LMFDB - Embedded newform 495.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} +1.00000 q^{11} -1.46410 q^{13} -3.46410 q^{14} -5.00000 q^{16} -1.46410 q^{19} +1.00000 q^{20} -1.73205 q^{22} +6.92820 q^{23} +1.00000 q^{25} +2.53590 q^{26} +2.00000 q^{28} -3.46410 q^{29} +2.92820 q^{31} +5.19615 q^{32} +2.00000 q^{35} +8.92820 q^{37} +2.53590 q^{38} +1.73205 q^{40} +3.46410 q^{41} +8.92820 q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -1.46410 q^{52} +12.9282 q^{53} +1.00000 q^{55} +3.46410 q^{56} +6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} -5.07180 q^{62} +1.00000 q^{64} -1.46410 q^{65} +8.00000 q^{67} -3.46410 q^{70} +13.8564 q^{71} +12.3923 q^{73} -15.4641 q^{74} -1.46410 q^{76} +2.00000 q^{77} -13.4641 q^{79} -5.00000 q^{80} -6.00000 q^{82} -15.4641 q^{83} -15.4641 q^{86} +1.73205 q^{88} +12.9282 q^{89} -2.92820 q^{91} +6.92820 q^{92} +12.0000 q^{94} -1.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7	$ 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} - 10 q^{16} + 4 q^{19} + 2 q^{20} + 2 q^{25} + 12 q^{26} + 4 q^{28} - 8 q^{31} + 4 q^{35} + 4 q^{37} + 12 q^{38} + 4 q^{43} + 2 q^{44} - 24 q^{46} - 6 q^{49} + 4 q^{52} + 12 q^{53} + 2 q^{55} + 12 q^{58} + 4 q^{61} - 24 q^{62} + 2 q^{64} + 4 q^{65} + 16 q^{67} + 4 q^{73} - 24 q^{74} + 4 q^{76} + 4 q^{77} - 20 q^{79} - 10 q^{80} - 12 q^{82} - 24 q^{83} - 24 q^{86} + 12 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^11 + 4 * q^13 - 10 * q^16 + 4 * q^19 + 2 * q^20 + 2 * q^25 + 12 * q^26 + 4 * q^28 - 8 * q^31 + 4 * q^35 + 4 * q^37 + 12 * q^38 + 4 * q^43 + 2 * q^44 - 24 * q^46 - 6 * q^49 + 4 * q^52 + 12 * q^53 + 2 * q^55 + 12 * q^58 + 4 * q^61 - 24 * q^62 + 2 * q^64 + 4 * q^65 + 16 * q^67 + 4 * q^73 - 24 * q^74 + 4 * q^76 + 4 * q^77 - 20 * q^79 - 10 * q^80 - 12 * q^82 - 24 * q^83 - 24 * q^86 + 12 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	−1.73205	−0.547723
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.46410	−0.335888	−0.167944	−	0.985797i	$-0.553713\pi$
$19$	−1.46410	−0.335888	−0.167944	+	0.985797i	$0.553713\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.73205	−0.369274
$23$	6.92820	1.44463	0.722315	−	0.691564i	$-0.243078\pi$
$23$	6.92820	1.44463	0.722315	+	0.691564i	$0.243078\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.53590	0.497331
$27$	0	0
$28$	2.00000	0.377964
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	2.53590	0.411377
$39$	0	0
$40$	1.73205	0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−12.0000	−1.76930
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.73205	−0.244949
$51$	0	0
$52$	−1.46410	−0.203034
$53$	12.9282	1.77583	0.887913	−	0.460012i	$-0.152155\pi$
$53$	12.9282	1.77583	0.887913	+	0.460012i	$0.152155\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	3.46410	0.462910
$57$	0	0
$58$	6.00000	0.787839
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−5.07180	−0.644119
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.46410	−0.181599
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	−3.46410	−0.414039
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	0	0
$73$	12.3923	1.45041	0.725205	−	0.688533i	$-0.241745\pi$
$73$	12.3923	1.45041	0.725205	+	0.688533i	$0.241745\pi$
$74$	−15.4641	−1.79767
$75$	0	0
$76$	−1.46410	−0.167944
$77$	2.00000	0.227921
$78$	0	0
$79$	−13.4641	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$79$	−13.4641	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$80$	−5.00000	−0.559017
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−15.4641	−1.69741	−0.848703	−	0.528870i	$-0.822616\pi$
$83$	−15.4641	−1.69741	−0.848703	+	0.528870i	$0.822616\pi$
$84$	0	0
$85$	0	0
$86$	−15.4641	−1.66754
$87$	0	0
$88$	1.73205	0.184637
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	−2.92820	−0.306959
$92$	6.92820	0.722315
$93$	0	0
$94$	12.0000	1.23771
$95$	−1.46410	−0.150214
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	5.19615	0.524891
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.53590	−0.248665
$105$	0	0
$106$	−22.3923	−2.17493
$107$	−15.4641	−1.49497	−0.747486	−	0.664278i	$-0.768739\pi$
$107$	−15.4641	−1.49497	−0.747486	+	0.664278i	$0.768739\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.73205	−0.165145
$111$	0	0
$112$	−10.0000	−0.944911
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	6.92820	0.646058
$116$	−3.46410	−0.321634
$117$	0	0
$118$	12.0000	1.10469
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.46410	−0.313625
$123$	0	0
$124$	2.92820	0.262960
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.92820	−0.437307	−0.218654	−	0.975803i	$-0.570166\pi$
$127$	−4.92820	−0.437307	−0.218654	+	0.975803i	$0.570166\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	2.53590	0.222413
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	−2.92820	−0.253907
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−24.0000	−2.01404
$143$	−1.46410	−0.122434
$144$	0	0
$145$	−3.46410	−0.287678
$146$	−21.4641	−1.77638
$147$	0	0
$148$	8.92820	0.733894
$149$	8.53590	0.699288	0.349644	−	0.936883i	$-0.386303\pi$
$149$	8.53590	0.699288	0.349644	+	0.936883i	$0.386303\pi$
$150$	0	0
$151$	0.392305	0.0319253	0.0159627	−	0.999873i	$-0.494919\pi$
$151$	0.392305	0.0319253	0.0159627	+	0.999873i	$0.494919\pi$
$152$	−2.53590	−0.205689
$153$	0	0
$154$	−3.46410	−0.279145
$155$	2.92820	0.235199
$156$	0	0
$157$	−16.9282	−1.35102	−0.675509	−	0.737352i	$-0.736076\pi$
$157$	−16.9282	−1.35102	−0.675509	+	0.737352i	$0.736076\pi$
$158$	23.3205	1.85528
$159$	0	0
$160$	5.19615	0.410792
$161$	13.8564	1.09204
$162$	0	0
$163$	−17.8564	−1.39862	−0.699311	−	0.714818i	$-0.746510\pi$
$163$	−17.8564	−1.39862	−0.699311	+	0.714818i	$0.746510\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	8.92820	0.680769
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−5.00000	−0.376889
$177$	0	0
$178$	−22.3923	−1.67837
$179$	6.92820	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	5.07180	0.375947
$183$	0	0
$184$	12.0000	0.884652
$185$	8.92820	0.656415
$186$	0	0
$187$	0	0
$188$	−6.92820	−0.505291
$189$	0	0
$190$	2.53590	0.183973
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	3.60770	0.259688	0.129844	−	0.991534i	$-0.458552\pi$
$193$	3.60770	0.259688	0.129844	+	0.991534i	$0.458552\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	16.7846	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$200$	1.73205	0.122474
$201$	0	0
$202$	18.0000	1.26648
$203$	−6.92820	−0.486265
$204$	0	0
$205$	3.46410	0.241943
$206$	−13.8564	−0.965422
$207$	0	0
$208$	7.32051	0.507586
$209$	−1.46410	−0.101274
$210$	0	0
$211$	12.3923	0.853121	0.426561	−	0.904459i	$-0.359725\pi$
$211$	12.3923	0.853121	0.426561	+	0.904459i	$0.359725\pi$
$212$	12.9282	0.887913
$213$	0	0
$214$	26.7846	1.83096
$215$	8.92820	0.608898
$216$	0	0
$217$	5.85641	0.397559
$218$	17.3205	1.17309
$219$	0	0
$220$	1.00000	0.0674200
$221$	0	0
$222$	0	0
$223$	−17.8564	−1.19575	−0.597877	−	0.801588i	$-0.703989\pi$
$223$	−17.8564	−1.19575	−0.597877	+	0.801588i	$0.703989\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	1.60770	0.106942
$227$	−8.53590	−0.566547	−0.283274	−	0.959039i	$-0.591420\pi$
$227$	−8.53590	−0.566547	−0.283274	+	0.959039i	$0.591420\pi$
$228$	0	0
$229$	3.85641	0.254839	0.127419	−	0.991849i	$-0.459331\pi$
$229$	3.85641	0.254839	0.127419	+	0.991849i	$0.459331\pi$
$230$	−12.0000	−0.791257
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	−6.92820	−0.451946
$236$	−6.92820	−0.450988
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	27.8564	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$241$	27.8564	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$242$	−1.73205	−0.111340
$243$	0	0
$244$	2.00000	0.128037
$245$	−3.00000	−0.191663
$246$	0	0
$247$	2.14359	0.136394
$248$	5.07180	0.322059
$249$	0	0
$250$	−1.73205	−0.109545
$251$	−25.8564	−1.63204	−0.816021	−	0.578022i	$-0.803825\pi$
$251$	−25.8564	−1.63204	−0.816021	+	0.578022i	$0.803825\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	8.53590	0.535590
$255$	0	0
$256$	19.0000	1.18750
$257$	−7.85641	−0.490069	−0.245035	−	0.969514i	$-0.578799\pi$
$257$	−7.85641	−0.490069	−0.245035	+	0.969514i	$0.578799\pi$
$258$	0	0
$259$	17.8564	1.10954
$260$	−1.46410	−0.0907997
$261$	0	0
$262$	8.78461	0.542715
$263$	−27.4641	−1.69351	−0.846755	−	0.531984i	$-0.821447\pi$
$263$	−27.4641	−1.69351	−0.846755	+	0.531984i	$0.821447\pi$
$264$	0	0
$265$	12.9282	0.794173
$266$	5.07180	0.310972
$267$	0	0
$268$	8.00000	0.488678
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	0	0
$271$	−32.3923	−1.96769	−0.983846	−	0.179016i	$-0.942709\pi$
$271$	−32.3923	−1.96769	−0.983846	+	0.179016i	$0.942709\pi$
$272$	0	0
$273$	0	0
$274$	−31.1769	−1.88347
$275$	1.00000	0.0603023
$276$	0	0
$277$	22.5359	1.35405	0.677025	−	0.735960i	$-0.263269\pi$
$277$	22.5359	1.35405	0.677025	+	0.735960i	$0.263269\pi$
$278$	14.5359	0.871805
$279$	0	0
$280$	3.46410	0.207020
$281$	3.46410	0.206651	0.103325	−	0.994648i	$-0.467052\pi$
$281$	3.46410	0.206651	0.103325	+	0.994648i	$0.467052\pi$
$282$	0	0
$283$	8.92820	0.530727	0.265363	−	0.964148i	$-0.414508\pi$
$283$	8.92820	0.530727	0.265363	+	0.964148i	$0.414508\pi$
$284$	13.8564	0.822226
$285$	0	0
$286$	2.53590	0.149951
$287$	6.92820	0.408959
$288$	0	0
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	0	0
$292$	12.3923	0.725205
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	−6.92820	−0.403376
$296$	15.4641	0.898833
$297$	0	0
$298$	−14.7846	−0.856449
$299$	−10.1436	−0.586619
$300$	0	0
$301$	17.8564	1.02923
$302$	−0.679492	−0.0391004
$303$	0	0
$304$	7.32051	0.419860
$305$	2.00000	0.114520
$306$	0	0
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	−5.07180	−0.288059
$311$	−18.9282	−1.07332	−0.536660	−	0.843799i	$-0.680314\pi$
$311$	−18.9282	−1.07332	−0.536660	+	0.843799i	$0.680314\pi$
$312$	0	0
$313$	7.07180	0.399722	0.199861	−	0.979824i	$-0.435951\pi$
$313$	7.07180	0.399722	0.199861	+	0.979824i	$0.435951\pi$
$314$	29.3205	1.65465
$315$	0	0
$316$	−13.4641	−0.757415
$317$	11.0718	0.621854	0.310927	−	0.950434i	$-0.399361\pi$
$317$	11.0718	0.621854	0.310927	+	0.950434i	$0.399361\pi$
$318$	0	0
$319$	−3.46410	−0.193952
$320$	1.00000	0.0559017
$321$	0	0
$322$	−24.0000	−1.33747
$323$	0	0
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	30.9282	1.71295
$327$	0	0
$328$	6.00000	0.331295
$329$	−13.8564	−0.763928
$330$	0	0
$331$	−17.8564	−0.981477	−0.490738	−	0.871307i	$-0.663273\pi$
$331$	−17.8564	−0.981477	−0.490738	+	0.871307i	$0.663273\pi$
$332$	−15.4641	−0.848703
$333$	0	0
$334$	18.0000	0.984916
$335$	8.00000	0.437087
$336$	0	0
$337$	−29.1769	−1.58937	−0.794684	−	0.607023i	$-0.792363\pi$
$337$	−29.1769	−1.58937	−0.794684	+	0.607023i	$0.792363\pi$
$338$	18.8038	1.02279
$339$	0	0
$340$	0	0
$341$	2.92820	0.158571
$342$	0	0
$343$	−20.0000	−1.07990
$344$	15.4641	0.833768
$345$	0	0
$346$	−20.7846	−1.11739
$347$	−1.60770	−0.0863056	−0.0431528	−	0.999068i	$-0.513740\pi$
$347$	−1.60770	−0.0863056	−0.0431528	+	0.999068i	$0.513740\pi$
$348$	0	0
$349$	−35.8564	−1.91935	−0.959675	−	0.281113i	$-0.909296\pi$
$349$	−35.8564	−1.91935	−0.959675	+	0.281113i	$0.909296\pi$
$350$	−3.46410	−0.185164
$351$	0	0
$352$	5.19615	0.276956
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	0	0
$355$	13.8564	0.735422
$356$	12.9282	0.685193
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−20.7846	−1.09697	−0.548485	−	0.836160i	$-0.684795\pi$
$359$	−20.7846	−1.09697	−0.548485	+	0.836160i	$0.684795\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	20.5359	1.07934
$363$	0	0
$364$	−2.92820	−0.153480
$365$	12.3923	0.648643
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	−34.6410	−1.80579
$369$	0	0
$370$	−15.4641	−0.803940
$371$	25.8564	1.34240
$372$	0	0
$373$	0.392305	0.0203128	0.0101564	−	0.999948i	$-0.496767\pi$
$373$	0.392305	0.0203128	0.0101564	+	0.999948i	$0.496767\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	5.07180	0.261211
$378$	0	0
$379$	9.85641	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$379$	9.85641	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$380$	−1.46410	−0.0751068
$381$	0	0
$382$	−8.78461	−0.449460
$383$	−13.8564	−0.708029	−0.354015	−	0.935240i	$-0.615184\pi$
$383$	−13.8564	−0.708029	−0.354015	+	0.935240i	$0.615184\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	−6.24871	−0.318051
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−24.9282	−1.26391	−0.631955	−	0.775005i	$-0.717747\pi$
$389$	−24.9282	−1.26391	−0.631955	+	0.775005i	$0.717747\pi$
$390$	0	0
$391$	0	0
$392$	−5.19615	−0.262445
$393$	0	0
$394$	−20.7846	−1.04711
$395$	−13.4641	−0.677452
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−29.0718	−1.45724
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−19.8564	−0.991582	−0.495791	−	0.868442i	$-0.665122\pi$
$401$	−19.8564	−0.991582	−0.495791	+	0.868442i	$0.665122\pi$
$402$	0	0
$403$	−4.28719	−0.213560
$404$	−10.3923	−0.517036
$405$	0	0
$406$	12.0000	0.595550
$407$	8.92820	0.442555
$408$	0	0
$409$	34.7846	1.71999	0.859994	−	0.510304i	$-0.170467\pi$
$409$	34.7846	1.71999	0.859994	+	0.510304i	$0.170467\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	8.00000	0.394132
$413$	−13.8564	−0.681829
$414$	0	0
$415$	−15.4641	−0.759103
$416$	−7.60770	−0.372998
$417$	0	0
$418$	2.53590	0.124035
$419$	−17.0718	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$419$	−17.0718	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−21.4641	−1.04486
$423$	0	0
$424$	22.3923	1.08747
$425$	0	0
$426$	0	0
$427$	4.00000	0.193574
$428$	−15.4641	−0.747486
$429$	0	0
$430$	−15.4641	−0.745745
$431$	32.7846	1.57918	0.789590	−	0.613635i	$-0.210294\pi$
$431$	32.7846	1.57918	0.789590	+	0.613635i	$0.210294\pi$
$432$	0	0
$433$	27.8564	1.33869	0.669347	−	0.742950i	$-0.266574\pi$
$433$	27.8564	1.33869	0.669347	+	0.742950i	$0.266574\pi$
$434$	−10.1436	−0.486908
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−10.1436	−0.485234
$438$	0	0
$439$	−29.1769	−1.39254	−0.696269	−	0.717781i	$-0.745158\pi$
$439$	−29.1769	−1.39254	−0.696269	+	0.717781i	$0.745158\pi$
$440$	1.73205	0.0825723
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	30.9282	1.46449
$447$	0	0
$448$	2.00000	0.0944911
$449$	−14.7846	−0.697729	−0.348864	−	0.937173i	$-0.613433\pi$
$449$	−14.7846	−0.697729	−0.348864	+	0.937173i	$0.613433\pi$
$450$	0	0
$451$	3.46410	0.163118
$452$	−0.928203	−0.0436590
$453$	0	0
$454$	14.7846	0.693876
$455$	−2.92820	−0.137276
$456$	0	0
$457$	−8.39230	−0.392575	−0.196288	−	0.980546i	$-0.562889\pi$
$457$	−8.39230	−0.392575	−0.196288	+	0.980546i	$0.562889\pi$
$458$	−6.67949	−0.312112
$459$	0	0
$460$	6.92820	0.323029
$461$	12.2487	0.570479	0.285240	−	0.958456i	$-0.407927\pi$
$461$	12.2487	0.570479	0.285240	+	0.958456i	$0.407927\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	17.3205	0.804084
$465$	0	0
$466$	20.7846	0.962828
$467$	−18.9282	−0.875893	−0.437946	−	0.899001i	$-0.644294\pi$
$467$	−18.9282	−0.875893	−0.437946	+	0.899001i	$0.644294\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	12.0000	0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	8.92820	0.410519
$474$	0	0
$475$	−1.46410	−0.0671776
$476$	0	0
$477$	0	0
$478$	−20.7846	−0.950666
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−13.0718	−0.596023
$482$	−48.2487	−2.19767
$483$	0	0
$484$	1.00000	0.0454545
$485$	−10.0000	−0.454077
$486$	0	0
$487$	23.7128	1.07453	0.537265	−	0.843413i	$-0.319457\pi$
$487$	23.7128	1.07453	0.537265	+	0.843413i	$0.319457\pi$
$488$	3.46410	0.156813
$489$	0	0
$490$	5.19615	0.234738
$491$	17.0718	0.770439	0.385220	−	0.922825i	$-0.374126\pi$
$491$	17.0718	0.770439	0.385220	+	0.922825i	$0.374126\pi$
$492$	0	0
$493$	0	0
$494$	−3.71281	−0.167047
$495$	0	0
$496$	−14.6410	−0.657401
$497$	27.7128	1.24309
$498$	0	0
$499$	−12.7846	−0.572318	−0.286159	−	0.958182i	$-0.592379\pi$
$499$	−12.7846	−0.572318	−0.286159	+	0.958182i	$0.592379\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	44.7846	1.99883
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	−10.3923	−0.462451
$506$	−12.0000	−0.533465
$507$	0	0
$508$	−4.92820	−0.218654
$509$	7.85641	0.348229	0.174115	−	0.984725i	$-0.444294\pi$
$509$	7.85641	0.348229	0.174115	+	0.984725i	$0.444294\pi$
$510$	0	0
$511$	24.7846	1.09641
$512$	−8.66025	−0.382733
$513$	0	0
$514$	13.6077	0.600210
$515$	8.00000	0.352522
$516$	0	0
$517$	−6.92820	−0.304702
$518$	−30.9282	−1.35891
$519$	0	0
$520$	−2.53590	−0.111207
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	−5.07180	−0.221562
$525$	0	0
$526$	47.5692	2.07412
$527$	0	0
$528$	0	0
$529$	25.0000	1.08696
$530$	−22.3923	−0.972660
$531$	0	0
$532$	−2.92820	−0.126954
$533$	−5.07180	−0.219684
$534$	0	0
$535$	−15.4641	−0.668571
$536$	13.8564	0.598506
$537$	0	0
$538$	−13.6077	−0.586669
$539$	−3.00000	−0.129219
$540$	0	0
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	56.1051	2.40992
$543$	0	0
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	−1.73205	−0.0738549
$551$	5.07180	0.216066
$552$	0	0
$553$	−26.9282	−1.14510
$554$	−39.0333	−1.65837
$555$	0	0
$556$	−8.39230	−0.355913
$557$	−44.7846	−1.89758	−0.948792	−	0.315900i	$-0.897694\pi$
$557$	−44.7846	−1.89758	−0.948792	+	0.315900i	$0.897694\pi$
$558$	0	0
$559$	−13.0718	−0.552878
$560$	−10.0000	−0.422577
$561$	0	0
$562$	−6.00000	−0.253095
$563$	10.3923	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$564$	0	0
$565$	−0.928203	−0.0390498
$566$	−15.4641	−0.650005
$567$	0	0
$568$	24.0000	1.00702
$569$	29.3205	1.22918	0.614590	−	0.788847i	$-0.289322\pi$
$569$	29.3205	1.22918	0.614590	+	0.788847i	$0.289322\pi$
$570$	0	0
$571$	24.3923	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$571$	24.3923	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$572$	−1.46410	−0.0612172
$573$	0	0
$574$	−12.0000	−0.500870
$575$	6.92820	0.288926
$576$	0	0
$577$	22.7846	0.948536	0.474268	−	0.880381i	$-0.342713\pi$
$577$	22.7846	0.948536	0.474268	+	0.880381i	$0.342713\pi$
$578$	29.4449	1.22474
$579$	0	0
$580$	−3.46410	−0.143839
$581$	−30.9282	−1.28312
$582$	0	0
$583$	12.9282	0.535431
$584$	21.4641	0.888191
$585$	0	0
$586$	24.0000	0.991431
$587$	5.07180	0.209335	0.104668	−	0.994507i	$-0.466622\pi$
$587$	5.07180	0.209335	0.104668	+	0.994507i	$0.466622\pi$
$588$	0	0
$589$	−4.28719	−0.176650
$590$	12.0000	0.494032
$591$	0	0
$592$	−44.6410	−1.83473
$593$	32.7846	1.34630	0.673151	−	0.739505i	$-0.264940\pi$
$593$	32.7846	1.34630	0.673151	+	0.739505i	$0.264940\pi$
$594$	0	0
$595$	0	0
$596$	8.53590	0.349644
$597$	0	0
$598$	17.5692	0.718459
$599$	10.1436	0.414456	0.207228	−	0.978293i	$-0.433556\pi$
$599$	10.1436	0.414456	0.207228	+	0.978293i	$0.433556\pi$
$600$	0	0
$601$	36.6410	1.49462	0.747309	−	0.664477i	$-0.231345\pi$
$601$	36.6410	1.49462	0.747309	+	0.664477i	$0.231345\pi$
$602$	−30.9282	−1.26054
$603$	0	0
$604$	0.392305	0.0159627
$605$	1.00000	0.0406558
$606$	0	0
$607$	22.7846	0.924799	0.462399	−	0.886672i	$-0.346989\pi$
$607$	22.7846	0.924799	0.462399	+	0.886672i	$0.346989\pi$
$608$	−7.60770	−0.308533
$609$	0	0
$610$	−3.46410	−0.140257
$611$	10.1436	0.410366
$612$	0	0
$613$	0.392305	0.0158450	0.00792252	−	0.999969i	$-0.497478\pi$
$613$	0.392305	0.0158450	0.00792252	+	0.999969i	$0.497478\pi$
$614$	−24.2487	−0.978598
$615$	0	0
$616$	3.46410	0.139573
$617$	23.0718	0.928836	0.464418	−	0.885616i	$-0.346264\pi$
$617$	23.0718	0.928836	0.464418	+	0.885616i	$0.346264\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	2.92820	0.117599
$621$	0	0
$622$	32.7846	1.31454
$623$	25.8564	1.03592
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.2487	−0.489557
$627$	0	0
$628$	−16.9282	−0.675509
$629$	0	0
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	−23.3205	−0.927640
$633$	0	0
$634$	−19.1769	−0.761613
$635$	−4.92820	−0.195570
$636$	0	0
$637$	4.39230	0.174029
$638$	6.00000	0.237542
$639$	0	0
$640$	−12.1244	−0.479257
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	0	0
$643$	−45.5692	−1.79707	−0.898537	−	0.438897i	$-0.855369\pi$
$643$	−45.5692	−1.79707	−0.898537	+	0.438897i	$0.855369\pi$
$644$	13.8564	0.546019
$645$	0	0
$646$	0	0
$647$	27.7128	1.08950	0.544752	−	0.838597i	$-0.316624\pi$
$647$	27.7128	1.08950	0.544752	+	0.838597i	$0.316624\pi$
$648$	0	0
$649$	−6.92820	−0.271956
$650$	2.53590	0.0994661
$651$	0	0
$652$	−17.8564	−0.699311
$653$	7.85641	0.307445	0.153722	−	0.988114i	$-0.450874\pi$
$653$	7.85641	0.307445	0.153722	+	0.988114i	$0.450874\pi$
$654$	0	0
$655$	−5.07180	−0.198171
$656$	−17.3205	−0.676252
$657$	0	0
$658$	24.0000	0.935617
$659$	−39.7128	−1.54699	−0.773496	−	0.633801i	$-0.781494\pi$
$659$	−39.7128	−1.54699	−0.773496	+	0.633801i	$0.781494\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	30.9282	1.20206
$663$	0	0
$664$	−26.7846	−1.03944
$665$	−2.92820	−0.113551
$666$	0	0
$667$	−24.0000	−0.929284
$668$	−10.3923	−0.402090
$669$	0	0
$670$	−13.8564	−0.535320
$671$	2.00000	0.0772091
$672$	0	0
$673$	31.3205	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$673$	31.3205	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$674$	50.5359	1.94657
$675$	0	0
$676$	−10.8564	−0.417554
$677$	32.7846	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$677$	32.7846	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	0	0
$682$	−5.07180	−0.194209
$683$	−8.78461	−0.336134	−0.168067	−	0.985776i	$-0.553752\pi$
$683$	−8.78461	−0.336134	−0.168067	+	0.985776i	$0.553752\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	34.6410	1.32260
$687$	0	0
$688$	−44.6410	−1.70192
$689$	−18.9282	−0.721107
$690$	0	0
$691$	−7.71281	−0.293409	−0.146705	−	0.989180i	$-0.546867\pi$
$691$	−7.71281	−0.293409	−0.146705	+	0.989180i	$0.546867\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	2.78461	0.105702
$695$	−8.39230	−0.318338
$696$	0	0
$697$	0	0
$698$	62.1051	2.35071
$699$	0	0
$700$	2.00000	0.0755929
$701$	−32.5359	−1.22886	−0.614432	−	0.788970i	$-0.710615\pi$
$701$	−32.5359	−1.22886	−0.614432	+	0.788970i	$0.710615\pi$
$702$	0	0
$703$	−13.0718	−0.493012
$704$	1.00000	0.0376889
$705$	0	0
$706$	−1.60770	−0.0605064
$707$	−20.7846	−0.781686
$708$	0	0
$709$	15.8564	0.595500	0.297750	−	0.954644i	$-0.403764\pi$
$709$	15.8564	0.595500	0.297750	+	0.954644i	$0.403764\pi$
$710$	−24.0000	−0.900704
$711$	0	0
$712$	22.3923	0.839187
$713$	20.2872	0.759761
$714$	0	0
$715$	−1.46410	−0.0547543
$716$	6.92820	0.258919
$717$	0	0
$718$	36.0000	1.34351
$719$	18.9282	0.705903	0.352951	−	0.935642i	$-0.385178\pi$
$719$	18.9282	0.705903	0.352951	+	0.935642i	$0.385178\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	29.1962	1.08657
$723$	0	0
$724$	−11.8564	−0.440640
$725$	−3.46410	−0.128654
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−5.07180	−0.187973
$729$	0	0
$730$	−21.4641	−0.794422
$731$	0	0
$732$	0	0
$733$	−49.9615	−1.84537	−0.922686	−	0.385553i	$-0.874011\pi$
$733$	−49.9615	−1.84537	−0.922686	+	0.385553i	$0.874011\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	8.00000	0.294684
$738$	0	0
$739$	10.5359	0.387569	0.193785	−	0.981044i	$-0.437924\pi$
$739$	10.5359	0.387569	0.193785	+	0.981044i	$0.437924\pi$
$740$	8.92820	0.328207
$741$	0	0
$742$	−44.7846	−1.64409
$743$	−46.3923	−1.70197	−0.850984	−	0.525191i	$-0.823994\pi$
$743$	−46.3923	−1.70197	−0.850984	+	0.525191i	$0.823994\pi$
$744$	0	0
$745$	8.53590	0.312731
$746$	−0.679492	−0.0248780
$747$	0	0
$748$	0	0
$749$	−30.9282	−1.13009
$750$	0	0
$751$	13.0718	0.476997	0.238498	−	0.971143i	$-0.423345\pi$
$751$	13.0718	0.476997	0.238498	+	0.971143i	$0.423345\pi$
$752$	34.6410	1.26323
$753$	0	0
$754$	−8.78461	−0.319917
$755$	0.392305	0.0142774
$756$	0	0
$757$	−6.78461	−0.246591	−0.123295	−	0.992370i	$-0.539346\pi$
$757$	−6.78461	−0.246591	−0.123295	+	0.992370i	$0.539346\pi$
$758$	−17.0718	−0.620076
$759$	0	0
$760$	−2.53590	−0.0919867
$761$	39.4641	1.43057	0.715286	−	0.698832i	$-0.246296\pi$
$761$	39.4641	1.43057	0.715286	+	0.698832i	$0.246296\pi$
$762$	0	0
$763$	−20.0000	−0.724049
$764$	5.07180	0.183491
$765$	0	0
$766$	24.0000	0.867155
$767$	10.1436	0.366264
$768$	0	0
$769$	−46.4974	−1.67674	−0.838370	−	0.545102i	$-0.816491\pi$
$769$	−46.4974	−1.67674	−0.838370	+	0.545102i	$0.816491\pi$
$770$	−3.46410	−0.124838
$771$	0	0
$772$	3.60770	0.129844
$773$	31.8564	1.14580	0.572898	−	0.819627i	$-0.305819\pi$
$773$	31.8564	1.14580	0.572898	+	0.819627i	$0.305819\pi$
$774$	0	0
$775$	2.92820	0.105184
$776$	−17.3205	−0.621770
$777$	0	0
$778$	43.1769	1.54797
$779$	−5.07180	−0.181716
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	0	0
$784$	15.0000	0.535714
$785$	−16.9282	−0.604193
$786$	0	0
$787$	−18.7846	−0.669599	−0.334800	−	0.942289i	$-0.608669\pi$
$787$	−18.7846	−0.669599	−0.334800	+	0.942289i	$0.608669\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	23.3205	0.829706
$791$	−1.85641	−0.0660062
$792$	0	0
$793$	−2.92820	−0.103984
$794$	−3.46410	−0.122936
$795$	0	0
$796$	16.7846	0.594915
$797$	−16.6410	−0.589455	−0.294728	−	0.955581i	$-0.595229\pi$
$797$	−16.6410	−0.589455	−0.294728	+	0.955581i	$0.595229\pi$
$798$	0	0
$799$	0	0
$800$	5.19615	0.183712
$801$	0	0
$802$	34.3923	1.21443
$803$	12.3923	0.437315
$804$	0	0
$805$	13.8564	0.488374
$806$	7.42563	0.261557
$807$	0	0
$808$	−18.0000	−0.633238
$809$	8.53590	0.300106	0.150053	−	0.988678i	$-0.452056\pi$
$809$	8.53590	0.300106	0.150053	+	0.988678i	$0.452056\pi$
$810$	0	0
$811$	−8.39230	−0.294694	−0.147347	−	0.989085i	$-0.547073\pi$
$811$	−8.39230	−0.294694	−0.147347	+	0.989085i	$0.547073\pi$
$812$	−6.92820	−0.243132
$813$	0	0
$814$	−15.4641	−0.542016
$815$	−17.8564	−0.625483
$816$	0	0
$817$	−13.0718	−0.457324
$818$	−60.2487	−2.10655
$819$	0	0
$820$	3.46410	0.120972
$821$	−27.4641	−0.958504	−0.479252	−	0.877677i	$-0.659092\pi$
$821$	−27.4641	−0.958504	−0.479252	+	0.877677i	$0.659092\pi$
$822$	0	0
$823$	49.5692	1.72787	0.863937	−	0.503600i	$-0.167991\pi$
$823$	49.5692	1.72787	0.863937	+	0.503600i	$0.167991\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	−1.60770	−0.0559050	−0.0279525	−	0.999609i	$-0.508899\pi$
$827$	−1.60770	−0.0559050	−0.0279525	+	0.999609i	$0.508899\pi$
$828$	0	0
$829$	−25.7128	−0.893043	−0.446521	−	0.894773i	$-0.647337\pi$
$829$	−25.7128	−0.893043	−0.446521	+	0.894773i	$0.647337\pi$
$830$	26.7846	0.929707
$831$	0	0
$832$	−1.46410	−0.0507586
$833$	0	0
$834$	0	0
$835$	−10.3923	−0.359641
$836$	−1.46410	−0.0506370
$837$	0	0
$838$	29.5692	1.02145
$839$	−15.2154	−0.525294	−0.262647	−	0.964892i	$-0.584595\pi$
$839$	−15.2154	−0.525294	−0.262647	+	0.964892i	$0.584595\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−3.46410	−0.119381
$843$	0	0
$844$	12.3923	0.426561
$845$	−10.8564	−0.373472
$846$	0	0
$847$	2.00000	0.0687208
$848$	−64.6410	−2.21978
$849$	0	0
$850$	0	0
$851$	61.8564	2.12041
$852$	0	0
$853$	24.3923	0.835177	0.417588	−	0.908636i	$-0.362875\pi$
$853$	24.3923	0.835177	0.417588	+	0.908636i	$0.362875\pi$
$854$	−6.92820	−0.237078
$855$	0	0
$856$	−26.7846	−0.915479
$857$	10.1436	0.346499	0.173249	−	0.984878i	$-0.444573\pi$
$857$	10.1436	0.346499	0.173249	+	0.984878i	$0.444573\pi$
$858$	0	0
$859$	47.7128	1.62794	0.813970	−	0.580907i	$-0.197302\pi$
$859$	47.7128	1.62794	0.813970	+	0.580907i	$0.197302\pi$
$860$	8.92820	0.304449
$861$	0	0
$862$	−56.7846	−1.93409
$863$	10.1436	0.345292	0.172646	−	0.984984i	$-0.444768\pi$
$863$	10.1436	0.345292	0.172646	+	0.984984i	$0.444768\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	−48.2487	−1.63956
$867$	0	0
$868$	5.85641	0.198779
$869$	−13.4641	−0.456738
$870$	0	0
$871$	−11.7128	−0.396874
$872$	−17.3205	−0.586546
$873$	0	0
$874$	17.5692	0.594288
$875$	2.00000	0.0676123
$876$	0	0
$877$	14.2487	0.481145	0.240572	−	0.970631i	$-0.422665\pi$
$877$	14.2487	0.481145	0.240572	+	0.970631i	$0.422665\pi$
$878$	50.5359	1.70550
$879$	0	0
$880$	−5.00000	−0.168550
$881$	−12.9282	−0.435562	−0.217781	−	0.975998i	$-0.569882\pi$
$881$	−12.9282	−0.435562	−0.217781	+	0.975998i	$0.569882\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	0	0
$886$	20.7846	0.698273
$887$	36.2487	1.21711	0.608556	−	0.793511i	$-0.291749\pi$
$887$	36.2487	1.21711	0.608556	+	0.793511i	$0.291749\pi$
$888$	0	0
$889$	−9.85641	−0.330573
$890$	−22.3923	−0.750592
$891$	0	0
$892$	−17.8564	−0.597877
$893$	10.1436	0.339442
$894$	0	0
$895$	6.92820	0.231584
$896$	−24.2487	−0.810093
$897$	0	0
$898$	25.6077	0.854540
$899$	−10.1436	−0.338308
$900$	0	0
$901$	0	0
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−1.60770	−0.0534711
$905$	−11.8564	−0.394120
$906$	0	0
$907$	45.8564	1.52264	0.761318	−	0.648378i	$-0.224552\pi$
$907$	45.8564	1.52264	0.761318	+	0.648378i	$0.224552\pi$
$908$	−8.53590	−0.283274
$909$	0	0
$910$	5.07180	0.168128
$911$	−5.07180	−0.168036	−0.0840181	−	0.996464i	$-0.526775\pi$
$911$	−5.07180	−0.168036	−0.0840181	+	0.996464i	$0.526775\pi$
$912$	0	0
$913$	−15.4641	−0.511787
$914$	14.5359	0.480805
$915$	0	0
$916$	3.85641	0.127419
$917$	−10.1436	−0.334971
$918$	0	0
$919$	−11.6077	−0.382903	−0.191451	−	0.981502i	$-0.561319\pi$
$919$	−11.6077	−0.382903	−0.191451	+	0.981502i	$0.561319\pi$
$920$	12.0000	0.395628
$921$	0	0
$922$	−21.2154	−0.698692
$923$	−20.2872	−0.667761
$924$	0	0
$925$	8.92820	0.293558
$926$	48.4974	1.59372
$927$	0	0
$928$	−18.0000	−0.590879
$929$	38.7846	1.27248	0.636241	−	0.771490i	$-0.280488\pi$
$929$	38.7846	1.27248	0.636241	+	0.771490i	$0.280488\pi$
$930$	0	0
$931$	4.39230	0.143952
$932$	−12.0000	−0.393073
$933$	0	0
$934$	32.7846	1.07275
$935$	0	0
$936$	0	0
$937$	0.392305	0.0128160	0.00640802	−	0.999979i	$-0.497960\pi$
$937$	0.392305	0.0128160	0.00640802	+	0.999979i	$0.497960\pi$
$938$	−27.7128	−0.904855
$939$	0	0
$940$	−6.92820	−0.225973
$941$	−20.5359	−0.669451	−0.334726	−	0.942316i	$-0.608644\pi$
$941$	−20.5359	−0.669451	−0.334726	+	0.942316i	$0.608644\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	34.6410	1.12747
$945$	0	0
$946$	−15.4641	−0.502781
$947$	5.07180	0.164811	0.0824056	−	0.996599i	$-0.473740\pi$
$947$	5.07180	0.164811	0.0824056	+	0.996599i	$0.473740\pi$
$948$	0	0
$949$	−18.1436	−0.588966
$950$	2.53590	0.0822754
$951$	0	0
$952$	0	0
$953$	44.7846	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$953$	44.7846	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$954$	0	0
$955$	5.07180	0.164119
$956$	12.0000	0.388108
$957$	0	0
$958$	20.7846	0.671520
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.4256	−0.723407
$962$	22.6410	0.729976
$963$	0	0
$964$	27.8564	0.897194
$965$	3.60770	0.116136
$966$	0	0
$967$	−18.7846	−0.604072	−0.302036	−	0.953296i	$-0.597666\pi$
$967$	−18.7846	−0.604072	−0.302036	+	0.953296i	$0.597666\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	17.3205	0.556128
$971$	1.85641	0.0595749	0.0297875	−	0.999556i	$-0.490517\pi$
$971$	1.85641	0.0595749	0.0297875	+	0.999556i	$0.490517\pi$
$972$	0	0
$973$	−16.7846	−0.538090
$974$	−41.0718	−1.31603
$975$	0	0
$976$	−10.0000	−0.320092
$977$	35.5692	1.13796	0.568980	−	0.822351i	$-0.307338\pi$
$977$	35.5692	1.13796	0.568980	+	0.822351i	$0.307338\pi$
$978$	0	0
$979$	12.9282	0.413187
$980$	−3.00000	−0.0958315
$981$	0	0
$982$	−29.5692	−0.943592
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	2.14359	0.0681968
$989$	61.8564	1.96692
$990$	0	0
$991$	−48.7846	−1.54969	−0.774847	−	0.632149i	$-0.782173\pi$
$991$	−48.7846	−1.54969	−0.774847	+	0.632149i	$0.782173\pi$
$992$	15.2154	0.483089
$993$	0	0
$994$	−48.0000	−1.52247
$995$	16.7846	0.532108
$996$	0	0
$997$	48.3923	1.53260	0.766300	−	0.642483i	$-0.222096\pi$
$997$	48.3923	1.53260	0.766300	+	0.642483i	$0.222096\pi$
$998$	22.1436	0.700943
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.a.c.1.1		2
3.2	odd	2		165.2.a.b.1.2	✓	2
4.3	odd	2		7920.2.a.bz.1.1		2
5.2	odd	4		2475.2.c.n.199.2		4
5.3	odd	4		2475.2.c.n.199.3		4
5.4	even	2		2475.2.a.r.1.2		2
11.10	odd	2		5445.2.a.s.1.2		2
12.11	even	2		2640.2.a.x.1.1		2
15.2	even	4		825.2.c.c.199.3		4
15.8	even	4		825.2.c.c.199.2		4
15.14	odd	2		825.2.a.e.1.1		2
21.20	even	2		8085.2.a.bd.1.2		2
33.32	even	2		1815.2.a.i.1.1		2
165.164	even	2		9075.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.2	✓	2	3.2	odd	2
495.2.a.c.1.1		2	1.1	even	1	trivial
825.2.a.e.1.1		2	15.14	odd	2
825.2.c.c.199.2		4	15.8	even	4
825.2.c.c.199.3		4	15.2	even	4
1815.2.a.i.1.1		2	33.32	even	2
2475.2.a.r.1.2		2	5.4	even	2
2475.2.c.n.199.2		4	5.2	odd	4
2475.2.c.n.199.3		4	5.3	odd	4
2640.2.a.x.1.1		2	12.11	even	2
5445.2.a.s.1.2		2	11.10	odd	2
7920.2.a.bz.1.1		2	4.3	odd	2
8085.2.a.bd.1.2		2	21.20	even	2
9075.2.a.bh.1.2		2	165.164	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} +1.00000 q^{11} -1.46410 q^{13} -3.46410 q^{14} -5.00000 q^{16} -1.46410 q^{19} +1.00000 q^{20} -1.73205 q^{22} +6.92820 q^{23} +1.00000 q^{25} +2.53590 q^{26} +2.00000 q^{28} -3.46410 q^{29} +2.92820 q^{31} +5.19615 q^{32} +2.00000 q^{35} +8.92820 q^{37} +2.53590 q^{38} +1.73205 q^{40} +3.46410 q^{41} +8.92820 q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -1.46410 q^{52} +12.9282 q^{53} +1.00000 q^{55} +3.46410 q^{56} +6.00000 q^{58} -6.92820 q^{59} +2.00000 q^{61} -5.07180 q^{62} +1.00000 q^{64} -1.46410 q^{65} +8.00000 q^{67} -3.46410 q^{70} +13.8564 q^{71} +12.3923 q^{73} -15.4641 q^{74} -1.46410 q^{76} +2.00000 q^{77} -13.4641 q^{79} -5.00000 q^{80} -6.00000 q^{82} -15.4641 q^{83} -15.4641 q^{86} +1.73205 q^{88} +12.9282 q^{89} -2.92820 q^{91} +6.92820 q^{92} +12.0000 q^{94} -1.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7	\( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} - 10 q^{16} + 4 q^{19} + 2 q^{20} + 2 q^{25} + 12 q^{26} + 4 q^{28} - 8 q^{31} + 4 q^{35} + 4 q^{37} + 12 q^{38} + 4 q^{43} + 2 q^{44} - 24 q^{46} - 6 q^{49} + 4 q^{52} + 12 q^{53} + 2 q^{55} + 12 q^{58} + 4 q^{61} - 24 q^{62} + 2 q^{64} + 4 q^{65} + 16 q^{67} + 4 q^{73} - 24 q^{74} + 4 q^{76} + 4 q^{77} - 20 q^{79} - 10 q^{80} - 12 q^{82} - 24 q^{83} - 24 q^{86} + 12 q^{89} + 8 q^{91} + 24 q^{94} + 4 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^11 + 4 * q^13 - 10 * q^16 + 4 * q^19 + 2 * q^20 + 2 * q^25 + 12 * q^26 + 4 * q^28 - 8 * q^31 + 4 * q^35 + 4 * q^37 + 12 * q^38 + 4 * q^43 + 2 * q^44 - 24 * q^46 - 6 * q^49 + 4 * q^52 + 12 * q^53 + 2 * q^55 + 12 * q^58 + 4 * q^61 - 24 * q^62 + 2 * q^64 + 4 * q^65 + 16 * q^67 + 4 * q^73 - 24 * q^74 + 4 * q^76 + 4 * q^77 - 20 * q^79 - 10 * q^80 - 12 * q^82 - 24 * q^83 - 24 * q^86 + 12 * q^89 + 8 * q^91 + 24 * q^94 + 4 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more