LMFDB - Embedded newform 2475.2.c.j.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:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.j.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{2} -0.438447 q^{4} +1.00000i q^{7} +2.43845i q^{8} +O(q^{10})$	$q+1.56155i q^{2} -0.438447 q^{4} +1.00000i q^{7} +2.43845i q^{8} +1.00000 q^{11} +4.56155i q^{13} -1.56155 q^{14} -4.68466 q^{16} -5.56155i q^{17} -3.00000 q^{19} +1.56155i q^{22} +2.43845i q^{23} -7.12311 q^{26} -0.438447i q^{28} -3.12311 q^{29} -7.68466 q^{31} -2.43845i q^{32} +8.68466 q^{34} +9.80776i q^{37} -4.68466i q^{38} -4.68466 q^{41} +7.68466i q^{43} -0.438447 q^{44} -3.80776 q^{46} -9.56155i q^{47} +6.00000 q^{49} -2.00000i q^{52} +7.12311i q^{53} -2.43845 q^{56} -4.87689i q^{58} +6.43845 q^{59} -3.43845 q^{61} -12.0000i q^{62} -5.56155 q^{64} +5.68466i q^{67} +2.43845i q^{68} +11.8078 q^{71} +0.246211i q^{73} -15.3153 q^{74} +1.31534 q^{76} +1.00000i q^{77} +3.31534 q^{79} -7.31534i q^{82} -12.0000 q^{86} +2.43845i q^{88} -8.87689 q^{89} -4.56155 q^{91} -1.06913i q^{92} +14.9309 q^{94} -6.12311i q^{97} +9.36932i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4}+O(q^{10}) $ 4 * q - 10 * q^4	$ 4 q - 10 q^{4} + 4 q^{11} + 2 q^{14} + 6 q^{16} - 12 q^{19} - 12 q^{26} + 4 q^{29} - 6 q^{31} + 10 q^{34} + 6 q^{41} - 10 q^{44} + 26 q^{46} + 24 q^{49} - 18 q^{56} + 34 q^{59} - 22 q^{61} - 14 q^{64} + 6 q^{71} - 86 q^{74} + 30 q^{76} + 38 q^{79} - 48 q^{86} - 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100}) $ 4 * q - 10 * q^4 + 4 * q^11 + 2 * q^14 + 6 * q^16 - 12 * q^19 - 12 * q^26 + 4 * q^29 - 6 * q^31 + 10 * q^34 + 6 * q^41 - 10 * q^44 + 26 * q^46 + 24 * q^49 - 18 * q^56 + 34 * q^59 - 22 * q^61 - 14 * q^64 + 6 * q^71 - 86 * q^74 + 30 * q^76 + 38 * q^79 - 48 * q^86 - 52 * q^89 - 10 * q^91 + 2 * 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 $

$2$	1.56155i	1.10418i	0.833783	+	0.552092i	$0.186170\pi$
$2$	1.56155i	1.10418i	−0.833783	+	0.552092i	$0.813830\pi$
$3$	0	0
$3$	0	0
$4$	−0.438447	−0.219224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	2.43845i	0.862121i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.56155i	1.26515i	0.774500	+	0.632574i	$0.218001\pi$
$13$	4.56155i	1.26515i	−0.774500	+	0.632574i	$0.781999\pi$
$14$	−1.56155	−0.417343
$15$	0	0
$16$	−4.68466	−1.17116
$17$	− 5.56155i	− 1.34887i	−0.738332	−	0.674437i	$-0.764386\pi$
$17$	− 5.56155i	− 1.34887i	0.738332	−	0.674437i	$-0.235614\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	1.56155i	0.332924i
$23$	2.43845i	0.508451i	0.967145	+	0.254226i	$0.0818206\pi$
$23$	2.43845i	0.508451i	−0.967145	+	0.254226i	$0.918179\pi$
$24$	0	0
$25$	0	0
$26$	−7.12311	−1.39696
$27$	0	0
$28$	− 0.438447i	− 0.0828587i
$29$	−3.12311	−0.579946	−0.289973	−	0.957035i	$-0.593646\pi$
$29$	−3.12311	−0.579946	−0.289973	+	0.957035i	$0.593646\pi$
$30$	0	0
$31$	−7.68466	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$31$	−7.68466	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$32$	− 2.43845i	− 0.431061i
$33$	0	0
$34$	8.68466	1.48941
$35$	0	0
$36$	0	0
$37$	9.80776i	1.61239i	0.591652	+	0.806193i	$0.298476\pi$
$37$	9.80776i	1.61239i	−0.591652	+	0.806193i	$0.701524\pi$
$38$	− 4.68466i	− 0.759952i
$39$	0	0
$40$	0	0
$41$	−4.68466	−0.731621	−0.365810	−	0.930689i	$-0.619208\pi$
$41$	−4.68466	−0.731621	−0.365810	+	0.930689i	$0.619208\pi$
$42$	0	0
$43$	7.68466i	1.17190i	0.810347	+	0.585950i	$0.199278\pi$
$43$	7.68466i	1.17190i	−0.810347	+	0.585950i	$0.800722\pi$
$44$	−0.438447	−0.0660984
$45$	0	0
$46$	−3.80776	−0.561424
$47$	− 9.56155i	− 1.39470i	−0.716733	−	0.697348i	$-0.754363\pi$
$47$	− 9.56155i	− 1.39470i	0.716733	−	0.697348i	$-0.245637\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	7.12311i	0.978434i	0.872162	+	0.489217i	$0.162717\pi$
$53$	7.12311i	0.978434i	−0.872162	+	0.489217i	$0.837283\pi$
$54$	0	0
$55$	0	0
$56$	−2.43845	−0.325851
$57$	0	0
$58$	− 4.87689i	− 0.640368i
$59$	6.43845	0.838214	0.419107	−	0.907937i	$-0.362343\pi$
$59$	6.43845	0.838214	0.419107	+	0.907937i	$0.362343\pi$
$60$	0	0
$61$	−3.43845	−0.440248	−0.220124	−	0.975472i	$-0.570646\pi$
$61$	−3.43845	−0.440248	−0.220124	+	0.975472i	$0.570646\pi$
$62$	− 12.0000i	− 1.52400i
$63$	0	0
$64$	−5.56155	−0.695194
$65$	0	0
$66$	0	0
$67$	5.68466i	0.694492i	0.937774	+	0.347246i	$0.112883\pi$
$67$	5.68466i	0.694492i	−0.937774	+	0.347246i	$0.887117\pi$
$68$	2.43845i	0.295705i
$69$	0	0
$70$	0	0
$71$	11.8078	1.40132	0.700662	−	0.713493i	$-0.252888\pi$
$71$	11.8078	1.40132	0.700662	+	0.713493i	$0.252888\pi$
$72$	0	0
$73$	0.246211i	0.0288168i	0.999896	+	0.0144084i	$0.00458650\pi$
$73$	0.246211i	0.0288168i	−0.999896	+	0.0144084i	$0.995413\pi$
$74$	−15.3153	−1.78037
$75$	0	0
$76$	1.31534	0.150880
$77$	1.00000i	0.113961i
$78$	0	0
$79$	3.31534	0.373005	0.186503	−	0.982454i	$-0.440285\pi$
$79$	3.31534	0.373005	0.186503	+	0.982454i	$0.440285\pi$
$80$	0	0
$81$	0	0
$82$	− 7.31534i	− 0.807844i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	2.43845i	0.259939i
$89$	−8.87689	−0.940949	−0.470474	−	0.882414i	$-0.655917\pi$
$89$	−8.87689	−0.940949	−0.470474	+	0.882414i	$0.655917\pi$
$90$	0	0
$91$	−4.56155	−0.478181
$92$	− 1.06913i	− 0.111465i
$93$	0	0
$94$	14.9309	1.54000
$95$	0	0
$96$	0	0
$97$	− 6.12311i	− 0.621707i	−0.950458	−	0.310854i	$-0.899385\pi$
$97$	− 6.12311i	− 0.621707i	0.950458	−	0.310854i	$-0.100615\pi$
$98$	9.36932i	0.946444i
$99$	0	0
$100$	0	0
$101$	−5.56155	−0.553395	−0.276698	−	0.960957i	$-0.589240\pi$
$101$	−5.56155	−0.553395	−0.276698	+	0.960957i	$0.589240\pi$
$102$	0	0
$103$	0.876894i	0.0864030i	0.999066	+	0.0432015i	$0.0137557\pi$
$103$	0.876894i	0.0864030i	−0.999066	+	0.0432015i	$0.986244\pi$
$104$	−11.1231	−1.09071
$105$	0	0
$106$	−11.1231	−1.08037
$107$	0.876894i	0.0847726i	0.999101	+	0.0423863i	$0.0134960\pi$
$107$	0.876894i	0.0847726i	−0.999101	+	0.0423863i	$0.986504\pi$
$108$	0	0
$109$	−12.8078	−1.22676	−0.613381	−	0.789787i	$-0.710191\pi$
$109$	−12.8078	−1.22676	−0.613381	+	0.789787i	$0.710191\pi$
$110$	0	0
$111$	0	0
$112$	− 4.68466i	− 0.442659i
$113$	− 0.876894i	− 0.0824913i	−0.999149	−	0.0412456i	$-0.986867\pi$
$113$	− 0.876894i	− 0.0824913i	0.999149	−	0.0412456i	$-0.0131326\pi$
$114$	0	0
$115$	0	0
$116$	1.36932	0.127138
$117$	0	0
$118$	10.0540i	0.925543i
$119$	5.56155	0.509827
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 5.36932i	− 0.486115i
$123$	0	0
$124$	3.36932	0.302574
$125$	0	0
$126$	0	0
$127$	15.8078i	1.40271i	0.712811	+	0.701356i	$0.247422\pi$
$127$	15.8078i	1.40271i	−0.712811	+	0.701356i	$0.752578\pi$
$128$	− 13.5616i	− 1.19868i
$129$	0	0
$130$	0	0
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	−8.87689	−0.766847
$135$	0	0
$136$	13.5616	1.16289
$137$	− 21.3693i	− 1.82570i	−0.408291	−	0.912852i	$-0.633875\pi$
$137$	− 21.3693i	− 1.82570i	0.408291	−	0.912852i	$-0.366125\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	18.4384i	1.54732i
$143$	4.56155i	0.381456i
$144$	0	0
$145$	0	0
$146$	−0.384472	−0.0318191
$147$	0	0
$148$	− 4.30019i	− 0.353473i
$149$	−11.8078	−0.967330	−0.483665	−	0.875253i	$-0.660695\pi$
$149$	−11.8078	−0.967330	−0.483665	+	0.875253i	$0.660695\pi$
$150$	0	0
$151$	−4.31534	−0.351178	−0.175589	−	0.984464i	$-0.556183\pi$
$151$	−4.31534	−0.351178	−0.175589	+	0.984464i	$0.556183\pi$
$152$	− 7.31534i	− 0.593353i
$153$	0	0
$154$	−1.56155	−0.125834
$155$	0	0
$156$	0	0
$157$	− 9.68466i	− 0.772920i	−0.922306	−	0.386460i	$-0.873698\pi$
$157$	− 9.68466i	− 0.772920i	0.922306	−	0.386460i	$-0.126302\pi$
$158$	5.17708i	0.411866i
$159$	0	0
$160$	0	0
$161$	−2.43845	−0.192177
$162$	0	0
$163$	17.6847i	1.38517i	0.721337	+	0.692585i	$0.243528\pi$
$163$	17.6847i	1.38517i	−0.721337	+	0.692585i	$0.756472\pi$
$164$	2.05398	0.160389
$165$	0	0
$166$	0	0
$167$	− 9.36932i	− 0.725020i	−0.931980	−	0.362510i	$-0.881920\pi$
$167$	− 9.36932i	− 0.725020i	0.931980	−	0.362510i	$-0.118080\pi$
$168$	0	0
$169$	−7.80776	−0.600597
$170$	0	0
$171$	0	0
$172$	− 3.36932i	− 0.256908i
$173$	17.5616i	1.33518i	0.744529	+	0.667590i	$0.232674\pi$
$173$	17.5616i	1.33518i	−0.744529	+	0.667590i	$0.767326\pi$
$174$	0	0
$175$	0	0
$176$	−4.68466	−0.353119
$177$	0	0
$178$	− 13.8617i	− 1.03898i
$179$	10.4384	0.780206	0.390103	−	0.920771i	$-0.372439\pi$
$179$	10.4384	0.780206	0.390103	+	0.920771i	$0.372439\pi$
$180$	0	0
$181$	20.1231	1.49574	0.747869	−	0.663846i	$-0.231077\pi$
$181$	20.1231	1.49574	0.747869	+	0.663846i	$0.231077\pi$
$182$	− 7.12311i	− 0.528000i
$183$	0	0
$184$	−5.94602	−0.438347
$185$	0	0
$186$	0	0
$187$	− 5.56155i	− 0.406701i
$188$	4.19224i	0.305750i
$189$	0	0
$190$	0	0
$191$	−26.9309	−1.94865	−0.974325	−	0.225148i	$-0.927714\pi$
$191$	−26.9309	−1.94865	−0.974325	+	0.225148i	$0.927714\pi$
$192$	0	0
$193$	− 13.4384i	− 0.967321i	−0.875256	−	0.483660i	$-0.839307\pi$
$193$	− 13.4384i	− 0.967321i	0.875256	−	0.483660i	$-0.160693\pi$
$194$	9.56155	0.686479
$195$	0	0
$196$	−2.63068	−0.187906
$197$	10.9309i	0.778792i	0.921070	+	0.389396i	$0.127316\pi$
$197$	10.9309i	0.778792i	−0.921070	+	0.389396i	$0.872684\pi$
$198$	0	0
$199$	−6.80776	−0.482590	−0.241295	−	0.970452i	$-0.577572\pi$
$199$	−6.80776	−0.482590	−0.241295	+	0.970452i	$0.577572\pi$
$200$	0	0
$201$	0	0
$202$	− 8.68466i	− 0.611050i
$203$	− 3.12311i	− 0.219199i
$204$	0	0
$205$	0	0
$206$	−1.36932	−0.0954048
$207$	0	0
$208$	− 21.3693i	− 1.48170i
$209$	−3.00000	−0.207514
$210$	0	0
$211$	15.6847	1.07978	0.539888	−	0.841737i	$-0.318467\pi$
$211$	15.6847	1.07978	0.539888	+	0.841737i	$0.318467\pi$
$212$	− 3.12311i	− 0.214496i
$213$	0	0
$214$	−1.36932	−0.0936046
$215$	0	0
$216$	0	0
$217$	− 7.68466i	− 0.521669i
$218$	− 20.0000i	− 1.35457i
$219$	0	0
$220$	0	0
$221$	25.3693	1.70652
$222$	0	0
$223$	− 1.19224i	− 0.0798380i	−0.999203	−	0.0399190i	$-0.987290\pi$
$223$	− 1.19224i	− 0.0798380i	0.999203	−	0.0399190i	$-0.0127100\pi$
$224$	2.43845	0.162926
$225$	0	0
$226$	1.36932	0.0910856
$227$	12.4924i	0.829151i	0.910015	+	0.414576i	$0.136070\pi$
$227$	12.4924i	0.829151i	−0.910015	+	0.414576i	$0.863930\pi$
$228$	0	0
$229$	−0.123106	−0.00813505	−0.00406752	−	0.999992i	$-0.501295\pi$
$229$	−0.123106	−0.00813505	−0.00406752	+	0.999992i	$0.501295\pi$
$230$	0	0
$231$	0	0
$232$	− 7.61553i	− 0.499984i
$233$	4.19224i	0.274643i	0.990527	+	0.137321i	$0.0438493\pi$
$233$	4.19224i	0.274643i	−0.990527	+	0.137321i	$0.956151\pi$
$234$	0	0
$235$	0	0
$236$	−2.82292	−0.183756
$237$	0	0
$238$	8.68466i	0.562943i
$239$	16.4924	1.06681	0.533403	−	0.845861i	$-0.320913\pi$
$239$	16.4924	1.06681	0.533403	+	0.845861i	$0.320913\pi$
$240$	0	0
$241$	23.0540	1.48504	0.742519	−	0.669826i	$-0.233631\pi$
$241$	23.0540	1.48504	0.742519	+	0.669826i	$0.233631\pi$
$242$	1.56155i	0.100380i
$243$	0	0
$244$	1.50758	0.0965128
$245$	0	0
$246$	0	0
$247$	− 13.6847i	− 0.870734i
$248$	− 18.7386i	− 1.18990i
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	2.43845i	0.153304i
$254$	−24.6847	−1.54885
$255$	0	0
$256$	10.0540	0.628373
$257$	− 1.75379i	− 0.109398i	−0.998503	−	0.0546992i	$-0.982580\pi$
$257$	− 1.75379i	− 0.109398i	0.998503	−	0.0546992i	$-0.0174200\pi$
$258$	0	0
$259$	−9.80776	−0.609425
$260$	0	0
$261$	0	0
$262$	− 25.7538i	− 1.59107i
$263$	21.3693i	1.31769i	0.752279	+	0.658844i	$0.228954\pi$
$263$	21.3693i	1.31769i	−0.752279	+	0.658844i	$0.771046\pi$
$264$	0	0
$265$	0	0
$266$	4.68466	0.287235
$267$	0	0
$268$	− 2.49242i	− 0.152249i
$269$	−4.49242	−0.273908	−0.136954	−	0.990577i	$-0.543731\pi$
$269$	−4.49242	−0.273908	−0.136954	+	0.990577i	$0.543731\pi$
$270$	0	0
$271$	17.1771	1.04343	0.521717	−	0.853119i	$-0.325292\pi$
$271$	17.1771	1.04343	0.521717	+	0.853119i	$0.325292\pi$
$272$	26.0540i	1.57975i
$273$	0	0
$274$	33.3693	2.01591
$275$	0	0
$276$	0	0
$277$	22.1771i	1.33249i	0.745732	+	0.666246i	$0.232100\pi$
$277$	22.1771i	1.33249i	−0.745732	+	0.666246i	$0.767900\pi$
$278$	− 12.4924i	− 0.749246i
$279$	0	0
$280$	0	0
$281$	23.8078	1.42025	0.710126	−	0.704075i	$-0.248638\pi$
$281$	23.8078	1.42025	0.710126	+	0.704075i	$0.248638\pi$
$282$	0	0
$283$	− 23.8769i	− 1.41933i	−0.704537	−	0.709667i	$-0.748845\pi$
$283$	− 23.8769i	− 1.41933i	0.704537	−	0.709667i	$-0.251155\pi$
$284$	−5.17708	−0.307203
$285$	0	0
$286$	−7.12311	−0.421198
$287$	− 4.68466i	− 0.276527i
$288$	0	0
$289$	−13.9309	−0.819463
$290$	0	0
$291$	0	0
$292$	− 0.107951i	− 0.00631733i
$293$	30.9309i	1.80700i	0.428587	+	0.903500i	$0.359011\pi$
$293$	30.9309i	1.80700i	−0.428587	+	0.903500i	$0.640989\pi$
$294$	0	0
$295$	0	0
$296$	−23.9157	−1.39007
$297$	0	0
$298$	− 18.4384i	− 1.06811i
$299$	−11.1231	−0.643266
$300$	0	0
$301$	−7.68466	−0.442936
$302$	− 6.73863i	− 0.387765i
$303$	0	0
$304$	14.0540	0.806051
$305$	0	0
$306$	0	0
$307$	5.93087i	0.338493i	0.985574	+	0.169246i	$0.0541333\pi$
$307$	5.93087i	0.338493i	−0.985574	+	0.169246i	$0.945867\pi$
$308$	− 0.438447i	− 0.0249828i
$309$	0	0
$310$	0	0
$311$	30.2462	1.71511	0.857553	−	0.514396i	$-0.171984\pi$
$311$	30.2462	1.71511	0.857553	+	0.514396i	$0.171984\pi$
$312$	0	0
$313$	14.7538i	0.833933i	0.908922	+	0.416967i	$0.136907\pi$
$313$	14.7538i	0.833933i	−0.908922	+	0.416967i	$0.863093\pi$
$314$	15.1231	0.853446
$315$	0	0
$316$	−1.45360	−0.0817715
$317$	− 15.1231i	− 0.849398i	−0.905335	−	0.424699i	$-0.860380\pi$
$317$	− 15.1231i	− 0.849398i	0.905335	−	0.424699i	$-0.139620\pi$
$318$	0	0
$319$	−3.12311	−0.174860
$320$	0	0
$321$	0	0
$322$	− 3.80776i	− 0.212198i
$323$	16.6847i	0.928359i
$324$	0	0
$325$	0	0
$326$	−27.6155	−1.52948
$327$	0	0
$328$	− 11.4233i	− 0.630746i
$329$	9.56155	0.527145
$330$	0	0
$331$	6.24621	0.343323	0.171661	−	0.985156i	$-0.445086\pi$
$331$	6.24621	0.343323	0.171661	+	0.985156i	$0.445086\pi$
$332$	0	0
$333$	0	0
$334$	14.6307	0.800555
$335$	0	0
$336$	0	0
$337$	1.68466i	0.0917692i	0.998947	+	0.0458846i	$0.0146107\pi$
$337$	1.68466i	0.0917692i	−0.998947	+	0.0458846i	$0.985389\pi$
$338$	− 12.1922i	− 0.663170i
$339$	0	0
$340$	0	0
$341$	−7.68466	−0.416148
$342$	0	0
$343$	13.0000i	0.701934i
$344$	−18.7386	−1.01032
$345$	0	0
$346$	−27.4233	−1.47429
$347$	2.63068i	0.141222i	0.997504	+	0.0706112i	$0.0224950\pi$
$347$	2.63068i	0.141222i	−0.997504	+	0.0706112i	$0.977505\pi$
$348$	0	0
$349$	−31.3693	−1.67916	−0.839581	−	0.543235i	$-0.817199\pi$
$349$	−31.3693	−1.67916	−0.839581	+	0.543235i	$0.817199\pi$
$350$	0	0
$351$	0	0
$352$	− 2.43845i	− 0.129970i
$353$	− 29.8617i	− 1.58938i	−0.607015	−	0.794690i	$-0.707633\pi$
$353$	− 29.8617i	− 1.58938i	0.607015	−	0.794690i	$-0.292367\pi$
$354$	0	0
$355$	0	0
$356$	3.89205	0.206278
$357$	0	0
$358$	16.3002i	0.861492i
$359$	9.75379	0.514785	0.257393	−	0.966307i	$-0.417137\pi$
$359$	9.75379	0.514785	0.257393	+	0.966307i	$0.417137\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	31.4233i	1.65157i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 23.0540i	− 1.20341i	−0.798719	−	0.601704i	$-0.794489\pi$
$367$	− 23.0540i	− 1.20341i	0.798719	−	0.601704i	$-0.205511\pi$
$368$	− 11.4233i	− 0.595480i
$369$	0	0
$370$	0	0
$371$	−7.12311	−0.369813
$372$	0	0
$373$	− 6.80776i	− 0.352493i	−0.984346	−	0.176246i	$-0.943604\pi$
$373$	− 6.80776i	− 0.352493i	0.984346	−	0.176246i	$-0.0563955\pi$
$374$	8.68466	0.449073
$375$	0	0
$376$	23.3153	1.20240
$377$	− 14.2462i	− 0.733717i
$378$	0	0
$379$	−29.3002	−1.50505	−0.752525	−	0.658564i	$-0.771164\pi$
$379$	−29.3002	−1.50505	−0.752525	+	0.658564i	$0.771164\pi$
$380$	0	0
$381$	0	0
$382$	− 42.0540i	− 2.15167i
$383$	5.75379i	0.294005i	0.989136	+	0.147002i	$0.0469625\pi$
$383$	5.75379i	0.294005i	−0.989136	+	0.147002i	$0.953037\pi$
$384$	0	0
$385$	0	0
$386$	20.9848	1.06810
$387$	0	0
$388$	2.68466i	0.136293i
$389$	33.8617	1.71686	0.858429	−	0.512932i	$-0.171441\pi$
$389$	33.8617	1.71686	0.858429	+	0.512932i	$0.171441\pi$
$390$	0	0
$391$	13.5616	0.685837
$392$	14.6307i	0.738961i
$393$	0	0
$394$	−17.0691	−0.859930
$395$	0	0
$396$	0	0
$397$	32.5616i	1.63422i	0.576484	+	0.817109i	$0.304424\pi$
$397$	32.5616i	1.63422i	−0.576484	+	0.817109i	$0.695576\pi$
$398$	− 10.6307i	− 0.532868i
$399$	0	0
$400$	0	0
$401$	35.6155	1.77855	0.889277	−	0.457368i	$-0.151208\pi$
$401$	35.6155	1.77855	0.889277	+	0.457368i	$0.151208\pi$
$402$	0	0
$403$	− 35.0540i	− 1.74616i
$404$	2.43845	0.121317
$405$	0	0
$406$	4.87689	0.242036
$407$	9.80776i	0.486153i
$408$	0	0
$409$	29.0540	1.43663	0.718313	−	0.695720i	$-0.244914\pi$
$409$	29.0540	1.43663	0.718313	+	0.695720i	$0.244914\pi$
$410$	0	0
$411$	0	0
$412$	− 0.384472i	− 0.0189416i
$413$	6.43845i	0.316815i
$414$	0	0
$415$	0	0
$416$	11.1231	0.545355
$417$	0	0
$418$	− 4.68466i	− 0.229134i
$419$	29.5616	1.44418	0.722088	−	0.691801i	$-0.243182\pi$
$419$	29.5616	1.44418	0.722088	+	0.691801i	$0.243182\pi$
$420$	0	0
$421$	20.0540	0.977371	0.488685	−	0.872460i	$-0.337477\pi$
$421$	20.0540	0.977371	0.488685	+	0.872460i	$0.337477\pi$
$422$	24.4924i	1.19227i
$423$	0	0
$424$	−17.3693	−0.843529
$425$	0	0
$426$	0	0
$427$	− 3.43845i	− 0.166398i
$428$	− 0.384472i	− 0.0185841i
$429$	0	0
$430$	0	0
$431$	38.2462	1.84226	0.921128	−	0.389261i	$-0.127270\pi$
$431$	38.2462	1.84226	0.921128	+	0.389261i	$0.127270\pi$
$432$	0	0
$433$	11.9309i	0.573361i	0.958026	+	0.286681i	$0.0925518\pi$
$433$	11.9309i	0.573361i	−0.958026	+	0.286681i	$0.907448\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	5.61553	0.268935
$437$	− 7.31534i	− 0.349940i
$438$	0	0
$439$	37.7386	1.80117	0.900583	−	0.434683i	$-0.143140\pi$
$439$	37.7386	1.80117	0.900583	+	0.434683i	$0.143140\pi$
$440$	0	0
$441$	0	0
$442$	39.6155i	1.88432i
$443$	− 28.3002i	− 1.34458i	−0.740287	−	0.672291i	$-0.765310\pi$
$443$	− 28.3002i	− 1.34458i	0.740287	−	0.672291i	$-0.234690\pi$
$444$	0	0
$445$	0	0
$446$	1.86174	0.0881559
$447$	0	0
$448$	− 5.56155i	− 0.262759i
$449$	4.49242	0.212011	0.106005	−	0.994366i	$-0.466194\pi$
$449$	4.49242	0.212011	0.106005	+	0.994366i	$0.466194\pi$
$450$	0	0
$451$	−4.68466	−0.220592
$452$	0.384472i	0.0180840i
$453$	0	0
$454$	−19.5076	−0.915536
$455$	0	0
$456$	0	0
$457$	− 11.3693i	− 0.531834i	−0.963996	−	0.265917i	$-0.914325\pi$
$457$	− 11.3693i	− 0.531834i	0.963996	−	0.265917i	$-0.0856747\pi$
$458$	− 0.192236i	− 0.00898260i
$459$	0	0
$460$	0	0
$461$	−41.8617	−1.94970	−0.974848	−	0.222872i	$-0.928457\pi$
$461$	−41.8617	−1.94970	−0.974848	+	0.222872i	$0.928457\pi$
$462$	0	0
$463$	18.2462i	0.847973i	0.905668	+	0.423987i	$0.139370\pi$
$463$	18.2462i	0.847973i	−0.905668	+	0.423987i	$0.860630\pi$
$464$	14.6307	0.679212
$465$	0	0
$466$	−6.54640	−0.303256
$467$	14.2462i	0.659236i	0.944114	+	0.329618i	$0.106920\pi$
$467$	14.2462i	0.659236i	−0.944114	+	0.329618i	$0.893080\pi$
$468$	0	0
$469$	−5.68466	−0.262493
$470$	0	0
$471$	0	0
$472$	15.6998i	0.722642i
$473$	7.68466i	0.353341i
$474$	0	0
$475$	0	0
$476$	−2.43845	−0.111766
$477$	0	0
$478$	25.7538i	1.17795i
$479$	−21.3693	−0.976389	−0.488195	−	0.872735i	$-0.662344\pi$
$479$	−21.3693	−0.976389	−0.488195	+	0.872735i	$0.662344\pi$
$480$	0	0
$481$	−44.7386	−2.03991
$482$	36.0000i	1.63976i
$483$	0	0
$484$	−0.438447	−0.0199294
$485$	0	0
$486$	0	0
$487$	25.0540i	1.13530i	0.823269	+	0.567652i	$0.192148\pi$
$487$	25.0540i	1.13530i	−0.823269	+	0.567652i	$0.807852\pi$
$488$	− 8.38447i	− 0.379547i
$489$	0	0
$490$	0	0
$491$	16.4924	0.744293	0.372146	−	0.928174i	$-0.378622\pi$
$491$	16.4924	0.744293	0.372146	+	0.928174i	$0.378622\pi$
$492$	0	0
$493$	17.3693i	0.782275i
$494$	21.3693	0.961451
$495$	0	0
$496$	36.0000	1.61645
$497$	11.8078i	0.529651i
$498$	0	0
$499$	16.8078	0.752419	0.376209	−	0.926535i	$-0.377227\pi$
$499$	16.8078	0.752419	0.376209	+	0.926535i	$0.377227\pi$
$500$	0	0
$501$	0	0
$502$	18.7386i	0.836346i
$503$	16.4924i	0.735361i	0.929952	+	0.367680i	$0.119848\pi$
$503$	16.4924i	0.735361i	−0.929952	+	0.367680i	$0.880152\pi$
$504$	0	0
$505$	0	0
$506$	−3.80776	−0.169276
$507$	0	0
$508$	− 6.93087i	− 0.307508i
$509$	−8.87689	−0.393461	−0.196731	−	0.980458i	$-0.563032\pi$
$509$	−8.87689	−0.393461	−0.196731	+	0.980458i	$0.563032\pi$
$510$	0	0
$511$	−0.246211	−0.0108917
$512$	− 11.4233i	− 0.504843i
$513$	0	0
$514$	2.73863	0.120796
$515$	0	0
$516$	0	0
$517$	− 9.56155i	− 0.420517i
$518$	− 15.3153i	− 0.672917i
$519$	0	0
$520$	0	0
$521$	−15.6155	−0.684129	−0.342064	−	0.939677i	$-0.611126\pi$
$521$	−15.6155	−0.684129	−0.342064	+	0.939677i	$0.611126\pi$
$522$	0	0
$523$	7.24621i	0.316855i	0.987371	+	0.158427i	$0.0506424\pi$
$523$	7.24621i	0.316855i	−0.987371	+	0.158427i	$0.949358\pi$
$524$	7.23106	0.315890
$525$	0	0
$526$	−33.3693	−1.45497
$527$	42.7386i	1.86172i
$528$	0	0
$529$	17.0540	0.741477
$530$	0	0
$531$	0	0
$532$	1.31534i	0.0570273i
$533$	− 21.3693i	− 0.925608i
$534$	0	0
$535$	0	0
$536$	−13.8617	−0.598736
$537$	0	0
$538$	− 7.01515i	− 0.302445i
$539$	6.00000	0.258438
$540$	0	0
$541$	−19.6847	−0.846310	−0.423155	−	0.906057i	$-0.639077\pi$
$541$	−19.6847	−0.846310	−0.423155	+	0.906057i	$0.639077\pi$
$542$	26.8229i	1.15214i
$543$	0	0
$544$	−13.5616	−0.581447
$545$	0	0
$546$	0	0
$547$	− 8.68466i	− 0.371329i	−0.982613	−	0.185665i	$-0.940556\pi$
$547$	− 8.68466i	− 0.371329i	0.982613	−	0.185665i	$-0.0594438\pi$
$548$	9.36932i	0.400237i
$549$	0	0
$550$	0	0
$551$	9.36932	0.399146
$552$	0	0
$553$	3.31534i	0.140983i
$554$	−34.6307	−1.47132
$555$	0	0
$556$	3.50758	0.148754
$557$	7.12311i	0.301816i	0.988548	+	0.150908i	$0.0482197\pi$
$557$	7.12311i	0.301816i	−0.988548	+	0.150908i	$0.951780\pi$
$558$	0	0
$559$	−35.0540	−1.48263
$560$	0	0
$561$	0	0
$562$	37.1771i	1.56822i
$563$	− 19.1231i	− 0.805943i	−0.915213	−	0.402971i	$-0.867977\pi$
$563$	− 19.1231i	− 0.805943i	0.915213	−	0.402971i	$-0.132023\pi$
$564$	0	0
$565$	0	0
$566$	37.2850	1.56721
$567$	0	0
$568$	28.7926i	1.20811i
$569$	9.06913	0.380198	0.190099	−	0.981765i	$-0.439119\pi$
$569$	9.06913	0.380198	0.190099	+	0.981765i	$0.439119\pi$
$570$	0	0
$571$	0.946025	0.0395899	0.0197950	−	0.999804i	$-0.493699\pi$
$571$	0.946025	0.0395899	0.0197950	+	0.999804i	$0.493699\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	7.31534	0.305336
$575$	0	0
$576$	0	0
$577$	− 3.63068i	− 0.151147i	−0.997140	−	0.0755737i	$-0.975921\pi$
$577$	− 3.63068i	− 0.151147i	0.997140	−	0.0755737i	$-0.0240788\pi$
$578$	− 21.7538i	− 0.904838i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.12311i	0.295009i
$584$	−0.600373	−0.0248436
$585$	0	0
$586$	−48.3002	−1.99526
$587$	47.8078i	1.97324i	0.163044	+	0.986619i	$0.447869\pi$
$587$	47.8078i	1.97324i	−0.163044	+	0.986619i	$0.552131\pi$
$588$	0	0
$589$	23.0540	0.949923
$590$	0	0
$591$	0	0
$592$	− 45.9460i	− 1.88837i
$593$	40.1080i	1.64704i	0.567290	+	0.823518i	$0.307992\pi$
$593$	40.1080i	1.64704i	−0.567290	+	0.823518i	$0.692008\pi$
$594$	0	0
$595$	0	0
$596$	5.17708	0.212061
$597$	0	0
$598$	− 17.3693i	− 0.710284i
$599$	−21.1771	−0.865272	−0.432636	−	0.901569i	$-0.642416\pi$
$599$	−21.1771	−0.865272	−0.432636	+	0.901569i	$0.642416\pi$
$600$	0	0
$601$	−24.4233	−0.996247	−0.498123	−	0.867106i	$-0.665977\pi$
$601$	−24.4233	−0.996247	−0.498123	+	0.867106i	$0.665977\pi$
$602$	− 12.0000i	− 0.489083i
$603$	0	0
$604$	1.89205	0.0769864
$605$	0	0
$606$	0	0
$607$	− 10.2462i	− 0.415881i	−0.978141	−	0.207940i	$-0.933324\pi$
$607$	− 10.2462i	− 0.415881i	0.978141	−	0.207940i	$-0.0666760\pi$
$608$	7.31534i	0.296676i
$609$	0	0
$610$	0	0
$611$	43.6155	1.76450
$612$	0	0
$613$	46.1080i	1.86228i	0.364659	+	0.931141i	$0.381186\pi$
$613$	46.1080i	1.86228i	−0.364659	+	0.931141i	$0.618814\pi$
$614$	−9.26137	−0.373758
$615$	0	0
$616$	−2.43845	−0.0982478
$617$	− 38.2462i	− 1.53973i	−0.638204	−	0.769867i	$-0.720322\pi$
$617$	− 38.2462i	− 1.53973i	0.638204	−	0.769867i	$-0.279678\pi$
$618$	0	0
$619$	38.1771	1.53447	0.767233	−	0.641368i	$-0.221633\pi$
$619$	38.1771	1.53447	0.767233	+	0.641368i	$0.221633\pi$
$620$	0	0
$621$	0	0
$622$	47.2311i	1.89379i
$623$	− 8.87689i	− 0.355645i
$624$	0	0
$625$	0	0
$626$	−23.0388	−0.920816
$627$	0	0
$628$	4.24621i	0.169442i
$629$	54.5464	2.17491
$630$	0	0
$631$	−10.3153	−0.410647	−0.205324	−	0.978694i	$-0.565825\pi$
$631$	−10.3153	−0.410647	−0.205324	+	0.978694i	$0.565825\pi$
$632$	8.08429i	0.321576i
$633$	0	0
$634$	23.6155	0.937892
$635$	0	0
$636$	0	0
$637$	27.3693i	1.08441i
$638$	− 4.87689i	− 0.193078i
$639$	0	0
$640$	0	0
$641$	6.73863	0.266160	0.133080	−	0.991105i	$-0.457513\pi$
$641$	6.73863	0.266160	0.133080	+	0.991105i	$0.457513\pi$
$642$	0	0
$643$	9.36932i	0.369490i	0.982787	+	0.184745i	$0.0591459\pi$
$643$	9.36932i	0.369490i	−0.982787	+	0.184745i	$0.940854\pi$
$644$	1.06913	0.0421296
$645$	0	0
$646$	−26.0540	−1.02508
$647$	20.1922i	0.793839i	0.917853	+	0.396919i	$0.129921\pi$
$647$	20.1922i	0.793839i	−0.917853	+	0.396919i	$0.870079\pi$
$648$	0	0
$649$	6.43845	0.252731
$650$	0	0
$651$	0	0
$652$	− 7.75379i	− 0.303662i
$653$	40.4924i	1.58459i	0.610138	+	0.792295i	$0.291114\pi$
$653$	40.4924i	1.58459i	−0.610138	+	0.792295i	$0.708886\pi$
$654$	0	0
$655$	0	0
$656$	21.9460	0.856848
$657$	0	0
$658$	14.9309i	0.582066i
$659$	33.3693	1.29988	0.649942	−	0.759984i	$-0.274793\pi$
$659$	33.3693	1.29988	0.649942	+	0.759984i	$0.274793\pi$
$660$	0	0
$661$	−13.8078	−0.537060	−0.268530	−	0.963271i	$-0.586538\pi$
$661$	−13.8078	−0.537060	−0.268530	+	0.963271i	$0.586538\pi$
$662$	9.75379i	0.379092i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.61553i	− 0.294874i
$668$	4.10795i	0.158941i
$669$	0	0
$670$	0	0
$671$	−3.43845	−0.132740
$672$	0	0
$673$	− 35.3693i	− 1.36339i	−0.731638	−	0.681693i	$-0.761244\pi$
$673$	− 35.3693i	− 1.36339i	0.731638	−	0.681693i	$-0.238756\pi$
$674$	−2.63068	−0.101330
$675$	0	0
$676$	3.42329	0.131665
$677$	− 20.8769i	− 0.802364i	−0.915998	−	0.401182i	$-0.868599\pi$
$677$	− 20.8769i	− 0.802364i	0.915998	−	0.401182i	$-0.131401\pi$
$678$	0	0
$679$	6.12311	0.234983
$680$	0	0
$681$	0	0
$682$	− 12.0000i	− 0.459504i
$683$	− 0.192236i	− 0.00735570i	−0.999993	−	0.00367785i	$-0.998829\pi$
$683$	− 0.192236i	− 0.00735570i	0.999993	−	0.00367785i	$-0.00117070\pi$
$684$	0	0
$685$	0	0
$686$	−20.3002	−0.775065
$687$	0	0
$688$	− 36.0000i	− 1.37249i
$689$	−32.4924	−1.23786
$690$	0	0
$691$	−20.4924	−0.779568	−0.389784	−	0.920906i	$-0.627450\pi$
$691$	−20.4924	−0.779568	−0.389784	+	0.920906i	$0.627450\pi$
$692$	− 7.69981i	− 0.292703i
$693$	0	0
$694$	−4.10795	−0.155936
$695$	0	0
$696$	0	0
$697$	26.0540i	0.986865i
$698$	− 48.9848i	− 1.85410i
$699$	0	0
$700$	0	0
$701$	−26.9309	−1.01716	−0.508582	−	0.861013i	$-0.669830\pi$
$701$	−26.9309	−1.01716	−0.508582	+	0.861013i	$0.669830\pi$
$702$	0	0
$703$	− 29.4233i	− 1.10972i
$704$	−5.56155	−0.209609
$705$	0	0
$706$	46.6307	1.75497
$707$	− 5.56155i	− 0.209164i
$708$	0	0
$709$	2.12311	0.0797349	0.0398675	−	0.999205i	$-0.487306\pi$
$709$	2.12311	0.0797349	0.0398675	+	0.999205i	$0.487306\pi$
$710$	0	0
$711$	0	0
$712$	− 21.6458i	− 0.811212i
$713$	− 18.7386i	− 0.701767i
$714$	0	0
$715$	0	0
$716$	−4.57671	−0.171040
$717$	0	0
$718$	15.2311i	0.568418i
$719$	32.9848	1.23013	0.615064	−	0.788478i	$-0.289130\pi$
$719$	32.9848	1.23013	0.615064	+	0.788478i	$0.289130\pi$
$720$	0	0
$721$	−0.876894	−0.0326573
$722$	− 15.6155i	− 0.581150i
$723$	0	0
$724$	−8.82292	−0.327901
$725$	0	0
$726$	0	0
$727$	7.05398i	0.261617i	0.991408	+	0.130809i	$0.0417574\pi$
$727$	7.05398i	0.261617i	−0.991408	+	0.130809i	$0.958243\pi$
$728$	− 11.1231i	− 0.412250i
$729$	0	0
$730$	0	0
$731$	42.7386	1.58075
$732$	0	0
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	5.94602	0.219173
$737$	5.68466i	0.209397i
$738$	0	0
$739$	−0.684658	−0.0251856	−0.0125928	−	0.999921i	$-0.504009\pi$
$739$	−0.684658	−0.0251856	−0.0125928	+	0.999921i	$0.504009\pi$
$740$	0	0
$741$	0	0
$742$	− 11.1231i	− 0.408342i
$743$	− 9.86174i	− 0.361792i	−0.983502	−	0.180896i	$-0.942100\pi$
$743$	− 9.86174i	− 0.361792i	0.983502	−	0.180896i	$-0.0578998\pi$
$744$	0	0
$745$	0	0
$746$	10.6307	0.389217
$747$	0	0
$748$	2.43845i	0.0891585i
$749$	−0.876894	−0.0320410
$750$	0	0
$751$	−22.7386	−0.829745	−0.414872	−	0.909880i	$-0.636174\pi$
$751$	−22.7386	−0.829745	−0.414872	+	0.909880i	$0.636174\pi$
$752$	44.7926i	1.63342i
$753$	0	0
$754$	22.2462	0.810159
$755$	0	0
$756$	0	0
$757$	− 47.3002i	− 1.71915i	−0.511006	−	0.859577i	$-0.670727\pi$
$757$	− 47.3002i	− 1.71915i	0.511006	−	0.859577i	$-0.329273\pi$
$758$	− 45.7538i	− 1.66185i
$759$	0	0
$760$	0	0
$761$	−45.3693	−1.64464	−0.822318	−	0.569028i	$-0.807320\pi$
$761$	−45.3693	−1.64464	−0.822318	+	0.569028i	$0.807320\pi$
$762$	0	0
$763$	− 12.8078i	− 0.463672i
$764$	11.8078	0.427190
$765$	0	0
$766$	−8.98485	−0.324636
$767$	29.3693i	1.06046i
$768$	0	0
$769$	31.0540	1.11983	0.559917	−	0.828548i	$-0.310833\pi$
$769$	31.0540	1.11983	0.559917	+	0.828548i	$0.310833\pi$
$770$	0	0
$771$	0	0
$772$	5.89205i	0.212059i
$773$	15.5076i	0.557769i	0.960325	+	0.278884i	$0.0899646\pi$
$773$	15.5076i	0.557769i	−0.960325	+	0.278884i	$0.910035\pi$
$774$	0	0
$775$	0	0
$776$	14.9309	0.535987
$777$	0	0
$778$	52.8769i	1.89573i
$779$	14.0540	0.503536
$780$	0	0
$781$	11.8078	0.422515
$782$	21.1771i	0.757291i
$783$	0	0
$784$	−28.1080	−1.00386
$785$	0	0
$786$	0	0
$787$	53.7386i	1.91558i	0.287476	+	0.957788i	$0.407184\pi$
$787$	53.7386i	1.91558i	−0.287476	+	0.957788i	$0.592816\pi$
$788$	− 4.79261i	− 0.170730i
$789$	0	0
$790$	0	0
$791$	0.876894	0.0311788
$792$	0	0
$793$	− 15.6847i	− 0.556979i
$794$	−50.8466	−1.80448
$795$	0	0
$796$	2.98485	0.105795
$797$	− 51.1231i	− 1.81087i	−0.424481	−	0.905437i	$-0.639544\pi$
$797$	− 51.1231i	− 1.81087i	0.424481	−	0.905437i	$-0.360456\pi$
$798$	0	0
$799$	−53.1771	−1.88127
$800$	0	0
$801$	0	0
$802$	55.6155i	1.96385i
$803$	0.246211i	0.00868861i
$804$	0	0
$805$	0	0
$806$	54.7386	1.92809
$807$	0	0
$808$	− 13.5616i	− 0.477094i
$809$	7.31534	0.257194	0.128597	−	0.991697i	$-0.458953\pi$
$809$	7.31534	0.257194	0.128597	+	0.991697i	$0.458953\pi$
$810$	0	0
$811$	−1.00000	−0.0351147	−0.0175574	−	0.999846i	$-0.505589\pi$
$811$	−1.00000	−0.0351147	−0.0175574	+	0.999846i	$0.505589\pi$
$812$	1.36932i	0.0480536i
$813$	0	0
$814$	−15.3153	−0.536802
$815$	0	0
$816$	0	0
$817$	− 23.0540i	− 0.806557i
$818$	45.3693i	1.58630i
$819$	0	0
$820$	0	0
$821$	16.8769	0.589008	0.294504	−	0.955650i	$-0.404846\pi$
$821$	16.8769	0.589008	0.294504	+	0.955650i	$0.404846\pi$
$822$	0	0
$823$	37.0540i	1.29162i	0.763498	+	0.645810i	$0.223480\pi$
$823$	37.0540i	1.29162i	−0.763498	+	0.645810i	$0.776520\pi$
$824$	−2.13826	−0.0744898
$825$	0	0
$826$	−10.0540	−0.349823
$827$	− 26.2462i	− 0.912670i	−0.889808	−	0.456335i	$-0.849162\pi$
$827$	− 26.2462i	− 0.912670i	0.889808	−	0.456335i	$-0.150838\pi$
$828$	0	0
$829$	0.738634	0.0256538	0.0128269	−	0.999918i	$-0.495917\pi$
$829$	0.738634	0.0256538	0.0128269	+	0.999918i	$0.495917\pi$
$830$	0	0
$831$	0	0
$832$	− 25.3693i	− 0.879523i
$833$	− 33.3693i	− 1.15618i
$834$	0	0
$835$	0	0
$836$	1.31534	0.0454920
$837$	0	0
$838$	46.1619i	1.59464i
$839$	−12.4924	−0.431286	−0.215643	−	0.976472i	$-0.569185\pi$
$839$	−12.4924	−0.431286	−0.215643	+	0.976472i	$0.569185\pi$
$840$	0	0
$841$	−19.2462	−0.663662
$842$	31.3153i	1.07920i
$843$	0	0
$844$	−6.87689	−0.236712
$845$	0	0
$846$	0	0
$847$	1.00000i	0.0343604i
$848$	− 33.3693i	− 1.14591i
$849$	0	0
$850$	0	0
$851$	−23.9157	−0.819820
$852$	0	0
$853$	− 26.4233i	− 0.904716i	−0.891836	−	0.452358i	$-0.850583\pi$
$853$	− 26.4233i	− 0.904716i	0.891836	−	0.452358i	$-0.149417\pi$
$854$	5.36932	0.183734
$855$	0	0
$856$	−2.13826	−0.0730842
$857$	− 9.56155i	− 0.326616i	−0.986575	−	0.163308i	$-0.947784\pi$
$857$	− 9.56155i	− 0.326616i	0.986575	−	0.163308i	$-0.0522165\pi$
$858$	0	0
$859$	34.3542	1.17215	0.586074	−	0.810257i	$-0.300673\pi$
$859$	34.3542	1.17215	0.586074	+	0.810257i	$0.300673\pi$
$860$	0	0
$861$	0	0
$862$	59.7235i	2.03419i
$863$	14.2462i	0.484947i	0.970158	+	0.242473i	$0.0779587\pi$
$863$	14.2462i	0.484947i	−0.970158	+	0.242473i	$0.922041\pi$
$864$	0	0
$865$	0	0
$866$	−18.6307	−0.633096
$867$	0	0
$868$	3.36932i	0.114362i
$869$	3.31534	0.112465
$870$	0	0
$871$	−25.9309	−0.878634
$872$	− 31.2311i	− 1.05762i
$873$	0	0
$874$	11.4233	0.386399
$875$	0	0
$876$	0	0
$877$	− 23.0540i	− 0.778477i	−0.921137	−	0.389239i	$-0.872738\pi$
$877$	− 23.0540i	− 0.778477i	0.921137	−	0.389239i	$-0.127262\pi$
$878$	58.9309i	1.98882i
$879$	0	0
$880$	0	0
$881$	26.2462	0.884257	0.442129	−	0.896952i	$-0.354223\pi$
$881$	26.2462	0.884257	0.442129	+	0.896952i	$0.354223\pi$
$882$	0	0
$883$	16.0691i	0.540769i	0.962752	+	0.270385i	$0.0871509\pi$
$883$	16.0691i	0.540769i	−0.962752	+	0.270385i	$0.912849\pi$
$884$	−11.1231	−0.374111
$885$	0	0
$886$	44.1922	1.48467
$887$	29.8617i	1.00266i	0.865256	+	0.501330i	$0.167156\pi$
$887$	29.8617i	1.00266i	−0.865256	+	0.501330i	$0.832844\pi$
$888$	0	0
$889$	−15.8078	−0.530175
$890$	0	0
$891$	0	0
$892$	0.522732i	0.0175024i
$893$	28.6847i	0.959895i
$894$	0	0
$895$	0	0
$896$	13.5616	0.453060
$897$	0	0
$898$	7.01515i	0.234099i
$899$	24.0000	0.800445
$900$	0	0
$901$	39.6155	1.31978
$902$	− 7.31534i	− 0.243574i
$903$	0	0
$904$	2.13826	0.0711175
$905$	0	0
$906$	0	0
$907$	− 28.1080i	− 0.933309i	−0.884440	−	0.466655i	$-0.845459\pi$
$907$	− 28.1080i	− 0.933309i	0.884440	−	0.466655i	$-0.154541\pi$
$908$	− 5.47727i	− 0.181770i
$909$	0	0
$910$	0	0
$911$	−54.1619	−1.79446	−0.897232	−	0.441559i	$-0.854426\pi$
$911$	−54.1619	−1.79446	−0.897232	+	0.441559i	$0.854426\pi$
$912$	0	0
$913$	0	0
$914$	17.7538	0.587243
$915$	0	0
$916$	0.0539753	0.00178339
$917$	− 16.4924i	− 0.544628i
$918$	0	0
$919$	33.1080	1.09213	0.546065	−	0.837743i	$-0.316125\pi$
$919$	33.1080	1.09213	0.546065	+	0.837743i	$0.316125\pi$
$920$	0	0
$921$	0	0
$922$	− 65.3693i	− 2.15282i
$923$	53.8617i	1.77288i
$924$	0	0
$925$	0	0
$926$	−28.4924	−0.936319
$927$	0	0
$928$	7.61553i	0.249992i
$929$	−59.2311	−1.94331	−0.971654	−	0.236408i	$-0.924030\pi$
$929$	−59.2311	−1.94331	−0.971654	+	0.236408i	$0.924030\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	− 1.83807i	− 0.0602081i
$933$	0	0
$934$	−22.2462	−0.727918
$935$	0	0
$936$	0	0
$937$	22.3153i	0.729010i	0.931201	+	0.364505i	$0.118762\pi$
$937$	22.3153i	0.729010i	−0.931201	+	0.364505i	$0.881238\pi$
$938$	− 8.87689i	− 0.289841i
$939$	0	0
$940$	0	0
$941$	−14.8229	−0.483213	−0.241607	−	0.970374i	$-0.577674\pi$
$941$	−14.8229	−0.483213	−0.241607	+	0.970374i	$0.577674\pi$
$942$	0	0
$943$	− 11.4233i	− 0.371994i
$944$	−30.1619	−0.981687
$945$	0	0
$946$	−12.0000	−0.390154
$947$	− 13.1771i	− 0.428198i	−0.976812	−	0.214099i	$-0.931319\pi$
$947$	− 13.1771i	− 0.428198i	0.976812	−	0.214099i	$-0.0686815\pi$
$948$	0	0
$949$	−1.12311	−0.0364576
$950$	0	0
$951$	0	0
$952$	13.5616i	0.439532i
$953$	15.8078i	0.512064i	0.966668	+	0.256032i	$0.0824152\pi$
$953$	15.8078i	0.512064i	−0.966668	+	0.256032i	$0.917585\pi$
$954$	0	0
$955$	0	0
$956$	−7.23106	−0.233869
$957$	0	0
$958$	− 33.3693i	− 1.07811i
$959$	21.3693	0.690051
$960$	0	0
$961$	28.0540	0.904967
$962$	− 69.8617i	− 2.25243i
$963$	0	0
$964$	−10.1080	−0.325555
$965$	0	0
$966$	0	0
$967$	44.0000i	1.41494i	0.706741	+	0.707472i	$0.250165\pi$
$967$	44.0000i	1.41494i	−0.706741	+	0.707472i	$0.749835\pi$
$968$	2.43845i	0.0783747i
$969$	0	0
$970$	0	0
$971$	37.1771	1.19307	0.596535	−	0.802587i	$-0.296544\pi$
$971$	37.1771	1.19307	0.596535	+	0.802587i	$0.296544\pi$
$972$	0	0
$973$	− 8.00000i	− 0.256468i
$974$	−39.1231	−1.25359
$975$	0	0
$976$	16.1080	0.515603
$977$	− 15.6155i	− 0.499585i	−0.968299	−	0.249793i	$-0.919638\pi$
$977$	− 15.6155i	− 0.499585i	0.968299	−	0.249793i	$-0.0803624\pi$
$978$	0	0
$979$	−8.87689	−0.283707
$980$	0	0
$981$	0	0
$982$	25.7538i	0.821836i
$983$	− 6.93087i	− 0.221060i	−0.993873	−	0.110530i	$-0.964745\pi$
$983$	− 6.93087i	− 0.221060i	0.993873	−	0.110530i	$-0.0352549\pi$
$984$	0	0
$985$	0	0
$986$	−27.1231	−0.863776
$987$	0	0
$988$	6.00000i	0.190885i
$989$	−18.7386	−0.595854
$990$	0	0
$991$	49.6847	1.57829	0.789143	−	0.614210i	$-0.210525\pi$
$991$	49.6847	1.57829	0.789143	+	0.614210i	$0.210525\pi$
$992$	18.7386i	0.594952i
$993$	0	0
$994$	−18.4384	−0.584832
$995$	0	0
$996$	0	0
$997$	− 28.7386i	− 0.910162i	−0.890450	−	0.455081i	$-0.849610\pi$
$997$	− 28.7386i	− 0.910162i	0.890450	−	0.455081i	$-0.150390\pi$
$998$	26.2462i	0.830809i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.j.199.3		4
3.2	odd	2		2475.2.c.i.199.2		4
5.2	odd	4		2475.2.a.u.1.1	yes	2
5.3	odd	4		2475.2.a.q.1.2	yes	2
5.4	even	2	inner	2475.2.c.j.199.2		4
15.2	even	4		2475.2.a.p.1.2	✓	2
15.8	even	4		2475.2.a.v.1.1	yes	2
15.14	odd	2		2475.2.c.i.199.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.2.a.p.1.2	✓	2	15.2	even	4
2475.2.a.q.1.2	yes	2	5.3	odd	4
2475.2.a.u.1.1	yes	2	5.2	odd	4
2475.2.a.v.1.1	yes	2	15.8	even	4
2475.2.c.i.199.2		4	3.2	odd	2
2475.2.c.i.199.3		4	15.14	odd	2
2475.2.c.j.199.2		4	5.4	even	2	inner
2475.2.c.j.199.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.56155i q^{2} -0.438447 q^{4} +1.00000i q^{7} +2.43845i q^{8} +O(q^{10})\)	\(q+1.56155i q^{2} -0.438447 q^{4} +1.00000i q^{7} +2.43845i q^{8} +1.00000 q^{11} +4.56155i q^{13} -1.56155 q^{14} -4.68466 q^{16} -5.56155i q^{17} -3.00000 q^{19} +1.56155i q^{22} +2.43845i q^{23} -7.12311 q^{26} -0.438447i q^{28} -3.12311 q^{29} -7.68466 q^{31} -2.43845i q^{32} +8.68466 q^{34} +9.80776i q^{37} -4.68466i q^{38} -4.68466 q^{41} +7.68466i q^{43} -0.438447 q^{44} -3.80776 q^{46} -9.56155i q^{47} +6.00000 q^{49} -2.00000i q^{52} +7.12311i q^{53} -2.43845 q^{56} -4.87689i q^{58} +6.43845 q^{59} -3.43845 q^{61} -12.0000i q^{62} -5.56155 q^{64} +5.68466i q^{67} +2.43845i q^{68} +11.8078 q^{71} +0.246211i q^{73} -15.3153 q^{74} +1.31534 q^{76} +1.00000i q^{77} +3.31534 q^{79} -7.31534i q^{82} -12.0000 q^{86} +2.43845i q^{88} -8.87689 q^{89} -4.56155 q^{91} -1.06913i q^{92} +14.9309 q^{94} -6.12311i q^{97} +9.36932i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4}+O(q^{10}) \) 4 * q - 10 * q^4	\( 4 q - 10 q^{4} + 4 q^{11} + 2 q^{14} + 6 q^{16} - 12 q^{19} - 12 q^{26} + 4 q^{29} - 6 q^{31} + 10 q^{34} + 6 q^{41} - 10 q^{44} + 26 q^{46} + 24 q^{49} - 18 q^{56} + 34 q^{59} - 22 q^{61} - 14 q^{64} + 6 q^{71} - 86 q^{74} + 30 q^{76} + 38 q^{79} - 48 q^{86} - 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100}) \) 4 * q - 10 * q^4 + 4 * q^11 + 2 * q^14 + 6 * q^16 - 12 * q^19 - 12 * q^26 + 4 * q^29 - 6 * q^31 + 10 * q^34 + 6 * q^41 - 10 * q^44 + 26 * q^46 + 24 * q^49 - 18 * q^56 + 34 * q^59 - 22 * q^61 - 14 * q^64 + 6 * q^71 - 86 * q^74 + 30 * q^76 + 38 * q^79 - 48 * q^86 - 52 * q^89 - 10 * q^91 + 2 * q^94

Introduction

L-functions

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Database

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