LMFDB - Embedded newform 6096.2.a.bm.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6096,2,Mod(1,6096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6096 = 2^{4} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6096.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:	$48.6768050722$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 32x^{8} + 48x^{7} + 358x^{6} - 312x^{5} - 1691x^{4} + 226x^{3} + 3044x^{2} + 1624x - 32 $ x^10 - 2x^9 - 32x^8 + 48x^7 + 358x^6 - 312x^5 - 1691x^4 + 226x^3 + 3044x^2 + 1624*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3048)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.819181$ of defining polynomial
Character	$\chi$	$=$	6096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.819181 q^{5} +3.00192 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.819181 q^{5} +3.00192 q^{7} +1.00000 q^{9} -5.86482 q^{11} -6.42798 q^{13} -0.819181 q^{15} +4.39811 q^{17} +2.60656 q^{19} +3.00192 q^{21} -3.34836 q^{23} -4.32894 q^{25} +1.00000 q^{27} -1.28618 q^{29} +8.44924 q^{31} -5.86482 q^{33} -2.45912 q^{35} +6.67849 q^{37} -6.42798 q^{39} -2.28698 q^{41} -0.110370 q^{43} -0.819181 q^{45} -0.758642 q^{47} +2.01155 q^{49} +4.39811 q^{51} +12.1610 q^{53} +4.80435 q^{55} +2.60656 q^{57} +10.8710 q^{59} +3.39305 q^{61} +3.00192 q^{63} +5.26568 q^{65} +7.59782 q^{67} -3.34836 q^{69} +9.63797 q^{71} +16.4757 q^{73} -4.32894 q^{75} -17.6057 q^{77} -3.23579 q^{79} +1.00000 q^{81} -11.1216 q^{83} -3.60285 q^{85} -1.28618 q^{87} +15.1666 q^{89} -19.2963 q^{91} +8.44924 q^{93} -2.13524 q^{95} +12.3545 q^{97} -5.86482 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} + 2 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 + 2 * q^5 - 2 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} + 2 q^{5} - 2 q^{7} + 10 q^{9} - 4 q^{11} + 12 q^{13} + 2 q^{15} + 4 q^{17} - 3 q^{19} - 2 q^{21} + 4 q^{23} + 18 q^{25} + 10 q^{27} + 9 q^{29} + 8 q^{31} - 4 q^{33} - 3 q^{35} + 30 q^{37} + 12 q^{39} - 2 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} + 4 q^{51} + 16 q^{53} + 7 q^{55} - 3 q^{57} + 10 q^{59} + 32 q^{61} - 2 q^{63} + 15 q^{65} - 8 q^{67} + 4 q^{69} + 20 q^{71} + 29 q^{73} + 18 q^{75} + 13 q^{77} + 17 q^{79} + 10 q^{81} - 12 q^{83} + 51 q^{85} + 9 q^{87} + 10 q^{89} - 30 q^{91} + 8 q^{93} + 9 q^{95} + 40 q^{97} - 4 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 + 2 * q^5 - 2 * q^7 + 10 * q^9 - 4 * q^11 + 12 * q^13 + 2 * q^15 + 4 * q^17 - 3 * q^19 - 2 * q^21 + 4 * q^23 + 18 * q^25 + 10 * q^27 + 9 * q^29 + 8 * q^31 - 4 * q^33 - 3 * q^35 + 30 * q^37 + 12 * q^39 - 2 * q^41 + 2 * q^43 + 2 * q^45 + 10 * q^47 + 14 * q^49 + 4 * q^51 + 16 * q^53 + 7 * q^55 - 3 * q^57 + 10 * q^59 + 32 * q^61 - 2 * q^63 + 15 * q^65 - 8 * q^67 + 4 * q^69 + 20 * q^71 + 29 * q^73 + 18 * q^75 + 13 * q^77 + 17 * q^79 + 10 * q^81 - 12 * q^83 + 51 * q^85 + 9 * q^87 + 10 * q^89 - 30 * q^91 + 8 * q^93 + 9 * q^95 + 40 * q^97 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.819181	−0.366349	−0.183174	−	0.983080i	$-0.558637\pi$
$5$	−0.819181	−0.366349	−0.183174	+	0.983080i	$0.558637\pi$
$6$	0	0
$7$	3.00192	1.13462	0.567310	−	0.823504i	$-0.307984\pi$
$7$	3.00192	1.13462	0.567310	+	0.823504i	$0.307984\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.86482	−1.76831	−0.884154	−	0.467195i	$-0.845265\pi$
$11$	−5.86482	−1.76831	−0.884154	+	0.467195i	$0.845265\pi$
$12$	0	0
$13$	−6.42798	−1.78280	−0.891400	−	0.453217i	$-0.850276\pi$
$13$	−6.42798	−1.78280	−0.891400	+	0.453217i	$0.850276\pi$
$14$	0	0
$15$	−0.819181	−0.211512
$16$	0	0
$17$	4.39811	1.06670	0.533349	−	0.845896i	$-0.320933\pi$
$17$	4.39811	1.06670	0.533349	+	0.845896i	$0.320933\pi$
$18$	0	0
$19$	2.60656	0.597986	0.298993	−	0.954255i	$-0.403349\pi$
$19$	2.60656	0.597986	0.298993	+	0.954255i	$0.403349\pi$
$20$	0	0
$21$	3.00192	0.655074
$22$	0	0
$23$	−3.34836	−0.698181	−0.349091	−	0.937089i	$-0.613509\pi$
$23$	−3.34836	−0.698181	−0.349091	+	0.937089i	$0.613509\pi$
$24$	0	0
$25$	−4.32894	−0.865788
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.28618	−0.238837	−0.119419	−	0.992844i	$-0.538103\pi$
$29$	−1.28618	−0.238837	−0.119419	+	0.992844i	$0.538103\pi$
$30$	0	0
$31$	8.44924	1.51753	0.758764	−	0.651366i	$-0.225804\pi$
$31$	8.44924	1.51753	0.758764	+	0.651366i	$0.225804\pi$
$32$	0	0
$33$	−5.86482	−1.02093
$34$	0	0
$35$	−2.45912	−0.415667
$36$	0	0
$37$	6.67849	1.09794	0.548968	−	0.835843i	$-0.315021\pi$
$37$	6.67849	1.09794	0.548968	+	0.835843i	$0.315021\pi$
$38$	0	0
$39$	−6.42798	−1.02930
$40$	0	0
$41$	−2.28698	−0.357166	−0.178583	−	0.983925i	$-0.557151\pi$
$41$	−2.28698	−0.357166	−0.178583	+	0.983925i	$0.557151\pi$
$42$	0	0
$43$	−0.110370	−0.0168313	−0.00841566	−	0.999965i	$-0.502679\pi$
$43$	−0.110370	−0.0168313	−0.00841566	+	0.999965i	$0.502679\pi$
$44$	0	0
$45$	−0.819181	−0.122116
$46$	0	0
$47$	−0.758642	−0.110659	−0.0553297	−	0.998468i	$-0.517621\pi$
$47$	−0.758642	−0.110659	−0.0553297	+	0.998468i	$0.517621\pi$
$48$	0	0
$49$	2.01155	0.287364
$50$	0	0
$51$	4.39811	0.615858
$52$	0	0
$53$	12.1610	1.67044	0.835220	−	0.549916i	$-0.185340\pi$
$53$	12.1610	1.67044	0.835220	+	0.549916i	$0.185340\pi$
$54$	0	0
$55$	4.80435	0.647818
$56$	0	0
$57$	2.60656	0.345247
$58$	0	0
$59$	10.8710	1.41528	0.707642	−	0.706571i	$-0.249759\pi$
$59$	10.8710	1.41528	0.707642	+	0.706571i	$0.249759\pi$
$60$	0	0
$61$	3.39305	0.434436	0.217218	−	0.976123i	$-0.430302\pi$
$61$	3.39305	0.434436	0.217218	+	0.976123i	$0.430302\pi$
$62$	0	0
$63$	3.00192	0.378207
$64$	0	0
$65$	5.26568	0.653127
$66$	0	0
$67$	7.59782	0.928221	0.464111	−	0.885777i	$-0.346374\pi$
$67$	7.59782	0.928221	0.464111	+	0.885777i	$0.346374\pi$
$68$	0	0
$69$	−3.34836	−0.403095
$70$	0	0
$71$	9.63797	1.14382	0.571908	−	0.820318i	$-0.306203\pi$
$71$	9.63797	1.14382	0.571908	+	0.820318i	$0.306203\pi$
$72$	0	0
$73$	16.4757	1.92834	0.964169	−	0.265290i	$-0.0854675\pi$
$73$	16.4757	1.92834	0.964169	+	0.265290i	$0.0854675\pi$
$74$	0	0
$75$	−4.32894	−0.499863
$76$	0	0
$77$	−17.6057	−2.00636
$78$	0	0
$79$	−3.23579	−0.364054	−0.182027	−	0.983293i	$-0.558266\pi$
$79$	−3.23579	−0.364054	−0.182027	+	0.983293i	$0.558266\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.1216	−1.22075	−0.610377	−	0.792111i	$-0.708982\pi$
$83$	−11.1216	−1.22075	−0.610377	+	0.792111i	$0.708982\pi$
$84$	0	0
$85$	−3.60285	−0.390783
$86$	0	0
$87$	−1.28618	−0.137893
$88$	0	0
$89$	15.1666	1.60765	0.803827	−	0.594863i	$-0.202794\pi$
$89$	15.1666	1.60765	0.803827	+	0.594863i	$0.202794\pi$
$90$	0	0
$91$	−19.2963	−2.02280
$92$	0	0
$93$	8.44924	0.876145
$94$	0	0
$95$	−2.13524	−0.219071
$96$	0	0
$97$	12.3545	1.25441	0.627204	−	0.778855i	$-0.284199\pi$
$97$	12.3545	1.25441	0.627204	+	0.778855i	$0.284199\pi$
$98$	0	0
$99$	−5.86482	−0.589436
$100$	0	0
$101$	−1.34836	−0.134167	−0.0670833	−	0.997747i	$-0.521369\pi$
$101$	−1.34836	−0.134167	−0.0670833	+	0.997747i	$0.521369\pi$
$102$	0	0
$103$	−1.60473	−0.158119	−0.0790594	−	0.996870i	$-0.525192\pi$
$103$	−1.60473	−0.158119	−0.0790594	+	0.996870i	$0.525192\pi$
$104$	0	0
$105$	−2.45912	−0.239986
$106$	0	0
$107$	−12.7416	−1.23177	−0.615887	−	0.787835i	$-0.711202\pi$
$107$	−12.7416	−1.23177	−0.615887	+	0.787835i	$0.711202\pi$
$108$	0	0
$109$	13.2485	1.26897	0.634487	−	0.772934i	$-0.281211\pi$
$109$	13.2485	1.26897	0.634487	+	0.772934i	$0.281211\pi$
$110$	0	0
$111$	6.67849	0.633894
$112$	0	0
$113$	−12.2405	−1.15149	−0.575744	−	0.817630i	$-0.695288\pi$
$113$	−12.2405	−1.15149	−0.575744	+	0.817630i	$0.695288\pi$
$114$	0	0
$115$	2.74291	0.255778
$116$	0	0
$117$	−6.42798	−0.594267
$118$	0	0
$119$	13.2028	1.21030
$120$	0	0
$121$	23.3961	2.12692
$122$	0	0
$123$	−2.28698	−0.206210
$124$	0	0
$125$	7.64209	0.683530
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	−0.110370	−0.00971757
$130$	0	0
$131$	1.57089	0.137250	0.0686248	−	0.997643i	$-0.478139\pi$
$131$	1.57089	0.137250	0.0686248	+	0.997643i	$0.478139\pi$
$132$	0	0
$133$	7.82469	0.678487
$134$	0	0
$135$	−0.819181	−0.0705039
$136$	0	0
$137$	16.4088	1.40190	0.700948	−	0.713212i	$-0.252760\pi$
$137$	16.4088	1.40190	0.700948	+	0.713212i	$0.252760\pi$
$138$	0	0
$139$	−1.21715	−0.103237	−0.0516185	−	0.998667i	$-0.516438\pi$
$139$	−1.21715	−0.103237	−0.0516185	+	0.998667i	$0.516438\pi$
$140$	0	0
$141$	−0.758642	−0.0638892
$142$	0	0
$143$	37.6989	3.15254
$144$	0	0
$145$	1.05361	0.0874978
$146$	0	0
$147$	2.01155	0.165910
$148$	0	0
$149$	2.39915	0.196546	0.0982731	−	0.995159i	$-0.468668\pi$
$149$	2.39915	0.196546	0.0982731	+	0.995159i	$0.468668\pi$
$150$	0	0
$151$	10.2935	0.837676	0.418838	−	0.908061i	$-0.362437\pi$
$151$	10.2935	0.837676	0.418838	+	0.908061i	$0.362437\pi$
$152$	0	0
$153$	4.39811	0.355566
$154$	0	0
$155$	−6.92146	−0.555945
$156$	0	0
$157$	12.8146	1.02272	0.511359	−	0.859367i	$-0.329142\pi$
$157$	12.8146	1.02272	0.511359	+	0.859367i	$0.329142\pi$
$158$	0	0
$159$	12.1610	0.964429
$160$	0	0
$161$	−10.0515	−0.792171
$162$	0	0
$163$	−9.04978	−0.708833	−0.354417	−	0.935088i	$-0.615321\pi$
$163$	−9.04978	−0.708833	−0.354417	+	0.935088i	$0.615321\pi$
$164$	0	0
$165$	4.80435	0.374018
$166$	0	0
$167$	−11.9387	−0.923844	−0.461922	−	0.886921i	$-0.652840\pi$
$167$	−11.9387	−0.923844	−0.461922	+	0.886921i	$0.652840\pi$
$168$	0	0
$169$	28.3189	2.17838
$170$	0	0
$171$	2.60656	0.199329
$172$	0	0
$173$	−7.89769	−0.600450	−0.300225	−	0.953868i	$-0.597062\pi$
$173$	−7.89769	−0.600450	−0.300225	+	0.953868i	$0.597062\pi$
$174$	0	0
$175$	−12.9952	−0.982342
$176$	0	0
$177$	10.8710	0.817114
$178$	0	0
$179$	−8.17803	−0.611255	−0.305627	−	0.952151i	$-0.598866\pi$
$179$	−8.17803	−0.611255	−0.305627	+	0.952151i	$0.598866\pi$
$180$	0	0
$181$	−9.85676	−0.732647	−0.366323	−	0.930488i	$-0.619384\pi$
$181$	−9.85676	−0.732647	−0.366323	+	0.930488i	$0.619384\pi$
$182$	0	0
$183$	3.39305	0.250822
$184$	0	0
$185$	−5.47089	−0.402228
$186$	0	0
$187$	−25.7941	−1.88625
$188$	0	0
$189$	3.00192	0.218358
$190$	0	0
$191$	3.86883	0.279939	0.139969	−	0.990156i	$-0.455300\pi$
$191$	3.86883	0.279939	0.139969	+	0.990156i	$0.455300\pi$
$192$	0	0
$193$	−20.1923	−1.45347	−0.726737	−	0.686916i	$-0.758964\pi$
$193$	−20.1923	−1.45347	−0.726737	+	0.686916i	$0.758964\pi$
$194$	0	0
$195$	5.26568	0.377083
$196$	0	0
$197$	9.01100	0.642007	0.321004	−	0.947078i	$-0.395980\pi$
$197$	9.01100	0.642007	0.321004	+	0.947078i	$0.395980\pi$
$198$	0	0
$199$	20.9535	1.48536	0.742678	−	0.669649i	$-0.233556\pi$
$199$	20.9535	1.48536	0.742678	+	0.669649i	$0.233556\pi$
$200$	0	0
$201$	7.59782	0.535909
$202$	0	0
$203$	−3.86101	−0.270990
$204$	0	0
$205$	1.87345	0.130847
$206$	0	0
$207$	−3.34836	−0.232727
$208$	0	0
$209$	−15.2870	−1.05742
$210$	0	0
$211$	−24.4188	−1.68106	−0.840529	−	0.541767i	$-0.817755\pi$
$211$	−24.4188	−1.68106	−0.840529	+	0.541767i	$0.817755\pi$
$212$	0	0
$213$	9.63797	0.660383
$214$	0	0
$215$	0.0904133	0.00616614
$216$	0	0
$217$	25.3640	1.72182
$218$	0	0
$219$	16.4757	1.11333
$220$	0	0
$221$	−28.2709	−1.90171
$222$	0	0
$223$	4.50624	0.301760	0.150880	−	0.988552i	$-0.451789\pi$
$223$	4.50624	0.301760	0.150880	+	0.988552i	$0.451789\pi$
$224$	0	0
$225$	−4.32894	−0.288596
$226$	0	0
$227$	−10.4658	−0.694640	−0.347320	−	0.937747i	$-0.612908\pi$
$227$	−10.4658	−0.694640	−0.347320	+	0.937747i	$0.612908\pi$
$228$	0	0
$229$	16.7010	1.10363	0.551817	−	0.833965i	$-0.313935\pi$
$229$	16.7010	1.10363	0.551817	+	0.833965i	$0.313935\pi$
$230$	0	0
$231$	−17.6057	−1.15837
$232$	0	0
$233$	−4.05582	−0.265706	−0.132853	−	0.991136i	$-0.542414\pi$
$233$	−4.05582	−0.265706	−0.132853	+	0.991136i	$0.542414\pi$
$234$	0	0
$235$	0.621466	0.0405399
$236$	0	0
$237$	−3.23579	−0.210187
$238$	0	0
$239$	−21.3032	−1.37799	−0.688995	−	0.724766i	$-0.741948\pi$
$239$	−21.3032	−1.37799	−0.688995	+	0.724766i	$0.741948\pi$
$240$	0	0
$241$	−14.6887	−0.946179	−0.473090	−	0.881014i	$-0.656861\pi$
$241$	−14.6887	−0.946179	−0.473090	+	0.881014i	$0.656861\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.64782	−0.105276
$246$	0	0
$247$	−16.7549	−1.06609
$248$	0	0
$249$	−11.1216	−0.704803
$250$	0	0
$251$	−30.8851	−1.94945	−0.974725	−	0.223406i	$-0.928282\pi$
$251$	−30.8851	−1.94945	−0.974725	+	0.223406i	$0.928282\pi$
$252$	0	0
$253$	19.6375	1.23460
$254$	0	0
$255$	−3.60285	−0.225619
$256$	0	0
$257$	23.5796	1.47085	0.735427	−	0.677603i	$-0.236981\pi$
$257$	23.5796	1.47085	0.735427	+	0.677603i	$0.236981\pi$
$258$	0	0
$259$	20.0483	1.24574
$260$	0	0
$261$	−1.28618	−0.0796124
$262$	0	0
$263$	−10.6931	−0.659364	−0.329682	−	0.944092i	$-0.606942\pi$
$263$	−10.6931	−0.659364	−0.329682	+	0.944092i	$0.606942\pi$
$264$	0	0
$265$	−9.96205	−0.611964
$266$	0	0
$267$	15.1666	0.928179
$268$	0	0
$269$	−20.1765	−1.23018	−0.615091	−	0.788456i	$-0.710881\pi$
$269$	−20.1765	−1.23018	−0.615091	+	0.788456i	$0.710881\pi$
$270$	0	0
$271$	13.7884	0.837588	0.418794	−	0.908081i	$-0.362453\pi$
$271$	13.7884	0.837588	0.418794	+	0.908081i	$0.362453\pi$
$272$	0	0
$273$	−19.2963	−1.16787
$274$	0	0
$275$	25.3885	1.53098
$276$	0	0
$277$	4.71059	0.283032	0.141516	−	0.989936i	$-0.454802\pi$
$277$	4.71059	0.283032	0.141516	+	0.989936i	$0.454802\pi$
$278$	0	0
$279$	8.44924	0.505843
$280$	0	0
$281$	−9.30986	−0.555380	−0.277690	−	0.960671i	$-0.589569\pi$
$281$	−9.30986	−0.555380	−0.277690	+	0.960671i	$0.589569\pi$
$282$	0	0
$283$	1.80749	0.107444	0.0537219	−	0.998556i	$-0.482892\pi$
$283$	1.80749	0.107444	0.0537219	+	0.998556i	$0.482892\pi$
$284$	0	0
$285$	−2.13524	−0.126481
$286$	0	0
$287$	−6.86533	−0.405248
$288$	0	0
$289$	2.34333	0.137843
$290$	0	0
$291$	12.3545	0.724233
$292$	0	0
$293$	−10.4660	−0.611432	−0.305716	−	0.952123i	$-0.598896\pi$
$293$	−10.4660	−0.611432	−0.305716	+	0.952123i	$0.598896\pi$
$294$	0	0
$295$	−8.90532	−0.518488
$296$	0	0
$297$	−5.86482	−0.340311
$298$	0	0
$299$	21.5232	1.24472
$300$	0	0
$301$	−0.331323	−0.0190972
$302$	0	0
$303$	−1.34836	−0.0774612
$304$	0	0
$305$	−2.77953	−0.159155
$306$	0	0
$307$	27.6039	1.57544	0.787718	−	0.616036i	$-0.211262\pi$
$307$	27.6039	1.57544	0.787718	+	0.616036i	$0.211262\pi$
$308$	0	0
$309$	−1.60473	−0.0912899
$310$	0	0
$311$	4.09446	0.232176	0.116088	−	0.993239i	$-0.462965\pi$
$311$	4.09446	0.232176	0.116088	+	0.993239i	$0.462965\pi$
$312$	0	0
$313$	−9.68712	−0.547548	−0.273774	−	0.961794i	$-0.588272\pi$
$313$	−9.68712	−0.547548	−0.273774	+	0.961794i	$0.588272\pi$
$314$	0	0
$315$	−2.45912	−0.138556
$316$	0	0
$317$	30.7287	1.72590	0.862948	−	0.505292i	$-0.168615\pi$
$317$	30.7287	1.72590	0.862948	+	0.505292i	$0.168615\pi$
$318$	0	0
$319$	7.54320	0.422338
$320$	0	0
$321$	−12.7416	−0.711165
$322$	0	0
$323$	11.4639	0.637870
$324$	0	0
$325$	27.8264	1.54353
$326$	0	0
$327$	13.2485	0.732642
$328$	0	0
$329$	−2.27739	−0.125556
$330$	0	0
$331$	0.0268906	0.00147804	0.000739020	−	1.00000i	$-0.499765\pi$
$331$	0.0268906	0.00147804	0.000739020		1.00000i	$0.499765\pi$
$332$	0	0
$333$	6.67849	0.365979
$334$	0	0
$335$	−6.22399	−0.340053
$336$	0	0
$337$	30.7074	1.67274	0.836369	−	0.548167i	$-0.184674\pi$
$337$	30.7074	1.67274	0.836369	+	0.548167i	$0.184674\pi$
$338$	0	0
$339$	−12.2405	−0.664812
$340$	0	0
$341$	−49.5532	−2.68346
$342$	0	0
$343$	−14.9749	−0.808571
$344$	0	0
$345$	2.74291	0.147673
$346$	0	0
$347$	−22.7508	−1.22133	−0.610663	−	0.791891i	$-0.709097\pi$
$347$	−22.7508	−1.22133	−0.610663	+	0.791891i	$0.709097\pi$
$348$	0	0
$349$	−26.7873	−1.43389	−0.716946	−	0.697128i	$-0.754461\pi$
$349$	−26.7873	−1.43389	−0.716946	+	0.697128i	$0.754461\pi$
$350$	0	0
$351$	−6.42798	−0.343100
$352$	0	0
$353$	9.39368	0.499975	0.249988	−	0.968249i	$-0.419573\pi$
$353$	9.39368	0.499975	0.249988	+	0.968249i	$0.419573\pi$
$354$	0	0
$355$	−7.89524	−0.419036
$356$	0	0
$357$	13.2028	0.698765
$358$	0	0
$359$	−9.13900	−0.482338	−0.241169	−	0.970483i	$-0.577531\pi$
$359$	−9.13900	−0.482338	−0.241169	+	0.970483i	$0.577531\pi$
$360$	0	0
$361$	−12.2059	−0.642413
$362$	0	0
$363$	23.3961	1.22798
$364$	0	0
$365$	−13.4966	−0.706444
$366$	0	0
$367$	28.7213	1.49924	0.749621	−	0.661868i	$-0.230236\pi$
$367$	28.7213	1.49924	0.749621	+	0.661868i	$0.230236\pi$
$368$	0	0
$369$	−2.28698	−0.119055
$370$	0	0
$371$	36.5063	1.89532
$372$	0	0
$373$	−14.7872	−0.765655	−0.382827	−	0.923820i	$-0.625050\pi$
$373$	−14.7872	−0.765655	−0.382827	+	0.923820i	$0.625050\pi$
$374$	0	0
$375$	7.64209	0.394636
$376$	0	0
$377$	8.26752	0.425799
$378$	0	0
$379$	−9.72204	−0.499387	−0.249694	−	0.968325i	$-0.580330\pi$
$379$	−9.72204	−0.499387	−0.249694	+	0.968325i	$0.580330\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	−16.9441	−0.865805	−0.432903	−	0.901441i	$-0.642511\pi$
$383$	−16.9441	−0.865805	−0.432903	+	0.901441i	$0.642511\pi$
$384$	0	0
$385$	14.4223	0.735028
$386$	0	0
$387$	−0.110370	−0.00561044
$388$	0	0
$389$	22.9489	1.16356	0.581779	−	0.813347i	$-0.302357\pi$
$389$	22.9489	1.16356	0.581779	+	0.813347i	$0.302357\pi$
$390$	0	0
$391$	−14.7264	−0.744748
$392$	0	0
$393$	1.57089	0.0792411
$394$	0	0
$395$	2.65070	0.133371
$396$	0	0
$397$	15.4423	0.775027	0.387513	−	0.921864i	$-0.373334\pi$
$397$	15.4423	0.775027	0.387513	+	0.921864i	$0.373334\pi$
$398$	0	0
$399$	7.82469	0.391725
$400$	0	0
$401$	−20.9415	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$401$	−20.9415	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$402$	0	0
$403$	−54.3115	−2.70545
$404$	0	0
$405$	−0.819181	−0.0407054
$406$	0	0
$407$	−39.1681	−1.94149
$408$	0	0
$409$	30.4468	1.50549	0.752747	−	0.658309i	$-0.228728\pi$
$409$	30.4468	1.50549	0.752747	+	0.658309i	$0.228728\pi$
$410$	0	0
$411$	16.4088	0.809385
$412$	0	0
$413$	32.6339	1.60581
$414$	0	0
$415$	9.11060	0.447222
$416$	0	0
$417$	−1.21715	−0.0596039
$418$	0	0
$419$	31.0975	1.51921	0.759607	−	0.650382i	$-0.225391\pi$
$419$	31.0975	1.51921	0.759607	+	0.650382i	$0.225391\pi$
$420$	0	0
$421$	−14.2813	−0.696028	−0.348014	−	0.937489i	$-0.613144\pi$
$421$	−14.2813	−0.696028	−0.348014	+	0.937489i	$0.613144\pi$
$422$	0	0
$423$	−0.758642	−0.0368865
$424$	0	0
$425$	−19.0391	−0.923534
$426$	0	0
$427$	10.1857	0.492920
$428$	0	0
$429$	37.6989	1.82012
$430$	0	0
$431$	34.2232	1.64848	0.824238	−	0.566244i	$-0.191604\pi$
$431$	34.2232	1.64848	0.824238	+	0.566244i	$0.191604\pi$
$432$	0	0
$433$	13.0159	0.625504	0.312752	−	0.949835i	$-0.398749\pi$
$433$	13.0159	0.625504	0.312752	+	0.949835i	$0.398749\pi$
$434$	0	0
$435$	1.05361	0.0505169
$436$	0	0
$437$	−8.72769	−0.417502
$438$	0	0
$439$	−2.11252	−0.100825	−0.0504126	−	0.998728i	$-0.516054\pi$
$439$	−2.11252	−0.100825	−0.0504126	+	0.998728i	$0.516054\pi$
$440$	0	0
$441$	2.01155	0.0957881
$442$	0	0
$443$	−29.2901	−1.39161	−0.695806	−	0.718229i	$-0.744953\pi$
$443$	−29.2901	−1.39161	−0.695806	+	0.718229i	$0.744953\pi$
$444$	0	0
$445$	−12.4242	−0.588962
$446$	0	0
$447$	2.39915	0.113476
$448$	0	0
$449$	34.8080	1.64269	0.821345	−	0.570432i	$-0.193224\pi$
$449$	34.8080	1.64269	0.821345	+	0.570432i	$0.193224\pi$
$450$	0	0
$451$	13.4127	0.631579
$452$	0	0
$453$	10.2935	0.483633
$454$	0	0
$455$	15.8072	0.741052
$456$	0	0
$457$	−24.0722	−1.12605	−0.563025	−	0.826440i	$-0.690363\pi$
$457$	−24.0722	−1.12605	−0.563025	+	0.826440i	$0.690363\pi$
$458$	0	0
$459$	4.39811	0.205286
$460$	0	0
$461$	33.7868	1.57361	0.786804	−	0.617204i	$-0.211735\pi$
$461$	33.7868	1.57361	0.786804	+	0.617204i	$0.211735\pi$
$462$	0	0
$463$	22.8895	1.06376	0.531882	−	0.846819i	$-0.321485\pi$
$463$	22.8895	1.06376	0.531882	+	0.846819i	$0.321485\pi$
$464$	0	0
$465$	−6.92146	−0.320975
$466$	0	0
$467$	−11.4191	−0.528414	−0.264207	−	0.964466i	$-0.585110\pi$
$467$	−11.4191	−0.528414	−0.264207	+	0.964466i	$0.585110\pi$
$468$	0	0
$469$	22.8081	1.05318
$470$	0	0
$471$	12.8146	0.590467
$472$	0	0
$473$	0.647302	0.0297630
$474$	0	0
$475$	−11.2836	−0.517729
$476$	0	0
$477$	12.1610	0.556813
$478$	0	0
$479$	12.6490	0.577946	0.288973	−	0.957337i	$-0.406686\pi$
$479$	12.6490	0.577946	0.288973	+	0.957337i	$0.406686\pi$
$480$	0	0
$481$	−42.9292	−1.95740
$482$	0	0
$483$	−10.0515	−0.457360
$484$	0	0
$485$	−10.1206	−0.459551
$486$	0	0
$487$	−18.0084	−0.816039	−0.408019	−	0.912973i	$-0.633780\pi$
$487$	−18.0084	−0.816039	−0.408019	+	0.912973i	$0.633780\pi$
$488$	0	0
$489$	−9.04978	−0.409245
$490$	0	0
$491$	−5.65947	−0.255408	−0.127704	−	0.991812i	$-0.540761\pi$
$491$	−5.65947	−0.255408	−0.127704	+	0.991812i	$0.540761\pi$
$492$	0	0
$493$	−5.65674	−0.254767
$494$	0	0
$495$	4.80435	0.215939
$496$	0	0
$497$	28.9324	1.29780
$498$	0	0
$499$	34.6962	1.55322	0.776608	−	0.629984i	$-0.216939\pi$
$499$	34.6962	1.55322	0.776608	+	0.629984i	$0.216939\pi$
$500$	0	0
$501$	−11.9387	−0.533382
$502$	0	0
$503$	33.9527	1.51387	0.756937	−	0.653487i	$-0.226695\pi$
$503$	33.9527	1.51387	0.756937	+	0.653487i	$0.226695\pi$
$504$	0	0
$505$	1.10455	0.0491518
$506$	0	0
$507$	28.3189	1.25769
$508$	0	0
$509$	5.46147	0.242075	0.121038	−	0.992648i	$-0.461378\pi$
$509$	5.46147	0.242075	0.121038	+	0.992648i	$0.461378\pi$
$510$	0	0
$511$	49.4589	2.18793
$512$	0	0
$513$	2.60656	0.115082
$514$	0	0
$515$	1.31456	0.0579266
$516$	0	0
$517$	4.44930	0.195680
$518$	0	0
$519$	−7.89769	−0.346670
$520$	0	0
$521$	41.8197	1.83216	0.916078	−	0.401001i	$-0.131338\pi$
$521$	41.8197	1.83216	0.916078	+	0.401001i	$0.131338\pi$
$522$	0	0
$523$	−18.3587	−0.802769	−0.401385	−	0.915910i	$-0.631471\pi$
$523$	−18.3587	−0.802769	−0.401385	+	0.915910i	$0.631471\pi$
$524$	0	0
$525$	−12.9952	−0.567155
$526$	0	0
$527$	37.1606	1.61874
$528$	0	0
$529$	−11.7885	−0.512543
$530$	0	0
$531$	10.8710	0.471761
$532$	0	0
$533$	14.7006	0.636755
$534$	0	0
$535$	10.4376	0.451259
$536$	0	0
$537$	−8.17803	−0.352908
$538$	0	0
$539$	−11.7974	−0.508149
$540$	0	0
$541$	35.4080	1.52231	0.761154	−	0.648572i	$-0.224633\pi$
$541$	35.4080	1.52231	0.761154	+	0.648572i	$0.224633\pi$
$542$	0	0
$543$	−9.85676	−0.422994
$544$	0	0
$545$	−10.8529	−0.464887
$546$	0	0
$547$	12.6294	0.539993	0.269996	−	0.962861i	$-0.412977\pi$
$547$	12.6294	0.539993	0.269996	+	0.962861i	$0.412977\pi$
$548$	0	0
$549$	3.39305	0.144812
$550$	0	0
$551$	−3.35250	−0.142821
$552$	0	0
$553$	−9.71359	−0.413064
$554$	0	0
$555$	−5.47089	−0.232226
$556$	0	0
$557$	−15.4603	−0.655073	−0.327536	−	0.944839i	$-0.606218\pi$
$557$	−15.4603	−0.655073	−0.327536	+	0.944839i	$0.606218\pi$
$558$	0	0
$559$	0.709458	0.0300069
$560$	0	0
$561$	−25.7941	−1.08903
$562$	0	0
$563$	9.24989	0.389836	0.194918	−	0.980820i	$-0.437556\pi$
$563$	9.24989	0.389836	0.194918	+	0.980820i	$0.437556\pi$
$564$	0	0
$565$	10.0272	0.421847
$566$	0	0
$567$	3.00192	0.126069
$568$	0	0
$569$	−22.8524	−0.958024	−0.479012	−	0.877808i	$-0.659005\pi$
$569$	−22.8524	−0.958024	−0.479012	+	0.877808i	$0.659005\pi$
$570$	0	0
$571$	12.2459	0.512476	0.256238	−	0.966614i	$-0.417517\pi$
$571$	12.2459	0.512476	0.256238	+	0.966614i	$0.417517\pi$
$572$	0	0
$573$	3.86883	0.161623
$574$	0	0
$575$	14.4949	0.604477
$576$	0	0
$577$	−10.0666	−0.419079	−0.209540	−	0.977800i	$-0.567197\pi$
$577$	−10.0666	−0.419079	−0.209540	+	0.977800i	$0.567197\pi$
$578$	0	0
$579$	−20.1923	−0.839163
$580$	0	0
$581$	−33.3862	−1.38509
$582$	0	0
$583$	−71.3219	−2.95385
$584$	0	0
$585$	5.26568	0.217709
$586$	0	0
$587$	21.6408	0.893210	0.446605	−	0.894731i	$-0.352633\pi$
$587$	21.6408	0.893210	0.446605	+	0.894731i	$0.352633\pi$
$588$	0	0
$589$	22.0234	0.907460
$590$	0	0
$591$	9.01100	0.370663
$592$	0	0
$593$	−34.5598	−1.41920	−0.709599	−	0.704605i	$-0.751124\pi$
$593$	−34.5598	−1.41920	−0.709599	+	0.704605i	$0.751124\pi$
$594$	0	0
$595$	−10.8155	−0.443391
$596$	0	0
$597$	20.9535	0.857570
$598$	0	0
$599$	−11.2305	−0.458864	−0.229432	−	0.973325i	$-0.573687\pi$
$599$	−11.2305	−0.458864	−0.229432	+	0.973325i	$0.573687\pi$
$600$	0	0
$601$	−24.6608	−1.00594	−0.502968	−	0.864305i	$-0.667759\pi$
$601$	−24.6608	−1.00594	−0.502968	+	0.864305i	$0.667759\pi$
$602$	0	0
$603$	7.59782	0.309407
$604$	0	0
$605$	−19.1656	−0.779193
$606$	0	0
$607$	19.6322	0.796847	0.398423	−	0.917202i	$-0.369557\pi$
$607$	19.6322	0.796847	0.398423	+	0.917202i	$0.369557\pi$
$608$	0	0
$609$	−3.86101	−0.156456
$610$	0	0
$611$	4.87654	0.197284
$612$	0	0
$613$	22.0468	0.890463	0.445232	−	0.895415i	$-0.353121\pi$
$613$	22.0468	0.890463	0.445232	+	0.895415i	$0.353121\pi$
$614$	0	0
$615$	1.87345	0.0755447
$616$	0	0
$617$	0.751100	0.0302381	0.0151191	−	0.999886i	$-0.495187\pi$
$617$	0.751100	0.0302381	0.0151191	+	0.999886i	$0.495187\pi$
$618$	0	0
$619$	24.3563	0.978961	0.489480	−	0.872014i	$-0.337186\pi$
$619$	24.3563	0.978961	0.489480	+	0.872014i	$0.337186\pi$
$620$	0	0
$621$	−3.34836	−0.134365
$622$	0	0
$623$	45.5289	1.82408
$624$	0	0
$625$	15.3845	0.615378
$626$	0	0
$627$	−15.2870	−0.610504
$628$	0	0
$629$	29.3727	1.17117
$630$	0	0
$631$	−38.8560	−1.54683	−0.773417	−	0.633898i	$-0.781454\pi$
$631$	−38.8560	−1.54683	−0.773417	+	0.633898i	$0.781454\pi$
$632$	0	0
$633$	−24.4188	−0.970559
$634$	0	0
$635$	0.819181	0.0325082
$636$	0	0
$637$	−12.9302	−0.512313
$638$	0	0
$639$	9.63797	0.381272
$640$	0	0
$641$	6.08970	0.240529	0.120264	−	0.992742i	$-0.461626\pi$
$641$	6.08970	0.240529	0.120264	+	0.992742i	$0.461626\pi$
$642$	0	0
$643$	−24.5149	−0.966773	−0.483387	−	0.875407i	$-0.660593\pi$
$643$	−24.5149	−0.966773	−0.483387	+	0.875407i	$0.660593\pi$
$644$	0	0
$645$	0.0904133	0.00356002
$646$	0	0
$647$	17.7165	0.696507	0.348254	−	0.937400i	$-0.386775\pi$
$647$	17.7165	0.696507	0.348254	+	0.937400i	$0.386775\pi$
$648$	0	0
$649$	−63.7564	−2.50266
$650$	0	0
$651$	25.3640	0.994092
$652$	0	0
$653$	12.2159	0.478044	0.239022	−	0.971014i	$-0.423173\pi$
$653$	12.2159	0.478044	0.239022	+	0.971014i	$0.423173\pi$
$654$	0	0
$655$	−1.28685	−0.0502813
$656$	0	0
$657$	16.4757	0.642779
$658$	0	0
$659$	0.869781	0.0338819	0.0169409	−	0.999856i	$-0.494607\pi$
$659$	0.869781	0.0338819	0.0169409	+	0.999856i	$0.494607\pi$
$660$	0	0
$661$	−23.7213	−0.922652	−0.461326	−	0.887231i	$-0.652626\pi$
$661$	−23.7213	−0.922652	−0.461326	+	0.887231i	$0.652626\pi$
$662$	0	0
$663$	−28.2709	−1.09795
$664$	0	0
$665$	−6.40984	−0.248563
$666$	0	0
$667$	4.30658	0.166752
$668$	0	0
$669$	4.50624	0.174221
$670$	0	0
$671$	−19.8996	−0.768217
$672$	0	0
$673$	−44.9102	−1.73116	−0.865580	−	0.500771i	$-0.833050\pi$
$673$	−44.9102	−1.73116	−0.865580	+	0.500771i	$0.833050\pi$
$674$	0	0
$675$	−4.32894	−0.166621
$676$	0	0
$677$	−45.8920	−1.76377	−0.881886	−	0.471462i	$-0.843726\pi$
$677$	−45.8920	−1.76377	−0.881886	+	0.471462i	$0.843726\pi$
$678$	0	0
$679$	37.0872	1.42328
$680$	0	0
$681$	−10.4658	−0.401050
$682$	0	0
$683$	16.1211	0.616855	0.308428	−	0.951248i	$-0.400197\pi$
$683$	16.1211	0.616855	0.308428	+	0.951248i	$0.400197\pi$
$684$	0	0
$685$	−13.4418	−0.513583
$686$	0	0
$687$	16.7010	0.637183
$688$	0	0
$689$	−78.1705	−2.97806
$690$	0	0
$691$	−17.2823	−0.657449	−0.328724	−	0.944426i	$-0.606619\pi$
$691$	−17.2823	−0.657449	−0.328724	+	0.944426i	$0.606619\pi$
$692$	0	0
$693$	−17.6057	−0.668787
$694$	0	0
$695$	0.997062	0.0378207
$696$	0	0
$697$	−10.0584	−0.380988
$698$	0	0
$699$	−4.05582	−0.153405
$700$	0	0
$701$	−12.8547	−0.485514	−0.242757	−	0.970087i	$-0.578052\pi$
$701$	−12.8547	−0.485514	−0.242757	+	0.970087i	$0.578052\pi$
$702$	0	0
$703$	17.4079	0.656550
$704$	0	0
$705$	0.621466	0.0234057
$706$	0	0
$707$	−4.04767	−0.152228
$708$	0	0
$709$	16.0091	0.601235	0.300618	−	0.953745i	$-0.402807\pi$
$709$	16.0091	0.601235	0.300618	+	0.953745i	$0.402807\pi$
$710$	0	0
$711$	−3.23579	−0.121351
$712$	0	0
$713$	−28.2911	−1.05951
$714$	0	0
$715$	−30.8822	−1.15493
$716$	0	0
$717$	−21.3032	−0.795583
$718$	0	0
$719$	13.5041	0.503618	0.251809	−	0.967777i	$-0.418975\pi$
$719$	13.5041	0.503618	0.251809	+	0.967777i	$0.418975\pi$
$720$	0	0
$721$	−4.81728	−0.179405
$722$	0	0
$723$	−14.6887	−0.546277
$724$	0	0
$725$	5.56779	0.206782
$726$	0	0
$727$	16.8178	0.623737	0.311869	−	0.950125i	$-0.399045\pi$
$727$	16.8178	0.623737	0.311869	+	0.950125i	$0.399045\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.485420	−0.0179539
$732$	0	0
$733$	35.9628	1.32832	0.664158	−	0.747592i	$-0.268790\pi$
$733$	35.9628	1.32832	0.664158	+	0.747592i	$0.268790\pi$
$734$	0	0
$735$	−1.64782	−0.0607809
$736$	0	0
$737$	−44.5598	−1.64138
$738$	0	0
$739$	−29.6987	−1.09248	−0.546242	−	0.837627i	$-0.683942\pi$
$739$	−29.6987	−1.09248	−0.546242	+	0.837627i	$0.683942\pi$
$740$	0	0
$741$	−16.7549	−0.615507
$742$	0	0
$743$	9.72723	0.356858	0.178429	−	0.983953i	$-0.442899\pi$
$743$	9.72723	0.356858	0.178429	+	0.983953i	$0.442899\pi$
$744$	0	0
$745$	−1.96534	−0.0720045
$746$	0	0
$747$	−11.1216	−0.406918
$748$	0	0
$749$	−38.2492	−1.39760
$750$	0	0
$751$	−7.97671	−0.291074	−0.145537	−	0.989353i	$-0.546491\pi$
$751$	−7.97671	−0.291074	−0.145537	+	0.989353i	$0.546491\pi$
$752$	0	0
$753$	−30.8851	−1.12552
$754$	0	0
$755$	−8.43227	−0.306882
$756$	0	0
$757$	14.0637	0.511153	0.255577	−	0.966789i	$-0.417735\pi$
$757$	14.0637	0.511153	0.255577	+	0.966789i	$0.417735\pi$
$758$	0	0
$759$	19.6375	0.712796
$760$	0	0
$761$	−0.0566492	−0.00205353	−0.00102677	−	0.999999i	$-0.500327\pi$
$761$	−0.0566492	−0.00205353	−0.00102677	+	0.999999i	$0.500327\pi$
$762$	0	0
$763$	39.7709	1.43980
$764$	0	0
$765$	−3.60285	−0.130261
$766$	0	0
$767$	−69.8786	−2.52317
$768$	0	0
$769$	37.5644	1.35461	0.677304	−	0.735703i	$-0.263148\pi$
$769$	37.5644	1.35461	0.677304	+	0.735703i	$0.263148\pi$
$770$	0	0
$771$	23.5796	0.849198
$772$	0	0
$773$	2.75785	0.0991930	0.0495965	−	0.998769i	$-0.484206\pi$
$773$	2.75785	0.0991930	0.0495965	+	0.998769i	$0.484206\pi$
$774$	0	0
$775$	−36.5763	−1.31386
$776$	0	0
$777$	20.0483	0.719229
$778$	0	0
$779$	−5.96114	−0.213580
$780$	0	0
$781$	−56.5249	−2.02262
$782$	0	0
$783$	−1.28618	−0.0459642
$784$	0	0
$785$	−10.4975	−0.374672
$786$	0	0
$787$	−6.88351	−0.245371	−0.122685	−	0.992446i	$-0.539151\pi$
$787$	−6.88351	−0.245371	−0.122685	+	0.992446i	$0.539151\pi$
$788$	0	0
$789$	−10.6931	−0.380684
$790$	0	0
$791$	−36.7450	−1.30650
$792$	0	0
$793$	−21.8105	−0.774513
$794$	0	0
$795$	−9.96205	−0.353317
$796$	0	0
$797$	−14.3258	−0.507445	−0.253723	−	0.967277i	$-0.581655\pi$
$797$	−14.3258	−0.507445	−0.253723	+	0.967277i	$0.581655\pi$
$798$	0	0
$799$	−3.33659	−0.118040
$800$	0	0
$801$	15.1666	0.535885
$802$	0	0
$803$	−96.6271	−3.40990
$804$	0	0
$805$	8.23402	0.290211
$806$	0	0
$807$	−20.1765	−0.710246
$808$	0	0
$809$	−31.5037	−1.10761	−0.553806	−	0.832646i	$-0.686825\pi$
$809$	−31.5037	−1.10761	−0.553806	+	0.832646i	$0.686825\pi$
$810$	0	0
$811$	−7.87689	−0.276595	−0.138298	−	0.990391i	$-0.544163\pi$
$811$	−7.87689	−0.276595	−0.138298	+	0.990391i	$0.544163\pi$
$812$	0	0
$813$	13.7884	0.483582
$814$	0	0
$815$	7.41341	0.259680
$816$	0	0
$817$	−0.287687	−0.0100649
$818$	0	0
$819$	−19.2963	−0.674268
$820$	0	0
$821$	−28.7577	−1.00365	−0.501825	−	0.864969i	$-0.667338\pi$
$821$	−28.7577	−1.00365	−0.501825	+	0.864969i	$0.667338\pi$
$822$	0	0
$823$	49.8328	1.73706	0.868530	−	0.495636i	$-0.165065\pi$
$823$	49.8328	1.73706	0.868530	+	0.495636i	$0.165065\pi$
$824$	0	0
$825$	25.3885	0.883912
$826$	0	0
$827$	−33.5452	−1.16648	−0.583240	−	0.812300i	$-0.698215\pi$
$827$	−33.5452	−1.16648	−0.583240	+	0.812300i	$0.698215\pi$
$828$	0	0
$829$	44.4712	1.54455	0.772274	−	0.635289i	$-0.219119\pi$
$829$	44.4712	1.54455	0.772274	+	0.635289i	$0.219119\pi$
$830$	0	0
$831$	4.71059	0.163409
$832$	0	0
$833$	8.84701	0.306531
$834$	0	0
$835$	9.77996	0.338449
$836$	0	0
$837$	8.44924	0.292048
$838$	0	0
$839$	20.0594	0.692527	0.346263	−	0.938137i	$-0.387450\pi$
$839$	20.0594	0.692527	0.346263	+	0.938137i	$0.387450\pi$
$840$	0	0
$841$	−27.3457	−0.942957
$842$	0	0
$843$	−9.30986	−0.320649
$844$	0	0
$845$	−23.1983	−0.798047
$846$	0	0
$847$	70.2332	2.41324
$848$	0	0
$849$	1.80749	0.0620327
$850$	0	0
$851$	−22.3620	−0.766559
$852$	0	0
$853$	2.23180	0.0764154	0.0382077	−	0.999270i	$-0.487835\pi$
$853$	2.23180	0.0764154	0.0382077	+	0.999270i	$0.487835\pi$
$854$	0	0
$855$	−2.13524	−0.0730238
$856$	0	0
$857$	55.6382	1.90056	0.950282	−	0.311392i	$-0.100795\pi$
$857$	55.6382	1.90056	0.950282	+	0.311392i	$0.100795\pi$
$858$	0	0
$859$	−42.2943	−1.44306	−0.721532	−	0.692381i	$-0.756562\pi$
$859$	−42.2943	−1.44306	−0.721532	+	0.692381i	$0.756562\pi$
$860$	0	0
$861$	−6.86533	−0.233970
$862$	0	0
$863$	−1.37763	−0.0468951	−0.0234475	−	0.999725i	$-0.507464\pi$
$863$	−1.37763	−0.0468951	−0.0234475	+	0.999725i	$0.507464\pi$
$864$	0	0
$865$	6.46964	0.219974
$866$	0	0
$867$	2.34333	0.0795837
$868$	0	0
$869$	18.9773	0.643761
$870$	0	0
$871$	−48.8386	−1.65483
$872$	0	0
$873$	12.3545	0.418136
$874$	0	0
$875$	22.9410	0.775547
$876$	0	0
$877$	49.2776	1.66399	0.831994	−	0.554785i	$-0.187199\pi$
$877$	49.2776	1.66399	0.831994	+	0.554785i	$0.187199\pi$
$878$	0	0
$879$	−10.4660	−0.353011
$880$	0	0
$881$	−15.9031	−0.535788	−0.267894	−	0.963448i	$-0.586328\pi$
$881$	−15.9031	−0.535788	−0.267894	+	0.963448i	$0.586328\pi$
$882$	0	0
$883$	2.18765	0.0736201	0.0368101	−	0.999322i	$-0.488280\pi$
$883$	2.18765	0.0736201	0.0368101	+	0.999322i	$0.488280\pi$
$884$	0	0
$885$	−8.90532	−0.299349
$886$	0	0
$887$	20.6930	0.694802	0.347401	−	0.937717i	$-0.387064\pi$
$887$	20.6930	0.694802	0.347401	+	0.937717i	$0.387064\pi$
$888$	0	0
$889$	−3.00192	−0.100681
$890$	0	0
$891$	−5.86482	−0.196479
$892$	0	0
$893$	−1.97745	−0.0661727
$894$	0	0
$895$	6.69929	0.223933
$896$	0	0
$897$	21.5232	0.718638
$898$	0	0
$899$	−10.8672	−0.362442
$900$	0	0
$901$	53.4853	1.78185
$902$	0	0
$903$	−0.331323	−0.0110258
$904$	0	0
$905$	8.07447	0.268404
$906$	0	0
$907$	1.95664	0.0649691	0.0324846	−	0.999472i	$-0.489658\pi$
$907$	1.95664	0.0649691	0.0324846	+	0.999472i	$0.489658\pi$
$908$	0	0
$909$	−1.34836	−0.0447222
$910$	0	0
$911$	18.4212	0.610320	0.305160	−	0.952301i	$-0.401290\pi$
$911$	18.4212	0.610320	0.305160	+	0.952301i	$0.401290\pi$
$912$	0	0
$913$	65.2261	2.15867
$914$	0	0
$915$	−2.77953	−0.0918883
$916$	0	0
$917$	4.71570	0.155726
$918$	0	0
$919$	−8.68813	−0.286595	−0.143298	−	0.989680i	$-0.545771\pi$
$919$	−8.68813	−0.286595	−0.143298	+	0.989680i	$0.545771\pi$
$920$	0	0
$921$	27.6039	0.909578
$922$	0	0
$923$	−61.9526	−2.03920
$924$	0	0
$925$	−28.9108	−0.950581
$926$	0	0
$927$	−1.60473	−0.0527062
$928$	0	0
$929$	15.3430	0.503389	0.251694	−	0.967807i	$-0.419012\pi$
$929$	15.3430	0.503389	0.251694	+	0.967807i	$0.419012\pi$
$930$	0	0
$931$	5.24322	0.171840
$932$	0	0
$933$	4.09446	0.134047
$934$	0	0
$935$	21.1300	0.691026
$936$	0	0
$937$	−7.53438	−0.246137	−0.123069	−	0.992398i	$-0.539274\pi$
$937$	−7.53438	−0.246137	−0.123069	+	0.992398i	$0.539274\pi$
$938$	0	0
$939$	−9.68712	−0.316127
$940$	0	0
$941$	42.4975	1.38538	0.692690	−	0.721236i	$-0.256426\pi$
$941$	42.4975	1.38538	0.692690	+	0.721236i	$0.256426\pi$
$942$	0	0
$943$	7.65762	0.249366
$944$	0	0
$945$	−2.45912	−0.0799952
$946$	0	0
$947$	20.2333	0.657495	0.328747	−	0.944418i	$-0.393374\pi$
$947$	20.2333	0.657495	0.328747	+	0.944418i	$0.393374\pi$
$948$	0	0
$949$	−105.906	−3.43784
$950$	0	0
$951$	30.7287	0.996447
$952$	0	0
$953$	−10.8350	−0.350981	−0.175491	−	0.984481i	$-0.556151\pi$
$953$	−10.8350	−0.350981	−0.175491	+	0.984481i	$0.556151\pi$
$954$	0	0
$955$	−3.16927	−0.102555
$956$	0	0
$957$	7.54320	0.243837
$958$	0	0
$959$	49.2579	1.59062
$960$	0	0
$961$	40.3896	1.30289
$962$	0	0
$963$	−12.7416	−0.410591
$964$	0	0
$965$	16.5412	0.532479
$966$	0	0
$967$	−12.6344	−0.406295	−0.203148	−	0.979148i	$-0.565117\pi$
$967$	−12.6344	−0.406295	−0.203148	+	0.979148i	$0.565117\pi$
$968$	0	0
$969$	11.4639	0.368274
$970$	0	0
$971$	20.1346	0.646150	0.323075	−	0.946373i	$-0.395283\pi$
$971$	20.1346	0.646150	0.323075	+	0.946373i	$0.395283\pi$
$972$	0	0
$973$	−3.65378	−0.117135
$974$	0	0
$975$	27.8264	0.891157
$976$	0	0
$977$	−25.8531	−0.827114	−0.413557	−	0.910478i	$-0.635714\pi$
$977$	−25.8531	−0.827114	−0.413557	+	0.910478i	$0.635714\pi$
$978$	0	0
$979$	−88.9492	−2.84283
$980$	0	0
$981$	13.2485	0.422991
$982$	0	0
$983$	8.49580	0.270974	0.135487	−	0.990779i	$-0.456740\pi$
$983$	8.49580	0.270974	0.135487	+	0.990779i	$0.456740\pi$
$984$	0	0
$985$	−7.38164	−0.235199
$986$	0	0
$987$	−2.27739	−0.0724900
$988$	0	0
$989$	0.369559	0.0117513
$990$	0	0
$991$	38.5504	1.22459	0.612296	−	0.790628i	$-0.290246\pi$
$991$	38.5504	1.22459	0.612296	+	0.790628i	$0.290246\pi$
$992$	0	0
$993$	0.0268906	0.000853347 0
$994$	0	0
$995$	−17.1647	−0.544158
$996$	0	0
$997$	29.7808	0.943168	0.471584	−	0.881821i	$-0.343682\pi$
$997$	29.7808	0.943168	0.471584	+	0.881821i	$0.343682\pi$
$998$	0	0
$999$	6.67849	0.211298

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.bm.1.5		10
4.3	odd	2		3048.2.a.m.1.5	✓	10
12.11	even	2		9144.2.a.ba.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3048.2.a.m.1.5	✓	10	4.3	odd	2
6096.2.a.bm.1.5		10	1.1	even	1	trivial
9144.2.a.ba.1.6		10	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.819181 q^{5} +3.00192 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.819181 q^{5} +3.00192 q^{7} +1.00000 q^{9} -5.86482 q^{11} -6.42798 q^{13} -0.819181 q^{15} +4.39811 q^{17} +2.60656 q^{19} +3.00192 q^{21} -3.34836 q^{23} -4.32894 q^{25} +1.00000 q^{27} -1.28618 q^{29} +8.44924 q^{31} -5.86482 q^{33} -2.45912 q^{35} +6.67849 q^{37} -6.42798 q^{39} -2.28698 q^{41} -0.110370 q^{43} -0.819181 q^{45} -0.758642 q^{47} +2.01155 q^{49} +4.39811 q^{51} +12.1610 q^{53} +4.80435 q^{55} +2.60656 q^{57} +10.8710 q^{59} +3.39305 q^{61} +3.00192 q^{63} +5.26568 q^{65} +7.59782 q^{67} -3.34836 q^{69} +9.63797 q^{71} +16.4757 q^{73} -4.32894 q^{75} -17.6057 q^{77} -3.23579 q^{79} +1.00000 q^{81} -11.1216 q^{83} -3.60285 q^{85} -1.28618 q^{87} +15.1666 q^{89} -19.2963 q^{91} +8.44924 q^{93} -2.13524 q^{95} +12.3545 q^{97} -5.86482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 2 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 + 2 * q^5 - 2 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} + 2 q^{5} - 2 q^{7} + 10 q^{9} - 4 q^{11} + 12 q^{13} + 2 q^{15} + 4 q^{17} - 3 q^{19} - 2 q^{21} + 4 q^{23} + 18 q^{25} + 10 q^{27} + 9 q^{29} + 8 q^{31} - 4 q^{33} - 3 q^{35} + 30 q^{37} + 12 q^{39} - 2 q^{41} + 2 q^{43} + 2 q^{45} + 10 q^{47} + 14 q^{49} + 4 q^{51} + 16 q^{53} + 7 q^{55} - 3 q^{57} + 10 q^{59} + 32 q^{61} - 2 q^{63} + 15 q^{65} - 8 q^{67} + 4 q^{69} + 20 q^{71} + 29 q^{73} + 18 q^{75} + 13 q^{77} + 17 q^{79} + 10 q^{81} - 12 q^{83} + 51 q^{85} + 9 q^{87} + 10 q^{89} - 30 q^{91} + 8 q^{93} + 9 q^{95} + 40 q^{97} - 4 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 + 2 * q^5 - 2 * q^7 + 10 * q^9 - 4 * q^11 + 12 * q^13 + 2 * q^15 + 4 * q^17 - 3 * q^19 - 2 * q^21 + 4 * q^23 + 18 * q^25 + 10 * q^27 + 9 * q^29 + 8 * q^31 - 4 * q^33 - 3 * q^35 + 30 * q^37 + 12 * q^39 - 2 * q^41 + 2 * q^43 + 2 * q^45 + 10 * q^47 + 14 * q^49 + 4 * q^51 + 16 * q^53 + 7 * q^55 - 3 * q^57 + 10 * q^59 + 32 * q^61 - 2 * q^63 + 15 * q^65 - 8 * q^67 + 4 * q^69 + 20 * q^71 + 29 * q^73 + 18 * q^75 + 13 * q^77 + 17 * q^79 + 10 * q^81 - 12 * q^83 + 51 * q^85 + 9 * q^87 + 10 * q^89 - 30 * q^91 + 8 * q^93 + 9 * q^95 + 40 * q^97 - 4 * q^99

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