LMFDB - Embedded newform 6044.2.a.a.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53490 q^{3} +4.14677 q^{5} +1.26365 q^{7} +3.42570 q^{9} +O(q^{10})$	$q-2.53490 q^{3} +4.14677 q^{5} +1.26365 q^{7} +3.42570 q^{9} -5.32535 q^{11} -7.04801 q^{13} -10.5116 q^{15} +0.539993 q^{17} -2.21920 q^{19} -3.20321 q^{21} +6.37959 q^{23} +12.1957 q^{25} -1.07910 q^{27} +2.89060 q^{29} +2.30370 q^{31} +13.4992 q^{33} +5.24006 q^{35} +0.219930 q^{37} +17.8660 q^{39} +4.67600 q^{41} -4.96533 q^{43} +14.2056 q^{45} -6.32496 q^{47} -5.40320 q^{49} -1.36883 q^{51} +1.55864 q^{53} -22.0830 q^{55} +5.62543 q^{57} -9.22048 q^{59} +11.8963 q^{61} +4.32887 q^{63} -29.2265 q^{65} +4.30232 q^{67} -16.1716 q^{69} +0.142413 q^{71} +6.03505 q^{73} -30.9149 q^{75} -6.72936 q^{77} -5.69624 q^{79} -7.54169 q^{81} -13.9501 q^{83} +2.23923 q^{85} -7.32738 q^{87} +10.4122 q^{89} -8.90619 q^{91} -5.83965 q^{93} -9.20250 q^{95} -1.70230 q^{97} -18.2430 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * 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
$3$	−2.53490	−1.46352	−0.731761	−	0.681561i	$-0.761301\pi$
$3$	−2.53490	−1.46352	−0.731761	+	0.681561i	$0.761301\pi$
$4$	0	0
$5$	4.14677	1.85449	0.927246	−	0.374452i	$-0.122169\pi$
$5$	4.14677	1.85449	0.927246	+	0.374452i	$0.122169\pi$
$6$	0	0
$7$	1.26365	0.477614	0.238807	−	0.971067i	$-0.423244\pi$
$7$	1.26365	0.477614	0.238807	+	0.971067i	$0.423244\pi$
$8$	0	0
$9$	3.42570	1.14190
$10$	0	0
$11$	−5.32535	−1.60565	−0.802827	−	0.596213i	$-0.796672\pi$
$11$	−5.32535	−1.60565	−0.802827	+	0.596213i	$0.796672\pi$
$12$	0	0
$13$	−7.04801	−1.95477	−0.977383	−	0.211478i	$-0.932172\pi$
$13$	−7.04801	−1.95477	−0.977383	+	0.211478i	$0.932172\pi$
$14$	0	0
$15$	−10.5116	−2.71409
$16$	0	0
$17$	0.539993	0.130968	0.0654838	−	0.997854i	$-0.479141\pi$
$17$	0.539993	0.130968	0.0654838	+	0.997854i	$0.479141\pi$
$18$	0	0
$19$	−2.21920	−0.509119	−0.254559	−	0.967057i	$-0.581930\pi$
$19$	−2.21920	−0.509119	−0.254559	+	0.967057i	$0.581930\pi$
$20$	0	0
$21$	−3.20321	−0.698999
$22$	0	0
$23$	6.37959	1.33024	0.665118	−	0.746738i	$-0.268381\pi$
$23$	6.37959	1.33024	0.665118	+	0.746738i	$0.268381\pi$
$24$	0	0
$25$	12.1957	2.43914
$26$	0	0
$27$	−1.07910	−0.207673
$28$	0	0
$29$	2.89060	0.536772	0.268386	−	0.963311i	$-0.413510\pi$
$29$	2.89060	0.536772	0.268386	+	0.963311i	$0.413510\pi$
$30$	0	0
$31$	2.30370	0.413758	0.206879	−	0.978367i	$-0.433669\pi$
$31$	2.30370	0.413758	0.206879	+	0.978367i	$0.433669\pi$
$32$	0	0
$33$	13.4992	2.34991
$34$	0	0
$35$	5.24006	0.885731
$36$	0	0
$37$	0.219930	0.0361562	0.0180781	−	0.999837i	$-0.494245\pi$
$37$	0.219930	0.0361562	0.0180781	+	0.999837i	$0.494245\pi$
$38$	0	0
$39$	17.8660	2.86084
$40$	0	0
$41$	4.67600	0.730268	0.365134	−	0.930955i	$-0.381023\pi$
$41$	4.67600	0.730268	0.365134	+	0.930955i	$0.381023\pi$
$42$	0	0
$43$	−4.96533	−0.757206	−0.378603	−	0.925559i	$-0.623595\pi$
$43$	−4.96533	−0.757206	−0.378603	+	0.925559i	$0.623595\pi$
$44$	0	0
$45$	14.2056	2.11764
$46$	0	0
$47$	−6.32496	−0.922591	−0.461295	−	0.887247i	$-0.652615\pi$
$47$	−6.32496	−0.922591	−0.461295	+	0.887247i	$0.652615\pi$
$48$	0	0
$49$	−5.40320	−0.771885
$50$	0	0
$51$	−1.36883	−0.191674
$52$	0	0
$53$	1.55864	0.214095	0.107048	−	0.994254i	$-0.465860\pi$
$53$	1.55864	0.214095	0.107048	+	0.994254i	$0.465860\pi$
$54$	0	0
$55$	−22.0830	−2.97767
$56$	0	0
$57$	5.62543	0.745107
$58$	0	0
$59$	−9.22048	−1.20040	−0.600202	−	0.799849i	$-0.704913\pi$
$59$	−9.22048	−1.20040	−0.600202	+	0.799849i	$0.704913\pi$
$60$	0	0
$61$	11.8963	1.52317	0.761584	−	0.648066i	$-0.224422\pi$
$61$	11.8963	1.52317	0.761584	+	0.648066i	$0.224422\pi$
$62$	0	0
$63$	4.32887	0.545387
$64$	0	0
$65$	−29.2265	−3.62510
$66$	0	0
$67$	4.30232	0.525613	0.262806	−	0.964849i	$-0.415352\pi$
$67$	4.30232	0.525613	0.262806	+	0.964849i	$0.415352\pi$
$68$	0	0
$69$	−16.1716	−1.94683
$70$	0	0
$71$	0.142413	0.0169013	0.00845065	−	0.999964i	$-0.497310\pi$
$71$	0.142413	0.0169013	0.00845065	+	0.999964i	$0.497310\pi$
$72$	0	0
$73$	6.03505	0.706349	0.353174	−	0.935558i	$-0.385102\pi$
$73$	6.03505	0.706349	0.353174	+	0.935558i	$0.385102\pi$
$74$	0	0
$75$	−30.9149	−3.56974
$76$	0	0
$77$	−6.72936	−0.766882
$78$	0	0
$79$	−5.69624	−0.640877	−0.320439	−	0.947269i	$-0.603830\pi$
$79$	−5.69624	−0.640877	−0.320439	+	0.947269i	$0.603830\pi$
$80$	0	0
$81$	−7.54169	−0.837965
$82$	0	0
$83$	−13.9501	−1.53122	−0.765612	−	0.643303i	$-0.777564\pi$
$83$	−13.9501	−1.53122	−0.765612	+	0.643303i	$0.777564\pi$
$84$	0	0
$85$	2.23923	0.242878
$86$	0	0
$87$	−7.32738	−0.785578
$88$	0	0
$89$	10.4122	1.10369	0.551845	−	0.833947i	$-0.313924\pi$
$89$	10.4122	1.10369	0.551845	+	0.833947i	$0.313924\pi$
$90$	0	0
$91$	−8.90619	−0.933623
$92$	0	0
$93$	−5.83965	−0.605544
$94$	0	0
$95$	−9.20250	−0.944157
$96$	0	0
$97$	−1.70230	−0.172842	−0.0864209	−	0.996259i	$-0.527543\pi$
$97$	−1.70230	−0.172842	−0.0864209	+	0.996259i	$0.527543\pi$
$98$	0	0
$99$	−18.2430	−1.83349
$100$	0	0
$101$	−6.06073	−0.603065	−0.301533	−	0.953456i	$-0.597498\pi$
$101$	−6.06073	−0.603065	−0.301533	+	0.953456i	$0.597498\pi$
$102$	0	0
$103$	3.49499	0.344371	0.172186	−	0.985065i	$-0.444917\pi$
$103$	3.49499	0.344371	0.172186	+	0.985065i	$0.444917\pi$
$104$	0	0
$105$	−13.2830	−1.29629
$106$	0	0
$107$	12.0537	1.16527	0.582636	−	0.812733i	$-0.302021\pi$
$107$	12.0537	1.16527	0.582636	+	0.812733i	$0.302021\pi$
$108$	0	0
$109$	−8.59268	−0.823029	−0.411515	−	0.911403i	$-0.635000\pi$
$109$	−8.59268	−0.823029	−0.411515	+	0.911403i	$0.635000\pi$
$110$	0	0
$111$	−0.557499	−0.0529154
$112$	0	0
$113$	−10.4746	−0.985368	−0.492684	−	0.870208i	$-0.663984\pi$
$113$	−10.4746	−0.985368	−0.492684	+	0.870208i	$0.663984\pi$
$114$	0	0
$115$	26.4547	2.46691
$116$	0	0
$117$	−24.1443	−2.23215
$118$	0	0
$119$	0.682361	0.0625519
$120$	0	0
$121$	17.3593	1.57812
$122$	0	0
$123$	−11.8532	−1.06876
$124$	0	0
$125$	29.8390	2.66888
$126$	0	0
$127$	−18.4276	−1.63518	−0.817591	−	0.575799i	$-0.804691\pi$
$127$	−18.4276	−1.63518	−0.817591	+	0.575799i	$0.804691\pi$
$128$	0	0
$129$	12.5866	1.10819
$130$	0	0
$131$	−17.0485	−1.48953	−0.744766	−	0.667326i	$-0.767439\pi$
$131$	−17.0485	−1.48953	−0.744766	+	0.667326i	$0.767439\pi$
$132$	0	0
$133$	−2.80428	−0.243162
$134$	0	0
$135$	−4.47478	−0.385127
$136$	0	0
$137$	0.398809	0.0340725	0.0170363	−	0.999855i	$-0.494577\pi$
$137$	0.398809	0.0340725	0.0170363	+	0.999855i	$0.494577\pi$
$138$	0	0
$139$	−16.8533	−1.42948	−0.714740	−	0.699390i	$-0.753455\pi$
$139$	−16.8533	−1.42948	−0.714740	+	0.699390i	$0.753455\pi$
$140$	0	0
$141$	16.0331	1.35023
$142$	0	0
$143$	37.5331	3.13868
$144$	0	0
$145$	11.9867	0.995439
$146$	0	0
$147$	13.6965	1.12967
$148$	0	0
$149$	−21.1359	−1.73152	−0.865761	−	0.500458i	$-0.833165\pi$
$149$	−21.1359	−1.73152	−0.865761	+	0.500458i	$0.833165\pi$
$150$	0	0
$151$	2.37191	0.193023	0.0965115	−	0.995332i	$-0.469232\pi$
$151$	2.37191	0.193023	0.0965115	+	0.995332i	$0.469232\pi$
$152$	0	0
$153$	1.84985	0.149552
$154$	0	0
$155$	9.55293	0.767310
$156$	0	0
$157$	−13.1584	−1.05015	−0.525076	−	0.851056i	$-0.675963\pi$
$157$	−13.1584	−1.05015	−0.525076	+	0.851056i	$0.675963\pi$
$158$	0	0
$159$	−3.95099	−0.313334
$160$	0	0
$161$	8.06155	0.635339
$162$	0	0
$163$	−11.7936	−0.923750	−0.461875	−	0.886945i	$-0.652823\pi$
$163$	−11.7936	−0.923750	−0.461875	+	0.886945i	$0.652823\pi$
$164$	0	0
$165$	55.9781	4.35789
$166$	0	0
$167$	5.64670	0.436955	0.218478	−	0.975842i	$-0.429891\pi$
$167$	5.64670	0.436955	0.218478	+	0.975842i	$0.429891\pi$
$168$	0	0
$169$	36.6744	2.82111
$170$	0	0
$171$	−7.60230	−0.581362
$172$	0	0
$173$	−12.5130	−0.951349	−0.475674	−	0.879621i	$-0.657796\pi$
$173$	−12.5130	−0.951349	−0.475674	+	0.879621i	$0.657796\pi$
$174$	0	0
$175$	15.4111	1.16497
$176$	0	0
$177$	23.3729	1.75682
$178$	0	0
$179$	−10.1473	−0.758445	−0.379222	−	0.925306i	$-0.623808\pi$
$179$	−10.1473	−0.758445	−0.379222	+	0.925306i	$0.623808\pi$
$180$	0	0
$181$	1.76227	0.130988	0.0654941	−	0.997853i	$-0.479138\pi$
$181$	1.76227	0.130988	0.0654941	+	0.997853i	$0.479138\pi$
$182$	0	0
$183$	−30.1559	−2.22919
$184$	0	0
$185$	0.911998	0.0670514
$186$	0	0
$187$	−2.87565	−0.210288
$188$	0	0
$189$	−1.36360	−0.0991873
$190$	0	0
$191$	−14.5195	−1.05059	−0.525296	−	0.850919i	$-0.676045\pi$
$191$	−14.5195	−1.05059	−0.525296	+	0.850919i	$0.676045\pi$
$192$	0	0
$193$	0.263937	0.0189986	0.00949929	−	0.999955i	$-0.496976\pi$
$193$	0.263937	0.0189986	0.00949929	+	0.999955i	$0.496976\pi$
$194$	0	0
$195$	74.0861	5.30541
$196$	0	0
$197$	−7.16372	−0.510394	−0.255197	−	0.966889i	$-0.582140\pi$
$197$	−7.16372	−0.510394	−0.255197	+	0.966889i	$0.582140\pi$
$198$	0	0
$199$	−7.87662	−0.558359	−0.279179	−	0.960239i	$-0.590062\pi$
$199$	−7.87662	−0.558359	−0.279179	+	0.960239i	$0.590062\pi$
$200$	0	0
$201$	−10.9059	−0.769246
$202$	0	0
$203$	3.65270	0.256370
$204$	0	0
$205$	19.3903	1.35428
$206$	0	0
$207$	21.8545	1.51900
$208$	0	0
$209$	11.8180	0.817468
$210$	0	0
$211$	−6.41618	−0.441708	−0.220854	−	0.975307i	$-0.570884\pi$
$211$	−6.41618	−0.441708	−0.220854	+	0.975307i	$0.570884\pi$
$212$	0	0
$213$	−0.361002	−0.0247354
$214$	0	0
$215$	−20.5901	−1.40423
$216$	0	0
$217$	2.91107	0.197616
$218$	0	0
$219$	−15.2982	−1.03376
$220$	0	0
$221$	−3.80587	−0.256011
$222$	0	0
$223$	21.3813	1.43180	0.715899	−	0.698203i	$-0.246017\pi$
$223$	21.3813	1.43180	0.715899	+	0.698203i	$0.246017\pi$
$224$	0	0
$225$	41.7788	2.78525
$226$	0	0
$227$	−19.2407	−1.27705	−0.638526	−	0.769601i	$-0.720455\pi$
$227$	−19.2407	−1.27705	−0.638526	+	0.769601i	$0.720455\pi$
$228$	0	0
$229$	10.7802	0.712374	0.356187	−	0.934415i	$-0.384077\pi$
$229$	10.7802	0.712374	0.356187	+	0.934415i	$0.384077\pi$
$230$	0	0
$231$	17.0582	1.12235
$232$	0	0
$233$	−22.0162	−1.44233	−0.721166	−	0.692763i	$-0.756393\pi$
$233$	−22.0162	−1.44233	−0.721166	+	0.692763i	$0.756393\pi$
$234$	0	0
$235$	−26.2282	−1.71094
$236$	0	0
$237$	14.4394	0.937938
$238$	0	0
$239$	12.4659	0.806349	0.403175	−	0.915123i	$-0.367907\pi$
$239$	12.4659	0.806349	0.403175	+	0.915123i	$0.367907\pi$
$240$	0	0
$241$	6.15037	0.396180	0.198090	−	0.980184i	$-0.436526\pi$
$241$	6.15037	0.396180	0.198090	+	0.980184i	$0.436526\pi$
$242$	0	0
$243$	22.3547	1.43405
$244$	0	0
$245$	−22.4058	−1.43146
$246$	0	0
$247$	15.6409	0.995208
$248$	0	0
$249$	35.3621	2.24098
$250$	0	0
$251$	3.32373	0.209792	0.104896	−	0.994483i	$-0.466549\pi$
$251$	3.32373	0.209792	0.104896	+	0.994483i	$0.466549\pi$
$252$	0	0
$253$	−33.9735	−2.13590
$254$	0	0
$255$	−5.67621	−0.355458
$256$	0	0
$257$	−8.75925	−0.546387	−0.273194	−	0.961959i	$-0.588080\pi$
$257$	−8.75925	−0.546387	−0.273194	+	0.961959i	$0.588080\pi$
$258$	0	0
$259$	0.277913	0.0172687
$260$	0	0
$261$	9.90234	0.612939
$262$	0	0
$263$	−12.4646	−0.768603	−0.384301	−	0.923208i	$-0.625558\pi$
$263$	−12.4646	−0.768603	−0.384301	+	0.923208i	$0.625558\pi$
$264$	0	0
$265$	6.46332	0.397038
$266$	0	0
$267$	−26.3938	−1.61528
$268$	0	0
$269$	16.0214	0.976843	0.488422	−	0.872608i	$-0.337573\pi$
$269$	16.0214	0.976843	0.488422	+	0.872608i	$0.337573\pi$
$270$	0	0
$271$	−31.4459	−1.91020	−0.955101	−	0.296279i	$-0.904254\pi$
$271$	−31.4459	−1.91020	−0.955101	+	0.296279i	$0.904254\pi$
$272$	0	0
$273$	22.5763	1.36638
$274$	0	0
$275$	−64.9464	−3.91642
$276$	0	0
$277$	6.21008	0.373127	0.186564	−	0.982443i	$-0.440265\pi$
$277$	6.21008	0.373127	0.186564	+	0.982443i	$0.440265\pi$
$278$	0	0
$279$	7.89179	0.472469
$280$	0	0
$281$	17.5888	1.04926	0.524631	−	0.851330i	$-0.324203\pi$
$281$	17.5888	1.04926	0.524631	+	0.851330i	$0.324203\pi$
$282$	0	0
$283$	9.19091	0.546343	0.273171	−	0.961965i	$-0.411927\pi$
$283$	9.19091	0.546343	0.273171	+	0.961965i	$0.411927\pi$
$284$	0	0
$285$	23.3274	1.38179
$286$	0	0
$287$	5.90881	0.348786
$288$	0	0
$289$	−16.7084	−0.982848
$290$	0	0
$291$	4.31514	0.252958
$292$	0	0
$293$	−28.3838	−1.65820	−0.829099	−	0.559102i	$-0.811146\pi$
$293$	−28.3838	−1.65820	−0.829099	+	0.559102i	$0.811146\pi$
$294$	0	0
$295$	−38.2352	−2.22614
$296$	0	0
$297$	5.74658	0.333450
$298$	0	0
$299$	−44.9634	−2.60030
$300$	0	0
$301$	−6.27443	−0.361652
$302$	0	0
$303$	15.3633	0.882600
$304$	0	0
$305$	49.3313	2.82470
$306$	0	0
$307$	31.4538	1.79516	0.897582	−	0.440847i	$-0.145322\pi$
$307$	31.4538	1.79516	0.897582	+	0.440847i	$0.145322\pi$
$308$	0	0
$309$	−8.85942	−0.503995
$310$	0	0
$311$	8.27407	0.469179	0.234590	−	0.972095i	$-0.424625\pi$
$311$	8.27407	0.469179	0.234590	+	0.972095i	$0.424625\pi$
$312$	0	0
$313$	−5.95356	−0.336515	−0.168258	−	0.985743i	$-0.553814\pi$
$313$	−5.95356	−0.336515	−0.168258	+	0.985743i	$0.553814\pi$
$314$	0	0
$315$	17.9508	1.01142
$316$	0	0
$317$	20.6380	1.15914	0.579572	−	0.814921i	$-0.303220\pi$
$317$	20.6380	1.15914	0.579572	+	0.814921i	$0.303220\pi$
$318$	0	0
$319$	−15.3935	−0.861869
$320$	0	0
$321$	−30.5548	−1.70540
$322$	0	0
$323$	−1.19835	−0.0666780
$324$	0	0
$325$	−85.9554	−4.76795
$326$	0	0
$327$	21.7815	1.20452
$328$	0	0
$329$	−7.99252	−0.440642
$330$	0	0
$331$	8.58637	0.471950	0.235975	−	0.971759i	$-0.424172\pi$
$331$	8.58637	0.471950	0.235975	+	0.971759i	$0.424172\pi$
$332$	0	0
$333$	0.753412	0.0412867
$334$	0	0
$335$	17.8408	0.974744
$336$	0	0
$337$	−2.59793	−0.141518	−0.0707592	−	0.997493i	$-0.522542\pi$
$337$	−2.59793	−0.141518	−0.0707592	+	0.997493i	$0.522542\pi$
$338$	0	0
$339$	26.5520	1.44211
$340$	0	0
$341$	−12.2680	−0.664351
$342$	0	0
$343$	−15.6733	−0.846277
$344$	0	0
$345$	−67.0599	−3.61038
$346$	0	0
$347$	18.5382	0.995185	0.497592	−	0.867411i	$-0.334218\pi$
$347$	18.5382	0.995185	0.497592	+	0.867411i	$0.334218\pi$
$348$	0	0
$349$	−20.2949	−1.08636	−0.543180	−	0.839616i	$-0.682780\pi$
$349$	−20.2949	−1.08636	−0.543180	+	0.839616i	$0.682780\pi$
$350$	0	0
$351$	7.60550	0.405951
$352$	0	0
$353$	25.6093	1.36305	0.681523	−	0.731797i	$-0.261318\pi$
$353$	25.6093	1.36305	0.681523	+	0.731797i	$0.261318\pi$
$354$	0	0
$355$	0.590554	0.0313433
$356$	0	0
$357$	−1.72971	−0.0915461
$358$	0	0
$359$	−24.0675	−1.27023	−0.635117	−	0.772416i	$-0.719048\pi$
$359$	−24.0675	−1.27023	−0.635117	+	0.772416i	$0.719048\pi$
$360$	0	0
$361$	−14.0752	−0.740798
$362$	0	0
$363$	−44.0041	−2.30962
$364$	0	0
$365$	25.0260	1.30992
$366$	0	0
$367$	18.3067	0.955602	0.477801	−	0.878468i	$-0.341434\pi$
$367$	18.3067	0.955602	0.477801	+	0.878468i	$0.341434\pi$
$368$	0	0
$369$	16.0186	0.833893
$370$	0	0
$371$	1.96957	0.102255
$372$	0	0
$373$	−35.3303	−1.82933	−0.914667	−	0.404209i	$-0.867547\pi$
$373$	−35.3303	−1.82933	−0.914667	+	0.404209i	$0.867547\pi$
$374$	0	0
$375$	−75.6386	−3.90596
$376$	0	0
$377$	−20.3730	−1.04926
$378$	0	0
$379$	−2.52014	−0.129451	−0.0647254	−	0.997903i	$-0.520617\pi$
$379$	−2.52014	−0.129451	−0.0647254	+	0.997903i	$0.520617\pi$
$380$	0	0
$381$	46.7120	2.39313
$382$	0	0
$383$	−25.4289	−1.29936	−0.649678	−	0.760210i	$-0.725096\pi$
$383$	−25.4289	−1.29936	−0.649678	+	0.760210i	$0.725096\pi$
$384$	0	0
$385$	−27.9051	−1.42218
$386$	0	0
$387$	−17.0097	−0.864653
$388$	0	0
$389$	27.0388	1.37092	0.685460	−	0.728111i	$-0.259601\pi$
$389$	27.0388	1.37092	0.685460	+	0.728111i	$0.259601\pi$
$390$	0	0
$391$	3.44493	0.174218
$392$	0	0
$393$	43.2161	2.17996
$394$	0	0
$395$	−23.6210	−1.18850
$396$	0	0
$397$	−2.60825	−0.130905	−0.0654523	−	0.997856i	$-0.520849\pi$
$397$	−2.60825	−0.130905	−0.0654523	+	0.997856i	$0.520849\pi$
$398$	0	0
$399$	7.10856	0.355873
$400$	0	0
$401$	−2.25760	−0.112739	−0.0563697	−	0.998410i	$-0.517953\pi$
$401$	−2.25760	−0.112739	−0.0563697	+	0.998410i	$0.517953\pi$
$402$	0	0
$403$	−16.2365	−0.808799
$404$	0	0
$405$	−31.2737	−1.55400
$406$	0	0
$407$	−1.17120	−0.0580543
$408$	0	0
$409$	−18.5033	−0.914931	−0.457466	−	0.889227i	$-0.651243\pi$
$409$	−18.5033	−0.914931	−0.457466	+	0.889227i	$0.651243\pi$
$410$	0	0
$411$	−1.01094	−0.0498659
$412$	0	0
$413$	−11.6514	−0.573329
$414$	0	0
$415$	−57.8479	−2.83964
$416$	0	0
$417$	42.7214	2.09208
$418$	0	0
$419$	20.4026	0.996734	0.498367	−	0.866966i	$-0.333933\pi$
$419$	20.4026	0.996734	0.498367	+	0.866966i	$0.333933\pi$
$420$	0	0
$421$	−15.0576	−0.733860	−0.366930	−	0.930248i	$-0.619591\pi$
$421$	−15.0576	−0.733860	−0.366930	+	0.930248i	$0.619591\pi$
$422$	0	0
$423$	−21.6674	−1.05351
$424$	0	0
$425$	6.58560	0.319448
$426$	0	0
$427$	15.0328	0.727486
$428$	0	0
$429$	−95.1425	−4.59352
$430$	0	0
$431$	8.24157	0.396982	0.198491	−	0.980103i	$-0.436396\pi$
$431$	8.24157	0.396982	0.198491	+	0.980103i	$0.436396\pi$
$432$	0	0
$433$	32.8348	1.57794	0.788970	−	0.614431i	$-0.210615\pi$
$433$	32.8348	1.57794	0.788970	+	0.614431i	$0.210615\pi$
$434$	0	0
$435$	−30.3850	−1.45685
$436$	0	0
$437$	−14.1576	−0.677248
$438$	0	0
$439$	−10.2611	−0.489737	−0.244869	−	0.969556i	$-0.578745\pi$
$439$	−10.2611	−0.489737	−0.244869	+	0.969556i	$0.578745\pi$
$440$	0	0
$441$	−18.5097	−0.881415
$442$	0	0
$443$	−35.9584	−1.70844	−0.854218	−	0.519916i	$-0.825963\pi$
$443$	−35.9584	−1.70844	−0.854218	+	0.519916i	$0.825963\pi$
$444$	0	0
$445$	43.1770	2.04678
$446$	0	0
$447$	53.5773	2.53412
$448$	0	0
$449$	−39.2998	−1.85467	−0.927337	−	0.374227i	$-0.877908\pi$
$449$	−39.2998	−1.85467	−0.927337	+	0.374227i	$0.877908\pi$
$450$	0	0
$451$	−24.9013	−1.17256
$452$	0	0
$453$	−6.01253	−0.282494
$454$	0	0
$455$	−36.9319	−1.73140
$456$	0	0
$457$	−7.42093	−0.347137	−0.173568	−	0.984822i	$-0.555530\pi$
$457$	−7.42093	−0.347137	−0.173568	+	0.984822i	$0.555530\pi$
$458$	0	0
$459$	−0.582706	−0.0271984
$460$	0	0
$461$	2.31131	0.107648	0.0538242	−	0.998550i	$-0.482859\pi$
$461$	2.31131	0.107648	0.0538242	+	0.998550i	$0.482859\pi$
$462$	0	0
$463$	−4.88347	−0.226954	−0.113477	−	0.993541i	$-0.536199\pi$
$463$	−4.88347	−0.226954	−0.113477	+	0.993541i	$0.536199\pi$
$464$	0	0
$465$	−24.2157	−1.12298
$466$	0	0
$467$	8.68135	0.401725	0.200862	−	0.979619i	$-0.435626\pi$
$467$	8.68135	0.401725	0.200862	+	0.979619i	$0.435626\pi$
$468$	0	0
$469$	5.43662	0.251040
$470$	0	0
$471$	33.3551	1.53692
$472$	0	0
$473$	26.4421	1.21581
$474$	0	0
$475$	−27.0647	−1.24181
$476$	0	0
$477$	5.33942	0.244475
$478$	0	0
$479$	−13.1544	−0.601041	−0.300520	−	0.953775i	$-0.597160\pi$
$479$	−13.1544	−0.601041	−0.300520	+	0.953775i	$0.597160\pi$
$480$	0	0
$481$	−1.55007	−0.0706769
$482$	0	0
$483$	−20.4352	−0.929833
$484$	0	0
$485$	−7.05903	−0.320534
$486$	0	0
$487$	39.5910	1.79404	0.897019	−	0.441992i	$-0.145728\pi$
$487$	39.5910	1.79404	0.897019	+	0.441992i	$0.145728\pi$
$488$	0	0
$489$	29.8957	1.35193
$490$	0	0
$491$	−10.3495	−0.467068	−0.233534	−	0.972349i	$-0.575029\pi$
$491$	−10.3495	−0.467068	−0.233534	+	0.972349i	$0.575029\pi$
$492$	0	0
$493$	1.56091	0.0702997
$494$	0	0
$495$	−75.6497	−3.40020
$496$	0	0
$497$	0.179960	0.00807230
$498$	0	0
$499$	−12.1488	−0.543856	−0.271928	−	0.962318i	$-0.587661\pi$
$499$	−12.1488	−0.543856	−0.271928	+	0.962318i	$0.587661\pi$
$500$	0	0
$501$	−14.3138	−0.639494
$502$	0	0
$503$	−40.5695	−1.80890	−0.904452	−	0.426576i	$-0.859720\pi$
$503$	−40.5695	−1.80890	−0.904452	+	0.426576i	$0.859720\pi$
$504$	0	0
$505$	−25.1325	−1.11838
$506$	0	0
$507$	−92.9658	−4.12876
$508$	0	0
$509$	40.0583	1.77555	0.887777	−	0.460275i	$-0.152249\pi$
$509$	40.0583	1.77555	0.887777	+	0.460275i	$0.152249\pi$
$510$	0	0
$511$	7.62617	0.337362
$512$	0	0
$513$	2.39473	0.105730
$514$	0	0
$515$	14.4929	0.638634
$516$	0	0
$517$	33.6826	1.48136
$518$	0	0
$519$	31.7193	1.39232
$520$	0	0
$521$	−43.1610	−1.89092	−0.945460	−	0.325739i	$-0.894387\pi$
$521$	−43.1610	−1.89092	−0.945460	+	0.325739i	$0.894387\pi$
$522$	0	0
$523$	19.3145	0.844566	0.422283	−	0.906464i	$-0.361229\pi$
$523$	19.3145	0.844566	0.422283	+	0.906464i	$0.361229\pi$
$524$	0	0
$525$	−39.0655	−1.70496
$526$	0	0
$527$	1.24398	0.0541888
$528$	0	0
$529$	17.6991	0.769527
$530$	0	0
$531$	−31.5866	−1.37074
$532$	0	0
$533$	−32.9565	−1.42750
$534$	0	0
$535$	49.9838	2.16099
$536$	0	0
$537$	25.7224	1.11000
$538$	0	0
$539$	28.7739	1.23938
$540$	0	0
$541$	−22.8683	−0.983185	−0.491592	−	0.870825i	$-0.663585\pi$
$541$	−22.8683	−0.983185	−0.491592	+	0.870825i	$0.663585\pi$
$542$	0	0
$543$	−4.46716	−0.191704
$544$	0	0
$545$	−35.6319	−1.52630
$546$	0	0
$547$	34.8627	1.49062	0.745311	−	0.666717i	$-0.232301\pi$
$547$	34.8627	1.49062	0.745311	+	0.666717i	$0.232301\pi$
$548$	0	0
$549$	40.7532	1.73930
$550$	0	0
$551$	−6.41482	−0.273281
$552$	0	0
$553$	−7.19804	−0.306092
$554$	0	0
$555$	−2.31182	−0.0981313
$556$	0	0
$557$	−11.9937	−0.508191	−0.254095	−	0.967179i	$-0.581778\pi$
$557$	−11.9937	−0.508191	−0.254095	+	0.967179i	$0.581778\pi$
$558$	0	0
$559$	34.9957	1.48016
$560$	0	0
$561$	7.28948	0.307762
$562$	0	0
$563$	25.5628	1.07735	0.538673	−	0.842515i	$-0.318926\pi$
$563$	25.5628	1.07735	0.538673	+	0.842515i	$0.318926\pi$
$564$	0	0
$565$	−43.4358	−1.82736
$566$	0	0
$567$	−9.53003	−0.400224
$568$	0	0
$569$	−0.0443215	−0.00185805	−0.000929026	−	1.00000i	$-0.500296\pi$
$569$	−0.0443215	−0.00185805	−0.000929026		1.00000i	$0.500296\pi$
$570$	0	0
$571$	−17.5742	−0.735456	−0.367728	−	0.929933i	$-0.619864\pi$
$571$	−17.5742	−0.735456	−0.367728	+	0.929933i	$0.619864\pi$
$572$	0	0
$573$	36.8054	1.53757
$574$	0	0
$575$	77.8036	3.24463
$576$	0	0
$577$	−29.0733	−1.21034	−0.605169	−	0.796097i	$-0.706894\pi$
$577$	−29.0733	−1.21034	−0.605169	+	0.796097i	$0.706894\pi$
$578$	0	0
$579$	−0.669052	−0.0278048
$580$	0	0
$581$	−17.6280	−0.731333
$582$	0	0
$583$	−8.30029	−0.343763
$584$	0	0
$585$	−100.121	−4.13950
$586$	0	0
$587$	22.4835	0.927994	0.463997	−	0.885837i	$-0.346415\pi$
$587$	22.4835	0.927994	0.463997	+	0.885837i	$0.346415\pi$
$588$	0	0
$589$	−5.11237	−0.210652
$590$	0	0
$591$	18.1593	0.746973
$592$	0	0
$593$	19.2986	0.792500	0.396250	−	0.918143i	$-0.370311\pi$
$593$	19.2986	0.792500	0.396250	+	0.918143i	$0.370311\pi$
$594$	0	0
$595$	2.82959	0.116002
$596$	0	0
$597$	19.9664	0.817171
$598$	0	0
$599$	45.5619	1.86161	0.930804	−	0.365518i	$-0.119108\pi$
$599$	45.5619	1.86161	0.930804	+	0.365518i	$0.119108\pi$
$600$	0	0
$601$	35.2031	1.43596	0.717981	−	0.696062i	$-0.245066\pi$
$601$	35.2031	1.43596	0.717981	+	0.696062i	$0.245066\pi$
$602$	0	0
$603$	14.7385	0.600197
$604$	0	0
$605$	71.9852	2.92662
$606$	0	0
$607$	2.23455	0.0906975	0.0453488	−	0.998971i	$-0.485560\pi$
$607$	2.23455	0.0906975	0.0453488	+	0.998971i	$0.485560\pi$
$608$	0	0
$609$	−9.25923	−0.375203
$610$	0	0
$611$	44.5784	1.80345
$612$	0	0
$613$	30.9164	1.24870	0.624350	−	0.781144i	$-0.285364\pi$
$613$	30.9164	1.24870	0.624350	+	0.781144i	$0.285364\pi$
$614$	0	0
$615$	−49.1524	−1.98202
$616$	0	0
$617$	−14.6103	−0.588190	−0.294095	−	0.955776i	$-0.595018\pi$
$617$	−14.6103	−0.588190	−0.294095	+	0.955776i	$0.595018\pi$
$618$	0	0
$619$	−9.16062	−0.368196	−0.184098	−	0.982908i	$-0.558936\pi$
$619$	−9.16062	−0.368196	−0.184098	+	0.982908i	$0.558936\pi$
$620$	0	0
$621$	−6.88420	−0.276254
$622$	0	0
$623$	13.1573	0.527138
$624$	0	0
$625$	62.7568	2.51027
$626$	0	0
$627$	−29.9574	−1.19638
$628$	0	0
$629$	0.118760	0.00473529
$630$	0	0
$631$	39.8808	1.58763	0.793814	−	0.608160i	$-0.208092\pi$
$631$	39.8808	1.58763	0.793814	+	0.608160i	$0.208092\pi$
$632$	0	0
$633$	16.2644	0.646450
$634$	0	0
$635$	−76.4149	−3.03243
$636$	0	0
$637$	38.0818	1.50885
$638$	0	0
$639$	0.487864	0.0192996
$640$	0	0
$641$	−43.4824	−1.71745	−0.858726	−	0.512434i	$-0.828744\pi$
$641$	−43.4824	−1.71745	−0.858726	+	0.512434i	$0.828744\pi$
$642$	0	0
$643$	−28.8722	−1.13861	−0.569305	−	0.822126i	$-0.692788\pi$
$643$	−28.8722	−1.13861	−0.569305	+	0.822126i	$0.692788\pi$
$644$	0	0
$645$	52.1938	2.05513
$646$	0	0
$647$	34.4157	1.35302	0.676511	−	0.736432i	$-0.263491\pi$
$647$	34.4157	1.35302	0.676511	+	0.736432i	$0.263491\pi$
$648$	0	0
$649$	49.1022	1.92743
$650$	0	0
$651$	−7.37926	−0.289216
$652$	0	0
$653$	14.6834	0.574605	0.287303	−	0.957840i	$-0.407241\pi$
$653$	14.6834	0.574605	0.287303	+	0.957840i	$0.407241\pi$
$654$	0	0
$655$	−70.6961	−2.76233
$656$	0	0
$657$	20.6742	0.806579
$658$	0	0
$659$	−33.8634	−1.31913	−0.659566	−	0.751646i	$-0.729260\pi$
$659$	−33.8634	−1.31913	−0.659566	+	0.751646i	$0.729260\pi$
$660$	0	0
$661$	9.97884	0.388132	0.194066	−	0.980988i	$-0.437832\pi$
$661$	9.97884	0.388132	0.194066	+	0.980988i	$0.437832\pi$
$662$	0	0
$663$	9.64750	0.374678
$664$	0	0
$665$	−11.6287	−0.450942
$666$	0	0
$667$	18.4409	0.714033
$668$	0	0
$669$	−54.1994	−2.09547
$670$	0	0
$671$	−63.3521	−2.44568
$672$	0	0
$673$	−0.0802736	−0.00309432	−0.00154716	−	0.999999i	$-0.500492\pi$
$673$	−0.0802736	−0.00309432	−0.00154716	+	0.999999i	$0.500492\pi$
$674$	0	0
$675$	−13.1604	−0.506543
$676$	0	0
$677$	23.6345	0.908346	0.454173	−	0.890914i	$-0.349935\pi$
$677$	23.6345	0.908346	0.454173	+	0.890914i	$0.349935\pi$
$678$	0	0
$679$	−2.15110	−0.0825517
$680$	0	0
$681$	48.7732	1.86899
$682$	0	0
$683$	5.51666	0.211089	0.105545	−	0.994415i	$-0.466341\pi$
$683$	5.51666	0.211089	0.105545	+	0.994415i	$0.466341\pi$
$684$	0	0
$685$	1.65377	0.0631873
$686$	0	0
$687$	−27.3266	−1.04258
$688$	0	0
$689$	−10.9853	−0.418506
$690$	0	0
$691$	−15.8226	−0.601919	−0.300959	−	0.953637i	$-0.597307\pi$
$691$	−15.8226	−0.601919	−0.300959	+	0.953637i	$0.597307\pi$
$692$	0	0
$693$	−23.0528	−0.875702
$694$	0	0
$695$	−69.8868	−2.65096
$696$	0	0
$697$	2.52501	0.0956415
$698$	0	0
$699$	55.8089	2.11088
$700$	0	0
$701$	9.37565	0.354113	0.177057	−	0.984201i	$-0.443342\pi$
$701$	9.37565	0.354113	0.177057	+	0.984201i	$0.443342\pi$
$702$	0	0
$703$	−0.488067	−0.0184078
$704$	0	0
$705$	66.4857	2.50400
$706$	0	0
$707$	−7.65862	−0.288032
$708$	0	0
$709$	−44.5915	−1.67467	−0.837334	−	0.546691i	$-0.815887\pi$
$709$	−44.5915	−1.67467	−0.837334	+	0.546691i	$0.815887\pi$
$710$	0	0
$711$	−19.5136	−0.731817
$712$	0	0
$713$	14.6967	0.550395
$714$	0	0
$715$	155.641	5.82065
$716$	0	0
$717$	−31.5996	−1.18011
$718$	0	0
$719$	−5.46428	−0.203783	−0.101892	−	0.994795i	$-0.532490\pi$
$719$	−5.46428	−0.203783	−0.101892	+	0.994795i	$0.532490\pi$
$720$	0	0
$721$	4.41643	0.164476
$722$	0	0
$723$	−15.5906	−0.579819
$724$	0	0
$725$	35.2530	1.30926
$726$	0	0
$727$	−11.2319	−0.416569	−0.208285	−	0.978068i	$-0.566788\pi$
$727$	−11.2319	−0.416569	−0.208285	+	0.978068i	$0.566788\pi$
$728$	0	0
$729$	−34.0418	−1.26081
$730$	0	0
$731$	−2.68125	−0.0991694
$732$	0	0
$733$	19.2689	0.711713	0.355856	−	0.934541i	$-0.384189\pi$
$733$	19.2689	0.711713	0.355856	+	0.934541i	$0.384189\pi$
$734$	0	0
$735$	56.7964	2.09497
$736$	0	0
$737$	−22.9114	−0.843951
$738$	0	0
$739$	−34.3992	−1.26540	−0.632698	−	0.774399i	$-0.718052\pi$
$739$	−34.3992	−1.26540	−0.632698	+	0.774399i	$0.718052\pi$
$740$	0	0
$741$	−39.6481	−1.45651
$742$	0	0
$743$	22.4043	0.821935	0.410967	−	0.911650i	$-0.365191\pi$
$743$	22.4043	0.821935	0.410967	+	0.911650i	$0.365191\pi$
$744$	0	0
$745$	−87.6458	−3.21109
$746$	0	0
$747$	−47.7889	−1.74850
$748$	0	0
$749$	15.2316	0.556550
$750$	0	0
$751$	1.35711	0.0495215	0.0247608	−	0.999693i	$-0.492118\pi$
$751$	1.35711	0.0495215	0.0247608	+	0.999693i	$0.492118\pi$
$752$	0	0
$753$	−8.42530	−0.307035
$754$	0	0
$755$	9.83575	0.357960
$756$	0	0
$757$	−8.93155	−0.324623	−0.162311	−	0.986740i	$-0.551895\pi$
$757$	−8.93155	−0.324623	−0.162311	+	0.986740i	$0.551895\pi$
$758$	0	0
$759$	86.1194	3.12593
$760$	0	0
$761$	−41.3706	−1.49968	−0.749841	−	0.661618i	$-0.769870\pi$
$761$	−41.3706	−1.49968	−0.749841	+	0.661618i	$0.769870\pi$
$762$	0	0
$763$	−10.8581	−0.393090
$764$	0	0
$765$	7.67092	0.277343
$766$	0	0
$767$	64.9860	2.34651
$768$	0	0
$769$	−0.0975584	−0.00351805	−0.00175902	−	0.999998i	$-0.500560\pi$
$769$	−0.0975584	−0.00351805	−0.00175902	+	0.999998i	$0.500560\pi$
$770$	0	0
$771$	22.2038	0.799650
$772$	0	0
$773$	−15.7210	−0.565444	−0.282722	−	0.959202i	$-0.591237\pi$
$773$	−15.7210	−0.565444	−0.282722	+	0.959202i	$0.591237\pi$
$774$	0	0
$775$	28.0953	1.00921
$776$	0	0
$777$	−0.704482	−0.0252731
$778$	0	0
$779$	−10.3770	−0.371793
$780$	0	0
$781$	−0.758399	−0.0271376
$782$	0	0
$783$	−3.11925	−0.111473
$784$	0	0
$785$	−54.5647	−1.94750
$786$	0	0
$787$	21.0492	0.750325	0.375162	−	0.926959i	$-0.377587\pi$
$787$	21.0492	0.750325	0.375162	+	0.926959i	$0.377587\pi$
$788$	0	0
$789$	31.5966	1.12487
$790$	0	0
$791$	−13.2362	−0.470626
$792$	0	0
$793$	−83.8454	−2.97744
$794$	0	0
$795$	−16.3838	−0.581075
$796$	0	0
$797$	16.8336	0.596278	0.298139	−	0.954522i	$-0.403634\pi$
$797$	16.8336	0.596278	0.298139	+	0.954522i	$0.403634\pi$
$798$	0	0
$799$	−3.41544	−0.120829
$800$	0	0
$801$	35.6690	1.26030
$802$	0	0
$803$	−32.1387	−1.13415
$804$	0	0
$805$	33.4294	1.17823
$806$	0	0
$807$	−40.6126	−1.42963
$808$	0	0
$809$	−26.6262	−0.936129	−0.468064	−	0.883694i	$-0.655048\pi$
$809$	−26.6262	−0.936129	−0.468064	+	0.883694i	$0.655048\pi$
$810$	0	0
$811$	1.57627	0.0553502	0.0276751	−	0.999617i	$-0.491190\pi$
$811$	1.57627	0.0553502	0.0276751	+	0.999617i	$0.491190\pi$
$812$	0	0
$813$	79.7121	2.79563
$814$	0	0
$815$	−48.9056	−1.71309
$816$	0	0
$817$	11.0191	0.385508
$818$	0	0
$819$	−30.5099	−1.06610
$820$	0	0
$821$	47.8462	1.66984	0.834922	−	0.550369i	$-0.185513\pi$
$821$	47.8462	1.66984	0.834922	+	0.550369i	$0.185513\pi$
$822$	0	0
$823$	−48.0923	−1.67639	−0.838196	−	0.545369i	$-0.816390\pi$
$823$	−48.0923	−1.67639	−0.838196	+	0.545369i	$0.816390\pi$
$824$	0	0
$825$	164.632	5.73176
$826$	0	0
$827$	17.9261	0.623351	0.311675	−	0.950189i	$-0.399110\pi$
$827$	17.9261	0.623351	0.311675	+	0.950189i	$0.399110\pi$
$828$	0	0
$829$	25.9409	0.900966	0.450483	−	0.892785i	$-0.351252\pi$
$829$	25.9409	0.900966	0.450483	+	0.892785i	$0.351252\pi$
$830$	0	0
$831$	−15.7419	−0.546081
$832$	0	0
$833$	−2.91769	−0.101092
$834$	0	0
$835$	23.4156	0.810330
$836$	0	0
$837$	−2.48592	−0.0859261
$838$	0	0
$839$	25.9279	0.895131	0.447565	−	0.894251i	$-0.352291\pi$
$839$	25.9279	0.895131	0.447565	+	0.894251i	$0.352291\pi$
$840$	0	0
$841$	−20.6444	−0.711876
$842$	0	0
$843$	−44.5859	−1.53562
$844$	0	0
$845$	152.080	5.23172
$846$	0	0
$847$	21.9361	0.753733
$848$	0	0
$849$	−23.2980	−0.799585
$850$	0	0
$851$	1.40306	0.0480963
$852$	0	0
$853$	−56.1489	−1.92250	−0.961251	−	0.275674i	$-0.911099\pi$
$853$	−56.1489	−1.92250	−0.961251	+	0.275674i	$0.911099\pi$
$854$	0	0
$855$	−31.5250	−1.07813
$856$	0	0
$857$	−23.7374	−0.810854	−0.405427	−	0.914127i	$-0.632877\pi$
$857$	−23.7374	−0.810854	−0.405427	+	0.914127i	$0.632877\pi$
$858$	0	0
$859$	−4.48566	−0.153049	−0.0765244	−	0.997068i	$-0.524382\pi$
$859$	−4.48566	−0.153049	−0.0765244	+	0.997068i	$0.524382\pi$
$860$	0	0
$861$	−14.9782	−0.510457
$862$	0	0
$863$	−40.5560	−1.38054	−0.690271	−	0.723551i	$-0.742509\pi$
$863$	−40.5560	−1.38054	−0.690271	+	0.723551i	$0.742509\pi$
$864$	0	0
$865$	−51.8887	−1.76427
$866$	0	0
$867$	42.3541	1.43842
$868$	0	0
$869$	30.3345	1.02903
$870$	0	0
$871$	−30.3228	−1.02745
$872$	0	0
$873$	−5.83155	−0.197368
$874$	0	0
$875$	37.7059	1.27469
$876$	0	0
$877$	25.9504	0.876284	0.438142	−	0.898906i	$-0.355637\pi$
$877$	25.9504	0.876284	0.438142	+	0.898906i	$0.355637\pi$
$878$	0	0
$879$	71.9499	2.42681
$880$	0	0
$881$	−51.3322	−1.72942	−0.864712	−	0.502267i	$-0.832499\pi$
$881$	−51.3322	−1.72942	−0.864712	+	0.502267i	$0.832499\pi$
$882$	0	0
$883$	46.5521	1.56660	0.783302	−	0.621641i	$-0.213534\pi$
$883$	46.5521	1.56660	0.783302	+	0.621641i	$0.213534\pi$
$884$	0	0
$885$	96.9223	3.25801
$886$	0	0
$887$	−17.0533	−0.572593	−0.286296	−	0.958141i	$-0.592424\pi$
$887$	−17.0533	−0.572593	−0.286296	+	0.958141i	$0.592424\pi$
$888$	0	0
$889$	−23.2859	−0.780985
$890$	0	0
$891$	40.1621	1.34548
$892$	0	0
$893$	14.0363	0.469708
$894$	0	0
$895$	−42.0785	−1.40653
$896$	0	0
$897$	113.977	3.80560
$898$	0	0
$899$	6.65910	0.222093
$900$	0	0
$901$	0.841654	0.0280396
$902$	0	0
$903$	15.9050	0.529286
$904$	0	0
$905$	7.30771	0.242917
$906$	0	0
$907$	37.3415	1.23990	0.619952	−	0.784640i	$-0.287152\pi$
$907$	37.3415	1.23990	0.619952	+	0.784640i	$0.287152\pi$
$908$	0	0
$909$	−20.7622	−0.688640
$910$	0	0
$911$	−8.66574	−0.287109	−0.143554	−	0.989642i	$-0.545853\pi$
$911$	−8.66574	−0.287109	−0.143554	+	0.989642i	$0.545853\pi$
$912$	0	0
$913$	74.2892	2.45861
$914$	0	0
$915$	−125.050	−4.13402
$916$	0	0
$917$	−21.5433	−0.711421
$918$	0	0
$919$	46.5696	1.53619	0.768094	−	0.640337i	$-0.221205\pi$
$919$	46.5696	1.53619	0.768094	+	0.640337i	$0.221205\pi$
$920$	0	0
$921$	−79.7322	−2.62726
$922$	0	0
$923$	−1.00373	−0.0330381
$924$	0	0
$925$	2.68220	0.0881901
$926$	0	0
$927$	11.9728	0.393237
$928$	0	0
$929$	15.2633	0.500774	0.250387	−	0.968146i	$-0.419442\pi$
$929$	15.2633	0.500774	0.250387	+	0.968146i	$0.419442\pi$
$930$	0	0
$931$	11.9908	0.392981
$932$	0	0
$933$	−20.9739	−0.686654
$934$	0	0
$935$	−11.9247	−0.389978
$936$	0	0
$937$	−3.49055	−0.114031	−0.0570156	−	0.998373i	$-0.518158\pi$
$937$	−3.49055	−0.114031	−0.0570156	+	0.998373i	$0.518158\pi$
$938$	0	0
$939$	15.0917	0.492498
$940$	0	0
$941$	43.2231	1.40903	0.704516	−	0.709688i	$-0.251164\pi$
$941$	43.2231	1.40903	0.704516	+	0.709688i	$0.251164\pi$
$942$	0	0
$943$	29.8309	0.971429
$944$	0	0
$945$	−5.65454	−0.183942
$946$	0	0
$947$	−49.5801	−1.61114	−0.805569	−	0.592503i	$-0.798140\pi$
$947$	−49.5801	−1.61114	−0.805569	+	0.592503i	$0.798140\pi$
$948$	0	0
$949$	−42.5351	−1.38075
$950$	0	0
$951$	−52.3151	−1.69643
$952$	0	0
$953$	15.7883	0.511433	0.255716	−	0.966752i	$-0.417689\pi$
$953$	15.7883	0.511433	0.255716	+	0.966752i	$0.417689\pi$
$954$	0	0
$955$	−60.2089	−1.94832
$956$	0	0
$957$	39.0209	1.26137
$958$	0	0
$959$	0.503954	0.0162735
$960$	0	0
$961$	−25.6929	−0.828805
$962$	0	0
$963$	41.2922	1.33062
$964$	0	0
$965$	1.09448	0.0352327
$966$	0	0
$967$	21.6588	0.696499	0.348250	−	0.937402i	$-0.386776\pi$
$967$	21.6588	0.696499	0.348250	+	0.937402i	$0.386776\pi$
$968$	0	0
$969$	3.03769	0.0975848
$970$	0	0
$971$	28.9634	0.929480	0.464740	−	0.885447i	$-0.346148\pi$
$971$	28.9634	0.929480	0.464740	+	0.885447i	$0.346148\pi$
$972$	0	0
$973$	−21.2966	−0.682739
$974$	0	0
$975$	217.888	6.97800
$976$	0	0
$977$	−16.7746	−0.536668	−0.268334	−	0.963326i	$-0.586473\pi$
$977$	−16.7746	−0.536668	−0.268334	+	0.963326i	$0.586473\pi$
$978$	0	0
$979$	−55.4486	−1.77214
$980$	0	0
$981$	−29.4359	−0.939816
$982$	0	0
$983$	−24.8162	−0.791514	−0.395757	−	0.918355i	$-0.629518\pi$
$983$	−24.8162	−0.791514	−0.395757	+	0.918355i	$0.629518\pi$
$984$	0	0
$985$	−29.7063	−0.946522
$986$	0	0
$987$	20.2602	0.644890
$988$	0	0
$989$	−31.6768	−1.00726
$990$	0	0
$991$	−1.38946	−0.0441377	−0.0220688	−	0.999756i	$-0.507025\pi$
$991$	−1.38946	−0.0441377	−0.0220688	+	0.999756i	$0.507025\pi$
$992$	0	0
$993$	−21.7656	−0.690709
$994$	0	0
$995$	−32.6625	−1.03547
$996$	0	0
$997$	−18.4408	−0.584026	−0.292013	−	0.956414i	$-0.594325\pi$
$997$	−18.4408	−0.584026	−0.292013	+	0.956414i	$0.594325\pi$
$998$	0	0
$999$	−0.237326	−0.00750866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.10	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.10	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53490 q^{3} +4.14677 q^{5} +1.26365 q^{7} +3.42570 q^{9} +O(q^{10})\)	\(q-2.53490 q^{3} +4.14677 q^{5} +1.26365 q^{7} +3.42570 q^{9} -5.32535 q^{11} -7.04801 q^{13} -10.5116 q^{15} +0.539993 q^{17} -2.21920 q^{19} -3.20321 q^{21} +6.37959 q^{23} +12.1957 q^{25} -1.07910 q^{27} +2.89060 q^{29} +2.30370 q^{31} +13.4992 q^{33} +5.24006 q^{35} +0.219930 q^{37} +17.8660 q^{39} +4.67600 q^{41} -4.96533 q^{43} +14.2056 q^{45} -6.32496 q^{47} -5.40320 q^{49} -1.36883 q^{51} +1.55864 q^{53} -22.0830 q^{55} +5.62543 q^{57} -9.22048 q^{59} +11.8963 q^{61} +4.32887 q^{63} -29.2265 q^{65} +4.30232 q^{67} -16.1716 q^{69} +0.142413 q^{71} +6.03505 q^{73} -30.9149 q^{75} -6.72936 q^{77} -5.69624 q^{79} -7.54169 q^{81} -13.9501 q^{83} +2.23923 q^{85} -7.32738 q^{87} +10.4122 q^{89} -8.90619 q^{91} -5.83965 q^{93} -9.20250 q^{95} -1.70230 q^{97} -18.2430 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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