LMFDB - Embedded newform 2475.2.a.m.1.1

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:	$\Q(\zeta_{8})^+$
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:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.41421 q^{2} +3.82843 q^{4} -0.828427 q^{7} -4.41421 q^{8} +O(q^{10})$	$q-2.41421 q^{2} +3.82843 q^{4} -0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{11} +5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -1.17157 q^{17} -6.82843 q^{19} -2.41421 q^{22} -4.00000 q^{23} -13.6569 q^{26} -3.17157 q^{28} +4.82843 q^{29} +1.58579 q^{32} +2.82843 q^{34} -11.6569 q^{37} +16.4853 q^{38} -4.82843 q^{41} +8.82843 q^{43} +3.82843 q^{44} +9.65685 q^{46} -4.00000 q^{47} -6.31371 q^{49} +21.6569 q^{52} +9.31371 q^{53} +3.65685 q^{56} -11.6569 q^{58} +4.00000 q^{59} -11.6569 q^{61} -9.82843 q^{64} +5.65685 q^{67} -4.48528 q^{68} -2.34315 q^{71} -11.3137 q^{73} +28.1421 q^{74} -26.1421 q^{76} -0.828427 q^{77} +8.48528 q^{79} +11.6569 q^{82} -10.0000 q^{83} -21.3137 q^{86} -4.41421 q^{88} -3.65685 q^{89} -4.68629 q^{91} -15.3137 q^{92} +9.65685 q^{94} -11.6569 q^{97} +15.2426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 2 q^{11} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 8 q^{19} - 2 q^{22} - 8 q^{23} - 16 q^{26} - 12 q^{28} + 4 q^{29} + 6 q^{32} - 12 q^{37} + 16 q^{38} - 4 q^{41} + 12 q^{43} + 2 q^{44} + 8 q^{46} - 8 q^{47} + 10 q^{49} + 32 q^{52} - 4 q^{53} - 4 q^{56} - 12 q^{58} + 8 q^{59} - 12 q^{61} - 14 q^{64} + 8 q^{68} - 16 q^{71} + 28 q^{74} - 24 q^{76} + 4 q^{77} + 12 q^{82} - 20 q^{83} - 20 q^{86} - 6 q^{88} + 4 q^{89} - 32 q^{91} - 8 q^{92} + 8 q^{94} - 12 q^{97} + 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 2 * q^11 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 8 * q^19 - 2 * q^22 - 8 * q^23 - 16 * q^26 - 12 * q^28 + 4 * q^29 + 6 * q^32 - 12 * q^37 + 16 * q^38 - 4 * q^41 + 12 * q^43 + 2 * q^44 + 8 * q^46 - 8 * q^47 + 10 * q^49 + 32 * q^52 - 4 * q^53 - 4 * q^56 - 12 * q^58 + 8 * q^59 - 12 * q^61 - 14 * q^64 + 8 * q^68 - 16 * q^71 + 28 * q^74 - 24 * q^76 + 4 * q^77 + 12 * q^82 - 20 * q^83 - 20 * q^86 - 6 * q^88 + 4 * q^89 - 32 * q^91 - 8 * q^92 + 8 * q^94 - 12 * q^97 + 22 * 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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	−4.41421	−1.56066
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	−1.17157	−0.284148	−0.142074	−	0.989856i	$-0.545377\pi$
$17$	−1.17157	−0.284148	−0.142074	+	0.989856i	$0.545377\pi$
$18$	0	0
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	0	0
$21$	0	0
$22$	−2.41421	−0.514712
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	−13.6569	−2.67833
$27$	0	0
$28$	−3.17157	−0.599371
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	2.82843	0.485071
$35$	0	0
$36$	0	0
$37$	−11.6569	−1.91638	−0.958188	−	0.286141i	$-0.907627\pi$
$37$	−11.6569	−1.91638	−0.958188	+	0.286141i	$0.907627\pi$
$38$	16.4853	2.67427
$39$	0	0
$40$	0	0
$41$	−4.82843	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$42$	0	0
$43$	8.82843	1.34632	0.673161	−	0.739496i	$-0.264936\pi$
$43$	8.82843	1.34632	0.673161	+	0.739496i	$0.264936\pi$
$44$	3.82843	0.577157
$45$	0	0
$46$	9.65685	1.42383
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	0	0
$52$	21.6569	3.00327
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	0	0
$56$	3.65685	0.488668
$57$	0	0
$58$	−11.6569	−1.53062
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−11.6569	−1.49251	−0.746254	−	0.665662i	$-0.768149\pi$
$61$	−11.6569	−1.49251	−0.746254	+	0.665662i	$0.768149\pi$
$62$	0	0
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	−4.48528	−0.543920
$69$	0	0
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−11.3137	−1.32417	−0.662085	−	0.749429i	$-0.730328\pi$
$73$	−11.3137	−1.32417	−0.662085	+	0.749429i	$0.730328\pi$
$74$	28.1421	3.27146
$75$	0	0
$76$	−26.1421	−2.99871
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	0	0
$82$	11.6569	1.28728
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	0	0
$86$	−21.3137	−2.29832
$87$	0	0
$88$	−4.41421	−0.470557
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−4.68629	−0.491257
$92$	−15.3137	−1.59656
$93$	0	0
$94$	9.65685	0.996028
$95$	0	0
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	15.2426	1.53974
$99$	0	0
$100$	0	0
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	0	0
$103$	3.31371	0.326509	0.163255	−	0.986584i	$-0.447801\pi$
$103$	3.31371	0.326509	0.163255	+	0.986584i	$0.447801\pi$
$104$	−24.9706	−2.44857
$105$	0	0
$106$	−22.4853	−2.18396
$107$	17.3137	1.67378	0.836890	−	0.547372i	$-0.184372\pi$
$107$	17.3137	1.67378	0.836890	+	0.547372i	$0.184372\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	0	0
$112$	−2.48528	−0.234837
$113$	−18.9706	−1.78460	−0.892300	−	0.451442i	$-0.850910\pi$
$113$	−18.9706	−1.78460	−0.892300	+	0.451442i	$0.850910\pi$
$114$	0	0
$115$	0	0
$116$	18.4853	1.71632
$117$	0	0
$118$	−9.65685	−0.888985
$119$	0.970563	0.0889713
$120$	0	0
$121$	1.00000	0.0909091
$122$	28.1421	2.54787
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.4853	1.28536	0.642680	−	0.766134i	$-0.277822\pi$
$127$	14.4853	1.28536	0.642680	+	0.766134i	$0.277822\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	0	0
$131$	−3.31371	−0.289520	−0.144760	−	0.989467i	$-0.546241\pi$
$131$	−3.31371	−0.289520	−0.144760	+	0.989467i	$0.546241\pi$
$132$	0	0
$133$	5.65685	0.490511
$134$	−13.6569	−1.17977
$135$	0	0
$136$	5.17157	0.443459
$137$	−13.3137	−1.13747	−0.568733	−	0.822522i	$-0.692566\pi$
$137$	−13.3137	−1.13747	−0.568733	+	0.822522i	$0.692566\pi$
$138$	0	0
$139$	0.485281	0.0411610	0.0205805	−	0.999788i	$-0.493449\pi$
$139$	0.485281	0.0411610	0.0205805	+	0.999788i	$0.493449\pi$
$140$	0	0
$141$	0	0
$142$	5.65685	0.474713
$143$	5.65685	0.473050
$144$	0	0
$145$	0	0
$146$	27.3137	2.26050
$147$	0	0
$148$	−44.6274	−3.66835
$149$	−1.51472	−0.124091	−0.0620453	−	0.998073i	$-0.519762\pi$
$149$	−1.51472	−0.124091	−0.0620453	+	0.998073i	$0.519762\pi$
$150$	0	0
$151$	16.4853	1.34155	0.670777	−	0.741659i	$-0.265961\pi$
$151$	16.4853	1.34155	0.670777	+	0.741659i	$0.265961\pi$
$152$	30.1421	2.44485
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−20.4853	−1.62972
$159$	0	0
$160$	0	0
$161$	3.31371	0.261157
$162$	0	0
$163$	7.31371	0.572854	0.286427	−	0.958102i	$-0.407532\pi$
$163$	7.31371	0.572854	0.286427	+	0.958102i	$0.407532\pi$
$164$	−18.4853	−1.44346
$165$	0	0
$166$	24.1421	1.87379
$167$	−13.3137	−1.03025	−0.515123	−	0.857116i	$-0.672254\pi$
$167$	−13.3137	−1.03025	−0.515123	+	0.857116i	$0.672254\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	0	0
$172$	33.7990	2.57715
$173$	2.82843	0.215041	0.107521	−	0.994203i	$-0.465709\pi$
$173$	2.82843	0.215041	0.107521	+	0.994203i	$0.465709\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	8.82843	0.661719
$179$	17.6569	1.31974	0.659868	−	0.751382i	$-0.270612\pi$
$179$	17.6569	1.31974	0.659868	+	0.751382i	$0.270612\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	11.3137	0.838628
$183$	0	0
$184$	17.6569	1.30168
$185$	0	0
$186$	0	0
$187$	−1.17157	−0.0856739
$188$	−15.3137	−1.11687
$189$	0	0
$190$	0	0
$191$	−5.65685	−0.409316	−0.204658	−	0.978834i	$-0.565608\pi$
$191$	−5.65685	−0.409316	−0.204658	+	0.978834i	$0.565608\pi$
$192$	0	0
$193$	−13.6569	−0.983042	−0.491521	−	0.870866i	$-0.663559\pi$
$193$	−13.6569	−0.983042	−0.491521	+	0.870866i	$0.663559\pi$
$194$	28.1421	2.02049
$195$	0	0
$196$	−24.1716	−1.72654
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	0	0
$202$	−2.00000	−0.140720
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	16.9706	1.17670
$209$	−6.82843	−0.472332
$210$	0	0
$211$	1.17157	0.0806544	0.0403272	−	0.999187i	$-0.487160\pi$
$211$	1.17157	0.0806544	0.0403272	+	0.999187i	$0.487160\pi$
$212$	35.6569	2.44892
$213$	0	0
$214$	−41.7990	−2.85732
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−41.7990	−2.83098
$219$	0	0
$220$	0	0
$221$	−6.62742	−0.445808
$222$	0	0
$223$	6.34315	0.424768	0.212384	−	0.977186i	$-0.431877\pi$
$223$	6.34315	0.424768	0.212384	+	0.977186i	$0.431877\pi$
$224$	−1.31371	−0.0877758
$225$	0	0
$226$	45.7990	3.04650
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	−21.3137	−1.39931
$233$	−18.8284	−1.23349	−0.616746	−	0.787163i	$-0.711549\pi$
$233$	−18.8284	−1.23349	−0.616746	+	0.787163i	$0.711549\pi$
$234$	0	0
$235$	0	0
$236$	15.3137	0.996838
$237$	0	0
$238$	−2.34315	−0.151884
$239$	−17.6569	−1.14213	−0.571063	−	0.820906i	$-0.693469\pi$
$239$	−17.6569	−1.14213	−0.571063	+	0.820906i	$0.693469\pi$
$240$	0	0
$241$	−12.3431	−0.795092	−0.397546	−	0.917582i	$-0.630138\pi$
$241$	−12.3431	−0.795092	−0.397546	+	0.917582i	$0.630138\pi$
$242$	−2.41421	−0.155192
$243$	0	0
$244$	−44.6274	−2.85698
$245$	0	0
$246$	0	0
$247$	−38.6274	−2.45780
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.9706	−1.32365	−0.661825	−	0.749658i	$-0.730218\pi$
$251$	−20.9706	−1.32365	−0.661825	+	0.749658i	$0.730218\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−34.9706	−2.19425
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−16.3431	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$257$	−16.3431	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$258$	0	0
$259$	9.65685	0.600048
$260$	0	0
$261$	0	0
$262$	8.00000	0.494242
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	−13.6569	−0.837355
$267$	0	0
$268$	21.6569	1.32290
$269$	−20.6274	−1.25768	−0.628838	−	0.777536i	$-0.716469\pi$
$269$	−20.6274	−1.25768	−0.628838	+	0.777536i	$0.716469\pi$
$270$	0	0
$271$	−11.7990	−0.716738	−0.358369	−	0.933580i	$-0.616667\pi$
$271$	−11.7990	−0.716738	−0.358369	+	0.933580i	$0.616667\pi$
$272$	−3.51472	−0.213111
$273$	0	0
$274$	32.1421	1.94178
$275$	0	0
$276$	0	0
$277$	2.34315	0.140786	0.0703930	−	0.997519i	$-0.477575\pi$
$277$	2.34315	0.140786	0.0703930	+	0.997519i	$0.477575\pi$
$278$	−1.17157	−0.0702663
$279$	0	0
$280$	0	0
$281$	11.1716	0.666440	0.333220	−	0.942849i	$-0.391865\pi$
$281$	11.1716	0.666440	0.333220	+	0.942849i	$0.391865\pi$
$282$	0	0
$283$	8.82843	0.524796	0.262398	−	0.964960i	$-0.415487\pi$
$283$	8.82843	0.524796	0.262398	+	0.964960i	$0.415487\pi$
$284$	−8.97056	−0.532305
$285$	0	0
$286$	−13.6569	−0.807547
$287$	4.00000	0.236113
$288$	0	0
$289$	−15.6274	−0.919260
$290$	0	0
$291$	0	0
$292$	−43.3137	−2.53474
$293$	−6.82843	−0.398921	−0.199460	−	0.979906i	$-0.563919\pi$
$293$	−6.82843	−0.398921	−0.199460	+	0.979906i	$0.563919\pi$
$294$	0	0
$295$	0	0
$296$	51.4558	2.99081
$297$	0	0
$298$	3.65685	0.211836
$299$	−22.6274	−1.30858
$300$	0	0
$301$	−7.31371	−0.421555
$302$	−39.7990	−2.29017
$303$	0	0
$304$	−20.4853	−1.17491
$305$	0	0
$306$	0	0
$307$	3.17157	0.181011	0.0905056	−	0.995896i	$-0.471152\pi$
$307$	3.17157	0.181011	0.0905056	+	0.995896i	$0.471152\pi$
$308$	−3.17157	−0.180717
$309$	0	0
$310$	0	0
$311$	3.31371	0.187903	0.0939516	−	0.995577i	$-0.470050\pi$
$311$	3.31371	0.187903	0.0939516	+	0.995577i	$0.470050\pi$
$312$	0	0
$313$	−15.6569	−0.884978	−0.442489	−	0.896774i	$-0.645904\pi$
$313$	−15.6569	−0.884978	−0.442489	+	0.896774i	$0.645904\pi$
$314$	43.4558	2.45236
$315$	0	0
$316$	32.4853	1.82744
$317$	−26.2843	−1.47627	−0.738136	−	0.674652i	$-0.764294\pi$
$317$	−26.2843	−1.47627	−0.738136	+	0.674652i	$0.764294\pi$
$318$	0	0
$319$	4.82843	0.270340
$320$	0	0
$321$	0	0
$322$	−8.00000	−0.445823
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	−17.6569	−0.977923
$327$	0	0
$328$	21.3137	1.17685
$329$	3.31371	0.182691
$330$	0	0
$331$	6.34315	0.348651	0.174325	−	0.984688i	$-0.444226\pi$
$331$	6.34315	0.348651	0.174325	+	0.984688i	$0.444226\pi$
$332$	−38.2843	−2.10112
$333$	0	0
$334$	32.1421	1.75874
$335$	0	0
$336$	0	0
$337$	−3.31371	−0.180509	−0.0902546	−	0.995919i	$-0.528768\pi$
$337$	−3.31371	−0.180509	−0.0902546	+	0.995919i	$0.528768\pi$
$338$	−45.8701	−2.49500
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	11.0294	0.595534
$344$	−38.9706	−2.10115
$345$	0	0
$346$	−6.82843	−0.367099
$347$	−29.3137	−1.57364	−0.786821	−	0.617181i	$-0.788275\pi$
$347$	−29.3137	−1.57364	−0.786821	+	0.617181i	$0.788275\pi$
$348$	0	0
$349$	10.9706	0.587241	0.293620	−	0.955922i	$-0.405140\pi$
$349$	10.9706	0.587241	0.293620	+	0.955922i	$0.405140\pi$
$350$	0	0
$351$	0	0
$352$	1.58579	0.0845227
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−42.6274	−2.25293
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	33.7990	1.77644
$363$	0	0
$364$	−17.9411	−0.940370
$365$	0	0
$366$	0	0
$367$	−9.65685	−0.504084	−0.252042	−	0.967716i	$-0.581102\pi$
$367$	−9.65685	−0.504084	−0.252042	+	0.967716i	$0.581102\pi$
$368$	−12.0000	−0.625543
$369$	0	0
$370$	0	0
$371$	−7.71573	−0.400581
$372$	0	0
$373$	−10.6274	−0.550267	−0.275133	−	0.961406i	$-0.588722\pi$
$373$	−10.6274	−0.550267	−0.275133	+	0.961406i	$0.588722\pi$
$374$	2.82843	0.146254
$375$	0	0
$376$	17.6569	0.910583
$377$	27.3137	1.40673
$378$	0	0
$379$	−23.3137	−1.19754	−0.598772	−	0.800919i	$-0.704345\pi$
$379$	−23.3137	−1.19754	−0.598772	+	0.800919i	$0.704345\pi$
$380$	0	0
$381$	0	0
$382$	13.6569	0.698745
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	32.9706	1.67816
$387$	0	0
$388$	−44.6274	−2.26561
$389$	23.6569	1.19945	0.599725	−	0.800206i	$-0.295277\pi$
$389$	23.6569	1.19945	0.599725	+	0.800206i	$0.295277\pi$
$390$	0	0
$391$	4.68629	0.236996
$392$	27.8701	1.40765
$393$	0	0
$394$	20.4853	1.03203
$395$	0	0
$396$	0	0
$397$	14.9706	0.751351	0.375676	−	0.926751i	$-0.377411\pi$
$397$	14.9706	0.751351	0.375676	+	0.926751i	$0.377411\pi$
$398$	52.2843	2.62077
$399$	0	0
$400$	0	0
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	0	0
$404$	3.17157	0.157792
$405$	0	0
$406$	9.65685	0.479262
$407$	−11.6569	−0.577809
$408$	0	0
$409$	−19.6569	−0.971969	−0.485984	−	0.873967i	$-0.661539\pi$
$409$	−19.6569	−0.971969	−0.485984	+	0.873967i	$0.661539\pi$
$410$	0	0
$411$	0	0
$412$	12.6863	0.625009
$413$	−3.31371	−0.163057
$414$	0	0
$415$	0	0
$416$	8.97056	0.439818
$417$	0	0
$418$	16.4853	0.806321
$419$	36.9706	1.80613	0.903065	−	0.429504i	$-0.141311\pi$
$419$	36.9706	1.80613	0.903065	+	0.429504i	$0.141311\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−2.82843	−0.137686
$423$	0	0
$424$	−41.1127	−1.99661
$425$	0	0
$426$	0	0
$427$	9.65685	0.467328
$428$	66.2843	3.20397
$429$	0	0
$430$	0	0
$431$	−21.6569	−1.04317	−0.521587	−	0.853198i	$-0.674660\pi$
$431$	−21.6569	−1.04317	−0.521587	+	0.853198i	$0.674660\pi$
$432$	0	0
$433$	15.6569	0.752420	0.376210	−	0.926534i	$-0.377227\pi$
$433$	15.6569	0.752420	0.376210	+	0.926534i	$0.377227\pi$
$434$	0	0
$435$	0	0
$436$	66.2843	3.17444
$437$	27.3137	1.30659
$438$	0	0
$439$	−20.4853	−0.977709	−0.488855	−	0.872365i	$-0.662585\pi$
$439$	−20.4853	−0.977709	−0.488855	+	0.872365i	$0.662585\pi$
$440$	0	0
$441$	0	0
$442$	16.0000	0.761042
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	0	0
$446$	−15.3137	−0.725125
$447$	0	0
$448$	8.14214	0.384680
$449$	−30.9706	−1.46159	−0.730796	−	0.682596i	$-0.760851\pi$
$449$	−30.9706	−1.46159	−0.730796	+	0.682596i	$0.760851\pi$
$450$	0	0
$451$	−4.82843	−0.227362
$452$	−72.6274	−3.41611
$453$	0	0
$454$	33.7990	1.58627
$455$	0	0
$456$	0	0
$457$	23.3137	1.09057	0.545285	−	0.838251i	$-0.316422\pi$
$457$	23.3137	1.09057	0.545285	+	0.838251i	$0.316422\pi$
$458$	4.82843	0.225618
$459$	0	0
$460$	0	0
$461$	−0.142136	−0.00661992	−0.00330996	−	0.999995i	$-0.501054\pi$
$461$	−0.142136	−0.00661992	−0.00330996	+	0.999995i	$0.501054\pi$
$462$	0	0
$463$	−4.97056	−0.231002	−0.115501	−	0.993307i	$-0.536847\pi$
$463$	−4.97056	−0.231002	−0.115501	+	0.993307i	$0.536847\pi$
$464$	14.4853	0.672462
$465$	0	0
$466$	45.4558	2.10570
$467$	−22.6274	−1.04707	−0.523536	−	0.852004i	$-0.675387\pi$
$467$	−22.6274	−1.04707	−0.523536	+	0.852004i	$0.675387\pi$
$468$	0	0
$469$	−4.68629	−0.216393
$470$	0	0
$471$	0	0
$472$	−17.6569	−0.812723
$473$	8.82843	0.405932
$474$	0	0
$475$	0	0
$476$	3.71573	0.170310
$477$	0	0
$478$	42.6274	1.94973
$479$	−36.9706	−1.68923	−0.844614	−	0.535376i	$-0.820170\pi$
$479$	−36.9706	−1.68923	−0.844614	+	0.535376i	$0.820170\pi$
$480$	0	0
$481$	−65.9411	−3.00666
$482$	29.7990	1.35731
$483$	0	0
$484$	3.82843	0.174019
$485$	0	0
$486$	0	0
$487$	12.9706	0.587752	0.293876	−	0.955844i	$-0.405055\pi$
$487$	12.9706	0.587752	0.293876	+	0.955844i	$0.405055\pi$
$488$	51.4558	2.32930
$489$	0	0
$490$	0	0
$491$	−14.3431	−0.647297	−0.323649	−	0.946177i	$-0.604910\pi$
$491$	−14.3431	−0.647297	−0.323649	+	0.946177i	$0.604910\pi$
$492$	0	0
$493$	−5.65685	−0.254772
$494$	93.2548	4.19573
$495$	0	0
$496$	0	0
$497$	1.94113	0.0870714
$498$	0	0
$499$	−22.3431	−1.00022	−0.500108	−	0.865963i	$-0.666706\pi$
$499$	−22.3431	−1.00022	−0.500108	+	0.865963i	$0.666706\pi$
$500$	0	0
$501$	0	0
$502$	50.6274	2.25961
$503$	17.3137	0.771980	0.385990	−	0.922503i	$-0.373860\pi$
$503$	17.3137	0.771980	0.385990	+	0.922503i	$0.373860\pi$
$504$	0	0
$505$	0	0
$506$	9.65685	0.429300
$507$	0	0
$508$	55.4558	2.46046
$509$	−18.6863	−0.828255	−0.414128	−	0.910219i	$-0.635913\pi$
$509$	−18.6863	−0.828255	−0.414128	+	0.910219i	$0.635913\pi$
$510$	0	0
$511$	9.37258	0.414619
$512$	31.2426	1.38074
$513$	0	0
$514$	39.4558	1.74032
$515$	0	0
$516$	0	0
$517$	−4.00000	−0.175920
$518$	−23.3137	−1.02435
$519$	0	0
$520$	0	0
$521$	32.6274	1.42943	0.714717	−	0.699414i	$-0.246556\pi$
$521$	32.6274	1.42943	0.714717	+	0.699414i	$0.246556\pi$
$522$	0	0
$523$	9.51472	0.416050	0.208025	−	0.978124i	$-0.433297\pi$
$523$	9.51472	0.416050	0.208025	+	0.978124i	$0.433297\pi$
$524$	−12.6863	−0.554203
$525$	0	0
$526$	−43.4558	−1.89476
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	21.6569	0.938944
$533$	−27.3137	−1.18309
$534$	0	0
$535$	0	0
$536$	−24.9706	−1.07856
$537$	0	0
$538$	49.7990	2.14699
$539$	−6.31371	−0.271951
$540$	0	0
$541$	17.3137	0.744374	0.372187	−	0.928158i	$-0.378608\pi$
$541$	17.3137	0.744374	0.372187	+	0.928158i	$0.378608\pi$
$542$	28.4853	1.22355
$543$	0	0
$544$	−1.85786	−0.0796553
$545$	0	0
$546$	0	0
$547$	8.14214	0.348133	0.174066	−	0.984734i	$-0.444309\pi$
$547$	8.14214	0.348133	0.174066	+	0.984734i	$0.444309\pi$
$548$	−50.9706	−2.17735
$549$	0	0
$550$	0	0
$551$	−32.9706	−1.40459
$552$	0	0
$553$	−7.02944	−0.298922
$554$	−5.65685	−0.240337
$555$	0	0
$556$	1.85786	0.0787910
$557$	5.17157	0.219127	0.109563	−	0.993980i	$-0.465055\pi$
$557$	5.17157	0.219127	0.109563	+	0.993980i	$0.465055\pi$
$558$	0	0
$559$	49.9411	2.11228
$560$	0	0
$561$	0	0
$562$	−26.9706	−1.13768
$563$	31.6569	1.33418	0.667089	−	0.744978i	$-0.267540\pi$
$563$	31.6569	1.33418	0.667089	+	0.744978i	$0.267540\pi$
$564$	0	0
$565$	0	0
$566$	−21.3137	−0.895882
$567$	0	0
$568$	10.3431	0.433989
$569$	35.4558	1.48639	0.743193	−	0.669077i	$-0.233310\pi$
$569$	35.4558	1.48639	0.743193	+	0.669077i	$0.233310\pi$
$570$	0	0
$571$	−16.4853	−0.689888	−0.344944	−	0.938623i	$-0.612102\pi$
$571$	−16.4853	−0.689888	−0.344944	+	0.938623i	$0.612102\pi$
$572$	21.6569	0.905519
$573$	0	0
$574$	−9.65685	−0.403069
$575$	0	0
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	37.7279	1.56927
$579$	0	0
$580$	0	0
$581$	8.28427	0.343689
$582$	0	0
$583$	9.31371	0.385734
$584$	49.9411	2.06658
$585$	0	0
$586$	16.4853	0.681001
$587$	14.6274	0.603738	0.301869	−	0.953349i	$-0.402389\pi$
$587$	14.6274	0.603738	0.301869	+	0.953349i	$0.402389\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−34.9706	−1.43728
$593$	−22.8284	−0.937451	−0.468726	−	0.883344i	$-0.655287\pi$
$593$	−22.8284	−0.937451	−0.468726	+	0.883344i	$0.655287\pi$
$594$	0	0
$595$	0	0
$596$	−5.79899	−0.237536
$597$	0	0
$598$	54.6274	2.23388
$599$	−27.3137	−1.11601	−0.558004	−	0.829838i	$-0.688433\pi$
$599$	−27.3137	−1.11601	−0.558004	+	0.829838i	$0.688433\pi$
$600$	0	0
$601$	−5.31371	−0.216751	−0.108375	−	0.994110i	$-0.534565\pi$
$601$	−5.31371	−0.216751	−0.108375	+	0.994110i	$0.534565\pi$
$602$	17.6569	0.719640
$603$	0	0
$604$	63.1127	2.56802
$605$	0	0
$606$	0	0
$607$	−1.51472	−0.0614805	−0.0307403	−	0.999527i	$-0.509786\pi$
$607$	−1.51472	−0.0614805	−0.0307403	+	0.999527i	$0.509786\pi$
$608$	−10.8284	−0.439151
$609$	0	0
$610$	0	0
$611$	−22.6274	−0.915407
$612$	0	0
$613$	45.9411	1.85554	0.927772	−	0.373147i	$-0.121721\pi$
$613$	45.9411	1.85554	0.927772	+	0.373147i	$0.121721\pi$
$614$	−7.65685	−0.309005
$615$	0	0
$616$	3.65685	0.147339
$617$	0.343146	0.0138145	0.00690726	−	0.999976i	$-0.497801\pi$
$617$	0.343146	0.0138145	0.00690726	+	0.999976i	$0.497801\pi$
$618$	0	0
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	3.02944	0.121372
$624$	0	0
$625$	0	0
$626$	37.7990	1.51075
$627$	0	0
$628$	−68.9117	−2.74988
$629$	13.6569	0.544534
$630$	0	0
$631$	45.6569	1.81757	0.908785	−	0.417264i	$-0.137011\pi$
$631$	45.6569	1.81757	0.908785	+	0.417264i	$0.137011\pi$
$632$	−37.4558	−1.48991
$633$	0	0
$634$	63.4558	2.52015
$635$	0	0
$636$	0	0
$637$	−35.7157	−1.41511
$638$	−11.6569	−0.461499
$639$	0	0
$640$	0	0
$641$	−6.97056	−0.275321	−0.137660	−	0.990479i	$-0.543958\pi$
$641$	−6.97056	−0.275321	−0.137660	+	0.990479i	$0.543958\pi$
$642$	0	0
$643$	−37.9411	−1.49625	−0.748126	−	0.663557i	$-0.769046\pi$
$643$	−37.9411	−1.49625	−0.748126	+	0.663557i	$0.769046\pi$
$644$	12.6863	0.499910
$645$	0	0
$646$	−19.3137	−0.759888
$647$	4.68629	0.184237	0.0921186	−	0.995748i	$-0.470636\pi$
$647$	4.68629	0.184237	0.0921186	+	0.995748i	$0.470636\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	28.0000	1.09656
$653$	6.97056	0.272779	0.136390	−	0.990655i	$-0.456450\pi$
$653$	6.97056	0.272779	0.136390	+	0.990655i	$0.456450\pi$
$654$	0	0
$655$	0	0
$656$	−14.4853	−0.565555
$657$	0	0
$658$	−8.00000	−0.311872
$659$	15.3137	0.596537	0.298269	−	0.954482i	$-0.403591\pi$
$659$	15.3137	0.596537	0.298269	+	0.954482i	$0.403591\pi$
$660$	0	0
$661$	9.31371	0.362261	0.181131	−	0.983459i	$-0.442024\pi$
$661$	9.31371	0.362261	0.181131	+	0.983459i	$0.442024\pi$
$662$	−15.3137	−0.595184
$663$	0	0
$664$	44.1421	1.71305
$665$	0	0
$666$	0	0
$667$	−19.3137	−0.747830
$668$	−50.9706	−1.97211
$669$	0	0
$670$	0	0
$671$	−11.6569	−0.450008
$672$	0	0
$673$	−18.3431	−0.707076	−0.353538	−	0.935420i	$-0.615022\pi$
$673$	−18.3431	−0.707076	−0.353538	+	0.935420i	$0.615022\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	72.7401	2.79770
$677$	29.4558	1.13208	0.566040	−	0.824378i	$-0.308475\pi$
$677$	29.4558	1.13208	0.566040	+	0.824378i	$0.308475\pi$
$678$	0	0
$679$	9.65685	0.370596
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	0	0
$686$	−26.6274	−1.01664
$687$	0	0
$688$	26.4853	1.00974
$689$	52.6863	2.00719
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	10.8284	0.411635
$693$	0	0
$694$	70.7696	2.68638
$695$	0	0
$696$	0	0
$697$	5.65685	0.214269
$698$	−26.4853	−1.00248
$699$	0	0
$700$	0	0
$701$	−36.1421	−1.36507	−0.682535	−	0.730853i	$-0.739122\pi$
$701$	−36.1421	−1.36507	−0.682535	+	0.730853i	$0.739122\pi$
$702$	0	0
$703$	79.5980	3.00209
$704$	−9.82843	−0.370423
$705$	0	0
$706$	−62.7696	−2.36236
$707$	−0.686292	−0.0258106
$708$	0	0
$709$	6.68629	0.251109	0.125554	−	0.992087i	$-0.459929\pi$
$709$	6.68629	0.251109	0.125554	+	0.992087i	$0.459929\pi$
$710$	0	0
$711$	0	0
$712$	16.1421	0.604952
$713$	0	0
$714$	0	0
$715$	0	0
$716$	67.5980	2.52626
$717$	0	0
$718$	28.9706	1.08117
$719$	47.5980	1.77511	0.887553	−	0.460706i	$-0.152404\pi$
$719$	47.5980	1.77511	0.887553	+	0.460706i	$0.152404\pi$
$720$	0	0
$721$	−2.74517	−0.102235
$722$	−66.6985	−2.48226
$723$	0	0
$724$	−53.5980	−1.99195
$725$	0	0
$726$	0	0
$727$	−33.9411	−1.25881	−0.629403	−	0.777079i	$-0.716701\pi$
$727$	−33.9411	−1.25881	−0.629403	+	0.777079i	$0.716701\pi$
$728$	20.6863	0.766685
$729$	0	0
$730$	0	0
$731$	−10.3431	−0.382555
$732$	0	0
$733$	6.34315	0.234289	0.117145	−	0.993115i	$-0.462626\pi$
$733$	6.34315	0.234289	0.117145	+	0.993115i	$0.462626\pi$
$734$	23.3137	0.860525
$735$	0	0
$736$	−6.34315	−0.233811
$737$	5.65685	0.208373
$738$	0	0
$739$	−15.1127	−0.555930	−0.277965	−	0.960591i	$-0.589660\pi$
$739$	−15.1127	−0.555930	−0.277965	+	0.960591i	$0.589660\pi$
$740$	0	0
$741$	0	0
$742$	18.6274	0.683834
$743$	36.3431	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$743$	36.3431	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$744$	0	0
$745$	0	0
$746$	25.6569	0.939364
$747$	0	0
$748$	−4.48528	−0.163998
$749$	−14.3431	−0.524087
$750$	0	0
$751$	20.2843	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$751$	20.2843	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	−65.9411	−2.40143
$755$	0	0
$756$	0	0
$757$	36.6274	1.33125	0.665623	−	0.746288i	$-0.268166\pi$
$757$	36.6274	1.33125	0.665623	+	0.746288i	$0.268166\pi$
$758$	56.2843	2.04434
$759$	0	0
$760$	0	0
$761$	28.8284	1.04503	0.522515	−	0.852630i	$-0.324994\pi$
$761$	28.8284	1.04503	0.522515	+	0.852630i	$0.324994\pi$
$762$	0	0
$763$	−14.3431	−0.519257
$764$	−21.6569	−0.783517
$765$	0	0
$766$	19.3137	0.697833
$767$	22.6274	0.817029
$768$	0	0
$769$	10.6863	0.385358	0.192679	−	0.981262i	$-0.438282\pi$
$769$	10.6863	0.385358	0.192679	+	0.981262i	$0.438282\pi$
$770$	0	0
$771$	0	0
$772$	−52.2843	−1.88175
$773$	3.65685	0.131528	0.0657640	−	0.997835i	$-0.479052\pi$
$773$	3.65685	0.131528	0.0657640	+	0.997835i	$0.479052\pi$
$774$	0	0
$775$	0	0
$776$	51.4558	1.84716
$777$	0	0
$778$	−57.1127	−2.04759
$779$	32.9706	1.18129
$780$	0	0
$781$	−2.34315	−0.0838443
$782$	−11.3137	−0.404577
$783$	0	0
$784$	−18.9411	−0.676469
$785$	0	0
$786$	0	0
$787$	20.1421	0.717990	0.358995	−	0.933340i	$-0.383120\pi$
$787$	20.1421	0.717990	0.358995	+	0.933340i	$0.383120\pi$
$788$	−32.4853	−1.15724
$789$	0	0
$790$	0	0
$791$	15.7157	0.558787
$792$	0	0
$793$	−65.9411	−2.34164
$794$	−36.1421	−1.28264
$795$	0	0
$796$	−82.9117	−2.93873
$797$	34.9706	1.23872	0.619360	−	0.785107i	$-0.287392\pi$
$797$	34.9706	1.23872	0.619360	+	0.785107i	$0.287392\pi$
$798$	0	0
$799$	4.68629	0.165789
$800$	0	0
$801$	0	0
$802$	−16.1421	−0.569999
$803$	−11.3137	−0.399252
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−3.65685	−0.128648
$809$	−28.4264	−0.999419	−0.499710	−	0.866193i	$-0.666560\pi$
$809$	−28.4264	−0.999419	−0.499710	+	0.866193i	$0.666560\pi$
$810$	0	0
$811$	0.485281	0.0170405	0.00852027	−	0.999964i	$-0.497288\pi$
$811$	0.485281	0.0170405	0.00852027	+	0.999964i	$0.497288\pi$
$812$	−15.3137	−0.537406
$813$	0	0
$814$	28.1421	0.986381
$815$	0	0
$816$	0	0
$817$	−60.2843	−2.10908
$818$	47.4558	1.65925
$819$	0	0
$820$	0	0
$821$	12.8284	0.447715	0.223858	−	0.974622i	$-0.428135\pi$
$821$	12.8284	0.447715	0.223858	+	0.974622i	$0.428135\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−14.6274	−0.509570
$825$	0	0
$826$	8.00000	0.278356
$827$	41.3137	1.43662	0.718309	−	0.695724i	$-0.244916\pi$
$827$	41.3137	1.43662	0.718309	+	0.695724i	$0.244916\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	0	0
$832$	−55.5980	−1.92751
$833$	7.39697	0.256290
$834$	0	0
$835$	0	0
$836$	−26.1421	−0.904145
$837$	0	0
$838$	−89.2548	−3.08326
$839$	22.6274	0.781185	0.390593	−	0.920564i	$-0.372270\pi$
$839$	22.6274	0.781185	0.390593	+	0.920564i	$0.372270\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	14.4853	0.499196
$843$	0	0
$844$	4.48528	0.154390
$845$	0	0
$846$	0	0
$847$	−0.828427	−0.0284651
$848$	27.9411	0.959502
$849$	0	0
$850$	0	0
$851$	46.6274	1.59837
$852$	0	0
$853$	8.68629	0.297413	0.148706	−	0.988881i	$-0.452489\pi$
$853$	8.68629	0.297413	0.148706	+	0.988881i	$0.452489\pi$
$854$	−23.3137	−0.797779
$855$	0	0
$856$	−76.4264	−2.61220
$857$	−28.4853	−0.973039	−0.486519	−	0.873670i	$-0.661734\pi$
$857$	−28.4853	−0.973039	−0.486519	+	0.873670i	$0.661734\pi$
$858$	0	0
$859$	−52.9706	−1.80733	−0.903666	−	0.428238i	$-0.859135\pi$
$859$	−52.9706	−1.80733	−0.903666	+	0.428238i	$0.859135\pi$
$860$	0	0
$861$	0	0
$862$	52.2843	1.78081
$863$	−20.6863	−0.704170	−0.352085	−	0.935968i	$-0.614527\pi$
$863$	−20.6863	−0.704170	−0.352085	+	0.935968i	$0.614527\pi$
$864$	0	0
$865$	0	0
$866$	−37.7990	−1.28446
$867$	0	0
$868$	0	0
$869$	8.48528	0.287843
$870$	0	0
$871$	32.0000	1.08428
$872$	−76.4264	−2.58812
$873$	0	0
$874$	−65.9411	−2.23049
$875$	0	0
$876$	0	0
$877$	−2.62742	−0.0887216	−0.0443608	−	0.999016i	$-0.514125\pi$
$877$	−2.62742	−0.0887216	−0.0443608	+	0.999016i	$0.514125\pi$
$878$	49.4558	1.66905
$879$	0	0
$880$	0	0
$881$	46.9706	1.58248	0.791239	−	0.611507i	$-0.209436\pi$
$881$	46.9706	1.58248	0.791239	+	0.611507i	$0.209436\pi$
$882$	0	0
$883$	5.37258	0.180802	0.0904009	−	0.995905i	$-0.471185\pi$
$883$	5.37258	0.180802	0.0904009	+	0.995905i	$0.471185\pi$
$884$	−25.3726	−0.853372
$885$	0	0
$886$	−28.9706	−0.973285
$887$	−15.6569	−0.525706	−0.262853	−	0.964836i	$-0.584663\pi$
$887$	−15.6569	−0.525706	−0.262853	+	0.964836i	$0.584663\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	24.2843	0.813098
$893$	27.3137	0.914018
$894$	0	0
$895$	0	0
$896$	−17.0294	−0.568914
$897$	0	0
$898$	74.7696	2.49509
$899$	0	0
$900$	0	0
$901$	−10.9117	−0.363521
$902$	11.6569	0.388131
$903$	0	0
$904$	83.7401	2.78515
$905$	0	0
$906$	0	0
$907$	−40.9706	−1.36041	−0.680203	−	0.733024i	$-0.738108\pi$
$907$	−40.9706	−1.36041	−0.680203	+	0.733024i	$0.738108\pi$
$908$	−53.5980	−1.77871
$909$	0	0
$910$	0	0
$911$	48.9706	1.62247	0.811234	−	0.584722i	$-0.198797\pi$
$911$	48.9706	1.62247	0.811234	+	0.584722i	$0.198797\pi$
$912$	0	0
$913$	−10.0000	−0.330952
$914$	−56.2843	−1.86172
$915$	0	0
$916$	−7.65685	−0.252990
$917$	2.74517	0.0906534
$918$	0	0
$919$	11.5147	0.379836	0.189918	−	0.981800i	$-0.439178\pi$
$919$	11.5147	0.379836	0.189918	+	0.981800i	$0.439178\pi$
$920$	0	0
$921$	0	0
$922$	0.343146	0.0113009
$923$	−13.2548	−0.436288
$924$	0	0
$925$	0	0
$926$	12.0000	0.394344
$927$	0	0
$928$	7.65685	0.251349
$929$	45.5980	1.49602	0.748011	−	0.663687i	$-0.231009\pi$
$929$	45.5980	1.49602	0.748011	+	0.663687i	$0.231009\pi$
$930$	0	0
$931$	43.1127	1.41296
$932$	−72.0833	−2.36117
$933$	0	0
$934$	54.6274	1.78746
$935$	0	0
$936$	0	0
$937$	−11.0294	−0.360316	−0.180158	−	0.983638i	$-0.557661\pi$
$937$	−11.0294	−0.360316	−0.180158	+	0.983638i	$0.557661\pi$
$938$	11.3137	0.369406
$939$	0	0
$940$	0	0
$941$	34.7696	1.13346	0.566728	−	0.823905i	$-0.308209\pi$
$941$	34.7696	1.13346	0.566728	+	0.823905i	$0.308209\pi$
$942$	0	0
$943$	19.3137	0.628941
$944$	12.0000	0.390567
$945$	0	0
$946$	−21.3137	−0.692968
$947$	6.62742	0.215362	0.107681	−	0.994185i	$-0.465657\pi$
$947$	6.62742	0.215362	0.107681	+	0.994185i	$0.465657\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	0	0
$951$	0	0
$952$	−4.28427	−0.138854
$953$	−11.7990	−0.382207	−0.191103	−	0.981570i	$-0.561207\pi$
$953$	−11.7990	−0.382207	−0.191103	+	0.981570i	$0.561207\pi$
$954$	0	0
$955$	0	0
$956$	−67.5980	−2.18627
$957$	0	0
$958$	89.2548	2.88369
$959$	11.0294	0.356159
$960$	0	0
$961$	−31.0000	−1.00000
$962$	159.196	5.13268
$963$	0	0
$964$	−47.2548	−1.52198
$965$	0	0
$966$	0	0
$967$	−11.4558	−0.368395	−0.184198	−	0.982889i	$-0.558969\pi$
$967$	−11.4558	−0.368395	−0.184198	+	0.982889i	$0.558969\pi$
$968$	−4.41421	−0.141878
$969$	0	0
$970$	0	0
$971$	34.6274	1.11125	0.555623	−	0.831434i	$-0.312480\pi$
$971$	34.6274	1.11125	0.555623	+	0.831434i	$0.312480\pi$
$972$	0	0
$973$	−0.402020	−0.0128882
$974$	−31.3137	−1.00336
$975$	0	0
$976$	−34.9706	−1.11938
$977$	2.68629	0.0859421	0.0429710	−	0.999076i	$-0.486318\pi$
$977$	2.68629	0.0859421	0.0429710	+	0.999076i	$0.486318\pi$
$978$	0	0
$979$	−3.65685	−0.116874
$980$	0	0
$981$	0	0
$982$	34.6274	1.10501
$983$	−30.6274	−0.976863	−0.488431	−	0.872602i	$-0.662431\pi$
$983$	−30.6274	−0.976863	−0.488431	+	0.872602i	$0.662431\pi$
$984$	0	0
$985$	0	0
$986$	13.6569	0.434923
$987$	0	0
$988$	−147.882	−4.70476
$989$	−35.3137	−1.12291
$990$	0	0
$991$	−30.6274	−0.972912	−0.486456	−	0.873705i	$-0.661711\pi$
$991$	−30.6274	−0.972912	−0.486456	+	0.873705i	$0.661711\pi$
$992$	0	0
$993$	0	0
$994$	−4.68629	−0.148640
$995$	0	0
$996$	0	0
$997$	39.3137	1.24508	0.622539	−	0.782589i	$-0.286101\pi$
$997$	39.3137	1.24508	0.622539	+	0.782589i	$0.286101\pi$
$998$	53.9411	1.70748
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.m.1.1		2
3.2	odd	2		825.2.a.g.1.2		2
5.2	odd	4		2475.2.c.m.199.1		4
5.3	odd	4		2475.2.c.m.199.4		4
5.4	even	2		495.2.a.d.1.2		2
15.2	even	4		825.2.c.e.199.4		4
15.8	even	4		825.2.c.e.199.1		4
15.14	odd	2		165.2.a.a.1.1	✓	2
20.19	odd	2		7920.2.a.cg.1.1		2
33.32	even	2		9075.2.a.v.1.1		2
55.54	odd	2		5445.2.a.m.1.1		2
60.59	even	2		2640.2.a.bb.1.1		2
105.104	even	2		8085.2.a.ba.1.1		2
165.164	even	2		1815.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.a.1.1	✓	2	15.14	odd	2
495.2.a.d.1.2		2	5.4	even	2
825.2.a.g.1.2		2	3.2	odd	2
825.2.c.e.199.1		4	15.8	even	4
825.2.c.e.199.4		4	15.2	even	4
1815.2.a.k.1.2		2	165.164	even	2
2475.2.a.m.1.1		2	1.1	even	1	trivial
2475.2.c.m.199.1		4	5.2	odd	4
2475.2.c.m.199.4		4	5.3	odd	4
2640.2.a.bb.1.1		2	60.59	even	2
5445.2.a.m.1.1		2	55.54	odd	2
7920.2.a.cg.1.1		2	20.19	odd	2
8085.2.a.ba.1.1		2	105.104	even	2
9075.2.a.v.1.1		2	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -0.828427 q^{7} -4.41421 q^{8} +O(q^{10})\)	\(q-2.41421 q^{2} +3.82843 q^{4} -0.828427 q^{7} -4.41421 q^{8} +1.00000 q^{11} +5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -1.17157 q^{17} -6.82843 q^{19} -2.41421 q^{22} -4.00000 q^{23} -13.6569 q^{26} -3.17157 q^{28} +4.82843 q^{29} +1.58579 q^{32} +2.82843 q^{34} -11.6569 q^{37} +16.4853 q^{38} -4.82843 q^{41} +8.82843 q^{43} +3.82843 q^{44} +9.65685 q^{46} -4.00000 q^{47} -6.31371 q^{49} +21.6569 q^{52} +9.31371 q^{53} +3.65685 q^{56} -11.6569 q^{58} +4.00000 q^{59} -11.6569 q^{61} -9.82843 q^{64} +5.65685 q^{67} -4.48528 q^{68} -2.34315 q^{71} -11.3137 q^{73} +28.1421 q^{74} -26.1421 q^{76} -0.828427 q^{77} +8.48528 q^{79} +11.6569 q^{82} -10.0000 q^{83} -21.3137 q^{86} -4.41421 q^{88} -3.65685 q^{89} -4.68629 q^{91} -15.3137 q^{92} +9.65685 q^{94} -11.6569 q^{97} +15.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 2 q^{11} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 8 q^{19} - 2 q^{22} - 8 q^{23} - 16 q^{26} - 12 q^{28} + 4 q^{29} + 6 q^{32} - 12 q^{37} + 16 q^{38} - 4 q^{41} + 12 q^{43} + 2 q^{44} + 8 q^{46} - 8 q^{47} + 10 q^{49} + 32 q^{52} - 4 q^{53} - 4 q^{56} - 12 q^{58} + 8 q^{59} - 12 q^{61} - 14 q^{64} + 8 q^{68} - 16 q^{71} + 28 q^{74} - 24 q^{76} + 4 q^{77} + 12 q^{82} - 20 q^{83} - 20 q^{86} - 6 q^{88} + 4 q^{89} - 32 q^{91} - 8 q^{92} + 8 q^{94} - 12 q^{97} + 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 6 * q^8 + 2 * q^11 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 8 * q^19 - 2 * q^22 - 8 * q^23 - 16 * q^26 - 12 * q^28 + 4 * q^29 + 6 * q^32 - 12 * q^37 + 16 * q^38 - 4 * q^41 + 12 * q^43 + 2 * q^44 + 8 * q^46 - 8 * q^47 + 10 * q^49 + 32 * q^52 - 4 * q^53 - 4 * q^56 - 12 * q^58 + 8 * q^59 - 12 * q^61 - 14 * q^64 + 8 * q^68 - 16 * q^71 + 28 * q^74 - 24 * q^76 + 4 * q^77 + 12 * q^82 - 20 * q^83 - 20 * q^86 - 6 * q^88 + 4 * q^89 - 32 * q^91 - 8 * q^92 + 8 * q^94 - 12 * q^97 + 22 * q^98

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