LMFDB - Embedded newform 2475.2.c.n.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.n.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.73205i q^{8} +O(q^{10})$	$q+1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.73205i q^{8} +1.00000 q^{11} -1.46410i q^{13} +3.46410 q^{14} -5.00000 q^{16} +1.46410 q^{19} +1.73205i q^{22} +6.92820i q^{23} +2.53590 q^{26} +2.00000i q^{28} +3.46410 q^{29} +2.92820 q^{31} -5.19615i q^{32} -8.92820i q^{37} +2.53590i q^{38} +3.46410 q^{41} +8.92820i q^{43} -1.00000 q^{44} -12.0000 q^{46} +6.92820i q^{47} +3.00000 q^{49} +1.46410i q^{52} +12.9282i q^{53} +3.46410 q^{56} +6.00000i q^{58} +6.92820 q^{59} +2.00000 q^{61} +5.07180i q^{62} -1.00000 q^{64} -8.00000i q^{67} +13.8564 q^{71} +12.3923i q^{73} +15.4641 q^{74} -1.46410 q^{76} -2.00000i q^{77} +13.4641 q^{79} +6.00000i q^{82} -15.4641i q^{83} -15.4641 q^{86} +1.73205i q^{88} -12.9282 q^{89} -2.92820 q^{91} -6.92820i q^{92} -12.0000 q^{94} +10.0000i q^{97} +5.19615i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{11} - 20 q^{16} - 8 q^{19} + 24 q^{26} - 16 q^{31} - 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} + 48 q^{74} + 8 q^{76} + 40 q^{79} - 48 q^{86} - 24 q^{89} + 16 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^11 - 20 * q^16 - 8 * q^19 + 24 * q^26 - 16 * q^31 - 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 + 48 * q^74 + 8 * q^76 + 40 * q^79 - 48 * q^86 - 24 * q^89 + 16 * q^91 - 48 * q^94

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.73205i	0.612372i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 1.46410i	− 0.406069i	−0.979172	−	0.203034i	$-0.934920\pi$
$13$	− 1.46410i	− 0.406069i	0.979172	−	0.203034i	$-0.0650803\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	0	0
$22$	1.73205i	0.369274i
$23$	6.92820i	1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$	6.92820i	1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$	0	0
$25$	0	0
$26$	2.53590	0.497331
$27$	0	0
$28$	2.00000i	0.377964i
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	− 5.19615i	− 0.918559i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.92820i	− 1.46779i	−0.679264	−	0.733894i	$-0.737701\pi$
$37$	− 8.92820i	− 1.46779i	0.679264	−	0.733894i	$-0.262299\pi$
$38$	2.53590i	0.411377i
$39$	0	0
$40$	0	0
$41$	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
$43$	8.92820i	1.36154i	0.732498	+	0.680769i	$0.238354\pi$
$43$	8.92820i	1.36154i	−0.732498	+	0.680769i	$0.761646\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−12.0000	−1.76930
$47$	6.92820i	1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$	6.92820i	1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	1.46410i	0.203034i
$53$	12.9282i	1.77583i	0.460012	+	0.887913i	$0.347845\pi$
$53$	12.9282i	1.77583i	−0.460012	+	0.887913i	$0.652155\pi$
$54$	0	0
$55$	0	0
$56$	3.46410	0.462910
$57$	0	0
$58$	6.00000i	0.787839i
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	5.07180i	0.644119i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	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
$73$	12.3923i	1.45041i	0.688533	+	0.725205i	$0.258255\pi$
$73$	12.3923i	1.45041i	−0.688533	+	0.725205i	$0.741745\pi$
$74$	15.4641	1.79767
$75$	0	0
$76$	−1.46410	−0.167944
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	13.4641	1.51483	0.757415	−	0.652934i	$-0.226462\pi$
$79$	13.4641	1.51483	0.757415	+	0.652934i	$0.226462\pi$
$80$	0	0
$81$	0	0
$82$	6.00000i	0.662589i
$83$	− 15.4641i	− 1.69741i	−0.528870	−	0.848703i	$-0.677384\pi$
$83$	− 15.4641i	− 1.69741i	0.528870	−	0.848703i	$-0.322616\pi$
$84$	0	0
$85$	0	0
$86$	−15.4641	−1.66754
$87$	0	0
$88$	1.73205i	0.184637i
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−2.92820	−0.306959
$92$	− 6.92820i	− 0.722315i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	5.19615i	0.524891i
$99$	0	0
$100$	0	0
$101$	−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
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.53590	0.248665
$105$	0	0
$106$	−22.3923	−2.17493
$107$	15.4641i	1.49497i	0.664278	+	0.747486i	$0.268739\pi$
$107$	15.4641i	1.49497i	−0.664278	+	0.747486i	$0.731261\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	10.0000i	0.944911i
$113$	− 0.928203i	− 0.0873180i	−0.999046	−	0.0436590i	$-0.986098\pi$
$113$	− 0.928203i	− 0.0873180i	0.999046	−	0.0436590i	$-0.0139015\pi$
$114$	0	0
$115$	0	0
$116$	−3.46410	−0.321634
$117$	0	0
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.46410i	0.313625i
$123$	0	0
$124$	−2.92820	−0.262960
$125$	0	0
$126$	0	0
$127$	4.92820i	0.437307i	0.975803	+	0.218654i	$0.0701665\pi$
$127$	4.92820i	0.437307i	−0.975803	+	0.218654i	$0.929834\pi$
$128$	− 12.1244i	− 1.07165i
$129$	0	0
$130$	0	0
$131$	−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
$133$	− 2.92820i	− 0.253907i
$134$	13.8564	1.19701
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	0	0
$142$	24.0000i	2.01404i
$143$	− 1.46410i	− 0.122434i
$144$	0	0
$145$	0	0
$146$	−21.4641	−1.77638
$147$	0	0
$148$	8.92820i	0.733894i
$149$	−8.53590	−0.699288	−0.349644	−	0.936883i	$-0.613697\pi$
$149$	−8.53590	−0.699288	−0.349644	+	0.936883i	$0.613697\pi$
$150$	0	0
$151$	0.392305	0.0319253	0.0159627	−	0.999873i	$-0.494919\pi$
$151$	0.392305	0.0319253	0.0159627	+	0.999873i	$0.494919\pi$
$152$	2.53590i	0.205689i
$153$	0	0
$154$	3.46410	0.279145
$155$	0	0
$156$	0	0
$157$	16.9282i	1.35102i	0.737352	+	0.675509i	$0.236076\pi$
$157$	16.9282i	1.35102i	−0.737352	+	0.675509i	$0.763924\pi$
$158$	23.3205i	1.85528i
$159$	0	0
$160$	0	0
$161$	13.8564	1.09204
$162$	0	0
$163$	− 17.8564i	− 1.39862i	−0.714818	−	0.699311i	$-0.753490\pi$
$163$	− 17.8564i	− 1.39862i	0.714818	−	0.699311i	$-0.246510\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	10.3923i	0.804181i	0.915600	+	0.402090i	$0.131716\pi$
$167$	10.3923i	0.804181i	−0.915600	+	0.402090i	$0.868284\pi$
$168$	0	0
$169$	10.8564	0.835108
$170$	0	0
$171$	0	0
$172$	− 8.92820i	− 0.680769i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	−5.00000	−0.376889
$177$	0	0
$178$	− 22.3923i	− 1.67837i
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	− 5.07180i	− 0.375947i
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 6.92820i	− 0.505291i
$189$	0	0
$190$	0	0
$191$	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
$193$	3.60770i	0.259688i	0.991534	+	0.129844i	$0.0414476\pi$
$193$	3.60770i	0.259688i	−0.991534	+	0.129844i	$0.958552\pi$
$194$	−17.3205	−1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	0	0
$202$	− 18.0000i	− 1.26648i
$203$	− 6.92820i	− 0.486265i
$204$	0	0
$205$	0	0
$206$	−13.8564	−0.965422
$207$	0	0
$208$	7.32051i	0.507586i
$209$	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	$0.359725\pi$
$212$	− 12.9282i	− 0.887913i
$213$	0	0
$214$	−26.7846	−1.83096
$215$	0	0
$216$	0	0
$217$	− 5.85641i	− 0.397559i
$218$	17.3205i	1.17309i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 17.8564i	− 1.19575i	−0.801588	−	0.597877i	$-0.796011\pi$
$223$	− 17.8564i	− 1.19575i	0.801588	−	0.597877i	$-0.203989\pi$
$224$	−10.3923	−0.694365
$225$	0	0
$226$	1.60770	0.106942
$227$	8.53590i	0.566547i	0.959039	+	0.283274i	$0.0914205\pi$
$227$	8.53590i	0.566547i	−0.959039	+	0.283274i	$0.908580\pi$
$228$	0	0
$229$	−3.85641	−0.254839	−0.127419	−	0.991849i	$-0.540669\pi$
$229$	−3.85641	−0.254839	−0.127419	+	0.991849i	$0.540669\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	0	0
$236$	−6.92820	−0.450988
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	27.8564	1.79439	0.897194	−	0.441636i	$-0.145602\pi$
$241$	27.8564	1.79439	0.897194	+	0.441636i	$0.145602\pi$
$242$	1.73205i	0.111340i
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	− 2.14359i	− 0.136394i
$248$	5.07180i	0.322059i
$249$	0	0
$250$	0	0
$251$	−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
$253$	6.92820i	0.435572i
$254$	−8.53590	−0.535590
$255$	0	0
$256$	19.0000	1.18750
$257$	7.85641i	0.490069i	0.969514	+	0.245035i	$0.0787993\pi$
$257$	7.85641i	0.490069i	−0.969514	+	0.245035i	$0.921201\pi$
$258$	0	0
$259$	−17.8564	−1.10954
$260$	0	0
$261$	0	0
$262$	− 8.78461i	− 0.542715i
$263$	− 27.4641i	− 1.69351i	−0.531984	−	0.846755i	$-0.678553\pi$
$263$	− 27.4641i	− 1.69351i	0.531984	−	0.846755i	$-0.321447\pi$
$264$	0	0
$265$	0	0
$266$	5.07180	0.310972
$267$	0	0
$268$	8.00000i	0.488678i
$269$	−7.85641	−0.479014	−0.239507	−	0.970895i	$-0.576986\pi$
$269$	−7.85641	−0.479014	−0.239507	+	0.970895i	$0.576986\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	− 22.5359i	− 1.35405i	−0.735960	−	0.677025i	$-0.763269\pi$
$277$	− 22.5359i	− 1.35405i	0.735960	−	0.677025i	$-0.236731\pi$
$278$	14.5359i	0.871805i
$279$	0	0
$280$	0	0
$281$	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
$283$	8.92820i	0.530727i	0.964148	+	0.265363i	$0.0854919\pi$
$283$	8.92820i	0.530727i	−0.964148	+	0.265363i	$0.914508\pi$
$284$	−13.8564	−0.822226
$285$	0	0
$286$	2.53590	0.149951
$287$	− 6.92820i	− 0.408959i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	− 12.3923i	− 0.725205i
$293$	− 13.8564i	− 0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$	− 13.8564i	− 0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$	0	0
$295$	0	0
$296$	15.4641	0.898833
$297$	0	0
$298$	− 14.7846i	− 0.856449i
$299$	10.1436	0.586619
$300$	0	0
$301$	17.8564	1.02923
$302$	0.679492i	0.0391004i
$303$	0	0
$304$	−7.32051	−0.419860
$305$	0	0
$306$	0	0
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	2.00000i	0.113961i
$309$	0	0
$310$	0	0
$311$	−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
$313$	7.07180i	0.399722i	0.979824	+	0.199861i	$0.0640490\pi$
$313$	7.07180i	0.399722i	−0.979824	+	0.199861i	$0.935951\pi$
$314$	−29.3205	−1.65465
$315$	0	0
$316$	−13.4641	−0.757415
$317$	− 11.0718i	− 0.621854i	−0.950434	−	0.310927i	$-0.899361\pi$
$317$	− 11.0718i	− 0.621854i	0.950434	−	0.310927i	$-0.100639\pi$
$318$	0	0
$319$	3.46410	0.193952
$320$	0	0
$321$	0	0
$322$	24.0000i	1.33747i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	30.9282	1.71295
$327$	0	0
$328$	6.00000i	0.331295i
$329$	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	$0.663273\pi$
$332$	15.4641i	0.848703i
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	29.1769i	1.58937i	0.607023	+	0.794684i	$0.292363\pi$
$337$	29.1769i	1.58937i	−0.607023	+	0.794684i	$0.707637\pi$
$338$	18.8038i	1.02279i
$339$	0	0
$340$	0	0
$341$	2.92820	0.158571
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	−15.4641	−0.833768
$345$	0	0
$346$	−20.7846	−1.11739
$347$	1.60770i	0.0863056i	0.999068	+	0.0431528i	$0.0137402\pi$
$347$	1.60770i	0.0863056i	−0.999068	+	0.0431528i	$0.986260\pi$
$348$	0	0
$349$	35.8564	1.91935	0.959675	−	0.281113i	$-0.0907035\pi$
$349$	35.8564	1.91935	0.959675	+	0.281113i	$0.0907035\pi$
$350$	0	0
$351$	0	0
$352$	− 5.19615i	− 0.276956i
$353$	0.928203i	0.0494033i	0.999695	+	0.0247016i	$0.00786357\pi$
$353$	0.928203i	0.0494033i	−0.999695	+	0.0247016i	$0.992136\pi$
$354$	0	0
$355$	0	0
$356$	12.9282	0.685193
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	20.7846	1.09697	0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846	1.09697	0.548485	+	0.836160i	$0.315205\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	− 20.5359i	− 1.07934i
$363$	0	0
$364$	2.92820	0.153480
$365$	0	0
$366$	0	0
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	− 34.6410i	− 1.80579i
$369$	0	0
$370$	0	0
$371$	25.8564	1.34240
$372$	0	0
$373$	0.392305i	0.0203128i	0.999948	+	0.0101564i	$0.00323293\pi$
$373$	0.392305i	0.0203128i	−0.999948	+	0.0101564i	$0.996767\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	− 5.07180i	− 0.261211i
$378$	0	0
$379$	−9.85641	−0.506290	−0.253145	−	0.967428i	$-0.581465\pi$
$379$	−9.85641	−0.506290	−0.253145	+	0.967428i	$0.581465\pi$
$380$	0	0
$381$	0	0
$382$	8.78461i	0.449460i
$383$	− 13.8564i	− 0.708029i	−0.935240	−	0.354015i	$-0.884816\pi$
$383$	− 13.8564i	− 0.708029i	0.935240	−	0.354015i	$-0.115184\pi$
$384$	0	0
$385$	0	0
$386$	−6.24871	−0.318051
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	24.9282	1.26391	0.631955	−	0.775005i	$-0.282253\pi$
$389$	24.9282	1.26391	0.631955	+	0.775005i	$0.282253\pi$
$390$	0	0
$391$	0	0
$392$	5.19615i	0.262445i
$393$	0	0
$394$	20.7846	1.04711
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 29.0718i	− 1.45724i
$399$	0	0
$400$	0	0
$401$	−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
$403$	− 4.28719i	− 0.213560i
$404$	10.3923	0.517036
$405$	0	0
$406$	12.0000	0.595550
$407$	− 8.92820i	− 0.442555i
$408$	0	0
$409$	−34.7846	−1.71999	−0.859994	−	0.510304i	$-0.829533\pi$
$409$	−34.7846	−1.71999	−0.859994	+	0.510304i	$0.829533\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	− 13.8564i	− 0.681829i
$414$	0	0
$415$	0	0
$416$	−7.60770	−0.372998
$417$	0	0
$418$	2.53590i	0.124035i
$419$	17.0718	0.834012	0.417006	−	0.908904i	$-0.363079\pi$
$419$	17.0718	0.834012	0.417006	+	0.908904i	$0.363079\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	21.4641i	1.04486i
$423$	0	0
$424$	−22.3923	−1.08747
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	− 15.4641i	− 0.747486i
$429$	0	0
$430$	0	0
$431$	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
$433$	27.8564i	1.33869i	0.742950	+	0.669347i	$0.233426\pi$
$433$	27.8564i	1.33869i	−0.742950	+	0.669347i	$0.766574\pi$
$434$	10.1436	0.486908
$435$	0	0
$436$	−10.0000	−0.478913
$437$	10.1436i	0.485234i
$438$	0	0
$439$	29.1769	1.39254	0.696269	−	0.717781i	$-0.254842\pi$
$439$	29.1769	1.39254	0.696269	+	0.717781i	$0.254842\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	30.9282	1.46449
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	14.7846	0.697729	0.348864	−	0.937173i	$-0.386567\pi$
$449$	14.7846	0.697729	0.348864	+	0.937173i	$0.386567\pi$
$450$	0	0
$451$	3.46410	0.163118
$452$	0.928203i	0.0436590i
$453$	0	0
$454$	−14.7846	−0.693876
$455$	0	0
$456$	0	0
$457$	8.39230i	0.392575i	0.980546	+	0.196288i	$0.0628887\pi$
$457$	8.39230i	0.392575i	−0.980546	+	0.196288i	$0.937111\pi$
$458$	− 6.67949i	− 0.312112i
$459$	0	0
$460$	0	0
$461$	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
$463$	− 28.0000i	− 1.30127i	−0.759390	−	0.650635i	$-0.774503\pi$
$463$	− 28.0000i	− 1.30127i	0.759390	−	0.650635i	$-0.225497\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	20.7846	0.962828
$467$	18.9282i	0.875893i	0.899001	+	0.437946i	$0.144294\pi$
$467$	18.9282i	0.875893i	−0.899001	+	0.437946i	$0.855706\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	8.92820i	0.410519i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	− 20.7846i	− 0.950666i
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−13.0718	−0.596023
$482$	48.2487i	2.19767i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 23.7128i	− 1.07453i	−0.843413	−	0.537265i	$-0.819457\pi$
$487$	− 23.7128i	− 1.07453i	0.843413	−	0.537265i	$-0.180543\pi$
$488$	3.46410i	0.156813i
$489$	0	0
$490$	0	0
$491$	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
$497$	− 27.7128i	− 1.24309i
$498$	0	0
$499$	12.7846	0.572318	0.286159	−	0.958182i	$-0.407621\pi$
$499$	12.7846	0.572318	0.286159	+	0.958182i	$0.407621\pi$
$500$	0	0
$501$	0	0
$502$	− 44.7846i	− 1.99883i
$503$	31.1769i	1.39011i	0.718957	+	0.695055i	$0.244620\pi$
$503$	31.1769i	1.39011i	−0.718957	+	0.695055i	$0.755380\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	− 4.92820i	− 0.218654i
$509$	−7.85641	−0.348229	−0.174115	−	0.984725i	$-0.555706\pi$
$509$	−7.85641	−0.348229	−0.174115	+	0.984725i	$0.555706\pi$
$510$	0	0
$511$	24.7846	1.09641
$512$	8.66025i	0.382733i
$513$	0	0
$514$	−13.6077	−0.600210
$515$	0	0
$516$	0	0
$517$	6.92820i	0.304702i
$518$	− 30.9282i	− 1.35891i
$519$	0	0
$520$	0	0
$521$	−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
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	5.07180	0.221562
$525$	0	0
$526$	47.5692	2.07412
$527$	0	0
$528$	0	0
$529$	−25.0000	−1.08696
$530$	0	0
$531$	0	0
$532$	2.92820i	0.126954i
$533$	− 5.07180i	− 0.219684i
$534$	0	0
$535$	0	0
$536$	13.8564	0.598506
$537$	0	0
$538$	− 13.6077i	− 0.586669i
$539$	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	$0.499017\pi$
$542$	− 56.1051i	− 2.40992i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	18.0000i	0.768922i
$549$	0	0
$550$	0	0
$551$	5.07180	0.216066
$552$	0	0
$553$	− 26.9282i	− 1.14510i
$554$	39.0333	1.65837
$555$	0	0
$556$	−8.39230	−0.355913
$557$	44.7846i	1.89758i	0.315900	+	0.948792i	$0.397694\pi$
$557$	44.7846i	1.89758i	−0.315900	+	0.948792i	$0.602306\pi$
$558$	0	0
$559$	13.0718	0.552878
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	10.3923i	0.437983i	0.975727	+	0.218992i	$0.0702768\pi$
$563$	10.3923i	0.437983i	−0.975727	+	0.218992i	$0.929723\pi$
$564$	0	0
$565$	0	0
$566$	−15.4641	−0.650005
$567$	0	0
$568$	24.0000i	1.00702i
$569$	−29.3205	−1.22918	−0.614590	−	0.788847i	$-0.710678\pi$
$569$	−29.3205	−1.22918	−0.614590	+	0.788847i	$0.710678\pi$
$570$	0	0
$571$	24.3923	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$571$	24.3923	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$572$	1.46410i	0.0612172i
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	− 22.7846i	− 0.948536i	−0.880381	−	0.474268i	$-0.842713\pi$
$577$	− 22.7846i	− 0.948536i	0.880381	−	0.474268i	$-0.157287\pi$
$578$	29.4449i	1.22474i
$579$	0	0
$580$	0	0
$581$	−30.9282	−1.28312
$582$	0	0
$583$	12.9282i	0.535431i
$584$	−21.4641	−0.888191
$585$	0	0
$586$	24.0000	0.991431
$587$	− 5.07180i	− 0.209335i	−0.994507	−	0.104668i	$-0.966622\pi$
$587$	− 5.07180i	− 0.209335i	0.994507	−	0.104668i	$-0.0333779\pi$
$588$	0	0
$589$	4.28719	0.176650
$590$	0	0
$591$	0	0
$592$	44.6410i	1.83473i
$593$	32.7846i	1.34630i	0.739505	+	0.673151i	$0.235060\pi$
$593$	32.7846i	1.34630i	−0.739505	+	0.673151i	$0.764940\pi$
$594$	0	0
$595$	0	0
$596$	8.53590	0.349644
$597$	0	0
$598$	17.5692i	0.718459i
$599$	−10.1436	−0.414456	−0.207228	−	0.978293i	$-0.566444\pi$
$599$	−10.1436	−0.414456	−0.207228	+	0.978293i	$0.566444\pi$
$600$	0	0
$601$	36.6410	1.49462	0.747309	−	0.664477i	$-0.231345\pi$
$601$	36.6410	1.49462	0.747309	+	0.664477i	$0.231345\pi$
$602$	30.9282i	1.26054i
$603$	0	0
$604$	−0.392305	−0.0159627
$605$	0	0
$606$	0	0
$607$	− 22.7846i	− 0.924799i	−0.886672	−	0.462399i	$-0.846989\pi$
$607$	− 22.7846i	− 0.924799i	0.886672	−	0.462399i	$-0.153011\pi$
$608$	− 7.60770i	− 0.308533i
$609$	0	0
$610$	0	0
$611$	10.1436	0.410366
$612$	0	0
$613$	0.392305i	0.0158450i	0.999969	+	0.00792252i	$0.00252184\pi$
$613$	0.392305i	0.0158450i	−0.999969	+	0.00792252i	$0.997478\pi$
$614$	24.2487	0.978598
$615$	0	0
$616$	3.46410	0.139573
$617$	− 23.0718i	− 0.928836i	−0.885616	−	0.464418i	$-0.846264\pi$
$617$	− 23.0718i	− 0.928836i	0.885616	−	0.464418i	$-0.153736\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	− 32.7846i	− 1.31454i
$623$	25.8564i	1.03592i
$624$	0	0
$625$	0	0
$626$	−12.2487	−0.489557
$627$	0	0
$628$	− 16.9282i	− 0.675509i
$629$	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	$0.744700\pi$
$632$	23.3205i	0.927640i
$633$	0	0
$634$	19.1769	0.761613
$635$	0	0
$636$	0	0
$637$	− 4.39230i	− 0.174029i
$638$	6.00000i	0.237542i
$639$	0	0
$640$	0	0
$641$	−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
$643$	− 45.5692i	− 1.79707i	−0.438897	−	0.898537i	$-0.644631\pi$
$643$	− 45.5692i	− 1.79707i	0.438897	−	0.898537i	$-0.355369\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	0	0
$647$	− 27.7128i	− 1.08950i	−0.838597	−	0.544752i	$-0.816624\pi$
$647$	− 27.7128i	− 1.08950i	0.838597	−	0.544752i	$-0.183376\pi$
$648$	0	0
$649$	6.92820	0.271956
$650$	0	0
$651$	0	0
$652$	17.8564i	0.699311i
$653$	7.85641i	0.307445i	0.988114	+	0.153722i	$0.0491262\pi$
$653$	7.85641i	0.307445i	−0.988114	+	0.153722i	$0.950874\pi$
$654$	0	0
$655$	0	0
$656$	−17.3205	−0.676252
$657$	0	0
$658$	24.0000i	0.935617i
$659$	39.7128	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$659$	39.7128	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	− 30.9282i	− 1.20206i
$663$	0	0
$664$	26.7846	1.03944
$665$	0	0
$666$	0	0
$667$	24.0000i	0.929284i
$668$	− 10.3923i	− 0.402090i
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	31.3205i	1.20732i	0.797243	+	0.603658i	$0.206291\pi$
$673$	31.3205i	1.20732i	−0.797243	+	0.603658i	$0.793709\pi$
$674$	−50.5359	−1.94657
$675$	0	0
$676$	−10.8564	−0.417554
$677$	− 32.7846i	− 1.26001i	−0.776589	−	0.630007i	$-0.783052\pi$
$677$	− 32.7846i	− 1.26001i	0.776589	−	0.630007i	$-0.216948\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	5.07180i	0.194209i
$683$	− 8.78461i	− 0.336134i	−0.985776	−	0.168067i	$-0.946248\pi$
$683$	− 8.78461i	− 0.336134i	0.985776	−	0.168067i	$-0.0537525\pi$
$684$	0	0
$685$	0	0
$686$	34.6410	1.32260
$687$	0	0
$688$	− 44.6410i	− 1.70192i
$689$	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	$0.546867\pi$
$692$	− 12.0000i	− 0.456172i
$693$	0	0
$694$	−2.78461	−0.105702
$695$	0	0
$696$	0	0
$697$	0	0
$698$	62.1051i	2.35071i
$699$	0	0
$700$	0	0
$701$	−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
$703$	− 13.0718i	− 0.493012i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−1.60770	−0.0605064
$707$	20.7846i	0.781686i
$708$	0	0
$709$	−15.8564	−0.595500	−0.297750	−	0.954644i	$-0.596236\pi$
$709$	−15.8564	−0.595500	−0.297750	+	0.954644i	$0.596236\pi$
$710$	0	0
$711$	0	0
$712$	− 22.3923i	− 0.839187i
$713$	20.2872i	0.759761i
$714$	0	0
$715$	0	0
$716$	6.92820	0.258919
$717$	0	0
$718$	36.0000i	1.34351i
$719$	−18.9282	−0.705903	−0.352951	−	0.935642i	$-0.614822\pi$
$719$	−18.9282	−0.705903	−0.352951	+	0.935642i	$0.614822\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 29.1962i	− 1.08657i
$723$	0	0
$724$	11.8564	0.440640
$725$	0	0
$726$	0	0
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	− 5.07180i	− 0.187973i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 49.9615i	− 1.84537i	−0.385553	−	0.922686i	$-0.625989\pi$
$733$	− 49.9615i	− 1.84537i	0.385553	−	0.922686i	$-0.374011\pi$
$734$	34.6410	1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−10.5359	−0.387569	−0.193785	−	0.981044i	$-0.562076\pi$
$739$	−10.5359	−0.387569	−0.193785	+	0.981044i	$0.562076\pi$
$740$	0	0
$741$	0	0
$742$	44.7846i	1.64409i
$743$	− 46.3923i	− 1.70197i	−0.525191	−	0.850984i	$-0.676006\pi$
$743$	− 46.3923i	− 1.70197i	0.525191	−	0.850984i	$-0.323994\pi$
$744$	0	0
$745$	0	0
$746$	−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	$0.423345\pi$
$752$	− 34.6410i	− 1.26323i
$753$	0	0
$754$	8.78461	0.319917
$755$	0	0
$756$	0	0
$757$	6.78461i	0.246591i	0.992370	+	0.123295i	$0.0393463\pi$
$757$	6.78461i	0.246591i	−0.992370	+	0.123295i	$0.960654\pi$
$758$	− 17.0718i	− 0.620076i
$759$	0	0
$760$	0	0
$761$	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
$763$	− 20.0000i	− 0.724049i
$764$	−5.07180	−0.183491
$765$	0	0
$766$	24.0000	0.867155
$767$	− 10.1436i	− 0.366264i
$768$	0	0
$769$	46.4974	1.67674	0.838370	−	0.545102i	$-0.183509\pi$
$769$	46.4974	1.67674	0.838370	+	0.545102i	$0.183509\pi$
$770$	0	0
$771$	0	0
$772$	− 3.60770i	− 0.129844i
$773$	31.8564i	1.14580i	0.819627	+	0.572898i	$0.194181\pi$
$773$	31.8564i	1.14580i	−0.819627	+	0.572898i	$0.805819\pi$
$774$	0	0
$775$	0	0
$776$	−17.3205	−0.621770
$777$	0	0
$778$	43.1769i	1.54797i
$779$	5.07180	0.181716
$780$	0	0
$781$	13.8564	0.495821
$782$	0	0
$783$	0	0
$784$	−15.0000	−0.535714
$785$	0	0
$786$	0	0
$787$	18.7846i	0.669599i	0.942289	+	0.334800i	$0.108669\pi$
$787$	18.7846i	0.669599i	−0.942289	+	0.334800i	$0.891331\pi$
$788$	12.0000i	0.427482i
$789$	0	0
$790$	0	0
$791$	−1.85641	−0.0660062
$792$	0	0
$793$	− 2.92820i	− 0.103984i
$794$	3.46410	0.122936
$795$	0	0
$796$	16.7846	0.594915
$797$	16.6410i	0.589455i	0.955581	+	0.294728i	$0.0952289\pi$
$797$	16.6410i	0.589455i	−0.955581	+	0.294728i	$0.904771\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 34.3923i	− 1.21443i
$803$	12.3923i	0.437315i
$804$	0	0
$805$	0	0
$806$	7.42563	0.261557
$807$	0	0
$808$	− 18.0000i	− 0.633238i
$809$	−8.53590	−0.300106	−0.150053	−	0.988678i	$-0.547944\pi$
$809$	−8.53590	−0.300106	−0.150053	+	0.988678i	$0.547944\pi$
$810$	0	0
$811$	−8.39230	−0.294694	−0.147347	−	0.989085i	$-0.547073\pi$
$811$	−8.39230	−0.294694	−0.147347	+	0.989085i	$0.547073\pi$
$812$	6.92820i	0.243132i
$813$	0	0
$814$	15.4641	0.542016
$815$	0	0
$816$	0	0
$817$	13.0718i	0.457324i
$818$	− 60.2487i	− 2.10655i
$819$	0	0
$820$	0	0
$821$	−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
$823$	49.5692i	1.72787i	0.503600	+	0.863937i	$0.332009\pi$
$823$	49.5692i	1.72787i	−0.503600	+	0.863937i	$0.667991\pi$
$824$	−13.8564	−0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	1.60770i	0.0559050i	0.999609	+	0.0279525i	$0.00889872\pi$
$827$	1.60770i	0.0559050i	−0.999609	+	0.0279525i	$0.991101\pi$
$828$	0	0
$829$	25.7128	0.893043	0.446521	−	0.894773i	$-0.352663\pi$
$829$	25.7128	0.893043	0.446521	+	0.894773i	$0.352663\pi$
$830$	0	0
$831$	0	0
$832$	1.46410i	0.0507586i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−1.46410	−0.0506370
$837$	0	0
$838$	29.5692i	1.02145i
$839$	15.2154	0.525294	0.262647	−	0.964892i	$-0.415405\pi$
$839$	15.2154	0.525294	0.262647	+	0.964892i	$0.415405\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	3.46410i	0.119381i
$843$	0	0
$844$	−12.3923	−0.426561
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	− 64.6410i	− 2.21978i
$849$	0	0
$850$	0	0
$851$	61.8564	2.12041
$852$	0	0
$853$	24.3923i	0.835177i	0.908636	+	0.417588i	$0.137125\pi$
$853$	24.3923i	0.835177i	−0.908636	+	0.417588i	$0.862875\pi$
$854$	6.92820	0.237078
$855$	0	0
$856$	−26.7846	−0.915479
$857$	− 10.1436i	− 0.346499i	−0.984878	−	0.173249i	$-0.944573\pi$
$857$	− 10.1436i	− 0.346499i	0.984878	−	0.173249i	$-0.0554266\pi$
$858$	0	0
$859$	−47.7128	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$859$	−47.7128	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$860$	0	0
$861$	0	0
$862$	56.7846i	1.93409i
$863$	10.1436i	0.345292i	0.984984	+	0.172646i	$0.0552317\pi$
$863$	10.1436i	0.345292i	−0.984984	+	0.172646i	$0.944768\pi$
$864$	0	0
$865$	0	0
$866$	−48.2487	−1.63956
$867$	0	0
$868$	5.85641i	0.198779i
$869$	13.4641	0.456738
$870$	0	0
$871$	−11.7128	−0.396874
$872$	17.3205i	0.586546i
$873$	0	0
$874$	−17.5692	−0.594288
$875$	0	0
$876$	0	0
$877$	− 14.2487i	− 0.481145i	−0.970631	−	0.240572i	$-0.922665\pi$
$877$	− 14.2487i	− 0.481145i	0.970631	−	0.240572i	$-0.0773351\pi$
$878$	50.5359i	1.70550i
$879$	0	0
$880$	0	0
$881$	−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
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	20.7846	0.698273
$887$	− 36.2487i	− 1.21711i	−0.793511	−	0.608556i	$-0.791749\pi$
$887$	− 36.2487i	− 1.21711i	0.793511	−	0.608556i	$-0.208251\pi$
$888$	0	0
$889$	9.85641	0.330573
$890$	0	0
$891$	0	0
$892$	17.8564i	0.597877i
$893$	10.1436i	0.339442i
$894$	0	0
$895$	0	0
$896$	−24.2487	−0.810093
$897$	0	0
$898$	25.6077i	0.854540i
$899$	10.1436	0.338308
$900$	0	0
$901$	0	0
$902$	6.00000i	0.199778i
$903$	0	0
$904$	1.60770	0.0534711
$905$	0	0
$906$	0	0
$907$	− 45.8564i	− 1.52264i	−0.648378	−	0.761318i	$-0.724552\pi$
$907$	− 45.8564i	− 1.52264i	0.648378	−	0.761318i	$-0.275448\pi$
$908$	− 8.53590i	− 0.283274i
$909$	0	0
$910$	0	0
$911$	−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
$913$	− 15.4641i	− 0.511787i
$914$	−14.5359	−0.480805
$915$	0	0
$916$	3.85641	0.127419
$917$	10.1436i	0.334971i
$918$	0	0
$919$	11.6077	0.382903	0.191451	−	0.981502i	$-0.438681\pi$
$919$	11.6077	0.382903	0.191451	+	0.981502i	$0.438681\pi$
$920$	0	0
$921$	0	0
$922$	21.2154i	0.698692i
$923$	− 20.2872i	− 0.667761i
$924$	0	0
$925$	0	0
$926$	48.4974	1.59372
$927$	0	0
$928$	− 18.0000i	− 0.590879i
$929$	−38.7846	−1.27248	−0.636241	−	0.771490i	$-0.719512\pi$
$929$	−38.7846	−1.27248	−0.636241	+	0.771490i	$0.719512\pi$
$930$	0	0
$931$	4.39230	0.143952
$932$	12.0000i	0.393073i
$933$	0	0
$934$	−32.7846	−1.07275
$935$	0	0
$936$	0	0
$937$	− 0.392305i	− 0.0128160i	−0.999979	−	0.00640802i	$-0.997960\pi$
$937$	− 0.392305i	− 0.0128160i	0.999979	−	0.00640802i	$-0.00203975\pi$
$938$	− 27.7128i	− 0.904855i
$939$	0	0
$940$	0	0
$941$	−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
$943$	24.0000i	0.781548i
$944$	−34.6410	−1.12747
$945$	0	0
$946$	−15.4641	−0.502781
$947$	− 5.07180i	− 0.164811i	−0.996599	−	0.0824056i	$-0.973740\pi$
$947$	− 5.07180i	− 0.164811i	0.996599	−	0.0824056i	$-0.0262603\pi$
$948$	0	0
$949$	18.1436	0.588966
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.7846i	1.45072i	0.688372	+	0.725358i	$0.258326\pi$
$953$	44.7846i	1.45072i	−0.688372	+	0.725358i	$0.741674\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	20.7846i	0.671520i
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−22.4256	−0.723407
$962$	− 22.6410i	− 0.729976i
$963$	0	0
$964$	−27.8564	−0.897194
$965$	0	0
$966$	0	0
$967$	18.7846i	0.604072i	0.953296	+	0.302036i	$0.0976663\pi$
$967$	18.7846i	0.604072i	−0.953296	+	0.302036i	$0.902334\pi$
$968$	1.73205i	0.0556702i
$969$	0	0
$970$	0	0
$971$	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
$973$	− 16.7846i	− 0.538090i
$974$	41.0718	1.31603
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 35.5692i	− 1.13796i	−0.822351	−	0.568980i	$-0.807338\pi$
$977$	− 35.5692i	− 1.13796i	0.822351	−	0.568980i	$-0.192662\pi$
$978$	0	0
$979$	−12.9282	−0.413187
$980$	0	0
$981$	0	0
$982$	29.5692i	0.943592i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	2.14359i	0.0681968i
$989$	−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	$0.782173\pi$
$992$	− 15.2154i	− 0.483089i
$993$	0	0
$994$	48.0000	1.52247
$995$	0	0
$996$	0	0
$997$	− 48.3923i	− 1.53260i	−0.642483	−	0.766300i	$-0.722096\pi$
$997$	− 48.3923i	− 1.53260i	0.642483	−	0.766300i	$-0.277904\pi$
$998$	22.1436i	0.700943i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.n.199.3		4
3.2	odd	2		825.2.c.c.199.2		4
5.2	odd	4		495.2.a.c.1.1		2
5.3	odd	4		2475.2.a.r.1.2		2
5.4	even	2	inner	2475.2.c.n.199.2		4
15.2	even	4		165.2.a.b.1.2	✓	2
15.8	even	4		825.2.a.e.1.1		2
15.14	odd	2		825.2.c.c.199.3		4
20.7	even	4		7920.2.a.bz.1.1		2
55.32	even	4		5445.2.a.s.1.2		2
60.47	odd	4		2640.2.a.x.1.1		2
105.62	odd	4		8085.2.a.bd.1.2		2
165.32	odd	4		1815.2.a.i.1.1		2
165.98	odd	4		9075.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.2	✓	2	15.2	even	4
495.2.a.c.1.1		2	5.2	odd	4
825.2.a.e.1.1		2	15.8	even	4
825.2.c.c.199.2		4	3.2	odd	2
825.2.c.c.199.3		4	15.14	odd	2
1815.2.a.i.1.1		2	165.32	odd	4
2475.2.a.r.1.2		2	5.3	odd	4
2475.2.c.n.199.2		4	5.4	even	2	inner
2475.2.c.n.199.3		4	1.1	even	1	trivial
2640.2.a.x.1.1		2	60.47	odd	4
5445.2.a.s.1.2		2	55.32	even	4
7920.2.a.bz.1.1		2	20.7	even	4
8085.2.a.bd.1.2		2	105.62	odd	4
9075.2.a.bh.1.2		2	165.98	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.73205i q^{8} +O(q^{10})\)	\(q+1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.73205i q^{8} +1.00000 q^{11} -1.46410i q^{13} +3.46410 q^{14} -5.00000 q^{16} +1.46410 q^{19} +1.73205i q^{22} +6.92820i q^{23} +2.53590 q^{26} +2.00000i q^{28} +3.46410 q^{29} +2.92820 q^{31} -5.19615i q^{32} -8.92820i q^{37} +2.53590i q^{38} +3.46410 q^{41} +8.92820i q^{43} -1.00000 q^{44} -12.0000 q^{46} +6.92820i q^{47} +3.00000 q^{49} +1.46410i q^{52} +12.9282i q^{53} +3.46410 q^{56} +6.00000i q^{58} +6.92820 q^{59} +2.00000 q^{61} +5.07180i q^{62} -1.00000 q^{64} -8.00000i q^{67} +13.8564 q^{71} +12.3923i q^{73} +15.4641 q^{74} -1.46410 q^{76} -2.00000i q^{77} +13.4641 q^{79} +6.00000i q^{82} -15.4641i q^{83} -15.4641 q^{86} +1.73205i q^{88} -12.9282 q^{89} -2.92820 q^{91} -6.92820i q^{92} -12.0000 q^{94} +10.0000i q^{97} +5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{11} - 20 q^{16} - 8 q^{19} + 24 q^{26} - 16 q^{31} - 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} + 48 q^{74} + 8 q^{76} + 40 q^{79} - 48 q^{86} - 24 q^{89} + 16 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^11 - 20 * q^16 - 8 * q^19 + 24 * q^26 - 16 * q^31 - 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 + 48 * q^74 + 8 * q^76 + 40 * q^79 - 48 * q^86 - 24 * q^89 + 16 * q^91 - 48 * q^94

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