LMFDB - Embedded newform 2475.4.a.bt.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,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, 4, names="a")

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.32942$ 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)

$f(q)$	$=$	$q+4.32942 q^{2} +10.7439 q^{4} +6.68577 q^{7} +11.8793 q^{8} +O(q^{10})$	$q+4.32942 q^{2} +10.7439 q^{4} +6.68577 q^{7} +11.8793 q^{8} -11.0000 q^{11} -59.6847 q^{13} +28.9455 q^{14} -34.5204 q^{16} +131.453 q^{17} +52.1020 q^{19} -47.6236 q^{22} -15.3496 q^{23} -258.400 q^{26} +71.8310 q^{28} -59.6632 q^{29} +229.334 q^{31} -244.488 q^{32} +569.117 q^{34} +241.772 q^{37} +225.571 q^{38} -196.739 q^{41} +195.563 q^{43} -118.182 q^{44} -66.4549 q^{46} +517.196 q^{47} -298.300 q^{49} -641.244 q^{52} -306.018 q^{53} +79.4223 q^{56} -258.307 q^{58} +763.034 q^{59} +269.264 q^{61} +992.885 q^{62} -782.326 q^{64} +895.972 q^{67} +1412.32 q^{68} +855.296 q^{71} -391.697 q^{73} +1046.73 q^{74} +559.776 q^{76} -73.5435 q^{77} +829.746 q^{79} -851.767 q^{82} -939.585 q^{83} +846.673 q^{86} -130.672 q^{88} -412.394 q^{89} -399.038 q^{91} -164.914 q^{92} +2239.16 q^{94} -1416.81 q^{97} -1291.47 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8}+O(q^{10}) $ 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8	$ 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8} - 77 q^{11} - 38 q^{13} - 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} - 55 q^{22} + 334 q^{23} + 372 q^{26} - 812 q^{28} + 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} - 466 q^{37} - 494 q^{38} + 258 q^{41} - 308 q^{43} - 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} - 512 q^{52} + 110 q^{53} - 20 q^{56} - 1362 q^{58} + 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} - 2268 q^{67} - 1186 q^{68} + 166 q^{71} - 200 q^{73} + 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1006 q^{82} + 370 q^{83} - 106 q^{86} - 495 q^{88} + 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} - 3698 q^{97} + 697 q^{98}+O(q^{100}) $ 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8 - 77 * q^11 - 38 * q^13 - 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 - 55 * q^22 + 334 * q^23 + 372 * q^26 - 812 * q^28 + 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 - 466 * q^37 - 494 * q^38 + 258 * q^41 - 308 * q^43 - 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 - 512 * q^52 + 110 * q^53 - 20 * q^56 - 1362 * q^58 + 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 - 2268 * q^67 - 1186 * q^68 + 166 * q^71 - 200 * q^73 + 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1006 * q^82 + 370 * q^83 - 106 * q^86 - 495 * q^88 + 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 - 3698 * q^97 + 697 * 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 $

$2$	4.32942	1.53068	0.765340	−	0.643626i	$-0.222571\pi$
$2$	4.32942	1.53068	0.765340	+	0.643626i	$0.222571\pi$
$3$	0	0
$3$	0	0
$4$	10.7439	1.34298
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.68577	0.360998	0.180499	−	0.983575i	$-0.442229\pi$
$7$	6.68577	0.360998	0.180499	+	0.983575i	$0.442229\pi$
$8$	11.8793	0.524996
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−59.6847	−1.27335	−0.636675	−	0.771132i	$-0.719691\pi$
$13$	−59.6847	−1.27335	−0.636675	+	0.771132i	$0.719691\pi$
$14$	28.9455	0.552572
$15$	0	0
$16$	−34.5204	−0.539381
$17$	131.453	1.87542	0.937710	−	0.347418i	$-0.112942\pi$
$17$	131.453	1.87542	0.937710	+	0.347418i	$0.112942\pi$
$18$	0	0
$19$	52.1020	0.629106	0.314553	−	0.949240i	$-0.398145\pi$
$19$	52.1020	0.629106	0.314553	+	0.949240i	$0.398145\pi$
$20$	0	0
$21$	0	0
$22$	−47.6236	−0.461517
$23$	−15.3496	−0.139157	−0.0695786	−	0.997576i	$-0.522165\pi$
$23$	−15.3496	−0.139157	−0.0695786	+	0.997576i	$0.522165\pi$
$24$	0	0
$25$	0	0
$26$	−258.400	−1.94909
$27$	0	0
$28$	71.8310	0.484813
$29$	−59.6632	−0.382041	−0.191020	−	0.981586i	$-0.561180\pi$
$29$	−59.6632	−0.382041	−0.191020	+	0.981586i	$0.561180\pi$
$30$	0	0
$31$	229.334	1.32870	0.664350	−	0.747422i	$-0.268709\pi$
$31$	229.334	1.32870	0.664350	+	0.747422i	$0.268709\pi$
$32$	−244.488	−1.35062
$33$	0	0
$34$	569.117	2.87067
$35$	0	0
$36$	0	0
$37$	241.772	1.07424	0.537122	−	0.843505i	$-0.319512\pi$
$37$	241.772	1.07424	0.537122	+	0.843505i	$0.319512\pi$
$38$	225.571	0.962960
$39$	0	0
$40$	0	0
$41$	−196.739	−0.749403	−0.374702	−	0.927145i	$-0.622255\pi$
$41$	−196.739	−0.749403	−0.374702	+	0.927145i	$0.622255\pi$
$42$	0	0
$43$	195.563	0.693559	0.346780	−	0.937947i	$-0.387275\pi$
$43$	195.563	0.693559	0.346780	+	0.937947i	$0.387275\pi$
$44$	−118.182	−0.404924
$45$	0	0
$46$	−66.4549	−0.213005
$47$	517.196	1.60512	0.802562	−	0.596569i	$-0.203470\pi$
$47$	517.196	1.60512	0.802562	+	0.596569i	$0.203470\pi$
$48$	0	0
$49$	−298.300	−0.869681
$50$	0	0
$51$	0	0
$52$	−641.244	−1.71009
$53$	−306.018	−0.793108	−0.396554	−	0.918011i	$-0.629794\pi$
$53$	−306.018	−0.793108	−0.396554	+	0.918011i	$0.629794\pi$
$54$	0	0
$55$	0	0
$56$	79.4223	0.189522
$57$	0	0
$58$	−258.307	−0.584782
$59$	763.034	1.68371	0.841853	−	0.539707i	$-0.181465\pi$
$59$	763.034	1.68371	0.841853	+	0.539707i	$0.181465\pi$
$60$	0	0
$61$	269.264	0.565176	0.282588	−	0.959241i	$-0.408807\pi$
$61$	269.264	0.565176	0.282588	+	0.959241i	$0.408807\pi$
$62$	992.885	2.03381
$63$	0	0
$64$	−782.326	−1.52798
$65$	0	0
$66$	0	0
$67$	895.972	1.63374	0.816868	−	0.576824i	$-0.195708\pi$
$67$	895.972	1.63374	0.816868	+	0.576824i	$0.195708\pi$
$68$	1412.32	2.51866
$69$	0	0
$70$	0	0
$71$	855.296	1.42965	0.714824	−	0.699305i	$-0.246507\pi$
$71$	855.296	1.42965	0.714824	+	0.699305i	$0.246507\pi$
$72$	0	0
$73$	−391.697	−0.628009	−0.314004	−	0.949422i	$-0.601671\pi$
$73$	−391.697	−0.628009	−0.314004	+	0.949422i	$0.601671\pi$
$74$	1046.73	1.64432
$75$	0	0
$76$	559.776	0.844878
$77$	−73.5435	−0.108845
$78$	0	0
$79$	829.746	1.18169	0.590846	−	0.806784i	$-0.298794\pi$
$79$	829.746	1.18169	0.590846	+	0.806784i	$0.298794\pi$
$80$	0	0
$81$	0	0
$82$	−851.767	−1.14710
$83$	−939.585	−1.24256	−0.621282	−	0.783587i	$-0.713388\pi$
$83$	−939.585	−1.24256	−0.621282	+	0.783587i	$0.713388\pi$
$84$	0	0
$85$	0	0
$86$	846.673	1.06162
$87$	0	0
$88$	−130.672	−0.158292
$89$	−412.394	−0.491165	−0.245583	−	0.969376i	$-0.578979\pi$
$89$	−412.394	−0.491165	−0.245583	+	0.969376i	$0.578979\pi$
$90$	0	0
$91$	−399.038	−0.459677
$92$	−164.914	−0.186886
$93$	0	0
$94$	2239.16	2.45693
$95$	0	0
$96$	0	0
$97$	−1416.81	−1.48304	−0.741520	−	0.670931i	$-0.765895\pi$
$97$	−1416.81	−1.48304	−0.741520	+	0.670931i	$0.765895\pi$
$98$	−1291.47	−1.33120
$99$	0	0
$100$	0	0
$101$	976.701	0.962231	0.481116	−	0.876657i	$-0.340232\pi$
$101$	976.701	0.962231	0.481116	+	0.876657i	$0.340232\pi$
$102$	0	0
$103$	175.786	0.168163	0.0840813	−	0.996459i	$-0.473204\pi$
$103$	175.786	0.168163	0.0840813	+	0.996459i	$0.473204\pi$
$104$	−709.013	−0.668504
$105$	0	0
$106$	−1324.88	−1.21400
$107$	224.503	0.202837	0.101418	−	0.994844i	$-0.467662\pi$
$107$	224.503	0.202837	0.101418	+	0.994844i	$0.467662\pi$
$108$	0	0
$109$	1765.83	1.55171	0.775855	−	0.630912i	$-0.217319\pi$
$109$	1765.83	1.55171	0.775855	+	0.630912i	$0.217319\pi$
$110$	0	0
$111$	0	0
$112$	−230.795	−0.194715
$113$	93.7154	0.0780177	0.0390089	−	0.999239i	$-0.487580\pi$
$113$	93.7154	0.0780177	0.0390089	+	0.999239i	$0.487580\pi$
$114$	0	0
$115$	0	0
$116$	−641.013	−0.513074
$117$	0	0
$118$	3303.49	2.57721
$119$	878.868	0.677022
$120$	0	0
$121$	121.000	0.0909091
$122$	1165.76	0.865104
$123$	0	0
$124$	2463.94	1.78442
$125$	0	0
$126$	0	0
$127$	−240.508	−0.168044	−0.0840220	−	0.996464i	$-0.526777\pi$
$127$	−240.508	−0.168044	−0.0840220	+	0.996464i	$0.526777\pi$
$128$	−1431.11	−0.988233
$129$	0	0
$130$	0	0
$131$	2451.63	1.63511	0.817556	−	0.575850i	$-0.195329\pi$
$131$	2451.63	1.63511	0.817556	+	0.575850i	$0.195329\pi$
$132$	0	0
$133$	348.342	0.227106
$134$	3879.04	2.50073
$135$	0	0
$136$	1561.58	0.984589
$137$	−1746.26	−1.08900	−0.544500	−	0.838761i	$-0.683280\pi$
$137$	−1746.26	−1.08900	−0.544500	+	0.838761i	$0.683280\pi$
$138$	0	0
$139$	1583.66	0.966361	0.483180	−	0.875521i	$-0.339482\pi$
$139$	1583.66	0.966361	0.483180	+	0.875521i	$0.339482\pi$
$140$	0	0
$141$	0	0
$142$	3702.93	2.18833
$143$	656.532	0.383930
$144$	0	0
$145$	0	0
$146$	−1695.82	−0.961281
$147$	0	0
$148$	2597.56	1.44269
$149$	2546.98	1.40038	0.700191	−	0.713956i	$-0.253098\pi$
$149$	2546.98	1.40038	0.700191	+	0.713956i	$0.253098\pi$
$150$	0	0
$151$	−646.407	−0.348370	−0.174185	−	0.984713i	$-0.555729\pi$
$151$	−646.407	−0.348370	−0.174185	+	0.984713i	$0.555729\pi$
$152$	618.935	0.330278
$153$	0	0
$154$	−318.400	−0.166607
$155$	0	0
$156$	0	0
$157$	−1765.10	−0.897266	−0.448633	−	0.893716i	$-0.648089\pi$
$157$	−1765.10	−0.897266	−0.448633	+	0.893716i	$0.648089\pi$
$158$	3592.32	1.80879
$159$	0	0
$160$	0	0
$161$	−102.624	−0.0502354
$162$	0	0
$163$	2422.10	1.16389	0.581943	−	0.813229i	$-0.302293\pi$
$163$	2422.10	1.16389	0.581943	+	0.813229i	$0.302293\pi$
$164$	−2113.74	−1.00644
$165$	0	0
$166$	−4067.86	−1.90197
$167$	80.2186	0.0371707	0.0185853	−	0.999827i	$-0.494084\pi$
$167$	80.2186	0.0371707	0.0185853	+	0.999827i	$0.494084\pi$
$168$	0	0
$169$	1365.26	0.621422
$170$	0	0
$171$	0	0
$172$	2101.10	0.931438
$173$	−2758.28	−1.21218	−0.606092	−	0.795395i	$-0.707264\pi$
$173$	−2758.28	−1.21218	−0.606092	+	0.795395i	$0.707264\pi$
$174$	0	0
$175$	0	0
$176$	379.724	0.162629
$177$	0	0
$178$	−1785.43	−0.751817
$179$	−2098.76	−0.876361	−0.438181	−	0.898887i	$-0.644377\pi$
$179$	−2098.76	−0.876361	−0.438181	+	0.898887i	$0.644377\pi$
$180$	0	0
$181$	−3838.01	−1.57612	−0.788058	−	0.615601i	$-0.788913\pi$
$181$	−3838.01	−1.57612	−0.788058	+	0.615601i	$0.788913\pi$
$182$	−1727.60	−0.703618
$183$	0	0
$184$	−182.343	−0.0730570
$185$	0	0
$186$	0	0
$187$	−1445.99	−0.565461
$188$	5556.68	2.15565
$189$	0	0
$190$	0	0
$191$	2593.98	0.982689	0.491344	−	0.870965i	$-0.336506\pi$
$191$	2593.98	0.982689	0.491344	+	0.870965i	$0.336506\pi$
$192$	0	0
$193$	2067.95	0.771267	0.385633	−	0.922652i	$-0.373983\pi$
$193$	2067.95	0.771267	0.385633	+	0.922652i	$0.373983\pi$
$194$	−6133.94	−2.27006
$195$	0	0
$196$	−3204.90	−1.16797
$197$	−2359.88	−0.853476	−0.426738	−	0.904375i	$-0.640337\pi$
$197$	−2359.88	−0.853476	−0.426738	+	0.904375i	$0.640337\pi$
$198$	0	0
$199$	−2418.59	−0.861556	−0.430778	−	0.902458i	$-0.641761\pi$
$199$	−2418.59	−0.861556	−0.430778	+	0.902458i	$0.641761\pi$
$200$	0	0
$201$	0	0
$202$	4228.55	1.47287
$203$	−398.894	−0.137916
$204$	0	0
$205$	0	0
$206$	761.053	0.257403
$207$	0	0
$208$	2060.34	0.686821
$209$	−573.122	−0.189683
$210$	0	0
$211$	1343.60	0.438376	0.219188	−	0.975683i	$-0.429659\pi$
$211$	1343.60	0.438376	0.219188	+	0.975683i	$0.429659\pi$
$212$	−3287.81	−1.06513
$213$	0	0
$214$	971.968	0.310478
$215$	0	0
$216$	0	0
$217$	1533.28	0.479657
$218$	7645.04	2.37517
$219$	0	0
$220$	0	0
$221$	−7845.76	−2.38807
$222$	0	0
$223$	−503.742	−0.151269	−0.0756347	−	0.997136i	$-0.524098\pi$
$223$	−503.742	−0.151269	−0.0756347	+	0.997136i	$0.524098\pi$
$224$	−1634.59	−0.487569
$225$	0	0
$226$	405.733	0.119420
$227$	−16.7023	−0.00488356	−0.00244178	−	0.999997i	$-0.500777\pi$
$227$	−16.7023	−0.00488356	−0.00244178	+	0.999997i	$0.500777\pi$
$228$	0	0
$229$	2489.06	0.718260	0.359130	−	0.933287i	$-0.383073\pi$
$229$	2489.06	0.718260	0.359130	+	0.933287i	$0.383073\pi$
$230$	0	0
$231$	0	0
$232$	−708.757	−0.200570
$233$	4313.16	1.21272	0.606361	−	0.795189i	$-0.292629\pi$
$233$	4313.16	1.21272	0.606361	+	0.795189i	$0.292629\pi$
$234$	0	0
$235$	0	0
$236$	8197.93	2.26119
$237$	0	0
$238$	3804.99	1.03630
$239$	−1933.96	−0.523420	−0.261710	−	0.965147i	$-0.584286\pi$
$239$	−1933.96	−0.523420	−0.261710	+	0.965147i	$0.584286\pi$
$240$	0	0
$241$	134.998	0.0360829	0.0180414	−	0.999837i	$-0.494257\pi$
$241$	134.998	0.0360829	0.0180414	+	0.999837i	$0.494257\pi$
$242$	523.860	0.139153
$243$	0	0
$244$	2892.94	0.759022
$245$	0	0
$246$	0	0
$247$	−3109.69	−0.801073
$248$	2724.33	0.697562
$249$	0	0
$250$	0	0
$251$	−880.954	−0.221535	−0.110768	−	0.993846i	$-0.535331\pi$
$251$	−880.954	−0.221535	−0.110768	+	0.993846i	$0.535331\pi$
$252$	0	0
$253$	168.846	0.0419575
$254$	−1041.26	−0.257222
$255$	0	0
$256$	62.7132	0.0153108
$257$	4955.50	1.20278	0.601392	−	0.798954i	$-0.294613\pi$
$257$	4955.50	1.20278	0.601392	+	0.798954i	$0.294613\pi$
$258$	0	0
$259$	1616.43	0.387799
$260$	0	0
$261$	0	0
$262$	10614.1	2.50283
$263$	−30.7319	−0.00720537	−0.00360269	−	0.999994i	$-0.501147\pi$
$263$	−30.7319	−0.00720537	−0.00360269	+	0.999994i	$0.501147\pi$
$264$	0	0
$265$	0	0
$266$	1508.12	0.347626
$267$	0	0
$268$	9626.19	2.19408
$269$	−1933.62	−0.438271	−0.219135	−	0.975694i	$-0.570324\pi$
$269$	−1933.62	−0.438271	−0.219135	+	0.975694i	$0.570324\pi$
$270$	0	0
$271$	−2560.68	−0.573987	−0.286994	−	0.957933i	$-0.592656\pi$
$271$	−2560.68	−0.573987	−0.286994	+	0.957933i	$0.592656\pi$
$272$	−4537.82	−1.01157
$273$	0	0
$274$	−7560.28	−1.66691
$275$	0	0
$276$	0	0
$277$	−6132.14	−1.33012	−0.665062	−	0.746788i	$-0.731595\pi$
$277$	−6132.14	−1.33012	−0.665062	+	0.746788i	$0.731595\pi$
$278$	6856.32	1.47919
$279$	0	0
$280$	0	0
$281$	6381.85	1.35484	0.677418	−	0.735598i	$-0.263099\pi$
$281$	6381.85	1.35484	0.677418	+	0.735598i	$0.263099\pi$
$282$	0	0
$283$	−7965.30	−1.67310	−0.836551	−	0.547890i	$-0.815431\pi$
$283$	−7965.30	−1.67310	−0.836551	+	0.547890i	$0.815431\pi$
$284$	9189.18	1.91999
$285$	0	0
$286$	2842.40	0.587674
$287$	−1315.35	−0.270533
$288$	0	0
$289$	12367.0	2.51720
$290$	0	0
$291$	0	0
$292$	−4208.33	−0.843405
$293$	−3638.26	−0.725424	−0.362712	−	0.931901i	$-0.618149\pi$
$293$	−3638.26	−0.725424	−0.362712	+	0.931901i	$0.618149\pi$
$294$	0	0
$295$	0	0
$296$	2872.08	0.563974
$297$	0	0
$298$	11026.9	2.14354
$299$	916.137	0.177196
$300$	0	0
$301$	1307.49	0.250373
$302$	−2798.57	−0.533243
$303$	0	0
$304$	−1798.58	−0.339328
$305$	0	0
$306$	0	0
$307$	308.188	0.0572938	0.0286469	−	0.999590i	$-0.490880\pi$
$307$	308.188	0.0572938	0.0286469	+	0.999590i	$0.490880\pi$
$308$	−790.141	−0.146177
$309$	0	0
$310$	0	0
$311$	6481.94	1.18186	0.590928	−	0.806724i	$-0.298762\pi$
$311$	6481.94	1.18186	0.590928	+	0.806724i	$0.298762\pi$
$312$	0	0
$313$	5974.54	1.07892	0.539459	−	0.842012i	$-0.318629\pi$
$313$	5974.54	1.07892	0.539459	+	0.842012i	$0.318629\pi$
$314$	−7641.88	−1.37343
$315$	0	0
$316$	8914.67	1.58699
$317$	−7710.57	−1.36615	−0.683074	−	0.730349i	$-0.739357\pi$
$317$	−7710.57	−1.36615	−0.683074	+	0.730349i	$0.739357\pi$
$318$	0	0
$319$	656.295	0.115190
$320$	0	0
$321$	0	0
$322$	−444.302	−0.0768944
$323$	6848.99	1.17984
$324$	0	0
$325$	0	0
$326$	10486.3	1.78154
$327$	0	0
$328$	−2337.13	−0.393434
$329$	3457.85	0.579446
$330$	0	0
$331$	−4495.09	−0.746442	−0.373221	−	0.927742i	$-0.621747\pi$
$331$	−4495.09	−0.746442	−0.373221	+	0.927742i	$0.621747\pi$
$332$	−10094.8	−1.66874
$333$	0	0
$334$	347.300	0.0568964
$335$	0	0
$336$	0	0
$337$	6468.60	1.04560	0.522800	−	0.852455i	$-0.324887\pi$
$337$	6468.60	1.04560	0.522800	+	0.852455i	$0.324887\pi$
$338$	5910.79	0.951198
$339$	0	0
$340$	0	0
$341$	−2522.68	−0.400618
$342$	0	0
$343$	−4287.59	−0.674950
$344$	2323.15	0.364116
$345$	0	0
$346$	−11941.7	−1.85547
$347$	1597.84	0.247194	0.123597	−	0.992332i	$-0.460557\pi$
$347$	1597.84	0.247194	0.123597	+	0.992332i	$0.460557\pi$
$348$	0	0
$349$	6182.32	0.948230	0.474115	−	0.880463i	$-0.342768\pi$
$349$	6182.32	0.948230	0.474115	+	0.880463i	$0.342768\pi$
$350$	0	0
$351$	0	0
$352$	2689.36	0.407226
$353$	8208.94	1.23773	0.618864	−	0.785498i	$-0.287593\pi$
$353$	8208.94	1.23773	0.618864	+	0.785498i	$0.287593\pi$
$354$	0	0
$355$	0	0
$356$	−4430.70	−0.659626
$357$	0	0
$358$	−9086.41	−1.34143
$359$	−234.639	−0.0344952	−0.0172476	−	0.999851i	$-0.505490\pi$
$359$	−234.639	−0.0344952	−0.0172476	+	0.999851i	$0.505490\pi$
$360$	0	0
$361$	−4144.38	−0.604226
$362$	−16616.3	−2.41253
$363$	0	0
$364$	−4287.21	−0.617337
$365$	0	0
$366$	0	0
$367$	−8607.23	−1.22423	−0.612117	−	0.790767i	$-0.709682\pi$
$367$	−8607.23	−1.22423	−0.612117	+	0.790767i	$0.709682\pi$
$368$	529.874	0.0750587
$369$	0	0
$370$	0	0
$371$	−2045.96	−0.286310
$372$	0	0
$373$	9551.32	1.32587	0.662934	−	0.748678i	$-0.269311\pi$
$373$	9551.32	1.32587	0.662934	+	0.748678i	$0.269311\pi$
$374$	−6260.29	−0.865539
$375$	0	0
$376$	6143.93	0.842683
$377$	3560.98	0.486472
$378$	0	0
$379$	−336.421	−0.0455957	−0.0227978	−	0.999740i	$-0.507257\pi$
$379$	−336.421	−0.0455957	−0.0227978	+	0.999740i	$0.507257\pi$
$380$	0	0
$381$	0	0
$382$	11230.4	1.50418
$383$	751.233	0.100225	0.0501125	−	0.998744i	$-0.484042\pi$
$383$	751.233	0.100225	0.0501125	+	0.998744i	$0.484042\pi$
$384$	0	0
$385$	0	0
$386$	8953.03	1.18056
$387$	0	0
$388$	−15222.0	−1.99170
$389$	−2801.12	−0.365096	−0.182548	−	0.983197i	$-0.558434\pi$
$389$	−2801.12	−0.365096	−0.182548	+	0.983197i	$0.558434\pi$
$390$	0	0
$391$	−2017.76	−0.260978
$392$	−3543.60	−0.456579
$393$	0	0
$394$	−10216.9	−1.30640
$395$	0	0
$396$	0	0
$397$	−4652.61	−0.588181	−0.294090	−	0.955778i	$-0.595017\pi$
$397$	−4652.61	−0.588181	−0.294090	+	0.955778i	$0.595017\pi$
$398$	−10471.1	−1.31877
$399$	0	0
$400$	0	0
$401$	−8362.58	−1.04142	−0.520708	−	0.853735i	$-0.674332\pi$
$401$	−8362.58	−1.04142	−0.520708	+	0.853735i	$0.674332\pi$
$402$	0	0
$403$	−13687.8	−1.69190
$404$	10493.5	1.29226
$405$	0	0
$406$	−1726.98	−0.211105
$407$	−2659.49	−0.323897
$408$	0	0
$409$	1873.28	0.226473	0.113237	−	0.993568i	$-0.463878\pi$
$409$	1873.28	0.226473	0.113237	+	0.993568i	$0.463878\pi$
$410$	0	0
$411$	0	0
$412$	1888.62	0.225839
$413$	5101.47	0.607814
$414$	0	0
$415$	0	0
$416$	14592.2	1.71981
$417$	0	0
$418$	−2481.28	−0.290343
$419$	14766.7	1.72172	0.860862	−	0.508839i	$-0.169925\pi$
$419$	14766.7	1.72172	0.860862	+	0.508839i	$0.169925\pi$
$420$	0	0
$421$	15269.0	1.76762	0.883810	−	0.467846i	$-0.154970\pi$
$421$	15269.0	1.76762	0.883810	+	0.467846i	$0.154970\pi$
$422$	5817.01	0.671013
$423$	0	0
$424$	−3635.28	−0.416379
$425$	0	0
$426$	0	0
$427$	1800.24	0.204027
$428$	2412.03	0.272406
$429$	0	0
$430$	0	0
$431$	−2262.39	−0.252844	−0.126422	−	0.991977i	$-0.540349\pi$
$431$	−2262.39	−0.252844	−0.126422	+	0.991977i	$0.540349\pi$
$432$	0	0
$433$	−13733.0	−1.52417	−0.762085	−	0.647477i	$-0.775824\pi$
$433$	−13733.0	−1.52417	−0.762085	+	0.647477i	$0.775824\pi$
$434$	6638.20	0.734202
$435$	0	0
$436$	18971.9	2.08392
$437$	−799.745	−0.0875446
$438$	0	0
$439$	−13290.9	−1.44496	−0.722480	−	0.691391i	$-0.756998\pi$
$439$	−13290.9	−1.44496	−0.722480	+	0.691391i	$0.756998\pi$
$440$	0	0
$441$	0	0
$442$	−33967.6	−3.65537
$443$	4855.33	0.520731	0.260365	−	0.965510i	$-0.416157\pi$
$443$	4855.33	0.520731	0.260365	+	0.965510i	$0.416157\pi$
$444$	0	0
$445$	0	0
$446$	−2180.91	−0.231545
$447$	0	0
$448$	−5230.45	−0.551597
$449$	−7615.60	−0.800451	−0.400226	−	0.916417i	$-0.631068\pi$
$449$	−7615.60	−0.800451	−0.400226	+	0.916417i	$0.631068\pi$
$450$	0	0
$451$	2164.13	0.225954
$452$	1006.87	0.104776
$453$	0	0
$454$	−72.3111	−0.00747518
$455$	0	0
$456$	0	0
$457$	−8573.64	−0.877588	−0.438794	−	0.898588i	$-0.644594\pi$
$457$	−8573.64	−0.877588	−0.438794	+	0.898588i	$0.644594\pi$
$458$	10776.2	1.09943
$459$	0	0
$460$	0	0
$461$	152.816	0.0154390	0.00771949	−	0.999970i	$-0.497543\pi$
$461$	152.816	0.0154390	0.00771949	+	0.999970i	$0.497543\pi$
$462$	0	0
$463$	8833.20	0.886639	0.443319	−	0.896364i	$-0.353801\pi$
$463$	8833.20	0.886639	0.443319	+	0.896364i	$0.353801\pi$
$464$	2059.60	0.206065
$465$	0	0
$466$	18673.5	1.85629
$467$	9924.92	0.983449	0.491724	−	0.870751i	$-0.336367\pi$
$467$	9924.92	0.983449	0.491724	+	0.870751i	$0.336367\pi$
$468$	0	0
$469$	5990.26	0.589775
$470$	0	0
$471$	0	0
$472$	9064.32	0.883939
$473$	−2151.19	−0.209116
$474$	0	0
$475$	0	0
$476$	9442.43	0.909229
$477$	0	0
$478$	−8372.91	−0.801189
$479$	−532.093	−0.0507557	−0.0253778	−	0.999678i	$-0.508079\pi$
$479$	−532.093	−0.0507557	−0.0253778	+	0.999678i	$0.508079\pi$
$480$	0	0
$481$	−14430.1	−1.36789
$482$	584.461	0.0552313
$483$	0	0
$484$	1300.01	0.122089
$485$	0	0
$486$	0	0
$487$	−12804.3	−1.19142	−0.595708	−	0.803201i	$-0.703128\pi$
$487$	−12804.3	−1.19142	−0.595708	+	0.803201i	$0.703128\pi$
$488$	3198.67	0.296715
$489$	0	0
$490$	0	0
$491$	12265.0	1.12731	0.563656	−	0.826009i	$-0.309394\pi$
$491$	12265.0	1.12731	0.563656	+	0.826009i	$0.309394\pi$
$492$	0	0
$493$	−7842.93	−0.716487
$494$	−13463.2	−1.22619
$495$	0	0
$496$	−7916.71	−0.716675
$497$	5718.31	0.516099
$498$	0	0
$499$	−13532.6	−1.21404	−0.607019	−	0.794688i	$-0.707635\pi$
$499$	−13532.6	−1.21404	−0.607019	+	0.794688i	$0.707635\pi$
$500$	0	0
$501$	0	0
$502$	−3814.02	−0.339100
$503$	10972.9	0.972679	0.486340	−	0.873770i	$-0.338332\pi$
$503$	10972.9	0.972679	0.486340	+	0.873770i	$0.338332\pi$
$504$	0	0
$505$	0	0
$506$	731.004	0.0642235
$507$	0	0
$508$	−2583.98	−0.225680
$509$	−4351.72	−0.378952	−0.189476	−	0.981885i	$-0.560679\pi$
$509$	−4351.72	−0.378952	−0.189476	+	0.981885i	$0.560679\pi$
$510$	0	0
$511$	−2618.79	−0.226710
$512$	11720.4	1.01167
$513$	0	0
$514$	21454.4	1.84108
$515$	0	0
$516$	0	0
$517$	−5689.16	−0.483963
$518$	6998.20	0.593597
$519$	0	0
$520$	0	0
$521$	10936.2	0.919626	0.459813	−	0.888016i	$-0.347916\pi$
$521$	10936.2	0.919626	0.459813	+	0.888016i	$0.347916\pi$
$522$	0	0
$523$	−21005.4	−1.75622	−0.878110	−	0.478458i	$-0.841196\pi$
$523$	−21005.4	−1.75622	−0.878110	+	0.478458i	$0.841196\pi$
$524$	26339.9	2.19593
$525$	0	0
$526$	−133.051	−0.0110291
$527$	30146.8	2.49187
$528$	0	0
$529$	−11931.4	−0.980635
$530$	0	0
$531$	0	0
$532$	3742.54	0.304999
$533$	11742.3	0.954253
$534$	0	0
$535$	0	0
$536$	10643.5	0.857706
$537$	0	0
$538$	−8371.44	−0.670852
$539$	3281.31	0.262219
$540$	0	0
$541$	5830.62	0.463361	0.231680	−	0.972792i	$-0.425578\pi$
$541$	5830.62	0.463361	0.231680	+	0.972792i	$0.425578\pi$
$542$	−11086.3	−0.878591
$543$	0	0
$544$	−32138.7	−2.53297
$545$	0	0
$546$	0	0
$547$	17033.9	1.33148	0.665738	−	0.746186i	$-0.268117\pi$
$547$	17033.9	1.33148	0.665738	+	0.746186i	$0.268117\pi$
$548$	−18761.5	−1.46251
$549$	0	0
$550$	0	0
$551$	−3108.57	−0.240344
$552$	0	0
$553$	5547.49	0.426588
$554$	−26548.6	−2.03599
$555$	0	0
$556$	17014.6	1.29781
$557$	−2472.67	−0.188098	−0.0940489	−	0.995568i	$-0.529981\pi$
$557$	−2472.67	−0.188098	−0.0940489	+	0.995568i	$0.529981\pi$
$558$	0	0
$559$	−11672.1	−0.883144
$560$	0	0
$561$	0	0
$562$	27629.7	2.07382
$563$	−17394.4	−1.30210	−0.651052	−	0.759033i	$-0.725672\pi$
$563$	−17394.4	−1.30210	−0.651052	+	0.759033i	$0.725672\pi$
$564$	0	0
$565$	0	0
$566$	−34485.1	−2.56098
$567$	0	0
$568$	10160.3	0.750559
$569$	−7210.07	−0.531216	−0.265608	−	0.964081i	$-0.585573\pi$
$569$	−7210.07	−0.531216	−0.265608	+	0.964081i	$0.585573\pi$
$570$	0	0
$571$	6428.19	0.471123	0.235562	−	0.971859i	$-0.424307\pi$
$571$	6428.19	0.471123	0.235562	+	0.971859i	$0.424307\pi$
$572$	7053.68	0.515611
$573$	0	0
$574$	−5694.72	−0.414099
$575$	0	0
$576$	0	0
$577$	−24629.9	−1.77704	−0.888522	−	0.458834i	$-0.848267\pi$
$577$	−24629.9	−1.77704	−0.888522	+	0.458834i	$0.848267\pi$
$578$	53542.0	3.85303
$579$	0	0
$580$	0	0
$581$	−6281.85	−0.448563
$582$	0	0
$583$	3366.19	0.239131
$584$	−4653.09	−0.329702
$585$	0	0
$586$	−15751.5	−1.11039
$587$	23437.4	1.64798	0.823991	−	0.566603i	$-0.191743\pi$
$587$	23437.4	1.64798	0.823991	+	0.566603i	$0.191743\pi$
$588$	0	0
$589$	11948.8	0.835893
$590$	0	0
$591$	0	0
$592$	−8346.05	−0.579427
$593$	−20520.6	−1.42105	−0.710523	−	0.703674i	$-0.751541\pi$
$593$	−20520.6	−1.42105	−0.710523	+	0.703674i	$0.751541\pi$
$594$	0	0
$595$	0	0
$596$	27364.4	1.88069
$597$	0	0
$598$	3966.34	0.271230
$599$	12162.6	0.829631	0.414816	−	0.909905i	$-0.363846\pi$
$599$	12162.6	0.829631	0.414816	+	0.909905i	$0.363846\pi$
$600$	0	0
$601$	−14899.2	−1.01123	−0.505617	−	0.862758i	$-0.668735\pi$
$601$	−14899.2	−1.01123	−0.505617	+	0.862758i	$0.668735\pi$
$602$	5660.66	0.383241
$603$	0	0
$604$	−6944.90	−0.467854
$605$	0	0
$606$	0	0
$607$	−5553.94	−0.371380	−0.185690	−	0.982608i	$-0.559452\pi$
$607$	−5553.94	−0.371380	−0.185690	+	0.982608i	$0.559452\pi$
$608$	−12738.3	−0.849681
$609$	0	0
$610$	0	0
$611$	−30868.7	−2.04388
$612$	0	0
$613$	−4810.63	−0.316965	−0.158482	−	0.987362i	$-0.550660\pi$
$613$	−4810.63	−0.316965	−0.158482	+	0.987362i	$0.550660\pi$
$614$	1334.27	0.0876985
$615$	0	0
$616$	−873.645	−0.0571431
$617$	−13948.4	−0.910114	−0.455057	−	0.890462i	$-0.650381\pi$
$617$	−13948.4	−0.910114	−0.455057	+	0.890462i	$0.650381\pi$
$618$	0	0
$619$	10652.1	0.691670	0.345835	−	0.938295i	$-0.387596\pi$
$619$	10652.1	0.691670	0.345835	+	0.938295i	$0.387596\pi$
$620$	0	0
$621$	0	0
$622$	28063.0	1.80904
$623$	−2757.17	−0.177309
$624$	0	0
$625$	0	0
$626$	25866.3	1.65148
$627$	0	0
$628$	−18964.0	−1.20501
$629$	31781.7	2.01466
$630$	0	0
$631$	−28298.9	−1.78536	−0.892680	−	0.450691i	$-0.851178\pi$
$631$	−28298.9	−1.78536	−0.892680	+	0.450691i	$0.851178\pi$
$632$	9856.80	0.620384
$633$	0	0
$634$	−33382.3	−2.09114
$635$	0	0
$636$	0	0
$637$	17804.0	1.10741
$638$	2841.38	0.176318
$639$	0	0
$640$	0	0
$641$	19838.5	1.22242	0.611211	−	0.791468i	$-0.290683\pi$
$641$	19838.5	1.22242	0.611211	+	0.791468i	$0.290683\pi$
$642$	0	0
$643$	23129.0	1.41854	0.709269	−	0.704938i	$-0.249025\pi$
$643$	23129.0	1.41854	0.709269	+	0.704938i	$0.249025\pi$
$644$	−1102.58	−0.0674653
$645$	0	0
$646$	29652.1	1.80596
$647$	3325.56	0.202073	0.101036	−	0.994883i	$-0.467784\pi$
$647$	3325.56	0.202073	0.101036	+	0.994883i	$0.467784\pi$
$648$	0	0
$649$	−8393.38	−0.507656
$650$	0	0
$651$	0	0
$652$	26022.7	1.56308
$653$	−22755.6	−1.36370	−0.681849	−	0.731493i	$-0.738824\pi$
$653$	−22755.6	−1.36370	−0.681849	+	0.731493i	$0.738824\pi$
$654$	0	0
$655$	0	0
$656$	6791.52	0.404214
$657$	0	0
$658$	14970.5	0.886946
$659$	−2456.94	−0.145233	−0.0726167	−	0.997360i	$-0.523135\pi$
$659$	−2456.94	−0.145233	−0.0726167	+	0.997360i	$0.523135\pi$
$660$	0	0
$661$	5689.88	0.334812	0.167406	−	0.985888i	$-0.446461\pi$
$661$	5689.88	0.334812	0.167406	+	0.985888i	$0.446461\pi$
$662$	−19461.1	−1.14256
$663$	0	0
$664$	−11161.6	−0.652341
$665$	0	0
$666$	0	0
$667$	915.807	0.0531637
$668$	861.858	0.0499196
$669$	0	0
$670$	0	0
$671$	−2961.91	−0.170407
$672$	0	0
$673$	−32352.3	−1.85303	−0.926515	−	0.376259i	$-0.877210\pi$
$673$	−32352.3	−1.85303	−0.926515	+	0.376259i	$0.877210\pi$
$674$	28005.3	1.60048
$675$	0	0
$676$	14668.2	0.834558
$677$	3956.14	0.224589	0.112295	−	0.993675i	$-0.464180\pi$
$677$	3956.14	0.224589	0.112295	+	0.993675i	$0.464180\pi$
$678$	0	0
$679$	−9472.43	−0.535374
$680$	0	0
$681$	0	0
$682$	−10921.7	−0.613218
$683$	−22638.3	−1.26827	−0.634136	−	0.773221i	$-0.718644\pi$
$683$	−22638.3	−1.26827	−0.634136	+	0.773221i	$0.718644\pi$
$684$	0	0
$685$	0	0
$686$	−18562.8	−1.03313
$687$	0	0
$688$	−6750.90	−0.374093
$689$	18264.6	1.00990
$690$	0	0
$691$	−9161.12	−0.504349	−0.252175	−	0.967682i	$-0.581146\pi$
$691$	−9161.12	−0.504349	−0.252175	+	0.967682i	$0.581146\pi$
$692$	−29634.5	−1.62794
$693$	0	0
$694$	6917.70	0.378375
$695$	0	0
$696$	0	0
$697$	−25862.1	−1.40545
$698$	26765.9	1.45144
$699$	0	0
$700$	0	0
$701$	−4396.09	−0.236859	−0.118429	−	0.992962i	$-0.537786\pi$
$701$	−4396.09	−0.236859	−0.118429	+	0.992962i	$0.537786\pi$
$702$	0	0
$703$	12596.8	0.675813
$704$	8605.59	0.460703
$705$	0	0
$706$	35539.9	1.89457
$707$	6530.00	0.347363
$708$	0	0
$709$	−35261.5	−1.86780	−0.933902	−	0.357529i	$-0.883619\pi$
$709$	−35261.5	−1.86780	−0.933902	+	0.357529i	$0.883619\pi$
$710$	0	0
$711$	0	0
$712$	−4898.96	−0.257860
$713$	−3520.19	−0.184898
$714$	0	0
$715$	0	0
$716$	−22548.8	−1.17694
$717$	0	0
$718$	−1015.85	−0.0528011
$719$	−28946.3	−1.50141	−0.750705	−	0.660638i	$-0.770286\pi$
$719$	−28946.3	−1.50141	−0.750705	+	0.660638i	$0.770286\pi$
$720$	0	0
$721$	1175.27	0.0607063
$722$	−17942.8	−0.924876
$723$	0	0
$724$	−41235.0	−2.11670
$725$	0	0
$726$	0	0
$727$	17366.2	0.885937	0.442969	−	0.896537i	$-0.353925\pi$
$727$	17366.2	0.885937	0.442969	+	0.896537i	$0.353925\pi$
$728$	−4740.30	−0.241328
$729$	0	0
$730$	0	0
$731$	25707.4	1.30072
$732$	0	0
$733$	−10063.4	−0.507095	−0.253548	−	0.967323i	$-0.581597\pi$
$733$	−10063.4	−0.507095	−0.253548	+	0.967323i	$0.581597\pi$
$734$	−37264.3	−1.87391
$735$	0	0
$736$	3752.79	0.187948
$737$	−9855.69	−0.492590
$738$	0	0
$739$	−29500.5	−1.46846	−0.734231	−	0.678900i	$-0.762457\pi$
$739$	−29500.5	−1.46846	−0.734231	+	0.678900i	$0.762457\pi$
$740$	0	0
$741$	0	0
$742$	−8857.83	−0.438249
$743$	−4857.99	−0.239868	−0.119934	−	0.992782i	$-0.538268\pi$
$743$	−4857.99	−0.239868	−0.119934	+	0.992782i	$0.538268\pi$
$744$	0	0
$745$	0	0
$746$	41351.7	2.02948
$747$	0	0
$748$	−15535.5	−0.759404
$749$	1500.98	0.0732236
$750$	0	0
$751$	2625.57	0.127574	0.0637871	−	0.997964i	$-0.479682\pi$
$751$	2625.57	0.127574	0.0637871	+	0.997964i	$0.479682\pi$
$752$	−17853.8	−0.865773
$753$	0	0
$754$	15417.0	0.744632
$755$	0	0
$756$	0	0
$757$	−29155.3	−1.39982	−0.699912	−	0.714229i	$-0.746778\pi$
$757$	−29155.3	−1.39982	−0.699912	+	0.714229i	$0.746778\pi$
$758$	−1456.51	−0.0697924
$759$	0	0
$760$	0	0
$761$	−10791.6	−0.514053	−0.257026	−	0.966404i	$-0.582743\pi$
$761$	−10791.6	−0.514053	−0.257026	+	0.966404i	$0.582743\pi$
$762$	0	0
$763$	11806.0	0.560163
$764$	27869.3	1.31973
$765$	0	0
$766$	3252.40	0.153413
$767$	−45541.5	−2.14395
$768$	0	0
$769$	35135.2	1.64760	0.823801	−	0.566879i	$-0.191849\pi$
$769$	35135.2	1.64760	0.823801	+	0.566879i	$0.191849\pi$
$770$	0	0
$771$	0	0
$772$	22217.8	1.03580
$773$	19630.6	0.913407	0.456704	−	0.889619i	$-0.349030\pi$
$773$	19630.6	0.913407	0.456704	+	0.889619i	$0.349030\pi$
$774$	0	0
$775$	0	0
$776$	−16830.7	−0.778590
$777$	0	0
$778$	−12127.2	−0.558845
$779$	−10250.5	−0.471454
$780$	0	0
$781$	−9408.25	−0.431055
$782$	−8735.72	−0.399474
$783$	0	0
$784$	10297.4	0.469089
$785$	0	0
$786$	0	0
$787$	23875.3	1.08140	0.540701	−	0.841215i	$-0.318159\pi$
$787$	23875.3	1.08140	0.540701	+	0.841215i	$0.318159\pi$
$788$	−25354.2	−1.14620
$789$	0	0
$790$	0	0
$791$	626.560	0.0281642
$792$	0	0
$793$	−16071.0	−0.719668
$794$	−20143.1	−0.900317
$795$	0	0
$796$	−25985.0	−1.15705
$797$	18553.7	0.824601	0.412301	−	0.911048i	$-0.364725\pi$
$797$	18553.7	0.824601	0.412301	+	0.911048i	$0.364725\pi$
$798$	0	0
$799$	67987.2	3.01028
$800$	0	0
$801$	0	0
$802$	−36205.1	−1.59407
$803$	4308.66	0.189352
$804$	0	0
$805$	0	0
$806$	−59260.0	−2.58976
$807$	0	0
$808$	11602.5	0.505168
$809$	12379.1	0.537980	0.268990	−	0.963143i	$-0.413310\pi$
$809$	12379.1	0.537980	0.268990	+	0.963143i	$0.413310\pi$
$810$	0	0
$811$	−43457.0	−1.88160	−0.940802	−	0.338958i	$-0.889926\pi$
$811$	−43457.0	−1.88160	−0.940802	+	0.338958i	$0.889926\pi$
$812$	−4285.66	−0.185218
$813$	0	0
$814$	−11514.0	−0.495782
$815$	0	0
$816$	0	0
$817$	10189.2	0.436322
$818$	8110.19	0.346658
$819$	0	0
$820$	0	0
$821$	11458.6	0.487097	0.243549	−	0.969889i	$-0.421688\pi$
$821$	11458.6	0.487097	0.243549	+	0.969889i	$0.421688\pi$
$822$	0	0
$823$	−3172.57	−0.134373	−0.0671865	−	0.997740i	$-0.521402\pi$
$823$	−3172.57	−0.134373	−0.0671865	+	0.997740i	$0.521402\pi$
$824$	2088.22	0.0882847
$825$	0	0
$826$	22086.4	0.930368
$827$	−13979.5	−0.587804	−0.293902	−	0.955836i	$-0.594954\pi$
$827$	−13979.5	−0.587804	−0.293902	+	0.955836i	$0.594954\pi$
$828$	0	0
$829$	−4477.01	−0.187567	−0.0937835	−	0.995593i	$-0.529896\pi$
$829$	−4477.01	−0.187567	−0.0937835	+	0.995593i	$0.529896\pi$
$830$	0	0
$831$	0	0
$832$	46692.9	1.94565
$833$	−39212.6	−1.63102
$834$	0	0
$835$	0	0
$836$	−6157.54	−0.254740
$837$	0	0
$838$	63931.4	2.63541
$839$	42031.0	1.72952	0.864761	−	0.502183i	$-0.167470\pi$
$839$	42031.0	1.72952	0.864761	+	0.502183i	$0.167470\pi$
$840$	0	0
$841$	−20829.3	−0.854045
$842$	66106.1	2.70566
$843$	0	0
$844$	14435.5	0.588731
$845$	0	0
$846$	0	0
$847$	808.978	0.0328180
$848$	10563.8	0.427788
$849$	0	0
$850$	0	0
$851$	−3711.10	−0.149489
$852$	0	0
$853$	40240.4	1.61524	0.807622	−	0.589700i	$-0.200754\pi$
$853$	40240.4	1.61524	0.807622	+	0.589700i	$0.200754\pi$
$854$	7793.99	0.312301
$855$	0	0
$856$	2666.94	0.106489
$857$	−7571.90	−0.301810	−0.150905	−	0.988548i	$-0.548219\pi$
$857$	−7571.90	−0.301810	−0.150905	+	0.988548i	$0.548219\pi$
$858$	0	0
$859$	8021.91	0.318631	0.159315	−	0.987228i	$-0.449071\pi$
$859$	8021.91	0.318631	0.159315	+	0.987228i	$0.449071\pi$
$860$	0	0
$861$	0	0
$862$	−9794.85	−0.387023
$863$	43916.3	1.73225	0.866124	−	0.499830i	$-0.166604\pi$
$863$	43916.3	1.73225	0.866124	+	0.499830i	$0.166604\pi$
$864$	0	0
$865$	0	0
$866$	−59455.9	−2.33302
$867$	0	0
$868$	16473.3	0.644171
$869$	−9127.20	−0.356294
$870$	0	0
$871$	−53475.8	−2.08032
$872$	20976.9	0.814641
$873$	0	0
$874$	−3462.43	−0.134003
$875$	0	0
$876$	0	0
$877$	26610.5	1.02460	0.512298	−	0.858808i	$-0.328794\pi$
$877$	26610.5	1.02460	0.512298	+	0.858808i	$0.328794\pi$
$878$	−57541.6	−2.21177
$879$	0	0
$880$	0	0
$881$	25186.7	0.963178	0.481589	−	0.876397i	$-0.340060\pi$
$881$	25186.7	0.963178	0.481589	+	0.876397i	$0.340060\pi$
$882$	0	0
$883$	−17849.8	−0.680288	−0.340144	−	0.940373i	$-0.610476\pi$
$883$	−17849.8	−0.680288	−0.340144	+	0.940373i	$0.610476\pi$
$884$	−84293.7	−3.20713
$885$	0	0
$886$	21020.8	0.797072
$887$	−20809.1	−0.787714	−0.393857	−	0.919172i	$-0.628859\pi$
$887$	−20809.1	−0.787714	−0.393857	+	0.919172i	$0.628859\pi$
$888$	0	0
$889$	−1607.98	−0.0606635
$890$	0	0
$891$	0	0
$892$	−5412.14	−0.203152
$893$	26946.9	1.00979
$894$	0	0
$895$	0	0
$896$	−9568.10	−0.356750
$897$	0	0
$898$	−32971.1	−1.22523
$899$	−13682.8	−0.507617
$900$	0	0
$901$	−40227.1	−1.48741
$902$	9369.44	0.345863
$903$	0	0
$904$	1113.27	0.0409590
$905$	0	0
$906$	0	0
$907$	−47377.1	−1.73443	−0.867217	−	0.497931i	$-0.834093\pi$
$907$	−47377.1	−1.73443	−0.867217	+	0.497931i	$0.834093\pi$
$908$	−179.447	−0.00655854
$909$	0	0
$910$	0	0
$911$	846.081	0.0307705	0.0153852	−	0.999882i	$-0.495103\pi$
$911$	846.081	0.0307705	0.0153852	+	0.999882i	$0.495103\pi$
$912$	0	0
$913$	10335.4	0.374647
$914$	−37118.9	−1.34331
$915$	0	0
$916$	26742.1	0.964611
$917$	16391.0	0.590271
$918$	0	0
$919$	−7957.57	−0.285632	−0.142816	−	0.989749i	$-0.545616\pi$
$919$	−7957.57	−0.285632	−0.142816	+	0.989749i	$0.545616\pi$
$920$	0	0
$921$	0	0
$922$	661.606	0.0236321
$923$	−51048.1	−1.82044
$924$	0	0
$925$	0	0
$926$	38242.6	1.35716
$927$	0	0
$928$	14586.9	0.515990
$929$	1869.40	0.0660207	0.0330103	−	0.999455i	$-0.489491\pi$
$929$	1869.40	0.0660207	0.0330103	+	0.999455i	$0.489491\pi$
$930$	0	0
$931$	−15542.0	−0.547121
$932$	46340.0	1.62866
$933$	0	0
$934$	42969.1	1.50535
$935$	0	0
$936$	0	0
$937$	−47095.1	−1.64197	−0.820987	−	0.570946i	$-0.806576\pi$
$937$	−47095.1	−1.64197	−0.820987	+	0.570946i	$0.806576\pi$
$938$	25934.3	0.902757
$939$	0	0
$940$	0	0
$941$	−42716.5	−1.47983	−0.739913	−	0.672702i	$-0.765134\pi$
$941$	−42716.5	−1.47983	−0.739913	+	0.672702i	$0.765134\pi$
$942$	0	0
$943$	3019.87	0.104285
$944$	−26340.2	−0.908159
$945$	0	0
$946$	−9313.40	−0.320090
$947$	56287.1	1.93145	0.965726	−	0.259565i	$-0.0835792\pi$
$947$	56287.1	1.93145	0.965726	+	0.259565i	$0.0835792\pi$
$948$	0	0
$949$	23378.3	0.799675
$950$	0	0
$951$	0	0
$952$	10440.3	0.355434
$953$	−1674.14	−0.0569054	−0.0284527	−	0.999595i	$-0.509058\pi$
$953$	−1674.14	−0.0569054	−0.0284527	+	0.999595i	$0.509058\pi$
$954$	0	0
$955$	0	0
$956$	−20778.2	−0.702944
$957$	0	0
$958$	−2303.65	−0.0776907
$959$	−11675.1	−0.393126
$960$	0	0
$961$	22803.3	0.765442
$962$	−62473.8	−2.09380
$963$	0	0
$964$	1450.40	0.0484586
$965$	0	0
$966$	0	0
$967$	−31471.5	−1.04659	−0.523296	−	0.852151i	$-0.675298\pi$
$967$	−31471.5	−1.04659	−0.523296	+	0.852151i	$0.675298\pi$
$968$	1437.40	0.0477269
$969$	0	0
$970$	0	0
$971$	10210.9	0.337471	0.168736	−	0.985661i	$-0.446032\pi$
$971$	10210.9	0.337471	0.168736	+	0.985661i	$0.446032\pi$
$972$	0	0
$973$	10588.0	0.348854
$974$	−55435.3	−1.82368
$975$	0	0
$976$	−9295.10	−0.304845
$977$	−39043.0	−1.27850	−0.639251	−	0.768998i	$-0.720756\pi$
$977$	−39043.0	−1.27850	−0.639251	+	0.768998i	$0.720756\pi$
$978$	0	0
$979$	4536.34	0.148092
$980$	0	0
$981$	0	0
$982$	53100.2	1.72556
$983$	12716.8	0.412618	0.206309	−	0.978487i	$-0.433855\pi$
$983$	12716.8	0.412618	0.206309	+	0.978487i	$0.433855\pi$
$984$	0	0
$985$	0	0
$986$	−33955.3	−1.09671
$987$	0	0
$988$	−33410.1	−1.07583
$989$	−3001.81	−0.0965138
$990$	0	0
$991$	−24150.0	−0.774118	−0.387059	−	0.922055i	$-0.626509\pi$
$991$	−24150.0	−0.774118	−0.387059	+	0.922055i	$0.626509\pi$
$992$	−56069.4	−1.79456
$993$	0	0
$994$	24757.0	0.789983
$995$	0	0
$996$	0	0
$997$	−7980.00	−0.253490	−0.126745	−	0.991935i	$-0.540453\pi$
$997$	−7980.00	−0.253490	−0.126745	+	0.991935i	$0.540453\pi$
$998$	−58588.5	−1.85830
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bt.1.6		7
3.2	odd	2		2475.4.a.bp.1.2		7
5.4	even	2		495.4.a.o.1.2	✓	7
15.14	odd	2		495.4.a.p.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.2	✓	7	5.4	even	2
495.4.a.p.1.6	yes	7	15.14	odd	2
2475.4.a.bp.1.2		7	3.2	odd	2
2475.4.a.bt.1.6		7	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+4.32942 q^{2} +10.7439 q^{4} +6.68577 q^{7} +11.8793 q^{8} +O(q^{10})\)	\(q+4.32942 q^{2} +10.7439 q^{4} +6.68577 q^{7} +11.8793 q^{8} -11.0000 q^{11} -59.6847 q^{13} +28.9455 q^{14} -34.5204 q^{16} +131.453 q^{17} +52.1020 q^{19} -47.6236 q^{22} -15.3496 q^{23} -258.400 q^{26} +71.8310 q^{28} -59.6632 q^{29} +229.334 q^{31} -244.488 q^{32} +569.117 q^{34} +241.772 q^{37} +225.571 q^{38} -196.739 q^{41} +195.563 q^{43} -118.182 q^{44} -66.4549 q^{46} +517.196 q^{47} -298.300 q^{49} -641.244 q^{52} -306.018 q^{53} +79.4223 q^{56} -258.307 q^{58} +763.034 q^{59} +269.264 q^{61} +992.885 q^{62} -782.326 q^{64} +895.972 q^{67} +1412.32 q^{68} +855.296 q^{71} -391.697 q^{73} +1046.73 q^{74} +559.776 q^{76} -73.5435 q^{77} +829.746 q^{79} -851.767 q^{82} -939.585 q^{83} +846.673 q^{86} -130.672 q^{88} -412.394 q^{89} -399.038 q^{91} -164.914 q^{92} +2239.16 q^{94} -1416.81 q^{97} -1291.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8}+O(q^{10}) \) 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8	\( 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8} - 77 q^{11} - 38 q^{13} - 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} - 55 q^{22} + 334 q^{23} + 372 q^{26} - 812 q^{28} + 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} - 466 q^{37} - 494 q^{38} + 258 q^{41} - 308 q^{43} - 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} - 512 q^{52} + 110 q^{53} - 20 q^{56} - 1362 q^{58} + 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} - 2268 q^{67} - 1186 q^{68} + 166 q^{71} - 200 q^{73} + 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1006 q^{82} + 370 q^{83} - 106 q^{86} - 495 q^{88} + 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} - 3698 q^{97} + 697 q^{98}+O(q^{100}) \) 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8 - 77 * q^11 - 38 * q^13 - 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 - 55 * q^22 + 334 * q^23 + 372 * q^26 - 812 * q^28 + 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 - 466 * q^37 - 494 * q^38 + 258 * q^41 - 308 * q^43 - 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 - 512 * q^52 + 110 * q^53 - 20 * q^56 - 1362 * q^58 + 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 - 2268 * q^67 - 1186 * q^68 + 166 * q^71 - 200 * q^73 + 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1006 * q^82 + 370 * q^83 - 106 * q^86 - 495 * q^88 + 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 - 3698 * q^97 + 697 * q^98

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