LMFDB - Embedded newform 7728.2.a.cg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
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} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 $ x^6 - x^5 - 13x^4 + 7x^3 + 31x^2 - 17x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.174924$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.39875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.39875 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.51973 q^{11} -5.51804 q^{13} +4.39875 q^{15} +3.11930 q^{17} +5.05863 q^{19} -1.00000 q^{21} -1.00000 q^{23} +14.3490 q^{25} -1.00000 q^{27} -6.66406 q^{29} -1.11930 q^{31} +3.51973 q^{33} -4.39875 q^{35} -6.66406 q^{37} +5.51804 q^{39} -10.6034 q^{41} -6.81835 q^{43} -4.39875 q^{45} -3.43403 q^{47} +1.00000 q^{49} -3.11930 q^{51} -9.54307 q^{53} +15.4824 q^{55} -5.05863 q^{57} -11.5431 q^{59} +13.9627 q^{61} +1.00000 q^{63} +24.2725 q^{65} -6.11258 q^{67} +1.00000 q^{69} -8.02334 q^{71} -2.95022 q^{73} -14.3490 q^{75} -3.51973 q^{77} -13.2171 q^{79} +1.00000 q^{81} +9.91679 q^{83} -13.7210 q^{85} +6.66406 q^{87} -4.31305 q^{89} -5.51804 q^{91} +1.11930 q^{93} -22.2516 q^{95} +16.2030 q^{97} -3.51973 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 - 5 * q^11 + 2 * q^13 + 10 * q^17 - 3 * q^19 - 6 * q^21 - 6 * q^23 + 18 * q^25 - 6 * q^27 + 3 * q^29 + 2 * q^31 + 5 * q^33 + 3 * q^37 - 2 * q^39 + 4 * q^41 - 6 * q^43 + 2 * q^47 + 6 * q^49 - 10 * q^51 - 4 * q^53 + 15 * q^55 + 3 * q^57 - 16 * q^59 + 22 * q^61 + 6 * q^63 + 35 * q^65 - 9 * q^67 + 6 * q^69 - 11 * q^71 + 24 * q^73 - 18 * q^75 - 5 * q^77 - 18 * q^79 + 6 * q^81 - 2 * q^83 + 13 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 - 2 * q^93 - 5 * q^95 + 37 * q^97 - 5 * 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
$5$	−4.39875	−1.96718	−0.983590	−	0.180421i	$-0.942254\pi$
$5$	−4.39875	−1.96718	−0.983590	+	0.180421i	$0.942254\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.51973	−1.06124	−0.530619	−	0.847610i	$-0.678041\pi$
$11$	−3.51973	−1.06124	−0.530619	+	0.847610i	$0.678041\pi$
$12$	0	0
$13$	−5.51804	−1.53043	−0.765215	−	0.643775i	$-0.777367\pi$
$13$	−5.51804	−1.53043	−0.765215	+	0.643775i	$0.777367\pi$
$14$	0	0
$15$	4.39875	1.13575
$16$	0	0
$17$	3.11930	0.756541	0.378270	−	0.925695i	$-0.376519\pi$
$17$	3.11930	0.756541	0.378270	+	0.925695i	$0.376519\pi$
$18$	0	0
$19$	5.05863	1.16053	0.580264	−	0.814428i	$-0.302949\pi$
$19$	5.05863	1.16053	0.580264	+	0.814428i	$0.302949\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	14.3490	2.86979
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.66406	−1.23748	−0.618742	−	0.785594i	$-0.712357\pi$
$29$	−6.66406	−1.23748	−0.618742	+	0.785594i	$0.712357\pi$
$30$	0	0
$31$	−1.11930	−0.201032	−0.100516	−	0.994935i	$-0.532049\pi$
$31$	−1.11930	−0.201032	−0.100516	+	0.994935i	$0.532049\pi$
$32$	0	0
$33$	3.51973	0.612706
$34$	0	0
$35$	−4.39875	−0.743524
$36$	0	0
$37$	−6.66406	−1.09556	−0.547782	−	0.836621i	$-0.684528\pi$
$37$	−6.66406	−1.09556	−0.547782	+	0.836621i	$0.684528\pi$
$38$	0	0
$39$	5.51804	0.883594
$40$	0	0
$41$	−10.6034	−1.65597	−0.827985	−	0.560750i	$-0.810513\pi$
$41$	−10.6034	−1.65597	−0.827985	+	0.560750i	$0.810513\pi$
$42$	0	0
$43$	−6.81835	−1.03979	−0.519894	−	0.854231i	$-0.674029\pi$
$43$	−6.81835	−1.03979	−0.519894	+	0.854231i	$0.674029\pi$
$44$	0	0
$45$	−4.39875	−0.655726
$46$	0	0
$47$	−3.43403	−0.500905	−0.250453	−	0.968129i	$-0.580579\pi$
$47$	−3.43403	−0.500905	−0.250453	+	0.968129i	$0.580579\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.11930	−0.436789
$52$	0	0
$53$	−9.54307	−1.31084	−0.655421	−	0.755264i	$-0.727509\pi$
$53$	−9.54307	−1.31084	−0.655421	+	0.755264i	$0.727509\pi$
$54$	0	0
$55$	15.4824	2.08765
$56$	0	0
$57$	−5.05863	−0.670032
$58$	0	0
$59$	−11.5431	−1.50278	−0.751390	−	0.659858i	$-0.770616\pi$
$59$	−11.5431	−1.50278	−0.751390	+	0.659858i	$0.770616\pi$
$60$	0	0
$61$	13.9627	1.78774	0.893868	−	0.448329i	$-0.147981\pi$
$61$	13.9627	1.78774	0.893868	+	0.448329i	$0.147981\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	24.2725	3.01063
$66$	0	0
$67$	−6.11258	−0.746771	−0.373385	−	0.927676i	$-0.621803\pi$
$67$	−6.11258	−0.746771	−0.373385	+	0.927676i	$0.621803\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.02334	−0.952195	−0.476098	−	0.879392i	$-0.657949\pi$
$71$	−8.02334	−0.952195	−0.476098	+	0.879392i	$0.657949\pi$
$72$	0	0
$73$	−2.95022	−0.345297	−0.172649	−	0.984983i	$-0.555233\pi$
$73$	−2.95022	−0.345297	−0.172649	+	0.984983i	$0.555233\pi$
$74$	0	0
$75$	−14.3490	−1.65688
$76$	0	0
$77$	−3.51973	−0.401111
$78$	0	0
$79$	−13.2171	−1.48704	−0.743519	−	0.668714i	$-0.766845\pi$
$79$	−13.2171	−1.48704	−0.743519	+	0.668714i	$0.766845\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.91679	1.08851	0.544255	−	0.838920i	$-0.316813\pi$
$83$	9.91679	1.08851	0.544255	+	0.838920i	$0.316813\pi$
$84$	0	0
$85$	−13.7210	−1.48825
$86$	0	0
$87$	6.66406	0.714462
$88$	0	0
$89$	−4.31305	−0.457182	−0.228591	−	0.973523i	$-0.573412\pi$
$89$	−4.31305	−0.457182	−0.228591	+	0.973523i	$0.573412\pi$
$90$	0	0
$91$	−5.51804	−0.578448
$92$	0	0
$93$	1.11930	0.116066
$94$	0	0
$95$	−22.2516	−2.28297
$96$	0	0
$97$	16.2030	1.64516	0.822580	−	0.568649i	$-0.192534\pi$
$97$	16.2030	1.64516	0.822580	+	0.568649i	$0.192534\pi$
$98$	0	0
$99$	−3.51973	−0.353746
$100$	0	0
$101$	1.44528	0.143811	0.0719053	−	0.997411i	$-0.477092\pi$
$101$	1.44528	0.143811	0.0719053	+	0.997411i	$0.477092\pi$
$102$	0	0
$103$	−10.9024	−1.07424	−0.537121	−	0.843505i	$-0.680488\pi$
$103$	−10.9024	−1.07424	−0.537121	+	0.843505i	$0.680488\pi$
$104$	0	0
$105$	4.39875	0.429274
$106$	0	0
$107$	−8.77415	−0.848229	−0.424115	−	0.905609i	$-0.639415\pi$
$107$	−8.77415	−0.848229	−0.424115	+	0.905609i	$0.639415\pi$
$108$	0	0
$109$	−7.47654	−0.716123	−0.358061	−	0.933698i	$-0.616562\pi$
$109$	−7.47654	−0.716123	−0.358061	+	0.933698i	$0.616562\pi$
$110$	0	0
$111$	6.66406	0.632524
$112$	0	0
$113$	6.90759	0.649811	0.324906	−	0.945746i	$-0.394667\pi$
$113$	6.90759	0.649811	0.324906	+	0.945746i	$0.394667\pi$
$114$	0	0
$115$	4.39875	0.410185
$116$	0	0
$117$	−5.51804	−0.510143
$118$	0	0
$119$	3.11930	0.285946
$120$	0	0
$121$	1.38851	0.126228
$122$	0	0
$123$	10.6034	0.956075
$124$	0	0
$125$	−41.1237	−3.67822
$126$	0	0
$127$	−17.5269	−1.55526	−0.777630	−	0.628722i	$-0.783578\pi$
$127$	−17.5269	−1.55526	−0.777630	+	0.628722i	$0.783578\pi$
$128$	0	0
$129$	6.81835	0.600322
$130$	0	0
$131$	9.00752	0.786990	0.393495	−	0.919327i	$-0.371266\pi$
$131$	9.00752	0.786990	0.393495	+	0.919327i	$0.371266\pi$
$132$	0	0
$133$	5.05863	0.438639
$134$	0	0
$135$	4.39875	0.378584
$136$	0	0
$137$	−2.72473	−0.232789	−0.116395	−	0.993203i	$-0.537134\pi$
$137$	−2.72473	−0.232789	−0.116395	+	0.993203i	$0.537134\pi$
$138$	0	0
$139$	5.43821	0.461263	0.230631	−	0.973041i	$-0.425921\pi$
$139$	5.43821	0.461263	0.230631	+	0.973041i	$0.425921\pi$
$140$	0	0
$141$	3.43403	0.289198
$142$	0	0
$143$	19.4220	1.62415
$144$	0	0
$145$	29.3135	2.43435
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	12.9788	1.06326	0.531632	−	0.846975i	$-0.321579\pi$
$149$	12.9788	1.06326	0.531632	+	0.846975i	$0.321579\pi$
$150$	0	0
$151$	−22.7393	−1.85050	−0.925249	−	0.379359i	$-0.876144\pi$
$151$	−22.7393	−1.85050	−0.925249	+	0.379359i	$0.876144\pi$
$152$	0	0
$153$	3.11930	0.252180
$154$	0	0
$155$	4.92350	0.395465
$156$	0	0
$157$	13.9393	1.11248	0.556240	−	0.831022i	$-0.312244\pi$
$157$	13.9393	1.11248	0.556240	+	0.831022i	$0.312244\pi$
$158$	0	0
$159$	9.54307	0.756815
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	1.82337	0.142818	0.0714088	−	0.997447i	$-0.477251\pi$
$163$	1.82337	0.142818	0.0714088	+	0.997447i	$0.477251\pi$
$164$	0	0
$165$	−15.4824	−1.20530
$166$	0	0
$167$	9.05863	0.700978	0.350489	−	0.936567i	$-0.386015\pi$
$167$	9.05863	0.700978	0.350489	+	0.936567i	$0.386015\pi$
$168$	0	0
$169$	17.4488	1.34222
$170$	0	0
$171$	5.05863	0.386843
$172$	0	0
$173$	8.71766	0.662791	0.331396	−	0.943492i	$-0.392481\pi$
$173$	8.71766	0.662791	0.331396	+	0.943492i	$0.392481\pi$
$174$	0	0
$175$	14.3490	1.08468
$176$	0	0
$177$	11.5431	0.867630
$178$	0	0
$179$	−1.59788	−0.119431	−0.0597155	−	0.998215i	$-0.519019\pi$
$179$	−1.59788	−0.119431	−0.0597155	+	0.998215i	$0.519019\pi$
$180$	0	0
$181$	16.7083	1.24191	0.620957	−	0.783844i	$-0.286744\pi$
$181$	16.7083	1.24191	0.620957	+	0.783844i	$0.286744\pi$
$182$	0	0
$183$	−13.9627	−1.03215
$184$	0	0
$185$	29.3135	2.15517
$186$	0	0
$187$	−10.9791	−0.802870
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	9.78971	0.708359	0.354179	−	0.935177i	$-0.384760\pi$
$191$	9.78971	0.708359	0.354179	+	0.935177i	$0.384760\pi$
$192$	0	0
$193$	6.13373	0.441516	0.220758	−	0.975329i	$-0.429147\pi$
$193$	6.13373	0.441516	0.220758	+	0.975329i	$0.429147\pi$
$194$	0	0
$195$	−24.2725	−1.73819
$196$	0	0
$197$	24.9157	1.77517	0.887584	−	0.460646i	$-0.152382\pi$
$197$	24.9157	1.77517	0.887584	+	0.460646i	$0.152382\pi$
$198$	0	0
$199$	−17.9499	−1.27243	−0.636216	−	0.771511i	$-0.719501\pi$
$199$	−17.9499	−1.27243	−0.636216	+	0.771511i	$0.719501\pi$
$200$	0	0
$201$	6.11258	0.431148
$202$	0	0
$203$	−6.66406	−0.467725
$204$	0	0
$205$	46.6416	3.25759
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−17.8050	−1.23160
$210$	0	0
$211$	3.51973	0.242308	0.121154	−	0.992634i	$-0.461340\pi$
$211$	3.51973	0.242308	0.121154	+	0.992634i	$0.461340\pi$
$212$	0	0
$213$	8.02334	0.549750
$214$	0	0
$215$	29.9922	2.04545
$216$	0	0
$217$	−1.11930	−0.0759829
$218$	0	0
$219$	2.95022	0.199357
$220$	0	0
$221$	−17.2124	−1.15783
$222$	0	0
$223$	15.1709	1.01592	0.507960	−	0.861381i	$-0.330400\pi$
$223$	15.1709	1.01592	0.507960	+	0.861381i	$0.330400\pi$
$224$	0	0
$225$	14.3490	0.956598
$226$	0	0
$227$	17.6039	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$227$	17.6039	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$228$	0	0
$229$	−14.4746	−0.956510	−0.478255	−	0.878221i	$-0.658731\pi$
$229$	−14.4746	−0.956510	−0.478255	+	0.878221i	$0.658731\pi$
$230$	0	0
$231$	3.51973	0.231581
$232$	0	0
$233$	−26.5474	−1.73918	−0.869588	−	0.493779i	$-0.835615\pi$
$233$	−26.5474	−1.73918	−0.869588	+	0.493779i	$0.835615\pi$
$234$	0	0
$235$	15.1054	0.985370
$236$	0	0
$237$	13.2171	0.858542
$238$	0	0
$239$	−4.35118	−0.281454	−0.140727	−	0.990048i	$-0.544944\pi$
$239$	−4.35118	−0.281454	−0.140727	+	0.990048i	$0.544944\pi$
$240$	0	0
$241$	20.0505	1.29157	0.645784	−	0.763520i	$-0.276531\pi$
$241$	20.0505	1.29157	0.645784	+	0.763520i	$0.276531\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.39875	−0.281026
$246$	0	0
$247$	−27.9137	−1.77611
$248$	0	0
$249$	−9.91679	−0.628451
$250$	0	0
$251$	−14.1426	−0.892672	−0.446336	−	0.894866i	$-0.647271\pi$
$251$	−14.1426	−0.892672	−0.446336	+	0.894866i	$0.647271\pi$
$252$	0	0
$253$	3.51973	0.221284
$254$	0	0
$255$	13.7210	0.859242
$256$	0	0
$257$	−22.8292	−1.42404	−0.712022	−	0.702157i	$-0.752221\pi$
$257$	−22.8292	−1.42404	−0.712022	+	0.702157i	$0.752221\pi$
$258$	0	0
$259$	−6.66406	−0.414084
$260$	0	0
$261$	−6.66406	−0.412495
$262$	0	0
$263$	−7.95143	−0.490306	−0.245153	−	0.969484i	$-0.578838\pi$
$263$	−7.95143	−0.490306	−0.245153	+	0.969484i	$0.578838\pi$
$264$	0	0
$265$	41.9776	2.57866
$266$	0	0
$267$	4.31305	0.263954
$268$	0	0
$269$	2.07445	0.126482	0.0632408	−	0.997998i	$-0.479856\pi$
$269$	2.07445	0.126482	0.0632408	+	0.997998i	$0.479856\pi$
$270$	0	0
$271$	18.7177	1.13702	0.568508	−	0.822678i	$-0.307521\pi$
$271$	18.7177	1.13702	0.568508	+	0.822678i	$0.307521\pi$
$272$	0	0
$273$	5.51804	0.333967
$274$	0	0
$275$	−50.5045	−3.04554
$276$	0	0
$277$	5.07342	0.304832	0.152416	−	0.988316i	$-0.451295\pi$
$277$	5.07342	0.304832	0.152416	+	0.988316i	$0.451295\pi$
$278$	0	0
$279$	−1.11930	−0.0670106
$280$	0	0
$281$	13.6688	0.815410	0.407705	−	0.913114i	$-0.366329\pi$
$281$	13.6688	0.815410	0.407705	+	0.913114i	$0.366329\pi$
$282$	0	0
$283$	−28.8011	−1.71205	−0.856024	−	0.516936i	$-0.827073\pi$
$283$	−28.8011	−1.71205	−0.856024	+	0.516936i	$0.827073\pi$
$284$	0	0
$285$	22.2516	1.31807
$286$	0	0
$287$	−10.6034	−0.625898
$288$	0	0
$289$	−7.26998	−0.427646
$290$	0	0
$291$	−16.2030	−0.949834
$292$	0	0
$293$	2.03833	0.119081	0.0595403	−	0.998226i	$-0.481037\pi$
$293$	2.03833	0.119081	0.0595403	+	0.998226i	$0.481037\pi$
$294$	0	0
$295$	50.7750	2.95624
$296$	0	0
$297$	3.51973	0.204235
$298$	0	0
$299$	5.51804	0.319117
$300$	0	0
$301$	−6.81835	−0.393003
$302$	0	0
$303$	−1.44528	−0.0830290
$304$	0	0
$305$	−61.4182	−3.51680
$306$	0	0
$307$	5.85313	0.334056	0.167028	−	0.985952i	$-0.446583\pi$
$307$	5.85313	0.334056	0.167028	+	0.985952i	$0.446583\pi$
$308$	0	0
$309$	10.9024	0.620213
$310$	0	0
$311$	0.298169	0.0169076	0.00845381	−	0.999964i	$-0.497309\pi$
$311$	0.298169	0.0169076	0.00845381	+	0.999964i	$0.497309\pi$
$312$	0	0
$313$	−23.7991	−1.34521	−0.672603	−	0.740004i	$-0.734824\pi$
$313$	−23.7991	−1.34521	−0.672603	+	0.740004i	$0.734824\pi$
$314$	0	0
$315$	−4.39875	−0.247841
$316$	0	0
$317$	2.90759	0.163306	0.0816532	−	0.996661i	$-0.473980\pi$
$317$	2.90759	0.163306	0.0816532	+	0.996661i	$0.473980\pi$
$318$	0	0
$319$	23.4557	1.31327
$320$	0	0
$321$	8.77415	0.489725
$322$	0	0
$323$	15.7794	0.877987
$324$	0	0
$325$	−79.1782	−4.39202
$326$	0	0
$327$	7.47654	0.413454
$328$	0	0
$329$	−3.43403	−0.189324
$330$	0	0
$331$	13.3354	0.732977	0.366489	−	0.930423i	$-0.380560\pi$
$331$	13.3354	0.732977	0.366489	+	0.930423i	$0.380560\pi$
$332$	0	0
$333$	−6.66406	−0.365188
$334$	0	0
$335$	26.8877	1.46903
$336$	0	0
$337$	27.5716	1.50192	0.750962	−	0.660346i	$-0.229590\pi$
$337$	27.5716	1.50192	0.750962	+	0.660346i	$0.229590\pi$
$338$	0	0
$339$	−6.90759	−0.375169
$340$	0	0
$341$	3.93963	0.213343
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.39875	−0.236821
$346$	0	0
$347$	27.1016	1.45489	0.727445	−	0.686166i	$-0.240707\pi$
$347$	27.1016	1.45489	0.727445	+	0.686166i	$0.240707\pi$
$348$	0	0
$349$	−30.8549	−1.65162	−0.825812	−	0.563946i	$-0.809283\pi$
$349$	−30.8549	−1.65162	−0.825812	+	0.563946i	$0.809283\pi$
$350$	0	0
$351$	5.51804	0.294531
$352$	0	0
$353$	−6.13344	−0.326450	−0.163225	−	0.986589i	$-0.552190\pi$
$353$	−6.13344	−0.326450	−0.163225	+	0.986589i	$0.552190\pi$
$354$	0	0
$355$	35.2926	1.87314
$356$	0	0
$357$	−3.11930	−0.165091
$358$	0	0
$359$	−24.6084	−1.29878	−0.649390	−	0.760456i	$-0.724976\pi$
$359$	−24.6084	−1.29878	−0.649390	+	0.760456i	$0.724976\pi$
$360$	0	0
$361$	6.58971	0.346827
$362$	0	0
$363$	−1.38851	−0.0728776
$364$	0	0
$365$	12.9773	0.679261
$366$	0	0
$367$	31.3384	1.63585	0.817924	−	0.575326i	$-0.195125\pi$
$367$	31.3384	1.63585	0.817924	+	0.575326i	$0.195125\pi$
$368$	0	0
$369$	−10.6034	−0.551990
$370$	0	0
$371$	−9.54307	−0.495452
$372$	0	0
$373$	27.7860	1.43871	0.719353	−	0.694644i	$-0.244438\pi$
$373$	27.7860	1.43871	0.719353	+	0.694644i	$0.244438\pi$
$374$	0	0
$375$	41.1237	2.12362
$376$	0	0
$377$	36.7726	1.89388
$378$	0	0
$379$	18.1449	0.932041	0.466020	−	0.884774i	$-0.345687\pi$
$379$	18.1449	0.932041	0.466020	+	0.884774i	$0.345687\pi$
$380$	0	0
$381$	17.5269	0.897930
$382$	0	0
$383$	−9.59881	−0.490476	−0.245238	−	0.969463i	$-0.578866\pi$
$383$	−9.59881	−0.490476	−0.245238	+	0.969463i	$0.578866\pi$
$384$	0	0
$385$	15.4824	0.789056
$386$	0	0
$387$	−6.81835	−0.346596
$388$	0	0
$389$	−19.9645	−1.01224	−0.506121	−	0.862463i	$-0.668921\pi$
$389$	−19.9645	−1.01224	−0.506121	+	0.862463i	$0.668921\pi$
$390$	0	0
$391$	−3.11930	−0.157750
$392$	0	0
$393$	−9.00752	−0.454369
$394$	0	0
$395$	58.1386	2.92527
$396$	0	0
$397$	15.1473	0.760222	0.380111	−	0.924941i	$-0.375886\pi$
$397$	15.1473	0.760222	0.380111	+	0.924941i	$0.375886\pi$
$398$	0	0
$399$	−5.05863	−0.253248
$400$	0	0
$401$	−23.6837	−1.18271	−0.591353	−	0.806413i	$-0.701406\pi$
$401$	−23.6837	−1.18271	−0.591353	+	0.806413i	$0.701406\pi$
$402$	0	0
$403$	6.17633	0.307665
$404$	0	0
$405$	−4.39875	−0.218575
$406$	0	0
$407$	23.4557	1.16266
$408$	0	0
$409$	24.2817	1.20065	0.600327	−	0.799755i	$-0.295037\pi$
$409$	24.2817	1.20065	0.600327	+	0.799755i	$0.295037\pi$
$410$	0	0
$411$	2.72473	0.134401
$412$	0	0
$413$	−11.5431	−0.567997
$414$	0	0
$415$	−43.6214	−2.14129
$416$	0	0
$417$	−5.43821	−0.266310
$418$	0	0
$419$	−23.0431	−1.12573	−0.562864	−	0.826549i	$-0.690301\pi$
$419$	−23.0431	−1.12573	−0.562864	+	0.826549i	$0.690301\pi$
$420$	0	0
$421$	1.77888	0.0866975	0.0433487	−	0.999060i	$-0.486197\pi$
$421$	1.77888	0.0866975	0.0433487	+	0.999060i	$0.486197\pi$
$422$	0	0
$423$	−3.43403	−0.166968
$424$	0	0
$425$	44.7587	2.17112
$426$	0	0
$427$	13.9627	0.675701
$428$	0	0
$429$	−19.4220	−0.937704
$430$	0	0
$431$	−0.691068	−0.0332876	−0.0166438	−	0.999861i	$-0.505298\pi$
$431$	−0.691068	−0.0332876	−0.0166438	+	0.999861i	$0.505298\pi$
$432$	0	0
$433$	−31.0340	−1.49140	−0.745699	−	0.666283i	$-0.767884\pi$
$433$	−31.0340	−1.49140	−0.745699	+	0.666283i	$0.767884\pi$
$434$	0	0
$435$	−29.3135	−1.40547
$436$	0	0
$437$	−5.05863	−0.241987
$438$	0	0
$439$	−10.3974	−0.496239	−0.248119	−	0.968729i	$-0.579813\pi$
$439$	−10.3974	−0.496239	−0.248119	+	0.968729i	$0.579813\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−9.20500	−0.437343	−0.218671	−	0.975799i	$-0.570172\pi$
$443$	−9.20500	−0.437343	−0.218671	+	0.975799i	$0.570172\pi$
$444$	0	0
$445$	18.9720	0.899359
$446$	0	0
$447$	−12.9788	−0.613876
$448$	0	0
$449$	20.2127	0.953896	0.476948	−	0.878931i	$-0.341743\pi$
$449$	20.2127	0.953896	0.476948	+	0.878931i	$0.341743\pi$
$450$	0	0
$451$	37.3211	1.75738
$452$	0	0
$453$	22.7393	1.06839
$454$	0	0
$455$	24.2725	1.13791
$456$	0	0
$457$	−31.8183	−1.48840	−0.744199	−	0.667958i	$-0.767169\pi$
$457$	−31.8183	−1.48840	−0.744199	+	0.667958i	$0.767169\pi$
$458$	0	0
$459$	−3.11930	−0.145596
$460$	0	0
$461$	30.9379	1.44092	0.720460	−	0.693496i	$-0.243930\pi$
$461$	30.9379	1.44092	0.720460	+	0.693496i	$0.243930\pi$
$462$	0	0
$463$	41.5016	1.92874	0.964371	−	0.264553i	$-0.0852245\pi$
$463$	41.5016	1.92874	0.964371	+	0.264553i	$0.0852245\pi$
$464$	0	0
$465$	−4.92350	−0.228322
$466$	0	0
$467$	−36.9392	−1.70934	−0.854672	−	0.519169i	$-0.826242\pi$
$467$	−36.9392	−1.70934	−0.854672	+	0.519169i	$0.826242\pi$
$468$	0	0
$469$	−6.11258	−0.282253
$470$	0	0
$471$	−13.9393	−0.642290
$472$	0	0
$473$	23.9987	1.10346
$474$	0	0
$475$	72.5861	3.33048
$476$	0	0
$477$	−9.54307	−0.436947
$478$	0	0
$479$	−2.97512	−0.135937	−0.0679684	−	0.997687i	$-0.521652\pi$
$479$	−2.97512	−0.135937	−0.0679684	+	0.997687i	$0.521652\pi$
$480$	0	0
$481$	36.7726	1.67668
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−71.2727	−3.23633
$486$	0	0
$487$	−18.3542	−0.831707	−0.415854	−	0.909432i	$-0.636517\pi$
$487$	−18.3542	−0.831707	−0.415854	+	0.909432i	$0.636517\pi$
$488$	0	0
$489$	−1.82337	−0.0824558
$490$	0	0
$491$	−8.65281	−0.390496	−0.195248	−	0.980754i	$-0.562551\pi$
$491$	−8.65281	−0.390496	−0.195248	+	0.980754i	$0.562551\pi$
$492$	0	0
$493$	−20.7872	−0.936207
$494$	0	0
$495$	15.4824	0.695882
$496$	0	0
$497$	−8.02334	−0.359896
$498$	0	0
$499$	−0.624797	−0.0279698	−0.0139849	−	0.999902i	$-0.504452\pi$
$499$	−0.624797	−0.0279698	−0.0139849	+	0.999902i	$0.504452\pi$
$500$	0	0
$501$	−9.05863	−0.404710
$502$	0	0
$503$	0.197295	0.00879695	0.00439848	−	0.999990i	$-0.498600\pi$
$503$	0.197295	0.00879695	0.00439848	+	0.999990i	$0.498600\pi$
$504$	0	0
$505$	−6.35741	−0.282901
$506$	0	0
$507$	−17.4488	−0.774929
$508$	0	0
$509$	−14.8988	−0.660378	−0.330189	−	0.943915i	$-0.607113\pi$
$509$	−14.8988	−0.660378	−0.330189	+	0.943915i	$0.607113\pi$
$510$	0	0
$511$	−2.95022	−0.130510
$512$	0	0
$513$	−5.05863	−0.223344
$514$	0	0
$515$	47.9567	2.11322
$516$	0	0
$517$	12.0869	0.531580
$518$	0	0
$519$	−8.71766	−0.382663
$520$	0	0
$521$	16.7786	0.735085	0.367542	−	0.930007i	$-0.380199\pi$
$521$	16.7786	0.735085	0.367542	+	0.930007i	$0.380199\pi$
$522$	0	0
$523$	−18.8854	−0.825800	−0.412900	−	0.910776i	$-0.635484\pi$
$523$	−18.8854	−0.825800	−0.412900	+	0.910776i	$0.635484\pi$
$524$	0	0
$525$	−14.3490	−0.626240
$526$	0	0
$527$	−3.49142	−0.152089
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.5431	−0.500927
$532$	0	0
$533$	58.5100	2.53435
$534$	0	0
$535$	38.5953	1.66862
$536$	0	0
$537$	1.59788	0.0689536
$538$	0	0
$539$	−3.51973	−0.151606
$540$	0	0
$541$	35.0510	1.50696	0.753479	−	0.657472i	$-0.228374\pi$
$541$	35.0510	1.50696	0.753479	+	0.657472i	$0.228374\pi$
$542$	0	0
$543$	−16.7083	−0.717020
$544$	0	0
$545$	32.8874	1.40874
$546$	0	0
$547$	17.8488	0.763161	0.381580	−	0.924336i	$-0.375380\pi$
$547$	17.8488	0.763161	0.381580	+	0.924336i	$0.375380\pi$
$548$	0	0
$549$	13.9627	0.595912
$550$	0	0
$551$	−33.7110	−1.43614
$552$	0	0
$553$	−13.2171	−0.562048
$554$	0	0
$555$	−29.3135	−1.24429
$556$	0	0
$557$	8.11388	0.343796	0.171898	−	0.985115i	$-0.445010\pi$
$557$	8.11388	0.343796	0.171898	+	0.985115i	$0.445010\pi$
$558$	0	0
$559$	37.6239	1.59132
$560$	0	0
$561$	10.9791	0.463537
$562$	0	0
$563$	36.4963	1.53813	0.769067	−	0.639168i	$-0.220721\pi$
$563$	36.4963	1.53813	0.769067	+	0.639168i	$0.220721\pi$
$564$	0	0
$565$	−30.3847	−1.27829
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	7.04084	0.295167	0.147584	−	0.989050i	$-0.452850\pi$
$569$	7.04084	0.295167	0.147584	+	0.989050i	$0.452850\pi$
$570$	0	0
$571$	8.90016	0.372460	0.186230	−	0.982506i	$-0.440373\pi$
$571$	8.90016	0.372460	0.186230	+	0.982506i	$0.440373\pi$
$572$	0	0
$573$	−9.78971	−0.408971
$574$	0	0
$575$	−14.3490	−0.598393
$576$	0	0
$577$	11.5829	0.482203	0.241102	−	0.970500i	$-0.422491\pi$
$577$	11.5829	0.482203	0.241102	+	0.970500i	$0.422491\pi$
$578$	0	0
$579$	−6.13373	−0.254909
$580$	0	0
$581$	9.91679	0.411418
$582$	0	0
$583$	33.5890	1.39112
$584$	0	0
$585$	24.2725	1.00354
$586$	0	0
$587$	38.7702	1.60022	0.800109	−	0.599855i	$-0.204775\pi$
$587$	38.7702	1.60022	0.800109	+	0.599855i	$0.204775\pi$
$588$	0	0
$589$	−5.66211	−0.233303
$590$	0	0
$591$	−24.9157	−1.02489
$592$	0	0
$593$	−29.8912	−1.22748	−0.613742	−	0.789506i	$-0.710337\pi$
$593$	−29.8912	−1.22748	−0.613742	+	0.789506i	$0.710337\pi$
$594$	0	0
$595$	−13.7210	−0.562506
$596$	0	0
$597$	17.9499	0.734639
$598$	0	0
$599$	−30.6861	−1.25380	−0.626899	−	0.779100i	$-0.715676\pi$
$599$	−30.6861	−1.25380	−0.626899	+	0.779100i	$0.715676\pi$
$600$	0	0
$601$	25.4323	1.03741	0.518703	−	0.854954i	$-0.326415\pi$
$601$	25.4323	1.03741	0.518703	+	0.854954i	$0.326415\pi$
$602$	0	0
$603$	−6.11258	−0.248924
$604$	0	0
$605$	−6.10768	−0.248313
$606$	0	0
$607$	1.11462	0.0452412	0.0226206	−	0.999744i	$-0.492799\pi$
$607$	1.11462	0.0452412	0.0226206	+	0.999744i	$0.492799\pi$
$608$	0	0
$609$	6.66406	0.270041
$610$	0	0
$611$	18.9491	0.766600
$612$	0	0
$613$	23.7284	0.958382	0.479191	−	0.877711i	$-0.340930\pi$
$613$	23.7284	0.958382	0.479191	+	0.877711i	$0.340930\pi$
$614$	0	0
$615$	−46.6416	−1.88077
$616$	0	0
$617$	4.48212	0.180444	0.0902218	−	0.995922i	$-0.471242\pi$
$617$	4.48212	0.180444	0.0902218	+	0.995922i	$0.471242\pi$
$618$	0	0
$619$	−45.2828	−1.82007	−0.910035	−	0.414531i	$-0.863946\pi$
$619$	−45.2828	−1.82007	−0.910035	+	0.414531i	$0.863946\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−4.31305	−0.172799
$624$	0	0
$625$	109.148	4.36592
$626$	0	0
$627$	17.8050	0.711064
$628$	0	0
$629$	−20.7872	−0.828839
$630$	0	0
$631$	41.5050	1.65229	0.826145	−	0.563458i	$-0.190529\pi$
$631$	41.5050	1.65229	0.826145	+	0.563458i	$0.190529\pi$
$632$	0	0
$633$	−3.51973	−0.139897
$634$	0	0
$635$	77.0963	3.05948
$636$	0	0
$637$	−5.51804	−0.218633
$638$	0	0
$639$	−8.02334	−0.317398
$640$	0	0
$641$	−23.0886	−0.911946	−0.455973	−	0.889994i	$-0.650709\pi$
$641$	−23.0886	−0.911946	−0.455973	+	0.889994i	$0.650709\pi$
$642$	0	0
$643$	25.0216	0.986755	0.493378	−	0.869815i	$-0.335762\pi$
$643$	25.0216	0.986755	0.493378	+	0.869815i	$0.335762\pi$
$644$	0	0
$645$	−29.9922	−1.18094
$646$	0	0
$647$	3.50932	0.137966	0.0689828	−	0.997618i	$-0.478025\pi$
$647$	3.50932	0.137966	0.0689828	+	0.997618i	$0.478025\pi$
$648$	0	0
$649$	40.6285	1.59481
$650$	0	0
$651$	1.11930	0.0438687
$652$	0	0
$653$	−11.2514	−0.440300	−0.220150	−	0.975466i	$-0.570655\pi$
$653$	−11.2514	−0.440300	−0.220150	+	0.975466i	$0.570655\pi$
$654$	0	0
$655$	−39.6218	−1.54815
$656$	0	0
$657$	−2.95022	−0.115099
$658$	0	0
$659$	−25.9614	−1.01131	−0.505656	−	0.862735i	$-0.668750\pi$
$659$	−25.9614	−1.01131	−0.505656	+	0.862735i	$0.668750\pi$
$660$	0	0
$661$	−1.56641	−0.0609264	−0.0304632	−	0.999536i	$-0.509698\pi$
$661$	−1.56641	−0.0609264	−0.0304632	+	0.999536i	$0.509698\pi$
$662$	0	0
$663$	17.2124	0.668475
$664$	0	0
$665$	−22.2516	−0.862881
$666$	0	0
$667$	6.66406	0.258033
$668$	0	0
$669$	−15.1709	−0.586541
$670$	0	0
$671$	−49.1448	−1.89722
$672$	0	0
$673$	6.85541	0.264257	0.132128	−	0.991233i	$-0.457819\pi$
$673$	6.85541	0.264257	0.132128	+	0.991233i	$0.457819\pi$
$674$	0	0
$675$	−14.3490	−0.552292
$676$	0	0
$677$	−0.443792	−0.0170563	−0.00852817	−	0.999964i	$-0.502715\pi$
$677$	−0.443792	−0.0170563	−0.00852817	+	0.999964i	$0.502715\pi$
$678$	0	0
$679$	16.2030	0.621812
$680$	0	0
$681$	−17.6039	−0.674583
$682$	0	0
$683$	19.7794	0.756836	0.378418	−	0.925635i	$-0.376468\pi$
$683$	19.7794	0.756836	0.378418	+	0.925635i	$0.376468\pi$
$684$	0	0
$685$	11.9854	0.457938
$686$	0	0
$687$	14.4746	0.552241
$688$	0	0
$689$	52.6591	2.00615
$690$	0	0
$691$	12.4590	0.473963	0.236981	−	0.971514i	$-0.423842\pi$
$691$	12.4590	0.473963	0.236981	+	0.971514i	$0.423842\pi$
$692$	0	0
$693$	−3.51973	−0.133704
$694$	0	0
$695$	−23.9213	−0.907386
$696$	0	0
$697$	−33.0751	−1.25281
$698$	0	0
$699$	26.5474	1.00411
$700$	0	0
$701$	14.2983	0.540038	0.270019	−	0.962855i	$-0.412970\pi$
$701$	14.2983	0.540038	0.270019	+	0.962855i	$0.412970\pi$
$702$	0	0
$703$	−33.7110	−1.27143
$704$	0	0
$705$	−15.1054	−0.568904
$706$	0	0
$707$	1.44528	0.0543553
$708$	0	0
$709$	3.46444	0.130110	0.0650550	−	0.997882i	$-0.479278\pi$
$709$	3.46444	0.130110	0.0650550	+	0.997882i	$0.479278\pi$
$710$	0	0
$711$	−13.2171	−0.495680
$712$	0	0
$713$	1.11930	0.0419180
$714$	0	0
$715$	−85.4326	−3.19500
$716$	0	0
$717$	4.35118	0.162498
$718$	0	0
$719$	25.8379	0.963593	0.481796	−	0.876283i	$-0.339985\pi$
$719$	25.8379	0.963593	0.481796	+	0.876283i	$0.339985\pi$
$720$	0	0
$721$	−10.9024	−0.406025
$722$	0	0
$723$	−20.0505	−0.745687
$724$	0	0
$725$	−95.6223	−3.55132
$726$	0	0
$727$	−22.7739	−0.844637	−0.422319	−	0.906447i	$-0.638784\pi$
$727$	−22.7739	−0.844637	−0.422319	+	0.906447i	$0.638784\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.2684	−0.786642
$732$	0	0
$733$	44.3651	1.63866	0.819331	−	0.573321i	$-0.194345\pi$
$733$	44.3651	1.63866	0.819331	+	0.573321i	$0.194345\pi$
$734$	0	0
$735$	4.39875	0.162250
$736$	0	0
$737$	21.5146	0.792502
$738$	0	0
$739$	−20.6877	−0.761011	−0.380505	−	0.924779i	$-0.624250\pi$
$739$	−20.6877	−0.761011	−0.380505	+	0.924779i	$0.624250\pi$
$740$	0	0
$741$	27.9137	1.02544
$742$	0	0
$743$	21.2056	0.777959	0.388980	−	0.921246i	$-0.372828\pi$
$743$	21.2056	0.777959	0.388980	+	0.921246i	$0.372828\pi$
$744$	0	0
$745$	−57.0904	−2.09163
$746$	0	0
$747$	9.91679	0.362836
$748$	0	0
$749$	−8.77415	−0.320601
$750$	0	0
$751$	35.6869	1.30223	0.651117	−	0.758977i	$-0.274301\pi$
$751$	35.6869	1.30223	0.651117	+	0.758977i	$0.274301\pi$
$752$	0	0
$753$	14.1426	0.515384
$754$	0	0
$755$	100.024	3.64026
$756$	0	0
$757$	−25.1596	−0.914440	−0.457220	−	0.889354i	$-0.651155\pi$
$757$	−25.1596	−0.914440	−0.457220	+	0.889354i	$0.651155\pi$
$758$	0	0
$759$	−3.51973	−0.127758
$760$	0	0
$761$	35.4096	1.28360	0.641799	−	0.766873i	$-0.278189\pi$
$761$	35.4096	1.28360	0.641799	+	0.766873i	$0.278189\pi$
$762$	0	0
$763$	−7.47654	−0.270669
$764$	0	0
$765$	−13.7210	−0.496084
$766$	0	0
$767$	63.6952	2.29990
$768$	0	0
$769$	8.97083	0.323496	0.161748	−	0.986832i	$-0.448287\pi$
$769$	8.97083	0.323496	0.161748	+	0.986832i	$0.448287\pi$
$770$	0	0
$771$	22.8292	0.822173
$772$	0	0
$773$	−15.2725	−0.549312	−0.274656	−	0.961543i	$-0.588564\pi$
$773$	−15.2725	−0.549312	−0.274656	+	0.961543i	$0.588564\pi$
$774$	0	0
$775$	−16.0608	−0.576920
$776$	0	0
$777$	6.66406	0.239072
$778$	0	0
$779$	−53.6386	−1.92180
$780$	0	0
$781$	28.2400	1.01051
$782$	0	0
$783$	6.66406	0.238154
$784$	0	0
$785$	−61.3156	−2.18845
$786$	0	0
$787$	−4.07137	−0.145129	−0.0725644	−	0.997364i	$-0.523118\pi$
$787$	−4.07137	−0.145129	−0.0725644	+	0.997364i	$0.523118\pi$
$788$	0	0
$789$	7.95143	0.283078
$790$	0	0
$791$	6.90759	0.245605
$792$	0	0
$793$	−77.0466	−2.73601
$794$	0	0
$795$	−41.9776	−1.48879
$796$	0	0
$797$	−2.06935	−0.0733002	−0.0366501	−	0.999328i	$-0.511669\pi$
$797$	−2.06935	−0.0733002	−0.0366501	+	0.999328i	$0.511669\pi$
$798$	0	0
$799$	−10.7118	−0.378955
$800$	0	0
$801$	−4.31305	−0.152394
$802$	0	0
$803$	10.3840	0.366443
$804$	0	0
$805$	4.39875	0.155035
$806$	0	0
$807$	−2.07445	−0.0730242
$808$	0	0
$809$	−3.73806	−0.131423	−0.0657117	−	0.997839i	$-0.520932\pi$
$809$	−3.73806	−0.131423	−0.0657117	+	0.997839i	$0.520932\pi$
$810$	0	0
$811$	−43.7178	−1.53514	−0.767571	−	0.640965i	$-0.778535\pi$
$811$	−43.7178	−1.53514	−0.767571	+	0.640965i	$0.778535\pi$
$812$	0	0
$813$	−18.7177	−0.656457
$814$	0	0
$815$	−8.02056	−0.280948
$816$	0	0
$817$	−34.4915	−1.20670
$818$	0	0
$819$	−5.51804	−0.192816
$820$	0	0
$821$	36.7480	1.28251	0.641257	−	0.767326i	$-0.278413\pi$
$821$	36.7480	1.28251	0.641257	+	0.767326i	$0.278413\pi$
$822$	0	0
$823$	−25.5698	−0.891306	−0.445653	−	0.895206i	$-0.647028\pi$
$823$	−25.5698	−0.891306	−0.445653	+	0.895206i	$0.647028\pi$
$824$	0	0
$825$	50.5045	1.75834
$826$	0	0
$827$	−2.15043	−0.0747776	−0.0373888	−	0.999301i	$-0.511904\pi$
$827$	−2.15043	−0.0747776	−0.0373888	+	0.999301i	$0.511904\pi$
$828$	0	0
$829$	−54.4745	−1.89198	−0.945989	−	0.324199i	$-0.894905\pi$
$829$	−54.4745	−1.89198	−0.945989	+	0.324199i	$0.894905\pi$
$830$	0	0
$831$	−5.07342	−0.175995
$832$	0	0
$833$	3.11930	0.108077
$834$	0	0
$835$	−39.8466	−1.37895
$836$	0	0
$837$	1.11930	0.0386886
$838$	0	0
$839$	−35.4836	−1.22503	−0.612514	−	0.790460i	$-0.709842\pi$
$839$	−35.4836	−1.22503	−0.612514	+	0.790460i	$0.709842\pi$
$840$	0	0
$841$	15.4097	0.531367
$842$	0	0
$843$	−13.6688	−0.470777
$844$	0	0
$845$	−76.7529	−2.64038
$846$	0	0
$847$	1.38851	0.0477096
$848$	0	0
$849$	28.8011	0.988452
$850$	0	0
$851$	6.66406	0.228441
$852$	0	0
$853$	8.84772	0.302940	0.151470	−	0.988462i	$-0.451599\pi$
$853$	8.84772	0.302940	0.151470	+	0.988462i	$0.451599\pi$
$854$	0	0
$855$	−22.2516	−0.760989
$856$	0	0
$857$	24.5225	0.837673	0.418836	−	0.908062i	$-0.362438\pi$
$857$	24.5225	0.837673	0.418836	+	0.908062i	$0.362438\pi$
$858$	0	0
$859$	−6.52138	−0.222507	−0.111253	−	0.993792i	$-0.535487\pi$
$859$	−6.52138	−0.222507	−0.111253	+	0.993792i	$0.535487\pi$
$860$	0	0
$861$	10.6034	0.361362
$862$	0	0
$863$	−3.21523	−0.109448	−0.0547238	−	0.998502i	$-0.517428\pi$
$863$	−3.21523	−0.109448	−0.0547238	+	0.998502i	$0.517428\pi$
$864$	0	0
$865$	−38.3468	−1.30383
$866$	0	0
$867$	7.26998	0.246902
$868$	0	0
$869$	46.5206	1.57810
$870$	0	0
$871$	33.7295	1.14288
$872$	0	0
$873$	16.2030	0.548387
$874$	0	0
$875$	−41.1237	−1.39024
$876$	0	0
$877$	12.8366	0.433462	0.216731	−	0.976231i	$-0.430460\pi$
$877$	12.8366	0.433462	0.216731	+	0.976231i	$0.430460\pi$
$878$	0	0
$879$	−2.03833	−0.0687512
$880$	0	0
$881$	34.4082	1.15924	0.579621	−	0.814886i	$-0.303200\pi$
$881$	34.4082	1.15924	0.579621	+	0.814886i	$0.303200\pi$
$882$	0	0
$883$	14.9055	0.501612	0.250806	−	0.968037i	$-0.419304\pi$
$883$	14.9055	0.501612	0.250806	+	0.968037i	$0.419304\pi$
$884$	0	0
$885$	−50.7750	−1.70678
$886$	0	0
$887$	−16.3556	−0.549169	−0.274584	−	0.961563i	$-0.588540\pi$
$887$	−16.3556	−0.549169	−0.274584	+	0.961563i	$0.588540\pi$
$888$	0	0
$889$	−17.5269	−0.587833
$890$	0	0
$891$	−3.51973	−0.117915
$892$	0	0
$893$	−17.3715	−0.581315
$894$	0	0
$895$	7.02866	0.234942
$896$	0	0
$897$	−5.51804	−0.184242
$898$	0	0
$899$	7.45906	0.248774
$900$	0	0
$901$	−29.7677	−0.991705
$902$	0	0
$903$	6.81835	0.226900
$904$	0	0
$905$	−73.4954	−2.44307
$906$	0	0
$907$	−32.6282	−1.08340	−0.541700	−	0.840572i	$-0.682219\pi$
$907$	−32.6282	−1.08340	−0.541700	+	0.840572i	$0.682219\pi$
$908$	0	0
$909$	1.44528	0.0479368
$910$	0	0
$911$	11.9478	0.395847	0.197923	−	0.980217i	$-0.436580\pi$
$911$	11.9478	0.395847	0.197923	+	0.980217i	$0.436580\pi$
$912$	0	0
$913$	−34.9044	−1.15517
$914$	0	0
$915$	61.4182	2.03042
$916$	0	0
$917$	9.00752	0.297454
$918$	0	0
$919$	−29.0429	−0.958037	−0.479019	−	0.877805i	$-0.659007\pi$
$919$	−29.0429	−0.958037	−0.479019	+	0.877805i	$0.659007\pi$
$920$	0	0
$921$	−5.85313	−0.192867
$922$	0	0
$923$	44.2731	1.45727
$924$	0	0
$925$	−95.6223	−3.14404
$926$	0	0
$927$	−10.9024	−0.358080
$928$	0	0
$929$	38.3200	1.25724	0.628619	−	0.777713i	$-0.283620\pi$
$929$	38.3200	1.25724	0.628619	+	0.777713i	$0.283620\pi$
$930$	0	0
$931$	5.05863	0.165790
$932$	0	0
$933$	−0.298169	−0.00976161
$934$	0	0
$935$	48.2942	1.57939
$936$	0	0
$937$	39.5517	1.29210	0.646049	−	0.763296i	$-0.276420\pi$
$937$	39.5517	1.29210	0.646049	+	0.763296i	$0.276420\pi$
$938$	0	0
$939$	23.7991	0.776655
$940$	0	0
$941$	45.1917	1.47321	0.736604	−	0.676324i	$-0.236428\pi$
$941$	45.1917	1.47321	0.736604	+	0.676324i	$0.236428\pi$
$942$	0	0
$943$	10.6034	0.345294
$944$	0	0
$945$	4.39875	0.143091
$946$	0	0
$947$	43.3517	1.40874	0.704370	−	0.709833i	$-0.251230\pi$
$947$	43.3517	1.40874	0.704370	+	0.709833i	$0.251230\pi$
$948$	0	0
$949$	16.2794	0.528453
$950$	0	0
$951$	−2.90759	−0.0942849
$952$	0	0
$953$	−22.7725	−0.737673	−0.368836	−	0.929494i	$-0.620244\pi$
$953$	−22.7725	−0.737673	−0.368836	+	0.929494i	$0.620244\pi$
$954$	0	0
$955$	−43.0625	−1.39347
$956$	0	0
$957$	−23.4557	−0.758215
$958$	0	0
$959$	−2.72473	−0.0879860
$960$	0	0
$961$	−29.7472	−0.959586
$962$	0	0
$963$	−8.77415	−0.282743
$964$	0	0
$965$	−26.9807	−0.868540
$966$	0	0
$967$	−6.09956	−0.196149	−0.0980743	−	0.995179i	$-0.531268\pi$
$967$	−6.09956	−0.196149	−0.0980743	+	0.995179i	$0.531268\pi$
$968$	0	0
$969$	−15.7794	−0.506906
$970$	0	0
$971$	29.6337	0.950991	0.475495	−	0.879718i	$-0.342269\pi$
$971$	29.6337	0.950991	0.475495	+	0.879718i	$0.342269\pi$
$972$	0	0
$973$	5.43821	0.174341
$974$	0	0
$975$	79.1782	2.53573
$976$	0	0
$977$	5.51346	0.176391	0.0881956	−	0.996103i	$-0.471890\pi$
$977$	5.51346	0.176391	0.0881956	+	0.996103i	$0.471890\pi$
$978$	0	0
$979$	15.1808	0.485179
$980$	0	0
$981$	−7.47654	−0.238708
$982$	0	0
$983$	6.35637	0.202737	0.101368	−	0.994849i	$-0.467678\pi$
$983$	6.35637	0.202737	0.101368	+	0.994849i	$0.467678\pi$
$984$	0	0
$985$	−109.598	−3.49207
$986$	0	0
$987$	3.43403	0.109306
$988$	0	0
$989$	6.81835	0.216811
$990$	0	0
$991$	5.45383	0.173247	0.0866233	−	0.996241i	$-0.472392\pi$
$991$	5.45383	0.173247	0.0866233	+	0.996241i	$0.472392\pi$
$992$	0	0
$993$	−13.3354	−0.423185
$994$	0	0
$995$	78.9569	2.50310
$996$	0	0
$997$	−27.2701	−0.863651	−0.431826	−	0.901957i	$-0.642130\pi$
$997$	−27.2701	−0.863651	−0.431826	+	0.901957i	$0.642130\pi$
$998$	0	0
$999$	6.66406	0.210841

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cg.1.1		6
4.3	odd	2		3864.2.a.x.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.x.1.1	✓	6	4.3	odd	2
7728.2.a.cg.1.1		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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.39875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.39875 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.51973 q^{11} -5.51804 q^{13} +4.39875 q^{15} +3.11930 q^{17} +5.05863 q^{19} -1.00000 q^{21} -1.00000 q^{23} +14.3490 q^{25} -1.00000 q^{27} -6.66406 q^{29} -1.11930 q^{31} +3.51973 q^{33} -4.39875 q^{35} -6.66406 q^{37} +5.51804 q^{39} -10.6034 q^{41} -6.81835 q^{43} -4.39875 q^{45} -3.43403 q^{47} +1.00000 q^{49} -3.11930 q^{51} -9.54307 q^{53} +15.4824 q^{55} -5.05863 q^{57} -11.5431 q^{59} +13.9627 q^{61} +1.00000 q^{63} +24.2725 q^{65} -6.11258 q^{67} +1.00000 q^{69} -8.02334 q^{71} -2.95022 q^{73} -14.3490 q^{75} -3.51973 q^{77} -13.2171 q^{79} +1.00000 q^{81} +9.91679 q^{83} -13.7210 q^{85} +6.66406 q^{87} -4.31305 q^{89} -5.51804 q^{91} +1.11930 q^{93} -22.2516 q^{95} +16.2030 q^{97} -3.51973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 - 5 * q^11 + 2 * q^13 + 10 * q^17 - 3 * q^19 - 6 * q^21 - 6 * q^23 + 18 * q^25 - 6 * q^27 + 3 * q^29 + 2 * q^31 + 5 * q^33 + 3 * q^37 - 2 * q^39 + 4 * q^41 - 6 * q^43 + 2 * q^47 + 6 * q^49 - 10 * q^51 - 4 * q^53 + 15 * q^55 + 3 * q^57 - 16 * q^59 + 22 * q^61 + 6 * q^63 + 35 * q^65 - 9 * q^67 + 6 * q^69 - 11 * q^71 + 24 * q^73 - 18 * q^75 - 5 * q^77 - 18 * q^79 + 6 * q^81 - 2 * q^83 + 13 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 - 2 * q^93 - 5 * q^95 + 37 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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