LMFDB - Embedded newform 2475.2.c.k.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.k.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833i q^{17} +1.00000 q^{19} +1.30278i q^{22} -7.30278i q^{23} +6.51388 q^{26} +0.211103i q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722i q^{32} +9.00000 q^{34} +2.39445i q^{37} +1.30278i q^{38} +5.60555 q^{41} -7.21110i q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000i q^{47} +6.51388 q^{49} -1.51388i q^{52} -1.30278i q^{53} -2.09167 q^{56} +1.18335i q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028i q^{62} -8.81665 q^{64} -4.00000i q^{67} -2.09167i q^{68} +2.60555 q^{71} +7.90833i q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224i q^{77} +10.9083 q^{79} +7.30278i q^{82} -3.51388i q^{83} +9.39445 q^{86} +3.00000i q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110i q^{92} -3.90833 q^{94} +15.3028i q^{97} +8.48612i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4}+O(q^{10}) $ 4 * q - 6 * q^4	$ 4 q - 6 q^{4} + 4 q^{11} + 18 q^{14} - 6 q^{16} + 4 q^{19} - 10 q^{26} - 18 q^{29} + 12 q^{31} + 36 q^{34} + 8 q^{41} - 6 q^{44} + 2 q^{46} - 10 q^{49} - 30 q^{56} - 28 q^{59} - 10 q^{61} + 8 q^{64} - 4 q^{71} + 38 q^{74} - 6 q^{76} + 22 q^{79} + 52 q^{86} + 14 q^{89} + 50 q^{91} + 6 q^{94}+O(q^{100}) $ 4 * q - 6 * q^4 + 4 * q^11 + 18 * q^14 - 6 * q^16 + 4 * q^19 - 10 * q^26 - 18 * q^29 + 12 * q^31 + 36 * q^34 + 8 * q^41 - 6 * q^44 + 2 * q^46 - 10 * q^49 - 30 * q^56 - 28 * q^59 - 10 * q^61 + 8 * q^64 - 4 * q^71 + 38 * q^74 - 6 * q^76 + 22 * q^79 + 52 * q^86 + 14 * q^89 + 50 * q^91 + 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.30278i	0.921201i	0.887607	+	0.460601i	$0.152366\pi$
$2$	1.30278i	0.921201i	−0.887607	+	0.460601i	$0.847634\pi$
$3$	0	0
$3$	0	0
$4$	0.302776	0.151388
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.697224i	0.263526i	0.991281	+	0.131763i	$0.0420638\pi$
$7$	0.697224i	0.263526i	−0.991281	+	0.131763i	$0.957936\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	−0.908327	−0.242761
$15$	0	0
$16$	−3.30278	−0.825694
$17$	− 6.90833i	− 1.67552i	−0.546042	−	0.837758i	$-0.683866\pi$
$17$	− 6.90833i	− 1.67552i	0.546042	−	0.837758i	$-0.316134\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	1.30278i	0.277753i
$23$	− 7.30278i	− 1.52273i	−0.648321	−	0.761367i	$-0.724529\pi$
$23$	− 7.30278i	− 1.52273i	0.648321	−	0.761367i	$-0.275471\pi$
$24$	0	0
$25$	0	0
$26$	6.51388	1.27748
$27$	0	0
$28$	0.211103i	0.0398946i
$29$	0.908327	0.168672	0.0843360	−	0.996437i	$-0.473123\pi$
$29$	0.908327	0.168672	0.0843360	+	0.996437i	$0.473123\pi$
$30$	0	0
$31$	10.2111	1.83397	0.916984	−	0.398924i	$-0.130616\pi$
$31$	10.2111	1.83397	0.916984	+	0.398924i	$0.130616\pi$
$32$	1.69722i	0.300030i
$33$	0	0
$34$	9.00000	1.54349
$35$	0	0
$36$	0	0
$37$	2.39445i	0.393645i	0.980439	+	0.196822i	$0.0630623\pi$
$37$	2.39445i	0.393645i	−0.980439	+	0.196822i	$0.936938\pi$
$38$	1.30278i	0.211338i
$39$	0	0
$40$	0	0
$41$	5.60555	0.875440	0.437720	−	0.899111i	$-0.355786\pi$
$41$	5.60555	0.875440	0.437720	+	0.899111i	$0.355786\pi$
$42$	0	0
$43$	− 7.21110i	− 1.09968i	−0.835269	−	0.549841i	$-0.814688\pi$
$43$	− 7.21110i	− 1.09968i	0.835269	−	0.549841i	$-0.185312\pi$
$44$	0.302776	0.0456451
$45$	0	0
$46$	9.51388	1.40274
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	6.51388	0.930554
$50$	0	0
$51$	0	0
$52$	− 1.51388i	− 0.209937i
$53$	− 1.30278i	− 0.178950i	−0.995989	−	0.0894750i	$-0.971481\pi$
$53$	− 1.30278i	− 0.178950i	0.995989	−	0.0894750i	$-0.0285189\pi$
$54$	0	0
$55$	0	0
$56$	−2.09167	−0.279512
$57$	0	0
$58$	1.18335i	0.155381i
$59$	−14.2111	−1.85013	−0.925064	−	0.379811i	$-0.875989\pi$
$59$	−14.2111	−1.85013	−0.925064	+	0.379811i	$0.875989\pi$
$60$	0	0
$61$	−7.90833	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$61$	−7.90833	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$62$	13.3028i	1.68945i
$63$	0	0
$64$	−8.81665	−1.10208
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 2.09167i	− 0.253653i
$69$	0	0
$70$	0	0
$71$	2.60555	0.309222	0.154611	−	0.987975i	$-0.450588\pi$
$71$	2.60555	0.309222	0.154611	+	0.987975i	$0.450588\pi$
$72$	0	0
$73$	7.90833i	0.925600i	0.886463	+	0.462800i	$0.153155\pi$
$73$	7.90833i	0.925600i	−0.886463	+	0.462800i	$0.846845\pi$
$74$	−3.11943	−0.362626
$75$	0	0
$76$	0.302776	0.0347307
$77$	0.697224i	0.0794561i
$78$	0	0
$79$	10.9083	1.22728	0.613641	−	0.789585i	$-0.289704\pi$
$79$	10.9083	1.22728	0.613641	+	0.789585i	$0.289704\pi$
$80$	0	0
$81$	0	0
$82$	7.30278i	0.806457i
$83$	− 3.51388i	− 0.385698i	−0.981228	−	0.192849i	$-0.938227\pi$
$83$	− 3.51388i	− 0.385698i	0.981228	−	0.192849i	$-0.0617728\pi$
$84$	0	0
$85$	0	0
$86$	9.39445	1.01303
$87$	0	0
$88$	3.00000i	0.319801i
$89$	1.69722	0.179905	0.0899527	−	0.995946i	$-0.471328\pi$
$89$	1.69722	0.179905	0.0899527	+	0.995946i	$0.471328\pi$
$90$	0	0
$91$	3.48612	0.365445
$92$	− 2.21110i	− 0.230523i
$93$	0	0
$94$	−3.90833	−0.403113
$95$	0	0
$96$	0	0
$97$	15.3028i	1.55376i	0.629648	+	0.776881i	$0.283199\pi$
$97$	15.3028i	1.55376i	−0.629648	+	0.776881i	$0.716801\pi$
$98$	8.48612i	0.857228i
$99$	0	0
$100$	0	0
$101$	0.513878	0.0511328	0.0255664	−	0.999673i	$-0.491861\pi$
$101$	0.513878	0.0511328	0.0255664	+	0.999673i	$0.491861\pi$
$102$	0	0
$103$	− 2.90833i	− 0.286566i	−0.989682	−	0.143283i	$-0.954234\pi$
$103$	− 2.90833i	− 0.286566i	0.989682	−	0.143283i	$-0.0457659\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	1.69722	0.164849
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	−11.5139	−1.10283	−0.551415	−	0.834231i	$-0.685912\pi$
$109$	−11.5139	−1.10283	−0.551415	+	0.834231i	$0.685912\pi$
$110$	0	0
$111$	0	0
$112$	− 2.30278i	− 0.217592i
$113$	− 10.8167i	− 1.01755i	−0.860901	−	0.508773i	$-0.830099\pi$
$113$	− 10.8167i	− 1.01755i	0.860901	−	0.508773i	$-0.169901\pi$
$114$	0	0
$115$	0	0
$116$	0.275019	0.0255349
$117$	0	0
$118$	− 18.5139i	− 1.70434i
$119$	4.81665	0.441542
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 10.3028i	− 0.932769i
$123$	0	0
$124$	3.09167	0.277640
$125$	0	0
$126$	0	0
$127$	8.11943i	0.720483i	0.932859	+	0.360241i	$0.117306\pi$
$127$	8.11943i	0.720483i	−0.932859	+	0.360241i	$0.882694\pi$
$128$	− 8.09167i	− 0.715210i
$129$	0	0
$130$	0	0
$131$	9.90833	0.865695	0.432847	−	0.901467i	$-0.357509\pi$
$131$	9.90833	0.865695	0.432847	+	0.901467i	$0.357509\pi$
$132$	0	0
$133$	0.697224i	0.0604570i
$134$	5.21110	0.450171
$135$	0	0
$136$	20.7250	1.77715
$137$	12.9083i	1.10283i	0.834230	+	0.551416i	$0.185912\pi$
$137$	12.9083i	1.10283i	−0.834230	+	0.551416i	$0.814088\pi$
$138$	0	0
$139$	6.21110	0.526819	0.263409	−	0.964684i	$-0.415153\pi$
$139$	6.21110	0.526819	0.263409	+	0.964684i	$0.415153\pi$
$140$	0	0
$141$	0	0
$142$	3.39445i	0.284856i
$143$	− 5.00000i	− 0.418121i
$144$	0	0
$145$	0	0
$146$	−10.3028	−0.852664
$147$	0	0
$148$	0.724981i	0.0595930i
$149$	17.2111	1.40999	0.704994	−	0.709213i	$-0.250950\pi$
$149$	17.2111	1.40999	0.704994	+	0.709213i	$0.250950\pi$
$150$	0	0
$151$	0.816654	0.0664583	0.0332292	−	0.999448i	$-0.489421\pi$
$151$	0.816654	0.0664583	0.0332292	+	0.999448i	$0.489421\pi$
$152$	3.00000i	0.243332i
$153$	0	0
$154$	−0.908327	−0.0731951
$155$	0	0
$156$	0	0
$157$	19.2111i	1.53321i	0.642117	+	0.766606i	$0.278056\pi$
$157$	19.2111i	1.53321i	−0.642117	+	0.766606i	$0.721944\pi$
$158$	14.2111i	1.13057i
$159$	0	0
$160$	0	0
$161$	5.09167	0.401280
$162$	0	0
$163$	− 9.30278i	− 0.728650i	−0.931272	−	0.364325i	$-0.881300\pi$
$163$	− 9.30278i	− 0.728650i	0.931272	−	0.364325i	$-0.118700\pi$
$164$	1.69722	0.132531
$165$	0	0
$166$	4.57779	0.355306
$167$	− 13.4222i	− 1.03864i	−0.854579	−	0.519321i	$-0.826185\pi$
$167$	− 13.4222i	− 1.03864i	0.854579	−	0.519321i	$-0.173815\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	− 2.18335i	− 0.166479i
$173$	− 4.81665i	− 0.366203i	−0.983094	−	0.183102i	$-0.941386\pi$
$173$	− 4.81665i	− 0.366203i	0.983094	−	0.183102i	$-0.0586138\pi$
$174$	0	0
$175$	0	0
$176$	−3.30278	−0.248956
$177$	0	0
$178$	2.21110i	0.165729i
$179$	12.5139	0.935331	0.467666	−	0.883905i	$-0.345095\pi$
$179$	12.5139	0.935331	0.467666	+	0.883905i	$0.345095\pi$
$180$	0	0
$181$	−19.9083	−1.47977	−0.739887	−	0.672731i	$-0.765121\pi$
$181$	−19.9083	−1.47977	−0.739887	+	0.672731i	$0.765121\pi$
$182$	4.54163i	0.336648i
$183$	0	0
$184$	21.9083	1.61510
$185$	0	0
$186$	0	0
$187$	− 6.90833i	− 0.505187i
$188$	0.908327i	0.0662465i
$189$	0	0
$190$	0	0
$191$	−10.3028	−0.745483	−0.372741	−	0.927935i	$-0.621582\pi$
$191$	−10.3028	−0.745483	−0.372741	+	0.927935i	$0.621582\pi$
$192$	0	0
$193$	− 13.2111i	− 0.950956i	−0.879728	−	0.475478i	$-0.842275\pi$
$193$	− 13.2111i	− 0.950956i	0.879728	−	0.475478i	$-0.157725\pi$
$194$	−19.9361	−1.43133
$195$	0	0
$196$	1.97224	0.140875
$197$	− 13.3028i	− 0.947784i	−0.880583	−	0.473892i	$-0.842849\pi$
$197$	− 13.3028i	− 0.947784i	0.880583	−	0.473892i	$-0.157151\pi$
$198$	0	0
$199$	6.48612	0.459789	0.229894	−	0.973216i	$-0.426162\pi$
$199$	6.48612	0.459789	0.229894	+	0.973216i	$0.426162\pi$
$200$	0	0
$201$	0	0
$202$	0.669468i	0.0471036i
$203$	0.633308i	0.0444495i
$204$	0	0
$205$	0	0
$206$	3.78890	0.263985
$207$	0	0
$208$	16.5139i	1.14503i
$209$	1.00000	0.0691714
$210$	0	0
$211$	−25.2389	−1.73751	−0.868757	−	0.495238i	$-0.835081\pi$
$211$	−25.2389	−1.73751	−0.868757	+	0.495238i	$0.835081\pi$
$212$	− 0.394449i	− 0.0270908i
$213$	0	0
$214$	−3.90833	−0.267168
$215$	0	0
$216$	0	0
$217$	7.11943i	0.483298i
$218$	− 15.0000i	− 1.01593i
$219$	0	0
$220$	0	0
$221$	−34.5416	−2.32352
$222$	0	0
$223$	22.6333i	1.51564i	0.652465	+	0.757819i	$0.273735\pi$
$223$	22.6333i	1.51564i	−0.652465	+	0.757819i	$0.726265\pi$
$224$	−1.18335	−0.0790656
$225$	0	0
$226$	14.0917	0.937364
$227$	− 1.69722i	− 0.112649i	−0.998413	−	0.0563244i	$-0.982062\pi$
$227$	− 1.69722i	− 0.112649i	0.998413	−	0.0563244i	$-0.0179381\pi$
$228$	0	0
$229$	18.7250	1.23738	0.618691	−	0.785635i	$-0.287663\pi$
$229$	18.7250	1.23738	0.618691	+	0.785635i	$0.287663\pi$
$230$	0	0
$231$	0	0
$232$	2.72498i	0.178904i
$233$	15.9083i	1.04219i	0.853499	+	0.521095i	$0.174476\pi$
$233$	15.9083i	1.04219i	−0.853499	+	0.521095i	$0.825524\pi$
$234$	0	0
$235$	0	0
$236$	−4.30278	−0.280087
$237$	0	0
$238$	6.27502i	0.406749i
$239$	21.1194	1.36610	0.683051	−	0.730371i	$-0.260653\pi$
$239$	21.1194	1.36610	0.683051	+	0.730371i	$0.260653\pi$
$240$	0	0
$241$	21.9361	1.41303	0.706514	−	0.707699i	$-0.250267\pi$
$241$	21.9361	1.41303	0.706514	+	0.707699i	$0.250267\pi$
$242$	1.30278i	0.0837456i
$243$	0	0
$244$	−2.39445	−0.153289
$245$	0	0
$246$	0	0
$247$	− 5.00000i	− 0.318142i
$248$	30.6333i	1.94522i
$249$	0	0
$250$	0	0
$251$	−6.90833	−0.436050	−0.218025	−	0.975943i	$-0.569961\pi$
$251$	−6.90833	−0.436050	−0.218025	+	0.975943i	$0.569961\pi$
$252$	0	0
$253$	− 7.30278i	− 0.459122i
$254$	−10.5778	−0.663710
$255$	0	0
$256$	−7.09167	−0.443230
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−1.66947	−0.103736
$260$	0	0
$261$	0	0
$262$	12.9083i	0.797479i
$263$	− 22.8167i	− 1.40694i	−0.710727	−	0.703468i	$-0.751634\pi$
$263$	− 22.8167i	− 1.40694i	0.710727	−	0.703468i	$-0.248366\pi$
$264$	0	0
$265$	0	0
$266$	−0.908327	−0.0556931
$267$	0	0
$268$	− 1.21110i	− 0.0739799i
$269$	8.72498	0.531971	0.265986	−	0.963977i	$-0.414303\pi$
$269$	8.72498	0.531971	0.265986	+	0.963977i	$0.414303\pi$
$270$	0	0
$271$	−0.211103	−0.0128236	−0.00641178	−	0.999979i	$-0.502041\pi$
$271$	−0.211103	−0.0128236	−0.00641178	+	0.999979i	$0.502041\pi$
$272$	22.8167i	1.38346i
$273$	0	0
$274$	−16.8167	−1.01593
$275$	0	0
$276$	0	0
$277$	14.3944i	0.864879i	0.901663	+	0.432439i	$0.142347\pi$
$277$	14.3944i	0.864879i	−0.901663	+	0.432439i	$0.857653\pi$
$278$	8.09167i	0.485306i
$279$	0	0
$280$	0	0
$281$	1.18335	0.0705925	0.0352963	−	0.999377i	$-0.488763\pi$
$281$	1.18335	0.0705925	0.0352963	+	0.999377i	$0.488763\pi$
$282$	0	0
$283$	− 6.30278i	− 0.374661i	−0.982297	−	0.187331i	$-0.940016\pi$
$283$	− 6.30278i	− 0.374661i	0.982297	−	0.187331i	$-0.0599836\pi$
$284$	0.788897	0.0468125
$285$	0	0
$286$	6.51388	0.385174
$287$	3.90833i	0.230701i
$288$	0	0
$289$	−30.7250	−1.80735
$290$	0	0
$291$	0	0
$292$	2.39445i	0.140125i
$293$	0.788897i	0.0460879i	0.999734	+	0.0230439i	$0.00733576\pi$
$293$	0.788897i	0.0460879i	−0.999734	+	0.0230439i	$0.992664\pi$
$294$	0	0
$295$	0	0
$296$	−7.18335	−0.417524
$297$	0	0
$298$	22.4222i	1.29888i
$299$	−36.5139	−2.11165
$300$	0	0
$301$	5.02776	0.289795
$302$	1.06392i	0.0612215i
$303$	0	0
$304$	−3.30278	−0.189427
$305$	0	0
$306$	0	0
$307$	− 16.9083i	− 0.965009i	−0.875893	−	0.482505i	$-0.839727\pi$
$307$	− 16.9083i	− 0.965009i	0.875893	−	0.482505i	$-0.160273\pi$
$308$	0.211103i	0.0120287i
$309$	0	0
$310$	0	0
$311$	−4.81665	−0.273127	−0.136564	−	0.990631i	$-0.543606\pi$
$311$	−4.81665	−0.273127	−0.136564	+	0.990631i	$0.543606\pi$
$312$	0	0
$313$	− 0.183346i	− 0.0103633i	−0.999987	−	0.00518167i	$-0.998351\pi$
$313$	− 0.183346i	− 0.0103633i	0.999987	−	0.00518167i	$-0.00164938\pi$
$314$	−25.0278	−1.41240
$315$	0	0
$316$	3.30278	0.185796
$317$	0.908327i	0.0510167i	0.999675	+	0.0255084i	$0.00812044\pi$
$317$	0.908327i	0.0510167i	−0.999675	+	0.0255084i	$0.991880\pi$
$318$	0	0
$319$	0.908327	0.0508565
$320$	0	0
$321$	0	0
$322$	6.63331i	0.369660i
$323$	− 6.90833i	− 0.384390i
$324$	0	0
$325$	0	0
$326$	12.1194	0.671233
$327$	0	0
$328$	16.8167i	0.928544i
$329$	−2.09167	−0.115318
$330$	0	0
$331$	−21.6056	−1.18755	−0.593774	−	0.804632i	$-0.702363\pi$
$331$	−21.6056	−1.18755	−0.593774	+	0.804632i	$0.702363\pi$
$332$	− 1.06392i	− 0.0583900i
$333$	0	0
$334$	17.4861	0.956798
$335$	0	0
$336$	0	0
$337$	− 30.8444i	− 1.68020i	−0.542430	−	0.840101i	$-0.682496\pi$
$337$	− 30.8444i	− 1.68020i	0.542430	−	0.840101i	$-0.317504\pi$
$338$	− 15.6333i	− 0.850340i
$339$	0	0
$340$	0	0
$341$	10.2111	0.552962
$342$	0	0
$343$	9.42221i	0.508751i
$344$	21.6333	1.16639
$345$	0	0
$346$	6.27502	0.337347
$347$	12.5139i	0.671780i	0.941901	+	0.335890i	$0.109037\pi$
$347$	12.5139i	0.671780i	−0.941901	+	0.335890i	$0.890963\pi$
$348$	0	0
$349$	5.18335	0.277458	0.138729	−	0.990330i	$-0.455698\pi$
$349$	5.18335	0.277458	0.138729	+	0.990330i	$0.455698\pi$
$350$	0	0
$351$	0	0
$352$	1.69722i	0.0904624i
$353$	18.6333i	0.991751i	0.868394	+	0.495875i	$0.165153\pi$
$353$	18.6333i	0.991751i	−0.868394	+	0.495875i	$0.834847\pi$
$354$	0	0
$355$	0	0
$356$	0.513878	0.0272355
$357$	0	0
$358$	16.3028i	0.861628i
$359$	0.788897	0.0416364	0.0208182	−	0.999783i	$-0.493373\pi$
$359$	0.788897	0.0416364	0.0208182	+	0.999783i	$0.493373\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	− 25.9361i	− 1.36317i
$363$	0	0
$364$	1.05551	0.0553239
$365$	0	0
$366$	0	0
$367$	− 20.6972i	− 1.08039i	−0.841541	−	0.540193i	$-0.818351\pi$
$367$	− 20.6972i	− 1.08039i	0.841541	−	0.540193i	$-0.181649\pi$
$368$	24.1194i	1.25731i
$369$	0	0
$370$	0	0
$371$	0.908327	0.0471580
$372$	0	0
$373$	− 27.4222i	− 1.41987i	−0.704268	−	0.709934i	$-0.748725\pi$
$373$	− 27.4222i	− 1.41987i	0.704268	−	0.709934i	$-0.251275\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	− 4.54163i	− 0.233906i
$378$	0	0
$379$	−3.18335	−0.163518	−0.0817588	−	0.996652i	$-0.526054\pi$
$379$	−3.18335	−0.163518	−0.0817588	+	0.996652i	$0.526054\pi$
$380$	0	0
$381$	0	0
$382$	− 13.4222i	− 0.686740i
$383$	21.6333i	1.10541i	0.833377	+	0.552705i	$0.186404\pi$
$383$	21.6333i	1.10541i	−0.833377	+	0.552705i	$0.813596\pi$
$384$	0	0
$385$	0	0
$386$	17.2111	0.876022
$387$	0	0
$388$	4.63331i	0.235221i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−50.4500	−2.55136
$392$	19.5416i	0.987002i
$393$	0	0
$394$	17.3305	0.873100
$395$	0	0
$396$	0	0
$397$	21.6972i	1.08895i	0.838776	+	0.544476i	$0.183272\pi$
$397$	21.6972i	1.08895i	−0.838776	+	0.544476i	$0.816728\pi$
$398$	8.44996i	0.423558i
$399$	0	0
$400$	0	0
$401$	12.7889	0.638647	0.319324	−	0.947646i	$-0.396544\pi$
$401$	12.7889	0.638647	0.319324	+	0.947646i	$0.396544\pi$
$402$	0	0
$403$	− 51.0555i	− 2.54326i
$404$	0.155590	0.00774088
$405$	0	0
$406$	−0.825058	−0.0409469
$407$	2.39445i	0.118688i
$408$	0	0
$409$	6.21110	0.307119	0.153560	−	0.988139i	$-0.450926\pi$
$409$	6.21110	0.307119	0.153560	+	0.988139i	$0.450926\pi$
$410$	0	0
$411$	0	0
$412$	− 0.880571i	− 0.0433826i
$413$	− 9.90833i	− 0.487557i
$414$	0	0
$415$	0	0
$416$	8.48612	0.416066
$417$	0	0
$418$	1.30278i	0.0637208i
$419$	6.39445	0.312389	0.156195	−	0.987726i	$-0.450077\pi$
$419$	6.39445	0.312389	0.156195	+	0.987726i	$0.450077\pi$
$420$	0	0
$421$	0.697224	0.0339806	0.0169903	−	0.999856i	$-0.494592\pi$
$421$	0.697224	0.0339806	0.0169903	+	0.999856i	$0.494592\pi$
$422$	− 32.8806i	− 1.60060i
$423$	0	0
$424$	3.90833	0.189805
$425$	0	0
$426$	0	0
$427$	− 5.51388i	− 0.266835i
$428$	0.908327i	0.0439056i
$429$	0	0
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	− 5.00000i	− 0.240285i	−0.992757	−	0.120142i	$-0.961665\pi$
$433$	− 5.00000i	− 0.240285i	0.992757	−	0.120142i	$-0.0383351\pi$
$434$	−9.27502	−0.445215
$435$	0	0
$436$	−3.48612	−0.166955
$437$	− 7.30278i	− 0.349339i
$438$	0	0
$439$	−24.3028	−1.15991	−0.579954	−	0.814649i	$-0.696930\pi$
$439$	−24.3028	−1.15991	−0.579954	+	0.814649i	$0.696930\pi$
$440$	0	0
$441$	0	0
$442$	− 45.0000i	− 2.14043i
$443$	− 8.60555i	− 0.408862i	−0.978881	−	0.204431i	$-0.934466\pi$
$443$	− 8.60555i	− 0.408862i	0.978881	−	0.204431i	$-0.0655344\pi$
$444$	0	0
$445$	0	0
$446$	−29.4861	−1.39621
$447$	0	0
$448$	− 6.14719i	− 0.290427i
$449$	23.4861	1.10838	0.554189	−	0.832391i	$-0.313028\pi$
$449$	23.4861	1.10838	0.554189	+	0.832391i	$0.313028\pi$
$450$	0	0
$451$	5.60555	0.263955
$452$	− 3.27502i	− 0.154044i
$453$	0	0
$454$	2.21110	0.103772
$455$	0	0
$456$	0	0
$457$	− 20.6972i	− 0.968175i	−0.875020	−	0.484088i	$-0.839152\pi$
$457$	− 20.6972i	− 0.968175i	0.875020	−	0.484088i	$-0.160848\pi$
$458$	24.3944i	1.13988i
$459$	0	0
$460$	0	0
$461$	−32.2111	−1.50022	−0.750110	−	0.661313i	$-0.770000\pi$
$461$	−32.2111	−1.50022	−0.750110	+	0.661313i	$0.770000\pi$
$462$	0	0
$463$	− 11.7889i	− 0.547877i	−0.961747	−	0.273938i	$-0.911674\pi$
$463$	− 11.7889i	− 0.547877i	0.961747	−	0.273938i	$-0.0883264\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−20.7250	−0.960066
$467$	18.6333i	0.862247i	0.902293	+	0.431123i	$0.141883\pi$
$467$	18.6333i	0.862247i	−0.902293	+	0.431123i	$0.858117\pi$
$468$	0	0
$469$	2.78890	0.128779
$470$	0	0
$471$	0	0
$472$	− 42.6333i	− 1.96236i
$473$	− 7.21110i	− 0.331567i
$474$	0	0
$475$	0	0
$476$	1.45837	0.0668441
$477$	0	0
$478$	27.5139i	1.25846i
$479$	−34.8167	−1.59081	−0.795407	−	0.606076i	$-0.792743\pi$
$479$	−34.8167	−1.59081	−0.795407	+	0.606076i	$0.792743\pi$
$480$	0	0
$481$	11.9722	0.545887
$482$	28.5778i	1.30168i
$483$	0	0
$484$	0.302776	0.0137625
$485$	0	0
$486$	0	0
$487$	4.21110i	0.190823i	0.995438	+	0.0954116i	$0.0304167\pi$
$487$	4.21110i	0.190823i	−0.995438	+	0.0954116i	$0.969583\pi$
$488$	− 23.7250i	− 1.07398i
$489$	0	0
$490$	0	0
$491$	9.78890	0.441767	0.220883	−	0.975300i	$-0.429106\pi$
$491$	9.78890	0.441767	0.220883	+	0.975300i	$0.429106\pi$
$492$	0	0
$493$	− 6.27502i	− 0.282613i
$494$	6.51388	0.293073
$495$	0	0
$496$	−33.7250	−1.51430
$497$	1.81665i	0.0814881i
$498$	0	0
$499$	3.48612	0.156060	0.0780301	−	0.996951i	$-0.475137\pi$
$499$	3.48612	0.156060	0.0780301	+	0.996951i	$0.475137\pi$
$500$	0	0
$501$	0	0
$502$	− 9.00000i	− 0.401690i
$503$	9.39445i	0.418878i	0.977822	+	0.209439i	$0.0671637\pi$
$503$	9.39445i	0.418878i	−0.977822	+	0.209439i	$0.932836\pi$
$504$	0	0
$505$	0	0
$506$	9.51388	0.422943
$507$	0	0
$508$	2.45837i	0.109072i
$509$	−22.6972	−1.00604	−0.503018	−	0.864276i	$-0.667777\pi$
$509$	−22.6972	−1.00604	−0.503018	+	0.864276i	$0.667777\pi$
$510$	0	0
$511$	−5.51388	−0.243920
$512$	− 25.4222i	− 1.12351i
$513$	0	0
$514$	−23.4500	−1.03433
$515$	0	0
$516$	0	0
$517$	3.00000i	0.131940i
$518$	− 2.17494i	− 0.0955615i
$519$	0	0
$520$	0	0
$521$	41.4500	1.81596	0.907978	−	0.419018i	$-0.137626\pi$
$521$	41.4500	1.81596	0.907978	+	0.419018i	$0.137626\pi$
$522$	0	0
$523$	32.4222i	1.41772i	0.705347	+	0.708862i	$0.250791\pi$
$523$	32.4222i	1.41772i	−0.705347	+	0.708862i	$0.749209\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	29.7250	1.29607
$527$	− 70.5416i	− 3.07284i
$528$	0	0
$529$	−30.3305	−1.31872
$530$	0	0
$531$	0	0
$532$	0.211103i	0.00915246i
$533$	− 28.0278i	− 1.21402i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	11.3667i	0.490053i
$539$	6.51388	0.280573
$540$	0	0
$541$	25.7250	1.10600	0.553002	−	0.833180i	$-0.313482\pi$
$541$	25.7250	1.10600	0.553002	+	0.833180i	$0.313482\pi$
$542$	− 0.275019i	− 0.0118131i
$543$	0	0
$544$	11.7250	0.502704
$545$	0	0
$546$	0	0
$547$	− 7.11943i	− 0.304405i	−0.988349	−	0.152202i	$-0.951363\pi$
$547$	− 7.11943i	− 0.304405i	0.988349	−	0.152202i	$-0.0486366\pi$
$548$	3.90833i	0.166955i
$549$	0	0
$550$	0	0
$551$	0.908327	0.0386960
$552$	0	0
$553$	7.60555i	0.323421i
$554$	−18.7527	−0.796727
$555$	0	0
$556$	1.88057	0.0797540
$557$	− 19.4222i	− 0.822945i	−0.911422	−	0.411473i	$-0.865015\pi$
$557$	− 19.4222i	− 0.822945i	0.911422	−	0.411473i	$-0.134985\pi$
$558$	0	0
$559$	−36.0555	−1.52499
$560$	0	0
$561$	0	0
$562$	1.54163i	0.0650299i
$563$	8.09167i	0.341023i	0.985356	+	0.170512i	$0.0545421\pi$
$563$	8.09167i	0.341023i	−0.985356	+	0.170512i	$0.945458\pi$
$564$	0	0
$565$	0	0
$566$	8.21110	0.345138
$567$	0	0
$568$	7.81665i	0.327980i
$569$	−46.1472	−1.93459	−0.967295	−	0.253653i	$-0.918368\pi$
$569$	−46.1472	−1.93459	−0.967295	+	0.253653i	$0.918368\pi$
$570$	0	0
$571$	22.3305	0.934504	0.467252	−	0.884124i	$-0.345244\pi$
$571$	22.3305	0.934504	0.467252	+	0.884124i	$0.345244\pi$
$572$	− 1.51388i	− 0.0632984i
$573$	0	0
$574$	−5.09167	−0.212522
$575$	0	0
$576$	0	0
$577$	44.3583i	1.84666i	0.384008	+	0.923330i	$0.374544\pi$
$577$	44.3583i	1.84666i	−0.384008	+	0.923330i	$0.625456\pi$
$578$	− 40.0278i	− 1.66494i
$579$	0	0
$580$	0	0
$581$	2.44996	0.101642
$582$	0	0
$583$	− 1.30278i	− 0.0539555i
$584$	−23.7250	−0.981747
$585$	0	0
$586$	−1.02776	−0.0424562
$587$	− 16.5416i	− 0.682746i	−0.939928	−	0.341373i	$-0.889108\pi$
$587$	− 16.5416i	− 0.682746i	0.939928	−	0.341373i	$-0.110892\pi$
$588$	0	0
$589$	10.2111	0.420741
$590$	0	0
$591$	0	0
$592$	− 7.90833i	− 0.325030i
$593$	− 6.39445i	− 0.262589i	−0.991343	−	0.131294i	$-0.958087\pi$
$593$	− 6.39445i	− 0.262589i	0.991343	−	0.131294i	$-0.0419133\pi$
$594$	0	0
$595$	0	0
$596$	5.21110	0.213455
$597$	0	0
$598$	− 47.5694i	− 1.94526i
$599$	−24.9083	−1.01773	−0.508863	−	0.860847i	$-0.669934\pi$
$599$	−24.9083	−1.01773	−0.508863	+	0.860847i	$0.669934\pi$
$600$	0	0
$601$	−1.90833	−0.0778423	−0.0389211	−	0.999242i	$-0.512392\pi$
$601$	−1.90833	−0.0778423	−0.0389211	+	0.999242i	$0.512392\pi$
$602$	6.55004i	0.266960i
$603$	0	0
$604$	0.247263	0.0100610
$605$	0	0
$606$	0	0
$607$	7.21110i	0.292690i	0.989234	+	0.146345i	$0.0467509\pi$
$607$	7.21110i	0.292690i	−0.989234	+	0.146345i	$0.953249\pi$
$608$	1.69722i	0.0688315i
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	15.8806i	0.641410i	0.947179	+	0.320705i	$0.103920\pi$
$613$	15.8806i	0.641410i	−0.947179	+	0.320705i	$0.896080\pi$
$614$	22.0278	0.888968
$615$	0	0
$616$	−2.09167	−0.0842759
$617$	− 3.39445i	− 0.136655i	−0.997663	−	0.0683277i	$-0.978234\pi$
$617$	− 3.39445i	− 0.136655i	0.997663	−	0.0683277i	$-0.0217663\pi$
$618$	0	0
$619$	11.4222	0.459097	0.229549	−	0.973297i	$-0.426275\pi$
$619$	11.4222	0.459097	0.229549	+	0.973297i	$0.426275\pi$
$620$	0	0
$621$	0	0
$622$	− 6.27502i	− 0.251605i
$623$	1.18335i	0.0474098i
$624$	0	0
$625$	0	0
$626$	0.238859	0.00954672
$627$	0	0
$628$	5.81665i	0.232110i
$629$	16.5416	0.659558
$630$	0	0
$631$	6.93608	0.276121	0.138061	−	0.990424i	$-0.455913\pi$
$631$	6.93608	0.276121	0.138061	+	0.990424i	$0.455913\pi$
$632$	32.7250i	1.30173i
$633$	0	0
$634$	−1.18335	−0.0469967
$635$	0	0
$636$	0	0
$637$	− 32.5694i	− 1.29045i
$638$	1.18335i	0.0468491i
$639$	0	0
$640$	0	0
$641$	−27.7889	−1.09760	−0.548798	−	0.835955i	$-0.684914\pi$
$641$	−27.7889	−1.09760	−0.548798	+	0.835955i	$0.684914\pi$
$642$	0	0
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	1.54163	0.0607489
$645$	0	0
$646$	9.00000	0.354100
$647$	33.2389i	1.30675i	0.757033	+	0.653377i	$0.226648\pi$
$647$	33.2389i	1.30675i	−0.757033	+	0.653377i	$0.773352\pi$
$648$	0	0
$649$	−14.2111	−0.557835
$650$	0	0
$651$	0	0
$652$	− 2.81665i	− 0.110309i
$653$	6.11943i	0.239472i	0.992806	+	0.119736i	$0.0382048\pi$
$653$	6.11943i	0.239472i	−0.992806	+	0.119736i	$0.961795\pi$
$654$	0	0
$655$	0	0
$656$	−18.5139	−0.722846
$657$	0	0
$658$	− 2.72498i	− 0.106231i
$659$	−30.9083	−1.20402	−0.602009	−	0.798489i	$-0.705633\pi$
$659$	−30.9083	−1.20402	−0.602009	+	0.798489i	$0.705633\pi$
$660$	0	0
$661$	−8.81665	−0.342928	−0.171464	−	0.985190i	$-0.554850\pi$
$661$	−8.81665	−0.342928	−0.171464	+	0.985190i	$0.554850\pi$
$662$	− 28.1472i	− 1.09397i
$663$	0	0
$664$	10.5416	0.409095
$665$	0	0
$666$	0	0
$667$	− 6.63331i	− 0.256843i
$668$	− 4.06392i	− 0.157238i
$669$	0	0
$670$	0	0
$671$	−7.90833	−0.305298
$672$	0	0
$673$	− 30.0278i	− 1.15748i	−0.815510	−	0.578742i	$-0.803544\pi$
$673$	− 30.0278i	− 1.15748i	0.815510	−	0.578742i	$-0.196456\pi$
$674$	40.1833	1.54780
$675$	0	0
$676$	−3.63331	−0.139743
$677$	− 24.2389i	− 0.931575i	−0.884897	−	0.465788i	$-0.845771\pi$
$677$	− 24.2389i	− 0.931575i	0.884897	−	0.465788i	$-0.154229\pi$
$678$	0	0
$679$	−10.6695	−0.409457
$680$	0	0
$681$	0	0
$682$	13.3028i	0.509390i
$683$	47.8444i	1.83072i	0.402642	+	0.915358i	$0.368092\pi$
$683$	47.8444i	1.83072i	−0.402642	+	0.915358i	$0.631908\pi$
$684$	0	0
$685$	0	0
$686$	−12.2750	−0.468662
$687$	0	0
$688$	23.8167i	0.908001i
$689$	−6.51388	−0.248159
$690$	0	0
$691$	27.5416	1.04773	0.523867	−	0.851800i	$-0.324489\pi$
$691$	27.5416	1.04773	0.523867	+	0.851800i	$0.324489\pi$
$692$	− 1.45837i	− 0.0554387i
$693$	0	0
$694$	−16.3028	−0.618845
$695$	0	0
$696$	0	0
$697$	− 38.7250i	− 1.46681i
$698$	6.75274i	0.255595i
$699$	0	0
$700$	0	0
$701$	41.2111	1.55652	0.778261	−	0.627941i	$-0.216102\pi$
$701$	41.2111	1.55652	0.778261	+	0.627941i	$0.216102\pi$
$702$	0	0
$703$	2.39445i	0.0903083i
$704$	−8.81665	−0.332290
$705$	0	0
$706$	−24.2750	−0.913602
$707$	0.358288i	0.0134748i
$708$	0	0
$709$	31.6333	1.18801	0.594007	−	0.804460i	$-0.297545\pi$
$709$	31.6333	1.18801	0.594007	+	0.804460i	$0.297545\pi$
$710$	0	0
$711$	0	0
$712$	5.09167i	0.190819i
$713$	− 74.5694i	− 2.79265i
$714$	0	0
$715$	0	0
$716$	3.78890	0.141598
$717$	0	0
$718$	1.02776i	0.0383555i
$719$	7.18335	0.267894	0.133947	−	0.990989i	$-0.457235\pi$
$719$	7.18335	0.267894	0.133947	+	0.990989i	$0.457235\pi$
$720$	0	0
$721$	2.02776	0.0755176
$722$	− 23.4500i	− 0.872717i
$723$	0	0
$724$	−6.02776	−0.224020
$725$	0	0
$726$	0	0
$727$	− 39.3305i	− 1.45869i	−0.684147	−	0.729344i	$-0.739825\pi$
$727$	− 39.3305i	− 1.45869i	0.684147	−	0.729344i	$-0.260175\pi$
$728$	10.4584i	0.387613i
$729$	0	0
$730$	0	0
$731$	−49.8167	−1.84254
$732$	0	0
$733$	− 19.6056i	− 0.724148i	−0.932149	−	0.362074i	$-0.882069\pi$
$733$	− 19.6056i	− 0.724148i	0.932149	−	0.362074i	$-0.117931\pi$
$734$	26.9638	0.995253
$735$	0	0
$736$	12.3944	0.456865
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−35.1194	−1.29189	−0.645945	−	0.763384i	$-0.723536\pi$
$739$	−35.1194	−1.29189	−0.645945	+	0.763384i	$0.723536\pi$
$740$	0	0
$741$	0	0
$742$	1.18335i	0.0434420i
$743$	40.6972i	1.49304i	0.665365	+	0.746518i	$0.268276\pi$
$743$	40.6972i	1.49304i	−0.665365	+	0.746518i	$0.731724\pi$
$744$	0	0
$745$	0	0
$746$	35.7250	1.30798
$747$	0	0
$748$	− 2.09167i	− 0.0764791i
$749$	−2.09167	−0.0764281
$750$	0	0
$751$	−45.3305	−1.65413	−0.827067	−	0.562103i	$-0.809992\pi$
$751$	−45.3305	−1.65413	−0.827067	+	0.562103i	$0.809992\pi$
$752$	− 9.90833i	− 0.361320i
$753$	0	0
$754$	5.91673	0.215475
$755$	0	0
$756$	0	0
$757$	49.0555i	1.78295i	0.453067	+	0.891476i	$0.350330\pi$
$757$	49.0555i	1.78295i	−0.453067	+	0.891476i	$0.649670\pi$
$758$	− 4.14719i	− 0.150633i
$759$	0	0
$760$	0	0
$761$	13.5778	0.492195	0.246097	−	0.969245i	$-0.420852\pi$
$761$	13.5778	0.492195	0.246097	+	0.969245i	$0.420852\pi$
$762$	0	0
$763$	− 8.02776i	− 0.290624i
$764$	−3.11943	−0.112857
$765$	0	0
$766$	−28.1833	−1.01831
$767$	71.0555i	2.56567i
$768$	0	0
$769$	5.18335	0.186916	0.0934581	−	0.995623i	$-0.470208\pi$
$769$	5.18335	0.186916	0.0934581	+	0.995623i	$0.470208\pi$
$770$	0	0
$771$	0	0
$772$	− 4.00000i	− 0.143963i
$773$	3.11943i	0.112198i	0.998425	+	0.0560990i	$0.0178663\pi$
$773$	3.11943i	0.112198i	−0.998425	+	0.0560990i	$0.982134\pi$
$774$	0	0
$775$	0	0
$776$	−45.9083	−1.64801
$777$	0	0
$778$	15.6333i	0.560481i
$779$	5.60555	0.200840
$780$	0	0
$781$	2.60555	0.0932340
$782$	− 65.7250i	− 2.35032i
$783$	0	0
$784$	−21.5139	−0.768353
$785$	0	0
$786$	0	0
$787$	10.2111i	0.363986i	0.983300	+	0.181993i	$0.0582549\pi$
$787$	10.2111i	0.363986i	−0.983300	+	0.181993i	$0.941745\pi$
$788$	− 4.02776i	− 0.143483i
$789$	0	0
$790$	0	0
$791$	7.54163	0.268150
$792$	0	0
$793$	39.5416i	1.40416i
$794$	−28.2666	−1.00314
$795$	0	0
$796$	1.96384	0.0696065
$797$	− 3.51388i	− 0.124468i	−0.998062	−	0.0622340i	$-0.980178\pi$
$797$	− 3.51388i	− 0.124468i	0.998062	−	0.0622340i	$-0.0198225\pi$
$798$	0	0
$799$	20.7250	0.733197
$800$	0	0
$801$	0	0
$802$	16.6611i	0.588323i
$803$	7.90833i	0.279079i
$804$	0	0
$805$	0	0
$806$	66.5139	2.34285
$807$	0	0
$808$	1.54163i	0.0542345i
$809$	39.6333	1.39343	0.696716	−	0.717347i	$-0.254644\pi$
$809$	39.6333	1.39343	0.696716	+	0.717347i	$0.254644\pi$
$810$	0	0
$811$	38.8722	1.36499	0.682493	−	0.730892i	$-0.260896\pi$
$811$	38.8722	1.36499	0.682493	+	0.730892i	$0.260896\pi$
$812$	0.191750i	0.00672911i
$813$	0	0
$814$	−3.11943	−0.109336
$815$	0	0
$816$	0	0
$817$	− 7.21110i	− 0.252285i
$818$	8.09167i	0.282919i
$819$	0	0
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	− 18.4222i	− 0.642158i	−0.947052	−	0.321079i	$-0.895955\pi$
$823$	− 18.4222i	− 0.642158i	0.947052	−	0.321079i	$-0.104045\pi$
$824$	8.72498	0.303949
$825$	0	0
$826$	12.9083	0.449138
$827$	− 13.8167i	− 0.480452i	−0.970717	−	0.240226i	$-0.922778\pi$
$827$	− 13.8167i	− 0.480452i	0.970717	−	0.240226i	$-0.0772216\pi$
$828$	0	0
$829$	−29.7527	−1.03336	−0.516678	−	0.856180i	$-0.672831\pi$
$829$	−29.7527	−1.03336	−0.516678	+	0.856180i	$0.672831\pi$
$830$	0	0
$831$	0	0
$832$	44.0833i	1.52831i
$833$	− 45.0000i	− 1.55916i
$834$	0	0
$835$	0	0
$836$	0.302776	0.0104717
$837$	0	0
$838$	8.33053i	0.287773i
$839$	−9.11943	−0.314838	−0.157419	−	0.987532i	$-0.550317\pi$
$839$	−9.11943	−0.314838	−0.157419	+	0.987532i	$0.550317\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	0.908327i	0.0313030i
$843$	0	0
$844$	−7.64171	−0.263039
$845$	0	0
$846$	0	0
$847$	0.697224i	0.0239569i
$848$	4.30278i	0.147758i
$849$	0	0
$850$	0	0
$851$	17.4861	0.599417
$852$	0	0
$853$	12.7250i	0.435695i	0.975983	+	0.217848i	$0.0699035\pi$
$853$	12.7250i	0.435695i	−0.975983	+	0.217848i	$0.930096\pi$
$854$	7.18335	0.245809
$855$	0	0
$856$	−9.00000	−0.307614
$857$	3.00000i	0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$	3.00000i	0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$	0	0
$859$	−41.3944	−1.41236	−0.706180	−	0.708032i	$-0.749583\pi$
$859$	−41.3944	−1.41236	−0.706180	+	0.708032i	$0.749583\pi$
$860$	0	0
$861$	0	0
$862$	− 42.9916i	− 1.46430i
$863$	− 12.3944i	− 0.421912i	−0.977496	−	0.210956i	$-0.932342\pi$
$863$	− 12.3944i	− 0.421912i	0.977496	−	0.210956i	$-0.0676577\pi$
$864$	0	0
$865$	0	0
$866$	6.51388	0.221351
$867$	0	0
$868$	2.15559i	0.0731655i
$869$	10.9083	0.370040
$870$	0	0
$871$	−20.0000	−0.677674
$872$	− 34.5416i	− 1.16973i
$873$	0	0
$874$	9.51388	0.321812
$875$	0	0
$876$	0	0
$877$	20.0000i	0.675352i	0.941262	+	0.337676i	$0.109641\pi$
$877$	20.0000i	0.675352i	−0.941262	+	0.337676i	$0.890359\pi$
$878$	− 31.6611i	− 1.06851i
$879$	0	0
$880$	0	0
$881$	−19.5416	−0.658374	−0.329187	−	0.944265i	$-0.606775\pi$
$881$	−19.5416	−0.658374	−0.329187	+	0.944265i	$0.606775\pi$
$882$	0	0
$883$	− 52.4500i	− 1.76508i	−0.470236	−	0.882541i	$-0.655831\pi$
$883$	− 52.4500i	− 1.76508i	0.470236	−	0.882541i	$-0.344169\pi$
$884$	−10.4584	−0.351753
$885$	0	0
$886$	11.2111	0.376644
$887$	3.23886i	0.108750i	0.998521	+	0.0543751i	$0.0173167\pi$
$887$	3.23886i	0.108750i	−0.998521	+	0.0543751i	$0.982683\pi$
$888$	0	0
$889$	−5.66106	−0.189866
$890$	0	0
$891$	0	0
$892$	6.85281i	0.229449i
$893$	3.00000i	0.100391i
$894$	0	0
$895$	0	0
$896$	5.64171	0.188476
$897$	0	0
$898$	30.5971i	1.02104i
$899$	9.27502	0.309339
$900$	0	0
$901$	−9.00000	−0.299833
$902$	7.30278i	0.243156i
$903$	0	0
$904$	32.4500	1.07927
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	− 0.513878i	− 0.0170536i
$909$	0	0
$910$	0	0
$911$	24.7889	0.821293	0.410646	−	0.911795i	$-0.365303\pi$
$911$	24.7889	0.821293	0.410646	+	0.911795i	$0.365303\pi$
$912$	0	0
$913$	− 3.51388i	− 0.116292i
$914$	26.9638	0.891885
$915$	0	0
$916$	5.66947	0.187324
$917$	6.90833i	0.228133i
$918$	0	0
$919$	−26.7889	−0.883684	−0.441842	−	0.897093i	$-0.645675\pi$
$919$	−26.7889	−0.883684	−0.441842	+	0.897093i	$0.645675\pi$
$920$	0	0
$921$	0	0
$922$	− 41.9638i	− 1.38201i
$923$	− 13.0278i	− 0.428814i
$924$	0	0
$925$	0	0
$926$	15.3583	0.504705
$927$	0	0
$928$	1.54163i	0.0506066i
$929$	53.6056	1.75874	0.879371	−	0.476138i	$-0.157964\pi$
$929$	53.6056	1.75874	0.879371	+	0.476138i	$0.157964\pi$
$930$	0	0
$931$	6.51388	0.213484
$932$	4.81665i	0.157775i
$933$	0	0
$934$	−24.2750	−0.794303
$935$	0	0
$936$	0	0
$937$	− 9.21110i	− 0.300914i	−0.988617	−	0.150457i	$-0.951926\pi$
$937$	− 9.21110i	− 0.300914i	0.988617	−	0.150457i	$-0.0480744\pi$
$938$	3.63331i	0.118632i
$939$	0	0
$940$	0	0
$941$	−59.6056	−1.94309	−0.971543	−	0.236864i	$-0.923880\pi$
$941$	−59.6056	−1.94309	−0.971543	+	0.236864i	$0.923880\pi$
$942$	0	0
$943$	− 40.9361i	− 1.33306i
$944$	46.9361	1.52764
$945$	0	0
$946$	9.39445	0.305440
$947$	− 6.63331i	− 0.215554i	−0.994175	−	0.107777i	$-0.965627\pi$
$947$	− 6.63331i	− 0.215554i	0.994175	−	0.107777i	$-0.0343732\pi$
$948$	0	0
$949$	39.5416	1.28358
$950$	0	0
$951$	0	0
$952$	14.4500i	0.468326i
$953$	37.2666i	1.20718i	0.797293	+	0.603592i	$0.206264\pi$
$953$	37.2666i	1.20718i	−0.797293	+	0.603592i	$0.793736\pi$
$954$	0	0
$955$	0	0
$956$	6.39445	0.206811
$957$	0	0
$958$	− 45.3583i	− 1.46546i
$959$	−9.00000	−0.290625
$960$	0	0
$961$	73.2666	2.36344
$962$	15.5971i	0.502872i
$963$	0	0
$964$	6.64171	0.213915
$965$	0	0
$966$	0	0
$967$	14.9083i	0.479419i	0.970845	+	0.239710i	$0.0770523\pi$
$967$	14.9083i	0.479419i	−0.970845	+	0.239710i	$0.922948\pi$
$968$	3.00000i	0.0964237i
$969$	0	0
$970$	0	0
$971$	−45.3583	−1.45562	−0.727808	−	0.685781i	$-0.759461\pi$
$971$	−45.3583	−1.45562	−0.727808	+	0.685781i	$0.759461\pi$
$972$	0	0
$973$	4.33053i	0.138830i
$974$	−5.48612	−0.175787
$975$	0	0
$976$	26.1194	0.836063
$977$	52.0278i	1.66452i	0.554389	+	0.832258i	$0.312952\pi$
$977$	52.0278i	1.66452i	−0.554389	+	0.832258i	$0.687048\pi$
$978$	0	0
$979$	1.69722	0.0542435
$980$	0	0
$981$	0	0
$982$	12.7527i	0.406956i
$983$	8.84441i	0.282093i	0.990003	+	0.141046i	$0.0450467\pi$
$983$	8.84441i	0.282093i	−0.990003	+	0.141046i	$0.954953\pi$
$984$	0	0
$985$	0	0
$986$	8.17494	0.260343
$987$	0	0
$988$	− 1.51388i	− 0.0481629i
$989$	−52.6611	−1.67452
$990$	0	0
$991$	−16.9083	−0.537111	−0.268555	−	0.963264i	$-0.586546\pi$
$991$	−16.9083	−0.537111	−0.268555	+	0.963264i	$0.586546\pi$
$992$	17.3305i	0.550245i
$993$	0	0
$994$	−2.36669	−0.0750669
$995$	0	0
$996$	0	0
$997$	46.7250i	1.47979i	0.672720	+	0.739897i	$0.265126\pi$
$997$	46.7250i	1.47979i	−0.672720	+	0.739897i	$0.734874\pi$
$998$	4.54163i	0.143763i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.k.199.3		4
3.2	odd	2		275.2.b.c.199.2		4
5.2	odd	4		2475.2.a.t.1.1		2
5.3	odd	4		2475.2.a.o.1.2		2
5.4	even	2	inner	2475.2.c.k.199.2		4
12.11	even	2		4400.2.b.y.4049.4		4
15.2	even	4		275.2.a.e.1.2	✓	2
15.8	even	4		275.2.a.f.1.1	yes	2
15.14	odd	2		275.2.b.c.199.3		4
60.23	odd	4		4400.2.a.bh.1.1		2
60.47	odd	4		4400.2.a.bs.1.2		2
60.59	even	2		4400.2.b.y.4049.1		4
165.32	odd	4		3025.2.a.n.1.1		2
165.98	odd	4		3025.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.2	✓	2	15.2	even	4
275.2.a.f.1.1	yes	2	15.8	even	4
275.2.b.c.199.2		4	3.2	odd	2
275.2.b.c.199.3		4	15.14	odd	2
2475.2.a.o.1.2		2	5.3	odd	4
2475.2.a.t.1.1		2	5.2	odd	4
2475.2.c.k.199.2		4	5.4	even	2	inner
2475.2.c.k.199.3		4	1.1	even	1	trivial
3025.2.a.h.1.2		2	165.98	odd	4
3025.2.a.n.1.1		2	165.32	odd	4
4400.2.a.bh.1.1		2	60.23	odd	4
4400.2.a.bs.1.2		2	60.47	odd	4
4400.2.b.y.4049.1		4	60.59	even	2
4400.2.b.y.4049.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833i q^{17} +1.00000 q^{19} +1.30278i q^{22} -7.30278i q^{23} +6.51388 q^{26} +0.211103i q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722i q^{32} +9.00000 q^{34} +2.39445i q^{37} +1.30278i q^{38} +5.60555 q^{41} -7.21110i q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000i q^{47} +6.51388 q^{49} -1.51388i q^{52} -1.30278i q^{53} -2.09167 q^{56} +1.18335i q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028i q^{62} -8.81665 q^{64} -4.00000i q^{67} -2.09167i q^{68} +2.60555 q^{71} +7.90833i q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224i q^{77} +10.9083 q^{79} +7.30278i q^{82} -3.51388i q^{83} +9.39445 q^{86} +3.00000i q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110i q^{92} -3.90833 q^{94} +15.3028i q^{97} +8.48612i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4}+O(q^{10}) \) 4 * q - 6 * q^4	\( 4 q - 6 q^{4} + 4 q^{11} + 18 q^{14} - 6 q^{16} + 4 q^{19} - 10 q^{26} - 18 q^{29} + 12 q^{31} + 36 q^{34} + 8 q^{41} - 6 q^{44} + 2 q^{46} - 10 q^{49} - 30 q^{56} - 28 q^{59} - 10 q^{61} + 8 q^{64} - 4 q^{71} + 38 q^{74} - 6 q^{76} + 22 q^{79} + 52 q^{86} + 14 q^{89} + 50 q^{91} + 6 q^{94}+O(q^{100}) \) 4 * q - 6 * q^4 + 4 * q^11 + 18 * q^14 - 6 * q^16 + 4 * q^19 - 10 * q^26 - 18 * q^29 + 12 * q^31 + 36 * q^34 + 8 * q^41 - 6 * q^44 + 2 * q^46 - 10 * q^49 - 30 * q^56 - 28 * q^59 - 10 * q^61 + 8 * q^64 - 4 * q^71 + 38 * q^74 - 6 * q^76 + 22 * q^79 + 52 * q^86 + 14 * q^89 + 50 * q^91 + 6 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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