LMFDB - Embedded newform 6975.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6975,2,Mod(1,6975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6975, 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("6975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.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:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -0.236068 q^{7} -2.23607 q^{8} -2.00000 q^{11} +3.23607 q^{13} -0.381966 q^{14} -4.85410 q^{16} +0.763932 q^{17} -2.23607 q^{19} -3.23607 q^{22} +5.70820 q^{23} +5.23607 q^{26} -0.145898 q^{28} -2.76393 q^{29} +1.00000 q^{31} -3.38197 q^{32} +1.23607 q^{34} +2.00000 q^{37} -3.61803 q^{38} -7.00000 q^{41} -1.23607 q^{43} -1.23607 q^{44} +9.23607 q^{46} +2.47214 q^{47} -6.94427 q^{49} +2.00000 q^{52} -10.4721 q^{53} +0.527864 q^{56} -4.47214 q^{58} -2.23607 q^{59} +8.18034 q^{61} +1.61803 q^{62} +4.23607 q^{64} -8.00000 q^{67} +0.472136 q^{68} +9.18034 q^{71} -8.47214 q^{73} +3.23607 q^{74} -1.38197 q^{76} +0.472136 q^{77} -11.7082 q^{79} -11.3262 q^{82} -14.9443 q^{83} -2.00000 q^{86} +4.47214 q^{88} -11.7082 q^{89} -0.763932 q^{91} +3.52786 q^{92} +4.00000 q^{94} +15.9443 q^{97} -11.2361 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 3 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 7 q^{28} - 10 q^{29} + 2 q^{31} - 9 q^{32} - 2 q^{34} + 4 q^{37} - 5 q^{38} - 14 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 3 * q^14 - 3 * q^16 + 6 * q^17 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 7 * q^28 - 10 * q^29 + 2 * q^31 - 9 * q^32 - 2 * q^34 + 4 * q^37 - 5 * q^38 - 14 * q^41 + 2 * q^43 + 2 * q^44 + 14 * q^46 - 4 * q^47 + 4 * q^49 + 4 * q^52 - 12 * q^53 + 10 * q^56 - 6 * q^61 + q^62 + 4 * q^64 - 16 * q^67 - 8 * q^68 - 4 * q^71 - 8 * q^73 + 2 * q^74 - 5 * q^76 - 8 * q^77 - 10 * q^79 - 7 * q^82 - 12 * q^83 - 4 * q^86 - 10 * q^89 - 6 * q^91 + 16 * q^92 + 8 * q^94 + 14 * q^97 - 18 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	−4.85410	−1.21353
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	−2.23607	−0.512989	−0.256495	−	0.966546i	$-0.582568\pi$
$19$	−2.23607	−0.512989	−0.256495	+	0.966546i	$0.582568\pi$
$20$	0	0
$21$	0	0
$22$	−3.23607	−0.689932
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	0	0
$26$	5.23607	1.02688
$27$	0	0
$28$	−0.145898	−0.0275721
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.38197	−0.597853
$33$	0	0
$34$	1.23607	0.211984
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−3.61803	−0.586923
$39$	0	0
$40$	0	0
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	−1.23607	−0.188499	−0.0942493	−	0.995549i	$-0.530045\pi$
$43$	−1.23607	−0.188499	−0.0942493	+	0.995549i	$0.530045\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	9.23607	1.36178
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	0	0
$51$	0	0
$52$	2.00000	0.277350
$53$	−10.4721	−1.43846	−0.719229	−	0.694773i	$-0.755505\pi$
$53$	−10.4721	−1.43846	−0.719229	+	0.694773i	$0.755505\pi$
$54$	0	0
$55$	0	0
$56$	0.527864	0.0705388
$57$	0	0
$58$	−4.47214	−0.587220
$59$	−2.23607	−0.291111	−0.145556	−	0.989350i	$-0.546497\pi$
$59$	−2.23607	−0.291111	−0.145556	+	0.989350i	$0.546497\pi$
$60$	0	0
$61$	8.18034	1.04739	0.523693	−	0.851907i	$-0.324554\pi$
$61$	8.18034	1.04739	0.523693	+	0.851907i	$0.324554\pi$
$62$	1.61803	0.205491
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0.472136	0.0572549
$69$	0	0
$70$	0	0
$71$	9.18034	1.08951	0.544753	−	0.838597i	$-0.316623\pi$
$71$	9.18034	1.08951	0.544753	+	0.838597i	$0.316623\pi$
$72$	0	0
$73$	−8.47214	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$73$	−8.47214	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$74$	3.23607	0.376185
$75$	0	0
$76$	−1.38197	−0.158522
$77$	0.472136	0.0538049
$78$	0	0
$79$	−11.7082	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$79$	−11.7082	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$80$	0	0
$81$	0	0
$82$	−11.3262	−1.25077
$83$	−14.9443	−1.64035	−0.820173	−	0.572115i	$-0.806123\pi$
$83$	−14.9443	−1.64035	−0.820173	+	0.572115i	$0.806123\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	4.47214	0.476731
$89$	−11.7082	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$89$	−11.7082	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	3.52786	0.367805
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	15.9443	1.61890	0.809448	−	0.587192i	$-0.199767\pi$
$97$	15.9443	1.61890	0.809448	+	0.587192i	$0.199767\pi$
$98$	−11.2361	−1.13501
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	−6.23607	−0.614458	−0.307229	−	0.951636i	$-0.599402\pi$
$103$	−6.23607	−0.614458	−0.307229	+	0.951636i	$0.599402\pi$
$104$	−7.23607	−0.709555
$105$	0	0
$106$	−16.9443	−1.64577
$107$	5.76393	0.557220	0.278610	−	0.960404i	$-0.410126\pi$
$107$	5.76393	0.557220	0.278610	+	0.960404i	$0.410126\pi$
$108$	0	0
$109$	−13.9443	−1.33562	−0.667810	−	0.744332i	$-0.732768\pi$
$109$	−13.9443	−1.33562	−0.667810	+	0.744332i	$0.732768\pi$
$110$	0	0
$111$	0	0
$112$	1.14590	0.108277
$113$	3.47214	0.326631	0.163316	−	0.986574i	$-0.447781\pi$
$113$	3.47214	0.326631	0.163316	+	0.986574i	$0.447781\pi$
$114$	0	0
$115$	0	0
$116$	−1.70820	−0.158603
$117$	0	0
$118$	−3.61803	−0.333067
$119$	−0.180340	−0.0165317
$120$	0	0
$121$	−7.00000	−0.636364
$122$	13.2361	1.19834
$123$	0	0
$124$	0.618034	0.0555011
$125$	0	0
$126$	0	0
$127$	−12.4721	−1.10672	−0.553362	−	0.832941i	$-0.686655\pi$
$127$	−12.4721	−1.10672	−0.553362	+	0.832941i	$0.686655\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0.527864	0.0457716
$134$	−12.9443	−1.11821
$135$	0	0
$136$	−1.70820	−0.146477
$137$	6.29180	0.537544	0.268772	−	0.963204i	$-0.413382\pi$
$137$	6.29180	0.537544	0.268772	+	0.963204i	$0.413382\pi$
$138$	0	0
$139$	13.4164	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$139$	13.4164	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$140$	0	0
$141$	0	0
$142$	14.8541	1.24653
$143$	−6.47214	−0.541227
$144$	0	0
$145$	0	0
$146$	−13.7082	−1.13450
$147$	0	0
$148$	1.23607	0.101604
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−14.1803	−1.15398	−0.576990	−	0.816751i	$-0.695773\pi$
$151$	−14.1803	−1.15398	−0.576990	+	0.816751i	$0.695773\pi$
$152$	5.00000	0.405554
$153$	0	0
$154$	0.763932	0.0615594
$155$	0	0
$156$	0	0
$157$	−20.8885	−1.66709	−0.833544	−	0.552454i	$-0.813692\pi$
$157$	−20.8885	−1.66709	−0.833544	+	0.552454i	$0.813692\pi$
$158$	−18.9443	−1.50713
$159$	0	0
$160$	0	0
$161$	−1.34752	−0.106200
$162$	0	0
$163$	−10.7082	−0.838731	−0.419366	−	0.907817i	$-0.637747\pi$
$163$	−10.7082	−0.838731	−0.419366	+	0.907817i	$0.637747\pi$
$164$	−4.32624	−0.337822
$165$	0	0
$166$	−24.1803	−1.87676
$167$	−6.47214	−0.500829	−0.250414	−	0.968139i	$-0.580567\pi$
$167$	−6.47214	−0.500829	−0.250414	+	0.968139i	$0.580567\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	0	0
$172$	−0.763932	−0.0582493
$173$	2.94427	0.223849	0.111924	−	0.993717i	$-0.464299\pi$
$173$	2.94427	0.223849	0.111924	+	0.993717i	$0.464299\pi$
$174$	0	0
$175$	0	0
$176$	9.70820	0.731783
$177$	0	0
$178$	−18.9443	−1.41993
$179$	−1.70820	−0.127677	−0.0638386	−	0.997960i	$-0.520334\pi$
$179$	−1.70820	−0.127677	−0.0638386	+	0.997960i	$0.520334\pi$
$180$	0	0
$181$	−4.18034	−0.310722	−0.155361	−	0.987858i	$-0.549654\pi$
$181$	−4.18034	−0.310722	−0.155361	+	0.987858i	$0.549654\pi$
$182$	−1.23607	−0.0916235
$183$	0	0
$184$	−12.7639	−0.940970
$185$	0	0
$186$	0	0
$187$	−1.52786	−0.111728
$188$	1.52786	0.111431
$189$	0	0
$190$	0	0
$191$	19.1803	1.38784	0.693920	−	0.720052i	$-0.255882\pi$
$191$	19.1803	1.38784	0.693920	+	0.720052i	$0.255882\pi$
$192$	0	0
$193$	−3.47214	−0.249930	−0.124965	−	0.992161i	$-0.539882\pi$
$193$	−3.47214	−0.249930	−0.124965	+	0.992161i	$0.539882\pi$
$194$	25.7984	1.85222
$195$	0	0
$196$	−4.29180	−0.306557
$197$	11.4164	0.813385	0.406693	−	0.913565i	$-0.366682\pi$
$197$	11.4164	0.813385	0.406693	+	0.913565i	$0.366682\pi$
$198$	0	0
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	0	0
$201$	0	0
$202$	4.85410	0.341533
$203$	0.652476	0.0457948
$204$	0	0
$205$	0	0
$206$	−10.0902	−0.703015
$207$	0	0
$208$	−15.7082	−1.08917
$209$	4.47214	0.309344
$210$	0	0
$211$	23.1803	1.59580	0.797900	−	0.602790i	$-0.205944\pi$
$211$	23.1803	1.59580	0.797900	+	0.602790i	$0.205944\pi$
$212$	−6.47214	−0.444508
$213$	0	0
$214$	9.32624	0.637528
$215$	0	0
$216$	0	0
$217$	−0.236068	−0.0160253
$218$	−22.5623	−1.52811
$219$	0	0
$220$	0	0
$221$	2.47214	0.166294
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0.798374	0.0533436
$225$	0	0
$226$	5.61803	0.373706
$227$	−6.47214	−0.429571	−0.214785	−	0.976661i	$-0.568905\pi$
$227$	−6.47214	−0.429571	−0.214785	+	0.976661i	$0.568905\pi$
$228$	0	0
$229$	−13.4164	−0.886581	−0.443291	−	0.896378i	$-0.646189\pi$
$229$	−13.4164	−0.886581	−0.443291	+	0.896378i	$0.646189\pi$
$230$	0	0
$231$	0	0
$232$	6.18034	0.405759
$233$	17.9443	1.17557	0.587784	−	0.809018i	$-0.300000\pi$
$233$	17.9443	1.17557	0.587784	+	0.809018i	$0.300000\pi$
$234$	0	0
$235$	0	0
$236$	−1.38197	−0.0899583
$237$	0	0
$238$	−0.291796	−0.0189143
$239$	11.7082	0.757341	0.378670	−	0.925532i	$-0.376381\pi$
$239$	11.7082	0.757341	0.378670	+	0.925532i	$0.376381\pi$
$240$	0	0
$241$	14.3607	0.925053	0.462526	−	0.886606i	$-0.346943\pi$
$241$	14.3607	0.925053	0.462526	+	0.886606i	$0.346943\pi$
$242$	−11.3262	−0.728078
$243$	0	0
$244$	5.05573	0.323660
$245$	0	0
$246$	0	0
$247$	−7.23607	−0.460420
$248$	−2.23607	−0.141990
$249$	0	0
$250$	0	0
$251$	1.81966	0.114856	0.0574280	−	0.998350i	$-0.481710\pi$
$251$	1.81966	0.114856	0.0574280	+	0.998350i	$0.481710\pi$
$252$	0	0
$253$	−11.4164	−0.717743
$254$	−20.1803	−1.26623
$255$	0	0
$256$	13.5623	0.847644
$257$	1.94427	0.121280	0.0606402	−	0.998160i	$-0.480686\pi$
$257$	1.94427	0.121280	0.0606402	+	0.998160i	$0.480686\pi$
$258$	0	0
$259$	−0.472136	−0.0293371
$260$	0	0
$261$	0	0
$262$	−19.4164	−1.19955
$263$	−23.2361	−1.43280	−0.716399	−	0.697691i	$-0.754211\pi$
$263$	−23.2361	−1.43280	−0.716399	+	0.697691i	$0.754211\pi$
$264$	0	0
$265$	0	0
$266$	0.854102	0.0523684
$267$	0	0
$268$	−4.94427	−0.302019
$269$	11.0557	0.674080	0.337040	−	0.941490i	$-0.390574\pi$
$269$	11.0557	0.674080	0.337040	+	0.941490i	$0.390574\pi$
$270$	0	0
$271$	−14.1803	−0.861394	−0.430697	−	0.902497i	$-0.641732\pi$
$271$	−14.1803	−0.861394	−0.430697	+	0.902497i	$0.641732\pi$
$272$	−3.70820	−0.224843
$273$	0	0
$274$	10.1803	0.615017
$275$	0	0
$276$	0	0
$277$	12.6525	0.760214	0.380107	−	0.924943i	$-0.375887\pi$
$277$	12.6525	0.760214	0.380107	+	0.924943i	$0.375887\pi$
$278$	21.7082	1.30197
$279$	0	0
$280$	0	0
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	0	0
$283$	13.8885	0.825588	0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885	0.825588	0.412794	+	0.910824i	$0.364553\pi$
$284$	5.67376	0.336676
$285$	0	0
$286$	−10.4721	−0.619230
$287$	1.65248	0.0975426
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	0	0
$292$	−5.23607	−0.306418
$293$	−0.472136	−0.0275825	−0.0137912	−	0.999905i	$-0.504390\pi$
$293$	−0.472136	−0.0275825	−0.0137912	+	0.999905i	$0.504390\pi$
$294$	0	0
$295$	0	0
$296$	−4.47214	−0.259938
$297$	0	0
$298$	−16.1803	−0.937302
$299$	18.4721	1.06827
$300$	0	0
$301$	0.291796	0.0168188
$302$	−22.9443	−1.32029
$303$	0	0
$304$	10.8541	0.622525
$305$	0	0
$306$	0	0
$307$	28.7082	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$307$	28.7082	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$308$	0.291796	0.0166266
$309$	0	0
$310$	0	0
$311$	29.1803	1.65467	0.827333	−	0.561712i	$-0.189857\pi$
$311$	29.1803	1.65467	0.827333	+	0.561712i	$0.189857\pi$
$312$	0	0
$313$	−16.7639	−0.947553	−0.473777	−	0.880645i	$-0.657110\pi$
$313$	−16.7639	−0.947553	−0.473777	+	0.880645i	$0.657110\pi$
$314$	−33.7984	−1.90735
$315$	0	0
$316$	−7.23607	−0.407061
$317$	4.05573	0.227792	0.113896	−	0.993493i	$-0.463667\pi$
$317$	4.05573	0.227792	0.113896	+	0.993493i	$0.463667\pi$
$318$	0	0
$319$	5.52786	0.309501
$320$	0	0
$321$	0	0
$322$	−2.18034	−0.121506
$323$	−1.70820	−0.0950470
$324$	0	0
$325$	0	0
$326$	−17.3262	−0.959612
$327$	0	0
$328$	15.6525	0.864263
$329$	−0.583592	−0.0321745
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	−9.23607	−0.506895
$333$	0	0
$334$	−10.4721	−0.573010
$335$	0	0
$336$	0	0
$337$	14.7639	0.804243	0.402121	−	0.915586i	$-0.368273\pi$
$337$	14.7639	0.804243	0.402121	+	0.915586i	$0.368273\pi$
$338$	−4.09017	−0.222476
$339$	0	0
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	3.29180	0.177740
$344$	2.76393	0.149021
$345$	0	0
$346$	4.76393	0.256111
$347$	24.1803	1.29807	0.649034	−	0.760759i	$-0.275173\pi$
$347$	24.1803	1.29807	0.649034	+	0.760759i	$0.275173\pi$
$348$	0	0
$349$	−7.88854	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$349$	−7.88854	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$350$	0	0
$351$	0	0
$352$	6.76393	0.360519
$353$	7.41641	0.394736	0.197368	−	0.980330i	$-0.436761\pi$
$353$	7.41641	0.394736	0.197368	+	0.980330i	$0.436761\pi$
$354$	0	0
$355$	0	0
$356$	−7.23607	−0.383511
$357$	0	0
$358$	−2.76393	−0.146078
$359$	−22.2361	−1.17357	−0.586787	−	0.809741i	$-0.699608\pi$
$359$	−22.2361	−1.17357	−0.586787	+	0.809741i	$0.699608\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	−6.76393	−0.355504
$363$	0	0
$364$	−0.472136	−0.0247466
$365$	0	0
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−27.7082	−1.44439
$369$	0	0
$370$	0	0
$371$	2.47214	0.128347
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	−2.47214	−0.127831
$375$	0	0
$376$	−5.52786	−0.285078
$377$	−8.94427	−0.460653
$378$	0	0
$379$	−2.11146	−0.108458	−0.0542291	−	0.998529i	$-0.517270\pi$
$379$	−2.11146	−0.108458	−0.0542291	+	0.998529i	$0.517270\pi$
$380$	0	0
$381$	0	0
$382$	31.0344	1.58786
$383$	−23.8885	−1.22065	−0.610324	−	0.792152i	$-0.708961\pi$
$383$	−23.8885	−1.22065	−0.610324	+	0.792152i	$0.708961\pi$
$384$	0	0
$385$	0	0
$386$	−5.61803	−0.285950
$387$	0	0
$388$	9.85410	0.500266
$389$	17.8885	0.906985	0.453493	−	0.891260i	$-0.350178\pi$
$389$	17.8885	0.906985	0.453493	+	0.891260i	$0.350178\pi$
$390$	0	0
$391$	4.36068	0.220529
$392$	15.5279	0.784276
$393$	0	0
$394$	18.4721	0.930613
$395$	0	0
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	−30.6525	−1.53647
$399$	0	0
$400$	0	0
$401$	−38.1803	−1.90664	−0.953318	−	0.301969i	$-0.902356\pi$
$401$	−38.1803	−1.90664	−0.953318	+	0.301969i	$0.902356\pi$
$402$	0	0
$403$	3.23607	0.161200
$404$	1.85410	0.0922450
$405$	0	0
$406$	1.05573	0.0523949
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−3.81966	−0.188870	−0.0944350	−	0.995531i	$-0.530104\pi$
$409$	−3.81966	−0.188870	−0.0944350	+	0.995531i	$0.530104\pi$
$410$	0	0
$411$	0	0
$412$	−3.85410	−0.189878
$413$	0.527864	0.0259745
$414$	0	0
$415$	0	0
$416$	−10.9443	−0.536587
$417$	0	0
$418$	7.23607	0.353928
$419$	10.1246	0.494620	0.247310	−	0.968936i	$-0.420453\pi$
$419$	10.1246	0.494620	0.247310	+	0.968936i	$0.420453\pi$
$420$	0	0
$421$	29.3607	1.43095	0.715476	−	0.698637i	$-0.246210\pi$
$421$	29.3607	1.43095	0.715476	+	0.698637i	$0.246210\pi$
$422$	37.5066	1.82579
$423$	0	0
$424$	23.4164	1.13720
$425$	0	0
$426$	0	0
$427$	−1.93112	−0.0934533
$428$	3.56231	0.172191
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−10.1803	−0.489236	−0.244618	−	0.969620i	$-0.578663\pi$
$433$	−10.1803	−0.489236	−0.244618	+	0.969620i	$0.578663\pi$
$434$	−0.381966	−0.0183350
$435$	0	0
$436$	−8.61803	−0.412729
$437$	−12.7639	−0.610582
$438$	0	0
$439$	−1.18034	−0.0563345	−0.0281673	−	0.999603i	$-0.508967\pi$
$439$	−1.18034	−0.0563345	−0.0281673	+	0.999603i	$0.508967\pi$
$440$	0	0
$441$	0	0
$442$	4.00000	0.190261
$443$	30.7082	1.45899	0.729495	−	0.683986i	$-0.239755\pi$
$443$	30.7082	1.45899	0.729495	+	0.683986i	$0.239755\pi$
$444$	0	0
$445$	0	0
$446$	−6.47214	−0.306465
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	31.3050	1.47737	0.738686	−	0.674050i	$-0.235447\pi$
$449$	31.3050	1.47737	0.738686	+	0.674050i	$0.235447\pi$
$450$	0	0
$451$	14.0000	0.659234
$452$	2.14590	0.100935
$453$	0	0
$454$	−10.4721	−0.491482
$455$	0	0
$456$	0	0
$457$	3.05573	0.142941	0.0714705	−	0.997443i	$-0.477231\pi$
$457$	3.05573	0.142941	0.0714705	+	0.997443i	$0.477231\pi$
$458$	−21.7082	−1.01436
$459$	0	0
$460$	0	0
$461$	−34.3607	−1.60034	−0.800168	−	0.599776i	$-0.795256\pi$
$461$	−34.3607	−1.60034	−0.800168	+	0.599776i	$0.795256\pi$
$462$	0	0
$463$	2.58359	0.120070	0.0600349	−	0.998196i	$-0.480879\pi$
$463$	2.58359	0.120070	0.0600349	+	0.998196i	$0.480879\pi$
$464$	13.4164	0.622841
$465$	0	0
$466$	29.0344	1.34499
$467$	4.70820	0.217870	0.108935	−	0.994049i	$-0.465256\pi$
$467$	4.70820	0.217870	0.108935	+	0.994049i	$0.465256\pi$
$468$	0	0
$469$	1.88854	0.0872049
$470$	0	0
$471$	0	0
$472$	5.00000	0.230144
$473$	2.47214	0.113669
$474$	0	0
$475$	0	0
$476$	−0.111456	−0.00510859
$477$	0	0
$478$	18.9443	0.866491
$479$	−23.2918	−1.06423	−0.532115	−	0.846672i	$-0.678602\pi$
$479$	−23.2918	−1.06423	−0.532115	+	0.846672i	$0.678602\pi$
$480$	0	0
$481$	6.47214	0.295104
$482$	23.2361	1.05837
$483$	0	0
$484$	−4.32624	−0.196647
$485$	0	0
$486$	0	0
$487$	19.2361	0.871669	0.435835	−	0.900027i	$-0.356453\pi$
$487$	19.2361	0.871669	0.435835	+	0.900027i	$0.356453\pi$
$488$	−18.2918	−0.828031
$489$	0	0
$490$	0	0
$491$	−4.36068	−0.196795	−0.0983974	−	0.995147i	$-0.531372\pi$
$491$	−4.36068	−0.196795	−0.0983974	+	0.995147i	$0.531372\pi$
$492$	0	0
$493$	−2.11146	−0.0950952
$494$	−11.7082	−0.526777
$495$	0	0
$496$	−4.85410	−0.217956
$497$	−2.16718	−0.0972115
$498$	0	0
$499$	−6.58359	−0.294722	−0.147361	−	0.989083i	$-0.547078\pi$
$499$	−6.58359	−0.294722	−0.147361	+	0.989083i	$0.547078\pi$
$500$	0	0
$501$	0	0
$502$	2.94427	0.131409
$503$	29.6525	1.32214	0.661069	−	0.750325i	$-0.270103\pi$
$503$	29.6525	1.32214	0.661069	+	0.750325i	$0.270103\pi$
$504$	0	0
$505$	0	0
$506$	−18.4721	−0.821187
$507$	0	0
$508$	−7.70820	−0.341996
$509$	−29.5967	−1.31185	−0.655926	−	0.754825i	$-0.727722\pi$
$509$	−29.5967	−1.31185	−0.655926	+	0.754825i	$0.727722\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−5.29180	−0.233867
$513$	0	0
$514$	3.14590	0.138760
$515$	0	0
$516$	0	0
$517$	−4.94427	−0.217449
$518$	−0.763932	−0.0335652
$519$	0	0
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	0	0
$523$	17.7082	0.774326	0.387163	−	0.922011i	$-0.373455\pi$
$523$	17.7082	0.774326	0.387163	+	0.922011i	$0.373455\pi$
$524$	−7.41641	−0.323987
$525$	0	0
$526$	−37.5967	−1.63930
$527$	0.763932	0.0332774
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	0	0
$532$	0.326238	0.0141442
$533$	−22.6525	−0.981188
$534$	0	0
$535$	0	0
$536$	17.8885	0.772667
$537$	0	0
$538$	17.8885	0.771230
$539$	13.8885	0.598222
$540$	0	0
$541$	−25.3607	−1.09034	−0.545170	−	0.838325i	$-0.683535\pi$
$541$	−25.3607	−1.09034	−0.545170	+	0.838325i	$0.683535\pi$
$542$	−22.9443	−0.985541
$543$	0	0
$544$	−2.58359	−0.110771
$545$	0	0
$546$	0	0
$547$	12.1246	0.518411	0.259205	−	0.965822i	$-0.416539\pi$
$547$	12.1246	0.518411	0.259205	+	0.965822i	$0.416539\pi$
$548$	3.88854	0.166110
$549$	0	0
$550$	0	0
$551$	6.18034	0.263291
$552$	0	0
$553$	2.76393	0.117534
$554$	20.4721	0.869778
$555$	0	0
$556$	8.29180	0.351650
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	−27.5066	−1.16029
$563$	27.5410	1.16072	0.580358	−	0.814362i	$-0.302913\pi$
$563$	27.5410	1.16072	0.580358	+	0.814362i	$0.302913\pi$
$564$	0	0
$565$	0	0
$566$	22.4721	0.944574
$567$	0	0
$568$	−20.5279	−0.861330
$569$	−5.52786	−0.231740	−0.115870	−	0.993264i	$-0.536966\pi$
$569$	−5.52786	−0.231740	−0.115870	+	0.993264i	$0.536966\pi$
$570$	0	0
$571$	28.1803	1.17931	0.589655	−	0.807655i	$-0.299264\pi$
$571$	28.1803	1.17931	0.589655	+	0.807655i	$0.299264\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	2.67376	0.111601
$575$	0	0
$576$	0	0
$577$	28.8328	1.20033	0.600163	−	0.799878i	$-0.295102\pi$
$577$	28.8328	1.20033	0.600163	+	0.799878i	$0.295102\pi$
$578$	−26.5623	−1.10485
$579$	0	0
$580$	0	0
$581$	3.52786	0.146360
$582$	0	0
$583$	20.9443	0.867423
$584$	18.9443	0.783920
$585$	0	0
$586$	−0.763932	−0.0315577
$587$	−6.47214	−0.267134	−0.133567	−	0.991040i	$-0.542643\pi$
$587$	−6.47214	−0.267134	−0.133567	+	0.991040i	$0.542643\pi$
$588$	0	0
$589$	−2.23607	−0.0921356
$590$	0	0
$591$	0	0
$592$	−9.70820	−0.399005
$593$	−6.52786	−0.268067	−0.134034	−	0.990977i	$-0.542793\pi$
$593$	−6.52786	−0.268067	−0.134034	+	0.990977i	$0.542793\pi$
$594$	0	0
$595$	0	0
$596$	−6.18034	−0.253157
$597$	0	0
$598$	29.8885	1.22223
$599$	14.5967	0.596407	0.298203	−	0.954502i	$-0.403613\pi$
$599$	14.5967	0.596407	0.298203	+	0.954502i	$0.403613\pi$
$600$	0	0
$601$	30.5410	1.24579	0.622897	−	0.782304i	$-0.285956\pi$
$601$	30.5410	1.24579	0.622897	+	0.782304i	$0.285956\pi$
$602$	0.472136	0.0192428
$603$	0	0
$604$	−8.76393	−0.356599
$605$	0	0
$606$	0	0
$607$	−22.4721	−0.912116	−0.456058	−	0.889950i	$-0.650739\pi$
$607$	−22.4721	−0.912116	−0.456058	+	0.889950i	$0.650739\pi$
$608$	7.56231	0.306692
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	43.8885	1.77264	0.886321	−	0.463072i	$-0.153253\pi$
$613$	43.8885	1.77264	0.886321	+	0.463072i	$0.153253\pi$
$614$	46.4508	1.87460
$615$	0	0
$616$	−1.05573	−0.0425365
$617$	32.4721	1.30728	0.653639	−	0.756806i	$-0.273241\pi$
$617$	32.4721	1.30728	0.653639	+	0.756806i	$0.273241\pi$
$618$	0	0
$619$	6.18034	0.248409	0.124204	−	0.992257i	$-0.460362\pi$
$619$	6.18034	0.248409	0.124204	+	0.992257i	$0.460362\pi$
$620$	0	0
$621$	0	0
$622$	47.2148	1.89314
$623$	2.76393	0.110735
$624$	0	0
$625$	0	0
$626$	−27.1246	−1.08412
$627$	0	0
$628$	−12.9098	−0.515158
$629$	1.52786	0.0609199
$630$	0	0
$631$	34.3607	1.36788	0.683939	−	0.729540i	$-0.260266\pi$
$631$	34.3607	1.36788	0.683939	+	0.729540i	$0.260266\pi$
$632$	26.1803	1.04140
$633$	0	0
$634$	6.56231	0.260622
$635$	0	0
$636$	0	0
$637$	−22.4721	−0.890378
$638$	8.94427	0.354107
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	−19.5279	−0.770104	−0.385052	−	0.922895i	$-0.625816\pi$
$643$	−19.5279	−0.770104	−0.385052	+	0.922895i	$0.625816\pi$
$644$	−0.832816	−0.0328175
$645$	0	0
$646$	−2.76393	−0.108745
$647$	−0.944272	−0.0371232	−0.0185616	−	0.999828i	$-0.505909\pi$
$647$	−0.944272	−0.0371232	−0.0185616	+	0.999828i	$0.505909\pi$
$648$	0	0
$649$	4.47214	0.175547
$650$	0	0
$651$	0	0
$652$	−6.61803	−0.259182
$653$	−47.3050	−1.85119	−0.925593	−	0.378521i	$-0.876433\pi$
$653$	−47.3050	−1.85119	−0.925593	+	0.378521i	$0.876433\pi$
$654$	0	0
$655$	0	0
$656$	33.9787	1.32665
$657$	0	0
$658$	−0.944272	−0.0368116
$659$	−25.6525	−0.999279	−0.499639	−	0.866234i	$-0.666534\pi$
$659$	−25.6525	−0.999279	−0.499639	+	0.866234i	$0.666534\pi$
$660$	0	0
$661$	−0.639320	−0.0248667	−0.0124333	−	0.999923i	$-0.503958\pi$
$661$	−0.639320	−0.0248667	−0.0124333	+	0.999923i	$0.503958\pi$
$662$	3.23607	0.125773
$663$	0	0
$664$	33.4164	1.29681
$665$	0	0
$666$	0	0
$667$	−15.7771	−0.610891
$668$	−4.00000	−0.154765
$669$	0	0
$670$	0	0
$671$	−16.3607	−0.631597
$672$	0	0
$673$	29.0132	1.11837	0.559187	−	0.829041i	$-0.311113\pi$
$673$	29.0132	1.11837	0.559187	+	0.829041i	$0.311113\pi$
$674$	23.8885	0.920152
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	−46.7214	−1.79565	−0.897824	−	0.440355i	$-0.854853\pi$
$677$	−46.7214	−1.79565	−0.897824	+	0.440355i	$0.854853\pi$
$678$	0	0
$679$	−3.76393	−0.144446
$680$	0	0
$681$	0	0
$682$	−3.23607	−0.123915
$683$	5.18034	0.198220	0.0991101	−	0.995076i	$-0.468400\pi$
$683$	5.18034	0.198220	0.0991101	+	0.995076i	$0.468400\pi$
$684$	0	0
$685$	0	0
$686$	5.32624	0.203357
$687$	0	0
$688$	6.00000	0.228748
$689$	−33.8885	−1.29105
$690$	0	0
$691$	3.18034	0.120986	0.0604929	−	0.998169i	$-0.480733\pi$
$691$	3.18034	0.120986	0.0604929	+	0.998169i	$0.480733\pi$
$692$	1.81966	0.0691731
$693$	0	0
$694$	39.1246	1.48515
$695$	0	0
$696$	0	0
$697$	−5.34752	−0.202552
$698$	−12.7639	−0.483122
$699$	0	0
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	−8.47214	−0.319306
$705$	0	0
$706$	12.0000	0.451626
$707$	−0.708204	−0.0266348
$708$	0	0
$709$	25.5279	0.958719	0.479360	−	0.877619i	$-0.340869\pi$
$709$	25.5279	0.958719	0.479360	+	0.877619i	$0.340869\pi$
$710$	0	0
$711$	0	0
$712$	26.1803	0.981150
$713$	5.70820	0.213774
$714$	0	0
$715$	0	0
$716$	−1.05573	−0.0394544
$717$	0	0
$718$	−35.9787	−1.34271
$719$	13.8197	0.515386	0.257693	−	0.966227i	$-0.417038\pi$
$719$	13.8197	0.515386	0.257693	+	0.966227i	$0.417038\pi$
$720$	0	0
$721$	1.47214	0.0548252
$722$	−22.6525	−0.843038
$723$	0	0
$724$	−2.58359	−0.0960184
$725$	0	0
$726$	0	0
$727$	44.2361	1.64062	0.820312	−	0.571916i	$-0.193800\pi$
$727$	44.2361	1.64062	0.820312	+	0.571916i	$0.193800\pi$
$728$	1.70820	0.0633102
$729$	0	0
$730$	0	0
$731$	−0.944272	−0.0349252
$732$	0	0
$733$	−3.47214	−0.128246	−0.0641231	−	0.997942i	$-0.520425\pi$
$733$	−3.47214	−0.128246	−0.0641231	+	0.997942i	$0.520425\pi$
$734$	−29.1246	−1.07501
$735$	0	0
$736$	−19.3050	−0.711590
$737$	16.0000	0.589368
$738$	0	0
$739$	6.18034	0.227347	0.113674	−	0.993518i	$-0.463738\pi$
$739$	6.18034	0.227347	0.113674	+	0.993518i	$0.463738\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	50.1803	1.84094	0.920469	−	0.390815i	$-0.127807\pi$
$743$	50.1803	1.84094	0.920469	+	0.390815i	$0.127807\pi$
$744$	0	0
$745$	0	0
$746$	−30.7426	−1.12557
$747$	0	0
$748$	−0.944272	−0.0345260
$749$	−1.36068	−0.0497182
$750$	0	0
$751$	−21.5410	−0.786043	−0.393021	−	0.919529i	$-0.628570\pi$
$751$	−21.5410	−0.786043	−0.393021	+	0.919529i	$0.628570\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−14.4721	−0.527044
$755$	0	0
$756$	0	0
$757$	−8.65248	−0.314480	−0.157240	−	0.987560i	$-0.550260\pi$
$757$	−8.65248	−0.314480	−0.157240	+	0.987560i	$0.550260\pi$
$758$	−3.41641	−0.124090
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	3.29180	0.119171
$764$	11.8541	0.428866
$765$	0	0
$766$	−38.6525	−1.39657
$767$	−7.23607	−0.261279
$768$	0	0
$769$	−47.3607	−1.70787	−0.853935	−	0.520380i	$-0.825790\pi$
$769$	−47.3607	−1.70787	−0.853935	+	0.520380i	$0.825790\pi$
$770$	0	0
$771$	0	0
$772$	−2.14590	−0.0772326
$773$	−11.1246	−0.400124	−0.200062	−	0.979783i	$-0.564114\pi$
$773$	−11.1246	−0.400124	−0.200062	+	0.979783i	$0.564114\pi$
$774$	0	0
$775$	0	0
$776$	−35.6525	−1.27985
$777$	0	0
$778$	28.9443	1.03770
$779$	15.6525	0.560808
$780$	0	0
$781$	−18.3607	−0.656997
$782$	7.05573	0.252312
$783$	0	0
$784$	33.7082	1.20386
$785$	0	0
$786$	0	0
$787$	−7.34752	−0.261911	−0.130955	−	0.991388i	$-0.541804\pi$
$787$	−7.34752	−0.261911	−0.130955	+	0.991388i	$0.541804\pi$
$788$	7.05573	0.251350
$789$	0	0
$790$	0	0
$791$	−0.819660	−0.0291438
$792$	0	0
$793$	26.4721	0.940053
$794$	11.3262	0.401953
$795$	0	0
$796$	−11.7082	−0.414986
$797$	−55.4164	−1.96295	−0.981475	−	0.191591i	$-0.938635\pi$
$797$	−55.4164	−1.96295	−0.981475	+	0.191591i	$0.938635\pi$
$798$	0	0
$799$	1.88854	0.0668119
$800$	0	0
$801$	0	0
$802$	−61.7771	−2.18142
$803$	16.9443	0.597950
$804$	0	0
$805$	0	0
$806$	5.23607	0.184433
$807$	0	0
$808$	−6.70820	−0.235994
$809$	−23.4164	−0.823277	−0.411639	−	0.911347i	$-0.635043\pi$
$809$	−23.4164	−0.823277	−0.411639	+	0.911347i	$0.635043\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0.403252	0.0141514
$813$	0	0
$814$	−6.47214	−0.226848
$815$	0	0
$816$	0	0
$817$	2.76393	0.0966977
$818$	−6.18034	−0.216091
$819$	0	0
$820$	0	0
$821$	−30.5410	−1.06589	−0.532944	−	0.846150i	$-0.678915\pi$
$821$	−30.5410	−1.06589	−0.532944	+	0.846150i	$0.678915\pi$
$822$	0	0
$823$	14.2918	0.498181	0.249090	−	0.968480i	$-0.419868\pi$
$823$	14.2918	0.498181	0.249090	+	0.968480i	$0.419868\pi$
$824$	13.9443	0.485772
$825$	0	0
$826$	0.854102	0.0297180
$827$	17.3475	0.603233	0.301616	−	0.953429i	$-0.402474\pi$
$827$	17.3475	0.603233	0.301616	+	0.953429i	$0.402474\pi$
$828$	0	0
$829$	16.8328	0.584628	0.292314	−	0.956322i	$-0.405575\pi$
$829$	16.8328	0.584628	0.292314	+	0.956322i	$0.405575\pi$
$830$	0	0
$831$	0	0
$832$	13.7082	0.475246
$833$	−5.30495	−0.183806
$834$	0	0
$835$	0	0
$836$	2.76393	0.0955926
$837$	0	0
$838$	16.3820	0.565906
$839$	−28.9443	−0.999267	−0.499634	−	0.866237i	$-0.666532\pi$
$839$	−28.9443	−0.999267	−0.499634	+	0.866237i	$0.666532\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	47.5066	1.63718
$843$	0	0
$844$	14.3262	0.493129
$845$	0	0
$846$	0	0
$847$	1.65248	0.0567797
$848$	50.8328	1.74561
$849$	0	0
$850$	0	0
$851$	11.4164	0.391349
$852$	0	0
$853$	−10.5836	−0.362375	−0.181188	−	0.983449i	$-0.557994\pi$
$853$	−10.5836	−0.362375	−0.181188	+	0.983449i	$0.557994\pi$
$854$	−3.12461	−0.106922
$855$	0	0
$856$	−12.8885	−0.440521
$857$	−55.6656	−1.90150	−0.950751	−	0.309956i	$-0.899686\pi$
$857$	−55.6656	−1.90150	−0.950751	+	0.309956i	$0.899686\pi$
$858$	0	0
$859$	2.11146	0.0720420	0.0360210	−	0.999351i	$-0.488532\pi$
$859$	2.11146	0.0720420	0.0360210	+	0.999351i	$0.488532\pi$
$860$	0	0
$861$	0	0
$862$	−19.4164	−0.661325
$863$	−9.81966	−0.334265	−0.167133	−	0.985934i	$-0.553451\pi$
$863$	−9.81966	−0.334265	−0.167133	+	0.985934i	$0.553451\pi$
$864$	0	0
$865$	0	0
$866$	−16.4721	−0.559746
$867$	0	0
$868$	−0.145898	−0.00495210
$869$	23.4164	0.794347
$870$	0	0
$871$	−25.8885	−0.877200
$872$	31.1803	1.05590
$873$	0	0
$874$	−20.6525	−0.698580
$875$	0	0
$876$	0	0
$877$	18.0557	0.609699	0.304849	−	0.952401i	$-0.401394\pi$
$877$	18.0557	0.609699	0.304849	+	0.952401i	$0.401394\pi$
$878$	−1.90983	−0.0644536
$879$	0	0
$880$	0	0
$881$	20.3607	0.685969	0.342984	−	0.939341i	$-0.388562\pi$
$881$	20.3607	0.685969	0.342984	+	0.939341i	$0.388562\pi$
$882$	0	0
$883$	31.7771	1.06938	0.534692	−	0.845047i	$-0.320428\pi$
$883$	31.7771	1.06938	0.534692	+	0.845047i	$0.320428\pi$
$884$	1.52786	0.0513876
$885$	0	0
$886$	49.6869	1.66926
$887$	27.0689	0.908884	0.454442	−	0.890776i	$-0.349839\pi$
$887$	27.0689	0.908884	0.454442	+	0.890776i	$0.349839\pi$
$888$	0	0
$889$	2.94427	0.0987477
$890$	0	0
$891$	0	0
$892$	−2.47214	−0.0827732
$893$	−5.52786	−0.184983
$894$	0	0
$895$	0	0
$896$	−3.21478	−0.107398
$897$	0	0
$898$	50.6525	1.69030
$899$	−2.76393	−0.0921823
$900$	0	0
$901$	−8.00000	−0.266519
$902$	22.6525	0.754245
$903$	0	0
$904$	−7.76393	−0.258225
$905$	0	0
$906$	0	0
$907$	24.2361	0.804745	0.402373	−	0.915476i	$-0.368186\pi$
$907$	24.2361	0.804745	0.402373	+	0.915476i	$0.368186\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	0	0
$911$	−18.1803	−0.602342	−0.301171	−	0.953570i	$-0.597377\pi$
$911$	−18.1803	−0.602342	−0.301171	+	0.953570i	$0.597377\pi$
$912$	0	0
$913$	29.8885	0.989166
$914$	4.94427	0.163542
$915$	0	0
$916$	−8.29180	−0.273969
$917$	2.83282	0.0935478
$918$	0	0
$919$	−14.4721	−0.477392	−0.238696	−	0.971094i	$-0.576720\pi$
$919$	−14.4721	−0.477392	−0.238696	+	0.971094i	$0.576720\pi$
$920$	0	0
$921$	0	0
$922$	−55.5967	−1.83098
$923$	29.7082	0.977857
$924$	0	0
$925$	0	0
$926$	4.18034	0.137374
$927$	0	0
$928$	9.34752	0.306848
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	15.5279	0.508905
$932$	11.0902	0.363271
$933$	0	0
$934$	7.61803	0.249270
$935$	0	0
$936$	0	0
$937$	−9.05573	−0.295838	−0.147919	−	0.988999i	$-0.547257\pi$
$937$	−9.05573	−0.295838	−0.147919	+	0.988999i	$0.547257\pi$
$938$	3.05573	0.0997731
$939$	0	0
$940$	0	0
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	0	0
$943$	−39.9574	−1.30119
$944$	10.8541	0.353271
$945$	0	0
$946$	4.00000	0.130051
$947$	−13.0557	−0.424254	−0.212127	−	0.977242i	$-0.568039\pi$
$947$	−13.0557	−0.424254	−0.212127	+	0.977242i	$0.568039\pi$
$948$	0	0
$949$	−27.4164	−0.889974
$950$	0	0
$951$	0	0
$952$	0.403252	0.0130695
$953$	45.7082	1.48063	0.740317	−	0.672258i	$-0.234675\pi$
$953$	45.7082	1.48063	0.740317	+	0.672258i	$0.234675\pi$
$954$	0	0
$955$	0	0
$956$	7.23607	0.234031
$957$	0	0
$958$	−37.6869	−1.21761
$959$	−1.48529	−0.0479626
$960$	0	0
$961$	1.00000	0.0322581
$962$	10.4721	0.337635
$963$	0	0
$964$	8.87539	0.285857
$965$	0	0
$966$	0	0
$967$	−60.3607	−1.94107	−0.970534	−	0.240963i	$-0.922537\pi$
$967$	−60.3607	−1.94107	−0.970534	+	0.240963i	$0.922537\pi$
$968$	15.6525	0.503090
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	−3.16718	−0.101535
$974$	31.1246	0.997297
$975$	0	0
$976$	−39.7082	−1.27103
$977$	−47.2492	−1.51164	−0.755818	−	0.654781i	$-0.772761\pi$
$977$	−47.2492	−1.51164	−0.755818	+	0.654781i	$0.772761\pi$
$978$	0	0
$979$	23.4164	0.748392
$980$	0	0
$981$	0	0
$982$	−7.05573	−0.225157
$983$	39.5279	1.26074	0.630372	−	0.776294i	$-0.282903\pi$
$983$	39.5279	1.26074	0.630372	+	0.776294i	$0.282903\pi$
$984$	0	0
$985$	0	0
$986$	−3.41641	−0.108801
$987$	0	0
$988$	−4.47214	−0.142278
$989$	−7.05573	−0.224359
$990$	0	0
$991$	−16.5410	−0.525443	−0.262721	−	0.964872i	$-0.584620\pi$
$991$	−16.5410	−0.525443	−0.262721	+	0.964872i	$0.584620\pi$
$992$	−3.38197	−0.107378
$993$	0	0
$994$	−3.50658	−0.111222
$995$	0	0
$996$	0	0
$997$	29.3607	0.929862	0.464931	−	0.885347i	$-0.346079\pi$
$997$	29.3607	0.929862	0.464931	+	0.885347i	$0.346079\pi$
$998$	−10.6525	−0.337198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.y.1.2		2
3.2	odd	2		775.2.a.d.1.1		2
5.4	even	2		279.2.a.a.1.1		2
15.2	even	4		775.2.b.d.249.1		4
15.8	even	4		775.2.b.d.249.4		4
15.14	odd	2		31.2.a.a.1.2	✓	2
20.19	odd	2		4464.2.a.bf.1.1		2
60.59	even	2		496.2.a.i.1.2		2
105.104	even	2		1519.2.a.a.1.2		2
120.29	odd	2		1984.2.a.r.1.2		2
120.59	even	2		1984.2.a.n.1.1		2
155.154	odd	2		8649.2.a.c.1.1		2
165.164	even	2		3751.2.a.b.1.1		2
195.194	odd	2		5239.2.a.f.1.1		2
255.254	odd	2		8959.2.a.b.1.2		2
465.14	odd	30		961.2.g.a.816.1		8
465.29	even	10		961.2.d.g.531.1		4
465.44	even	30		961.2.g.e.448.1		8
465.59	odd	30		961.2.g.h.846.1		8
465.74	even	30		961.2.g.e.547.1		8
465.89	even	10		961.2.d.a.388.1		4
465.104	even	30		961.2.g.d.338.1		8
465.119	even	6		961.2.c.c.521.2		4
465.134	odd	30		961.2.g.h.844.1		8
465.149	odd	6		961.2.c.e.439.2		4
465.164	odd	30		961.2.g.a.732.1		8
465.179	even	30		961.2.g.d.235.1		8
465.194	odd	10		961.2.d.c.374.1		4
465.209	even	10		961.2.d.a.374.1		4
465.224	odd	30		961.2.g.a.235.1		8
465.239	even	30		961.2.g.d.732.1		8
465.254	even	6		961.2.c.c.439.2		4
465.269	even	30		961.2.g.e.844.1		8
465.284	odd	6		961.2.c.e.521.2		4
465.299	odd	30		961.2.g.a.338.1		8
465.314	odd	10		961.2.d.c.388.1		4
465.329	odd	30		961.2.g.h.547.1		8
465.344	even	30		961.2.g.e.846.1		8
465.359	odd	30		961.2.g.h.448.1		8
465.374	odd	10		961.2.d.d.531.1		4
465.389	even	30		961.2.g.d.816.1		8
465.419	odd	10		961.2.d.d.628.1		4
465.449	even	10		961.2.d.g.628.1		4
465.464	even	2		961.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.a.a.1.2	✓	2	15.14	odd	2
279.2.a.a.1.1		2	5.4	even	2
496.2.a.i.1.2		2	60.59	even	2
775.2.a.d.1.1		2	3.2	odd	2
775.2.b.d.249.1		4	15.2	even	4
775.2.b.d.249.4		4	15.8	even	4
961.2.a.f.1.2		2	465.464	even	2
961.2.c.c.439.2		4	465.254	even	6
961.2.c.c.521.2		4	465.119	even	6
961.2.c.e.439.2		4	465.149	odd	6
961.2.c.e.521.2		4	465.284	odd	6
961.2.d.a.374.1		4	465.209	even	10
961.2.d.a.388.1		4	465.89	even	10
961.2.d.c.374.1		4	465.194	odd	10
961.2.d.c.388.1		4	465.314	odd	10
961.2.d.d.531.1		4	465.374	odd	10
961.2.d.d.628.1		4	465.419	odd	10
961.2.d.g.531.1		4	465.29	even	10
961.2.d.g.628.1		4	465.449	even	10
961.2.g.a.235.1		8	465.224	odd	30
961.2.g.a.338.1		8	465.299	odd	30
961.2.g.a.732.1		8	465.164	odd	30
961.2.g.a.816.1		8	465.14	odd	30
961.2.g.d.235.1		8	465.179	even	30
961.2.g.d.338.1		8	465.104	even	30
961.2.g.d.732.1		8	465.239	even	30
961.2.g.d.816.1		8	465.389	even	30
961.2.g.e.448.1		8	465.44	even	30
961.2.g.e.547.1		8	465.74	even	30
961.2.g.e.844.1		8	465.269	even	30
961.2.g.e.846.1		8	465.344	even	30
961.2.g.h.448.1		8	465.359	odd	30
961.2.g.h.547.1		8	465.329	odd	30
961.2.g.h.844.1		8	465.134	odd	30
961.2.g.h.846.1		8	465.59	odd	30
1519.2.a.a.1.2		2	105.104	even	2
1984.2.a.n.1.1		2	120.59	even	2
1984.2.a.r.1.2		2	120.29	odd	2
3751.2.a.b.1.1		2	165.164	even	2
4464.2.a.bf.1.1		2	20.19	odd	2
5239.2.a.f.1.1		2	195.194	odd	2
6975.2.a.y.1.2		2	1.1	even	1	trivial
8649.2.a.c.1.1		2	155.154	odd	2
8959.2.a.b.1.2		2	255.254	odd	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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -0.236068 q^{7} -2.23607 q^{8} -2.00000 q^{11} +3.23607 q^{13} -0.381966 q^{14} -4.85410 q^{16} +0.763932 q^{17} -2.23607 q^{19} -3.23607 q^{22} +5.70820 q^{23} +5.23607 q^{26} -0.145898 q^{28} -2.76393 q^{29} +1.00000 q^{31} -3.38197 q^{32} +1.23607 q^{34} +2.00000 q^{37} -3.61803 q^{38} -7.00000 q^{41} -1.23607 q^{43} -1.23607 q^{44} +9.23607 q^{46} +2.47214 q^{47} -6.94427 q^{49} +2.00000 q^{52} -10.4721 q^{53} +0.527864 q^{56} -4.47214 q^{58} -2.23607 q^{59} +8.18034 q^{61} +1.61803 q^{62} +4.23607 q^{64} -8.00000 q^{67} +0.472136 q^{68} +9.18034 q^{71} -8.47214 q^{73} +3.23607 q^{74} -1.38197 q^{76} +0.472136 q^{77} -11.7082 q^{79} -11.3262 q^{82} -14.9443 q^{83} -2.00000 q^{86} +4.47214 q^{88} -11.7082 q^{89} -0.763932 q^{91} +3.52786 q^{92} +4.00000 q^{94} +15.9443 q^{97} -11.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 3 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 7 q^{28} - 10 q^{29} + 2 q^{31} - 9 q^{32} - 2 q^{34} + 4 q^{37} - 5 q^{38} - 14 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 3 * q^14 - 3 * q^16 + 6 * q^17 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 7 * q^28 - 10 * q^29 + 2 * q^31 - 9 * q^32 - 2 * q^34 + 4 * q^37 - 5 * q^38 - 14 * q^41 + 2 * q^43 + 2 * q^44 + 14 * q^46 - 4 * q^47 + 4 * q^49 + 4 * q^52 - 12 * q^53 + 10 * q^56 - 6 * q^61 + q^62 + 4 * q^64 - 16 * q^67 - 8 * q^68 - 4 * q^71 - 8 * q^73 + 2 * q^74 - 5 * q^76 - 8 * q^77 - 10 * q^79 - 7 * q^82 - 12 * q^83 - 4 * q^86 - 10 * q^89 - 6 * q^91 + 16 * q^92 + 8 * q^94 + 14 * q^97 - 18 * q^98

Introduction

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