LMFDB - Embedded newform 4056.2.c.n.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.n.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.554958i q^{5} +3.04892i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.554958i q^{5} +3.04892i q^{7} +1.00000 q^{9} -1.80194i q^{11} +0.554958i q^{15} -1.24698 q^{17} +2.35690i q^{19} -3.04892i q^{21} +0.554958 q^{23} +4.69202 q^{25} -1.00000 q^{27} -6.18598 q^{29} +8.67994i q^{31} +1.80194i q^{33} +1.69202 q^{35} -0.960771i q^{37} -2.47219i q^{41} -0.384043 q^{43} -0.554958i q^{45} -9.96077i q^{47} -2.29590 q^{49} +1.24698 q^{51} +6.02177 q^{53} -1.00000 q^{55} -2.35690i q^{57} +7.30559i q^{59} -12.0586 q^{61} +3.04892i q^{63} +13.6528i q^{67} -0.554958 q^{69} -4.58211i q^{71} -1.04892i q^{73} -4.69202 q^{75} +5.49396 q^{77} +6.64071 q^{79} +1.00000 q^{81} +6.24698i q^{83} +0.692021i q^{85} +6.18598 q^{87} +13.4330i q^{89} -8.67994i q^{93} +1.30798 q^{95} -12.1099i q^{97} -1.80194i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} + 2 q^{17} + 4 q^{23} + 18 q^{25} - 6 q^{27} - 8 q^{29} + 18 q^{43} + 14 q^{49} - 2 q^{51} + 30 q^{53} - 6 q^{55} - 10 q^{61} - 4 q^{69} - 18 q^{75} + 14 q^{77} - 34 q^{79} + 6 q^{81} + 8 q^{87} + 18 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 + 2 * q^17 + 4 * q^23 + 18 * q^25 - 6 * q^27 - 8 * q^29 + 18 * q^43 + 14 * q^49 - 2 * q^51 + 30 * q^53 - 6 * q^55 - 10 * q^61 - 4 * q^69 - 18 * q^75 + 14 * q^77 - 34 * q^79 + 6 * q^81 + 8 * q^87 + 18 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	− 0.554958i	− 0.248185i	−0.992271	−	0.124092i	$-0.960398\pi$
$5$	− 0.554958i	− 0.248185i	0.992271	−	0.124092i	$-0.0396019\pi$
$6$	0	0
$7$	3.04892i	1.15238i	0.817315	+	0.576191i	$0.195462\pi$
$7$	3.04892i	1.15238i	−0.817315	+	0.576191i	$0.804538\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 1.80194i	− 0.543305i	−0.962395	−	0.271652i	$-0.912430\pi$
$11$	− 1.80194i	− 0.543305i	0.962395	−	0.271652i	$-0.0875701\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.554958i	0.143290i
$16$	0	0
$17$	−1.24698	−0.302437	−0.151218	−	0.988500i	$-0.548320\pi$
$17$	−1.24698	−0.302437	−0.151218	+	0.988500i	$0.548320\pi$
$18$	0	0
$19$	2.35690i	0.540709i	0.962761	+	0.270354i	$0.0871409\pi$
$19$	2.35690i	0.540709i	−0.962761	+	0.270354i	$0.912859\pi$
$20$	0	0
$21$	− 3.04892i	− 0.665328i
$22$	0	0
$23$	0.554958	0.115717	0.0578584	−	0.998325i	$-0.481573\pi$
$23$	0.554958	0.115717	0.0578584	+	0.998325i	$0.481573\pi$
$24$	0	0
$25$	4.69202	0.938404
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.18598	−1.14871	−0.574354	−	0.818607i	$-0.694747\pi$
$29$	−6.18598	−1.14871	−0.574354	+	0.818607i	$0.694747\pi$
$30$	0	0
$31$	8.67994i	1.55896i	0.626425	+	0.779482i	$0.284517\pi$
$31$	8.67994i	1.55896i	−0.626425	+	0.779482i	$0.715483\pi$
$32$	0	0
$33$	1.80194i	0.313677i
$34$	0	0
$35$	1.69202	0.286004
$36$	0	0
$37$	− 0.960771i	− 0.157950i	−0.996877	−	0.0789749i	$-0.974835\pi$
$37$	− 0.960771i	− 0.157950i	0.996877	−	0.0789749i	$-0.0251647\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.47219i	− 0.386091i	−0.981190	−	0.193046i	$-0.938163\pi$
$41$	− 2.47219i	− 0.386091i	0.981190	−	0.193046i	$-0.0618365\pi$
$42$	0	0
$43$	−0.384043	−0.0585660	−0.0292830	−	0.999571i	$-0.509322\pi$
$43$	−0.384043	−0.0585660	−0.0292830	+	0.999571i	$0.509322\pi$
$44$	0	0
$45$	− 0.554958i	− 0.0827283i
$46$	0	0
$47$	− 9.96077i	− 1.45293i	−0.687205	−	0.726464i	$-0.741162\pi$
$47$	− 9.96077i	− 1.45293i	0.687205	−	0.726464i	$-0.258838\pi$
$48$	0	0
$49$	−2.29590	−0.327985
$50$	0	0
$51$	1.24698	0.174612
$52$	0	0
$53$	6.02177	0.827154	0.413577	−	0.910469i	$-0.364279\pi$
$53$	6.02177	0.827154	0.413577	+	0.910469i	$0.364279\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	− 2.35690i	− 0.312178i
$58$	0	0
$59$	7.30559i	0.951106i	0.879687	+	0.475553i	$0.157752\pi$
$59$	7.30559i	0.951106i	−0.879687	+	0.475553i	$0.842248\pi$
$60$	0	0
$61$	−12.0586	−1.54395	−0.771973	−	0.635655i	$-0.780730\pi$
$61$	−12.0586	−1.54395	−0.771973	+	0.635655i	$0.780730\pi$
$62$	0	0
$63$	3.04892i	0.384127i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.6528i	1.66795i	0.551799	+	0.833977i	$0.313942\pi$
$67$	13.6528i	1.66795i	−0.551799	+	0.833977i	$0.686058\pi$
$68$	0	0
$69$	−0.554958	−0.0668091
$70$	0	0
$71$	− 4.58211i	− 0.543796i	−0.962326	−	0.271898i	$-0.912349\pi$
$71$	− 4.58211i	− 0.543796i	0.962326	−	0.271898i	$-0.0876513\pi$
$72$	0	0
$73$	− 1.04892i	− 0.122766i	−0.998114	−	0.0613832i	$-0.980449\pi$
$73$	− 1.04892i	− 0.122766i	0.998114	−	0.0613832i	$-0.0195512\pi$
$74$	0	0
$75$	−4.69202	−0.541788
$76$	0	0
$77$	5.49396	0.626095
$78$	0	0
$79$	6.64071	0.747138	0.373569	−	0.927602i	$-0.378134\pi$
$79$	6.64071	0.747138	0.373569	+	0.927602i	$0.378134\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.24698i	0.685695i	0.939391	+	0.342848i	$0.111391\pi$
$83$	6.24698i	0.685695i	−0.939391	+	0.342848i	$0.888609\pi$
$84$	0	0
$85$	0.692021i	0.0750603i
$86$	0	0
$87$	6.18598	0.663207
$88$	0	0
$89$	13.4330i	1.42389i	0.702235	+	0.711945i	$0.252186\pi$
$89$	13.4330i	1.42389i	−0.702235	+	0.711945i	$0.747814\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 8.67994i	− 0.900068i
$94$	0	0
$95$	1.30798	0.134196
$96$	0	0
$97$	− 12.1099i	− 1.22958i	−0.788693	−	0.614788i	$-0.789242\pi$
$97$	− 12.1099i	− 1.22958i	0.788693	−	0.614788i	$-0.210758\pi$
$98$	0	0
$99$	− 1.80194i	− 0.181102i
$100$	0	0
$101$	−19.0248	−1.89303	−0.946517	−	0.322654i	$-0.895425\pi$
$101$	−19.0248	−1.89303	−0.946517	+	0.322654i	$0.895425\pi$
$102$	0	0
$103$	10.5483	1.03935	0.519675	−	0.854364i	$-0.326053\pi$
$103$	10.5483	1.03935	0.519675	+	0.854364i	$0.326053\pi$
$104$	0	0
$105$	−1.69202	−0.165124
$106$	0	0
$107$	−11.1075	−1.07380	−0.536902	−	0.843644i	$-0.680406\pi$
$107$	−11.1075	−1.07380	−0.536902	+	0.843644i	$0.680406\pi$
$108$	0	0
$109$	3.85623i	0.369360i	0.982799	+	0.184680i	$0.0591249\pi$
$109$	3.85623i	0.369360i	−0.982799	+	0.184680i	$0.940875\pi$
$110$	0	0
$111$	0.960771i	0.0911924i
$112$	0	0
$113$	−12.2905	−1.15619	−0.578097	−	0.815968i	$-0.696205\pi$
$113$	−12.2905	−1.15619	−0.578097	+	0.815968i	$0.696205\pi$
$114$	0	0
$115$	− 0.307979i	− 0.0287191i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3.80194i	− 0.348523i
$120$	0	0
$121$	7.75302	0.704820
$122$	0	0
$123$	2.47219i	0.222910i
$124$	0	0
$125$	− 5.37867i	− 0.481083i
$126$	0	0
$127$	−4.42758	−0.392885	−0.196442	−	0.980515i	$-0.562939\pi$
$127$	−4.42758	−0.392885	−0.196442	+	0.980515i	$0.562939\pi$
$128$	0	0
$129$	0.384043	0.0338131
$130$	0	0
$131$	−9.86831	−0.862199	−0.431099	−	0.902305i	$-0.641874\pi$
$131$	−9.86831	−0.862199	−0.431099	+	0.902305i	$0.641874\pi$
$132$	0	0
$133$	−7.18598	−0.623104
$134$	0	0
$135$	0.554958i	0.0477632i
$136$	0	0
$137$	− 3.94438i	− 0.336991i	−0.985702	−	0.168495i	$-0.946109\pi$
$137$	− 3.94438i	− 0.336991i	0.985702	−	0.168495i	$-0.0538909\pi$
$138$	0	0
$139$	−19.7235	−1.67292	−0.836462	−	0.548025i	$-0.815380\pi$
$139$	−19.7235	−1.67292	−0.836462	+	0.548025i	$0.815380\pi$
$140$	0	0
$141$	9.96077i	0.838848i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.43296i	0.285092i
$146$	0	0
$147$	2.29590	0.189362
$148$	0	0
$149$	− 18.6843i	− 1.53067i	−0.643630	−	0.765337i	$-0.722572\pi$
$149$	− 18.6843i	− 1.53067i	0.643630	−	0.765337i	$-0.277428\pi$
$150$	0	0
$151$	− 1.74333i	− 0.141870i	−0.997481	−	0.0709352i	$-0.977402\pi$
$151$	− 1.74333i	− 0.141870i	0.997481	−	0.0709352i	$-0.0225983\pi$
$152$	0	0
$153$	−1.24698	−0.100812
$154$	0	0
$155$	4.81700	0.386911
$156$	0	0
$157$	−17.8116	−1.42152	−0.710761	−	0.703433i	$-0.751649\pi$
$157$	−17.8116	−1.42152	−0.710761	+	0.703433i	$0.751649\pi$
$158$	0	0
$159$	−6.02177	−0.477557
$160$	0	0
$161$	1.69202i	0.133350i
$162$	0	0
$163$	18.3424i	1.43669i	0.695687	+	0.718345i	$0.255100\pi$
$163$	18.3424i	1.43669i	−0.695687	+	0.718345i	$0.744900\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	19.4155i	1.50242i	0.660065	+	0.751208i	$0.270529\pi$
$167$	19.4155i	1.50242i	−0.660065	+	0.751208i	$0.729471\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	2.35690i	0.180236i
$172$	0	0
$173$	−1.33944	−0.101836	−0.0509178	−	0.998703i	$-0.516215\pi$
$173$	−1.33944	−0.101836	−0.0509178	+	0.998703i	$0.516215\pi$
$174$	0	0
$175$	14.3056i	1.08140i
$176$	0	0
$177$	− 7.30559i	− 0.549121i
$178$	0	0
$179$	11.6649	0.871874	0.435937	−	0.899977i	$-0.356417\pi$
$179$	11.6649	0.871874	0.435937	+	0.899977i	$0.356417\pi$
$180$	0	0
$181$	5.85325	0.435068	0.217534	−	0.976053i	$-0.430199\pi$
$181$	5.85325	0.435068	0.217534	+	0.976053i	$0.430199\pi$
$182$	0	0
$183$	12.0586	0.891398
$184$	0	0
$185$	−0.533188	−0.0392008
$186$	0	0
$187$	2.24698i	0.164315i
$188$	0	0
$189$	− 3.04892i	− 0.221776i
$190$	0	0
$191$	18.6950	1.35272	0.676362	−	0.736570i	$-0.263556\pi$
$191$	18.6950	1.35272	0.676362	+	0.736570i	$0.263556\pi$
$192$	0	0
$193$	11.5483i	0.831261i	0.909533	+	0.415631i	$0.136439\pi$
$193$	11.5483i	0.831261i	−0.909533	+	0.415631i	$0.863561\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.3884i	0.882634i	0.897351	+	0.441317i	$0.145489\pi$
$197$	12.3884i	0.882634i	−0.897351	+	0.441317i	$0.854511\pi$
$198$	0	0
$199$	−5.13467	−0.363987	−0.181994	−	0.983300i	$-0.558255\pi$
$199$	−5.13467	−0.363987	−0.181994	+	0.983300i	$0.558255\pi$
$200$	0	0
$201$	− 13.6528i	− 0.962994i
$202$	0	0
$203$	− 18.8605i	− 1.32375i
$204$	0	0
$205$	−1.37196	−0.0958219
$206$	0	0
$207$	0.554958	0.0385723
$208$	0	0
$209$	4.24698	0.293770
$210$	0	0
$211$	−19.8877	−1.36913	−0.684563	−	0.728954i	$-0.740007\pi$
$211$	−19.8877	−1.36913	−0.684563	+	0.728954i	$0.740007\pi$
$212$	0	0
$213$	4.58211i	0.313961i
$214$	0	0
$215$	0.213128i	0.0145352i
$216$	0	0
$217$	−26.4644	−1.79652
$218$	0	0
$219$	1.04892i	0.0708793i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.67994i	0.380357i	0.981750	+	0.190178i	$0.0609067\pi$
$223$	5.67994i	0.380357i	−0.981750	+	0.190178i	$0.939093\pi$
$224$	0	0
$225$	4.69202	0.312801
$226$	0	0
$227$	24.5555i	1.62981i	0.579595	+	0.814905i	$0.303211\pi$
$227$	24.5555i	1.62981i	−0.579595	+	0.814905i	$0.696789\pi$
$228$	0	0
$229$	10.2838i	0.679574i	0.940502	+	0.339787i	$0.110355\pi$
$229$	10.2838i	0.679574i	−0.940502	+	0.339787i	$0.889645\pi$
$230$	0	0
$231$	−5.49396	−0.361476
$232$	0	0
$233$	−8.23490	−0.539486	−0.269743	−	0.962932i	$-0.586939\pi$
$233$	−8.23490	−0.539486	−0.269743	+	0.962932i	$0.586939\pi$
$234$	0	0
$235$	−5.52781	−0.360595
$236$	0	0
$237$	−6.64071	−0.431361
$238$	0	0
$239$	10.9608i	0.708993i	0.935057	+	0.354497i	$0.115348\pi$
$239$	10.9608i	0.708993i	−0.935057	+	0.354497i	$0.884652\pi$
$240$	0	0
$241$	30.4644i	1.96239i	0.193030	+	0.981193i	$0.438169\pi$
$241$	30.4644i	1.96239i	−0.193030	+	0.981193i	$0.561831\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.27413i	0.0814010i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 6.24698i	− 0.395886i
$250$	0	0
$251$	14.0586	0.887371	0.443686	−	0.896182i	$-0.353671\pi$
$251$	14.0586	0.887371	0.443686	+	0.896182i	$0.353671\pi$
$252$	0	0
$253$	− 1.00000i	− 0.0628695i
$254$	0	0
$255$	− 0.692021i	− 0.0433361i
$256$	0	0
$257$	−18.0248	−1.12435	−0.562177	−	0.827017i	$-0.690036\pi$
$257$	−18.0248	−1.12435	−0.562177	+	0.827017i	$0.690036\pi$
$258$	0	0
$259$	2.92931	0.182019
$260$	0	0
$261$	−6.18598	−0.382903
$262$	0	0
$263$	−27.2760	−1.68191	−0.840957	−	0.541103i	$-0.818007\pi$
$263$	−27.2760	−1.68191	−0.840957	+	0.541103i	$0.818007\pi$
$264$	0	0
$265$	− 3.34183i	− 0.205287i
$266$	0	0
$267$	− 13.4330i	− 0.822084i
$268$	0	0
$269$	−3.42327	−0.208721	−0.104360	−	0.994540i	$-0.533280\pi$
$269$	−3.42327	−0.208721	−0.104360	+	0.994540i	$0.533280\pi$
$270$	0	0
$271$	− 6.64310i	− 0.403540i	−0.979433	−	0.201770i	$-0.935331\pi$
$271$	− 6.64310i	− 0.403540i	0.979433	−	0.201770i	$-0.0646693\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8.45473i	− 0.509839i
$276$	0	0
$277$	−20.6329	−1.23971	−0.619857	−	0.784715i	$-0.712809\pi$
$277$	−20.6329	−1.23971	−0.619857	+	0.784715i	$0.712809\pi$
$278$	0	0
$279$	8.67994i	0.519654i
$280$	0	0
$281$	12.0175i	0.716901i	0.933549	+	0.358451i	$0.116695\pi$
$281$	12.0175i	0.716901i	−0.933549	+	0.358451i	$0.883305\pi$
$282$	0	0
$283$	19.5754	1.16364	0.581818	−	0.813319i	$-0.302341\pi$
$283$	19.5754	1.16364	0.581818	+	0.813319i	$0.302341\pi$
$284$	0	0
$285$	−1.30798	−0.0774780
$286$	0	0
$287$	7.53750	0.444925
$288$	0	0
$289$	−15.4450	−0.908532
$290$	0	0
$291$	12.1099i	0.709896i
$292$	0	0
$293$	− 15.5429i	− 0.908025i	−0.890995	−	0.454012i	$-0.849992\pi$
$293$	− 15.5429i	− 0.908025i	0.890995	−	0.454012i	$-0.150008\pi$
$294$	0	0
$295$	4.05429	0.236050
$296$	0	0
$297$	1.80194i	0.104559i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1.17092i	− 0.0674904i
$302$	0	0
$303$	19.0248	1.09294
$304$	0	0
$305$	6.69202i	0.383184i
$306$	0	0
$307$	− 0.606268i	− 0.0346016i	−0.999850	−	0.0173008i	$-0.994493\pi$
$307$	− 0.606268i	− 0.0346016i	0.999850	−	0.0173008i	$-0.00550728\pi$
$308$	0	0
$309$	−10.5483	−0.600069
$310$	0	0
$311$	9.38942	0.532425	0.266213	−	0.963914i	$-0.414228\pi$
$311$	9.38942	0.532425	0.266213	+	0.963914i	$0.414228\pi$
$312$	0	0
$313$	−26.9138	−1.52126	−0.760628	−	0.649188i	$-0.775109\pi$
$313$	−26.9138	−1.52126	−0.760628	+	0.649188i	$0.775109\pi$
$314$	0	0
$315$	1.69202	0.0953346
$316$	0	0
$317$	24.0790i	1.35241i	0.736712	+	0.676207i	$0.236378\pi$
$317$	24.0790i	1.35241i	−0.736712	+	0.676207i	$0.763622\pi$
$318$	0	0
$319$	11.1468i	0.624098i
$320$	0	0
$321$	11.1075	0.619961
$322$	0	0
$323$	− 2.93900i	− 0.163530i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.85623i	− 0.213250i
$328$	0	0
$329$	30.3696	1.67433
$330$	0	0
$331$	19.6799i	1.08171i	0.841117	+	0.540854i	$0.181899\pi$
$331$	19.6799i	1.08171i	−0.841117	+	0.540854i	$0.818101\pi$
$332$	0	0
$333$	− 0.960771i	− 0.0526499i
$334$	0	0
$335$	7.57673	0.413961
$336$	0	0
$337$	−5.27173	−0.287170	−0.143585	−	0.989638i	$-0.545863\pi$
$337$	−5.27173	−0.287170	−0.143585	+	0.989638i	$0.545863\pi$
$338$	0	0
$339$	12.2905	0.667529
$340$	0	0
$341$	15.6407	0.846992
$342$	0	0
$343$	14.3424i	0.774418i
$344$	0	0
$345$	0.307979i	0.0165810i
$346$	0	0
$347$	1.47889	0.0793912	0.0396956	−	0.999212i	$-0.487361\pi$
$347$	1.47889	0.0793912	0.0396956	+	0.999212i	$0.487361\pi$
$348$	0	0
$349$	− 1.64742i	− 0.0881842i	−0.999027	−	0.0440921i	$-0.985961\pi$
$349$	− 1.64742i	− 0.0881842i	0.999027	−	0.0440921i	$-0.0140395\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.41013i	0.447626i	0.974632	+	0.223813i	$0.0718505\pi$
$353$	8.41013i	0.447626i	−0.974632	+	0.223813i	$0.928150\pi$
$354$	0	0
$355$	−2.54288	−0.134962
$356$	0	0
$357$	3.80194i	0.201220i
$358$	0	0
$359$	− 11.1347i	− 0.587665i	−0.955857	−	0.293833i	$-0.905069\pi$
$359$	− 11.1347i	− 0.587665i	0.955857	−	0.293833i	$-0.0949309\pi$
$360$	0	0
$361$	13.4450	0.707634
$362$	0	0
$363$	−7.75302	−0.406928
$364$	0	0
$365$	−0.582105	−0.0304688
$366$	0	0
$367$	−22.9390	−1.19741	−0.598703	−	0.800971i	$-0.704317\pi$
$367$	−22.9390	−1.19741	−0.598703	+	0.800971i	$0.704317\pi$
$368$	0	0
$369$	− 2.47219i	− 0.128697i
$370$	0	0
$371$	18.3599i	0.953197i
$372$	0	0
$373$	−8.35019	−0.432357	−0.216178	−	0.976354i	$-0.569359\pi$
$373$	−8.35019	−0.432357	−0.216178	+	0.976354i	$0.569359\pi$
$374$	0	0
$375$	5.37867i	0.277753i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11.4789i	0.589631i	0.955554	+	0.294816i	$0.0952582\pi$
$379$	11.4789i	0.589631i	−0.955554	+	0.294816i	$0.904742\pi$
$380$	0	0
$381$	4.42758	0.226832
$382$	0	0
$383$	− 6.17092i	− 0.315319i	−0.987494	−	0.157660i	$-0.949605\pi$
$383$	− 6.17092i	− 0.315319i	0.987494	−	0.157660i	$-0.0503948\pi$
$384$	0	0
$385$	− 3.04892i	− 0.155387i
$386$	0	0
$387$	−0.384043	−0.0195220
$388$	0	0
$389$	19.5375	0.990591	0.495295	−	0.868725i	$-0.335060\pi$
$389$	19.5375	0.990591	0.495295	+	0.868725i	$0.335060\pi$
$390$	0	0
$391$	−0.692021	−0.0349970
$392$	0	0
$393$	9.86831	0.497791
$394$	0	0
$395$	− 3.68532i	− 0.185428i
$396$	0	0
$397$	− 7.63235i	− 0.383057i	−0.981487	−	0.191528i	$-0.938656\pi$
$397$	− 7.63235i	− 0.383057i	0.981487	−	0.191528i	$-0.0613444\pi$
$398$	0	0
$399$	7.18598	0.359749
$400$	0	0
$401$	33.6872i	1.68226i	0.540833	+	0.841130i	$0.318109\pi$
$401$	33.6872i	1.68226i	−0.540833	+	0.841130i	$0.681891\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 0.554958i	− 0.0275761i
$406$	0	0
$407$	−1.73125	−0.0858149
$408$	0	0
$409$	− 18.4873i	− 0.914136i	−0.889432	−	0.457068i	$-0.848900\pi$
$409$	− 18.4873i	− 0.914136i	0.889432	−	0.457068i	$-0.151100\pi$
$410$	0	0
$411$	3.94438i	0.194562i
$412$	0	0
$413$	−22.2741	−1.09604
$414$	0	0
$415$	3.46681	0.170179
$416$	0	0
$417$	19.7235	0.965863
$418$	0	0
$419$	0.516794	0.0252471	0.0126235	−	0.999920i	$-0.495982\pi$
$419$	0.516794	0.0252471	0.0126235	+	0.999920i	$0.495982\pi$
$420$	0	0
$421$	− 34.9138i	− 1.70159i	−0.525495	−	0.850797i	$-0.676120\pi$
$421$	− 34.9138i	− 1.70159i	0.525495	−	0.850797i	$-0.323880\pi$
$422$	0	0
$423$	− 9.96077i	− 0.484309i
$424$	0	0
$425$	−5.85086	−0.283808
$426$	0	0
$427$	− 36.7657i	− 1.77922i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 22.7821i	− 1.09737i	−0.836028	−	0.548687i	$-0.815128\pi$
$431$	− 22.7821i	− 1.09737i	0.836028	−	0.548687i	$-0.184872\pi$
$432$	0	0
$433$	−30.8810	−1.48405	−0.742023	−	0.670375i	$-0.766133\pi$
$433$	−30.8810	−1.48405	−0.742023	+	0.670375i	$0.766133\pi$
$434$	0	0
$435$	− 3.43296i	− 0.164598i
$436$	0	0
$437$	1.30798i	0.0625691i
$438$	0	0
$439$	26.2161	1.25123	0.625613	−	0.780133i	$-0.284849\pi$
$439$	26.2161	1.25123	0.625613	+	0.780133i	$0.284849\pi$
$440$	0	0
$441$	−2.29590	−0.109328
$442$	0	0
$443$	15.3884	0.731123	0.365561	−	0.930787i	$-0.380877\pi$
$443$	15.3884	0.731123	0.365561	+	0.930787i	$0.380877\pi$
$444$	0	0
$445$	7.45473	0.353388
$446$	0	0
$447$	18.6843i	0.883735i
$448$	0	0
$449$	− 34.3629i	− 1.62168i	−0.585265	−	0.810842i	$-0.699010\pi$
$449$	− 34.3629i	− 1.62168i	0.585265	−	0.810842i	$-0.300990\pi$
$450$	0	0
$451$	−4.45473	−0.209765
$452$	0	0
$453$	1.74333i	0.0819089i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.7808i	0.878527i	0.898358	+	0.439263i	$0.144761\pi$
$457$	18.7808i	0.878527i	−0.898358	+	0.439263i	$0.855239\pi$
$458$	0	0
$459$	1.24698	0.0582040
$460$	0	0
$461$	− 4.52542i	− 0.210770i	−0.994432	−	0.105385i	$-0.966393\pi$
$461$	− 4.52542i	− 0.210770i	0.994432	−	0.105385i	$-0.0336074\pi$
$462$	0	0
$463$	24.5870i	1.14266i	0.820722	+	0.571328i	$0.193571\pi$
$463$	24.5870i	1.14266i	−0.820722	+	0.571328i	$0.806429\pi$
$464$	0	0
$465$	−4.81700	−0.223383
$466$	0	0
$467$	−25.8901	−1.19805	−0.599025	−	0.800730i	$-0.704445\pi$
$467$	−25.8901	−1.19805	−0.599025	+	0.800730i	$0.704445\pi$
$468$	0	0
$469$	−41.6262	−1.92212
$470$	0	0
$471$	17.8116	0.820716
$472$	0	0
$473$	0.692021i	0.0318192i
$474$	0	0
$475$	11.0586i	0.507404i
$476$	0	0
$477$	6.02177	0.275718
$478$	0	0
$479$	− 5.78315i	− 0.264239i	−0.991234	−	0.132119i	$-0.957822\pi$
$479$	− 5.78315i	− 0.264239i	0.991234	−	0.132119i	$-0.0421783\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 1.69202i	− 0.0769896i
$484$	0	0
$485$	−6.72050	−0.305162
$486$	0	0
$487$	− 4.12067i	− 0.186725i	−0.995632	−	0.0933627i	$-0.970238\pi$
$487$	− 4.12067i	− 0.186725i	0.995632	−	0.0933627i	$-0.0297616\pi$
$488$	0	0
$489$	− 18.3424i	− 0.829473i
$490$	0	0
$491$	−36.6329	−1.65322	−0.826611	−	0.562774i	$-0.809734\pi$
$491$	−36.6329	−1.65322	−0.826611	+	0.562774i	$0.809734\pi$
$492$	0	0
$493$	7.71379	0.347412
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	13.9705	0.626661
$498$	0	0
$499$	− 5.83446i	− 0.261186i	−0.991436	−	0.130593i	$-0.958312\pi$
$499$	− 5.83446i	− 0.261186i	0.991436	−	0.130593i	$-0.0416882\pi$
$500$	0	0
$501$	− 19.4155i	− 0.867421i
$502$	0	0
$503$	35.0683	1.56362	0.781809	−	0.623518i	$-0.214297\pi$
$503$	35.0683	1.56362	0.781809	+	0.623518i	$0.214297\pi$
$504$	0	0
$505$	10.5579i	0.469822i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 21.3357i	− 0.945689i	−0.881146	−	0.472845i	$-0.843227\pi$
$509$	− 21.3357i	− 0.945689i	0.881146	−	0.472845i	$-0.156773\pi$
$510$	0	0
$511$	3.19806	0.141474
$512$	0	0
$513$	− 2.35690i	− 0.104059i
$514$	0	0
$515$	− 5.85384i	− 0.257951i
$516$	0	0
$517$	−17.9487	−0.789382
$518$	0	0
$519$	1.33944	0.0587948
$520$	0	0
$521$	3.52888	0.154603	0.0773014	−	0.997008i	$-0.475370\pi$
$521$	3.52888	0.154603	0.0773014	+	0.997008i	$0.475370\pi$
$522$	0	0
$523$	9.41013	0.411476	0.205738	−	0.978607i	$-0.434041\pi$
$523$	9.41013	0.411476	0.205738	+	0.978607i	$0.434041\pi$
$524$	0	0
$525$	− 14.3056i	− 0.624347i
$526$	0	0
$527$	− 10.8237i	− 0.471488i
$528$	0	0
$529$	−22.6920	−0.986610
$530$	0	0
$531$	7.30559i	0.317035i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.16421i	0.266502i
$536$	0	0
$537$	−11.6649	−0.503376
$538$	0	0
$539$	4.13706i	0.178196i
$540$	0	0
$541$	5.15106i	0.221462i	0.993850	+	0.110731i	$0.0353191\pi$
$541$	5.15106i	0.221462i	−0.993850	+	0.110731i	$0.964681\pi$
$542$	0	0
$543$	−5.85325	−0.251187
$544$	0	0
$545$	2.14005	0.0916696
$546$	0	0
$547$	−7.02848	−0.300516	−0.150258	−	0.988647i	$-0.548010\pi$
$547$	−7.02848	−0.300516	−0.150258	+	0.988647i	$0.548010\pi$
$548$	0	0
$549$	−12.0586	−0.514649
$550$	0	0
$551$	− 14.5797i	− 0.621117i
$552$	0	0
$553$	20.2470i	0.860989i
$554$	0	0
$555$	0.533188	0.0226326
$556$	0	0
$557$	− 31.8442i	− 1.34928i	−0.738147	−	0.674640i	$-0.764299\pi$
$557$	− 31.8442i	− 1.34928i	0.738147	−	0.674640i	$-0.235701\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 2.24698i	− 0.0948676i
$562$	0	0
$563$	−14.1357	−0.595750	−0.297875	−	0.954605i	$-0.596278\pi$
$563$	−14.1357	−0.595750	−0.297875	+	0.954605i	$0.596278\pi$
$564$	0	0
$565$	6.82072i	0.286950i
$566$	0	0
$567$	3.04892i	0.128042i
$568$	0	0
$569$	43.5967	1.82767	0.913834	−	0.406087i	$-0.133107\pi$
$569$	43.5967	1.82767	0.913834	+	0.406087i	$0.133107\pi$
$570$	0	0
$571$	−2.09113	−0.0875111	−0.0437555	−	0.999042i	$-0.513932\pi$
$571$	−2.09113	−0.0875111	−0.0437555	+	0.999042i	$0.513932\pi$
$572$	0	0
$573$	−18.6950	−0.780995
$574$	0	0
$575$	2.60388	0.108589
$576$	0	0
$577$	− 46.6926i	− 1.94384i	−0.235314	−	0.971919i	$-0.575612\pi$
$577$	− 46.6926i	− 1.94384i	0.235314	−	0.971919i	$-0.424388\pi$
$578$	0	0
$579$	− 11.5483i	− 0.479929i
$580$	0	0
$581$	−19.0465	−0.790183
$582$	0	0
$583$	− 10.8509i	− 0.449396i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11.7995i	− 0.487019i	−0.969898	−	0.243510i	$-0.921701\pi$
$587$	− 11.7995i	− 0.487019i	0.969898	−	0.243510i	$-0.0782988\pi$
$588$	0	0
$589$	−20.4577	−0.842945
$590$	0	0
$591$	− 12.3884i	− 0.509589i
$592$	0	0
$593$	22.2392i	0.913255i	0.889658	+	0.456627i	$0.150943\pi$
$593$	22.2392i	0.913255i	−0.889658	+	0.456627i	$0.849057\pi$
$594$	0	0
$595$	−2.10992	−0.0864981
$596$	0	0
$597$	5.13467	0.210148
$598$	0	0
$599$	−5.09592	−0.208213	−0.104107	−	0.994566i	$-0.533198\pi$
$599$	−5.09592	−0.208213	−0.104107	+	0.994566i	$0.533198\pi$
$600$	0	0
$601$	37.6789	1.53695	0.768477	−	0.639878i	$-0.221015\pi$
$601$	37.6789	1.53695	0.768477	+	0.639878i	$0.221015\pi$
$602$	0	0
$603$	13.6528i	0.555985i
$604$	0	0
$605$	− 4.30260i	− 0.174926i
$606$	0	0
$607$	−28.6752	−1.16389	−0.581944	−	0.813229i	$-0.697708\pi$
$607$	−28.6752	−1.16389	−0.581944	+	0.813229i	$0.697708\pi$
$608$	0	0
$609$	18.8605i	0.764268i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 7.45473i	− 0.301094i	−0.988603	−	0.150547i	$-0.951897\pi$
$613$	− 7.45473i	− 0.301094i	0.988603	−	0.150547i	$-0.0481034\pi$
$614$	0	0
$615$	1.37196	0.0553228
$616$	0	0
$617$	− 42.1957i	− 1.69873i	−0.527803	−	0.849367i	$-0.676984\pi$
$617$	− 42.1957i	− 1.69873i	0.527803	−	0.849367i	$-0.323016\pi$
$618$	0	0
$619$	2.87023i	0.115364i	0.998335	+	0.0576822i	$0.0183710\pi$
$619$	2.87023i	0.115364i	−0.998335	+	0.0576822i	$0.981629\pi$
$620$	0	0
$621$	−0.554958	−0.0222697
$622$	0	0
$623$	−40.9560	−1.64087
$624$	0	0
$625$	20.4752	0.819007
$626$	0	0
$627$	−4.24698	−0.169608
$628$	0	0
$629$	1.19806i	0.0477699i
$630$	0	0
$631$	31.5090i	1.25435i	0.778877	+	0.627177i	$0.215790\pi$
$631$	31.5090i	1.25435i	−0.778877	+	0.627177i	$0.784210\pi$
$632$	0	0
$633$	19.8877	0.790465
$634$	0	0
$635$	2.45712i	0.0975080i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 4.58211i	− 0.181265i
$640$	0	0
$641$	39.3376	1.55374	0.776872	−	0.629659i	$-0.216805\pi$
$641$	39.3376	1.55374	0.776872	+	0.629659i	$0.216805\pi$
$642$	0	0
$643$	− 10.3327i	− 0.407483i	−0.979025	−	0.203742i	$-0.934690\pi$
$643$	− 10.3327i	− 0.407483i	0.979025	−	0.203742i	$-0.0653102\pi$
$644$	0	0
$645$	− 0.213128i	− 0.00839190i
$646$	0	0
$647$	17.7972	0.699678	0.349839	−	0.936810i	$-0.386236\pi$
$647$	17.7972	0.699678	0.349839	+	0.936810i	$0.386236\pi$
$648$	0	0
$649$	13.1642	0.516740
$650$	0	0
$651$	26.4644	1.03722
$652$	0	0
$653$	48.3726	1.89296	0.946482	−	0.322756i	$-0.104609\pi$
$653$	48.3726	1.89296	0.946482	+	0.322756i	$0.104609\pi$
$654$	0	0
$655$	5.47650i	0.213985i
$656$	0	0
$657$	− 1.04892i	− 0.0409222i
$658$	0	0
$659$	40.0062	1.55842	0.779211	−	0.626762i	$-0.215620\pi$
$659$	40.0062	1.55842	0.779211	+	0.626762i	$0.215620\pi$
$660$	0	0
$661$	− 38.5816i	− 1.50065i	−0.661068	−	0.750326i	$-0.729897\pi$
$661$	− 38.5816i	− 1.50065i	0.661068	−	0.750326i	$-0.270103\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.98792i	0.154645i
$666$	0	0
$667$	−3.43296	−0.132925
$668$	0	0
$669$	− 5.67994i	− 0.219599i
$670$	0	0
$671$	21.7289i	0.838833i
$672$	0	0
$673$	28.3260	1.09189	0.545944	−	0.837822i	$-0.316171\pi$
$673$	28.3260	1.09189	0.545944	+	0.837822i	$0.316171\pi$
$674$	0	0
$675$	−4.69202	−0.180596
$676$	0	0
$677$	37.2892	1.43314	0.716570	−	0.697515i	$-0.245711\pi$
$677$	37.2892	1.43314	0.716570	+	0.697515i	$0.245711\pi$
$678$	0	0
$679$	36.9221	1.41694
$680$	0	0
$681$	− 24.5555i	− 0.940971i
$682$	0	0
$683$	11.3714i	0.435113i	0.976048	+	0.217557i	$0.0698087\pi$
$683$	11.3714i	0.435113i	−0.976048	+	0.217557i	$0.930191\pi$
$684$	0	0
$685$	−2.18896	−0.0836360
$686$	0	0
$687$	− 10.2838i	− 0.392352i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	17.7450i	0.675051i	0.941316	+	0.337526i	$0.109590\pi$
$691$	17.7450i	0.675051i	−0.941316	+	0.337526i	$0.890410\pi$
$692$	0	0
$693$	5.49396	0.208698
$694$	0	0
$695$	10.9457i	0.415194i
$696$	0	0
$697$	3.08277i	0.116768i
$698$	0	0
$699$	8.23490	0.311472
$700$	0	0
$701$	21.0183	0.793851	0.396925	−	0.917851i	$-0.370077\pi$
$701$	21.0183	0.793851	0.396925	+	0.917851i	$0.370077\pi$
$702$	0	0
$703$	2.26444	0.0854049
$704$	0	0
$705$	5.52781	0.208189
$706$	0	0
$707$	− 58.0049i	− 2.18150i
$708$	0	0
$709$	− 19.8829i	− 0.746718i	−0.927687	−	0.373359i	$-0.878206\pi$
$709$	− 19.8829i	− 0.746718i	0.927687	−	0.373359i	$-0.121794\pi$
$710$	0	0
$711$	6.64071	0.249046
$712$	0	0
$713$	4.81700i	0.180398i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.9608i	− 0.409338i
$718$	0	0
$719$	−39.7560	−1.48265	−0.741324	−	0.671147i	$-0.765802\pi$
$719$	−39.7560	−1.48265	−0.741324	+	0.671147i	$0.765802\pi$
$720$	0	0
$721$	32.1608i	1.19773i
$722$	0	0
$723$	− 30.4644i	− 1.13298i
$724$	0	0
$725$	−29.0248	−1.07795
$726$	0	0
$727$	−18.8194	−0.697973	−0.348986	−	0.937128i	$-0.613474\pi$
$727$	−18.8194	−0.697973	−0.348986	+	0.937128i	$0.613474\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.478894	0.0177125
$732$	0	0
$733$	24.9148i	0.920251i	0.887854	+	0.460125i	$0.152196\pi$
$733$	24.9148i	0.920251i	−0.887854	+	0.460125i	$0.847804\pi$
$734$	0	0
$735$	− 1.27413i	− 0.0469969i
$736$	0	0
$737$	24.6015	0.906207
$738$	0	0
$739$	− 42.1637i	− 1.55102i	−0.631336	−	0.775509i	$-0.717493\pi$
$739$	− 42.1637i	− 1.55102i	0.631336	−	0.775509i	$-0.282507\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.5633i	0.534276i	0.963658	+	0.267138i	$0.0860780\pi$
$743$	14.5633i	0.534276i	−0.963658	+	0.267138i	$0.913922\pi$
$744$	0	0
$745$	−10.3690	−0.379890
$746$	0	0
$747$	6.24698i	0.228565i
$748$	0	0
$749$	− 33.8659i	− 1.23743i
$750$	0	0
$751$	26.2083	0.956356	0.478178	−	0.878263i	$-0.341297\pi$
$751$	26.2083	0.956356	0.478178	+	0.878263i	$0.341297\pi$
$752$	0	0
$753$	−14.0586	−0.512324
$754$	0	0
$755$	−0.967476	−0.0352101
$756$	0	0
$757$	−2.39373	−0.0870017	−0.0435008	−	0.999053i	$-0.513851\pi$
$757$	−2.39373	−0.0870017	−0.0435008	+	0.999053i	$0.513851\pi$
$758$	0	0
$759$	1.00000i	0.0362977i
$760$	0	0
$761$	− 12.3338i	− 0.447100i	−0.974692	−	0.223550i	$-0.928235\pi$
$761$	− 12.3338i	− 0.447100i	0.974692	−	0.223550i	$-0.0717646\pi$
$762$	0	0
$763$	−11.7573	−0.425644
$764$	0	0
$765$	0.692021i	0.0250201i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.7982i	1.11061i	0.831646	+	0.555306i	$0.187399\pi$
$769$	30.7982i	1.11061i	−0.831646	+	0.555306i	$0.812601\pi$
$770$	0	0
$771$	18.0248	0.649146
$772$	0	0
$773$	4.71187i	0.169474i	0.996403	+	0.0847371i	$0.0270051\pi$
$773$	4.71187i	0.169474i	−0.996403	+	0.0847371i	$0.972995\pi$
$774$	0	0
$775$	40.7265i	1.46294i
$776$	0	0
$777$	−2.92931	−0.105088
$778$	0	0
$779$	5.82669	0.208763
$780$	0	0
$781$	−8.25667	−0.295447
$782$	0	0
$783$	6.18598	0.221069
$784$	0	0
$785$	9.88471i	0.352800i
$786$	0	0
$787$	− 33.9627i	− 1.21064i	−0.795983	−	0.605320i	$-0.793045\pi$
$787$	− 33.9627i	− 1.21064i	0.795983	−	0.605320i	$-0.206955\pi$
$788$	0	0
$789$	27.2760	0.971053
$790$	0	0
$791$	− 37.4728i	− 1.33238i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	3.34183i	0.118523i
$796$	0	0
$797$	−13.2405	−0.469004	−0.234502	−	0.972116i	$-0.575346\pi$
$797$	−13.2405	−0.469004	−0.234502	+	0.972116i	$0.575346\pi$
$798$	0	0
$799$	12.4209i	0.439419i
$800$	0	0
$801$	13.4330i	0.474630i
$802$	0	0
$803$	−1.89008	−0.0666996
$804$	0	0
$805$	0.939001	0.0330954
$806$	0	0
$807$	3.42327	0.120505
$808$	0	0
$809$	41.8939	1.47291	0.736456	−	0.676486i	$-0.236498\pi$
$809$	41.8939	1.47291	0.736456	+	0.676486i	$0.236498\pi$
$810$	0	0
$811$	26.9245i	0.945448i	0.881211	+	0.472724i	$0.156729\pi$
$811$	26.9245i	0.945448i	−0.881211	+	0.472724i	$0.843271\pi$
$812$	0	0
$813$	6.64310i	0.232984i
$814$	0	0
$815$	10.1793	0.356564
$816$	0	0
$817$	− 0.905149i	− 0.0316672i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 12.2282i	− 0.426767i	−0.976969	−	0.213383i	$-0.931552\pi$
$821$	− 12.2282i	− 0.426767i	0.976969	−	0.213383i	$-0.0684484\pi$
$822$	0	0
$823$	−13.0054	−0.453339	−0.226669	−	0.973972i	$-0.572784\pi$
$823$	−13.0054	−0.453339	−0.226669	+	0.973972i	$0.572784\pi$
$824$	0	0
$825$	8.45473i	0.294356i
$826$	0	0
$827$	14.5714i	0.506696i	0.967375	+	0.253348i	$0.0815317\pi$
$827$	14.5714i	0.506696i	−0.967375	+	0.253348i	$0.918468\pi$
$828$	0	0
$829$	34.6045	1.20186	0.600931	−	0.799301i	$-0.294796\pi$
$829$	34.6045	1.20186	0.600931	+	0.799301i	$0.294796\pi$
$830$	0	0
$831$	20.6329	0.715749
$832$	0	0
$833$	2.86294	0.0991949
$834$	0	0
$835$	10.7748	0.372877
$836$	0	0
$837$	− 8.67994i	− 0.300023i
$838$	0	0
$839$	9.81641i	0.338900i	0.985539	+	0.169450i	$0.0541992\pi$
$839$	9.81641i	0.338900i	−0.985539	+	0.169450i	$0.945801\pi$
$840$	0	0
$841$	9.26636	0.319530
$842$	0	0
$843$	− 12.0175i	− 0.413903i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	23.6383i	0.812222i
$848$	0	0
$849$	−19.5754	−0.671826
$850$	0	0
$851$	− 0.533188i	− 0.0182774i
$852$	0	0
$853$	39.3752i	1.34818i	0.738649	+	0.674091i	$0.235464\pi$
$853$	39.3752i	1.34818i	−0.738649	+	0.674091i	$0.764536\pi$
$854$	0	0
$855$	1.30798	0.0447319
$856$	0	0
$857$	−43.9081	−1.49987	−0.749937	−	0.661510i	$-0.769916\pi$
$857$	−43.9081	−1.49987	−0.749937	+	0.661510i	$0.769916\pi$
$858$	0	0
$859$	41.8471	1.42781	0.713903	−	0.700245i	$-0.246926\pi$
$859$	41.8471	1.42781	0.713903	+	0.700245i	$0.246926\pi$
$860$	0	0
$861$	−7.53750	−0.256877
$862$	0	0
$863$	− 22.7590i	− 0.774725i	−0.921927	−	0.387362i	$-0.873386\pi$
$863$	− 22.7590i	− 0.774725i	0.921927	−	0.387362i	$-0.126614\pi$
$864$	0	0
$865$	0.743332i	0.0252740i
$866$	0	0
$867$	15.4450	0.524541
$868$	0	0
$869$	− 11.9661i	− 0.405924i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 12.1099i	− 0.409859i
$874$	0	0
$875$	16.3991	0.554391
$876$	0	0
$877$	10.2513i	0.346162i	0.984908	+	0.173081i	$0.0553722\pi$
$877$	10.2513i	0.346162i	−0.984908	+	0.173081i	$0.944628\pi$
$878$	0	0
$879$	15.5429i	0.524248i
$880$	0	0
$881$	27.1745	0.915532	0.457766	−	0.889073i	$-0.348650\pi$
$881$	27.1745	0.915532	0.457766	+	0.889073i	$0.348650\pi$
$882$	0	0
$883$	−19.7469	−0.664536	−0.332268	−	0.943185i	$-0.607814\pi$
$883$	−19.7469	−0.664536	−0.332268	+	0.943185i	$0.607814\pi$
$884$	0	0
$885$	−4.05429	−0.136284
$886$	0	0
$887$	−48.0544	−1.61351	−0.806755	−	0.590887i	$-0.798778\pi$
$887$	−48.0544	−1.61351	−0.806755	+	0.590887i	$0.798778\pi$
$888$	0	0
$889$	− 13.4993i	− 0.452753i
$890$	0	0
$891$	− 1.80194i	− 0.0603672i
$892$	0	0
$893$	23.4765	0.785611
$894$	0	0
$895$	− 6.47352i	− 0.216386i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 53.6939i	− 1.79079i
$900$	0	0
$901$	−7.50902	−0.250162
$902$	0	0
$903$	1.17092i	0.0389656i
$904$	0	0
$905$	− 3.24831i	− 0.107977i
$906$	0	0
$907$	23.0398	0.765025	0.382512	−	0.923950i	$-0.375059\pi$
$907$	23.0398	0.765025	0.382512	+	0.923950i	$0.375059\pi$
$908$	0	0
$909$	−19.0248	−0.631011
$910$	0	0
$911$	0.771807	0.0255711	0.0127855	−	0.999918i	$-0.495930\pi$
$911$	0.771807	0.0255711	0.0127855	+	0.999918i	$0.495930\pi$
$912$	0	0
$913$	11.2567	0.372541
$914$	0	0
$915$	− 6.69202i	− 0.221231i
$916$	0	0
$917$	− 30.0877i	− 0.993582i
$918$	0	0
$919$	−21.0312	−0.693755	−0.346878	−	0.937910i	$-0.612758\pi$
$919$	−21.0312	−0.693755	−0.346878	+	0.937910i	$0.612758\pi$
$920$	0	0
$921$	0.606268i	0.0199772i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.50796i	− 0.148221i
$926$	0	0
$927$	10.5483	0.346450
$928$	0	0
$929$	36.1836i	1.18715i	0.804781	+	0.593573i	$0.202283\pi$
$929$	36.1836i	1.18715i	−0.804781	+	0.593573i	$0.797717\pi$
$930$	0	0
$931$	− 5.41119i	− 0.177345i
$932$	0	0
$933$	−9.38942	−0.307396
$934$	0	0
$935$	1.24698	0.0407806
$936$	0	0
$937$	44.5730	1.45614	0.728068	−	0.685505i	$-0.240418\pi$
$937$	44.5730	1.45614	0.728068	+	0.685505i	$0.240418\pi$
$938$	0	0
$939$	26.9138	0.878298
$940$	0	0
$941$	42.5593i	1.38739i	0.720268	+	0.693696i	$0.244019\pi$
$941$	42.5593i	1.38739i	−0.720268	+	0.693696i	$0.755981\pi$
$942$	0	0
$943$	− 1.37196i	− 0.0446772i
$944$	0	0
$945$	−1.69202	−0.0550415
$946$	0	0
$947$	27.4166i	0.890919i	0.895302	+	0.445459i	$0.146960\pi$
$947$	27.4166i	0.890919i	−0.895302	+	0.445459i	$0.853040\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 24.0790i	− 0.780817i
$952$	0	0
$953$	36.4898	1.18202	0.591010	−	0.806664i	$-0.298729\pi$
$953$	36.4898	1.18202	0.591010	+	0.806664i	$0.298729\pi$
$954$	0	0
$955$	− 10.3749i	− 0.335725i
$956$	0	0
$957$	− 11.1468i	− 0.360323i
$958$	0	0
$959$	12.0261	0.388342
$960$	0	0
$961$	−44.3414	−1.43037
$962$	0	0
$963$	−11.1075	−0.357935
$964$	0	0
$965$	6.40880	0.206306
$966$	0	0
$967$	− 19.1457i	− 0.615684i	−0.951438	−	0.307842i	$-0.900393\pi$
$967$	− 19.1457i	− 0.615684i	0.951438	−	0.307842i	$-0.0996068\pi$
$968$	0	0
$969$	2.93900i	0.0944143i
$970$	0	0
$971$	53.9181	1.73031	0.865157	−	0.501501i	$-0.167219\pi$
$971$	53.9181	1.73031	0.865157	+	0.501501i	$0.167219\pi$
$972$	0	0
$973$	− 60.1353i	− 1.92785i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.6504i	1.07657i	0.842762	+	0.538286i	$0.180928\pi$
$977$	33.6504i	1.07657i	−0.842762	+	0.538286i	$0.819072\pi$
$978$	0	0
$979$	24.2054	0.773607
$980$	0	0
$981$	3.85623i	0.123120i
$982$	0	0
$983$	− 38.0200i	− 1.21265i	−0.795217	−	0.606324i	$-0.792643\pi$
$983$	− 38.0200i	− 1.21265i	0.795217	−	0.606324i	$-0.207357\pi$
$984$	0	0
$985$	6.87502	0.219056
$986$	0	0
$987$	−30.3696	−0.966674
$988$	0	0
$989$	−0.213128	−0.00677707
$990$	0	0
$991$	46.9130	1.49024	0.745121	−	0.666929i	$-0.232392\pi$
$991$	46.9130	1.49024	0.745121	+	0.666929i	$0.232392\pi$
$992$	0	0
$993$	− 19.6799i	− 0.624524i
$994$	0	0
$995$	2.84953i	0.0903361i
$996$	0	0
$997$	25.6179	0.811326	0.405663	−	0.914023i	$-0.367041\pi$
$997$	25.6179	0.811326	0.405663	+	0.914023i	$0.367041\pi$
$998$	0	0
$999$	0.960771i	0.0303975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.n.337.3		6
13.5	odd	4		4056.2.a.x.1.2	✓	3
13.8	odd	4		4056.2.a.ba.1.2	yes	3
13.12	even	2	inner	4056.2.c.n.337.4		6
52.31	even	4		8112.2.a.ci.1.2		3
52.47	even	4		8112.2.a.cn.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.x.1.2	✓	3	13.5	odd	4
4056.2.a.ba.1.2	yes	3	13.8	odd	4
4056.2.c.n.337.3		6	1.1	even	1	trivial
4056.2.c.n.337.4		6	13.12	even	2	inner
8112.2.a.ci.1.2		3	52.31	even	4
8112.2.a.cn.1.2		3	52.47	even	4

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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.554958i q^{5} +3.04892i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.554958i q^{5} +3.04892i q^{7} +1.00000 q^{9} -1.80194i q^{11} +0.554958i q^{15} -1.24698 q^{17} +2.35690i q^{19} -3.04892i q^{21} +0.554958 q^{23} +4.69202 q^{25} -1.00000 q^{27} -6.18598 q^{29} +8.67994i q^{31} +1.80194i q^{33} +1.69202 q^{35} -0.960771i q^{37} -2.47219i q^{41} -0.384043 q^{43} -0.554958i q^{45} -9.96077i q^{47} -2.29590 q^{49} +1.24698 q^{51} +6.02177 q^{53} -1.00000 q^{55} -2.35690i q^{57} +7.30559i q^{59} -12.0586 q^{61} +3.04892i q^{63} +13.6528i q^{67} -0.554958 q^{69} -4.58211i q^{71} -1.04892i q^{73} -4.69202 q^{75} +5.49396 q^{77} +6.64071 q^{79} +1.00000 q^{81} +6.24698i q^{83} +0.692021i q^{85} +6.18598 q^{87} +13.4330i q^{89} -8.67994i q^{93} +1.30798 q^{95} -12.1099i q^{97} -1.80194i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} + 2 q^{17} + 4 q^{23} + 18 q^{25} - 6 q^{27} - 8 q^{29} + 18 q^{43} + 14 q^{49} - 2 q^{51} + 30 q^{53} - 6 q^{55} - 10 q^{61} - 4 q^{69} - 18 q^{75} + 14 q^{77} - 34 q^{79} + 6 q^{81} + 8 q^{87} + 18 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 + 2 * q^17 + 4 * q^23 + 18 * q^25 - 6 * q^27 - 8 * q^29 + 18 * q^43 + 14 * q^49 - 2 * q^51 + 30 * q^53 - 6 * q^55 - 10 * q^61 - 4 * q^69 - 18 * q^75 + 14 * q^77 - 34 * q^79 + 6 * q^81 + 8 * q^87 + 18 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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