LMFDB - Embedded newform 882.3.c.f.685.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.c (of order $2$, degree $1$, 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:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.4
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.f.685.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +3.16693i q^{5} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} +3.16693i q^{5} +2.82843 q^{8} +4.47871i q^{10} +13.2426 q^{11} -5.49333i q^{13} +4.00000 q^{16} +13.5592i q^{17} +0.717439i q^{19} +6.33386i q^{20} +18.7279 q^{22} +2.27208 q^{23} +14.9706 q^{25} -7.76874i q^{26} -20.4853 q^{29} +24.6180i q^{31} +5.65685 q^{32} +19.1757i q^{34} +64.9411 q^{37} +1.01461i q^{38} +8.95743i q^{40} +21.0308i q^{41} +6.48528 q^{43} +26.4853 q^{44} +3.21320 q^{46} +47.7800i q^{47} +21.1716 q^{50} -10.9867i q^{52} -22.0294 q^{53} +41.9385i q^{55} -28.9706 q^{58} +83.7539i q^{59} -66.2593i q^{61} +34.8151i q^{62} +8.00000 q^{64} +17.3970 q^{65} +92.6396 q^{67} +27.1185i q^{68} +48.4264 q^{71} +130.991i q^{73} +91.8406 q^{74} +1.43488i q^{76} -76.2132 q^{79} +12.6677i q^{80} +29.7420i q^{82} -107.981i q^{83} -42.9411 q^{85} +9.17157 q^{86} +37.4558 q^{88} -167.907i q^{89} +4.54416 q^{92} +67.5711i q^{94} -2.27208 q^{95} +25.5816i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 36 q^{11} + 16 q^{16} + 24 q^{22} + 60 q^{23} - 8 q^{25} - 48 q^{29} + 124 q^{37} - 8 q^{43} + 72 q^{44} - 72 q^{46} + 96 q^{50} - 156 q^{53} - 48 q^{58} + 32 q^{64} - 168 q^{65} + 116 q^{67} + 24 q^{71} + 192 q^{74} - 220 q^{79} - 36 q^{85} + 48 q^{86} + 48 q^{88} + 120 q^{92} - 60 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 + 36 * q^11 + 16 * q^16 + 24 * q^22 + 60 * q^23 - 8 * q^25 - 48 * q^29 + 124 * q^37 - 8 * q^43 + 72 * q^44 - 72 * q^46 + 96 * q^50 - 156 * q^53 - 48 * q^58 + 32 * q^64 - 168 * q^65 + 116 * q^67 + 24 * q^71 + 192 * q^74 - 220 * q^79 - 36 * q^85 + 48 * q^86 + 48 * q^88 + 120 * q^92 - 60 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	3.16693i	0.633386i	0.948528	+	0.316693i	$0.102572\pi$
$5$	3.16693i	0.633386i	−0.948528	+	0.316693i	$0.897428\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	0	0
$10$	4.47871i	0.447871i
$11$	13.2426	1.20388	0.601938	−	0.798543i	$-0.294395\pi$
$11$	13.2426	1.20388	0.601938	+	0.798543i	$0.294395\pi$
$12$	0	0
$13$	− 5.49333i	− 0.422563i	−0.977425	−	0.211282i	$-0.932236\pi$
$13$	− 5.49333i	− 0.422563i	0.977425	−	0.211282i	$-0.0677638\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	13.5592i	0.797602i	0.917037	+	0.398801i	$0.130574\pi$
$17$	13.5592i	0.797602i	−0.917037	+	0.398801i	$0.869426\pi$
$18$	0	0
$19$	0.717439i	0.0377599i	0.999822	+	0.0188800i	$0.00601004\pi$
$19$	0.717439i	0.0377599i	−0.999822	+	0.0188800i	$0.993990\pi$
$20$	6.33386i	0.316693i
$21$	0	0
$22$	18.7279	0.851269
$23$	2.27208	0.0987860	0.0493930	−	0.998779i	$-0.484271\pi$
$23$	2.27208	0.0987860	0.0493930	+	0.998779i	$0.484271\pi$
$24$	0	0
$25$	14.9706	0.598823
$26$	− 7.76874i	− 0.298798i
$27$	0	0
$28$	0	0
$29$	−20.4853	−0.706389	−0.353195	−	0.935550i	$-0.614905\pi$
$29$	−20.4853	−0.706389	−0.353195	+	0.935550i	$0.614905\pi$
$30$	0	0
$31$	24.6180i	0.794129i	0.917791	+	0.397064i	$0.129971\pi$
$31$	24.6180i	0.794129i	−0.917791	+	0.397064i	$0.870029\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	19.1757i	0.563990i
$35$	0	0
$36$	0	0
$37$	64.9411	1.75517	0.877583	−	0.479425i	$-0.159155\pi$
$37$	64.9411	1.75517	0.877583	+	0.479425i	$0.159155\pi$
$38$	1.01461i	0.0267003i
$39$	0	0
$40$	8.95743i	0.223936i
$41$	21.0308i	0.512946i	0.966551	+	0.256473i	$0.0825605\pi$
$41$	21.0308i	0.512946i	−0.966551	+	0.256473i	$0.917439\pi$
$42$	0	0
$43$	6.48528	0.150820	0.0754102	−	0.997153i	$-0.475973\pi$
$43$	6.48528	0.150820	0.0754102	+	0.997153i	$0.475973\pi$
$44$	26.4853	0.601938
$45$	0	0
$46$	3.21320	0.0698522
$47$	47.7800i	1.01660i	0.861181	+	0.508298i	$0.169725\pi$
$47$	47.7800i	1.01660i	−0.861181	+	0.508298i	$0.830275\pi$
$48$	0	0
$49$	0	0
$50$	21.1716	0.423431
$51$	0	0
$52$	− 10.9867i	− 0.211282i
$53$	−22.0294	−0.415650	−0.207825	−	0.978166i	$-0.566638\pi$
$53$	−22.0294	−0.415650	−0.207825	+	0.978166i	$0.566638\pi$
$54$	0	0
$55$	41.9385i	0.762518i
$56$	0	0
$57$	0	0
$58$	−28.9706	−0.499492
$59$	83.7539i	1.41956i	0.704425	+	0.709779i	$0.251205\pi$
$59$	83.7539i	1.41956i	−0.704425	+	0.709779i	$0.748795\pi$
$60$	0	0
$61$	− 66.2593i	− 1.08622i	−0.839662	−	0.543109i	$-0.817247\pi$
$61$	− 66.2593i	− 1.08622i	0.839662	−	0.543109i	$-0.182753\pi$
$62$	34.8151i	0.561534i
$63$	0	0
$64$	8.00000	0.125000
$65$	17.3970	0.267646
$66$	0	0
$67$	92.6396	1.38268	0.691340	−	0.722529i	$-0.257021\pi$
$67$	92.6396	1.38268	0.691340	+	0.722529i	$0.257021\pi$
$68$	27.1185i	0.398801i
$69$	0	0
$70$	0	0
$71$	48.4264	0.682062	0.341031	−	0.940052i	$-0.389224\pi$
$71$	48.4264	0.682062	0.341031	+	0.940052i	$0.389224\pi$
$72$	0	0
$73$	130.991i	1.79439i	0.441634	+	0.897195i	$0.354399\pi$
$73$	130.991i	1.79439i	−0.441634	+	0.897195i	$0.645601\pi$
$74$	91.8406	1.24109
$75$	0	0
$76$	1.43488i	0.0188800i
$77$	0	0
$78$	0	0
$79$	−76.2132	−0.964724	−0.482362	−	0.875972i	$-0.660221\pi$
$79$	−76.2132	−0.964724	−0.482362	+	0.875972i	$0.660221\pi$
$80$	12.6677i	0.158346i
$81$	0	0
$82$	29.7420i	0.362708i
$83$	− 107.981i	− 1.30098i	−0.759514	−	0.650491i	$-0.774563\pi$
$83$	− 107.981i	− 1.30098i	0.759514	−	0.650491i	$-0.225437\pi$
$84$	0	0
$85$	−42.9411	−0.505190
$86$	9.17157	0.106646
$87$	0	0
$88$	37.4558	0.425635
$89$	− 167.907i	− 1.88659i	−0.331949	−	0.943297i	$-0.607706\pi$
$89$	− 167.907i	− 1.88659i	0.331949	−	0.943297i	$-0.392294\pi$
$90$	0	0
$91$	0	0
$92$	4.54416	0.0493930
$93$	0	0
$94$	67.5711i	0.718841i
$95$	−2.27208	−0.0239166
$96$	0	0
$97$	25.5816i	0.263728i	0.991268	+	0.131864i	$0.0420962\pi$
$97$	25.5816i	0.263728i	−0.991268	+	0.131864i	$0.957904\pi$
$98$	0	0
$99$	0	0
$100$	29.9411	0.299411
$101$	− 28.5024i	− 0.282202i	−0.989995	−	0.141101i	$-0.954936\pi$
$101$	− 28.5024i	− 0.282202i	0.989995	−	0.141101i	$-0.0450642\pi$
$102$	0	0
$103$	− 56.4912i	− 0.548458i	−0.961664	−	0.274229i	$-0.911577\pi$
$103$	− 56.4912i	− 0.548458i	0.961664	−	0.274229i	$-0.0884227\pi$
$104$	− 15.5375i	− 0.149399i
$105$	0	0
$106$	−31.1543	−0.293909
$107$	−47.6102	−0.444955	−0.222477	−	0.974938i	$-0.571414\pi$
$107$	−47.6102	−0.444955	−0.222477	+	0.974938i	$0.571414\pi$
$108$	0	0
$109$	75.3087	0.690905	0.345453	−	0.938436i	$-0.387725\pi$
$109$	75.3087	0.690905	0.345453	+	0.938436i	$0.387725\pi$
$110$	59.3100i	0.539182i
$111$	0	0
$112$	0	0
$113$	−85.4558	−0.756246	−0.378123	−	0.925755i	$-0.623430\pi$
$113$	−85.4558	−0.756246	−0.378123	+	0.925755i	$0.623430\pi$
$114$	0	0
$115$	7.19551i	0.0625696i
$116$	−40.9706	−0.353195
$117$	0	0
$118$	118.446i	1.00378i
$119$	0	0
$120$	0	0
$121$	54.3675	0.449318
$122$	− 93.7048i	− 0.768072i
$123$	0	0
$124$	49.2360i	0.397064i
$125$	126.584i	1.01267i
$126$	0	0
$127$	−60.6619	−0.477653	−0.238826	−	0.971062i	$-0.576763\pi$
$127$	−60.6619	−0.477653	−0.238826	+	0.971062i	$0.576763\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	24.6030	0.189254
$131$	− 132.948i	− 1.01487i	−0.861691	−	0.507434i	$-0.830594\pi$
$131$	− 132.948i	− 1.01487i	0.861691	−	0.507434i	$-0.169406\pi$
$132$	0	0
$133$	0	0
$134$	131.012	0.977703
$135$	0	0
$136$	38.3513i	0.281995i
$137$	117.426	0.857127	0.428564	−	0.903512i	$-0.359020\pi$
$137$	117.426	0.857127	0.428564	+	0.903512i	$0.359020\pi$
$138$	0	0
$139$	68.5857i	0.493422i	0.969089	+	0.246711i	$0.0793499\pi$
$139$	68.5857i	0.493422i	−0.969089	+	0.246711i	$0.920650\pi$
$140$	0	0
$141$	0	0
$142$	68.4853	0.482291
$143$	− 72.7461i	− 0.508714i
$144$	0	0
$145$	− 64.8754i	− 0.447417i
$146$	185.249i	1.26883i
$147$	0	0
$148$	129.882	0.877583
$149$	26.3970	0.177161	0.0885804	−	0.996069i	$-0.471767\pi$
$149$	26.3970	0.177161	0.0885804	+	0.996069i	$0.471767\pi$
$150$	0	0
$151$	−134.213	−0.888829	−0.444415	−	0.895821i	$-0.646588\pi$
$151$	−134.213	−0.888829	−0.444415	+	0.895821i	$0.646588\pi$
$152$	2.02922i	0.0133502i
$153$	0	0
$154$	0	0
$155$	−77.9634	−0.502990
$156$	0	0
$157$	− 226.695i	− 1.44392i	−0.691937	−	0.721958i	$-0.743243\pi$
$157$	− 226.695i	− 1.44392i	0.691937	−	0.721958i	$-0.256757\pi$
$158$	−107.782	−0.682163
$159$	0	0
$160$	17.9149i	0.111968i
$161$	0	0
$162$	0	0
$163$	−91.9777	−0.564280	−0.282140	−	0.959373i	$-0.591044\pi$
$163$	−91.9777	−0.564280	−0.282140	+	0.959373i	$0.591044\pi$
$164$	42.0616i	0.256473i
$165$	0	0
$166$	− 152.709i	− 0.919933i
$167$	− 203.482i	− 1.21845i	−0.792996	−	0.609227i	$-0.791480\pi$
$167$	− 203.482i	− 1.21845i	0.792996	−	0.609227i	$-0.208520\pi$
$168$	0	0
$169$	138.823	0.821440
$170$	−60.7279	−0.357223
$171$	0	0
$172$	12.9706	0.0754102
$173$	70.8101i	0.409307i	0.978834	+	0.204654i	$0.0656068\pi$
$173$	70.8101i	0.409307i	−0.978834	+	0.204654i	$0.934393\pi$
$174$	0	0
$175$	0	0
$176$	52.9706	0.300969
$177$	0	0
$178$	− 237.456i	− 1.33402i
$179$	−108.816	−0.607912	−0.303956	−	0.952686i	$-0.598308\pi$
$179$	−108.816	−0.607912	−0.303956	+	0.952686i	$0.598308\pi$
$180$	0	0
$181$	− 99.6607i	− 0.550611i	−0.961357	−	0.275306i	$-0.911221\pi$
$181$	− 99.6607i	− 0.550611i	0.961357	−	0.275306i	$-0.0887791\pi$
$182$	0	0
$183$	0	0
$184$	6.42641	0.0349261
$185$	205.664i	1.11170i
$186$	0	0
$187$	179.560i	0.960214i
$188$	95.5600i	0.508298i
$189$	0	0
$190$	−3.21320	−0.0169116
$191$	−69.9045	−0.365992	−0.182996	−	0.983114i	$-0.558580\pi$
$191$	−69.9045	−0.365992	−0.182996	+	0.983114i	$0.558580\pi$
$192$	0	0
$193$	−32.3381	−0.167555	−0.0837774	−	0.996484i	$-0.526698\pi$
$193$	−32.3381	−0.167555	−0.0837774	+	0.996484i	$0.526698\pi$
$194$	36.1779i	0.186484i
$195$	0	0
$196$	0	0
$197$	−277.103	−1.40661	−0.703306	−	0.710887i	$-0.748294\pi$
$197$	−277.103	−1.40661	−0.703306	+	0.710887i	$0.748294\pi$
$198$	0	0
$199$	167.444i	0.841429i	0.907193	+	0.420715i	$0.138221\pi$
$199$	167.444i	0.841429i	−0.907193	+	0.420715i	$0.861779\pi$
$200$	42.3431	0.211716
$201$	0	0
$202$	− 40.3084i	− 0.199547i
$203$	0	0
$204$	0	0
$205$	−66.6030	−0.324893
$206$	− 79.8907i	− 0.387819i
$207$	0	0
$208$	− 21.9733i	− 0.105641i
$209$	9.50079i	0.0454583i
$210$	0	0
$211$	−128.073	−0.606982	−0.303491	−	0.952834i	$-0.598152\pi$
$211$	−128.073	−0.606982	−0.303491	+	0.952834i	$0.598152\pi$
$212$	−44.0589	−0.207825
$213$	0	0
$214$	−67.3310	−0.314631
$215$	20.5384i	0.0955276i
$216$	0	0
$217$	0	0
$218$	106.503	0.488544
$219$	0	0
$220$	83.8770i	0.381259i
$221$	74.4853	0.337037
$222$	0	0
$223$	− 417.169i	− 1.87071i	−0.353705	−	0.935357i	$-0.615078\pi$
$223$	− 417.169i	− 1.87071i	0.353705	−	0.935357i	$-0.384922\pi$
$224$	0	0
$225$	0	0
$226$	−120.853	−0.534747
$227$	− 232.260i	− 1.02317i	−0.859232	−	0.511586i	$-0.829058\pi$
$227$	− 232.260i	− 1.02317i	0.859232	−	0.511586i	$-0.170942\pi$
$228$	0	0
$229$	83.6221i	0.365162i	0.983191	+	0.182581i	$0.0584452\pi$
$229$	83.6221i	0.365162i	−0.983191	+	0.182581i	$0.941555\pi$
$230$	10.1760i	0.0442434i
$231$	0	0
$232$	−57.9411	−0.249746
$233$	−219.073	−0.940228	−0.470114	−	0.882606i	$-0.655787\pi$
$233$	−219.073	−0.940228	−0.470114	+	0.882606i	$0.655787\pi$
$234$	0	0
$235$	−151.316	−0.643897
$236$	167.508i	0.709779i
$237$	0	0
$238$	0	0
$239$	−193.103	−0.807961	−0.403980	−	0.914768i	$-0.632374\pi$
$239$	−193.103	−0.807961	−0.403980	+	0.914768i	$0.632374\pi$
$240$	0	0
$241$	49.5332i	0.205532i	0.994706	+	0.102766i	$0.0327692\pi$
$241$	49.5332i	0.205532i	−0.994706	+	0.102766i	$0.967231\pi$
$242$	76.8873	0.317716
$243$	0	0
$244$	− 132.519i	− 0.543109i
$245$	0	0
$246$	0	0
$247$	3.94113	0.0159560
$248$	69.6302i	0.280767i
$249$	0	0
$250$	179.017i	0.716067i
$251$	− 162.524i	− 0.647507i	−0.946141	−	0.323754i	$-0.895055\pi$
$251$	− 162.524i	− 0.647507i	0.946141	−	0.323754i	$-0.104945\pi$
$252$	0	0
$253$	30.0883	0.118926
$254$	−85.7889	−0.337752
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 99.1595i	− 0.385835i	−0.981215	−	0.192917i	$-0.938205\pi$
$257$	− 99.1595i	− 0.385835i	0.981215	−	0.192917i	$-0.0617949\pi$
$258$	0	0
$259$	0	0
$260$	34.7939	0.133823
$261$	0	0
$262$	− 188.016i	− 0.717620i
$263$	−434.345	−1.65150	−0.825751	−	0.564034i	$-0.809249\pi$
$263$	−434.345	−1.65150	−0.825751	+	0.564034i	$0.809249\pi$
$264$	0	0
$265$	− 69.7657i	− 0.263267i
$266$	0	0
$267$	0	0
$268$	185.279	0.691340
$269$	91.4083i	0.339808i	0.985461	+	0.169904i	$0.0543457\pi$
$269$	91.4083i	0.339808i	−0.985461	+	0.169904i	$0.945654\pi$
$270$	0	0
$271$	− 17.0954i	− 0.0630828i	−0.999502	−	0.0315414i	$-0.989958\pi$
$271$	− 17.0954i	− 0.0630828i	0.999502	−	0.0315414i	$-0.0100416\pi$
$272$	54.2369i	0.199400i
$273$	0	0
$274$	166.066	0.606080
$275$	198.250	0.720908
$276$	0	0
$277$	−400.411	−1.44553	−0.722764	−	0.691095i	$-0.757129\pi$
$277$	−400.411	−1.44553	−0.722764	+	0.691095i	$0.757129\pi$
$278$	96.9948i	0.348902i
$279$	0	0
$280$	0	0
$281$	538.690	1.91705	0.958524	−	0.285012i	$-0.0919976\pi$
$281$	538.690	1.91705	0.958524	+	0.285012i	$0.0919976\pi$
$282$	0	0
$283$	− 309.209i	− 1.09261i	−0.837586	−	0.546306i	$-0.816034\pi$
$283$	− 309.209i	− 1.09261i	0.837586	−	0.546306i	$-0.183966\pi$
$284$	96.8528	0.341031
$285$	0	0
$286$	− 102.879i	− 0.359715i
$287$	0	0
$288$	0	0
$289$	105.147	0.363831
$290$	− 91.7477i	− 0.316371i
$291$	0	0
$292$	261.981i	0.897195i
$293$	− 327.391i	− 1.11738i	−0.829378	−	0.558688i	$-0.811305\pi$
$293$	− 327.391i	− 1.11738i	0.829378	−	0.558688i	$-0.188695\pi$
$294$	0	0
$295$	−265.243	−0.899128
$296$	183.681	0.620545
$297$	0	0
$298$	37.3310	0.125272
$299$	− 12.4813i	− 0.0417434i
$300$	0	0
$301$	0	0
$302$	−189.806	−0.628497
$303$	0	0
$304$	2.86976i	0.00943999i
$305$	209.839	0.687995
$306$	0	0
$307$	256.140i	0.834331i	0.908831	+	0.417165i	$0.136976\pi$
$307$	256.140i	0.834331i	−0.908831	+	0.417165i	$0.863024\pi$
$308$	0	0
$309$	0	0
$310$	−110.257	−0.355668
$311$	− 216.332i	− 0.695602i	−0.937568	−	0.347801i	$-0.886928\pi$
$311$	− 216.332i	− 0.695602i	0.937568	−	0.347801i	$-0.113072\pi$
$312$	0	0
$313$	− 156.818i	− 0.501017i	−0.968114	−	0.250509i	$-0.919402\pi$
$313$	− 156.818i	− 0.501017i	0.968114	−	0.250509i	$-0.0805978\pi$
$314$	− 320.595i	− 1.02100i
$315$	0	0
$316$	−152.426	−0.482362
$317$	448.029	1.41334	0.706671	−	0.707542i	$-0.250196\pi$
$317$	448.029	1.41334	0.706671	+	0.707542i	$0.250196\pi$
$318$	0	0
$319$	−271.279	−0.850405
$320$	25.3354i	0.0791732i
$321$	0	0
$322$	0	0
$323$	−9.72792	−0.0301174
$324$	0	0
$325$	− 82.2382i	− 0.253041i
$326$	−130.076	−0.399006
$327$	0	0
$328$	59.4841i	0.181354i
$329$	0	0
$330$	0	0
$331$	−55.0071	−0.166185	−0.0830924	−	0.996542i	$-0.526480\pi$
$331$	−55.0071	−0.166185	−0.0830924	+	0.996542i	$0.526480\pi$
$332$	− 215.963i	− 0.650491i
$333$	0	0
$334$	− 287.767i	− 0.861577i
$335$	293.383i	0.875770i
$336$	0	0
$337$	−111.632	−0.331254	−0.165627	−	0.986189i	$-0.552965\pi$
$337$	−111.632	−0.331254	−0.165627	+	0.986189i	$0.552965\pi$
$338$	196.326	0.580846
$339$	0	0
$340$	−85.8823	−0.252595
$341$	326.007i	0.956033i
$342$	0	0
$343$	0	0
$344$	18.3431	0.0533231
$345$	0	0
$346$	100.141i	0.289424i
$347$	−377.257	−1.08720	−0.543598	−	0.839346i	$-0.682938\pi$
$347$	−377.257	−1.08720	−0.543598	+	0.839346i	$0.682938\pi$
$348$	0	0
$349$	− 204.034i	− 0.584624i	−0.956323	−	0.292312i	$-0.905575\pi$
$349$	− 204.034i	− 0.584624i	0.956323	−	0.292312i	$-0.0944246\pi$
$350$	0	0
$351$	0	0
$352$	74.9117	0.212817
$353$	417.076i	1.18152i	0.806848	+	0.590759i	$0.201172\pi$
$353$	417.076i	1.18152i	−0.806848	+	0.590759i	$0.798828\pi$
$354$	0	0
$355$	153.363i	0.432008i
$356$	− 335.814i	− 0.943297i
$357$	0	0
$358$	−153.889	−0.429859
$359$	178.831	0.498135	0.249068	−	0.968486i	$-0.419876\pi$
$359$	178.831	0.498135	0.249068	+	0.968486i	$0.419876\pi$
$360$	0	0
$361$	360.485	0.998574
$362$	− 140.941i	− 0.389341i
$363$	0	0
$364$	0	0
$365$	−414.838	−1.13654
$366$	0	0
$367$	− 628.993i	− 1.71388i	−0.515418	−	0.856939i	$-0.672363\pi$
$367$	− 628.993i	− 1.71388i	0.515418	−	0.856939i	$-0.327637\pi$
$368$	9.08831	0.0246965
$369$	0	0
$370$	290.853i	0.786088i
$371$	0	0
$372$	0	0
$373$	−255.558	−0.685143	−0.342572	−	0.939492i	$-0.611298\pi$
$373$	−255.558	−0.685143	−0.342572	+	0.939492i	$0.611298\pi$
$374$	253.936i	0.678974i
$375$	0	0
$376$	135.142i	0.359421i
$377$	112.532i	0.298494i
$378$	0	0
$379$	219.750	0.579816	0.289908	−	0.957055i	$-0.406375\pi$
$379$	219.750	0.579816	0.289908	+	0.957055i	$0.406375\pi$
$380$	−4.54416	−0.0119583
$381$	0	0
$382$	−98.8600	−0.258796
$383$	− 17.0357i	− 0.0444797i	−0.999753	−	0.0222398i	$-0.992920\pi$
$383$	− 17.0357i	− 0.0444797i	0.999753	−	0.0222398i	$-0.00707974\pi$
$384$	0	0
$385$	0	0
$386$	−45.7330	−0.118479
$387$	0	0
$388$	51.1632i	0.131864i
$389$	152.220	0.391312	0.195656	−	0.980673i	$-0.437316\pi$
$389$	152.220	0.391312	0.195656	+	0.980673i	$0.437316\pi$
$390$	0	0
$391$	30.8076i	0.0787919i
$392$	0	0
$393$	0	0
$394$	−391.882	−0.994625
$395$	− 241.362i	− 0.611042i
$396$	0	0
$397$	372.722i	0.938845i	0.882974	+	0.469423i	$0.155538\pi$
$397$	372.722i	0.938845i	−0.882974	+	0.469423i	$0.844462\pi$
$398$	236.802i	0.594980i
$399$	0	0
$400$	59.8823	0.149706
$401$	−651.573	−1.62487	−0.812435	−	0.583052i	$-0.801858\pi$
$401$	−651.573	−1.62487	−0.812435	+	0.583052i	$0.801858\pi$
$402$	0	0
$403$	135.235	0.335570
$404$	− 57.0047i	− 0.141101i
$405$	0	0
$406$	0	0
$407$	859.992	2.11300
$408$	0	0
$409$	533.565i	1.30456i	0.757978	+	0.652280i	$0.226187\pi$
$409$	533.565i	1.30456i	−0.757978	+	0.652280i	$0.773813\pi$
$410$	−94.1909	−0.229734
$411$	0	0
$412$	− 112.982i	− 0.274229i
$413$	0	0
$414$	0	0
$415$	341.970	0.824023
$416$	− 31.0749i	− 0.0746994i
$417$	0	0
$418$	13.4361i	0.0321439i
$419$	− 534.252i	− 1.27507i	−0.770423	−	0.637533i	$-0.779955\pi$
$419$	− 534.252i	− 1.27507i	0.770423	−	0.637533i	$-0.220045\pi$
$420$	0	0
$421$	157.220	0.373445	0.186723	−	0.982413i	$-0.440213\pi$
$421$	157.220	0.373445	0.186723	+	0.982413i	$0.440213\pi$
$422$	−181.123	−0.429201
$423$	0	0
$424$	−62.3087	−0.146954
$425$	202.989i	0.477622i
$426$	0	0
$427$	0	0
$428$	−95.2203	−0.222477
$429$	0	0
$430$	29.0457i	0.0675482i
$431$	228.536	0.530246	0.265123	−	0.964215i	$-0.414587\pi$
$431$	228.536	0.530246	0.265123	+	0.964215i	$0.414587\pi$
$432$	0	0
$433$	− 47.5549i	− 0.109827i	−0.998491	−	0.0549133i	$-0.982512\pi$
$433$	− 47.5549i	− 0.109827i	0.998491	−	0.0549133i	$-0.0174882\pi$
$434$	0	0
$435$	0	0
$436$	150.617	0.345453
$437$	1.63008i	0.00373015i
$438$	0	0
$439$	73.8540i	0.168232i	0.996456	+	0.0841161i	$0.0268067\pi$
$439$	73.8540i	0.168232i	−0.996456	+	0.0841161i	$0.973193\pi$
$440$	118.620i	0.269591i
$441$	0	0
$442$	105.338	0.238321
$443$	−234.640	−0.529661	−0.264830	−	0.964295i	$-0.585316\pi$
$443$	−234.640	−0.529661	−0.264830	+	0.964295i	$0.585316\pi$
$444$	0	0
$445$	531.749	1.19494
$446$	− 589.966i	− 1.32279i
$447$	0	0
$448$	0	0
$449$	255.161	0.568288	0.284144	−	0.958782i	$-0.408291\pi$
$449$	255.161	0.568288	0.284144	+	0.958782i	$0.408291\pi$
$450$	0	0
$451$	278.503i	0.617524i
$452$	−170.912	−0.378123
$453$	0	0
$454$	− 328.465i	− 0.723492i
$455$	0	0
$456$	0	0
$457$	−145.735	−0.318895	−0.159448	−	0.987206i	$-0.550971\pi$
$457$	−145.735	−0.318895	−0.159448	+	0.987206i	$0.550971\pi$
$458$	118.259i	0.258208i
$459$	0	0
$460$	14.3910i	0.0312848i
$461$	888.329i	1.92696i	0.267777	+	0.963481i	$0.413711\pi$
$461$	888.329i	1.92696i	−0.267777	+	0.963481i	$0.586289\pi$
$462$	0	0
$463$	234.014	0.505430	0.252715	−	0.967541i	$-0.418676\pi$
$463$	234.014	0.505430	0.252715	+	0.967541i	$0.418676\pi$
$464$	−81.9411	−0.176597
$465$	0	0
$466$	−309.816	−0.664842
$467$	786.618i	1.68441i	0.539159	+	0.842204i	$0.318742\pi$
$467$	786.618i	1.68441i	−0.539159	+	0.842204i	$0.681258\pi$
$468$	0	0
$469$	0	0
$470$	−213.993	−0.455304
$471$	0	0
$472$	236.892i	0.501889i
$473$	85.8823	0.181569
$474$	0	0
$475$	10.7405i	0.0226115i
$476$	0	0
$477$	0	0
$478$	−273.088	−0.571314
$479$	− 736.932i	− 1.53848i	−0.638960	−	0.769240i	$-0.720635\pi$
$479$	− 736.932i	− 1.53848i	0.638960	−	0.769240i	$-0.279365\pi$
$480$	0	0
$481$	− 356.743i	− 0.741669i
$482$	70.0505i	0.145333i
$483$	0	0
$484$	108.735	0.224659
$485$	−81.0152	−0.167042
$486$	0	0
$487$	270.698	0.555849	0.277925	−	0.960603i	$-0.410353\pi$
$487$	270.698	0.555849	0.277925	+	0.960603i	$0.410353\pi$
$488$	− 187.410i	− 0.384036i
$489$	0	0
$490$	0	0
$491$	−760.161	−1.54819	−0.774094	−	0.633070i	$-0.781794\pi$
$491$	−760.161	−1.54819	−0.774094	+	0.633070i	$0.781794\pi$
$492$	0	0
$493$	− 277.765i	− 0.563417i
$494$	5.57359	0.0112826
$495$	0	0
$496$	98.4720i	0.198532i
$497$	0	0
$498$	0	0
$499$	125.492	0.251488	0.125744	−	0.992063i	$-0.459868\pi$
$499$	125.492	0.251488	0.125744	+	0.992063i	$0.459868\pi$
$500$	253.168i	0.506336i
$501$	0	0
$502$	− 229.844i	− 0.457857i
$503$	117.083i	0.232770i	0.993204	+	0.116385i	$0.0371306\pi$
$503$	117.083i	0.232770i	−0.993204	+	0.116385i	$0.962869\pi$
$504$	0	0
$505$	90.2649	0.178742
$506$	42.5513	0.0840935
$507$	0	0
$508$	−121.324	−0.238826
$509$	− 662.925i	− 1.30241i	−0.758903	−	0.651204i	$-0.774264\pi$
$509$	− 662.925i	− 1.30241i	0.758903	−	0.651204i	$-0.225736\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	0.0441942
$513$	0	0
$514$	− 140.233i	− 0.272826i
$515$	178.904	0.347386
$516$	0	0
$517$	632.733i	1.22386i
$518$	0	0
$519$	0	0
$520$	49.2061	0.0946270
$521$	47.1383i	0.0904765i	0.998976	+	0.0452383i	$0.0144047\pi$
$521$	47.1383i	0.0904765i	−0.998976	+	0.0452383i	$0.985595\pi$
$522$	0	0
$523$	− 499.471i	− 0.955011i	−0.878629	−	0.477506i	$-0.841541\pi$
$523$	− 499.471i	− 0.955011i	0.878629	−	0.477506i	$-0.158459\pi$
$524$	− 265.895i	− 0.507434i
$525$	0	0
$526$	−614.257	−1.16779
$527$	−333.801	−0.633399
$528$	0	0
$529$	−523.838	−0.990241
$530$	− 98.6635i	− 0.186158i
$531$	0	0
$532$	0	0
$533$	115.529	0.216752
$534$	0	0
$535$	− 150.778i	− 0.281828i
$536$	262.024	0.488851
$537$	0	0
$538$	129.271i	0.240280i
$539$	0	0
$540$	0	0
$541$	498.809	0.922013	0.461007	−	0.887397i	$-0.347488\pi$
$541$	498.809	0.922013	0.461007	+	0.887397i	$0.347488\pi$
$542$	− 24.1766i	− 0.0446063i
$543$	0	0
$544$	76.7026i	0.140997i
$545$	238.497i	0.437609i
$546$	0	0
$547$	−279.897	−0.511694	−0.255847	−	0.966717i	$-0.582354\pi$
$547$	−279.897	−0.511694	−0.255847	+	0.966717i	$0.582354\pi$
$548$	234.853	0.428564
$549$	0	0
$550$	280.368	0.509759
$551$	− 14.6969i	− 0.0266732i
$552$	0	0
$553$	0	0
$554$	−566.267	−1.02214
$555$	0	0
$556$	137.171i	0.246711i
$557$	261.780	0.469981	0.234991	−	0.971998i	$-0.424494\pi$
$557$	261.780	0.469981	0.234991	+	0.971998i	$0.424494\pi$
$558$	0	0
$559$	− 35.6258i	− 0.0637312i
$560$	0	0
$561$	0	0
$562$	761.823	1.35556
$563$	485.062i	0.861567i	0.902455	+	0.430784i	$0.141763\pi$
$563$	485.062i	0.861567i	−0.902455	+	0.430784i	$0.858237\pi$
$564$	0	0
$565$	− 270.633i	− 0.478996i
$566$	− 437.287i	− 0.772593i
$567$	0	0
$568$	136.971	0.241145
$569$	453.999	0.797890	0.398945	−	0.916975i	$-0.369377\pi$
$569$	453.999	0.797890	0.398945	+	0.916975i	$0.369377\pi$
$570$	0	0
$571$	−231.537	−0.405494	−0.202747	−	0.979231i	$-0.564987\pi$
$571$	−231.537	−0.405494	−0.202747	+	0.979231i	$0.564987\pi$
$572$	− 145.492i	− 0.254357i
$573$	0	0
$574$	0	0
$575$	34.0143	0.0591553
$576$	0	0
$577$	651.267i	1.12871i	0.825531	+	0.564356i	$0.190876\pi$
$577$	651.267i	1.12871i	−0.825531	+	0.564356i	$0.809124\pi$
$578$	148.701	0.257267
$579$	0	0
$580$	− 129.751i	− 0.223708i
$581$	0	0
$582$	0	0
$583$	−291.728	−0.500391
$584$	370.497i	0.634413i
$585$	0	0
$586$	− 463.001i	− 0.790104i
$587$	823.029i	1.40209i	0.713116	+	0.701046i	$0.247283\pi$
$587$	823.029i	1.40209i	−0.713116	+	0.701046i	$0.752717\pi$
$588$	0	0
$589$	−17.6619	−0.0299863
$590$	−375.110	−0.635779
$591$	0	0
$592$	259.765	0.438791
$593$	− 808.418i	− 1.36327i	−0.731694	−	0.681634i	$-0.761270\pi$
$593$	− 808.418i	− 1.36327i	0.731694	−	0.681634i	$-0.238730\pi$
$594$	0	0
$595$	0	0
$596$	52.7939	0.0885804
$597$	0	0
$598$	− 17.6512i	− 0.0295170i
$599$	−530.845	−0.886218	−0.443109	−	0.896468i	$-0.646125\pi$
$599$	−530.845	−0.886218	−0.443109	+	0.896468i	$0.646125\pi$
$600$	0	0
$601$	936.503i	1.55824i	0.626874	+	0.779121i	$0.284334\pi$
$601$	936.503i	1.55824i	−0.626874	+	0.779121i	$0.715666\pi$
$602$	0	0
$603$	0	0
$604$	−268.426	−0.444415
$605$	172.178i	0.284592i
$606$	0	0
$607$	− 602.121i	− 0.991962i	−0.868333	−	0.495981i	$-0.834809\pi$
$607$	− 602.121i	− 0.991962i	0.868333	−	0.495981i	$-0.165191\pi$
$608$	4.05845i	0.00667508i
$609$	0	0
$610$	296.756	0.486486
$611$	262.471	0.429576
$612$	0	0
$613$	1096.90	1.78939	0.894695	−	0.446677i	$-0.147393\pi$
$613$	1096.90	1.78939	0.894695	+	0.446677i	$0.147393\pi$
$614$	362.236i	0.589961i
$615$	0	0
$616$	0	0
$617$	432.956	0.701712	0.350856	−	0.936429i	$-0.385891\pi$
$617$	432.956	0.701712	0.350856	+	0.936429i	$0.385891\pi$
$618$	0	0
$619$	225.110i	0.363668i	0.983329	+	0.181834i	$0.0582034\pi$
$619$	225.110i	0.363668i	−0.983329	+	0.181834i	$0.941797\pi$
$620$	−155.927	−0.251495
$621$	0	0
$622$	− 305.940i	− 0.491865i
$623$	0	0
$624$	0	0
$625$	−26.6182	−0.0425891
$626$	− 221.775i	− 0.354273i
$627$	0	0
$628$	− 453.389i	− 0.721958i
$629$	880.552i	1.39992i
$630$	0	0
$631$	750.514	1.18940	0.594702	−	0.803946i	$-0.297270\pi$
$631$	750.514	1.18940	0.594702	+	0.803946i	$0.297270\pi$
$632$	−215.563	−0.341081
$633$	0	0
$634$	633.609	0.999384
$635$	− 192.112i	− 0.302538i
$636$	0	0
$637$	0	0
$638$	−383.647	−0.601327
$639$	0	0
$640$	35.8297i	0.0559839i
$641$	1161.85	1.81256	0.906281	−	0.422675i	$-0.138909\pi$
$641$	1161.85	1.81256	0.906281	+	0.422675i	$0.138909\pi$
$642$	0	0
$643$	− 121.957i	− 0.189669i	−0.995493	−	0.0948347i	$-0.969768\pi$
$643$	− 121.957i	− 0.189669i	0.995493	−	0.0948347i	$-0.0302322\pi$
$644$	0	0
$645$	0	0
$646$	−13.7574	−0.0212962
$647$	158.775i	0.245403i	0.992444	+	0.122701i	$0.0391557\pi$
$647$	158.775i	0.245403i	−0.992444	+	0.122701i	$0.960844\pi$
$648$	0	0
$649$	1109.12i	1.70897i
$650$	− 116.302i	− 0.178927i
$651$	0	0
$652$	−183.955	−0.282140
$653$	390.941	0.598685	0.299342	−	0.954146i	$-0.403233\pi$
$653$	390.941	0.598685	0.299342	+	0.954146i	$0.403233\pi$
$654$	0	0
$655$	421.036	0.642803
$656$	84.1232i	0.128237i
$657$	0	0
$658$	0	0
$659$	331.955	0.503726	0.251863	−	0.967763i	$-0.418957\pi$
$659$	331.955	0.503726	0.251863	+	0.967763i	$0.418957\pi$
$660$	0	0
$661$	647.820i	0.980061i	0.871705	+	0.490031i	$0.163014\pi$
$661$	647.820i	0.980061i	−0.871705	+	0.490031i	$0.836986\pi$
$662$	−77.7918	−0.117510
$663$	0	0
$664$	− 305.418i	− 0.459967i
$665$	0	0
$666$	0	0
$667$	−46.5442	−0.0697813
$668$	− 406.963i	− 0.609227i
$669$	0	0
$670$	414.906i	0.619263i
$671$	− 877.448i	− 1.30767i
$672$	0	0
$673$	100.956	0.150009	0.0750047	−	0.997183i	$-0.476103\pi$
$673$	100.956	0.150009	0.0750047	+	0.997183i	$0.476103\pi$
$674$	−157.872	−0.234232
$675$	0	0
$676$	277.647	0.410720
$677$	743.177i	1.09775i	0.835905	+	0.548875i	$0.184944\pi$
$677$	743.177i	1.09775i	−0.835905	+	0.548875i	$0.815056\pi$
$678$	0	0
$679$	0	0
$680$	−121.456	−0.178612
$681$	0	0
$682$	461.044i	0.676017i
$683$	−4.43442	−0.00649256	−0.00324628	−	0.999995i	$-0.501033\pi$
$683$	−4.43442	−0.00649256	−0.00324628	+	0.999995i	$0.501033\pi$
$684$	0	0
$685$	371.881i	0.542892i
$686$	0	0
$687$	0	0
$688$	25.9411	0.0377051
$689$	121.015i	0.175638i
$690$	0	0
$691$	977.169i	1.41414i	0.707145	+	0.707069i	$0.249983\pi$
$691$	977.169i	1.41414i	−0.707145	+	0.707069i	$0.750017\pi$
$692$	141.620i	0.204654i
$693$	0	0
$694$	−533.522	−0.768763
$695$	−217.206	−0.312527
$696$	0	0
$697$	−285.161	−0.409127
$698$	− 288.547i	− 0.413392i
$699$	0	0
$700$	0	0
$701$	840.177	1.19854	0.599270	−	0.800547i	$-0.295458\pi$
$701$	840.177	1.19854	0.599270	+	0.800547i	$0.295458\pi$
$702$	0	0
$703$	46.5913i	0.0662750i
$704$	105.941	0.150485
$705$	0	0
$706$	589.835i	0.835460i
$707$	0	0
$708$	0	0
$709$	682.558	0.962705	0.481352	−	0.876527i	$-0.340146\pi$
$709$	682.558	0.962705	0.481352	+	0.876527i	$0.340146\pi$
$710$	216.888i	0.305476i
$711$	0	0
$712$	− 474.913i	− 0.667012i
$713$	55.9340i	0.0784488i
$714$	0	0
$715$	230.382	0.322212
$716$	−217.632	−0.303956
$717$	0	0
$718$	252.905	0.352235
$719$	137.625i	0.191412i	0.995410	+	0.0957060i	$0.0305109\pi$
$719$	137.625i	0.191412i	−0.995410	+	0.0957060i	$0.969489\pi$
$720$	0	0
$721$	0	0
$722$	509.803	0.706099
$723$	0	0
$724$	− 199.321i	− 0.275306i
$725$	−306.676	−0.423002
$726$	0	0
$727$	− 264.137i	− 0.363325i	−0.983361	−	0.181662i	$-0.941852\pi$
$727$	− 264.137i	− 0.363325i	0.983361	−	0.181662i	$-0.0581478\pi$
$728$	0	0
$729$	0	0
$730$	−586.669	−0.803656
$731$	87.9354i	0.120295i
$732$	0	0
$733$	− 579.319i	− 0.790340i	−0.918608	−	0.395170i	$-0.870686\pi$
$733$	− 579.319i	− 0.790340i	0.918608	−	0.395170i	$-0.129314\pi$
$734$	− 889.530i	− 1.21189i
$735$	0	0
$736$	12.8528	0.0174631
$737$	1226.79	1.66458
$738$	0	0
$739$	−198.095	−0.268059	−0.134029	−	0.990977i	$-0.542792\pi$
$739$	−198.095	−0.268059	−0.134029	+	0.990977i	$0.542792\pi$
$740$	411.328i	0.555848i
$741$	0	0
$742$	0	0
$743$	−976.690	−1.31452	−0.657261	−	0.753663i	$-0.728285\pi$
$743$	−976.690	−1.31452	−0.657261	+	0.753663i	$0.728285\pi$
$744$	0	0
$745$	83.5973i	0.112211i
$746$	−361.414	−0.484469
$747$	0	0
$748$	359.120i	0.480107i
$749$	0	0
$750$	0	0
$751$	−835.330	−1.11229	−0.556145	−	0.831085i	$-0.687720\pi$
$751$	−835.330	−1.11229	−0.556145	+	0.831085i	$0.687720\pi$
$752$	191.120i	0.254149i
$753$	0	0
$754$	159.145i	0.211067i
$755$	− 425.044i	− 0.562972i
$756$	0	0
$757$	104.221	0.137677	0.0688383	−	0.997628i	$-0.478071\pi$
$757$	104.221	0.137677	0.0688383	+	0.997628i	$0.478071\pi$
$758$	310.774	0.409992
$759$	0	0
$760$	−6.42641	−0.00845580
$761$	− 547.080i	− 0.718897i	−0.933165	−	0.359448i	$-0.882965\pi$
$761$	− 547.080i	− 0.718897i	0.933165	−	0.359448i	$-0.117035\pi$
$762$	0	0
$763$	0	0
$764$	−139.809	−0.182996
$765$	0	0
$766$	− 24.0921i	− 0.0314519i
$767$	460.087	0.599853
$768$	0	0
$769$	341.205i	0.443700i	0.975081	+	0.221850i	$0.0712095\pi$
$769$	341.205i	0.443700i	−0.975081	+	0.221850i	$0.928790\pi$
$770$	0	0
$771$	0	0
$772$	−64.6762	−0.0837774
$773$	490.993i	0.635179i	0.948228	+	0.317590i	$0.102873\pi$
$773$	490.993i	0.635179i	−0.948228	+	0.317590i	$0.897127\pi$
$774$	0	0
$775$	368.545i	0.475542i
$776$	72.3557i	0.0932419i
$777$	0	0
$778$	215.272	0.276699
$779$	−15.0883	−0.0193688
$780$	0	0
$781$	641.294	0.821118
$782$	43.5686i	0.0557143i
$783$	0	0
$784$	0	0
$785$	717.926	0.914555
$786$	0	0
$787$	300.455i	0.381773i	0.981612	+	0.190887i	$0.0611363\pi$
$787$	300.455i	0.381773i	−0.981612	+	0.190887i	$0.938864\pi$
$788$	−554.205	−0.703306
$789$	0	0
$790$	− 341.337i	− 0.432072i
$791$	0	0
$792$	0	0
$793$	−363.984	−0.458996
$794$	527.108i	0.663864i
$795$	0	0
$796$	334.889i	0.420715i
$797$	− 370.072i	− 0.464331i	−0.972676	−	0.232165i	$-0.925419\pi$
$797$	− 370.072i	− 0.464331i	0.972676	−	0.232165i	$-0.0745811\pi$
$798$	0	0
$799$	−647.860	−0.810838
$800$	84.6863	0.105858
$801$	0	0
$802$	−921.463	−1.14896
$803$	1734.66i	2.16022i
$804$	0	0
$805$	0	0
$806$	191.251	0.237284
$807$	0	0
$808$	− 80.6168i	− 0.0997733i
$809$	−491.235	−0.607213	−0.303607	−	0.952797i	$-0.598191\pi$
$809$	−491.235	−0.607213	−0.303607	+	0.952797i	$0.598191\pi$
$810$	0	0
$811$	− 156.802i	− 0.193344i	−0.995316	−	0.0966722i	$-0.969180\pi$
$811$	− 156.802i	− 0.193344i	0.995316	−	0.0966722i	$-0.0308199\pi$
$812$	0	0
$813$	0	0
$814$	1216.21	1.49412
$815$	− 291.287i	− 0.357407i
$816$	0	0
$817$	4.65279i	0.00569497i
$818$	754.575i	0.922463i
$819$	0	0
$820$	−133.206	−0.162446
$821$	430.632	0.524522	0.262261	−	0.964997i	$-0.415532\pi$
$821$	430.632	0.524522	0.262261	+	0.964997i	$0.415532\pi$
$822$	0	0
$823$	−708.741	−0.861168	−0.430584	−	0.902550i	$-0.641692\pi$
$823$	−708.741	−0.861168	−0.430584	+	0.902550i	$0.641692\pi$
$824$	− 159.781i	− 0.193909i
$825$	0	0
$826$	0	0
$827$	1460.10	1.76554	0.882770	−	0.469805i	$-0.155676\pi$
$827$	1460.10	1.76554	0.882770	+	0.469805i	$0.155676\pi$
$828$	0	0
$829$	− 257.608i	− 0.310745i	−0.987856	−	0.155373i	$-0.950342\pi$
$829$	− 257.608i	− 0.310745i	0.987856	−	0.155373i	$-0.0496578\pi$
$830$	483.618	0.582673
$831$	0	0
$832$	− 43.9466i	− 0.0528204i
$833$	0	0
$834$	0	0
$835$	644.412	0.771751
$836$	19.0016i	0.0227292i
$837$	0	0
$838$	− 755.547i	− 0.901607i
$839$	− 213.621i	− 0.254613i	−0.991863	−	0.127307i	$-0.959367\pi$
$839$	− 213.621i	− 0.254613i	0.991863	−	0.127307i	$-0.0406332\pi$
$840$	0	0
$841$	−421.353	−0.501015
$842$	222.343	0.264065
$843$	0	0
$844$	−256.146	−0.303491
$845$	439.644i	0.520288i
$846$	0	0
$847$	0	0
$848$	−88.1177	−0.103912
$849$	0	0
$850$	287.070i	0.337730i
$851$	147.551	0.173386
$852$	0	0
$853$	1127.37i	1.32165i	0.750539	+	0.660826i	$0.229794\pi$
$853$	1127.37i	1.32165i	−0.750539	+	0.660826i	$0.770206\pi$
$854$	0	0
$855$	0	0
$856$	−134.662	−0.157315
$857$	− 1270.42i	− 1.48241i	−0.671280	−	0.741204i	$-0.734255\pi$
$857$	− 1270.42i	− 1.48241i	0.671280	−	0.741204i	$-0.265745\pi$
$858$	0	0
$859$	255.753i	0.297733i	0.988857	+	0.148867i	$0.0475625\pi$
$859$	255.753i	0.297733i	−0.988857	+	0.148867i	$0.952438\pi$
$860$	41.0768i	0.0477638i
$861$	0	0
$862$	323.199	0.374941
$863$	−1114.73	−1.29169	−0.645844	−	0.763469i	$-0.723494\pi$
$863$	−1114.73	−1.29169	−0.645844	+	0.763469i	$0.723494\pi$
$864$	0	0
$865$	−224.251	−0.259249
$866$	− 67.2528i	− 0.0776591i
$867$	0	0
$868$	0	0
$869$	−1009.26	−1.16141
$870$	0	0
$871$	− 508.900i	− 0.584270i
$872$	213.005	0.244272
$873$	0	0
$874$	2.30528i	0.00263762i
$875$	0	0
$876$	0	0
$877$	−1101.81	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$877$	−1101.81	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$878$	104.445i	0.118958i
$879$	0	0
$880$	167.754i	0.190630i
$881$	217.067i	0.246387i	0.992383	+	0.123194i	$0.0393136\pi$
$881$	217.067i	0.246387i	−0.992383	+	0.123194i	$0.960686\pi$
$882$	0	0
$883$	−516.544	−0.584988	−0.292494	−	0.956267i	$-0.594485\pi$
$883$	−516.544	−0.584988	−0.292494	+	0.956267i	$0.594485\pi$
$884$	148.971	0.168519
$885$	0	0
$886$	−331.831	−0.374527
$887$	− 1129.81i	− 1.27374i	−0.770970	−	0.636872i	$-0.780228\pi$
$887$	− 1129.81i	− 1.27374i	0.770970	−	0.636872i	$-0.219772\pi$
$888$	0	0
$889$	0	0
$890$	752.007	0.844952
$891$	0	0
$892$	− 834.339i	− 0.935357i
$893$	−34.2792	−0.0383866
$894$	0	0
$895$	− 344.613i	− 0.385043i
$896$	0	0
$897$	0	0
$898$	360.853	0.401841
$899$	− 504.306i	− 0.560964i
$900$	0	0
$901$	− 298.702i	− 0.331523i
$902$	393.863i	0.436655i
$903$	0	0
$904$	−241.706	−0.267373
$905$	315.618	0.348749
$906$	0	0
$907$	60.0223	0.0661767	0.0330884	−	0.999452i	$-0.489466\pi$
$907$	60.0223	0.0661767	0.0330884	+	0.999452i	$0.489466\pi$
$908$	− 464.520i	− 0.511586i
$909$	0	0
$910$	0	0
$911$	−1422.25	−1.56120	−0.780598	−	0.625033i	$-0.785085\pi$
$911$	−1422.25	−1.56120	−0.780598	+	0.625033i	$0.785085\pi$
$912$	0	0
$913$	− 1429.96i	− 1.56622i
$914$	−206.101	−0.225493
$915$	0	0
$916$	167.244i	0.182581i
$917$	0	0
$918$	0	0
$919$	1669.70	1.81686	0.908432	−	0.418033i	$-0.137280\pi$
$919$	1669.70	1.81686	0.908432	+	0.418033i	$0.137280\pi$
$920$	20.3520i	0.0221217i
$921$	0	0
$922$	1256.29i	1.36257i
$923$	− 266.022i	− 0.288215i
$924$	0	0
$925$	972.205	1.05103
$926$	330.946	0.357393
$927$	0	0
$928$	−115.882	−0.124873
$929$	968.860i	1.04291i	0.853280	+	0.521453i	$0.174610\pi$
$929$	968.860i	1.04291i	−0.853280	+	0.521453i	$0.825390\pi$
$930$	0	0
$931$	0	0
$932$	−438.146	−0.470114
$933$	0	0
$934$	1112.45i	1.19106i
$935$	−568.654	−0.608186
$936$	0	0
$937$	− 1212.57i	− 1.29410i	−0.762449	−	0.647049i	$-0.776003\pi$
$937$	− 1212.57i	− 1.29410i	0.762449	−	0.647049i	$-0.223997\pi$
$938$	0	0
$939$	0	0
$940$	−302.632	−0.321949
$941$	1494.06i	1.58774i	0.608087	+	0.793870i	$0.291937\pi$
$941$	1494.06i	1.58774i	−0.608087	+	0.793870i	$0.708063\pi$
$942$	0	0
$943$	47.7836i	0.0506719i
$944$	335.016i	0.354889i
$945$	0	0
$946$	121.456	0.128389
$947$	−775.462	−0.818862	−0.409431	−	0.912341i	$-0.634273\pi$
$947$	−775.462	−0.818862	−0.409431	+	0.912341i	$0.634273\pi$
$948$	0	0
$949$	719.574	0.758244
$950$	15.1893i	0.0159887i
$951$	0	0
$952$	0	0
$953$	−1055.40	−1.10745	−0.553723	−	0.832701i	$-0.686794\pi$
$953$	−1055.40	−1.10745	−0.553723	+	0.832701i	$0.686794\pi$
$954$	0	0
$955$	− 221.383i	− 0.231814i
$956$	−386.205	−0.403980
$957$	0	0
$958$	− 1042.18i	− 1.08787i
$959$	0	0
$960$	0	0
$961$	354.955	0.369360
$962$	− 504.510i	− 0.524439i
$963$	0	0
$964$	99.0663i	0.102766i
$965$	− 102.412i	− 0.106127i
$966$	0	0
$967$	1221.63	1.26332	0.631661	−	0.775245i	$-0.282373\pi$
$967$	1221.63	1.26332	0.631661	+	0.775245i	$0.282373\pi$
$968$	153.775	0.158858
$969$	0	0
$970$	−114.573	−0.118116
$971$	− 526.259i	− 0.541976i	−0.962583	−	0.270988i	$-0.912650\pi$
$971$	− 526.259i	− 0.541976i	0.962583	−	0.270988i	$-0.0873504\pi$
$972$	0	0
$973$	0	0
$974$	382.825	0.393045
$975$	0	0
$976$	− 265.037i	− 0.271555i
$977$	1000.10	1.02365	0.511823	−	0.859091i	$-0.328970\pi$
$977$	1000.10	1.02365	0.511823	+	0.859091i	$0.328970\pi$
$978$	0	0
$979$	− 2223.53i	− 2.27123i
$980$	0	0
$981$	0	0
$982$	−1075.03	−1.09473
$983$	− 1075.70i	− 1.09430i	−0.837033	−	0.547152i	$-0.815712\pi$
$983$	− 1075.70i	− 1.09430i	0.837033	−	0.547152i	$-0.184288\pi$
$984$	0	0
$985$	− 877.564i	− 0.890928i
$986$	− 392.819i	− 0.398396i
$987$	0	0
$988$	7.88225	0.00797799
$989$	14.7351	0.0148990
$990$	0	0
$991$	−1876.03	−1.89307	−0.946536	−	0.322597i	$-0.895444\pi$
$991$	−1876.03	−1.89307	−0.946536	+	0.322597i	$0.895444\pi$
$992$	139.260i	0.140383i
$993$	0	0
$994$	0	0
$995$	−530.285	−0.532949
$996$	0	0
$997$	582.224i	0.583976i	0.956422	+	0.291988i	$0.0943167\pi$
$997$	582.224i	0.583976i	−0.956422	+	0.291988i	$0.905683\pi$
$998$	177.473	0.177829
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.f.685.4		4
3.2	odd	2		98.3.b.b.97.1		4
7.2	even	3		882.3.n.b.325.1		4
7.3	odd	6		882.3.n.b.19.1		4
7.4	even	3		126.3.n.c.19.1		4
7.5	odd	6		126.3.n.c.73.1		4
7.6	odd	2	inner	882.3.c.f.685.3		4
12.11	even	2		784.3.c.e.97.4		4
21.2	odd	6		98.3.d.a.31.2		4
21.5	even	6		14.3.d.a.3.2	✓	4
21.11	odd	6		14.3.d.a.5.2	yes	4
21.17	even	6		98.3.d.a.19.2		4
21.20	even	2		98.3.b.b.97.2		4
28.11	odd	6		1008.3.cg.l.145.1		4
28.19	even	6		1008.3.cg.l.577.1		4
84.11	even	6		112.3.s.b.33.2		4
84.23	even	6		784.3.s.c.129.1		4
84.47	odd	6		112.3.s.b.17.2		4
84.59	odd	6		784.3.s.c.705.1		4
84.83	odd	2		784.3.c.e.97.1		4
105.32	even	12		350.3.i.a.299.4		8
105.47	odd	12		350.3.i.a.199.1		8
105.53	even	12		350.3.i.a.299.1		8
105.68	odd	12		350.3.i.a.199.4		8
105.74	odd	6		350.3.k.a.201.1		4
105.89	even	6		350.3.k.a.101.1		4
168.5	even	6		448.3.s.d.129.2		4
168.11	even	6		448.3.s.c.257.1		4
168.53	odd	6		448.3.s.d.257.2		4
168.131	odd	6		448.3.s.c.129.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.3.d.a.3.2	✓	4	21.5	even	6
14.3.d.a.5.2	yes	4	21.11	odd	6
98.3.b.b.97.1		4	3.2	odd	2
98.3.b.b.97.2		4	21.20	even	2
98.3.d.a.19.2		4	21.17	even	6
98.3.d.a.31.2		4	21.2	odd	6
112.3.s.b.17.2		4	84.47	odd	6
112.3.s.b.33.2		4	84.11	even	6
126.3.n.c.19.1		4	7.4	even	3
126.3.n.c.73.1		4	7.5	odd	6
350.3.i.a.199.1		8	105.47	odd	12
350.3.i.a.199.4		8	105.68	odd	12
350.3.i.a.299.1		8	105.53	even	12
350.3.i.a.299.4		8	105.32	even	12
350.3.k.a.101.1		4	105.89	even	6
350.3.k.a.201.1		4	105.74	odd	6
448.3.s.c.129.1		4	168.131	odd	6
448.3.s.c.257.1		4	168.11	even	6
448.3.s.d.129.2		4	168.5	even	6
448.3.s.d.257.2		4	168.53	odd	6
784.3.c.e.97.1		4	84.83	odd	2
784.3.c.e.97.4		4	12.11	even	2
784.3.s.c.129.1		4	84.23	even	6
784.3.s.c.705.1		4	84.59	odd	6
882.3.c.f.685.3		4	7.6	odd	2	inner
882.3.c.f.685.4		4	1.1	even	1	trivial
882.3.n.b.19.1		4	7.3	odd	6
882.3.n.b.325.1		4	7.2	even	3
1008.3.cg.l.145.1		4	28.11	odd	6
1008.3.cg.l.577.1		4	28.19	even	6

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +3.16693i q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +3.16693i q^{5} +2.82843 q^{8} +4.47871i q^{10} +13.2426 q^{11} -5.49333i q^{13} +4.00000 q^{16} +13.5592i q^{17} +0.717439i q^{19} +6.33386i q^{20} +18.7279 q^{22} +2.27208 q^{23} +14.9706 q^{25} -7.76874i q^{26} -20.4853 q^{29} +24.6180i q^{31} +5.65685 q^{32} +19.1757i q^{34} +64.9411 q^{37} +1.01461i q^{38} +8.95743i q^{40} +21.0308i q^{41} +6.48528 q^{43} +26.4853 q^{44} +3.21320 q^{46} +47.7800i q^{47} +21.1716 q^{50} -10.9867i q^{52} -22.0294 q^{53} +41.9385i q^{55} -28.9706 q^{58} +83.7539i q^{59} -66.2593i q^{61} +34.8151i q^{62} +8.00000 q^{64} +17.3970 q^{65} +92.6396 q^{67} +27.1185i q^{68} +48.4264 q^{71} +130.991i q^{73} +91.8406 q^{74} +1.43488i q^{76} -76.2132 q^{79} +12.6677i q^{80} +29.7420i q^{82} -107.981i q^{83} -42.9411 q^{85} +9.17157 q^{86} +37.4558 q^{88} -167.907i q^{89} +4.54416 q^{92} +67.5711i q^{94} -2.27208 q^{95} +25.5816i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 36 q^{11} + 16 q^{16} + 24 q^{22} + 60 q^{23} - 8 q^{25} - 48 q^{29} + 124 q^{37} - 8 q^{43} + 72 q^{44} - 72 q^{46} + 96 q^{50} - 156 q^{53} - 48 q^{58} + 32 q^{64} - 168 q^{65} + 116 q^{67} + 24 q^{71} + 192 q^{74} - 220 q^{79} - 36 q^{85} + 48 q^{86} + 48 q^{88} + 120 q^{92} - 60 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 + 36 * q^11 + 16 * q^16 + 24 * q^22 + 60 * q^23 - 8 * q^25 - 48 * q^29 + 124 * q^37 - 8 * q^43 + 72 * q^44 - 72 * q^46 + 96 * q^50 - 156 * q^53 - 48 * q^58 + 32 * q^64 - 168 * q^65 + 116 * q^67 + 24 * q^71 + 192 * q^74 - 220 * q^79 - 36 * q^85 + 48 * q^86 + 48 * q^88 + 120 * q^92 - 60 * q^95

Introduction

L-functions

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