LMFDB - Embedded newform 2925.2.c.u.2224.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.u.2224.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{7} +1.58579i q^{8} +O(q^{10})$	$q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{7} +1.58579i q^{8} +2.00000 q^{11} -1.00000i q^{13} +1.17157 q^{14} +3.00000 q^{16} +7.65685i q^{17} +2.82843 q^{19} +0.828427i q^{22} +4.00000i q^{23} +0.414214 q^{26} -5.17157i q^{28} +2.00000 q^{29} -1.17157 q^{31} +4.41421i q^{32} -3.17157 q^{34} +7.65685i q^{37} +1.17157i q^{38} -5.17157 q^{41} -1.65685i q^{43} +3.65685 q^{44} -1.65685 q^{46} -11.6569i q^{47} -1.00000 q^{49} -1.82843i q^{52} +2.00000i q^{53} +4.48528 q^{56} +0.828427i q^{58} +7.65685 q^{59} +13.3137 q^{61} -0.485281i q^{62} +4.17157 q^{64} -6.82843i q^{67} +14.0000i q^{68} -2.00000 q^{71} +0.343146i q^{73} -3.17157 q^{74} +5.17157 q^{76} -5.65685i q^{77} +11.3137 q^{79} -2.14214i q^{82} -3.65685i q^{83} +0.686292 q^{86} +3.17157i q^{88} +14.8284 q^{89} -2.82843 q^{91} +7.31371i q^{92} +4.82843 q^{94} -3.65685i q^{97} -0.414214i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 8 q^{11} + 16 q^{14} + 12 q^{16} - 4 q^{26} + 8 q^{29} - 16 q^{31} - 24 q^{34} - 32 q^{41} - 8 q^{44} + 16 q^{46} - 4 q^{49} - 16 q^{56} + 8 q^{59} + 8 q^{61} + 28 q^{64} - 8 q^{71} - 24 q^{74} + 32 q^{76} + 48 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^11 + 16 * q^14 + 12 * q^16 - 4 * q^26 + 8 * q^29 - 16 * q^31 - 24 * q^34 - 32 * q^41 - 8 * q^44 + 16 * q^46 - 4 * q^49 - 16 * q^56 + 8 * q^59 + 8 * q^61 + 28 * q^64 - 8 * q^71 - 24 * q^74 + 32 * q^76 + 48 * q^86 + 48 * q^89 + 8 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	0	0
$3$	0	0
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	1.58579i	0.560660i
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	1.17157	0.313116
$15$	0	0
$16$	3.00000	0.750000
$17$	7.65685i	1.85706i	0.371257	+	0.928530i	$0.378927\pi$
$17$	7.65685i	1.85706i	−0.371257	+	0.928530i	$0.621073\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0.828427i	0.176621i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0.414214	0.0812340
$27$	0	0
$28$	− 5.17157i	− 0.977335i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	4.41421i	0.780330i
$33$	0	0
$34$	−3.17157	−0.543920
$35$	0	0
$36$	0	0
$37$	7.65685i	1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$37$	7.65685i	1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$38$	1.17157i	0.190054i
$39$	0	0
$40$	0	0
$41$	−5.17157	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$41$	−5.17157	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$42$	0	0
$43$	− 1.65685i	− 0.252668i	−0.991988	−	0.126334i	$-0.959679\pi$
$43$	− 1.65685i	− 0.252668i	0.991988	−	0.126334i	$-0.0403211\pi$
$44$	3.65685	0.551292
$45$	0	0
$46$	−1.65685	−0.244290
$47$	− 11.6569i	− 1.70033i	−0.526519	−	0.850163i	$-0.676503\pi$
$47$	− 11.6569i	− 1.70033i	0.526519	−	0.850163i	$-0.323497\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 1.82843i	− 0.253557i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	4.48528	0.599371
$57$	0	0
$58$	0.828427i	0.108778i
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	− 0.485281i	− 0.0616308i
$63$	0	0
$64$	4.17157	0.521447
$65$	0	0
$66$	0	0
$67$	− 6.82843i	− 0.834225i	−0.908855	−	0.417113i	$-0.863042\pi$
$67$	− 6.82843i	− 0.834225i	0.908855	−	0.417113i	$-0.136958\pi$
$68$	14.0000i	1.69775i
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	0.343146i	0.0401622i	0.999798	+	0.0200811i	$0.00639244\pi$
$73$	0.343146i	0.0401622i	−0.999798	+	0.0200811i	$0.993608\pi$
$74$	−3.17157	−0.368688
$75$	0	0
$76$	5.17157	0.593220
$77$	− 5.65685i	− 0.644658i
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	0	0
$82$	− 2.14214i	− 0.236559i
$83$	− 3.65685i	− 0.401392i	−0.979654	−	0.200696i	$-0.935680\pi$
$83$	− 3.65685i	− 0.401392i	0.979654	−	0.200696i	$-0.0643203\pi$
$84$	0	0
$85$	0	0
$86$	0.686292	0.0740047
$87$	0	0
$88$	3.17157i	0.338091i
$89$	14.8284	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$89$	14.8284	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	7.31371i	0.762507i
$93$	0	0
$94$	4.82843	0.498014
$95$	0	0
$96$	0	0
$97$	− 3.65685i	− 0.371297i	−0.982616	−	0.185649i	$-0.940561\pi$
$97$	− 3.65685i	− 0.371297i	0.982616	−	0.185649i	$-0.0594386\pi$
$98$	− 0.414214i	− 0.0418419i
$99$	0	0
$100$	0	0
$101$	−7.65685	−0.761885	−0.380943	−	0.924599i	$-0.624401\pi$
$101$	−7.65685	−0.761885	−0.380943	+	0.924599i	$0.624401\pi$
$102$	0	0
$103$	2.34315i	0.230877i	0.993315	+	0.115439i	$0.0368273\pi$
$103$	2.34315i	0.230877i	−0.993315	+	0.115439i	$0.963173\pi$
$104$	1.58579	0.155499
$105$	0	0
$106$	−0.828427	−0.0804640
$107$	− 11.3137i	− 1.09374i	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	− 11.3137i	− 1.09374i	0.837218	−	0.546869i	$-0.184180\pi$
$108$	0	0
$109$	−5.31371	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$109$	−5.31371	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$110$	0	0
$111$	0	0
$112$	− 8.48528i	− 0.801784i
$113$	5.31371i	0.499872i	0.968262	+	0.249936i	$0.0804095\pi$
$113$	5.31371i	0.499872i	−0.968262	+	0.249936i	$0.919590\pi$
$114$	0	0
$115$	0	0
$116$	3.65685	0.339530
$117$	0	0
$118$	3.17157i	0.291967i
$119$	21.6569	1.98528
$120$	0	0
$121$	−7.00000	−0.636364
$122$	5.51472i	0.499279i
$123$	0	0
$124$	−2.14214	−0.192369
$125$	0	0
$126$	0	0
$127$	− 5.65685i	− 0.501965i	−0.967992	−	0.250982i	$-0.919246\pi$
$127$	− 5.65685i	− 0.501965i	0.967992	−	0.250982i	$-0.0807536\pi$
$128$	10.5563i	0.933058i
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	2.82843	0.244339
$135$	0	0
$136$	−12.1421	−1.04118
$137$	− 10.8284i	− 0.925135i	−0.886584	−	0.462567i	$-0.846928\pi$
$137$	− 10.8284i	− 0.925135i	0.886584	−	0.462567i	$-0.153072\pi$
$138$	0	0
$139$	7.31371	0.620341	0.310170	−	0.950681i	$-0.399614\pi$
$139$	7.31371	0.620341	0.310170	+	0.950681i	$0.399614\pi$
$140$	0	0
$141$	0	0
$142$	− 0.828427i	− 0.0695201i
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	−0.142136	−0.0117632
$147$	0	0
$148$	14.0000i	1.15079i
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	0	0
$151$	−3.51472	−0.286024	−0.143012	−	0.989721i	$-0.545679\pi$
$151$	−3.51472	−0.286024	−0.143012	+	0.989721i	$0.545679\pi$
$152$	4.48528i	0.363804i
$153$	0	0
$154$	2.34315	0.188816
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	4.68629i	0.372821i
$159$	0	0
$160$	0	0
$161$	11.3137	0.891645
$162$	0	0
$163$	18.8284i	1.47476i	0.675480	+	0.737378i	$0.263936\pi$
$163$	18.8284i	1.47476i	−0.675480	+	0.737378i	$0.736064\pi$
$164$	−9.45584	−0.738377
$165$	0	0
$166$	1.51472	0.117565
$167$	− 3.65685i	− 0.282976i	−0.989940	−	0.141488i	$-0.954811\pi$
$167$	− 3.65685i	− 0.282976i	0.989940	−	0.141488i	$-0.0451886\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 3.02944i	− 0.230992i
$173$	11.6569i	0.886254i	0.896459	+	0.443127i	$0.146131\pi$
$173$	11.6569i	0.886254i	−0.896459	+	0.443127i	$0.853869\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	6.14214i	0.460373i
$179$	−23.3137	−1.74255	−0.871274	−	0.490797i	$-0.836706\pi$
$179$	−23.3137	−1.74255	−0.871274	+	0.490797i	$0.836706\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	− 1.17157i	− 0.0868428i
$183$	0	0
$184$	−6.34315	−0.467623
$185$	0	0
$186$	0	0
$187$	15.3137i	1.11985i
$188$	− 21.3137i	− 1.55446i
$189$	0	0
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	5.31371i	0.382489i	0.981542	+	0.191245i	$0.0612524\pi$
$193$	5.31371i	0.382489i	−0.981542	+	0.191245i	$0.938748\pi$
$194$	1.51472	0.108750
$195$	0	0
$196$	−1.82843	−0.130602
$197$	0.485281i	0.0345749i	0.999851	+	0.0172874i	$0.00550304\pi$
$197$	0.485281i	0.0345749i	−0.999851	+	0.0172874i	$0.994497\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	0	0
$202$	− 3.17157i	− 0.223151i
$203$	− 5.65685i	− 0.397033i
$204$	0	0
$205$	0	0
$206$	−0.970563	−0.0676223
$207$	0	0
$208$	− 3.00000i	− 0.208013i
$209$	5.65685	0.391293
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	3.65685i	0.251154i
$213$	0	0
$214$	4.68629	0.320348
$215$	0	0
$216$	0	0
$217$	3.31371i	0.224949i
$218$	− 2.20101i	− 0.149071i
$219$	0	0
$220$	0	0
$221$	7.65685	0.515056
$222$	0	0
$223$	− 12.4853i	− 0.836076i	−0.908429	−	0.418038i	$-0.862718\pi$
$223$	− 12.4853i	− 0.836076i	0.908429	−	0.418038i	$-0.137282\pi$
$224$	12.4853	0.834208
$225$	0	0
$226$	−2.20101	−0.146409
$227$	− 17.3137i	− 1.14915i	−0.818452	−	0.574576i	$-0.805167\pi$
$227$	− 17.3137i	− 1.14915i	0.818452	−	0.574576i	$-0.194833\pi$
$228$	0	0
$229$	1.31371	0.0868123	0.0434062	−	0.999058i	$-0.486179\pi$
$229$	1.31371	0.0868123	0.0434062	+	0.999058i	$0.486179\pi$
$230$	0	0
$231$	0	0
$232$	3.17157i	0.208224i
$233$	− 6.97056i	− 0.456657i	−0.973584	−	0.228328i	$-0.926674\pi$
$233$	− 6.97056i	− 0.456657i	0.973584	−	0.228328i	$-0.0733260\pi$
$234$	0	0
$235$	0	0
$236$	14.0000	0.911322
$237$	0	0
$238$	8.97056i	0.581475i
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	0.343146	0.0221040	0.0110520	−	0.999939i	$-0.496482\pi$
$241$	0.343146	0.0221040	0.0110520	+	0.999939i	$0.496482\pi$
$242$	− 2.89949i	− 0.186387i
$243$	0	0
$244$	24.3431	1.55841
$245$	0	0
$246$	0	0
$247$	− 2.82843i	− 0.179969i
$248$	− 1.85786i	− 0.117975i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	2.34315	0.147022
$255$	0	0
$256$	3.97056	0.248160
$257$	− 4.34315i	− 0.270918i	−0.990783	−	0.135459i	$-0.956749\pi$
$257$	− 4.34315i	− 0.270918i	0.990783	−	0.135459i	$-0.0432509\pi$
$258$	0	0
$259$	21.6569	1.34569
$260$	0	0
$261$	0	0
$262$	3.31371i	0.204722i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	3.31371	0.203177
$267$	0	0
$268$	− 12.4853i	− 0.762660i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−27.7990	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$271$	−27.7990	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$272$	22.9706i	1.39279i
$273$	0	0
$274$	4.48528	0.270966
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	3.02944i	0.181694i
$279$	0	0
$280$	0	0
$281$	−21.1716	−1.26299	−0.631495	−	0.775380i	$-0.717558\pi$
$281$	−21.1716	−1.26299	−0.631495	+	0.775380i	$0.717558\pi$
$282$	0	0
$283$	28.9706i	1.72212i	0.508502	+	0.861061i	$0.330199\pi$
$283$	28.9706i	1.72212i	−0.508502	+	0.861061i	$0.669801\pi$
$284$	−3.65685	−0.216994
$285$	0	0
$286$	0.828427	0.0489859
$287$	14.6274i	0.863429i
$288$	0	0
$289$	−41.6274	−2.44867
$290$	0	0
$291$	0	0
$292$	0.627417i	0.0367168i
$293$	− 2.14214i	− 0.125145i	−0.998040	−	0.0625724i	$-0.980070\pi$
$293$	− 2.14214i	− 0.125145i	0.998040	−	0.0625724i	$-0.0199304\pi$
$294$	0	0
$295$	0	0
$296$	−12.1421	−0.705747
$297$	0	0
$298$	− 3.79899i	− 0.220070i
$299$	4.00000	0.231326
$300$	0	0
$301$	−4.68629	−0.270113
$302$	− 1.45584i	− 0.0837744i
$303$	0	0
$304$	8.48528	0.486664
$305$	0	0
$306$	0	0
$307$	22.8284i	1.30289i	0.758697	+	0.651444i	$0.225836\pi$
$307$	22.8284i	1.30289i	−0.758697	+	0.651444i	$0.774164\pi$
$308$	− 10.3431i	− 0.589355i
$309$	0	0
$310$	0	0
$311$	10.6274	0.602626	0.301313	−	0.953525i	$-0.402575\pi$
$311$	10.6274	0.602626	0.301313	+	0.953525i	$0.402575\pi$
$312$	0	0
$313$	6.00000i	0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$	6.00000i	0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	−4.14214	−0.233754
$315$	0	0
$316$	20.6863	1.16369
$317$	8.48528i	0.476581i	0.971194	+	0.238290i	$0.0765870\pi$
$317$	8.48528i	0.476581i	−0.971194	+	0.238290i	$0.923413\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	4.68629i	0.261157i
$323$	21.6569i	1.20502i
$324$	0	0
$325$	0	0
$326$	−7.79899	−0.431946
$327$	0	0
$328$	− 8.20101i	− 0.452825i
$329$	−32.9706	−1.81773
$330$	0	0
$331$	26.1421	1.43690	0.718451	−	0.695578i	$-0.244852\pi$
$331$	26.1421	1.43690	0.718451	+	0.695578i	$0.244852\pi$
$332$	− 6.68629i	− 0.366958i
$333$	0	0
$334$	1.51472	0.0828817
$335$	0	0
$336$	0	0
$337$	− 9.31371i	− 0.507350i	−0.967290	−	0.253675i	$-0.918361\pi$
$337$	− 9.31371i	− 0.507350i	0.967290	−	0.253675i	$-0.0816394\pi$
$338$	− 0.414214i	− 0.0225302i
$339$	0	0
$340$	0	0
$341$	−2.34315	−0.126888
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	2.62742	0.141661
$345$	0	0
$346$	−4.82843	−0.259578
$347$	− 8.68629i	− 0.466305i	−0.972440	−	0.233152i	$-0.925096\pi$
$347$	− 8.68629i	− 0.466305i	0.972440	−	0.233152i	$-0.0749041\pi$
$348$	0	0
$349$	−3.65685	−0.195747	−0.0978735	−	0.995199i	$-0.531204\pi$
$349$	−3.65685	−0.195747	−0.0978735	+	0.995199i	$0.531204\pi$
$350$	0	0
$351$	0	0
$352$	8.82843i	0.470557i
$353$	33.4558i	1.78067i	0.455301	+	0.890337i	$0.349532\pi$
$353$	33.4558i	1.78067i	−0.455301	+	0.890337i	$0.650468\pi$
$354$	0	0
$355$	0	0
$356$	27.1127	1.43697
$357$	0	0
$358$	− 9.65685i	− 0.510381i
$359$	34.9706	1.84568	0.922838	−	0.385189i	$-0.125864\pi$
$359$	34.9706	1.84568	0.922838	+	0.385189i	$0.125864\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	5.79899i	0.304788i
$363$	0	0
$364$	−5.17157	−0.271064
$365$	0	0
$366$	0	0
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	12.0000i	0.625543i
$369$	0	0
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	−6.34315	−0.327996
$375$	0	0
$376$	18.4853	0.953306
$377$	− 2.00000i	− 0.103005i
$378$	0	0
$379$	−0.485281	−0.0249272	−0.0124636	−	0.999922i	$-0.503967\pi$
$379$	−0.485281	−0.0249272	−0.0124636	+	0.999922i	$0.503967\pi$
$380$	0	0
$381$	0	0
$382$	− 1.37258i	− 0.0702275i
$383$	− 30.9706i	− 1.58252i	−0.611479	−	0.791261i	$-0.709425\pi$
$383$	− 30.9706i	− 1.58252i	0.611479	−	0.791261i	$-0.290575\pi$
$384$	0	0
$385$	0	0
$386$	−2.20101	−0.112028
$387$	0	0
$388$	− 6.68629i	− 0.339445i
$389$	−26.9706	−1.36746	−0.683731	−	0.729734i	$-0.739644\pi$
$389$	−26.9706	−1.36746	−0.683731	+	0.729734i	$0.739644\pi$
$390$	0	0
$391$	−30.6274	−1.54890
$392$	− 1.58579i	− 0.0800943i
$393$	0	0
$394$	−0.201010	−0.0101267
$395$	0	0
$396$	0	0
$397$	− 30.9706i	− 1.55437i	−0.629273	−	0.777184i	$-0.716647\pi$
$397$	− 30.9706i	− 1.55437i	0.629273	−	0.777184i	$-0.283353\pi$
$398$	− 8.97056i	− 0.449654i
$399$	0	0
$400$	0	0
$401$	−26.1421	−1.30548	−0.652738	−	0.757584i	$-0.726380\pi$
$401$	−26.1421	−1.30548	−0.652738	+	0.757584i	$0.726380\pi$
$402$	0	0
$403$	1.17157i	0.0583602i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	2.34315	0.116288
$407$	15.3137i	0.759072i
$408$	0	0
$409$	34.9706	1.72918	0.864592	−	0.502475i	$-0.167577\pi$
$409$	34.9706	1.72918	0.864592	+	0.502475i	$0.167577\pi$
$410$	0	0
$411$	0	0
$412$	4.28427i	0.211071i
$413$	− 21.6569i	− 1.06566i
$414$	0	0
$415$	0	0
$416$	4.41421	0.216425
$417$	0	0
$418$	2.34315i	0.114607i
$419$	14.6274	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$419$	14.6274	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$420$	0	0
$421$	37.3137	1.81856	0.909279	−	0.416186i	$-0.136634\pi$
$421$	37.3137	1.81856	0.909279	+	0.416186i	$0.136634\pi$
$422$	− 4.97056i	− 0.241963i
$423$	0	0
$424$	−3.17157	−0.154025
$425$	0	0
$426$	0	0
$427$	− 37.6569i	− 1.82234i
$428$	− 20.6863i	− 0.999910i
$429$	0	0
$430$	0	0
$431$	−8.34315	−0.401875	−0.200938	−	0.979604i	$-0.564399\pi$
$431$	−8.34315	−0.401875	−0.200938	+	0.979604i	$0.564399\pi$
$432$	0	0
$433$	− 21.3137i	− 1.02427i	−0.858905	−	0.512136i	$-0.828854\pi$
$433$	− 21.3137i	− 1.02427i	0.858905	−	0.512136i	$-0.171146\pi$
$434$	−1.37258	−0.0658861
$435$	0	0
$436$	−9.71573	−0.465299
$437$	11.3137i	0.541208i
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	0	0
$442$	3.17157i	0.150856i
$443$	25.9411i	1.23250i	0.787551	+	0.616250i	$0.211349\pi$
$443$	25.9411i	1.23250i	−0.787551	+	0.616250i	$0.788651\pi$
$444$	0	0
$445$	0	0
$446$	5.17157	0.244881
$447$	0	0
$448$	− 11.7990i	− 0.557450i
$449$	−31.7990	−1.50069	−0.750344	−	0.661048i	$-0.770112\pi$
$449$	−31.7990	−1.50069	−0.750344	+	0.661048i	$0.770112\pi$
$450$	0	0
$451$	−10.3431	−0.487040
$452$	9.71573i	0.456989i
$453$	0	0
$454$	7.17157	0.336579
$455$	0	0
$456$	0	0
$457$	7.65685i	0.358173i	0.983833	+	0.179086i	$0.0573141\pi$
$457$	7.65685i	0.358173i	−0.983833	+	0.179086i	$0.942686\pi$
$458$	0.544156i	0.0254267i
$459$	0	0
$460$	0	0
$461$	−5.17157	−0.240864	−0.120432	−	0.992722i	$-0.538428\pi$
$461$	−5.17157	−0.240864	−0.120432	+	0.992722i	$0.538428\pi$
$462$	0	0
$463$	− 24.4853i	− 1.13793i	−0.822363	−	0.568964i	$-0.807344\pi$
$463$	− 24.4853i	− 1.13793i	0.822363	−	0.568964i	$-0.192656\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	2.88730	0.133752
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	0	0
$469$	−19.3137	−0.891824
$470$	0	0
$471$	0	0
$472$	12.1421i	0.558887i
$473$	− 3.31371i	− 0.152364i
$474$	0	0
$475$	0	0
$476$	39.5980	1.81497
$477$	0	0
$478$	0.828427i	0.0378914i
$479$	25.3137	1.15661	0.578306	−	0.815820i	$-0.303714\pi$
$479$	25.3137	1.15661	0.578306	+	0.815820i	$0.303714\pi$
$480$	0	0
$481$	7.65685	0.349123
$482$	0.142136i	0.00647410i
$483$	0	0
$484$	−12.7990	−0.581772
$485$	0	0
$486$	0	0
$487$	7.79899i	0.353406i	0.984264	+	0.176703i	$0.0565432\pi$
$487$	7.79899i	0.353406i	−0.984264	+	0.176703i	$0.943457\pi$
$488$	21.1127i	0.955727i
$489$	0	0
$490$	0	0
$491$	−30.6274	−1.38220	−0.691098	−	0.722761i	$-0.742873\pi$
$491$	−30.6274	−1.38220	−0.691098	+	0.722761i	$0.742873\pi$
$492$	0	0
$493$	15.3137i	0.689695i
$494$	1.17157	0.0527116
$495$	0	0
$496$	−3.51472	−0.157816
$497$	5.65685i	0.253745i
$498$	0	0
$499$	−26.1421	−1.17028	−0.585141	−	0.810931i	$-0.698961\pi$
$499$	−26.1421	−1.17028	−0.585141	+	0.810931i	$0.698961\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.31371i	0.326102i	0.986618	+	0.163051i	$0.0521335\pi$
$503$	7.31371i	0.326102i	−0.986618	+	0.163051i	$0.947866\pi$
$504$	0	0
$505$	0	0
$506$	−3.31371	−0.147312
$507$	0	0
$508$	− 10.3431i	− 0.458903i
$509$	11.7990	0.522981	0.261491	−	0.965206i	$-0.415786\pi$
$509$	11.7990	0.522981	0.261491	+	0.965206i	$0.415786\pi$
$510$	0	0
$511$	0.970563	0.0429352
$512$	22.7574i	1.00574i
$513$	0	0
$514$	1.79899	0.0793500
$515$	0	0
$516$	0	0
$517$	− 23.3137i	− 1.02534i
$518$	8.97056i	0.394144i
$519$	0	0
$520$	0	0
$521$	−25.3137	−1.10901	−0.554507	−	0.832179i	$-0.687093\pi$
$521$	−25.3137	−1.10901	−0.554507	+	0.832179i	$0.687093\pi$
$522$	0	0
$523$	− 15.3137i	− 0.669622i	−0.942285	−	0.334811i	$-0.891328\pi$
$523$	− 15.3137i	− 0.669622i	0.942285	−	0.334811i	$-0.108672\pi$
$524$	14.6274	0.639002
$525$	0	0
$526$	4.97056	0.216727
$527$	− 8.97056i	− 0.390764i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	− 14.6274i	− 0.634179i
$533$	5.17157i	0.224006i
$534$	0	0
$535$	0	0
$536$	10.8284	0.467717
$537$	0	0
$538$	7.45584i	0.321444i
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	− 11.5147i	− 0.494600i
$543$	0	0
$544$	−33.7990	−1.44912
$545$	0	0
$546$	0	0
$547$	− 23.3137i	− 0.996822i	−0.866941	−	0.498411i	$-0.833917\pi$
$547$	− 23.3137i	− 0.996822i	0.866941	−	0.498411i	$-0.166083\pi$
$548$	− 19.7990i	− 0.845771i
$549$	0	0
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	−0.828427	−0.0351965
$555$	0	0
$556$	13.3726	0.567124
$557$	− 7.79899i	− 0.330454i	−0.986256	−	0.165227i	$-0.947164\pi$
$557$	− 7.79899i	− 0.330454i	0.986256	−	0.165227i	$-0.0528356\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	0	0
$562$	− 8.76955i	− 0.369921i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	− 3.17157i	− 0.133076i
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	− 3.65685i	− 0.152901i
$573$	0	0
$574$	−6.05887	−0.252893
$575$	0	0
$576$	0	0
$577$	31.9411i	1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$577$	31.9411i	1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$578$	− 17.2426i	− 0.717199i
$579$	0	0
$580$	0	0
$581$	−10.3431	−0.429106
$582$	0	0
$583$	4.00000i	0.165663i
$584$	−0.544156	−0.0225173
$585$	0	0
$586$	0.887302	0.0366541
$587$	− 10.9706i	− 0.452804i	−0.974034	−	0.226402i	$-0.927304\pi$
$587$	− 10.9706i	− 0.452804i	0.974034	−	0.226402i	$-0.0726962\pi$
$588$	0	0
$589$	−3.31371	−0.136539
$590$	0	0
$591$	0	0
$592$	22.9706i	0.944084i
$593$	20.4853i	0.841230i	0.907239	+	0.420615i	$0.138186\pi$
$593$	20.4853i	0.841230i	−0.907239	+	0.420615i	$0.861814\pi$
$594$	0	0
$595$	0	0
$596$	−16.7696	−0.686908
$597$	0	0
$598$	1.65685i	0.0677538i
$599$	−23.3137	−0.952572	−0.476286	−	0.879290i	$-0.658017\pi$
$599$	−23.3137	−0.952572	−0.476286	+	0.879290i	$0.658017\pi$
$600$	0	0
$601$	−0.627417	−0.0255929	−0.0127964	−	0.999918i	$-0.504073\pi$
$601$	−0.627417	−0.0255929	−0.0127964	+	0.999918i	$0.504073\pi$
$602$	− 1.94113i	− 0.0791144i
$603$	0	0
$604$	−6.42641	−0.261487
$605$	0	0
$606$	0	0
$607$	− 41.9411i	− 1.70234i	−0.524892	−	0.851169i	$-0.675894\pi$
$607$	− 41.9411i	− 1.70234i	0.524892	−	0.851169i	$-0.324106\pi$
$608$	12.4853i	0.506345i
$609$	0	0
$610$	0	0
$611$	−11.6569	−0.471586
$612$	0	0
$613$	− 47.6569i	− 1.92484i	−0.271561	−	0.962421i	$-0.587540\pi$
$613$	− 47.6569i	− 1.92484i	0.271561	−	0.962421i	$-0.412460\pi$
$614$	−9.45584	−0.381607
$615$	0	0
$616$	8.97056	0.361434
$617$	− 34.8284i	− 1.40214i	−0.713093	−	0.701070i	$-0.752706\pi$
$617$	− 34.8284i	− 1.40214i	0.713093	−	0.701070i	$-0.247294\pi$
$618$	0	0
$619$	−23.7990	−0.956562	−0.478281	−	0.878207i	$-0.658740\pi$
$619$	−23.7990	−0.956562	−0.478281	+	0.878207i	$0.658740\pi$
$620$	0	0
$621$	0	0
$622$	4.40202i	0.176505i
$623$	− 41.9411i	− 1.68034i
$624$	0	0
$625$	0	0
$626$	−2.48528	−0.0993318
$627$	0	0
$628$	18.2843i	0.729622i
$629$	−58.6274	−2.33763
$630$	0	0
$631$	43.1127	1.71629	0.858145	−	0.513408i	$-0.171617\pi$
$631$	43.1127	1.71629	0.858145	+	0.513408i	$0.171617\pi$
$632$	17.9411i	0.713660i
$633$	0	0
$634$	−3.51472	−0.139587
$635$	0	0
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	1.65685i	0.0655955i
$639$	0	0
$640$	0	0
$641$	−30.2843	−1.19616	−0.598078	−	0.801438i	$-0.704069\pi$
$641$	−30.2843	−1.19616	−0.598078	+	0.801438i	$0.704069\pi$
$642$	0	0
$643$	22.8284i	0.900265i	0.892962	+	0.450133i	$0.148623\pi$
$643$	22.8284i	0.900265i	−0.892962	+	0.450133i	$0.851377\pi$
$644$	20.6863	0.815154
$645$	0	0
$646$	−8.97056	−0.352942
$647$	11.3137i	0.444788i	0.974957	+	0.222394i	$0.0713871\pi$
$647$	11.3137i	0.444788i	−0.974957	+	0.222394i	$0.928613\pi$
$648$	0	0
$649$	15.3137	0.601116
$650$	0	0
$651$	0	0
$652$	34.4264i	1.34824i
$653$	25.3137i	0.990602i	0.868721	+	0.495301i	$0.164942\pi$
$653$	25.3137i	0.990602i	−0.868721	+	0.495301i	$0.835058\pi$
$654$	0	0
$655$	0	0
$656$	−15.5147	−0.605748
$657$	0	0
$658$	− 13.6569i	− 0.532400i
$659$	−47.3137	−1.84308	−0.921540	−	0.388283i	$-0.873068\pi$
$659$	−47.3137	−1.84308	−0.921540	+	0.388283i	$0.873068\pi$
$660$	0	0
$661$	−34.9706	−1.36020	−0.680099	−	0.733121i	$-0.738063\pi$
$661$	−34.9706	−1.36020	−0.680099	+	0.733121i	$0.738063\pi$
$662$	10.8284i	0.420859i
$663$	0	0
$664$	5.79899	0.225044
$665$	0	0
$666$	0	0
$667$	8.00000i	0.309761i
$668$	− 6.68629i	− 0.258700i
$669$	0	0
$670$	0	0
$671$	26.6274	1.02794
$672$	0	0
$673$	16.6274i	0.640940i	0.947259	+	0.320470i	$0.103841\pi$
$673$	16.6274i	0.640940i	−0.947259	+	0.320470i	$0.896159\pi$
$674$	3.85786	0.148599
$675$	0	0
$676$	−1.82843	−0.0703241
$677$	26.6863i	1.02564i	0.858497	+	0.512819i	$0.171399\pi$
$677$	26.6863i	1.02564i	−0.858497	+	0.512819i	$0.828601\pi$
$678$	0	0
$679$	−10.3431	−0.396934
$680$	0	0
$681$	0	0
$682$	− 0.970563i	− 0.0371648i
$683$	− 47.9411i	− 1.83442i	−0.398408	−	0.917208i	$-0.630437\pi$
$683$	− 47.9411i	− 1.83442i	0.398408	−	0.917208i	$-0.369563\pi$
$684$	0	0
$685$	0	0
$686$	7.02944	0.268385
$687$	0	0
$688$	− 4.97056i	− 0.189501i
$689$	2.00000	0.0761939
$690$	0	0
$691$	−5.85786	−0.222844	−0.111422	−	0.993773i	$-0.535540\pi$
$691$	−5.85786	−0.222844	−0.111422	+	0.993773i	$0.535540\pi$
$692$	21.3137i	0.810226i
$693$	0	0
$694$	3.59798	0.136577
$695$	0	0
$696$	0	0
$697$	− 39.5980i	− 1.49988i
$698$	− 1.51472i	− 0.0573329i
$699$	0	0
$700$	0	0
$701$	−5.02944	−0.189959	−0.0949796	−	0.995479i	$-0.530279\pi$
$701$	−5.02944	−0.189959	−0.0949796	+	0.995479i	$0.530279\pi$
$702$	0	0
$703$	21.6569i	0.816804i
$704$	8.34315	0.314444
$705$	0	0
$706$	−13.8579	−0.521548
$707$	21.6569i	0.814490i
$708$	0	0
$709$	4.62742	0.173786	0.0868931	−	0.996218i	$-0.472306\pi$
$709$	4.62742	0.173786	0.0868931	+	0.996218i	$0.472306\pi$
$710$	0	0
$711$	0	0
$712$	23.5147i	0.881251i
$713$	− 4.68629i	− 0.175503i
$714$	0	0
$715$	0	0
$716$	−42.6274	−1.59306
$717$	0	0
$718$	14.4853i	0.540586i
$719$	−29.9411	−1.11662	−0.558308	−	0.829634i	$-0.688549\pi$
$719$	−29.9411	−1.11662	−0.558308	+	0.829634i	$0.688549\pi$
$720$	0	0
$721$	6.62742	0.246818
$722$	− 4.55635i	− 0.169570i
$723$	0	0
$724$	25.5980	0.951341
$725$	0	0
$726$	0	0
$727$	10.3431i	0.383606i	0.981433	+	0.191803i	$0.0614335\pi$
$727$	10.3431i	0.383606i	−0.981433	+	0.191803i	$0.938567\pi$
$728$	− 4.48528i	− 0.166236i
$729$	0	0
$730$	0	0
$731$	12.6863	0.469219
$732$	0	0
$733$	− 36.6274i	− 1.35286i	−0.736505	−	0.676432i	$-0.763525\pi$
$733$	− 36.6274i	− 1.35286i	0.736505	−	0.676432i	$-0.236475\pi$
$734$	−9.94113	−0.366934
$735$	0	0
$736$	−17.6569	−0.650840
$737$	− 13.6569i	− 0.503057i
$738$	0	0
$739$	18.1421	0.667369	0.333685	−	0.942685i	$-0.391708\pi$
$739$	18.1421	0.667369	0.333685	+	0.942685i	$0.391708\pi$
$740$	0	0
$741$	0	0
$742$	2.34315i	0.0860196i
$743$	− 2.00000i	− 0.0733729i	−0.999327	−	0.0366864i	$-0.988320\pi$
$743$	− 2.00000i	− 0.0733729i	0.999327	−	0.0366864i	$-0.0116803\pi$
$744$	0	0
$745$	0	0
$746$	−4.14214	−0.151654
$747$	0	0
$748$	28.0000i	1.02378i
$749$	−32.0000	−1.16925
$750$	0	0
$751$	−0.970563	−0.0354163	−0.0177082	−	0.999843i	$-0.505637\pi$
$751$	−0.970563	−0.0354163	−0.0177082	+	0.999843i	$0.505637\pi$
$752$	− 34.9706i	− 1.27525i
$753$	0	0
$754$	0.828427	0.0301695
$755$	0	0
$756$	0	0
$757$	− 51.9411i	− 1.88783i	−0.330185	−	0.943916i	$-0.607111\pi$
$757$	− 51.9411i	− 1.88783i	0.330185	−	0.943916i	$-0.392889\pi$
$758$	− 0.201010i	− 0.00730102i
$759$	0	0
$760$	0	0
$761$	−32.4853	−1.17759	−0.588795	−	0.808282i	$-0.700398\pi$
$761$	−32.4853	−1.17759	−0.588795	+	0.808282i	$0.700398\pi$
$762$	0	0
$763$	15.0294i	0.544102i
$764$	−6.05887	−0.219202
$765$	0	0
$766$	12.8284	0.463510
$767$	− 7.65685i	− 0.276473i
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	0	0
$772$	9.71573i	0.349677i
$773$	− 34.1421i	− 1.22801i	−0.789303	−	0.614004i	$-0.789558\pi$
$773$	− 34.1421i	− 1.22801i	0.789303	−	0.614004i	$-0.210442\pi$
$774$	0	0
$775$	0	0
$776$	5.79899	0.208172
$777$	0	0
$778$	− 11.1716i	− 0.400520i
$779$	−14.6274	−0.524082
$780$	0	0
$781$	−4.00000	−0.143131
$782$	− 12.6863i	− 0.453661i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	40.7696i	1.45328i	0.687020	+	0.726639i	$0.258919\pi$
$787$	40.7696i	1.45328i	−0.687020	+	0.726639i	$0.741081\pi$
$788$	0.887302i	0.0316088i
$789$	0	0
$790$	0	0
$791$	15.0294	0.534385
$792$	0	0
$793$	− 13.3137i	− 0.472784i
$794$	12.8284	0.455264
$795$	0	0
$796$	−39.5980	−1.40351
$797$	24.3431i	0.862278i	0.902285	+	0.431139i	$0.141888\pi$
$797$	24.3431i	0.862278i	−0.902285	+	0.431139i	$0.858112\pi$
$798$	0	0
$799$	89.2548	3.15761
$800$	0	0
$801$	0	0
$802$	− 10.8284i	− 0.382365i
$803$	0.686292i	0.0242187i
$804$	0	0
$805$	0	0
$806$	−0.485281	−0.0170933
$807$	0	0
$808$	− 12.1421i	− 0.427159i
$809$	18.6863	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$809$	18.6863	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$810$	0	0
$811$	−30.1421	−1.05843	−0.529217	−	0.848487i	$-0.677514\pi$
$811$	−30.1421	−1.05843	−0.529217	+	0.848487i	$0.677514\pi$
$812$	− 10.3431i	− 0.362973i
$813$	0	0
$814$	−6.34315	−0.222327
$815$	0	0
$816$	0	0
$817$	− 4.68629i	− 0.163953i
$818$	14.4853i	0.506466i
$819$	0	0
$820$	0	0
$821$	23.7990	0.830590	0.415295	−	0.909687i	$-0.363678\pi$
$821$	23.7990	0.830590	0.415295	+	0.909687i	$0.363678\pi$
$822$	0	0
$823$	15.0294i	0.523893i	0.965082	+	0.261947i	$0.0843644\pi$
$823$	15.0294i	0.523893i	−0.965082	+	0.261947i	$0.915636\pi$
$824$	−3.71573	−0.129444
$825$	0	0
$826$	8.97056	0.312126
$827$	− 26.0000i	− 0.904109i	−0.891990	−	0.452054i	$-0.850691\pi$
$827$	− 26.0000i	− 0.904109i	0.891990	−	0.452054i	$-0.149309\pi$
$828$	0	0
$829$	17.3137	0.601330	0.300665	−	0.953730i	$-0.402791\pi$
$829$	17.3137	0.601330	0.300665	+	0.953730i	$0.402791\pi$
$830$	0	0
$831$	0	0
$832$	− 4.17157i	− 0.144623i
$833$	− 7.65685i	− 0.265294i
$834$	0	0
$835$	0	0
$836$	10.3431	0.357725
$837$	0	0
$838$	6.05887i	0.209300i
$839$	43.2548	1.49332	0.746661	−	0.665204i	$-0.231656\pi$
$839$	43.2548	1.49332	0.746661	+	0.665204i	$0.231656\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	15.4558i	0.532644i
$843$	0	0
$844$	−21.9411	−0.755245
$845$	0	0
$846$	0	0
$847$	19.7990i	0.680301i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−30.6274	−1.04989
$852$	0	0
$853$	3.65685i	0.125208i	0.998038	+	0.0626042i	$0.0199406\pi$
$853$	3.65685i	0.125208i	−0.998038	+	0.0626042i	$0.980059\pi$
$854$	15.5980	0.533752
$855$	0	0
$856$	17.9411	0.613215
$857$	− 49.5980i	− 1.69423i	−0.531406	−	0.847117i	$-0.678336\pi$
$857$	− 49.5980i	− 1.69423i	0.531406	−	0.847117i	$-0.321664\pi$
$858$	0	0
$859$	0.686292	0.0234160	0.0117080	−	0.999931i	$-0.496273\pi$
$859$	0.686292	0.0234160	0.0117080	+	0.999931i	$0.496273\pi$
$860$	0	0
$861$	0	0
$862$	− 3.45584i	− 0.117707i
$863$	28.3431i	0.964812i	0.875948	+	0.482406i	$0.160237\pi$
$863$	28.3431i	0.964812i	−0.875948	+	0.482406i	$0.839763\pi$
$864$	0	0
$865$	0	0
$866$	8.82843	0.300002
$867$	0	0
$868$	6.05887i	0.205652i
$869$	22.6274	0.767583
$870$	0	0
$871$	−6.82843	−0.231372
$872$	− 8.42641i	− 0.285354i
$873$	0	0
$874$	−4.68629	−0.158516
$875$	0	0
$876$	0	0
$877$	− 42.2843i	− 1.42784i	−0.700228	−	0.713919i	$-0.746918\pi$
$877$	− 42.2843i	− 1.42784i	0.700228	−	0.713919i	$-0.253082\pi$
$878$	− 7.02944i	− 0.237232i
$879$	0	0
$880$	0	0
$881$	25.5980	0.862418	0.431209	−	0.902252i	$-0.358087\pi$
$881$	25.5980	0.862418	0.431209	+	0.902252i	$0.358087\pi$
$882$	0	0
$883$	27.5980i	0.928746i	0.885640	+	0.464373i	$0.153720\pi$
$883$	27.5980i	0.928746i	−0.885640	+	0.464373i	$0.846280\pi$
$884$	14.0000	0.470871
$885$	0	0
$886$	−10.7452	−0.360991
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	− 22.8284i	− 0.764352i
$893$	− 32.9706i	− 1.10332i
$894$	0	0
$895$	0	0
$896$	29.8579	0.997481
$897$	0	0
$898$	− 13.1716i	− 0.439541i
$899$	−2.34315	−0.0781483
$900$	0	0
$901$	−15.3137	−0.510174
$902$	− 4.28427i	− 0.142651i
$903$	0	0
$904$	−8.42641	−0.280258
$905$	0	0
$906$	0	0
$907$	12.9706i	0.430680i	0.976539	+	0.215340i	$0.0690860\pi$
$907$	12.9706i	0.430680i	−0.976539	+	0.215340i	$0.930914\pi$
$908$	− 31.6569i	− 1.05057i
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	− 7.31371i	− 0.242048i
$914$	−3.17157	−0.104906
$915$	0	0
$916$	2.40202	0.0793650
$917$	− 22.6274i	− 0.747223i
$918$	0	0
$919$	−3.31371	−0.109309	−0.0546546	−	0.998505i	$-0.517406\pi$
$919$	−3.31371	−0.109309	−0.0546546	+	0.998505i	$0.517406\pi$
$920$	0	0
$921$	0	0
$922$	− 2.14214i	− 0.0705475i
$923$	2.00000i	0.0658308i
$924$	0	0
$925$	0	0
$926$	10.1421	0.333291
$927$	0	0
$928$	8.82843i	0.289807i
$929$	11.7990	0.387112	0.193556	−	0.981089i	$-0.437998\pi$
$929$	11.7990	0.387112	0.193556	+	0.981089i	$0.437998\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	− 12.7452i	− 0.417482i
$933$	0	0
$934$	3.31371	0.108428
$935$	0	0
$936$	0	0
$937$	21.3137i	0.696289i	0.937441	+	0.348144i	$0.113188\pi$
$937$	21.3137i	0.696289i	−0.937441	+	0.348144i	$0.886812\pi$
$938$	− 8.00000i	− 0.261209i
$939$	0	0
$940$	0	0
$941$	−34.1421	−1.11300	−0.556501	−	0.830847i	$-0.687856\pi$
$941$	−34.1421	−1.11300	−0.556501	+	0.830847i	$0.687856\pi$
$942$	0	0
$943$	− 20.6863i	− 0.673638i
$944$	22.9706	0.747628
$945$	0	0
$946$	1.37258	0.0446265
$947$	− 21.0294i	− 0.683365i	−0.939815	−	0.341682i	$-0.889003\pi$
$947$	− 21.0294i	− 0.683365i	0.939815	−	0.341682i	$-0.110997\pi$
$948$	0	0
$949$	0.343146	0.0111390
$950$	0	0
$951$	0	0
$952$	34.3431i	1.11307i
$953$	− 40.3431i	− 1.30684i	−0.756994	−	0.653421i	$-0.773333\pi$
$953$	− 40.3431i	− 1.30684i	0.756994	−	0.653421i	$-0.226667\pi$
$954$	0	0
$955$	0	0
$956$	3.65685	0.118271
$957$	0	0
$958$	10.4853i	0.338764i
$959$	−30.6274	−0.989011
$960$	0	0
$961$	−29.6274	−0.955723
$962$	3.17157i	0.102256i
$963$	0	0
$964$	0.627417	0.0202077
$965$	0	0
$966$	0	0
$967$	− 18.1421i	− 0.583412i	−0.956508	−	0.291706i	$-0.905777\pi$
$967$	− 18.1421i	− 0.583412i	0.956508	−	0.291706i	$-0.0942228\pi$
$968$	− 11.1005i	− 0.356784i
$969$	0	0
$970$	0	0
$971$	15.3137	0.491440	0.245720	−	0.969341i	$-0.420976\pi$
$971$	15.3137	0.491440	0.245720	+	0.969341i	$0.420976\pi$
$972$	0	0
$973$	− 20.6863i	− 0.663172i
$974$	−3.23045	−0.103510
$975$	0	0
$976$	39.9411	1.27848
$977$	42.1421i	1.34825i	0.738619	+	0.674123i	$0.235478\pi$
$977$	42.1421i	1.34825i	−0.738619	+	0.674123i	$0.764522\pi$
$978$	0	0
$979$	29.6569	0.947837
$980$	0	0
$981$	0	0
$982$	− 12.6863i	− 0.404836i
$983$	− 25.3137i	− 0.807382i	−0.914895	−	0.403691i	$-0.867727\pi$
$983$	− 25.3137i	− 0.807382i	0.914895	−	0.403691i	$-0.132273\pi$
$984$	0	0
$985$	0	0
$986$	−6.34315	−0.202007
$987$	0	0
$988$	− 5.17157i	− 0.164530i
$989$	6.62742	0.210740
$990$	0	0
$991$	4.68629	0.148865	0.0744325	−	0.997226i	$-0.476285\pi$
$991$	4.68629	0.148865	0.0744325	+	0.997226i	$0.476285\pi$
$992$	− 5.17157i	− 0.164198i
$993$	0	0
$994$	−2.34315	−0.0743201
$995$	0	0
$996$	0	0
$997$	39.2548i	1.24321i	0.783330	+	0.621607i	$0.213520\pi$
$997$	39.2548i	1.24321i	−0.783330	+	0.621607i	$0.786480\pi$
$998$	− 10.8284i	− 0.342768i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.u.2224.3		4
3.2	odd	2		975.2.c.h.274.2		4
5.2	odd	4		117.2.a.c.1.1		2
5.3	odd	4		2925.2.a.v.1.2		2
5.4	even	2	inner	2925.2.c.u.2224.2		4
15.2	even	4		39.2.a.b.1.2	✓	2
15.8	even	4		975.2.a.l.1.1		2
15.14	odd	2		975.2.c.h.274.3		4
20.7	even	4		1872.2.a.w.1.2		2
35.27	even	4		5733.2.a.u.1.1		2
40.27	even	4		7488.2.a.co.1.1		2
40.37	odd	4		7488.2.a.cl.1.1		2
45.2	even	12		1053.2.e.m.352.1		4
45.7	odd	12		1053.2.e.e.352.2		4
45.22	odd	12		1053.2.e.e.703.2		4
45.32	even	12		1053.2.e.m.703.1		4
60.47	odd	4		624.2.a.k.1.1		2
65.12	odd	4		1521.2.a.f.1.2		2
65.47	even	4		1521.2.b.j.1351.2		4
65.57	even	4		1521.2.b.j.1351.3		4
105.62	odd	4		1911.2.a.h.1.2		2
120.77	even	4		2496.2.a.bf.1.2		2
120.107	odd	4		2496.2.a.bi.1.2		2
165.32	odd	4		4719.2.a.p.1.1		2
195.2	odd	12		507.2.j.f.316.3		8
195.17	even	12		507.2.e.d.484.2		4
195.32	odd	12		507.2.j.f.361.2		8
195.47	odd	4		507.2.b.e.337.3		4
195.62	even	12		507.2.e.d.22.2		4
195.77	even	4		507.2.a.h.1.1		2
195.107	even	12		507.2.e.h.22.1		4
195.122	odd	4		507.2.b.e.337.2		4
195.137	odd	12		507.2.j.f.361.3		8
195.152	even	12		507.2.e.h.484.1		4
195.167	odd	12		507.2.j.f.316.2		8
780.467	odd	4		8112.2.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.2	✓	2	15.2	even	4
117.2.a.c.1.1		2	5.2	odd	4
507.2.a.h.1.1		2	195.77	even	4
507.2.b.e.337.2		4	195.122	odd	4
507.2.b.e.337.3		4	195.47	odd	4
507.2.e.d.22.2		4	195.62	even	12
507.2.e.d.484.2		4	195.17	even	12
507.2.e.h.22.1		4	195.107	even	12
507.2.e.h.484.1		4	195.152	even	12
507.2.j.f.316.2		8	195.167	odd	12
507.2.j.f.316.3		8	195.2	odd	12
507.2.j.f.361.2		8	195.32	odd	12
507.2.j.f.361.3		8	195.137	odd	12
624.2.a.k.1.1		2	60.47	odd	4
975.2.a.l.1.1		2	15.8	even	4
975.2.c.h.274.2		4	3.2	odd	2
975.2.c.h.274.3		4	15.14	odd	2
1053.2.e.e.352.2		4	45.7	odd	12
1053.2.e.e.703.2		4	45.22	odd	12
1053.2.e.m.352.1		4	45.2	even	12
1053.2.e.m.703.1		4	45.32	even	12
1521.2.a.f.1.2		2	65.12	odd	4
1521.2.b.j.1351.2		4	65.47	even	4
1521.2.b.j.1351.3		4	65.57	even	4
1872.2.a.w.1.2		2	20.7	even	4
1911.2.a.h.1.2		2	105.62	odd	4
2496.2.a.bf.1.2		2	120.77	even	4
2496.2.a.bi.1.2		2	120.107	odd	4
2925.2.a.v.1.2		2	5.3	odd	4
2925.2.c.u.2224.2		4	5.4	even	2	inner
2925.2.c.u.2224.3		4	1.1	even	1	trivial
4719.2.a.p.1.1		2	165.32	odd	4
5733.2.a.u.1.1		2	35.27	even	4
7488.2.a.cl.1.1		2	40.37	odd	4
7488.2.a.co.1.1		2	40.27	even	4
8112.2.a.bm.1.2		2	780.467	odd	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{7} +1.58579i q^{8} +O(q^{10})\)	\(q+0.414214i q^{2} +1.82843 q^{4} -2.82843i q^{7} +1.58579i q^{8} +2.00000 q^{11} -1.00000i q^{13} +1.17157 q^{14} +3.00000 q^{16} +7.65685i q^{17} +2.82843 q^{19} +0.828427i q^{22} +4.00000i q^{23} +0.414214 q^{26} -5.17157i q^{28} +2.00000 q^{29} -1.17157 q^{31} +4.41421i q^{32} -3.17157 q^{34} +7.65685i q^{37} +1.17157i q^{38} -5.17157 q^{41} -1.65685i q^{43} +3.65685 q^{44} -1.65685 q^{46} -11.6569i q^{47} -1.00000 q^{49} -1.82843i q^{52} +2.00000i q^{53} +4.48528 q^{56} +0.828427i q^{58} +7.65685 q^{59} +13.3137 q^{61} -0.485281i q^{62} +4.17157 q^{64} -6.82843i q^{67} +14.0000i q^{68} -2.00000 q^{71} +0.343146i q^{73} -3.17157 q^{74} +5.17157 q^{76} -5.65685i q^{77} +11.3137 q^{79} -2.14214i q^{82} -3.65685i q^{83} +0.686292 q^{86} +3.17157i q^{88} +14.8284 q^{89} -2.82843 q^{91} +7.31371i q^{92} +4.82843 q^{94} -3.65685i q^{97} -0.414214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 8 q^{11} + 16 q^{14} + 12 q^{16} - 4 q^{26} + 8 q^{29} - 16 q^{31} - 24 q^{34} - 32 q^{41} - 8 q^{44} + 16 q^{46} - 4 q^{49} - 16 q^{56} + 8 q^{59} + 8 q^{61} + 28 q^{64} - 8 q^{71} - 24 q^{74} + 32 q^{76} + 48 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^11 + 16 * q^14 + 12 * q^16 - 4 * q^26 + 8 * q^29 - 16 * q^31 - 24 * q^34 - 32 * q^41 - 8 * q^44 + 16 * q^46 - 4 * q^49 - 16 * q^56 + 8 * q^59 + 8 * q^61 + 28 * q^64 - 8 * q^71 - 24 * q^74 + 32 * q^76 + 48 * q^86 + 48 * q^89 + 8 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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