LMFDB - Embedded newform 225.3.c.a.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(26,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.3.c.a.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +2.00000 q^{4} -9.00000 q^{7} -8.48528i q^{8} +O(q^{10})$	$q-1.41421i q^{2} +2.00000 q^{4} -9.00000 q^{7} -8.48528i q^{8} -18.3848i q^{11} +1.00000 q^{13} +12.7279i q^{14} -4.00000 q^{16} -26.8701i q^{17} +25.0000 q^{19} -26.0000 q^{22} +4.24264i q^{23} -1.41421i q^{26} -18.0000 q^{28} +26.8701i q^{29} -39.0000 q^{31} -28.2843i q^{32} -38.0000 q^{34} +32.0000 q^{37} -35.3553i q^{38} +5.65685i q^{41} -23.0000 q^{43} -36.7696i q^{44} +6.00000 q^{46} +32.5269i q^{47} +32.0000 q^{49} +2.00000 q^{52} +96.1665i q^{53} +76.3675i q^{56} +38.0000 q^{58} -9.89949i q^{59} +73.0000 q^{61} +55.1543i q^{62} -56.0000 q^{64} +63.0000 q^{67} -53.7401i q^{68} -62.2254i q^{71} +136.000 q^{73} -45.2548i q^{74} +50.0000 q^{76} +165.463i q^{77} -24.0000 q^{79} +8.00000 q^{82} +46.6690i q^{83} +32.5269i q^{86} -156.000 q^{88} -101.823i q^{89} -9.00000 q^{91} +8.48528i q^{92} +46.0000 q^{94} -7.00000 q^{97} -45.2548i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 18 q^{7}+O(q^{10}) $ 2 * q + 4 * q^4 - 18 * q^7	$ 2 q + 4 q^{4} - 18 q^{7} + 2 q^{13} - 8 q^{16} + 50 q^{19} - 52 q^{22} - 36 q^{28} - 78 q^{31} - 76 q^{34} + 64 q^{37} - 46 q^{43} + 12 q^{46} + 64 q^{49} + 4 q^{52} + 76 q^{58} + 146 q^{61} - 112 q^{64} + 126 q^{67} + 272 q^{73} + 100 q^{76} - 48 q^{79} + 16 q^{82} - 312 q^{88} - 18 q^{91} + 92 q^{94} - 14 q^{97}+O(q^{100}) $ 2 * q + 4 * q^4 - 18 * q^7 + 2 * q^13 - 8 * q^16 + 50 * q^19 - 52 * q^22 - 36 * q^28 - 78 * q^31 - 76 * q^34 + 64 * q^37 - 46 * q^43 + 12 * q^46 + 64 * q^49 + 4 * q^52 + 76 * q^58 + 146 * q^61 - 112 * q^64 + 126 * q^67 + 272 * q^73 + 100 * q^76 - 48 * q^79 + 16 * q^82 - 312 * q^88 - 18 * q^91 + 92 * q^94 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.41421i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−9.00000	−1.28571	−0.642857	−	0.765986i	$-0.722251\pi$
$7$	−9.00000	−1.28571	−0.642857	+	0.765986i	$0.722251\pi$
$8$	− 8.48528i	− 1.06066i
$9$	0	0
$10$	0	0
$11$	− 18.3848i	− 1.67134i	−0.549229	−	0.835672i	$-0.685079\pi$
$11$	− 18.3848i	− 1.67134i	0.549229	−	0.835672i	$-0.314921\pi$
$12$	0	0
$13$	1.00000	0.0769231	0.0384615	−	0.999260i	$-0.487754\pi$
$13$	1.00000	0.0769231	0.0384615	+	0.999260i	$0.487754\pi$
$14$	12.7279i	0.909137i
$15$	0	0
$16$	−4.00000	−0.250000
$17$	− 26.8701i	− 1.58059i	−0.612725	−	0.790296i	$-0.709927\pi$
$17$	− 26.8701i	− 1.58059i	0.612725	−	0.790296i	$-0.290073\pi$
$18$	0	0
$19$	25.0000	1.31579	0.657895	−	0.753110i	$-0.271447\pi$
$19$	25.0000	1.31579	0.657895	+	0.753110i	$0.271447\pi$
$20$	0	0
$21$	0	0
$22$	−26.0000	−1.18182
$23$	4.24264i	0.184463i	0.995738	+	0.0922313i	$0.0293999\pi$
$23$	4.24264i	0.184463i	−0.995738	+	0.0922313i	$0.970600\pi$
$24$	0	0
$25$	0	0
$26$	− 1.41421i	− 0.0543928i
$27$	0	0
$28$	−18.0000	−0.642857
$29$	26.8701i	0.926554i	0.886214	+	0.463277i	$0.153326\pi$
$29$	26.8701i	0.926554i	−0.886214	+	0.463277i	$0.846674\pi$
$30$	0	0
$31$	−39.0000	−1.25806	−0.629032	−	0.777379i	$-0.716549\pi$
$31$	−39.0000	−1.25806	−0.629032	+	0.777379i	$0.716549\pi$
$32$	− 28.2843i	− 0.883883i
$33$	0	0
$34$	−38.0000	−1.11765
$35$	0	0
$36$	0	0
$37$	32.0000	0.864865	0.432432	−	0.901666i	$-0.357655\pi$
$37$	32.0000	0.864865	0.432432	+	0.901666i	$0.357655\pi$
$38$	− 35.3553i	− 0.930404i
$39$	0	0
$40$	0	0
$41$	5.65685i	0.137972i	0.997618	+	0.0689860i	$0.0219764\pi$
$41$	5.65685i	0.137972i	−0.997618	+	0.0689860i	$0.978024\pi$
$42$	0	0
$43$	−23.0000	−0.534884	−0.267442	−	0.963574i	$-0.586178\pi$
$43$	−23.0000	−0.534884	−0.267442	+	0.963574i	$0.586178\pi$
$44$	− 36.7696i	− 0.835672i
$45$	0	0
$46$	6.00000	0.130435
$47$	32.5269i	0.692062i	0.938223	+	0.346031i	$0.112471\pi$
$47$	32.5269i	0.692062i	−0.938223	+	0.346031i	$0.887529\pi$
$48$	0	0
$49$	32.0000	0.653061
$50$	0	0
$51$	0	0
$52$	2.00000	0.0384615
$53$	96.1665i	1.81446i	0.420632	+	0.907231i	$0.361808\pi$
$53$	96.1665i	1.81446i	−0.420632	+	0.907231i	$0.638192\pi$
$54$	0	0
$55$	0	0
$56$	76.3675i	1.36371i
$57$	0	0
$58$	38.0000	0.655172
$59$	− 9.89949i	− 0.167788i	−0.996475	−	0.0838940i	$-0.973264\pi$
$59$	− 9.89949i	− 0.167788i	0.996475	−	0.0838940i	$-0.0267357\pi$
$60$	0	0
$61$	73.0000	1.19672	0.598361	−	0.801227i	$-0.295819\pi$
$61$	73.0000	1.19672	0.598361	+	0.801227i	$0.295819\pi$
$62$	55.1543i	0.889586i
$63$	0	0
$64$	−56.0000	−0.875000
$65$	0	0
$66$	0	0
$67$	63.0000	0.940299	0.470149	−	0.882587i	$-0.344200\pi$
$67$	63.0000	0.940299	0.470149	+	0.882587i	$0.344200\pi$
$68$	− 53.7401i	− 0.790296i
$69$	0	0
$70$	0	0
$71$	− 62.2254i	− 0.876414i	−0.898874	−	0.438207i	$-0.855614\pi$
$71$	− 62.2254i	− 0.876414i	0.898874	−	0.438207i	$-0.144386\pi$
$72$	0	0
$73$	136.000	1.86301	0.931507	−	0.363724i	$-0.118495\pi$
$73$	136.000	1.86301	0.931507	+	0.363724i	$0.118495\pi$
$74$	− 45.2548i	− 0.611552i
$75$	0	0
$76$	50.0000	0.657895
$77$	165.463i	2.14887i
$78$	0	0
$79$	−24.0000	−0.303797	−0.151899	−	0.988396i	$-0.548539\pi$
$79$	−24.0000	−0.303797	−0.151899	+	0.988396i	$0.548539\pi$
$80$	0	0
$81$	0	0
$82$	8.00000	0.0975610
$83$	46.6690i	0.562278i	0.959667	+	0.281139i	$0.0907121\pi$
$83$	46.6690i	0.562278i	−0.959667	+	0.281139i	$0.909288\pi$
$84$	0	0
$85$	0	0
$86$	32.5269i	0.378220i
$87$	0	0
$88$	−156.000	−1.77273
$89$	− 101.823i	− 1.14408i	−0.820225	−	0.572041i	$-0.806152\pi$
$89$	− 101.823i	− 1.14408i	0.820225	−	0.572041i	$-0.193848\pi$
$90$	0	0
$91$	−9.00000	−0.0989011
$92$	8.48528i	0.0922313i
$93$	0	0
$94$	46.0000	0.489362
$95$	0	0
$96$	0	0
$97$	−7.00000	−0.0721649	−0.0360825	−	0.999349i	$-0.511488\pi$
$97$	−7.00000	−0.0721649	−0.0360825	+	0.999349i	$0.511488\pi$
$98$	− 45.2548i	− 0.461784i
$99$	0	0
$100$	0	0
$101$	50.9117i	0.504076i	0.967717	+	0.252038i	$0.0811008\pi$
$101$	50.9117i	0.504076i	−0.967717	+	0.252038i	$0.918899\pi$
$102$	0	0
$103$	110.000	1.06796	0.533981	−	0.845497i	$-0.320696\pi$
$103$	110.000	1.06796	0.533981	+	0.845497i	$0.320696\pi$
$104$	− 8.48528i	− 0.0815892i
$105$	0	0
$106$	136.000	1.28302
$107$	− 118.794i	− 1.11022i	−0.831776	−	0.555112i	$-0.812675\pi$
$107$	− 118.794i	− 1.11022i	0.831776	−	0.555112i	$-0.187325\pi$
$108$	0	0
$109$	−25.0000	−0.229358	−0.114679	−	0.993403i	$-0.536584\pi$
$109$	−25.0000	−0.229358	−0.114679	+	0.993403i	$0.536584\pi$
$110$	0	0
$111$	0	0
$112$	36.0000	0.321429
$113$	45.2548i	0.400485i	0.979746	+	0.200243i	$0.0641730\pi$
$113$	45.2548i	0.400485i	−0.979746	+	0.200243i	$0.935827\pi$
$114$	0	0
$115$	0	0
$116$	53.7401i	0.463277i
$117$	0	0
$118$	−14.0000	−0.118644
$119$	241.831i	2.03219i
$120$	0	0
$121$	−217.000	−1.79339
$122$	− 103.238i	− 0.846210i
$123$	0	0
$124$	−78.0000	−0.629032
$125$	0	0
$126$	0	0
$127$	−80.0000	−0.629921	−0.314961	−	0.949105i	$-0.601991\pi$
$127$	−80.0000	−0.629921	−0.314961	+	0.949105i	$0.601991\pi$
$128$	− 33.9411i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	− 84.8528i	− 0.647731i	−0.946103	−	0.323866i	$-0.895017\pi$
$131$	− 84.8528i	− 0.647731i	0.946103	−	0.323866i	$-0.104983\pi$
$132$	0	0
$133$	−225.000	−1.69173
$134$	− 89.0955i	− 0.664891i
$135$	0	0
$136$	−228.000	−1.67647
$137$	21.2132i	0.154841i	0.996999	+	0.0774205i	$0.0246684\pi$
$137$	21.2132i	0.154841i	−0.996999	+	0.0774205i	$0.975332\pi$
$138$	0	0
$139$	104.000	0.748201	0.374101	−	0.927388i	$-0.377951\pi$
$139$	104.000	0.748201	0.374101	+	0.927388i	$0.377951\pi$
$140$	0	0
$141$	0	0
$142$	−88.0000	−0.619718
$143$	− 18.3848i	− 0.128565i
$144$	0	0
$145$	0	0
$146$	− 192.333i	− 1.31735i
$147$	0	0
$148$	64.0000	0.432432
$149$	171.120i	1.14846i	0.818696	+	0.574228i	$0.194698\pi$
$149$	171.120i	1.14846i	−0.818696	+	0.574228i	$0.805302\pi$
$150$	0	0
$151$	−17.0000	−0.112583	−0.0562914	−	0.998414i	$-0.517928\pi$
$151$	−17.0000	−0.112583	−0.0562914	+	0.998414i	$0.517928\pi$
$152$	− 212.132i	− 1.39561i
$153$	0	0
$154$	234.000	1.51948
$155$	0	0
$156$	0	0
$157$	65.0000	0.414013	0.207006	−	0.978340i	$-0.433628\pi$
$157$	65.0000	0.414013	0.207006	+	0.978340i	$0.433628\pi$
$158$	33.9411i	0.214817i
$159$	0	0
$160$	0	0
$161$	− 38.1838i	− 0.237166i
$162$	0	0
$163$	17.0000	0.104294	0.0521472	−	0.998639i	$-0.483393\pi$
$163$	17.0000	0.104294	0.0521472	+	0.998639i	$0.483393\pi$
$164$	11.3137i	0.0689860i
$165$	0	0
$166$	66.0000	0.397590
$167$	− 254.558i	− 1.52430i	−0.647399	−	0.762151i	$-0.724143\pi$
$167$	− 254.558i	− 1.52430i	0.647399	−	0.762151i	$-0.275857\pi$
$168$	0	0
$169$	−168.000	−0.994083
$170$	0	0
$171$	0	0
$172$	−46.0000	−0.267442
$173$	− 173.948i	− 1.00548i	−0.864437	−	0.502741i	$-0.832325\pi$
$173$	− 173.948i	− 1.00548i	0.864437	−	0.502741i	$-0.167675\pi$
$174$	0	0
$175$	0	0
$176$	73.5391i	0.417836i
$177$	0	0
$178$	−144.000	−0.808989
$179$	125.865i	0.703156i	0.936159	+	0.351578i	$0.114355\pi$
$179$	125.865i	0.703156i	−0.936159	+	0.351578i	$0.885645\pi$
$180$	0	0
$181$	119.000	0.657459	0.328729	−	0.944424i	$-0.393380\pi$
$181$	119.000	0.657459	0.328729	+	0.944424i	$0.393380\pi$
$182$	12.7279i	0.0699336i
$183$	0	0
$184$	36.0000	0.195652
$185$	0	0
$186$	0	0
$187$	−494.000	−2.64171
$188$	65.0538i	0.346031i
$189$	0	0
$190$	0	0
$191$	− 15.5563i	− 0.0814469i	−0.999170	−	0.0407234i	$-0.987034\pi$
$191$	− 15.5563i	− 0.0814469i	0.999170	−	0.0407234i	$-0.0129663\pi$
$192$	0	0
$193$	281.000	1.45596	0.727979	−	0.685599i	$-0.240460\pi$
$193$	281.000	1.45596	0.727979	+	0.685599i	$0.240460\pi$
$194$	9.89949i	0.0510283i
$195$	0	0
$196$	64.0000	0.326531
$197$	224.860i	1.14142i	0.821151	+	0.570711i	$0.193332\pi$
$197$	224.860i	1.14142i	−0.821151	+	0.570711i	$0.806668\pi$
$198$	0	0
$199$	175.000	0.879397	0.439698	−	0.898145i	$-0.355085\pi$
$199$	175.000	0.879397	0.439698	+	0.898145i	$0.355085\pi$
$200$	0	0
$201$	0	0
$202$	72.0000	0.356436
$203$	− 241.831i	− 1.19128i
$204$	0	0
$205$	0	0
$206$	− 155.563i	− 0.755163i
$207$	0	0
$208$	−4.00000	−0.0192308
$209$	− 459.619i	− 2.19914i
$210$	0	0
$211$	−49.0000	−0.232227	−0.116114	−	0.993236i	$-0.537044\pi$
$211$	−49.0000	−0.232227	−0.116114	+	0.993236i	$0.537044\pi$
$212$	192.333i	0.907231i
$213$	0	0
$214$	−168.000	−0.785047
$215$	0	0
$216$	0	0
$217$	351.000	1.61751
$218$	35.3553i	0.162180i
$219$	0	0
$220$	0	0
$221$	− 26.8701i	− 0.121584i
$222$	0	0
$223$	−265.000	−1.18834	−0.594170	−	0.804339i	$-0.702520\pi$
$223$	−265.000	−1.18834	−0.594170	+	0.804339i	$0.702520\pi$
$224$	254.558i	1.13642i
$225$	0	0
$226$	64.0000	0.283186
$227$	− 130.108i	− 0.573161i	−0.958056	−	0.286581i	$-0.907481\pi$
$227$	− 130.108i	− 0.573161i	0.958056	−	0.286581i	$-0.0925187\pi$
$228$	0	0
$229$	−87.0000	−0.379913	−0.189956	−	0.981793i	$-0.560835\pi$
$229$	−87.0000	−0.379913	−0.189956	+	0.981793i	$0.560835\pi$
$230$	0	0
$231$	0	0
$232$	228.000	0.982759
$233$	152.735i	0.655515i	0.944762	+	0.327758i	$0.106293\pi$
$233$	152.735i	0.655515i	−0.944762	+	0.327758i	$0.893707\pi$
$234$	0	0
$235$	0	0
$236$	− 19.7990i	− 0.0838940i
$237$	0	0
$238$	342.000	1.43697
$239$	− 277.186i	− 1.15977i	−0.814697	−	0.579887i	$-0.803097\pi$
$239$	− 277.186i	− 1.15977i	0.814697	−	0.579887i	$-0.196903\pi$
$240$	0	0
$241$	415.000	1.72199	0.860996	−	0.508612i	$-0.169841\pi$
$241$	415.000	1.72199	0.860996	+	0.508612i	$0.169841\pi$
$242$	306.884i	1.26812i
$243$	0	0
$244$	146.000	0.598361
$245$	0	0
$246$	0	0
$247$	25.0000	0.101215
$248$	330.926i	1.33438i
$249$	0	0
$250$	0	0
$251$	373.352i	1.48746i	0.668480	+	0.743730i	$0.266945\pi$
$251$	373.352i	1.48746i	−0.668480	+	0.743730i	$0.733055\pi$
$252$	0	0
$253$	78.0000	0.308300
$254$	113.137i	0.445422i
$255$	0	0
$256$	−272.000	−1.06250
$257$	271.529i	1.05653i	0.849079	+	0.528267i	$0.177158\pi$
$257$	271.529i	1.05653i	−0.849079	+	0.528267i	$0.822842\pi$
$258$	0	0
$259$	−288.000	−1.11197
$260$	0	0
$261$	0	0
$262$	−120.000	−0.458015
$263$	209.304i	0.795831i	0.917422	+	0.397916i	$0.130266\pi$
$263$	209.304i	0.795831i	−0.917422	+	0.397916i	$0.869734\pi$
$264$	0	0
$265$	0	0
$266$	318.198i	1.19623i
$267$	0	0
$268$	126.000	0.470149
$269$	− 94.7523i	− 0.352239i	−0.984369	−	0.176120i	$-0.943645\pi$
$269$	− 94.7523i	− 0.352239i	0.984369	−	0.176120i	$-0.0563545\pi$
$270$	0	0
$271$	−176.000	−0.649446	−0.324723	−	0.945809i	$-0.605271\pi$
$271$	−176.000	−0.649446	−0.324723	+	0.945809i	$0.605271\pi$
$272$	107.480i	0.395148i
$273$	0	0
$274$	30.0000	0.109489
$275$	0	0
$276$	0	0
$277$	303.000	1.09386	0.546931	−	0.837177i	$-0.315796\pi$
$277$	303.000	1.09386	0.546931	+	0.837177i	$0.315796\pi$
$278$	− 147.078i	− 0.529058i
$279$	0	0
$280$	0	0
$281$	513.360i	1.82690i	0.406948	+	0.913451i	$0.366593\pi$
$281$	513.360i	1.82690i	−0.406948	+	0.913451i	$0.633407\pi$
$282$	0	0
$283$	−393.000	−1.38869	−0.694346	−	0.719641i	$-0.744306\pi$
$283$	−393.000	−1.38869	−0.694346	+	0.719641i	$0.744306\pi$
$284$	− 124.451i	− 0.438207i
$285$	0	0
$286$	−26.0000	−0.0909091
$287$	− 50.9117i	− 0.177393i
$288$	0	0
$289$	−433.000	−1.49827
$290$	0	0
$291$	0	0
$292$	272.000	0.931507
$293$	− 46.6690i	− 0.159280i	−0.996824	−	0.0796400i	$-0.974623\pi$
$293$	− 46.6690i	− 0.159280i	0.996824	−	0.0796400i	$-0.0253771\pi$
$294$	0	0
$295$	0	0
$296$	− 271.529i	− 0.917328i
$297$	0	0
$298$	242.000	0.812081
$299$	4.24264i	0.0141894i
$300$	0	0
$301$	207.000	0.687708
$302$	24.0416i	0.0796080i
$303$	0	0
$304$	−100.000	−0.328947
$305$	0	0
$306$	0	0
$307$	289.000	0.941368	0.470684	−	0.882302i	$-0.344007\pi$
$307$	289.000	0.941368	0.470684	+	0.882302i	$0.344007\pi$
$308$	330.926i	1.07443i
$309$	0	0
$310$	0	0
$311$	− 462.448i	− 1.48697i	−0.668752	−	0.743485i	$-0.733171\pi$
$311$	− 462.448i	− 1.48697i	0.668752	−	0.743485i	$-0.266829\pi$
$312$	0	0
$313$	−481.000	−1.53674	−0.768371	−	0.640005i	$-0.778932\pi$
$313$	−481.000	−1.53674	−0.768371	+	0.640005i	$0.778932\pi$
$314$	− 91.9239i	− 0.292751i
$315$	0	0
$316$	−48.0000	−0.151899
$317$	− 11.3137i	− 0.0356899i	−0.999841	−	0.0178450i	$-0.994319\pi$
$317$	− 11.3137i	− 0.0356899i	0.999841	−	0.0178450i	$-0.00568053\pi$
$318$	0	0
$319$	494.000	1.54859
$320$	0	0
$321$	0	0
$322$	−54.0000	−0.167702
$323$	− 671.751i	− 2.07973i
$324$	0	0
$325$	0	0
$326$	− 24.0416i	− 0.0737473i
$327$	0	0
$328$	48.0000	0.146341
$329$	− 292.742i	− 0.889794i
$330$	0	0
$331$	−72.0000	−0.217523	−0.108761	−	0.994068i	$-0.534688\pi$
$331$	−72.0000	−0.217523	−0.108761	+	0.994068i	$0.534688\pi$
$332$	93.3381i	0.281139i
$333$	0	0
$334$	−360.000	−1.07784
$335$	0	0
$336$	0	0
$337$	−335.000	−0.994065	−0.497033	−	0.867732i	$-0.665577\pi$
$337$	−335.000	−0.994065	−0.497033	+	0.867732i	$0.665577\pi$
$338$	237.588i	0.702923i
$339$	0	0
$340$	0	0
$341$	717.006i	2.10266i
$342$	0	0
$343$	153.000	0.446064
$344$	195.161i	0.567330i
$345$	0	0
$346$	−246.000	−0.710983
$347$	− 383.252i	− 1.10447i	−0.833688	−	0.552236i	$-0.813775\pi$
$347$	− 383.252i	− 1.10447i	0.833688	−	0.552236i	$-0.186225\pi$
$348$	0	0
$349$	568.000	1.62751	0.813754	−	0.581210i	$-0.197421\pi$
$349$	568.000	1.62751	0.813754	+	0.581210i	$0.197421\pi$
$350$	0	0
$351$	0	0
$352$	−520.000	−1.47727
$353$	131.522i	0.372583i	0.982495	+	0.186292i	$0.0596469\pi$
$353$	131.522i	0.372583i	−0.982495	+	0.186292i	$0.940353\pi$
$354$	0	0
$355$	0	0
$356$	− 203.647i	− 0.572041i
$357$	0	0
$358$	178.000	0.497207
$359$	424.264i	1.18179i	0.806747	+	0.590897i	$0.201226\pi$
$359$	424.264i	1.18179i	−0.806747	+	0.590897i	$0.798774\pi$
$360$	0	0
$361$	264.000	0.731302
$362$	− 168.291i	− 0.464893i
$363$	0	0
$364$	−18.0000	−0.0494505
$365$	0	0
$366$	0	0
$367$	−369.000	−1.00545	−0.502725	−	0.864447i	$-0.667669\pi$
$367$	−369.000	−1.00545	−0.502725	+	0.864447i	$0.667669\pi$
$368$	− 16.9706i	− 0.0461157i
$369$	0	0
$370$	0	0
$371$	− 865.499i	− 2.33288i
$372$	0	0
$373$	313.000	0.839142	0.419571	−	0.907723i	$-0.362181\pi$
$373$	313.000	0.839142	0.419571	+	0.907723i	$0.362181\pi$
$374$	698.621i	1.86797i
$375$	0	0
$376$	276.000	0.734043
$377$	26.8701i	0.0712734i
$378$	0	0
$379$	−47.0000	−0.124011	−0.0620053	−	0.998076i	$-0.519750\pi$
$379$	−47.0000	−0.124011	−0.0620053	+	0.998076i	$0.519750\pi$
$380$	0	0
$381$	0	0
$382$	−22.0000	−0.0575916
$383$	− 373.352i	− 0.974810i	−0.873176	−	0.487405i	$-0.837943\pi$
$383$	− 373.352i	− 0.974810i	0.873176	−	0.487405i	$-0.162057\pi$
$384$	0	0
$385$	0	0
$386$	− 397.394i	− 1.02952i
$387$	0	0
$388$	−14.0000	−0.0360825
$389$	− 390.323i	− 1.00340i	−0.865041	−	0.501700i	$-0.832708\pi$
$389$	− 390.323i	− 1.00340i	0.865041	−	0.501700i	$-0.167292\pi$
$390$	0	0
$391$	114.000	0.291560
$392$	− 271.529i	− 0.692676i
$393$	0	0
$394$	318.000	0.807107
$395$	0	0
$396$	0	0
$397$	−47.0000	−0.118388	−0.0591940	−	0.998247i	$-0.518853\pi$
$397$	−47.0000	−0.118388	−0.0591940	+	0.998247i	$0.518853\pi$
$398$	− 247.487i	− 0.621828i
$399$	0	0
$400$	0	0
$401$	− 152.735i	− 0.380885i	−0.981698	−	0.190443i	$-0.939008\pi$
$401$	− 152.735i	− 0.380885i	0.981698	−	0.190443i	$-0.0609923\pi$
$402$	0	0
$403$	−39.0000	−0.0967742
$404$	101.823i	0.252038i
$405$	0	0
$406$	−342.000	−0.842365
$407$	− 588.313i	− 1.44549i
$408$	0	0
$409$	−103.000	−0.251834	−0.125917	−	0.992041i	$-0.540187\pi$
$409$	−103.000	−0.251834	−0.125917	+	0.992041i	$0.540187\pi$
$410$	0	0
$411$	0	0
$412$	220.000	0.533981
$413$	89.0955i	0.215727i
$414$	0	0
$415$	0	0
$416$	− 28.2843i	− 0.0679910i
$417$	0	0
$418$	−650.000	−1.55502
$419$	39.5980i	0.0945059i	0.998883	+	0.0472530i	$0.0150467\pi$
$419$	39.5980i	0.0945059i	−0.998883	+	0.0472530i	$0.984953\pi$
$420$	0	0
$421$	−560.000	−1.33017	−0.665083	−	0.746769i	$-0.731604\pi$
$421$	−560.000	−1.33017	−0.665083	+	0.746769i	$0.731604\pi$
$422$	69.2965i	0.164210i
$423$	0	0
$424$	816.000	1.92453
$425$	0	0
$426$	0	0
$427$	−657.000	−1.53864
$428$	− 237.588i	− 0.555112i
$429$	0	0
$430$	0	0
$431$	− 479.418i	− 1.11234i	−0.831069	−	0.556170i	$-0.812270\pi$
$431$	− 479.418i	− 1.11234i	0.831069	−	0.556170i	$-0.187730\pi$
$432$	0	0
$433$	−257.000	−0.593533	−0.296767	−	0.954950i	$-0.595908\pi$
$433$	−257.000	−0.593533	−0.296767	+	0.954950i	$0.595908\pi$
$434$	− 496.389i	− 1.14375i
$435$	0	0
$436$	−50.0000	−0.114679
$437$	106.066i	0.242714i
$438$	0	0
$439$	−167.000	−0.380410	−0.190205	−	0.981744i	$-0.560915\pi$
$439$	−167.000	−0.380410	−0.190205	+	0.981744i	$0.560915\pi$
$440$	0	0
$441$	0	0
$442$	−38.0000	−0.0859729
$443$	− 169.706i	− 0.383083i	−0.981485	−	0.191541i	$-0.938651\pi$
$443$	− 169.706i	− 0.383083i	0.981485	−	0.191541i	$-0.0613486\pi$
$444$	0	0
$445$	0	0
$446$	374.767i	0.840284i
$447$	0	0
$448$	504.000	1.12500
$449$	690.136i	1.53705i	0.639819	+	0.768526i	$0.279009\pi$
$449$	690.136i	1.53705i	−0.639819	+	0.768526i	$0.720991\pi$
$450$	0	0
$451$	104.000	0.230599
$452$	90.5097i	0.200243i
$453$	0	0
$454$	−184.000	−0.405286
$455$	0	0
$456$	0	0
$457$	−50.0000	−0.109409	−0.0547046	−	0.998503i	$-0.517422\pi$
$457$	−50.0000	−0.109409	−0.0547046	+	0.998503i	$0.517422\pi$
$458$	123.037i	0.268639i
$459$	0	0
$460$	0	0
$461$	152.735i	0.331313i	0.986184	+	0.165656i	$0.0529742\pi$
$461$	152.735i	0.331313i	−0.986184	+	0.165656i	$0.947026\pi$
$462$	0	0
$463$	−240.000	−0.518359	−0.259179	−	0.965829i	$-0.583452\pi$
$463$	−240.000	−0.518359	−0.259179	+	0.965829i	$0.583452\pi$
$464$	− 107.480i	− 0.231638i
$465$	0	0
$466$	216.000	0.463519
$467$	− 592.555i	− 1.26886i	−0.772982	−	0.634428i	$-0.781236\pi$
$467$	− 592.555i	− 1.26886i	0.772982	−	0.634428i	$-0.218764\pi$
$468$	0	0
$469$	−567.000	−1.20896
$470$	0	0
$471$	0	0
$472$	−84.0000	−0.177966
$473$	422.850i	0.893974i
$474$	0	0
$475$	0	0
$476$	483.661i	1.01609i
$477$	0	0
$478$	−392.000	−0.820084
$479$	496.389i	1.03630i	0.855289	+	0.518151i	$0.173380\pi$
$479$	496.389i	1.03630i	−0.855289	+	0.518151i	$0.826620\pi$
$480$	0	0
$481$	32.0000	0.0665281
$482$	− 586.899i	− 1.21763i
$483$	0	0
$484$	−434.000	−0.896694
$485$	0	0
$486$	0	0
$487$	545.000	1.11910	0.559548	−	0.828798i	$-0.310975\pi$
$487$	545.000	1.11910	0.559548	+	0.828798i	$0.310975\pi$
$488$	− 619.426i	− 1.26931i
$489$	0	0
$490$	0	0
$491$	905.097i	1.84337i	0.387934	+	0.921687i	$0.373189\pi$
$491$	905.097i	1.84337i	−0.387934	+	0.921687i	$0.626811\pi$
$492$	0	0
$493$	722.000	1.46450
$494$	− 35.3553i	− 0.0715695i
$495$	0	0
$496$	156.000	0.314516
$497$	560.029i	1.12682i
$498$	0	0
$499$	7.00000	0.0140281	0.00701403	−	0.999975i	$-0.497767\pi$
$499$	7.00000	0.0140281	0.00701403	+	0.999975i	$0.497767\pi$
$500$	0	0
$501$	0	0
$502$	528.000	1.05179
$503$	786.303i	1.56323i	0.623764	+	0.781613i	$0.285603\pi$
$503$	786.303i	1.56323i	−0.623764	+	0.781613i	$0.714397\pi$
$504$	0	0
$505$	0	0
$506$	− 110.309i	− 0.218001i
$507$	0	0
$508$	−160.000	−0.314961
$509$	− 578.413i	− 1.13637i	−0.822900	−	0.568186i	$-0.807645\pi$
$509$	− 578.413i	− 1.13637i	0.822900	−	0.568186i	$-0.192355\pi$
$510$	0	0
$511$	−1224.00	−2.39530
$512$	248.902i	0.486136i
$513$	0	0
$514$	384.000	0.747082
$515$	0	0
$516$	0	0
$517$	598.000	1.15667
$518$	407.294i	0.786281i
$519$	0	0
$520$	0	0
$521$	− 414.365i	− 0.795325i	−0.917532	−	0.397663i	$-0.869821\pi$
$521$	− 414.365i	− 0.795325i	0.917532	−	0.397663i	$-0.130179\pi$
$522$	0	0
$523$	−263.000	−0.502868	−0.251434	−	0.967874i	$-0.580902\pi$
$523$	−263.000	−0.502868	−0.251434	+	0.967874i	$0.580902\pi$
$524$	− 169.706i	− 0.323866i
$525$	0	0
$526$	296.000	0.562738
$527$	1047.93i	1.98849i
$528$	0	0
$529$	511.000	0.965974
$530$	0	0
$531$	0	0
$532$	−450.000	−0.845865
$533$	5.65685i	0.0106132i
$534$	0	0
$535$	0	0
$536$	− 534.573i	− 0.997337i
$537$	0	0
$538$	−134.000	−0.249071
$539$	− 588.313i	− 1.09149i
$540$	0	0
$541$	489.000	0.903882	0.451941	−	0.892048i	$-0.350732\pi$
$541$	489.000	0.903882	0.451941	+	0.892048i	$0.350732\pi$
$542$	248.902i	0.459228i
$543$	0	0
$544$	−760.000	−1.39706
$545$	0	0
$546$	0	0
$547$	10.0000	0.0182815	0.00914077	−	0.999958i	$-0.497090\pi$
$547$	10.0000	0.0182815	0.00914077	+	0.999958i	$0.497090\pi$
$548$	42.4264i	0.0774205i
$549$	0	0
$550$	0	0
$551$	671.751i	1.21915i
$552$	0	0
$553$	216.000	0.390597
$554$	− 428.507i	− 0.773478i
$555$	0	0
$556$	208.000	0.374101
$557$	623.668i	1.11969i	0.828597	+	0.559846i	$0.189140\pi$
$557$	623.668i	1.11969i	−0.828597	+	0.559846i	$0.810860\pi$
$558$	0	0
$559$	−23.0000	−0.0411449
$560$	0	0
$561$	0	0
$562$	726.000	1.29181
$563$	− 63.6396i	− 0.113037i	−0.998402	−	0.0565183i	$-0.982000\pi$
$563$	− 63.6396i	− 0.113037i	0.998402	−	0.0565183i	$-0.0179999\pi$
$564$	0	0
$565$	0	0
$566$	555.786i	0.981954i
$567$	0	0
$568$	−528.000	−0.929577
$569$	810.344i	1.42416i	0.702101	+	0.712078i	$0.252246\pi$
$569$	810.344i	1.42416i	−0.702101	+	0.712078i	$0.747754\pi$
$570$	0	0
$571$	377.000	0.660245	0.330123	−	0.943938i	$-0.392910\pi$
$571$	377.000	0.660245	0.330123	+	0.943938i	$0.392910\pi$
$572$	− 36.7696i	− 0.0642824i
$573$	0	0
$574$	−72.0000	−0.125436
$575$	0	0
$576$	0	0
$577$	791.000	1.37088	0.685442	−	0.728127i	$-0.259609\pi$
$577$	791.000	1.37088	0.685442	+	0.728127i	$0.259609\pi$
$578$	612.354i	1.05944i
$579$	0	0
$580$	0	0
$581$	− 420.021i	− 0.722928i
$582$	0	0
$583$	1768.00	3.03259
$584$	− 1154.00i	− 1.97602i
$585$	0	0
$586$	−66.0000	−0.112628
$587$	356.382i	0.607124i	0.952812	+	0.303562i	$0.0981760\pi$
$587$	356.382i	0.607124i	−0.952812	+	0.303562i	$0.901824\pi$
$588$	0	0
$589$	−975.000	−1.65535
$590$	0	0
$591$	0	0
$592$	−128.000	−0.216216
$593$	− 435.578i	− 0.734533i	−0.930116	−	0.367266i	$-0.880294\pi$
$593$	− 435.578i	− 0.734533i	0.930116	−	0.367266i	$-0.119706\pi$
$594$	0	0
$595$	0	0
$596$	342.240i	0.574228i
$597$	0	0
$598$	6.00000	0.0100334
$599$	1108.74i	1.85099i	0.378759	+	0.925495i	$0.376351\pi$
$599$	1108.74i	1.85099i	−0.378759	+	0.925495i	$0.623649\pi$
$600$	0	0
$601$	233.000	0.387687	0.193844	−	0.981032i	$-0.437905\pi$
$601$	233.000	0.387687	0.193844	+	0.981032i	$0.437905\pi$
$602$	− 292.742i	− 0.486283i
$603$	0	0
$604$	−34.0000	−0.0562914
$605$	0	0
$606$	0	0
$607$	−776.000	−1.27842	−0.639209	−	0.769033i	$-0.720738\pi$
$607$	−776.000	−1.27842	−0.639209	+	0.769033i	$0.720738\pi$
$608$	− 707.107i	− 1.16300i
$609$	0	0
$610$	0	0
$611$	32.5269i	0.0532355i
$612$	0	0
$613$	376.000	0.613377	0.306688	−	0.951810i	$-0.400779\pi$
$613$	376.000	0.613377	0.306688	+	0.951810i	$0.400779\pi$
$614$	− 408.708i	− 0.665648i
$615$	0	0
$616$	1404.00	2.27922
$617$	593.970i	0.962674i	0.876536	+	0.481337i	$0.159849\pi$
$617$	593.970i	0.962674i	−0.876536	+	0.481337i	$0.840151\pi$
$618$	0	0
$619$	1031.00	1.66559	0.832795	−	0.553582i	$-0.186739\pi$
$619$	1031.00	1.66559	0.832795	+	0.553582i	$0.186739\pi$
$620$	0	0
$621$	0	0
$622$	−654.000	−1.05145
$623$	916.410i	1.47096i
$624$	0	0
$625$	0	0
$626$	680.237i	1.08664i
$627$	0	0
$628$	130.000	0.207006
$629$	− 859.842i	− 1.36700i
$630$	0	0
$631$	745.000	1.18067	0.590333	−	0.807160i	$-0.298997\pi$
$631$	745.000	1.18067	0.590333	+	0.807160i	$0.298997\pi$
$632$	203.647i	0.322226i
$633$	0	0
$634$	−16.0000	−0.0252366
$635$	0	0
$636$	0	0
$637$	32.0000	0.0502355
$638$	− 698.621i	− 1.09502i
$639$	0	0
$640$	0	0
$641$	− 203.647i	− 0.317702i	−0.987303	−	0.158851i	$-0.949221\pi$
$641$	− 203.647i	− 0.317702i	0.987303	−	0.158851i	$-0.0507789\pi$
$642$	0	0
$643$	−1032.00	−1.60498	−0.802488	−	0.596668i	$-0.796491\pi$
$643$	−1032.00	−1.60498	−0.802488	+	0.596668i	$0.796491\pi$
$644$	− 76.3675i	− 0.118583i
$645$	0	0
$646$	−950.000	−1.47059
$647$	− 56.5685i	− 0.0874321i	−0.999044	−	0.0437160i	$-0.986080\pi$
$647$	− 56.5685i	− 0.0874321i	0.999044	−	0.0437160i	$-0.0139197\pi$
$648$	0	0
$649$	−182.000	−0.280431
$650$	0	0
$651$	0	0
$652$	34.0000	0.0521472
$653$	272.943i	0.417983i	0.977917	+	0.208992i	$0.0670182\pi$
$653$	272.943i	0.417983i	−0.977917	+	0.208992i	$0.932982\pi$
$654$	0	0
$655$	0	0
$656$	− 22.6274i	− 0.0344930i
$657$	0	0
$658$	−414.000	−0.629179
$659$	− 379.009i	− 0.575128i	−0.957761	−	0.287564i	$-0.907155\pi$
$659$	− 379.009i	− 0.575128i	0.957761	−	0.287564i	$-0.0928454\pi$
$660$	0	0
$661$	−440.000	−0.665658	−0.332829	−	0.942987i	$-0.608003\pi$
$661$	−440.000	−0.665658	−0.332829	+	0.942987i	$0.608003\pi$
$662$	101.823i	0.153812i
$663$	0	0
$664$	396.000	0.596386
$665$	0	0
$666$	0	0
$667$	−114.000	−0.170915
$668$	− 509.117i	− 0.762151i
$669$	0	0
$670$	0	0
$671$	− 1342.09i	− 2.00013i
$672$	0	0
$673$	−264.000	−0.392273	−0.196137	−	0.980577i	$-0.562840\pi$
$673$	−264.000	−0.392273	−0.196137	+	0.980577i	$0.562840\pi$
$674$	473.762i	0.702910i
$675$	0	0
$676$	−336.000	−0.497041
$677$	− 538.815i	− 0.795887i	−0.917410	−	0.397943i	$-0.869724\pi$
$677$	− 538.815i	− 0.795887i	0.917410	−	0.397943i	$-0.130276\pi$
$678$	0	0
$679$	63.0000	0.0927835
$680$	0	0
$681$	0	0
$682$	1014.00	1.48680
$683$	576.999i	0.844801i	0.906409	+	0.422401i	$0.138812\pi$
$683$	576.999i	0.844801i	−0.906409	+	0.422401i	$0.861188\pi$
$684$	0	0
$685$	0	0
$686$	− 216.375i	− 0.315415i
$687$	0	0
$688$	92.0000	0.133721
$689$	96.1665i	0.139574i
$690$	0	0
$691$	−182.000	−0.263386	−0.131693	−	0.991291i	$-0.542041\pi$
$691$	−182.000	−0.263386	−0.131693	+	0.991291i	$0.542041\pi$
$692$	− 347.897i	− 0.502741i
$693$	0	0
$694$	−542.000	−0.780980
$695$	0	0
$696$	0	0
$697$	152.000	0.218077
$698$	− 803.273i	− 1.15082i
$699$	0	0
$700$	0	0
$701$	− 558.614i	− 0.796882i	−0.917194	−	0.398441i	$-0.869551\pi$
$701$	− 558.614i	− 0.796882i	0.917194	−	0.398441i	$-0.130449\pi$
$702$	0	0
$703$	800.000	1.13798
$704$	1029.55i	1.46243i
$705$	0	0
$706$	186.000	0.263456
$707$	− 458.205i	− 0.648098i
$708$	0	0
$709$	607.000	0.856135	0.428068	−	0.903747i	$-0.359194\pi$
$709$	607.000	0.856135	0.428068	+	0.903747i	$0.359194\pi$
$710$	0	0
$711$	0	0
$712$	−864.000	−1.21348
$713$	− 165.463i	− 0.232066i
$714$	0	0
$715$	0	0
$716$	251.730i	0.351578i
$717$	0	0
$718$	600.000	0.835655
$719$	106.066i	0.147519i	0.997276	+	0.0737594i	$0.0234997\pi$
$719$	106.066i	0.147519i	−0.997276	+	0.0737594i	$0.976500\pi$
$720$	0	0
$721$	−990.000	−1.37309
$722$	− 373.352i	− 0.517109i
$723$	0	0
$724$	238.000	0.328729
$725$	0	0
$726$	0	0
$727$	71.0000	0.0976616	0.0488308	−	0.998807i	$-0.484450\pi$
$727$	71.0000	0.0976616	0.0488308	+	0.998807i	$0.484450\pi$
$728$	76.3675i	0.104900i
$729$	0	0
$730$	0	0
$731$	618.011i	0.845433i
$732$	0	0
$733$	−904.000	−1.23329	−0.616644	−	0.787242i	$-0.711508\pi$
$733$	−904.000	−1.23329	−0.616644	+	0.787242i	$0.711508\pi$
$734$	521.845i	0.710960i
$735$	0	0
$736$	120.000	0.163043
$737$	− 1158.24i	− 1.57156i
$738$	0	0
$739$	−800.000	−1.08254	−0.541272	−	0.840848i	$-0.682057\pi$
$739$	−800.000	−1.08254	−0.541272	+	0.840848i	$0.682057\pi$
$740$	0	0
$741$	0	0
$742$	−1224.00	−1.64960
$743$	418.607i	0.563401i	0.959502	+	0.281701i	$0.0908985\pi$
$743$	418.607i	0.563401i	−0.959502	+	0.281701i	$0.909101\pi$
$744$	0	0
$745$	0	0
$746$	− 442.649i	− 0.593363i
$747$	0	0
$748$	−988.000	−1.32086
$749$	1069.15i	1.42743i
$750$	0	0
$751$	−704.000	−0.937417	−0.468708	−	0.883353i	$-0.655280\pi$
$751$	−704.000	−0.937417	−0.468708	+	0.883353i	$0.655280\pi$
$752$	− 130.108i	− 0.173015i
$753$	0	0
$754$	38.0000	0.0503979
$755$	0	0
$756$	0	0
$757$	−977.000	−1.29062	−0.645310	−	0.763920i	$-0.723272\pi$
$757$	−977.000	−1.29062	−0.645310	+	0.763920i	$0.723272\pi$
$758$	66.4680i	0.0876887i
$759$	0	0
$760$	0	0
$761$	746.705i	0.981215i	0.871381	+	0.490608i	$0.163225\pi$
$761$	746.705i	0.981215i	−0.871381	+	0.490608i	$0.836775\pi$
$762$	0	0
$763$	225.000	0.294889
$764$	− 31.1127i	− 0.0407234i
$765$	0	0
$766$	−528.000	−0.689295
$767$	− 9.89949i	− 0.0129068i
$768$	0	0
$769$	1337.00	1.73862	0.869311	−	0.494266i	$-0.164563\pi$
$769$	1337.00	1.73862	0.869311	+	0.494266i	$0.164563\pi$
$770$	0	0
$771$	0	0
$772$	562.000	0.727979
$773$	1170.97i	1.51484i	0.652930	+	0.757418i	$0.273540\pi$
$773$	1170.97i	1.51484i	−0.652930	+	0.757418i	$0.726460\pi$
$774$	0	0
$775$	0	0
$776$	59.3970i	0.0765425i
$777$	0	0
$778$	−552.000	−0.709512
$779$	141.421i	0.181542i
$780$	0	0
$781$	−1144.00	−1.46479
$782$	− 161.220i	− 0.206164i
$783$	0	0
$784$	−128.000	−0.163265
$785$	0	0
$786$	0	0
$787$	703.000	0.893266	0.446633	−	0.894717i	$-0.352623\pi$
$787$	703.000	0.893266	0.446633	+	0.894717i	$0.352623\pi$
$788$	449.720i	0.570711i
$789$	0	0
$790$	0	0
$791$	− 407.294i	− 0.514910i
$792$	0	0
$793$	73.0000	0.0920555
$794$	66.4680i	0.0837129i
$795$	0	0
$796$	350.000	0.439698
$797$	− 1538.66i	− 1.93057i	−0.261199	−	0.965285i	$-0.584118\pi$
$797$	− 1538.66i	− 1.93057i	0.261199	−	0.965285i	$-0.415882\pi$
$798$	0	0
$799$	874.000	1.09387
$800$	0	0
$801$	0	0
$802$	−216.000	−0.269327
$803$	− 2500.33i	− 3.11374i
$804$	0	0
$805$	0	0
$806$	55.1543i	0.0684297i
$807$	0	0
$808$	432.000	0.534653
$809$	73.5391i	0.0909012i	0.998967	+	0.0454506i	$0.0144724\pi$
$809$	73.5391i	0.0909012i	−0.998967	+	0.0454506i	$0.985528\pi$
$810$	0	0
$811$	1095.00	1.35018	0.675092	−	0.737733i	$-0.264104\pi$
$811$	1095.00	1.35018	0.675092	+	0.737733i	$0.264104\pi$
$812$	− 483.661i	− 0.595642i
$813$	0	0
$814$	−832.000	−1.02211
$815$	0	0
$816$	0	0
$817$	−575.000	−0.703794
$818$	145.664i	0.178073i
$819$	0	0
$820$	0	0
$821$	− 1166.73i	− 1.42110i	−0.703645	−	0.710552i	$-0.748445\pi$
$821$	− 1166.73i	− 1.42110i	0.703645	−	0.710552i	$-0.251555\pi$
$822$	0	0
$823$	−1015.00	−1.23329	−0.616646	−	0.787240i	$-0.711509\pi$
$823$	−1015.00	−1.23329	−0.616646	+	0.787240i	$0.711509\pi$
$824$	− 933.381i	− 1.13274i
$825$	0	0
$826$	126.000	0.152542
$827$	− 379.009i	− 0.458294i	−0.973392	−	0.229147i	$-0.926406\pi$
$827$	− 379.009i	− 0.458294i	0.973392	−	0.229147i	$-0.0735937\pi$
$828$	0	0
$829$	150.000	0.180941	0.0904704	−	0.995899i	$-0.471163\pi$
$829$	150.000	0.180941	0.0904704	+	0.995899i	$0.471163\pi$
$830$	0	0
$831$	0	0
$832$	−56.0000	−0.0673077
$833$	− 859.842i	− 1.03222i
$834$	0	0
$835$	0	0
$836$	− 919.239i	− 1.09957i
$837$	0	0
$838$	56.0000	0.0668258
$839$	− 660.438i	− 0.787173i	−0.919288	−	0.393586i	$-0.871234\pi$
$839$	− 660.438i	− 0.787173i	0.919288	−	0.393586i	$-0.128766\pi$
$840$	0	0
$841$	119.000	0.141498
$842$	791.960i	0.940570i
$843$	0	0
$844$	−98.0000	−0.116114
$845$	0	0
$846$	0	0
$847$	1953.00	2.30579
$848$	− 384.666i	− 0.453616i
$849$	0	0
$850$	0	0
$851$	135.765i	0.159535i
$852$	0	0
$853$	−297.000	−0.348183	−0.174091	−	0.984729i	$-0.555699\pi$
$853$	−297.000	−0.348183	−0.174091	+	0.984729i	$0.555699\pi$
$854$	929.138i	1.08798i
$855$	0	0
$856$	−1008.00	−1.17757
$857$	1165.31i	1.35976i	0.733325	+	0.679879i	$0.237968\pi$
$857$	1165.31i	1.35976i	−0.733325	+	0.679879i	$0.762032\pi$
$858$	0	0
$859$	−1286.00	−1.49709	−0.748545	−	0.663084i	$-0.769247\pi$
$859$	−1286.00	−1.49709	−0.748545	+	0.663084i	$0.769247\pi$
$860$	0	0
$861$	0	0
$862$	−678.000	−0.786543
$863$	633.568i	0.734146i	0.930192	+	0.367073i	$0.119640\pi$
$863$	633.568i	0.734146i	−0.930192	+	0.367073i	$0.880360\pi$
$864$	0	0
$865$	0	0
$866$	363.453i	0.419692i
$867$	0	0
$868$	702.000	0.808756
$869$	441.235i	0.507750i
$870$	0	0
$871$	63.0000	0.0723307
$872$	212.132i	0.243271i
$873$	0	0
$874$	150.000	0.171625
$875$	0	0
$876$	0	0
$877$	−1599.00	−1.82326	−0.911631	−	0.411011i	$-0.865176\pi$
$877$	−1599.00	−1.82326	−0.911631	+	0.411011i	$0.865176\pi$
$878$	236.174i	0.268991i
$879$	0	0
$880$	0	0
$881$	961.665i	1.09156i	0.837928	+	0.545780i	$0.183767\pi$
$881$	961.665i	1.09156i	−0.837928	+	0.545780i	$0.816233\pi$
$882$	0	0
$883$	−193.000	−0.218573	−0.109287	−	0.994010i	$-0.534857\pi$
$883$	−193.000	−0.218573	−0.109287	+	0.994010i	$0.534857\pi$
$884$	− 53.7401i	− 0.0607920i
$885$	0	0
$886$	−240.000	−0.270880
$887$	700.036i	0.789217i	0.918849	+	0.394609i	$0.129120\pi$
$887$	700.036i	0.789217i	−0.918849	+	0.394609i	$0.870880\pi$
$888$	0	0
$889$	720.000	0.809899
$890$	0	0
$891$	0	0
$892$	−530.000	−0.594170
$893$	813.173i	0.910608i
$894$	0	0
$895$	0	0
$896$	305.470i	0.340926i
$897$	0	0
$898$	976.000	1.08686
$899$	− 1047.93i	− 1.16566i
$900$	0	0
$901$	2584.00	2.86792
$902$	− 147.078i	− 0.163058i
$903$	0	0
$904$	384.000	0.424779
$905$	0	0
$906$	0	0
$907$	−240.000	−0.264609	−0.132304	−	0.991209i	$-0.542238\pi$
$907$	−240.000	−0.264609	−0.132304	+	0.991209i	$0.542238\pi$
$908$	− 260.215i	− 0.286581i
$909$	0	0
$910$	0	0
$911$	− 171.120i	− 0.187837i	−0.995580	−	0.0939187i	$-0.970061\pi$
$911$	− 171.120i	− 0.187837i	0.995580	−	0.0939187i	$-0.0299394\pi$
$912$	0	0
$913$	858.000	0.939759
$914$	70.7107i	0.0773640i
$915$	0	0
$916$	−174.000	−0.189956
$917$	763.675i	0.832798i
$918$	0	0
$919$	343.000	0.373232	0.186616	−	0.982433i	$-0.440248\pi$
$919$	343.000	0.373232	0.186616	+	0.982433i	$0.440248\pi$
$920$	0	0
$921$	0	0
$922$	216.000	0.234273
$923$	− 62.2254i	− 0.0674165i
$924$	0	0
$925$	0	0
$926$	339.411i	0.366535i
$927$	0	0
$928$	760.000	0.818966
$929$	− 719.835i	− 0.774849i	−0.921901	−	0.387424i	$-0.873365\pi$
$929$	− 719.835i	− 0.774849i	0.921901	−	0.387424i	$-0.126635\pi$
$930$	0	0
$931$	800.000	0.859291
$932$	305.470i	0.327758i
$933$	0	0
$934$	−838.000	−0.897216
$935$	0	0
$936$	0	0
$937$	609.000	0.649947	0.324973	−	0.945723i	$-0.394645\pi$
$937$	609.000	0.649947	0.324973	+	0.945723i	$0.394645\pi$
$938$	801.859i	0.854860i
$939$	0	0
$940$	0	0
$941$	111.723i	0.118728i	0.998236	+	0.0593639i	$0.0189072\pi$
$941$	111.723i	0.118728i	−0.998236	+	0.0593639i	$0.981093\pi$
$942$	0	0
$943$	−24.0000	−0.0254507
$944$	39.5980i	0.0419470i
$945$	0	0
$946$	598.000	0.632135
$947$	1022.48i	1.07970i	0.841761	+	0.539850i	$0.181519\pi$
$947$	1022.48i	1.07970i	−0.841761	+	0.539850i	$0.818481\pi$
$948$	0	0
$949$	136.000	0.143309
$950$	0	0
$951$	0	0
$952$	2052.00	2.15546
$953$	− 1380.27i	− 1.44834i	−0.689619	−	0.724172i	$-0.742222\pi$
$953$	− 1380.27i	− 1.44834i	0.689619	−	0.724172i	$-0.257778\pi$
$954$	0	0
$955$	0	0
$956$	− 554.372i	− 0.579887i
$957$	0	0
$958$	702.000	0.732777
$959$	− 190.919i	− 0.199081i
$960$	0	0
$961$	560.000	0.582726
$962$	− 45.2548i	− 0.0470424i
$963$	0	0
$964$	830.000	0.860996
$965$	0	0
$966$	0	0
$967$	584.000	0.603930	0.301965	−	0.953319i	$-0.402357\pi$
$967$	584.000	0.603930	0.301965	+	0.953319i	$0.402357\pi$
$968$	1841.31i	1.90218i
$969$	0	0
$970$	0	0
$971$	1030.96i	1.06175i	0.847449	+	0.530876i	$0.178137\pi$
$971$	1030.96i	1.06175i	−0.847449	+	0.530876i	$0.821863\pi$
$972$	0	0
$973$	−936.000	−0.961973
$974$	− 770.746i	− 0.791321i
$975$	0	0
$976$	−292.000	−0.299180
$977$	− 1028.13i	− 1.05234i	−0.850380	−	0.526169i	$-0.823628\pi$
$977$	− 1028.13i	− 1.05234i	0.850380	−	0.526169i	$-0.176372\pi$
$978$	0	0
$979$	−1872.00	−1.91216
$980$	0	0
$981$	0	0
$982$	1280.00	1.30346
$983$	1323.70i	1.34660i	0.739371	+	0.673298i	$0.235123\pi$
$983$	1323.70i	1.34660i	−0.739371	+	0.673298i	$0.764877\pi$
$984$	0	0
$985$	0	0
$986$	− 1021.06i	− 1.03556i
$987$	0	0
$988$	50.0000	0.0506073
$989$	− 97.5807i	− 0.0986661i
$990$	0	0
$991$	−961.000	−0.969728	−0.484864	−	0.874590i	$-0.661131\pi$
$991$	−961.000	−0.969728	−0.484864	+	0.874590i	$0.661131\pi$
$992$	1103.09i	1.11198i
$993$	0	0
$994$	792.000	0.796781
$995$	0	0
$996$	0	0
$997$	−600.000	−0.601805	−0.300903	−	0.953655i	$-0.597288\pi$
$997$	−600.000	−0.601805	−0.300903	+	0.953655i	$0.597288\pi$
$998$	− 9.89949i	− 0.00991933i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.c.a.26.1	✓	2
3.2	odd	2	inner	225.3.c.a.26.2	yes	2
4.3	odd	2		3600.3.l.j.1601.2		2
5.2	odd	4		225.3.d.a.224.3		4
5.3	odd	4		225.3.d.a.224.2		4
5.4	even	2		225.3.c.b.26.2	yes	2
12.11	even	2		3600.3.l.j.1601.1		2
15.2	even	4		225.3.d.a.224.1		4
15.8	even	4		225.3.d.a.224.4		4
15.14	odd	2		225.3.c.b.26.1	yes	2
20.3	even	4		3600.3.c.e.449.2		4
20.7	even	4		3600.3.c.e.449.4		4
20.19	odd	2		3600.3.l.b.1601.2		2
60.23	odd	4		3600.3.c.e.449.1		4
60.47	odd	4		3600.3.c.e.449.3		4
60.59	even	2		3600.3.l.b.1601.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.3.c.a.26.1	✓	2	1.1	even	1	trivial
225.3.c.a.26.2	yes	2	3.2	odd	2	inner
225.3.c.b.26.1	yes	2	15.14	odd	2
225.3.c.b.26.2	yes	2	5.4	even	2
225.3.d.a.224.1		4	15.2	even	4
225.3.d.a.224.2		4	5.3	odd	4
225.3.d.a.224.3		4	5.2	odd	4
225.3.d.a.224.4		4	15.8	even	4
3600.3.c.e.449.1		4	60.23	odd	4
3600.3.c.e.449.2		4	20.3	even	4
3600.3.c.e.449.3		4	60.47	odd	4
3600.3.c.e.449.4		4	20.7	even	4
3600.3.l.b.1601.1		2	60.59	even	2
3600.3.l.b.1601.2		2	20.19	odd	2
3600.3.l.j.1601.1		2	12.11	even	2
3600.3.l.j.1601.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +2.00000 q^{4} -9.00000 q^{7} -8.48528i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} +2.00000 q^{4} -9.00000 q^{7} -8.48528i q^{8} -18.3848i q^{11} +1.00000 q^{13} +12.7279i q^{14} -4.00000 q^{16} -26.8701i q^{17} +25.0000 q^{19} -26.0000 q^{22} +4.24264i q^{23} -1.41421i q^{26} -18.0000 q^{28} +26.8701i q^{29} -39.0000 q^{31} -28.2843i q^{32} -38.0000 q^{34} +32.0000 q^{37} -35.3553i q^{38} +5.65685i q^{41} -23.0000 q^{43} -36.7696i q^{44} +6.00000 q^{46} +32.5269i q^{47} +32.0000 q^{49} +2.00000 q^{52} +96.1665i q^{53} +76.3675i q^{56} +38.0000 q^{58} -9.89949i q^{59} +73.0000 q^{61} +55.1543i q^{62} -56.0000 q^{64} +63.0000 q^{67} -53.7401i q^{68} -62.2254i q^{71} +136.000 q^{73} -45.2548i q^{74} +50.0000 q^{76} +165.463i q^{77} -24.0000 q^{79} +8.00000 q^{82} +46.6690i q^{83} +32.5269i q^{86} -156.000 q^{88} -101.823i q^{89} -9.00000 q^{91} +8.48528i q^{92} +46.0000 q^{94} -7.00000 q^{97} -45.2548i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 18 q^{7}+O(q^{10}) \) 2 * q + 4 * q^4 - 18 * q^7	\( 2 q + 4 q^{4} - 18 q^{7} + 2 q^{13} - 8 q^{16} + 50 q^{19} - 52 q^{22} - 36 q^{28} - 78 q^{31} - 76 q^{34} + 64 q^{37} - 46 q^{43} + 12 q^{46} + 64 q^{49} + 4 q^{52} + 76 q^{58} + 146 q^{61} - 112 q^{64} + 126 q^{67} + 272 q^{73} + 100 q^{76} - 48 q^{79} + 16 q^{82} - 312 q^{88} - 18 q^{91} + 92 q^{94} - 14 q^{97}+O(q^{100}) \) 2 * q + 4 * q^4 - 18 * q^7 + 2 * q^13 - 8 * q^16 + 50 * q^19 - 52 * q^22 - 36 * q^28 - 78 * q^31 - 76 * q^34 + 64 * q^37 - 46 * q^43 + 12 * q^46 + 64 * q^49 + 4 * q^52 + 76 * q^58 + 146 * q^61 - 112 * q^64 + 126 * q^67 + 272 * q^73 + 100 * q^76 - 48 * q^79 + 16 * q^82 - 312 * q^88 - 18 * q^91 + 92 * q^94 - 14 * q^97

Introduction

L-functions

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