0.528543\pi$
$967$	−5385.55	−0.179098	−0.0895489	+	0.995982i	$0.528543\pi$
$968$	0	0
$969$	−4030.88	−0.133633
$970$	0	0
$971$	−40623.0	−1.34259	−0.671295	−	0.741190i	$-0.734262\pi$
$971$	−40623.0	−1.34259	−0.671295	+	0.741190i	$0.734262\pi$
$972$	0	0
$973$	5310.86	0.174983
$974$	0	0
$975$	1305.05	0.0428668
$976$	0	0
$977$	−41568.9	−1.36122	−0.680608	−	0.732648i	$-0.738284\pi$
$977$	−41568.9	−1.36122	−0.680608	+	0.732648i	$0.738284\pi$
$978$	0	0
$979$	−13902.2	−0.453848
$980$	0	0
$981$	2039.21	0.0663681
$982$	0	0
$983$	−23635.3	−0.766885	−0.383442	−	0.923565i	$-0.625261\pi$
$983$	−23635.3	−0.766885	−0.383442	+	0.923565i	$0.625261\pi$
$984$	0	0
$985$	−2544.31	−0.0823030
$986$	0	0
$987$	18692.8	0.602834
$988$	0	0
$989$	−1269.20	−0.0408071
$990$	0	0
$991$	−15052.9	−0.482514	−0.241257	−	0.970461i	$-0.577560\pi$
$991$	−15052.9	−0.482514	−0.241257	+	0.970461i	$0.577560\pi$
$992$	0	0
$993$	2264.75	0.0723762
$994$	0	0
$995$	13022.8	0.414925
$996$	0	0
$997$	−32416.5	−1.02973	−0.514865	−	0.857271i	$-0.672158\pi$
$997$	−32416.5	−1.02973	−0.514865	+	0.857271i	$0.672158\pi$
$998$	0	0
$999$	13450.5	0.425981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.ba.1.2		6
4.3	odd	2		616.4.a.h.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.h.1.5	✓	6	4.3	odd	2
1232.4.a.ba.1.2		6	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-4.18220 q^{3} -10.8550 q^{5} -7.00000 q^{7} -9.50917 q^{9} +O(q^{10})\)	\(q-4.18220 q^{3} -10.8550 q^{5} -7.00000 q^{7} -9.50917 q^{9} +11.0000 q^{11} +43.5213 q^{13} +45.3976 q^{15} -10.4078 q^{17} -92.6056 q^{19} +29.2754 q^{21} +102.063 q^{23} -7.17004 q^{25} +152.689 q^{27} +49.3034 q^{29} +256.365 q^{31} -46.0042 q^{33} +75.9847 q^{35} +88.0910 q^{37} -182.015 q^{39} -409.140 q^{41} -12.4355 q^{43} +103.222 q^{45} +638.513 q^{47} +49.0000 q^{49} +43.5274 q^{51} +51.5874 q^{53} -119.404 q^{55} +387.296 q^{57} +355.930 q^{59} -472.606 q^{61} +66.5642 q^{63} -472.421 q^{65} +1067.21 q^{67} -426.847 q^{69} +8.37831 q^{71} +339.261 q^{73} +29.9866 q^{75} -77.0000 q^{77} -989.847 q^{79} -381.828 q^{81} -1393.36 q^{83} +112.976 q^{85} -206.197 q^{87} -1263.84 q^{89} -304.649 q^{91} -1072.17 q^{93} +1005.23 q^{95} +303.261 q^{97} -104.601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{5} - 42 q^{7} + 28 q^{9}+O(q^{10}) \) 6 * q - 14 * q^5 - 42 * q^7 + 28 * q^9	\( 6 q - 14 q^{5} - 42 q^{7} + 28 q^{9} + 66 q^{11} - 70 q^{13} + 50 q^{15} - 102 q^{17} + 136 q^{19} + 146 q^{23} + 44 q^{25} + 30 q^{27} - 148 q^{29} - 308 q^{31} + 98 q^{35} - 6 q^{37} + 540 q^{39} - 90 q^{41} + 396 q^{43} - 812 q^{45} + 346 q^{47} + 294 q^{49} + 600 q^{51} - 984 q^{53} - 154 q^{55} - 1580 q^{57} + 44 q^{59} - 1414 q^{61} - 196 q^{63} - 1208 q^{65} + 926 q^{67} - 2550 q^{69} + 1478 q^{71} - 678 q^{73} + 1350 q^{75} - 462 q^{77} + 700 q^{79} - 966 q^{81} + 1192 q^{83} - 1596 q^{85} - 1900 q^{87} - 3314 q^{89} + 490 q^{91} - 2330 q^{93} - 668 q^{95} - 346 q^{97} + 308 q^{99}+O(q^{100}) \) 6 * q - 14 * q^5 - 42 * q^7 + 28 * q^9 + 66 * q^11 - 70 * q^13 + 50 * q^15 - 102 * q^17 + 136 * q^19 + 146 * q^23 + 44 * q^25 + 30 * q^27 - 148 * q^29 - 308 * q^31 + 98 * q^35 - 6 * q^37 + 540 * q^39 - 90 * q^41 + 396 * q^43 - 812 * q^45 + 346 * q^47 + 294 * q^49 + 600 * q^51 - 984 * q^53 - 154 * q^55 - 1580 * q^57 + 44 * q^59 - 1414 * q^61 - 196 * q^63 - 1208 * q^65 + 926 * q^67 - 2550 * q^69 + 1478 * q^71 - 678 * q^73 + 1350 * q^75 - 462 * q^77 + 700 * q^79 - 966 * q^81 + 1192 * q^83 - 1596 * q^85 - 1900 * q^87 - 3314 * q^89 + 490 * q^91 - 2330 * q^93 - 668 * q^95 - 346 * q^97 + 308 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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