LMFDB - Embedded newform 2475.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +1.00000 q^{11} +5.00000 q^{13} +0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} -1.00000 q^{19} +1.30278 q^{22} +7.30278 q^{23} +6.51388 q^{26} -0.211103 q^{28} -0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} -9.00000 q^{34} +2.39445 q^{37} -1.30278 q^{38} +5.60555 q^{41} +7.21110 q^{43} -0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} -6.51388 q^{49} -1.51388 q^{52} +1.30278 q^{53} -2.09167 q^{56} -1.18335 q^{58} +14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} +8.81665 q^{64} -4.00000 q^{67} +2.09167 q^{68} +2.60555 q^{71} -7.90833 q^{73} +3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} -10.9083 q^{79} +7.30278 q^{82} +3.51388 q^{83} +9.39445 q^{86} -3.00000 q^{88} -1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} +3.90833 q^{94} +15.3028 q^{97} -8.48612 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + 2 q^{11} + 10 q^{13} - 9 q^{14} - 3 q^{16} - 3 q^{17} - 2 q^{19} - q^{22} + 11 q^{23} - 5 q^{26} + 14 q^{28} + 9 q^{29} + 6 q^{31} + 7 q^{32} - 18 q^{34} + 12 q^{37}+ \cdots - 35 q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 + 2 * q^11 + 10 * q^13 - 9 * q^14 - 3 * q^16 - 3 * q^17 - 2 * q^19 - q^22 + 11 * q^23 - 5 * q^26 + 14 * q^28 + 9 * q^29 + 6 * q^31 + 7 * q^32 - 18 * q^34 + 12 * q^37 + q^38 + 4 * q^41 + 3 * q^44 + q^46 + 6 * q^47 + 5 * q^49 + 15 * q^52 - q^53 - 15 * q^56 - 24 * q^58 + 14 * q^59 - 5 * q^61 + 23 * q^62 - 4 * q^64 - 8 * q^67 + 15 * q^68 - 2 * q^71 - 5 * q^73 - 19 * q^74 - 3 * q^76 + 5 * q^77 - 11 * q^79 + 11 * q^82 - 11 * q^83 + 26 * q^86 - 6 * q^88 - 7 * q^89 + 25 * q^91 + 10 * q^92 - 3 * q^94 + 27 * q^97 - 35 * q^98

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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.697224	0.263526	0.131763	−	0.991281i	$-0.457936\pi$
$7$	0.697224	0.263526	0.131763	+	0.991281i	$0.457936\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0.908327	0.242761
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−6.90833	−1.67552	−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833	−1.67552	−0.837758	+	0.546042i	$0.816134\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	1.30278	0.277753
$23$	7.30278	1.52273	0.761367	−	0.648321i	$-0.224529\pi$
$23$	7.30278	1.52273	0.761367	+	0.648321i	$0.224529\pi$
$24$	0	0
$25$	0	0
$26$	6.51388	1.27748
$27$	0	0
$28$	−0.211103	−0.0398946
$29$	−0.908327	−0.168672	−0.0843360	−	0.996437i	$-0.526877\pi$
$29$	−0.908327	−0.168672	−0.0843360	+	0.996437i	$0.526877\pi$
$30$	0	0
$31$	10.2111	1.83397	0.916984	−	0.398924i	$-0.130616\pi$
$31$	10.2111	1.83397	0.916984	+	0.398924i	$0.130616\pi$
$32$	1.69722	0.300030
$33$	0	0
$34$	−9.00000	−1.54349
$35$	0	0
$36$	0	0
$37$	2.39445	0.393645	0.196822	−	0.980439i	$-0.436938\pi$
$37$	2.39445	0.393645	0.196822	+	0.980439i	$0.436938\pi$
$38$	−1.30278	−0.211338
$39$	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
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	−0.302776	−0.0456451
$45$	0	0
$46$	9.51388	1.40274
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	−6.51388	−0.930554
$50$	0	0
$51$	0	0
$52$	−1.51388	−0.209937
$53$	1.30278	0.178950	0.0894750	−	0.995989i	$-0.471481\pi$
$53$	1.30278	0.178950	0.0894750	+	0.995989i	$0.471481\pi$
$54$	0	0
$55$	0	0
$56$	−2.09167	−0.279512
$57$	0	0
$58$	−1.18335	−0.155381
$59$	14.2111	1.85013	0.925064	−	0.379811i	$-0.124011\pi$
$59$	14.2111	1.85013	0.925064	+	0.379811i	$0.124011\pi$
$60$	0	0
$61$	−7.90833	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$61$	−7.90833	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$62$	13.3028	1.68945
$63$	0	0
$64$	8.81665	1.10208
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.09167	0.253653
$69$	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
$73$	−7.90833	−0.925600	−0.462800	−	0.886463i	$-0.653155\pi$
$73$	−7.90833	−0.925600	−0.462800	+	0.886463i	$0.653155\pi$
$74$	3.11943	0.362626
$75$	0	0
$76$	0.302776	0.0347307
$77$	0.697224	0.0794561
$78$	0	0
$79$	−10.9083	−1.22728	−0.613641	−	0.789585i	$-0.710296\pi$
$79$	−10.9083	−1.22728	−0.613641	+	0.789585i	$0.710296\pi$
$80$	0	0
$81$	0	0
$82$	7.30278	0.806457
$83$	3.51388	0.385698	0.192849	−	0.981228i	$-0.438227\pi$
$83$	3.51388	0.385698	0.192849	+	0.981228i	$0.438227\pi$
$84$	0	0
$85$	0	0
$86$	9.39445	1.01303
$87$	0	0
$88$	−3.00000	−0.319801
$89$	−1.69722	−0.179905	−0.0899527	−	0.995946i	$-0.528672\pi$
$89$	−1.69722	−0.179905	−0.0899527	+	0.995946i	$0.528672\pi$
$90$	0	0
$91$	3.48612	0.365445
$92$	−2.21110	−0.230523
$93$	0	0
$94$	3.90833	0.403113
$95$	0	0
$96$	0	0
$97$	15.3028	1.55376	0.776881	−	0.629648i	$-0.216801\pi$
$97$	15.3028	1.55376	0.776881	+	0.629648i	$0.216801\pi$
$98$	−8.48612	−0.857228
$99$	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
$103$	2.90833	0.286566	0.143283	−	0.989682i	$-0.454234\pi$
$103$	2.90833	0.286566	0.143283	+	0.989682i	$0.454234\pi$
$104$	−15.0000	−1.47087
$105$	0	0
$106$	1.69722	0.164849
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	11.5139	1.10283	0.551415	−	0.834231i	$-0.314088\pi$
$109$	11.5139	1.10283	0.551415	+	0.834231i	$0.314088\pi$
$110$	0	0
$111$	0	0
$112$	−2.30278	−0.217592
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	0	0
$115$	0	0
$116$	0.275019	0.0255349
$117$	0	0
$118$	18.5139	1.70434
$119$	−4.81665	−0.441542
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.3028	−0.932769
$123$	0	0
$124$	−3.09167	−0.277640
$125$	0	0
$126$	0	0
$127$	8.11943	0.720483	0.360241	−	0.932859i	$-0.382694\pi$
$127$	8.11943	0.720483	0.360241	+	0.932859i	$0.382694\pi$
$128$	8.09167	0.715210
$129$	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
$133$	−0.697224	−0.0604570
$134$	−5.21110	−0.450171
$135$	0	0
$136$	20.7250	1.77715
$137$	12.9083	1.10283	0.551416	−	0.834230i	$-0.314088\pi$
$137$	12.9083	1.10283	0.551416	+	0.834230i	$0.314088\pi$
$138$	0	0
$139$	−6.21110	−0.526819	−0.263409	−	0.964684i	$-0.584847\pi$
$139$	−6.21110	−0.526819	−0.263409	+	0.964684i	$0.584847\pi$
$140$	0	0
$141$	0	0
$142$	3.39445	0.284856
$143$	5.00000	0.418121
$144$	0	0
$145$	0	0
$146$	−10.3028	−0.852664
$147$	0	0
$148$	−0.724981	−0.0595930
$149$	−17.2111	−1.40999	−0.704994	−	0.709213i	$-0.749050\pi$
$149$	−17.2111	−1.40999	−0.704994	+	0.709213i	$0.749050\pi$
$150$	0	0
$151$	0.816654	0.0664583	0.0332292	−	0.999448i	$-0.489421\pi$
$151$	0.816654	0.0664583	0.0332292	+	0.999448i	$0.489421\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	0.908327	0.0731951
$155$	0	0
$156$	0	0
$157$	19.2111	1.53321	0.766606	−	0.642117i	$-0.221944\pi$
$157$	19.2111	1.53321	0.766606	+	0.642117i	$0.221944\pi$
$158$	−14.2111	−1.13057
$159$	0	0
$160$	0	0
$161$	5.09167	0.401280
$162$	0	0
$163$	9.30278	0.728650	0.364325	−	0.931272i	$-0.381300\pi$
$163$	9.30278	0.728650	0.364325	+	0.931272i	$0.381300\pi$
$164$	−1.69722	−0.132531
$165$	0	0
$166$	4.57779	0.355306
$167$	−13.4222	−1.03864	−0.519321	−	0.854579i	$-0.673815\pi$
$167$	−13.4222	−1.03864	−0.519321	+	0.854579i	$0.673815\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	−2.18335	−0.166479
$173$	4.81665	0.366203	0.183102	−	0.983094i	$-0.441386\pi$
$173$	4.81665	0.366203	0.183102	+	0.983094i	$0.441386\pi$
$174$	0	0
$175$	0	0
$176$	−3.30278	−0.248956
$177$	0	0
$178$	−2.21110	−0.165729
$179$	−12.5139	−0.935331	−0.467666	−	0.883905i	$-0.654905\pi$
$179$	−12.5139	−0.935331	−0.467666	+	0.883905i	$0.654905\pi$
$180$	0	0
$181$	−19.9083	−1.47977	−0.739887	−	0.672731i	$-0.765121\pi$
$181$	−19.9083	−1.47977	−0.739887	+	0.672731i	$0.765121\pi$
$182$	4.54163	0.336648
$183$	0	0
$184$	−21.9083	−1.61510
$185$	0	0
$186$	0	0
$187$	−6.90833	−0.505187
$188$	−0.908327	−0.0662465
$189$	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
$193$	13.2111	0.950956	0.475478	−	0.879728i	$-0.342275\pi$
$193$	13.2111	0.950956	0.475478	+	0.879728i	$0.342275\pi$
$194$	19.9361	1.43133
$195$	0	0
$196$	1.97224	0.140875
$197$	−13.3028	−0.947784	−0.473892	−	0.880583i	$-0.657151\pi$
$197$	−13.3028	−0.947784	−0.473892	+	0.880583i	$0.657151\pi$
$198$	0	0
$199$	−6.48612	−0.459789	−0.229894	−	0.973216i	$-0.573838\pi$
$199$	−6.48612	−0.459789	−0.229894	+	0.973216i	$0.573838\pi$
$200$	0	0
$201$	0	0
$202$	0.669468	0.0471036
$203$	−0.633308	−0.0444495
$204$	0	0
$205$	0	0
$206$	3.78890	0.263985
$207$	0	0
$208$	−16.5139	−1.14503
$209$	−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	$0.835081\pi$
$212$	−0.394449	−0.0270908
$213$	0	0
$214$	3.90833	0.267168
$215$	0	0
$216$	0	0
$217$	7.11943	0.483298
$218$	15.0000	1.01593
$219$	0	0
$220$	0	0
$221$	−34.5416	−2.32352
$222$	0	0
$223$	−22.6333	−1.51564	−0.757819	−	0.652465i	$-0.773735\pi$
$223$	−22.6333	−1.51564	−0.757819	+	0.652465i	$0.773735\pi$
$224$	1.18335	0.0790656
$225$	0	0
$226$	14.0917	0.937364
$227$	−1.69722	−0.112649	−0.0563244	−	0.998413i	$-0.517938\pi$
$227$	−1.69722	−0.112649	−0.0563244	+	0.998413i	$0.517938\pi$
$228$	0	0
$229$	−18.7250	−1.23738	−0.618691	−	0.785635i	$-0.712337\pi$
$229$	−18.7250	−1.23738	−0.618691	+	0.785635i	$0.712337\pi$
$230$	0	0
$231$	0	0
$232$	2.72498	0.178904
$233$	−15.9083	−1.04219	−0.521095	−	0.853499i	$-0.674476\pi$
$233$	−15.9083	−1.04219	−0.521095	+	0.853499i	$0.674476\pi$
$234$	0	0
$235$	0	0
$236$	−4.30278	−0.280087
$237$	0	0
$238$	−6.27502	−0.406749
$239$	−21.1194	−1.36610	−0.683051	−	0.730371i	$-0.739347\pi$
$239$	−21.1194	−1.36610	−0.683051	+	0.730371i	$0.739347\pi$
$240$	0	0
$241$	21.9361	1.41303	0.706514	−	0.707699i	$-0.250267\pi$
$241$	21.9361	1.41303	0.706514	+	0.707699i	$0.250267\pi$
$242$	1.30278	0.0837456
$243$	0	0
$244$	2.39445	0.153289
$245$	0	0
$246$	0	0
$247$	−5.00000	−0.318142
$248$	−30.6333	−1.94522
$249$	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
$253$	7.30278	0.459122
$254$	10.5778	0.663710
$255$	0	0
$256$	−7.09167	−0.443230
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	1.66947	0.103736
$260$	0	0
$261$	0	0
$262$	12.9083	0.797479
$263$	22.8167	1.40694	0.703468	−	0.710727i	$-0.251634\pi$
$263$	22.8167	1.40694	0.703468	+	0.710727i	$0.251634\pi$
$264$	0	0
$265$	0	0
$266$	−0.908327	−0.0556931
$267$	0	0
$268$	1.21110	0.0739799
$269$	−8.72498	−0.531971	−0.265986	−	0.963977i	$-0.585697\pi$
$269$	−8.72498	−0.531971	−0.265986	+	0.963977i	$0.585697\pi$
$270$	0	0
$271$	−0.211103	−0.0128236	−0.00641178	−	0.999979i	$-0.502041\pi$
$271$	−0.211103	−0.0128236	−0.00641178	+	0.999979i	$0.502041\pi$
$272$	22.8167	1.38346
$273$	0	0
$274$	16.8167	1.01593
$275$	0	0
$276$	0	0
$277$	14.3944	0.864879	0.432439	−	0.901663i	$-0.357653\pi$
$277$	14.3944	0.864879	0.432439	+	0.901663i	$0.357653\pi$
$278$	−8.09167	−0.485306
$279$	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
$283$	6.30278	0.374661	0.187331	−	0.982297i	$-0.440016\pi$
$283$	6.30278	0.374661	0.187331	+	0.982297i	$0.440016\pi$
$284$	−0.788897	−0.0468125
$285$	0	0
$286$	6.51388	0.385174
$287$	3.90833	0.230701
$288$	0	0
$289$	30.7250	1.80735
$290$	0	0
$291$	0	0
$292$	2.39445	0.140125
$293$	−0.788897	−0.0460879	−0.0230439	−	0.999734i	$-0.507336\pi$
$293$	−0.788897	−0.0460879	−0.0230439	+	0.999734i	$0.507336\pi$
$294$	0	0
$295$	0	0
$296$	−7.18335	−0.417524
$297$	0	0
$298$	−22.4222	−1.29888
$299$	36.5139	2.11165
$300$	0	0
$301$	5.02776	0.289795
$302$	1.06392	0.0612215
$303$	0	0
$304$	3.30278	0.189427
$305$	0	0
$306$	0	0
$307$	−16.9083	−0.965009	−0.482505	−	0.875893i	$-0.660273\pi$
$307$	−16.9083	−0.965009	−0.482505	+	0.875893i	$0.660273\pi$
$308$	−0.211103	−0.0120287
$309$	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
$313$	0.183346	0.0103633	0.00518167	−	0.999987i	$-0.498351\pi$
$313$	0.183346	0.0103633	0.00518167	+	0.999987i	$0.498351\pi$
$314$	25.0278	1.41240
$315$	0	0
$316$	3.30278	0.185796
$317$	0.908327	0.0510167	0.0255084	−	0.999675i	$-0.491880\pi$
$317$	0.908327	0.0510167	0.0255084	+	0.999675i	$0.491880\pi$
$318$	0	0
$319$	−0.908327	−0.0508565
$320$	0	0
$321$	0	0
$322$	6.63331	0.369660
$323$	6.90833	0.384390
$324$	0	0
$325$	0	0
$326$	12.1194	0.671233
$327$	0	0
$328$	−16.8167	−0.928544
$329$	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	$0.702363\pi$
$332$	−1.06392	−0.0583900
$333$	0	0
$334$	−17.4861	−0.956798
$335$	0	0
$336$	0	0
$337$	−30.8444	−1.68020	−0.840101	−	0.542430i	$-0.817504\pi$
$337$	−30.8444	−1.68020	−0.840101	+	0.542430i	$0.817504\pi$
$338$	15.6333	0.850340
$339$	0	0
$340$	0	0
$341$	10.2111	0.552962
$342$	0	0
$343$	−9.42221	−0.508751
$344$	−21.6333	−1.16639
$345$	0	0
$346$	6.27502	0.337347
$347$	12.5139	0.671780	0.335890	−	0.941901i	$-0.390963\pi$
$347$	12.5139	0.671780	0.335890	+	0.941901i	$0.390963\pi$
$348$	0	0
$349$	−5.18335	−0.277458	−0.138729	−	0.990330i	$-0.544302\pi$
$349$	−5.18335	−0.277458	−0.138729	+	0.990330i	$0.544302\pi$
$350$	0	0
$351$	0	0
$352$	1.69722	0.0904624
$353$	−18.6333	−0.991751	−0.495875	−	0.868394i	$-0.665153\pi$
$353$	−18.6333	−0.991751	−0.495875	+	0.868394i	$0.665153\pi$
$354$	0	0
$355$	0	0
$356$	0.513878	0.0272355
$357$	0	0
$358$	−16.3028	−0.861628
$359$	−0.788897	−0.0416364	−0.0208182	−	0.999783i	$-0.506627\pi$
$359$	−0.788897	−0.0416364	−0.0208182	+	0.999783i	$0.506627\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−25.9361	−1.36317
$363$	0	0
$364$	−1.05551	−0.0553239
$365$	0	0
$366$	0	0
$367$	−20.6972	−1.08039	−0.540193	−	0.841541i	$-0.681649\pi$
$367$	−20.6972	−1.08039	−0.540193	+	0.841541i	$0.681649\pi$
$368$	−24.1194	−1.25731
$369$	0	0
$370$	0	0
$371$	0.908327	0.0471580
$372$	0	0
$373$	27.4222	1.41987	0.709934	−	0.704268i	$-0.248725\pi$
$373$	27.4222	1.41987	0.709934	+	0.704268i	$0.248725\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	−4.54163	−0.233906
$378$	0	0
$379$	3.18335	0.163518	0.0817588	−	0.996652i	$-0.473946\pi$
$379$	3.18335	0.163518	0.0817588	+	0.996652i	$0.473946\pi$
$380$	0	0
$381$	0	0
$382$	−13.4222	−0.686740
$383$	−21.6333	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$383$	−21.6333	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$384$	0	0
$385$	0	0
$386$	17.2111	0.876022
$387$	0	0
$388$	−4.63331	−0.235221
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−50.4500	−2.55136
$392$	19.5416	0.987002
$393$	0	0
$394$	−17.3305	−0.873100
$395$	0	0
$396$	0	0
$397$	21.6972	1.08895	0.544476	−	0.838776i	$-0.316728\pi$
$397$	21.6972	1.08895	0.544476	+	0.838776i	$0.316728\pi$
$398$	−8.44996	−0.423558
$399$	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
$403$	51.0555	2.54326
$404$	−0.155590	−0.00774088
$405$	0	0
$406$	−0.825058	−0.0409469
$407$	2.39445	0.118688
$408$	0	0
$409$	−6.21110	−0.307119	−0.153560	−	0.988139i	$-0.549074\pi$
$409$	−6.21110	−0.307119	−0.153560	+	0.988139i	$0.549074\pi$
$410$	0	0
$411$	0	0
$412$	−0.880571	−0.0433826
$413$	9.90833	0.487557
$414$	0	0
$415$	0	0
$416$	8.48612	0.416066
$417$	0	0
$418$	−1.30278	−0.0637208
$419$	−6.39445	−0.312389	−0.156195	−	0.987726i	$-0.549923\pi$
$419$	−6.39445	−0.312389	−0.156195	+	0.987726i	$0.549923\pi$
$420$	0	0
$421$	0.697224	0.0339806	0.0169903	−	0.999856i	$-0.494592\pi$
$421$	0.697224	0.0339806	0.0169903	+	0.999856i	$0.494592\pi$
$422$	−32.8806	−1.60060
$423$	0	0
$424$	−3.90833	−0.189805
$425$	0	0
$426$	0	0
$427$	−5.51388	−0.266835
$428$	−0.908327	−0.0439056
$429$	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
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	9.27502	0.445215
$435$	0	0
$436$	−3.48612	−0.166955
$437$	−7.30278	−0.349339
$438$	0	0
$439$	24.3028	1.15991	0.579954	−	0.814649i	$-0.303070\pi$
$439$	24.3028	1.15991	0.579954	+	0.814649i	$0.303070\pi$
$440$	0	0
$441$	0	0
$442$	−45.0000	−2.14043
$443$	8.60555	0.408862	0.204431	−	0.978881i	$-0.434466\pi$
$443$	8.60555	0.408862	0.204431	+	0.978881i	$0.434466\pi$
$444$	0	0
$445$	0	0
$446$	−29.4861	−1.39621
$447$	0	0
$448$	6.14719	0.290427
$449$	−23.4861	−1.10838	−0.554189	−	0.832391i	$-0.686972\pi$
$449$	−23.4861	−1.10838	−0.554189	+	0.832391i	$0.686972\pi$
$450$	0	0
$451$	5.60555	0.263955
$452$	−3.27502	−0.154044
$453$	0	0
$454$	−2.21110	−0.103772
$455$	0	0
$456$	0	0
$457$	−20.6972	−0.968175	−0.484088	−	0.875020i	$-0.660848\pi$
$457$	−20.6972	−0.968175	−0.484088	+	0.875020i	$0.660848\pi$
$458$	−24.3944	−1.13988
$459$	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
$463$	11.7889	0.547877	0.273938	−	0.961747i	$-0.411674\pi$
$463$	11.7889	0.547877	0.273938	+	0.961747i	$0.411674\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−20.7250	−0.960066
$467$	18.6333	0.862247	0.431123	−	0.902293i	$-0.358117\pi$
$467$	18.6333	0.862247	0.431123	+	0.902293i	$0.358117\pi$
$468$	0	0
$469$	−2.78890	−0.128779
$470$	0	0
$471$	0	0
$472$	−42.6333	−1.96236
$473$	7.21110	0.331567
$474$	0	0
$475$	0	0
$476$	1.45837	0.0668441
$477$	0	0
$478$	−27.5139	−1.25846
$479$	34.8167	1.59081	0.795407	−	0.606076i	$-0.207257\pi$
$479$	34.8167	1.59081	0.795407	+	0.606076i	$0.207257\pi$
$480$	0	0
$481$	11.9722	0.545887
$482$	28.5778	1.30168
$483$	0	0
$484$	−0.302776	−0.0137625
$485$	0	0
$486$	0	0
$487$	4.21110	0.190823	0.0954116	−	0.995438i	$-0.469583\pi$
$487$	4.21110	0.190823	0.0954116	+	0.995438i	$0.469583\pi$
$488$	23.7250	1.07398
$489$	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
$493$	6.27502	0.282613
$494$	−6.51388	−0.293073
$495$	0	0
$496$	−33.7250	−1.51430
$497$	1.81665	0.0814881
$498$	0	0
$499$	−3.48612	−0.156060	−0.0780301	−	0.996951i	$-0.524863\pi$
$499$	−3.48612	−0.156060	−0.0780301	+	0.996951i	$0.524863\pi$
$500$	0	0
$501$	0	0
$502$	−9.00000	−0.401690
$503$	−9.39445	−0.418878	−0.209439	−	0.977822i	$-0.567164\pi$
$503$	−9.39445	−0.418878	−0.209439	+	0.977822i	$0.567164\pi$
$504$	0	0
$505$	0	0
$506$	9.51388	0.422943
$507$	0	0
$508$	−2.45837	−0.109072
$509$	22.6972	1.00604	0.503018	−	0.864276i	$-0.332223\pi$
$509$	22.6972	1.00604	0.503018	+	0.864276i	$0.332223\pi$
$510$	0	0
$511$	−5.51388	−0.243920
$512$	−25.4222	−1.12351
$513$	0	0
$514$	23.4500	1.03433
$515$	0	0
$516$	0	0
$517$	3.00000	0.131940
$518$	2.17494	0.0955615
$519$	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
$523$	−32.4222	−1.41772	−0.708862	−	0.705347i	$-0.750791\pi$
$523$	−32.4222	−1.41772	−0.708862	+	0.705347i	$0.750791\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	29.7250	1.29607
$527$	−70.5416	−3.07284
$528$	0	0
$529$	30.3305	1.31872
$530$	0	0
$531$	0	0
$532$	0.211103	0.00915246
$533$	28.0278	1.21402
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	−11.3667	−0.490053
$539$	−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	$0.313482\pi$
$542$	−0.275019	−0.0118131
$543$	0	0
$544$	−11.7250	−0.502704
$545$	0	0
$546$	0	0
$547$	−7.11943	−0.304405	−0.152202	−	0.988349i	$-0.548637\pi$
$547$	−7.11943	−0.304405	−0.152202	+	0.988349i	$0.548637\pi$
$548$	−3.90833	−0.166955
$549$	0	0
$550$	0	0
$551$	0.908327	0.0386960
$552$	0	0
$553$	−7.60555	−0.323421
$554$	18.7527	0.796727
$555$	0	0
$556$	1.88057	0.0797540
$557$	−19.4222	−0.822945	−0.411473	−	0.911422i	$-0.634985\pi$
$557$	−19.4222	−0.822945	−0.411473	+	0.911422i	$0.634985\pi$
$558$	0	0
$559$	36.0555	1.52499
$560$	0	0
$561$	0	0
$562$	1.54163	0.0650299
$563$	−8.09167	−0.341023	−0.170512	−	0.985356i	$-0.554542\pi$
$563$	−8.09167	−0.341023	−0.170512	+	0.985356i	$0.554542\pi$
$564$	0	0
$565$	0	0
$566$	8.21110	0.345138
$567$	0	0
$568$	−7.81665	−0.327980
$569$	46.1472	1.93459	0.967295	−	0.253653i	$-0.0816321\pi$
$569$	46.1472	1.93459	0.967295	+	0.253653i	$0.0816321\pi$
$570$	0	0
$571$	22.3305	0.934504	0.467252	−	0.884124i	$-0.345244\pi$
$571$	22.3305	0.934504	0.467252	+	0.884124i	$0.345244\pi$
$572$	−1.51388	−0.0632984
$573$	0	0
$574$	5.09167	0.212522
$575$	0	0
$576$	0	0
$577$	44.3583	1.84666	0.923330	−	0.384008i	$-0.125456\pi$
$577$	44.3583	1.84666	0.923330	+	0.384008i	$0.125456\pi$
$578$	40.0278	1.66494
$579$	0	0
$580$	0	0
$581$	2.44996	0.101642
$582$	0	0
$583$	1.30278	0.0539555
$584$	23.7250	0.981747
$585$	0	0
$586$	−1.02776	−0.0424562
$587$	−16.5416	−0.682746	−0.341373	−	0.939928i	$-0.610892\pi$
$587$	−16.5416	−0.682746	−0.341373	+	0.939928i	$0.610892\pi$
$588$	0	0
$589$	−10.2111	−0.420741
$590$	0	0
$591$	0	0
$592$	−7.90833	−0.325030
$593$	6.39445	0.262589	0.131294	−	0.991343i	$-0.458087\pi$
$593$	6.39445	0.262589	0.131294	+	0.991343i	$0.458087\pi$
$594$	0	0
$595$	0	0
$596$	5.21110	0.213455
$597$	0	0
$598$	47.5694	1.94526
$599$	24.9083	1.01773	0.508863	−	0.860847i	$-0.330066\pi$
$599$	24.9083	1.01773	0.508863	+	0.860847i	$0.330066\pi$
$600$	0	0
$601$	−1.90833	−0.0778423	−0.0389211	−	0.999242i	$-0.512392\pi$
$601$	−1.90833	−0.0778423	−0.0389211	+	0.999242i	$0.512392\pi$
$602$	6.55004	0.266960
$603$	0	0
$604$	−0.247263	−0.0100610
$605$	0	0
$606$	0	0
$607$	7.21110	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$607$	7.21110	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$608$	−1.69722	−0.0688315
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	−15.8806	−0.641410	−0.320705	−	0.947179i	$-0.603920\pi$
$613$	−15.8806	−0.641410	−0.320705	+	0.947179i	$0.603920\pi$
$614$	−22.0278	−0.888968
$615$	0	0
$616$	−2.09167	−0.0842759
$617$	−3.39445	−0.136655	−0.0683277	−	0.997663i	$-0.521766\pi$
$617$	−3.39445	−0.136655	−0.0683277	+	0.997663i	$0.521766\pi$
$618$	0	0
$619$	−11.4222	−0.459097	−0.229549	−	0.973297i	$-0.573725\pi$
$619$	−11.4222	−0.459097	−0.229549	+	0.973297i	$0.573725\pi$
$620$	0	0
$621$	0	0
$622$	−6.27502	−0.251605
$623$	−1.18335	−0.0474098
$624$	0	0
$625$	0	0
$626$	0.238859	0.00954672
$627$	0	0
$628$	−5.81665	−0.232110
$629$	−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	$0.455913\pi$
$632$	32.7250	1.30173
$633$	0	0
$634$	1.18335	0.0469967
$635$	0	0
$636$	0	0
$637$	−32.5694	−1.29045
$638$	−1.18335	−0.0468491
$639$	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
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	−1.54163	−0.0607489
$645$	0	0
$646$	9.00000	0.354100
$647$	33.2389	1.30675	0.653377	−	0.757033i	$-0.273352\pi$
$647$	33.2389	1.30675	0.653377	+	0.757033i	$0.273352\pi$
$648$	0	0
$649$	14.2111	0.557835
$650$	0	0
$651$	0	0
$652$	−2.81665	−0.110309
$653$	−6.11943	−0.239472	−0.119736	−	0.992806i	$-0.538205\pi$
$653$	−6.11943	−0.239472	−0.119736	+	0.992806i	$0.538205\pi$
$654$	0	0
$655$	0	0
$656$	−18.5139	−0.722846
$657$	0	0
$658$	2.72498	0.106231
$659$	30.9083	1.20402	0.602009	−	0.798489i	$-0.294367\pi$
$659$	30.9083	1.20402	0.602009	+	0.798489i	$0.294367\pi$
$660$	0	0
$661$	−8.81665	−0.342928	−0.171464	−	0.985190i	$-0.554850\pi$
$661$	−8.81665	−0.342928	−0.171464	+	0.985190i	$0.554850\pi$
$662$	−28.1472	−1.09397
$663$	0	0
$664$	−10.5416	−0.409095
$665$	0	0
$666$	0	0
$667$	−6.63331	−0.256843
$668$	4.06392	0.157238
$669$	0	0
$670$	0	0
$671$	−7.90833	−0.305298
$672$	0	0
$673$	30.0278	1.15748	0.578742	−	0.815510i	$-0.303544\pi$
$673$	30.0278	1.15748	0.578742	+	0.815510i	$0.303544\pi$
$674$	−40.1833	−1.54780
$675$	0	0
$676$	−3.63331	−0.139743
$677$	−24.2389	−0.931575	−0.465788	−	0.884897i	$-0.654229\pi$
$677$	−24.2389	−0.931575	−0.465788	+	0.884897i	$0.654229\pi$
$678$	0	0
$679$	10.6695	0.409457
$680$	0	0
$681$	0	0
$682$	13.3028	0.509390
$683$	−47.8444	−1.83072	−0.915358	−	0.402642i	$-0.868092\pi$
$683$	−47.8444	−1.83072	−0.915358	+	0.402642i	$0.868092\pi$
$684$	0	0
$685$	0	0
$686$	−12.2750	−0.468662
$687$	0	0
$688$	−23.8167	−0.908001
$689$	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	$0.324489\pi$
$692$	−1.45837	−0.0554387
$693$	0	0
$694$	16.3028	0.618845
$695$	0	0
$696$	0	0
$697$	−38.7250	−1.46681
$698$	−6.75274	−0.255595
$699$	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
$703$	−2.39445	−0.0903083
$704$	8.81665	0.332290
$705$	0	0
$706$	−24.2750	−0.913602
$707$	0.358288	0.0134748
$708$	0	0
$709$	−31.6333	−1.18801	−0.594007	−	0.804460i	$-0.702455\pi$
$709$	−31.6333	−1.18801	−0.594007	+	0.804460i	$0.702455\pi$
$710$	0	0
$711$	0	0
$712$	5.09167	0.190819
$713$	74.5694	2.79265
$714$	0	0
$715$	0	0
$716$	3.78890	0.141598
$717$	0	0
$718$	−1.02776	−0.0383555
$719$	−7.18335	−0.267894	−0.133947	−	0.990989i	$-0.542765\pi$
$719$	−7.18335	−0.267894	−0.133947	+	0.990989i	$0.542765\pi$
$720$	0	0
$721$	2.02776	0.0755176
$722$	−23.4500	−0.872717
$723$	0	0
$724$	6.02776	0.224020
$725$	0	0
$726$	0	0
$727$	−39.3305	−1.45869	−0.729344	−	0.684147i	$-0.760175\pi$
$727$	−39.3305	−1.45869	−0.729344	+	0.684147i	$0.760175\pi$
$728$	−10.4584	−0.387613
$729$	0	0
$730$	0	0
$731$	−49.8167	−1.84254
$732$	0	0
$733$	19.6056	0.724148	0.362074	−	0.932149i	$-0.382069\pi$
$733$	19.6056	0.724148	0.362074	+	0.932149i	$0.382069\pi$
$734$	−26.9638	−0.995253
$735$	0	0
$736$	12.3944	0.456865
$737$	−4.00000	−0.147342
$738$	0	0
$739$	35.1194	1.29189	0.645945	−	0.763384i	$-0.276464\pi$
$739$	35.1194	1.29189	0.645945	+	0.763384i	$0.276464\pi$
$740$	0	0
$741$	0	0
$742$	1.18335	0.0434420
$743$	−40.6972	−1.49304	−0.746518	−	0.665365i	$-0.768276\pi$
$743$	−40.6972	−1.49304	−0.746518	+	0.665365i	$0.768276\pi$
$744$	0	0
$745$	0	0
$746$	35.7250	1.30798
$747$	0	0
$748$	2.09167	0.0764791
$749$	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	$0.809992\pi$
$752$	−9.90833	−0.361320
$753$	0	0
$754$	−5.91673	−0.215475
$755$	0	0
$756$	0	0
$757$	49.0555	1.78295	0.891476	−	0.453067i	$-0.149670\pi$
$757$	49.0555	1.78295	0.891476	+	0.453067i	$0.149670\pi$
$758$	4.14719	0.150633
$759$	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
$763$	8.02776	0.290624
$764$	3.11943	0.112857
$765$	0	0
$766$	−28.1833	−1.01831
$767$	71.0555	2.56567
$768$	0	0
$769$	−5.18335	−0.186916	−0.0934581	−	0.995623i	$-0.529792\pi$
$769$	−5.18335	−0.186916	−0.0934581	+	0.995623i	$0.529792\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−3.11943	−0.112198	−0.0560990	−	0.998425i	$-0.517866\pi$
$773$	−3.11943	−0.112198	−0.0560990	+	0.998425i	$0.517866\pi$
$774$	0	0
$775$	0	0
$776$	−45.9083	−1.64801
$777$	0	0
$778$	−15.6333	−0.560481
$779$	−5.60555	−0.200840
$780$	0	0
$781$	2.60555	0.0932340
$782$	−65.7250	−2.35032
$783$	0	0
$784$	21.5139	0.768353
$785$	0	0
$786$	0	0
$787$	10.2111	0.363986	0.181993	−	0.983300i	$-0.441745\pi$
$787$	10.2111	0.363986	0.181993	+	0.983300i	$0.441745\pi$
$788$	4.02776	0.143483
$789$	0	0
$790$	0	0
$791$	7.54163	0.268150
$792$	0	0
$793$	−39.5416	−1.40416
$794$	28.2666	1.00314
$795$	0	0
$796$	1.96384	0.0696065
$797$	−3.51388	−0.124468	−0.0622340	−	0.998062i	$-0.519822\pi$
$797$	−3.51388	−0.124468	−0.0622340	+	0.998062i	$0.519822\pi$
$798$	0	0
$799$	−20.7250	−0.733197
$800$	0	0
$801$	0	0
$802$	16.6611	0.588323
$803$	−7.90833	−0.279079
$804$	0	0
$805$	0	0
$806$	66.5139	2.34285
$807$	0	0
$808$	−1.54163	−0.0542345
$809$	−39.6333	−1.39343	−0.696716	−	0.717347i	$-0.745356\pi$
$809$	−39.6333	−1.39343	−0.696716	+	0.717347i	$0.745356\pi$
$810$	0	0
$811$	38.8722	1.36499	0.682493	−	0.730892i	$-0.260896\pi$
$811$	38.8722	1.36499	0.682493	+	0.730892i	$0.260896\pi$
$812$	0.191750	0.00672911
$813$	0	0
$814$	3.11943	0.109336
$815$	0	0
$816$	0	0
$817$	−7.21110	−0.252285
$818$	−8.09167	−0.282919
$819$	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
$823$	18.4222	0.642158	0.321079	−	0.947052i	$-0.395955\pi$
$823$	18.4222	0.642158	0.321079	+	0.947052i	$0.395955\pi$
$824$	−8.72498	−0.303949
$825$	0	0
$826$	12.9083	0.449138
$827$	−13.8167	−0.480452	−0.240226	−	0.970717i	$-0.577222\pi$
$827$	−13.8167	−0.480452	−0.240226	+	0.970717i	$0.577222\pi$
$828$	0	0
$829$	29.7527	1.03336	0.516678	−	0.856180i	$-0.327169\pi$
$829$	29.7527	1.03336	0.516678	+	0.856180i	$0.327169\pi$
$830$	0	0
$831$	0	0
$832$	44.0833	1.52831
$833$	45.0000	1.55916
$834$	0	0
$835$	0	0
$836$	0.302776	0.0104717
$837$	0	0
$838$	−8.33053	−0.287773
$839$	9.11943	0.314838	0.157419	−	0.987532i	$-0.449683\pi$
$839$	9.11943	0.314838	0.157419	+	0.987532i	$0.449683\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	0.908327	0.0313030
$843$	0	0
$844$	7.64171	0.263039
$845$	0	0
$846$	0	0
$847$	0.697224	0.0239569
$848$	−4.30278	−0.147758
$849$	0	0
$850$	0	0
$851$	17.4861	0.599417
$852$	0	0
$853$	−12.7250	−0.435695	−0.217848	−	0.975983i	$-0.569904\pi$
$853$	−12.7250	−0.435695	−0.217848	+	0.975983i	$0.569904\pi$
$854$	−7.18335	−0.245809
$855$	0	0
$856$	−9.00000	−0.307614
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	41.3944	1.41236	0.706180	−	0.708032i	$-0.250417\pi$
$859$	41.3944	1.41236	0.706180	+	0.708032i	$0.250417\pi$
$860$	0	0
$861$	0	0
$862$	−42.9916	−1.46430
$863$	12.3944	0.421912	0.210956	−	0.977496i	$-0.432342\pi$
$863$	12.3944	0.421912	0.210956	+	0.977496i	$0.432342\pi$
$864$	0	0
$865$	0	0
$866$	6.51388	0.221351
$867$	0	0
$868$	−2.15559	−0.0731655
$869$	−10.9083	−0.370040
$870$	0	0
$871$	−20.0000	−0.677674
$872$	−34.5416	−1.16973
$873$	0	0
$874$	−9.51388	−0.321812
$875$	0	0
$876$	0	0
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	31.6611	1.06851
$879$	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
$883$	52.4500	1.76508	0.882541	−	0.470236i	$-0.155831\pi$
$883$	52.4500	1.76508	0.882541	+	0.470236i	$0.155831\pi$
$884$	10.4584	0.351753
$885$	0	0
$886$	11.2111	0.376644
$887$	3.23886	0.108750	0.0543751	−	0.998521i	$-0.482683\pi$
$887$	3.23886	0.108750	0.0543751	+	0.998521i	$0.482683\pi$
$888$	0	0
$889$	5.66106	0.189866
$890$	0	0
$891$	0	0
$892$	6.85281	0.229449
$893$	−3.00000	−0.100391
$894$	0	0
$895$	0	0
$896$	5.64171	0.188476
$897$	0	0
$898$	−30.5971	−1.02104
$899$	−9.27502	−0.309339
$900$	0	0
$901$	−9.00000	−0.299833
$902$	7.30278	0.243156
$903$	0	0
$904$	−32.4500	−1.07927
$905$	0	0
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0.513878	0.0170536
$909$	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
$913$	3.51388	0.116292
$914$	−26.9638	−0.891885
$915$	0	0
$916$	5.66947	0.187324
$917$	6.90833	0.228133
$918$	0	0
$919$	26.7889	0.883684	0.441842	−	0.897093i	$-0.354325\pi$
$919$	26.7889	0.883684	0.441842	+	0.897093i	$0.354325\pi$
$920$	0	0
$921$	0	0
$922$	−41.9638	−1.38201
$923$	13.0278	0.428814
$924$	0	0
$925$	0	0
$926$	15.3583	0.504705
$927$	0	0
$928$	−1.54163	−0.0506066
$929$	−53.6056	−1.75874	−0.879371	−	0.476138i	$-0.842036\pi$
$929$	−53.6056	−1.75874	−0.879371	+	0.476138i	$0.842036\pi$
$930$	0	0
$931$	6.51388	0.213484
$932$	4.81665	0.157775
$933$	0	0
$934$	24.2750	0.794303
$935$	0	0
$936$	0	0
$937$	−9.21110	−0.300914	−0.150457	−	0.988617i	$-0.548074\pi$
$937$	−9.21110	−0.300914	−0.150457	+	0.988617i	$0.548074\pi$
$938$	−3.63331	−0.118632
$939$	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
$943$	40.9361	1.33306
$944$	−46.9361	−1.52764
$945$	0	0
$946$	9.39445	0.305440
$947$	−6.63331	−0.215554	−0.107777	−	0.994175i	$-0.534373\pi$
$947$	−6.63331	−0.215554	−0.107777	+	0.994175i	$0.534373\pi$
$948$	0	0
$949$	−39.5416	−1.28358
$950$	0	0
$951$	0	0
$952$	14.4500	0.468326
$953$	−37.2666	−1.20718	−0.603592	−	0.797293i	$-0.706264\pi$
$953$	−37.2666	−1.20718	−0.603592	+	0.797293i	$0.706264\pi$
$954$	0	0
$955$	0	0
$956$	6.39445	0.206811
$957$	0	0
$958$	45.3583	1.46546
$959$	9.00000	0.290625
$960$	0	0
$961$	73.2666	2.36344
$962$	15.5971	0.502872
$963$	0	0
$964$	−6.64171	−0.213915
$965$	0	0
$966$	0	0
$967$	14.9083	0.479419	0.239710	−	0.970845i	$-0.422948\pi$
$967$	14.9083	0.479419	0.239710	+	0.970845i	$0.422948\pi$
$968$	−3.00000	−0.0964237
$969$	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
$973$	−4.33053	−0.138830
$974$	5.48612	0.175787
$975$	0	0
$976$	26.1194	0.836063
$977$	52.0278	1.66452	0.832258	−	0.554389i	$-0.187048\pi$
$977$	52.0278	1.66452	0.832258	+	0.554389i	$0.187048\pi$
$978$	0	0
$979$	−1.69722	−0.0542435
$980$	0	0
$981$	0	0
$982$	12.7527	0.406956
$983$	−8.84441	−0.282093	−0.141046	−	0.990003i	$-0.545047\pi$
$983$	−8.84441	−0.282093	−0.141046	+	0.990003i	$0.545047\pi$
$984$	0	0
$985$	0	0
$986$	8.17494	0.260343
$987$	0	0
$988$	1.51388	0.0481629
$989$	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	$0.586546\pi$
$992$	17.3305	0.550245
$993$	0	0
$994$	2.36669	0.0750669
$995$	0	0
$996$	0	0
$997$	46.7250	1.47979	0.739897	−	0.672720i	$-0.234874\pi$
$997$	46.7250	1.47979	0.739897	+	0.672720i	$0.234874\pi$
$998$	−4.54163	−0.143763
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.2	✓	2	15.14	odd	2
275.2.a.f.1.1	yes	2	3.2	odd	2
275.2.b.c.199.2		4	15.2	even	4
275.2.b.c.199.3		4	15.8	even	4
2475.2.a.o.1.2		2	1.1	even	1	trivial
2475.2.a.t.1.1		2	5.4	even	2
2475.2.c.k.199.2		4	5.3	odd	4
2475.2.c.k.199.3		4	5.2	odd	4
3025.2.a.h.1.2		2	33.32	even	2
3025.2.a.n.1.1		2	165.164	even	2
4400.2.a.bh.1.1		2	12.11	even	2
4400.2.a.bs.1.2		2	60.59	even	2
4400.2.b.y.4049.1		4	60.23	odd	4
4400.2.b.y.4049.4		4	60.47	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +1.00000 q^{11} +5.00000 q^{13} +0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} -1.00000 q^{19} +1.30278 q^{22} +7.30278 q^{23} +6.51388 q^{26} -0.211103 q^{28} -0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} -9.00000 q^{34} +2.39445 q^{37} -1.30278 q^{38} +5.60555 q^{41} +7.21110 q^{43} -0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} -6.51388 q^{49} -1.51388 q^{52} +1.30278 q^{53} -2.09167 q^{56} -1.18335 q^{58} +14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} +8.81665 q^{64} -4.00000 q^{67} +2.09167 q^{68} +2.60555 q^{71} -7.90833 q^{73} +3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} -10.9083 q^{79} +7.30278 q^{82} +3.51388 q^{83} +9.39445 q^{86} -3.00000 q^{88} -1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} +3.90833 q^{94} +15.3028 q^{97} -8.48612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + 2 q^{11} + 10 q^{13} - 9 q^{14} - 3 q^{16} - 3 q^{17} - 2 q^{19} - q^{22} + 11 q^{23} - 5 q^{26} + 14 q^{28} + 9 q^{29} + 6 q^{31} + 7 q^{32} - 18 q^{34} + 12 q^{37}+ \cdots - 35 q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 + 2 * q^11 + 10 * q^13 - 9 * q^14 - 3 * q^16 - 3 * q^17 - 2 * q^19 - q^22 + 11 * q^23 - 5 * q^26 + 14 * q^28 + 9 * q^29 + 6 * q^31 + 7 * q^32 - 18 * q^34 + 12 * q^37 + q^38 + 4 * q^41 + 3 * q^44 + q^46 + 6 * q^47 + 5 * q^49 + 15 * q^52 - q^53 - 15 * q^56 - 24 * q^58 + 14 * q^59 - 5 * q^61 + 23 * q^62 - 4 * q^64 - 8 * q^67 + 15 * q^68 - 2 * q^71 - 5 * q^73 - 19 * q^74 - 3 * q^76 + 5 * q^77 - 11 * q^79 + 11 * q^82 - 11 * q^83 + 26 * q^86 - 6 * q^88 - 7 * q^89 + 25 * q^91 + 10 * q^92 - 3 * q^94 + 27 * q^97 - 35 * q^98

Introduction

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