LMFDB - Embedded newform 2475.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{7} -2.43845 q^{8} +O(q^{10})$	$q+1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{7} -2.43845 q^{8} +1.00000 q^{11} -4.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} -5.56155 q^{17} +3.00000 q^{19} +1.56155 q^{22} -2.43845 q^{23} -7.12311 q^{26} +0.438447 q^{28} +3.12311 q^{29} -7.68466 q^{31} -2.43845 q^{32} -8.68466 q^{34} +9.80776 q^{37} +4.68466 q^{38} -4.68466 q^{41} -7.68466 q^{43} +0.438447 q^{44} -3.80776 q^{46} -9.56155 q^{47} -6.00000 q^{49} -2.00000 q^{52} -7.12311 q^{53} -2.43845 q^{56} +4.87689 q^{58} -6.43845 q^{59} -3.43845 q^{61} -12.0000 q^{62} +5.56155 q^{64} +5.68466 q^{67} -2.43845 q^{68} +11.8078 q^{71} -0.246211 q^{73} +15.3153 q^{74} +1.31534 q^{76} +1.00000 q^{77} -3.31534 q^{79} -7.31534 q^{82} -12.0000 q^{86} -2.43845 q^{88} +8.87689 q^{89} -4.56155 q^{91} -1.06913 q^{92} -14.9309 q^{94} -6.12311 q^{97} -9.36932 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8}+O(q^{10}) $ 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8	$ 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} - 5 q^{13} - q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{19} - q^{22} - 9 q^{23} - 6 q^{26} + 5 q^{28} - 2 q^{29} - 3 q^{31} - 9 q^{32} - 5 q^{34} - q^{37} - 3 q^{38} + 3 q^{41} - 3 q^{43} + 5 q^{44} + 13 q^{46} - 15 q^{47} - 12 q^{49} - 4 q^{52} - 6 q^{53} - 9 q^{56} + 18 q^{58} - 17 q^{59} - 11 q^{61} - 24 q^{62} + 7 q^{64} - q^{67} - 9 q^{68} + 3 q^{71} + 16 q^{73} + 43 q^{74} + 15 q^{76} + 2 q^{77} - 19 q^{79} - 27 q^{82} - 24 q^{86} - 9 q^{88} + 26 q^{89} - 5 q^{91} - 31 q^{92} - q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 - 5 * q^13 - q^14 + 3 * q^16 - 7 * q^17 + 6 * q^19 - q^22 - 9 * q^23 - 6 * q^26 + 5 * q^28 - 2 * q^29 - 3 * q^31 - 9 * q^32 - 5 * q^34 - q^37 - 3 * q^38 + 3 * q^41 - 3 * q^43 + 5 * q^44 + 13 * q^46 - 15 * q^47 - 12 * q^49 - 4 * q^52 - 6 * q^53 - 9 * q^56 + 18 * q^58 - 17 * q^59 - 11 * q^61 - 24 * q^62 + 7 * q^64 - q^67 - 9 * q^68 + 3 * q^71 + 16 * q^73 + 43 * q^74 + 15 * q^76 + 2 * q^77 - 19 * q^79 - 27 * q^82 - 24 * q^86 - 9 * q^88 + 26 * q^89 - 5 * q^91 - 31 * q^92 - q^94 - 4 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.56155	1.10418	0.552092	−	0.833783i	$-0.313830\pi$
$2$	1.56155	1.10418	0.552092	+	0.833783i	$0.313830\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−2.43845	−0.862121
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.56155	−1.26515	−0.632574	−	0.774500i	$-0.718001\pi$
$13$	−4.56155	−1.26515	−0.632574	+	0.774500i	$0.718001\pi$
$14$	1.56155	0.417343
$15$	0	0
$16$	−4.68466	−1.17116
$17$	−5.56155	−1.34887	−0.674437	−	0.738332i	$-0.735614\pi$
$17$	−5.56155	−1.34887	−0.674437	+	0.738332i	$0.735614\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	1.56155	0.332924
$23$	−2.43845	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$23$	−2.43845	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$24$	0	0
$25$	0	0
$26$	−7.12311	−1.39696
$27$	0	0
$28$	0.438447	0.0828587
$29$	3.12311	0.579946	0.289973	−	0.957035i	$-0.406354\pi$
$29$	3.12311	0.579946	0.289973	+	0.957035i	$0.406354\pi$
$30$	0	0
$31$	−7.68466	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$31$	−7.68466	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$32$	−2.43845	−0.431061
$33$	0	0
$34$	−8.68466	−1.48941
$35$	0	0
$36$	0	0
$37$	9.80776	1.61239	0.806193	−	0.591652i	$-0.201524\pi$
$37$	9.80776	1.61239	0.806193	+	0.591652i	$0.201524\pi$
$38$	4.68466	0.759952
$39$	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
$43$	−7.68466	−1.17190	−0.585950	−	0.810347i	$-0.699278\pi$
$43$	−7.68466	−1.17190	−0.585950	+	0.810347i	$0.699278\pi$
$44$	0.438447	0.0660984
$45$	0	0
$46$	−3.80776	−0.561424
$47$	−9.56155	−1.39470	−0.697348	−	0.716733i	$-0.745637\pi$
$47$	−9.56155	−1.39470	−0.697348	+	0.716733i	$0.745637\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−7.12311	−0.978434	−0.489217	−	0.872162i	$-0.662717\pi$
$53$	−7.12311	−0.978434	−0.489217	+	0.872162i	$0.662717\pi$
$54$	0	0
$55$	0	0
$56$	−2.43845	−0.325851
$57$	0	0
$58$	4.87689	0.640368
$59$	−6.43845	−0.838214	−0.419107	−	0.907937i	$-0.637657\pi$
$59$	−6.43845	−0.838214	−0.419107	+	0.907937i	$0.637657\pi$
$60$	0	0
$61$	−3.43845	−0.440248	−0.220124	−	0.975472i	$-0.570646\pi$
$61$	−3.43845	−0.440248	−0.220124	+	0.975472i	$0.570646\pi$
$62$	−12.0000	−1.52400
$63$	0	0
$64$	5.56155	0.695194
$65$	0	0
$66$	0	0
$67$	5.68466	0.694492	0.347246	−	0.937774i	$-0.387117\pi$
$67$	5.68466	0.694492	0.347246	+	0.937774i	$0.387117\pi$
$68$	−2.43845	−0.295705
$69$	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
$73$	−0.246211	−0.0288168	−0.0144084	−	0.999896i	$-0.504587\pi$
$73$	−0.246211	−0.0288168	−0.0144084	+	0.999896i	$0.504587\pi$
$74$	15.3153	1.78037
$75$	0	0
$76$	1.31534	0.150880
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.31534	−0.373005	−0.186503	−	0.982454i	$-0.559715\pi$
$79$	−3.31534	−0.373005	−0.186503	+	0.982454i	$0.559715\pi$
$80$	0	0
$81$	0	0
$82$	−7.31534	−0.807844
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	−2.43845	−0.259939
$89$	8.87689	0.940949	0.470474	−	0.882414i	$-0.344083\pi$
$89$	8.87689	0.940949	0.470474	+	0.882414i	$0.344083\pi$
$90$	0	0
$91$	−4.56155	−0.478181
$92$	−1.06913	−0.111465
$93$	0	0
$94$	−14.9309	−1.54000
$95$	0	0
$96$	0	0
$97$	−6.12311	−0.621707	−0.310854	−	0.950458i	$-0.600615\pi$
$97$	−6.12311	−0.621707	−0.310854	+	0.950458i	$0.600615\pi$
$98$	−9.36932	−0.946444
$99$	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
$103$	−0.876894	−0.0864030	−0.0432015	−	0.999066i	$-0.513756\pi$
$103$	−0.876894	−0.0864030	−0.0432015	+	0.999066i	$0.513756\pi$
$104$	11.1231	1.09071
$105$	0	0
$106$	−11.1231	−1.08037
$107$	0.876894	0.0847726	0.0423863	−	0.999101i	$-0.486504\pi$
$107$	0.876894	0.0847726	0.0423863	+	0.999101i	$0.486504\pi$
$108$	0	0
$109$	12.8078	1.22676	0.613381	−	0.789787i	$-0.289809\pi$
$109$	12.8078	1.22676	0.613381	+	0.789787i	$0.289809\pi$
$110$	0	0
$111$	0	0
$112$	−4.68466	−0.442659
$113$	0.876894	0.0824913	0.0412456	−	0.999149i	$-0.486867\pi$
$113$	0.876894	0.0824913	0.0412456	+	0.999149i	$0.486867\pi$
$114$	0	0
$115$	0	0
$116$	1.36932	0.127138
$117$	0	0
$118$	−10.0540	−0.925543
$119$	−5.56155	−0.509827
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.36932	−0.486115
$123$	0	0
$124$	−3.36932	−0.302574
$125$	0	0
$126$	0	0
$127$	15.8078	1.40271	0.701356	−	0.712811i	$-0.252578\pi$
$127$	15.8078	1.40271	0.701356	+	0.712811i	$0.252578\pi$
$128$	13.5616	1.19868
$129$	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
$133$	3.00000	0.260133
$134$	8.87689	0.766847
$135$	0	0
$136$	13.5616	1.16289
$137$	−21.3693	−1.82570	−0.912852	−	0.408291i	$-0.866125\pi$
$137$	−21.3693	−1.82570	−0.912852	+	0.408291i	$0.866125\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	18.4384	1.54732
$143$	−4.56155	−0.381456
$144$	0	0
$145$	0	0
$146$	−0.384472	−0.0318191
$147$	0	0
$148$	4.30019	0.353473
$149$	11.8078	0.967330	0.483665	−	0.875253i	$-0.339305\pi$
$149$	11.8078	0.967330	0.483665	+	0.875253i	$0.339305\pi$
$150$	0	0
$151$	−4.31534	−0.351178	−0.175589	−	0.984464i	$-0.556183\pi$
$151$	−4.31534	−0.351178	−0.175589	+	0.984464i	$0.556183\pi$
$152$	−7.31534	−0.593353
$153$	0	0
$154$	1.56155	0.125834
$155$	0	0
$156$	0	0
$157$	−9.68466	−0.772920	−0.386460	−	0.922306i	$-0.626302\pi$
$157$	−9.68466	−0.772920	−0.386460	+	0.922306i	$0.626302\pi$
$158$	−5.17708	−0.411866
$159$	0	0
$160$	0	0
$161$	−2.43845	−0.192177
$162$	0	0
$163$	−17.6847	−1.38517	−0.692585	−	0.721337i	$-0.743528\pi$
$163$	−17.6847	−1.38517	−0.692585	+	0.721337i	$0.743528\pi$
$164$	−2.05398	−0.160389
$165$	0	0
$166$	0	0
$167$	−9.36932	−0.725020	−0.362510	−	0.931980i	$-0.618080\pi$
$167$	−9.36932	−0.725020	−0.362510	+	0.931980i	$0.618080\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	0	0
$171$	0	0
$172$	−3.36932	−0.256908
$173$	−17.5616	−1.33518	−0.667590	−	0.744529i	$-0.732674\pi$
$173$	−17.5616	−1.33518	−0.667590	+	0.744529i	$0.732674\pi$
$174$	0	0
$175$	0	0
$176$	−4.68466	−0.353119
$177$	0	0
$178$	13.8617	1.03898
$179$	−10.4384	−0.780206	−0.390103	−	0.920771i	$-0.627561\pi$
$179$	−10.4384	−0.780206	−0.390103	+	0.920771i	$0.627561\pi$
$180$	0	0
$181$	20.1231	1.49574	0.747869	−	0.663846i	$-0.231077\pi$
$181$	20.1231	1.49574	0.747869	+	0.663846i	$0.231077\pi$
$182$	−7.12311	−0.528000
$183$	0	0
$184$	5.94602	0.438347
$185$	0	0
$186$	0	0
$187$	−5.56155	−0.406701
$188$	−4.19224	−0.305750
$189$	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
$193$	13.4384	0.967321	0.483660	−	0.875256i	$-0.339307\pi$
$193$	13.4384	0.967321	0.483660	+	0.875256i	$0.339307\pi$
$194$	−9.56155	−0.686479
$195$	0	0
$196$	−2.63068	−0.187906
$197$	10.9309	0.778792	0.389396	−	0.921070i	$-0.372684\pi$
$197$	10.9309	0.778792	0.389396	+	0.921070i	$0.372684\pi$
$198$	0	0
$199$	6.80776	0.482590	0.241295	−	0.970452i	$-0.422428\pi$
$199$	6.80776	0.482590	0.241295	+	0.970452i	$0.422428\pi$
$200$	0	0
$201$	0	0
$202$	−8.68466	−0.611050
$203$	3.12311	0.219199
$204$	0	0
$205$	0	0
$206$	−1.36932	−0.0954048
$207$	0	0
$208$	21.3693	1.48170
$209$	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	$0.318467\pi$
$212$	−3.12311	−0.214496
$213$	0	0
$214$	1.36932	0.0936046
$215$	0	0
$216$	0	0
$217$	−7.68466	−0.521669
$218$	20.0000	1.35457
$219$	0	0
$220$	0	0
$221$	25.3693	1.70652
$222$	0	0
$223$	1.19224	0.0798380	0.0399190	−	0.999203i	$-0.487290\pi$
$223$	1.19224	0.0798380	0.0399190	+	0.999203i	$0.487290\pi$
$224$	−2.43845	−0.162926
$225$	0	0
$226$	1.36932	0.0910856
$227$	12.4924	0.829151	0.414576	−	0.910015i	$-0.363930\pi$
$227$	12.4924	0.829151	0.414576	+	0.910015i	$0.363930\pi$
$228$	0	0
$229$	0.123106	0.00813505	0.00406752	−	0.999992i	$-0.498705\pi$
$229$	0.123106	0.00813505	0.00406752	+	0.999992i	$0.498705\pi$
$230$	0	0
$231$	0	0
$232$	−7.61553	−0.499984
$233$	−4.19224	−0.274643	−0.137321	−	0.990527i	$-0.543849\pi$
$233$	−4.19224	−0.274643	−0.137321	+	0.990527i	$0.543849\pi$
$234$	0	0
$235$	0	0
$236$	−2.82292	−0.183756
$237$	0	0
$238$	−8.68466	−0.562943
$239$	−16.4924	−1.06681	−0.533403	−	0.845861i	$-0.679087\pi$
$239$	−16.4924	−1.06681	−0.533403	+	0.845861i	$0.679087\pi$
$240$	0	0
$241$	23.0540	1.48504	0.742519	−	0.669826i	$-0.233631\pi$
$241$	23.0540	1.48504	0.742519	+	0.669826i	$0.233631\pi$
$242$	1.56155	0.100380
$243$	0	0
$244$	−1.50758	−0.0965128
$245$	0	0
$246$	0	0
$247$	−13.6847	−0.870734
$248$	18.7386	1.18990
$249$	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
$253$	−2.43845	−0.153304
$254$	24.6847	1.54885
$255$	0	0
$256$	10.0540	0.628373
$257$	−1.75379	−0.109398	−0.0546992	−	0.998503i	$-0.517420\pi$
$257$	−1.75379	−0.109398	−0.0546992	+	0.998503i	$0.517420\pi$
$258$	0	0
$259$	9.80776	0.609425
$260$	0	0
$261$	0	0
$262$	−25.7538	−1.59107
$263$	−21.3693	−1.31769	−0.658844	−	0.752279i	$-0.728954\pi$
$263$	−21.3693	−1.31769	−0.658844	+	0.752279i	$0.728954\pi$
$264$	0	0
$265$	0	0
$266$	4.68466	0.287235
$267$	0	0
$268$	2.49242	0.152249
$269$	4.49242	0.273908	0.136954	−	0.990577i	$-0.456269\pi$
$269$	4.49242	0.273908	0.136954	+	0.990577i	$0.456269\pi$
$270$	0	0
$271$	17.1771	1.04343	0.521717	−	0.853119i	$-0.325292\pi$
$271$	17.1771	1.04343	0.521717	+	0.853119i	$0.325292\pi$
$272$	26.0540	1.57975
$273$	0	0
$274$	−33.3693	−2.01591
$275$	0	0
$276$	0	0
$277$	22.1771	1.33249	0.666246	−	0.745732i	$-0.267900\pi$
$277$	22.1771	1.33249	0.666246	+	0.745732i	$0.267900\pi$
$278$	12.4924	0.749246
$279$	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
$283$	23.8769	1.41933	0.709667	−	0.704537i	$-0.248845\pi$
$283$	23.8769	1.41933	0.709667	+	0.704537i	$0.248845\pi$
$284$	5.17708	0.307203
$285$	0	0
$286$	−7.12311	−0.421198
$287$	−4.68466	−0.276527
$288$	0	0
$289$	13.9309	0.819463
$290$	0	0
$291$	0	0
$292$	−0.107951	−0.00631733
$293$	−30.9309	−1.80700	−0.903500	−	0.428587i	$-0.859011\pi$
$293$	−30.9309	−1.80700	−0.903500	+	0.428587i	$0.859011\pi$
$294$	0	0
$295$	0	0
$296$	−23.9157	−1.39007
$297$	0	0
$298$	18.4384	1.06811
$299$	11.1231	0.643266
$300$	0	0
$301$	−7.68466	−0.442936
$302$	−6.73863	−0.387765
$303$	0	0
$304$	−14.0540	−0.806051
$305$	0	0
$306$	0	0
$307$	5.93087	0.338493	0.169246	−	0.985574i	$-0.445867\pi$
$307$	5.93087	0.338493	0.169246	+	0.985574i	$0.445867\pi$
$308$	0.438447	0.0249828
$309$	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
$313$	−14.7538	−0.833933	−0.416967	−	0.908922i	$-0.636907\pi$
$313$	−14.7538	−0.833933	−0.416967	+	0.908922i	$0.636907\pi$
$314$	−15.1231	−0.853446
$315$	0	0
$316$	−1.45360	−0.0817715
$317$	−15.1231	−0.849398	−0.424699	−	0.905335i	$-0.639620\pi$
$317$	−15.1231	−0.849398	−0.424699	+	0.905335i	$0.639620\pi$
$318$	0	0
$319$	3.12311	0.174860
$320$	0	0
$321$	0	0
$322$	−3.80776	−0.212198
$323$	−16.6847	−0.928359
$324$	0	0
$325$	0	0
$326$	−27.6155	−1.52948
$327$	0	0
$328$	11.4233	0.630746
$329$	−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
$337$	1.68466	0.0917692	0.0458846	−	0.998947i	$-0.485389\pi$
$337$	1.68466	0.0917692	0.0458846	+	0.998947i	$0.485389\pi$
$338$	12.1922	0.663170
$339$	0	0
$340$	0	0
$341$	−7.68466	−0.416148
$342$	0	0
$343$	−13.0000	−0.701934
$344$	18.7386	1.01032
$345$	0	0
$346$	−27.4233	−1.47429
$347$	2.63068	0.141222	0.0706112	−	0.997504i	$-0.477505\pi$
$347$	2.63068	0.141222	0.0706112	+	0.997504i	$0.477505\pi$
$348$	0	0
$349$	31.3693	1.67916	0.839581	−	0.543235i	$-0.182801\pi$
$349$	31.3693	1.67916	0.839581	+	0.543235i	$0.182801\pi$
$350$	0	0
$351$	0	0
$352$	−2.43845	−0.129970
$353$	29.8617	1.58938	0.794690	−	0.607015i	$-0.207633\pi$
$353$	29.8617	1.58938	0.794690	+	0.607015i	$0.207633\pi$
$354$	0	0
$355$	0	0
$356$	3.89205	0.206278
$357$	0	0
$358$	−16.3002	−0.861492
$359$	−9.75379	−0.514785	−0.257393	−	0.966307i	$-0.582863\pi$
$359$	−9.75379	−0.514785	−0.257393	+	0.966307i	$0.582863\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	31.4233	1.65157
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	−23.0540	−1.20341	−0.601704	−	0.798719i	$-0.705511\pi$
$367$	−23.0540	−1.20341	−0.601704	+	0.798719i	$0.705511\pi$
$368$	11.4233	0.595480
$369$	0	0
$370$	0	0
$371$	−7.12311	−0.369813
$372$	0	0
$373$	6.80776	0.352493	0.176246	−	0.984346i	$-0.443604\pi$
$373$	6.80776	0.352493	0.176246	+	0.984346i	$0.443604\pi$
$374$	−8.68466	−0.449073
$375$	0	0
$376$	23.3153	1.20240
$377$	−14.2462	−0.733717
$378$	0	0
$379$	29.3002	1.50505	0.752525	−	0.658564i	$-0.228836\pi$
$379$	29.3002	1.50505	0.752525	+	0.658564i	$0.228836\pi$
$380$	0	0
$381$	0	0
$382$	−42.0540	−2.15167
$383$	−5.75379	−0.294005	−0.147002	−	0.989136i	$-0.546963\pi$
$383$	−5.75379	−0.294005	−0.147002	+	0.989136i	$0.546963\pi$
$384$	0	0
$385$	0	0
$386$	20.9848	1.06810
$387$	0	0
$388$	−2.68466	−0.136293
$389$	−33.8617	−1.71686	−0.858429	−	0.512932i	$-0.828559\pi$
$389$	−33.8617	−1.71686	−0.858429	+	0.512932i	$0.828559\pi$
$390$	0	0
$391$	13.5616	0.685837
$392$	14.6307	0.738961
$393$	0	0
$394$	17.0691	0.859930
$395$	0	0
$396$	0	0
$397$	32.5616	1.63422	0.817109	−	0.576484i	$-0.195576\pi$
$397$	32.5616	1.63422	0.817109	+	0.576484i	$0.195576\pi$
$398$	10.6307	0.532868
$399$	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
$403$	35.0540	1.74616
$404$	−2.43845	−0.121317
$405$	0	0
$406$	4.87689	0.242036
$407$	9.80776	0.486153
$408$	0	0
$409$	−29.0540	−1.43663	−0.718313	−	0.695720i	$-0.755086\pi$
$409$	−29.0540	−1.43663	−0.718313	+	0.695720i	$0.755086\pi$
$410$	0	0
$411$	0	0
$412$	−0.384472	−0.0189416
$413$	−6.43845	−0.316815
$414$	0	0
$415$	0	0
$416$	11.1231	0.545355
$417$	0	0
$418$	4.68466	0.229134
$419$	−29.5616	−1.44418	−0.722088	−	0.691801i	$-0.756818\pi$
$419$	−29.5616	−1.44418	−0.722088	+	0.691801i	$0.756818\pi$
$420$	0	0
$421$	20.0540	0.977371	0.488685	−	0.872460i	$-0.337477\pi$
$421$	20.0540	0.977371	0.488685	+	0.872460i	$0.337477\pi$
$422$	24.4924	1.19227
$423$	0	0
$424$	17.3693	0.843529
$425$	0	0
$426$	0	0
$427$	−3.43845	−0.166398
$428$	0.384472	0.0185841
$429$	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
$433$	−11.9309	−0.573361	−0.286681	−	0.958026i	$-0.592552\pi$
$433$	−11.9309	−0.573361	−0.286681	+	0.958026i	$0.592552\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	5.61553	0.268935
$437$	−7.31534	−0.349940
$438$	0	0
$439$	−37.7386	−1.80117	−0.900583	−	0.434683i	$-0.856860\pi$
$439$	−37.7386	−1.80117	−0.900583	+	0.434683i	$0.856860\pi$
$440$	0	0
$441$	0	0
$442$	39.6155	1.88432
$443$	28.3002	1.34458	0.672291	−	0.740287i	$-0.265310\pi$
$443$	28.3002	1.34458	0.672291	+	0.740287i	$0.265310\pi$
$444$	0	0
$445$	0	0
$446$	1.86174	0.0881559
$447$	0	0
$448$	5.56155	0.262759
$449$	−4.49242	−0.212011	−0.106005	−	0.994366i	$-0.533806\pi$
$449$	−4.49242	−0.212011	−0.106005	+	0.994366i	$0.533806\pi$
$450$	0	0
$451$	−4.68466	−0.220592
$452$	0.384472	0.0180840
$453$	0	0
$454$	19.5076	0.915536
$455$	0	0
$456$	0	0
$457$	−11.3693	−0.531834	−0.265917	−	0.963996i	$-0.585675\pi$
$457$	−11.3693	−0.531834	−0.265917	+	0.963996i	$0.585675\pi$
$458$	0.192236	0.00898260
$459$	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
$463$	−18.2462	−0.847973	−0.423987	−	0.905668i	$-0.639370\pi$
$463$	−18.2462	−0.847973	−0.423987	+	0.905668i	$0.639370\pi$
$464$	−14.6307	−0.679212
$465$	0	0
$466$	−6.54640	−0.303256
$467$	14.2462	0.659236	0.329618	−	0.944114i	$-0.393080\pi$
$467$	14.2462	0.659236	0.329618	+	0.944114i	$0.393080\pi$
$468$	0	0
$469$	5.68466	0.262493
$470$	0	0
$471$	0	0
$472$	15.6998	0.722642
$473$	−7.68466	−0.353341
$474$	0	0
$475$	0	0
$476$	−2.43845	−0.111766
$477$	0	0
$478$	−25.7538	−1.17795
$479$	21.3693	0.976389	0.488195	−	0.872735i	$-0.337656\pi$
$479$	21.3693	0.976389	0.488195	+	0.872735i	$0.337656\pi$
$480$	0	0
$481$	−44.7386	−2.03991
$482$	36.0000	1.63976
$483$	0	0
$484$	0.438447	0.0199294
$485$	0	0
$486$	0	0
$487$	25.0540	1.13530	0.567652	−	0.823269i	$-0.307852\pi$
$487$	25.0540	1.13530	0.567652	+	0.823269i	$0.307852\pi$
$488$	8.38447	0.379547
$489$	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
$493$	−17.3693	−0.782275
$494$	−21.3693	−0.961451
$495$	0	0
$496$	36.0000	1.61645
$497$	11.8078	0.529651
$498$	0	0
$499$	−16.8078	−0.752419	−0.376209	−	0.926535i	$-0.622773\pi$
$499$	−16.8078	−0.752419	−0.376209	+	0.926535i	$0.622773\pi$
$500$	0	0
$501$	0	0
$502$	18.7386	0.836346
$503$	−16.4924	−0.735361	−0.367680	−	0.929952i	$-0.619848\pi$
$503$	−16.4924	−0.735361	−0.367680	+	0.929952i	$0.619848\pi$
$504$	0	0
$505$	0	0
$506$	−3.80776	−0.169276
$507$	0	0
$508$	6.93087	0.307508
$509$	8.87689	0.393461	0.196731	−	0.980458i	$-0.436968\pi$
$509$	8.87689	0.393461	0.196731	+	0.980458i	$0.436968\pi$
$510$	0	0
$511$	−0.246211	−0.0108917
$512$	−11.4233	−0.504843
$513$	0	0
$514$	−2.73863	−0.120796
$515$	0	0
$516$	0	0
$517$	−9.56155	−0.420517
$518$	15.3153	0.672917
$519$	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
$523$	−7.24621	−0.316855	−0.158427	−	0.987371i	$-0.550642\pi$
$523$	−7.24621	−0.316855	−0.158427	+	0.987371i	$0.550642\pi$
$524$	−7.23106	−0.315890
$525$	0	0
$526$	−33.3693	−1.45497
$527$	42.7386	1.86172
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	0	0
$532$	1.31534	0.0570273
$533$	21.3693	0.925608
$534$	0	0
$535$	0	0
$536$	−13.8617	−0.598736
$537$	0	0
$538$	7.01515	0.302445
$539$	−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	$0.639077\pi$
$542$	26.8229	1.15214
$543$	0	0
$544$	13.5616	0.581447
$545$	0	0
$546$	0	0
$547$	−8.68466	−0.371329	−0.185665	−	0.982613i	$-0.559444\pi$
$547$	−8.68466	−0.371329	−0.185665	+	0.982613i	$0.559444\pi$
$548$	−9.36932	−0.400237
$549$	0	0
$550$	0	0
$551$	9.36932	0.399146
$552$	0	0
$553$	−3.31534	−0.140983
$554$	34.6307	1.47132
$555$	0	0
$556$	3.50758	0.148754
$557$	7.12311	0.301816	0.150908	−	0.988548i	$-0.451780\pi$
$557$	7.12311	0.301816	0.150908	+	0.988548i	$0.451780\pi$
$558$	0	0
$559$	35.0540	1.48263
$560$	0	0
$561$	0	0
$562$	37.1771	1.56822
$563$	19.1231	0.805943	0.402971	−	0.915213i	$-0.367977\pi$
$563$	19.1231	0.805943	0.402971	+	0.915213i	$0.367977\pi$
$564$	0	0
$565$	0	0
$566$	37.2850	1.56721
$567$	0	0
$568$	−28.7926	−1.20811
$569$	−9.06913	−0.380198	−0.190099	−	0.981765i	$-0.560881\pi$
$569$	−9.06913	−0.380198	−0.190099	+	0.981765i	$0.560881\pi$
$570$	0	0
$571$	0.946025	0.0395899	0.0197950	−	0.999804i	$-0.493699\pi$
$571$	0.946025	0.0395899	0.0197950	+	0.999804i	$0.493699\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	−7.31534	−0.305336
$575$	0	0
$576$	0	0
$577$	−3.63068	−0.151147	−0.0755737	−	0.997140i	$-0.524079\pi$
$577$	−3.63068	−0.151147	−0.0755737	+	0.997140i	$0.524079\pi$
$578$	21.7538	0.904838
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.12311	−0.295009
$584$	0.600373	0.0248436
$585$	0	0
$586$	−48.3002	−1.99526
$587$	47.8078	1.97324	0.986619	−	0.163044i	$-0.0521313\pi$
$587$	47.8078	1.97324	0.986619	+	0.163044i	$0.0521313\pi$
$588$	0	0
$589$	−23.0540	−0.949923
$590$	0	0
$591$	0	0
$592$	−45.9460	−1.88837
$593$	−40.1080	−1.64704	−0.823518	−	0.567290i	$-0.807992\pi$
$593$	−40.1080	−1.64704	−0.823518	+	0.567290i	$0.807992\pi$
$594$	0	0
$595$	0	0
$596$	5.17708	0.212061
$597$	0	0
$598$	17.3693	0.710284
$599$	21.1771	0.865272	0.432636	−	0.901569i	$-0.357584\pi$
$599$	21.1771	0.865272	0.432636	+	0.901569i	$0.357584\pi$
$600$	0	0
$601$	−24.4233	−0.996247	−0.498123	−	0.867106i	$-0.665977\pi$
$601$	−24.4233	−0.996247	−0.498123	+	0.867106i	$0.665977\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	−1.89205	−0.0769864
$605$	0	0
$606$	0	0
$607$	−10.2462	−0.415881	−0.207940	−	0.978141i	$-0.566676\pi$
$607$	−10.2462	−0.415881	−0.207940	+	0.978141i	$0.566676\pi$
$608$	−7.31534	−0.296676
$609$	0	0
$610$	0	0
$611$	43.6155	1.76450
$612$	0	0
$613$	−46.1080	−1.86228	−0.931141	−	0.364659i	$-0.881186\pi$
$613$	−46.1080	−1.86228	−0.931141	+	0.364659i	$0.881186\pi$
$614$	9.26137	0.373758
$615$	0	0
$616$	−2.43845	−0.0982478
$617$	−38.2462	−1.53973	−0.769867	−	0.638204i	$-0.779678\pi$
$617$	−38.2462	−1.53973	−0.769867	+	0.638204i	$0.779678\pi$
$618$	0	0
$619$	−38.1771	−1.53447	−0.767233	−	0.641368i	$-0.778367\pi$
$619$	−38.1771	−1.53447	−0.767233	+	0.641368i	$0.778367\pi$
$620$	0	0
$621$	0	0
$622$	47.2311	1.89379
$623$	8.87689	0.355645
$624$	0	0
$625$	0	0
$626$	−23.0388	−0.920816
$627$	0	0
$628$	−4.24621	−0.169442
$629$	−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	$0.565825\pi$
$632$	8.08429	0.321576
$633$	0	0
$634$	−23.6155	−0.937892
$635$	0	0
$636$	0	0
$637$	27.3693	1.08441
$638$	4.87689	0.193078
$639$	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
$643$	−9.36932	−0.369490	−0.184745	−	0.982787i	$-0.559146\pi$
$643$	−9.36932	−0.369490	−0.184745	+	0.982787i	$0.559146\pi$
$644$	−1.06913	−0.0421296
$645$	0	0
$646$	−26.0540	−1.02508
$647$	20.1922	0.793839	0.396919	−	0.917853i	$-0.370079\pi$
$647$	20.1922	0.793839	0.396919	+	0.917853i	$0.370079\pi$
$648$	0	0
$649$	−6.43845	−0.252731
$650$	0	0
$651$	0	0
$652$	−7.75379	−0.303662
$653$	−40.4924	−1.58459	−0.792295	−	0.610138i	$-0.791114\pi$
$653$	−40.4924	−1.58459	−0.792295	+	0.610138i	$0.791114\pi$
$654$	0	0
$655$	0	0
$656$	21.9460	0.856848
$657$	0	0
$658$	−14.9309	−0.582066
$659$	−33.3693	−1.29988	−0.649942	−	0.759984i	$-0.725207\pi$
$659$	−33.3693	−1.29988	−0.649942	+	0.759984i	$0.725207\pi$
$660$	0	0
$661$	−13.8078	−0.537060	−0.268530	−	0.963271i	$-0.586538\pi$
$661$	−13.8078	−0.537060	−0.268530	+	0.963271i	$0.586538\pi$
$662$	9.75379	0.379092
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.61553	−0.294874
$668$	−4.10795	−0.158941
$669$	0	0
$670$	0	0
$671$	−3.43845	−0.132740
$672$	0	0
$673$	35.3693	1.36339	0.681693	−	0.731638i	$-0.261244\pi$
$673$	35.3693	1.36339	0.681693	+	0.731638i	$0.261244\pi$
$674$	2.63068	0.101330
$675$	0	0
$676$	3.42329	0.131665
$677$	−20.8769	−0.802364	−0.401182	−	0.915998i	$-0.631401\pi$
$677$	−20.8769	−0.802364	−0.401182	+	0.915998i	$0.631401\pi$
$678$	0	0
$679$	−6.12311	−0.234983
$680$	0	0
$681$	0	0
$682$	−12.0000	−0.459504
$683$	0.192236	0.00735570	0.00367785	−	0.999993i	$-0.498829\pi$
$683$	0.192236	0.00735570	0.00367785	+	0.999993i	$0.498829\pi$
$684$	0	0
$685$	0	0
$686$	−20.3002	−0.775065
$687$	0	0
$688$	36.0000	1.37249
$689$	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	$0.627450\pi$
$692$	−7.69981	−0.292703
$693$	0	0
$694$	4.10795	0.155936
$695$	0	0
$696$	0	0
$697$	26.0540	0.986865
$698$	48.9848	1.85410
$699$	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
$703$	29.4233	1.10972
$704$	5.56155	0.209609
$705$	0	0
$706$	46.6307	1.75497
$707$	−5.56155	−0.209164
$708$	0	0
$709$	−2.12311	−0.0797349	−0.0398675	−	0.999205i	$-0.512694\pi$
$709$	−2.12311	−0.0797349	−0.0398675	+	0.999205i	$0.512694\pi$
$710$	0	0
$711$	0	0
$712$	−21.6458	−0.811212
$713$	18.7386	0.701767
$714$	0	0
$715$	0	0
$716$	−4.57671	−0.171040
$717$	0	0
$718$	−15.2311	−0.568418
$719$	−32.9848	−1.23013	−0.615064	−	0.788478i	$-0.710870\pi$
$719$	−32.9848	−1.23013	−0.615064	+	0.788478i	$0.710870\pi$
$720$	0	0
$721$	−0.876894	−0.0326573
$722$	−15.6155	−0.581150
$723$	0	0
$724$	8.82292	0.327901
$725$	0	0
$726$	0	0
$727$	7.05398	0.261617	0.130809	−	0.991408i	$-0.458243\pi$
$727$	7.05398	0.261617	0.130809	+	0.991408i	$0.458243\pi$
$728$	11.1231	0.412250
$729$	0	0
$730$	0	0
$731$	42.7386	1.58075
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−36.0000	−1.32878
$735$	0	0
$736$	5.94602	0.219173
$737$	5.68466	0.209397
$738$	0	0
$739$	0.684658	0.0251856	0.0125928	−	0.999921i	$-0.495991\pi$
$739$	0.684658	0.0251856	0.0125928	+	0.999921i	$0.495991\pi$
$740$	0	0
$741$	0	0
$742$	−11.1231	−0.408342
$743$	9.86174	0.361792	0.180896	−	0.983502i	$-0.442100\pi$
$743$	9.86174	0.361792	0.180896	+	0.983502i	$0.442100\pi$
$744$	0	0
$745$	0	0
$746$	10.6307	0.389217
$747$	0	0
$748$	−2.43845	−0.0891585
$749$	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	$0.636174\pi$
$752$	44.7926	1.63342
$753$	0	0
$754$	−22.2462	−0.810159
$755$	0	0
$756$	0	0
$757$	−47.3002	−1.71915	−0.859577	−	0.511006i	$-0.829273\pi$
$757$	−47.3002	−1.71915	−0.859577	+	0.511006i	$0.829273\pi$
$758$	45.7538	1.66185
$759$	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
$763$	12.8078	0.463672
$764$	−11.8078	−0.427190
$765$	0	0
$766$	−8.98485	−0.324636
$767$	29.3693	1.06046
$768$	0	0
$769$	−31.0540	−1.11983	−0.559917	−	0.828548i	$-0.689167\pi$
$769$	−31.0540	−1.11983	−0.559917	+	0.828548i	$0.689167\pi$
$770$	0	0
$771$	0	0
$772$	5.89205	0.212059
$773$	−15.5076	−0.557769	−0.278884	−	0.960325i	$-0.589965\pi$
$773$	−15.5076	−0.557769	−0.278884	+	0.960325i	$0.589965\pi$
$774$	0	0
$775$	0	0
$776$	14.9309	0.535987
$777$	0	0
$778$	−52.8769	−1.89573
$779$	−14.0540	−0.503536
$780$	0	0
$781$	11.8078	0.422515
$782$	21.1771	0.757291
$783$	0	0
$784$	28.1080	1.00386
$785$	0	0
$786$	0	0
$787$	53.7386	1.91558	0.957788	−	0.287476i	$-0.0928163\pi$
$787$	53.7386	1.91558	0.957788	+	0.287476i	$0.0928163\pi$
$788$	4.79261	0.170730
$789$	0	0
$790$	0	0
$791$	0.876894	0.0311788
$792$	0	0
$793$	15.6847	0.556979
$794$	50.8466	1.80448
$795$	0	0
$796$	2.98485	0.105795
$797$	−51.1231	−1.81087	−0.905437	−	0.424481i	$-0.860456\pi$
$797$	−51.1231	−1.81087	−0.905437	+	0.424481i	$0.860456\pi$
$798$	0	0
$799$	53.1771	1.88127
$800$	0	0
$801$	0	0
$802$	55.6155	1.96385
$803$	−0.246211	−0.00868861
$804$	0	0
$805$	0	0
$806$	54.7386	1.92809
$807$	0	0
$808$	13.5616	0.477094
$809$	−7.31534	−0.257194	−0.128597	−	0.991697i	$-0.541047\pi$
$809$	−7.31534	−0.257194	−0.128597	+	0.991697i	$0.541047\pi$
$810$	0	0
$811$	−1.00000	−0.0351147	−0.0175574	−	0.999846i	$-0.505589\pi$
$811$	−1.00000	−0.0351147	−0.0175574	+	0.999846i	$0.505589\pi$
$812$	1.36932	0.0480536
$813$	0	0
$814$	15.3153	0.536802
$815$	0	0
$816$	0	0
$817$	−23.0540	−0.806557
$818$	−45.3693	−1.58630
$819$	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
$823$	−37.0540	−1.29162	−0.645810	−	0.763498i	$-0.723480\pi$
$823$	−37.0540	−1.29162	−0.645810	+	0.763498i	$0.723480\pi$
$824$	2.13826	0.0744898
$825$	0	0
$826$	−10.0540	−0.349823
$827$	−26.2462	−0.912670	−0.456335	−	0.889808i	$-0.650838\pi$
$827$	−26.2462	−0.912670	−0.456335	+	0.889808i	$0.650838\pi$
$828$	0	0
$829$	−0.738634	−0.0256538	−0.0128269	−	0.999918i	$-0.504083\pi$
$829$	−0.738634	−0.0256538	−0.0128269	+	0.999918i	$0.504083\pi$
$830$	0	0
$831$	0	0
$832$	−25.3693	−0.879523
$833$	33.3693	1.15618
$834$	0	0
$835$	0	0
$836$	1.31534	0.0454920
$837$	0	0
$838$	−46.1619	−1.59464
$839$	12.4924	0.431286	0.215643	−	0.976472i	$-0.430815\pi$
$839$	12.4924	0.431286	0.215643	+	0.976472i	$0.430815\pi$
$840$	0	0
$841$	−19.2462	−0.663662
$842$	31.3153	1.07920
$843$	0	0
$844$	6.87689	0.236712
$845$	0	0
$846$	0	0
$847$	1.00000	0.0343604
$848$	33.3693	1.14591
$849$	0	0
$850$	0	0
$851$	−23.9157	−0.819820
$852$	0	0
$853$	26.4233	0.904716	0.452358	−	0.891836i	$-0.350583\pi$
$853$	26.4233	0.904716	0.452358	+	0.891836i	$0.350583\pi$
$854$	−5.36932	−0.183734
$855$	0	0
$856$	−2.13826	−0.0730842
$857$	−9.56155	−0.326616	−0.163308	−	0.986575i	$-0.552216\pi$
$857$	−9.56155	−0.326616	−0.163308	+	0.986575i	$0.552216\pi$
$858$	0	0
$859$	−34.3542	−1.17215	−0.586074	−	0.810257i	$-0.699327\pi$
$859$	−34.3542	−1.17215	−0.586074	+	0.810257i	$0.699327\pi$
$860$	0	0
$861$	0	0
$862$	59.7235	2.03419
$863$	−14.2462	−0.484947	−0.242473	−	0.970158i	$-0.577959\pi$
$863$	−14.2462	−0.484947	−0.242473	+	0.970158i	$0.577959\pi$
$864$	0	0
$865$	0	0
$866$	−18.6307	−0.633096
$867$	0	0
$868$	−3.36932	−0.114362
$869$	−3.31534	−0.112465
$870$	0	0
$871$	−25.9309	−0.878634
$872$	−31.2311	−1.05762
$873$	0	0
$874$	−11.4233	−0.386399
$875$	0	0
$876$	0	0
$877$	−23.0540	−0.778477	−0.389239	−	0.921137i	$-0.627262\pi$
$877$	−23.0540	−0.778477	−0.389239	+	0.921137i	$0.627262\pi$
$878$	−58.9309	−1.98882
$879$	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
$883$	−16.0691	−0.540769	−0.270385	−	0.962752i	$-0.587151\pi$
$883$	−16.0691	−0.540769	−0.270385	+	0.962752i	$0.587151\pi$
$884$	11.1231	0.374111
$885$	0	0
$886$	44.1922	1.48467
$887$	29.8617	1.00266	0.501330	−	0.865256i	$-0.332844\pi$
$887$	29.8617	1.00266	0.501330	+	0.865256i	$0.332844\pi$
$888$	0	0
$889$	15.8078	0.530175
$890$	0	0
$891$	0	0
$892$	0.522732	0.0175024
$893$	−28.6847	−0.959895
$894$	0	0
$895$	0	0
$896$	13.5616	0.453060
$897$	0	0
$898$	−7.01515	−0.234099
$899$	−24.0000	−0.800445
$900$	0	0
$901$	39.6155	1.31978
$902$	−7.31534	−0.243574
$903$	0	0
$904$	−2.13826	−0.0711175
$905$	0	0
$906$	0	0
$907$	−28.1080	−0.933309	−0.466655	−	0.884440i	$-0.654541\pi$
$907$	−28.1080	−0.933309	−0.466655	+	0.884440i	$0.654541\pi$
$908$	5.47727	0.181770
$909$	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
$917$	−16.4924	−0.544628
$918$	0	0
$919$	−33.1080	−1.09213	−0.546065	−	0.837743i	$-0.683875\pi$
$919$	−33.1080	−1.09213	−0.546065	+	0.837743i	$0.683875\pi$
$920$	0	0
$921$	0	0
$922$	−65.3693	−2.15282
$923$	−53.8617	−1.77288
$924$	0	0
$925$	0	0
$926$	−28.4924	−0.936319
$927$	0	0
$928$	−7.61553	−0.249992
$929$	59.2311	1.94331	0.971654	−	0.236408i	$-0.0759701\pi$
$929$	59.2311	1.94331	0.971654	+	0.236408i	$0.0759701\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−1.83807	−0.0602081
$933$	0	0
$934$	22.2462	0.727918
$935$	0	0
$936$	0	0
$937$	22.3153	0.729010	0.364505	−	0.931201i	$-0.381238\pi$
$937$	22.3153	0.729010	0.364505	+	0.931201i	$0.381238\pi$
$938$	8.87689	0.289841
$939$	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
$943$	11.4233	0.371994
$944$	30.1619	0.981687
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−13.1771	−0.428198	−0.214099	−	0.976812i	$-0.568681\pi$
$947$	−13.1771	−0.428198	−0.214099	+	0.976812i	$0.568681\pi$
$948$	0	0
$949$	1.12311	0.0364576
$950$	0	0
$951$	0	0
$952$	13.5616	0.439532
$953$	−15.8078	−0.512064	−0.256032	−	0.966668i	$-0.582415\pi$
$953$	−15.8078	−0.512064	−0.256032	+	0.966668i	$0.582415\pi$
$954$	0	0
$955$	0	0
$956$	−7.23106	−0.233869
$957$	0	0
$958$	33.3693	1.07811
$959$	−21.3693	−0.690051
$960$	0	0
$961$	28.0540	0.904967
$962$	−69.8617	−2.25243
$963$	0	0
$964$	10.1080	0.325555
$965$	0	0
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	−2.43845	−0.0783747
$969$	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
$973$	8.00000	0.256468
$974$	39.1231	1.25359
$975$	0	0
$976$	16.1080	0.515603
$977$	−15.6155	−0.499585	−0.249793	−	0.968299i	$-0.580362\pi$
$977$	−15.6155	−0.499585	−0.249793	+	0.968299i	$0.580362\pi$
$978$	0	0
$979$	8.87689	0.283707
$980$	0	0
$981$	0	0
$982$	25.7538	0.821836
$983$	6.93087	0.221060	0.110530	−	0.993873i	$-0.464745\pi$
$983$	6.93087	0.221060	0.110530	+	0.993873i	$0.464745\pi$
$984$	0	0
$985$	0	0
$986$	−27.1231	−0.863776
$987$	0	0
$988$	−6.00000	−0.190885
$989$	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	$0.210525\pi$
$992$	18.7386	0.594952
$993$	0	0
$994$	18.4384	0.584832
$995$	0	0
$996$	0	0
$997$	−28.7386	−0.910162	−0.455081	−	0.890450i	$-0.650390\pi$
$997$	−28.7386	−0.910162	−0.455081	+	0.890450i	$0.650390\pi$
$998$	−26.2462	−0.830809
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{7} -2.43845 q^{8} +O(q^{10})\)	\(q+1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{7} -2.43845 q^{8} +1.00000 q^{11} -4.56155 q^{13} +1.56155 q^{14} -4.68466 q^{16} -5.56155 q^{17} +3.00000 q^{19} +1.56155 q^{22} -2.43845 q^{23} -7.12311 q^{26} +0.438447 q^{28} +3.12311 q^{29} -7.68466 q^{31} -2.43845 q^{32} -8.68466 q^{34} +9.80776 q^{37} +4.68466 q^{38} -4.68466 q^{41} -7.68466 q^{43} +0.438447 q^{44} -3.80776 q^{46} -9.56155 q^{47} -6.00000 q^{49} -2.00000 q^{52} -7.12311 q^{53} -2.43845 q^{56} +4.87689 q^{58} -6.43845 q^{59} -3.43845 q^{61} -12.0000 q^{62} +5.56155 q^{64} +5.68466 q^{67} -2.43845 q^{68} +11.8078 q^{71} -0.246211 q^{73} +15.3153 q^{74} +1.31534 q^{76} +1.00000 q^{77} -3.31534 q^{79} -7.31534 q^{82} -12.0000 q^{86} -2.43845 q^{88} +8.87689 q^{89} -4.56155 q^{91} -1.06913 q^{92} -14.9309 q^{94} -6.12311 q^{97} -9.36932 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8}+O(q^{10}) \) 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8	\( 2 q - q^{2} + 5 q^{4} + 2 q^{7} - 9 q^{8} + 2 q^{11} - 5 q^{13} - q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{19} - q^{22} - 9 q^{23} - 6 q^{26} + 5 q^{28} - 2 q^{29} - 3 q^{31} - 9 q^{32} - 5 q^{34} - q^{37} - 3 q^{38} + 3 q^{41} - 3 q^{43} + 5 q^{44} + 13 q^{46} - 15 q^{47} - 12 q^{49} - 4 q^{52} - 6 q^{53} - 9 q^{56} + 18 q^{58} - 17 q^{59} - 11 q^{61} - 24 q^{62} + 7 q^{64} - q^{67} - 9 q^{68} + 3 q^{71} + 16 q^{73} + 43 q^{74} + 15 q^{76} + 2 q^{77} - 19 q^{79} - 27 q^{82} - 24 q^{86} - 9 q^{88} + 26 q^{89} - 5 q^{91} - 31 q^{92} - q^{94} - 4 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 + 2 * q^7 - 9 * q^8 + 2 * q^11 - 5 * q^13 - q^14 + 3 * q^16 - 7 * q^17 + 6 * q^19 - q^22 - 9 * q^23 - 6 * q^26 + 5 * q^28 - 2 * q^29 - 3 * q^31 - 9 * q^32 - 5 * q^34 - q^37 - 3 * q^38 + 3 * q^41 - 3 * q^43 + 5 * q^44 + 13 * q^46 - 15 * q^47 - 12 * q^49 - 4 * q^52 - 6 * q^53 - 9 * q^56 + 18 * q^58 - 17 * q^59 - 11 * q^61 - 24 * q^62 + 7 * q^64 - q^67 - 9 * q^68 + 3 * q^71 + 16 * q^73 + 43 * q^74 + 15 * q^76 + 2 * q^77 - 19 * q^79 - 27 * q^82 - 24 * q^86 - 9 * q^88 + 26 * q^89 - 5 * q^91 - 31 * q^92 - q^94 - 4 * q^97 + 6 * q^98

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