LMFDB - Embedded newform 2475.4.a.v.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.76300$ 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)

$f(q)$	$=$	$q+3.76300 q^{2} +6.16019 q^{4} +23.6321 q^{7} -6.92320 q^{8} +O(q^{10})$	$q+3.76300 q^{2} +6.16019 q^{4} +23.6321 q^{7} -6.92320 q^{8} -11.0000 q^{11} -7.39511 q^{13} +88.9277 q^{14} -75.3336 q^{16} +8.68828 q^{17} -69.7002 q^{19} -41.3930 q^{22} -10.1643 q^{23} -27.8278 q^{26} +145.578 q^{28} +73.2803 q^{29} -290.970 q^{31} -228.095 q^{32} +32.6940 q^{34} +105.618 q^{37} -262.282 q^{38} -40.5108 q^{41} +77.3133 q^{43} -67.7621 q^{44} -38.2485 q^{46} -472.981 q^{47} +215.477 q^{49} -45.5553 q^{52} +205.479 q^{53} -163.610 q^{56} +275.754 q^{58} -330.261 q^{59} -931.991 q^{61} -1094.92 q^{62} -255.653 q^{64} +418.252 q^{67} +53.5215 q^{68} +506.329 q^{71} -612.497 q^{73} +397.441 q^{74} -429.367 q^{76} -259.953 q^{77} +54.6657 q^{79} -152.442 q^{82} +538.964 q^{83} +290.930 q^{86} +76.1552 q^{88} -781.086 q^{89} -174.762 q^{91} -62.6144 q^{92} -1779.83 q^{94} +531.762 q^{97} +810.839 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8	$ 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) $ 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8 - 33 * q^11 + 45 * q^13 + 116 * q^14 - 85 * q^16 + 58 * q^17 - 169 * q^19 + 11 * q^22 - 155 * q^23 - 167 * q^26 + 100 * q^28 + 277 * q^29 - 173 * q^31 - 97 * q^32 - 146 * q^34 + 60 * q^37 - 169 * q^38 - 44 * q^41 - 109 * q^43 - 77 * q^44 + 425 * q^46 - 270 * q^47 - 427 * q^49 + 45 * q^52 + 148 * q^53 - 168 * q^56 - 783 * q^58 + 684 * q^59 - 1038 * q^61 - 953 * q^62 - 1129 * q^64 + 314 * q^67 + 366 * q^68 + 1459 * q^71 - 1170 * q^73 + 1764 * q^74 + 211 * q^76 - 176 * q^77 - 506 * q^79 - 1040 * q^82 + 347 * q^83 + 527 * q^86 - 33 * q^88 + 607 * q^89 - 398 * q^91 - 687 * q^92 - 1822 * q^94 + 1263 * q^97 + 2273 * 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 $

$2$	3.76300	1.33042	0.665211	−	0.746655i	$-0.268341\pi$
$2$	3.76300	1.33042	0.665211	+	0.746655i	$0.268341\pi$
$3$	0	0
$3$	0	0
$4$	6.16019	0.770024
$5$	0	0
$5$	0	0
$6$	0	0
$7$	23.6321	1.27601	0.638007	−	0.770031i	$-0.279759\pi$
$7$	23.6321	1.27601	0.638007	+	0.770031i	$0.279759\pi$
$8$	−6.92320	−0.305965
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−7.39511	−0.157772	−0.0788860	−	0.996884i	$-0.525136\pi$
$13$	−7.39511	−0.157772	−0.0788860	+	0.996884i	$0.525136\pi$
$14$	88.9277	1.69764
$15$	0	0
$16$	−75.3336	−1.17709
$17$	8.68828	0.123954	0.0619770	−	0.998078i	$-0.480259\pi$
$17$	8.68828	0.123954	0.0619770	+	0.998078i	$0.480259\pi$
$18$	0	0
$19$	−69.7002	−0.841597	−0.420798	−	0.907154i	$-0.638250\pi$
$19$	−69.7002	−0.841597	−0.420798	+	0.907154i	$0.638250\pi$
$20$	0	0
$21$	0	0
$22$	−41.3930	−0.401137
$23$	−10.1643	−0.0921484	−0.0460742	−	0.998938i	$-0.514671\pi$
$23$	−10.1643	−0.0921484	−0.0460742	+	0.998938i	$0.514671\pi$
$24$	0	0
$25$	0	0
$26$	−27.8278	−0.209903
$27$	0	0
$28$	145.578	0.982562
$29$	73.2803	0.469235	0.234617	−	0.972088i	$-0.424616\pi$
$29$	73.2803	0.469235	0.234617	+	0.972088i	$0.424616\pi$
$30$	0	0
$31$	−290.970	−1.68580	−0.842898	−	0.538073i	$-0.819153\pi$
$31$	−290.970	−1.68580	−0.842898	+	0.538073i	$0.819153\pi$
$32$	−228.095	−1.26006
$33$	0	0
$34$	32.6940	0.164911
$35$	0	0
$36$	0	0
$37$	105.618	0.469284	0.234642	−	0.972082i	$-0.424608\pi$
$37$	105.618	0.469284	0.234642	+	0.972082i	$0.424608\pi$
$38$	−262.282	−1.11968
$39$	0	0
$40$	0	0
$41$	−40.5108	−0.154310	−0.0771551	−	0.997019i	$-0.524584\pi$
$41$	−40.5108	−0.154310	−0.0771551	+	0.997019i	$0.524584\pi$
$42$	0	0
$43$	77.3133	0.274190	0.137095	−	0.990558i	$-0.456223\pi$
$43$	77.3133	0.274190	0.137095	+	0.990558i	$0.456223\pi$
$44$	−67.7621	−0.232171
$45$	0	0
$46$	−38.2485	−0.122596
$47$	−472.981	−1.46790	−0.733951	−	0.679202i	$-0.762326\pi$
$47$	−472.981	−1.46790	−0.733951	+	0.679202i	$0.762326\pi$
$48$	0	0
$49$	215.477	0.628212
$50$	0	0
$51$	0	0
$52$	−45.5553	−0.121488
$53$	205.479	0.532541	0.266271	−	0.963898i	$-0.414208\pi$
$53$	205.479	0.532541	0.266271	+	0.963898i	$0.414208\pi$
$54$	0	0
$55$	0	0
$56$	−163.610	−0.390416
$57$	0	0
$58$	275.754	0.624281
$59$	−330.261	−0.728751	−0.364376	−	0.931252i	$-0.618718\pi$
$59$	−330.261	−0.728751	−0.364376	+	0.931252i	$0.618718\pi$
$60$	0	0
$61$	−931.991	−1.95622	−0.978109	−	0.208094i	$-0.933274\pi$
$61$	−931.991	−1.95622	−0.978109	+	0.208094i	$0.933274\pi$
$62$	−1094.92	−2.24282
$63$	0	0
$64$	−255.653	−0.499323
$65$	0	0
$66$	0	0
$67$	418.252	0.762652	0.381326	−	0.924441i	$-0.375468\pi$
$67$	418.252	0.762652	0.381326	+	0.924441i	$0.375468\pi$
$68$	53.5215	0.0954475
$69$	0	0
$70$	0	0
$71$	506.329	0.846340	0.423170	−	0.906050i	$-0.360917\pi$
$71$	506.329	0.846340	0.423170	+	0.906050i	$0.360917\pi$
$72$	0	0
$73$	−612.497	−0.982018	−0.491009	−	0.871154i	$-0.663372\pi$
$73$	−612.497	−0.982018	−0.491009	+	0.871154i	$0.663372\pi$
$74$	397.441	0.624346
$75$	0	0
$76$	−429.367	−0.648050
$77$	−259.953	−0.384733
$78$	0	0
$79$	54.6657	0.0778528	0.0389264	−	0.999242i	$-0.487606\pi$
$79$	54.6657	0.0778528	0.0389264	+	0.999242i	$0.487606\pi$
$80$	0	0
$81$	0	0
$82$	−152.442	−0.205298
$83$	538.964	0.712759	0.356379	−	0.934341i	$-0.384011\pi$
$83$	538.964	0.712759	0.356379	+	0.934341i	$0.384011\pi$
$84$	0	0
$85$	0	0
$86$	290.930	0.364788
$87$	0	0
$88$	76.1552	0.0922519
$89$	−781.086	−0.930280	−0.465140	−	0.885237i	$-0.653996\pi$
$89$	−781.086	−0.930280	−0.465140	+	0.885237i	$0.653996\pi$
$90$	0	0
$91$	−174.762	−0.201319
$92$	−62.6144	−0.0709565
$93$	0	0
$94$	−1779.83	−1.95293
$95$	0	0
$96$	0	0
$97$	531.762	0.556621	0.278311	−	0.960491i	$-0.410226\pi$
$97$	531.762	0.556621	0.278311	+	0.960491i	$0.410226\pi$
$98$	810.839	0.835787
$99$	0	0
$100$	0	0
$101$	−1898.16	−1.87004	−0.935022	−	0.354591i	$-0.884620\pi$
$101$	−1898.16	−1.87004	−0.935022	+	0.354591i	$0.884620\pi$
$102$	0	0
$103$	−1308.66	−1.25190	−0.625952	−	0.779861i	$-0.715290\pi$
$103$	−1308.66	−1.25190	−0.625952	+	0.779861i	$0.715290\pi$
$104$	51.1978	0.0482727
$105$	0	0
$106$	773.218	0.708505
$107$	206.655	0.186711	0.0933554	−	0.995633i	$-0.470241\pi$
$107$	206.655	0.186711	0.0933554	+	0.995633i	$0.470241\pi$
$108$	0	0
$109$	595.701	0.523466	0.261733	−	0.965140i	$-0.415706\pi$
$109$	595.701	0.523466	0.261733	+	0.965140i	$0.415706\pi$
$110$	0	0
$111$	0	0
$112$	−1780.29	−1.50198
$113$	1513.00	1.25957	0.629785	−	0.776769i	$-0.283143\pi$
$113$	1513.00	1.25957	0.629785	+	0.776769i	$0.283143\pi$
$114$	0	0
$115$	0	0
$116$	451.421	0.361322
$117$	0	0
$118$	−1242.77	−0.969547
$119$	205.322	0.158167
$120$	0	0
$121$	121.000	0.0909091
$122$	−3507.09	−2.60260
$123$	0	0
$124$	−1792.43	−1.29810
$125$	0	0
$126$	0	0
$127$	−77.9297	−0.0544499	−0.0272250	−	0.999629i	$-0.508667\pi$
$127$	−77.9297	−0.0544499	−0.0272250	+	0.999629i	$0.508667\pi$
$128$	862.735	0.595748
$129$	0	0
$130$	0	0
$131$	8.60213	0.00573719	0.00286859	−	0.999996i	$-0.499087\pi$
$131$	8.60213	0.00573719	0.00286859	+	0.999996i	$0.499087\pi$
$132$	0	0
$133$	−1647.16	−1.07389
$134$	1573.89	1.01465
$135$	0	0
$136$	−60.1507	−0.0379256
$137$	−665.161	−0.414807	−0.207403	−	0.978255i	$-0.566501\pi$
$137$	−665.161	−0.414807	−0.207403	+	0.978255i	$0.566501\pi$
$138$	0	0
$139$	745.658	0.455006	0.227503	−	0.973777i	$-0.426944\pi$
$139$	745.658	0.455006	0.227503	+	0.973777i	$0.426944\pi$
$140$	0	0
$141$	0	0
$142$	1905.32	1.12599
$143$	81.3462	0.0475700
$144$	0	0
$145$	0	0
$146$	−2304.83	−1.30650
$147$	0	0
$148$	650.628	0.361360
$149$	2398.49	1.31874	0.659370	−	0.751819i	$-0.270823\pi$
$149$	2398.49	1.31874	0.659370	+	0.751819i	$0.270823\pi$
$150$	0	0
$151$	−3254.28	−1.75384	−0.876920	−	0.480637i	$-0.840405\pi$
$151$	−3254.28	−1.75384	−0.876920	+	0.480637i	$0.840405\pi$
$152$	482.549	0.257499
$153$	0	0
$154$	−978.205	−0.511857
$155$	0	0
$156$	0	0
$157$	−2492.15	−1.26685	−0.633424	−	0.773805i	$-0.718351\pi$
$157$	−2492.15	−1.26685	−0.633424	+	0.773805i	$0.718351\pi$
$158$	205.707	0.103577
$159$	0	0
$160$	0	0
$161$	−240.205	−0.117583
$162$	0	0
$163$	807.550	0.388050	0.194025	−	0.980997i	$-0.437846\pi$
$163$	807.550	0.388050	0.194025	+	0.980997i	$0.437846\pi$
$164$	−249.554	−0.118823
$165$	0	0
$166$	2028.12	0.948270
$167$	1833.28	0.849484	0.424742	−	0.905314i	$-0.360365\pi$
$167$	1833.28	0.849484	0.424742	+	0.905314i	$0.360365\pi$
$168$	0	0
$169$	−2142.31	−0.975108
$170$	0	0
$171$	0	0
$172$	476.265	0.211133
$173$	−1046.74	−0.460011	−0.230005	−	0.973189i	$-0.573874\pi$
$173$	−1046.74	−0.460011	−0.230005	+	0.973189i	$0.573874\pi$
$174$	0	0
$175$	0	0
$176$	828.669	0.354905
$177$	0	0
$178$	−2939.23	−1.23767
$179$	−1880.80	−0.785348	−0.392674	−	0.919678i	$-0.628450\pi$
$179$	−1880.80	−0.785348	−0.392674	+	0.919678i	$0.628450\pi$
$180$	0	0
$181$	−1921.82	−0.789216	−0.394608	−	0.918850i	$-0.629120\pi$
$181$	−1921.82	−0.789216	−0.394608	+	0.918850i	$0.629120\pi$
$182$	−657.630	−0.267840
$183$	0	0
$184$	70.3698	0.0281942
$185$	0	0
$186$	0	0
$187$	−95.5711	−0.0373735
$188$	−2913.66	−1.13032
$189$	0	0
$190$	0	0
$191$	−203.716	−0.0771747	−0.0385874	−	0.999255i	$-0.512286\pi$
$191$	−203.716	−0.0771747	−0.0385874	+	0.999255i	$0.512286\pi$
$192$	0	0
$193$	−2858.36	−1.06606	−0.533030	−	0.846096i	$-0.678947\pi$
$193$	−2858.36	−1.06606	−0.533030	+	0.846096i	$0.678947\pi$
$194$	2001.02	0.740542
$195$	0	0
$196$	1327.38	0.483738
$197$	−1140.57	−0.412499	−0.206250	−	0.978499i	$-0.566126\pi$
$197$	−1140.57	−0.412499	−0.206250	+	0.978499i	$0.566126\pi$
$198$	0	0
$199$	−3223.56	−1.14830	−0.574151	−	0.818749i	$-0.694668\pi$
$199$	−3223.56	−1.14830	−0.574151	+	0.818749i	$0.694668\pi$
$200$	0	0
$201$	0	0
$202$	−7142.80	−2.48795
$203$	1731.77	0.598750
$204$	0	0
$205$	0	0
$206$	−4924.50	−1.66556
$207$	0	0
$208$	557.100	0.185711
$209$	766.703	0.253751
$210$	0	0
$211$	1679.38	0.547930	0.273965	−	0.961740i	$-0.411665\pi$
$211$	1679.38	0.547930	0.273965	+	0.961740i	$0.411665\pi$
$212$	1265.79	0.410070
$213$	0	0
$214$	777.642	0.248404
$215$	0	0
$216$	0	0
$217$	−6876.23	−2.15110
$218$	2241.62	0.696431
$219$	0	0
$220$	0	0
$221$	−64.2508	−0.0195565
$222$	0	0
$223$	6121.75	1.83831	0.919154	−	0.393898i	$-0.128874\pi$
$223$	6121.75	1.83831	0.919154	+	0.393898i	$0.128874\pi$
$224$	−5390.36	−1.60785
$225$	0	0
$226$	5693.44	1.67576
$227$	−1760.23	−0.514671	−0.257335	−	0.966322i	$-0.582845\pi$
$227$	−1760.23	−0.514671	−0.257335	+	0.966322i	$0.582845\pi$
$228$	0	0
$229$	−5780.58	−1.66808	−0.834042	−	0.551700i	$-0.813979\pi$
$229$	−5780.58	−1.66808	−0.834042	+	0.551700i	$0.813979\pi$
$230$	0	0
$231$	0	0
$232$	−507.334	−0.143569
$233$	5433.94	1.52785	0.763925	−	0.645305i	$-0.223270\pi$
$233$	5433.94	1.52785	0.763925	+	0.645305i	$0.223270\pi$
$234$	0	0
$235$	0	0
$236$	−2034.47	−0.561156
$237$	0	0
$238$	772.629	0.210429
$239$	−583.827	−0.158011	−0.0790055	−	0.996874i	$-0.525174\pi$
$239$	−583.827	−0.158011	−0.0790055	+	0.996874i	$0.525174\pi$
$240$	0	0
$241$	4217.86	1.12737	0.563686	−	0.825989i	$-0.309383\pi$
$241$	4217.86	1.12737	0.563686	+	0.825989i	$0.309383\pi$
$242$	455.323	0.120948
$243$	0	0
$244$	−5741.25	−1.50633
$245$	0	0
$246$	0	0
$247$	515.441	0.132780
$248$	2014.44	0.515795
$249$	0	0
$250$	0	0
$251$	4836.75	1.21631	0.608153	−	0.793820i	$-0.291911\pi$
$251$	4836.75	1.21631	0.608153	+	0.793820i	$0.291911\pi$
$252$	0	0
$253$	111.808	0.0277838
$254$	−293.250	−0.0724414
$255$	0	0
$256$	5291.70	1.29192
$257$	−4654.14	−1.12964	−0.564819	−	0.825215i	$-0.691054\pi$
$257$	−4654.14	−1.12964	−0.564819	+	0.825215i	$0.691054\pi$
$258$	0	0
$259$	2495.98	0.598813
$260$	0	0
$261$	0	0
$262$	32.3698	0.00763288
$263$	5102.11	1.19624	0.598118	−	0.801408i	$-0.295916\pi$
$263$	5102.11	1.19624	0.598118	+	0.801408i	$0.295916\pi$
$264$	0	0
$265$	0	0
$266$	−6198.28	−1.42873
$267$	0	0
$268$	2576.52	0.587260
$269$	−4475.15	−1.01433	−0.507165	−	0.861849i	$-0.669306\pi$
$269$	−4475.15	−1.01433	−0.507165	+	0.861849i	$0.669306\pi$
$270$	0	0
$271$	−7829.40	−1.75499	−0.877495	−	0.479586i	$-0.840787\pi$
$271$	−7829.40	−1.75499	−0.877495	+	0.479586i	$0.840787\pi$
$272$	−654.519	−0.145905
$273$	0	0
$274$	−2503.00	−0.551868
$275$	0	0
$276$	0	0
$277$	−3692.51	−0.800943	−0.400472	−	0.916309i	$-0.631154\pi$
$277$	−3692.51	−0.800943	−0.400472	+	0.916309i	$0.631154\pi$
$278$	2805.91	0.605351
$279$	0	0
$280$	0	0
$281$	1296.21	0.275179	0.137589	−	0.990489i	$-0.456065\pi$
$281$	1296.21	0.275179	0.137589	+	0.990489i	$0.456065\pi$
$282$	0	0
$283$	−1168.50	−0.245442	−0.122721	−	0.992441i	$-0.539162\pi$
$283$	−1168.50	−0.245442	−0.122721	+	0.992441i	$0.539162\pi$
$284$	3119.08	0.651703
$285$	0	0
$286$	306.106	0.0632882
$287$	−957.355	−0.196902
$288$	0	0
$289$	−4837.51	−0.984635
$290$	0	0
$291$	0	0
$292$	−3773.10	−0.756178
$293$	−8879.43	−1.77045	−0.885224	−	0.465164i	$-0.845995\pi$
$293$	−8879.43	−1.77045	−0.885224	+	0.465164i	$0.845995\pi$
$294$	0	0
$295$	0	0
$296$	−731.215	−0.143584
$297$	0	0
$298$	9025.54	1.75448
$299$	75.1665	0.0145384
$300$	0	0
$301$	1827.08	0.349870
$302$	−12245.9	−2.33335
$303$	0	0
$304$	5250.77	0.990632
$305$	0	0
$306$	0	0
$307$	4724.14	0.878244	0.439122	−	0.898427i	$-0.355290\pi$
$307$	4724.14	0.878244	0.439122	+	0.898427i	$0.355290\pi$
$308$	−1601.36	−0.296253
$309$	0	0
$310$	0	0
$311$	−53.7156	−0.00979399	−0.00489700	−	0.999988i	$-0.501559\pi$
$311$	−53.7156	−0.00979399	−0.00489700	+	0.999988i	$0.501559\pi$
$312$	0	0
$313$	4663.77	0.842210	0.421105	−	0.907012i	$-0.361642\pi$
$313$	4663.77	0.842210	0.421105	+	0.907012i	$0.361642\pi$
$314$	−9377.96	−1.68544
$315$	0	0
$316$	336.751	0.0599486
$317$	8664.97	1.53525	0.767624	−	0.640901i	$-0.221439\pi$
$317$	8664.97	1.53525	0.767624	+	0.640901i	$0.221439\pi$
$318$	0	0
$319$	−806.084	−0.141480
$320$	0	0
$321$	0	0
$322$	−903.892	−0.156435
$323$	−605.575	−0.104319
$324$	0	0
$325$	0	0
$326$	3038.81	0.516271
$327$	0	0
$328$	280.464	0.0472135
$329$	−11177.5	−1.87306
$330$	0	0
$331$	−3422.54	−0.568337	−0.284169	−	0.958774i	$-0.591718\pi$
$331$	−3422.54	−0.568337	−0.284169	+	0.958774i	$0.591718\pi$
$332$	3320.12	0.548841
$333$	0	0
$334$	6898.66	1.13017
$335$	0	0
$336$	0	0
$337$	−118.863	−0.0192134	−0.00960668	−	0.999954i	$-0.503058\pi$
$337$	−118.863	−0.0192134	−0.00960668	+	0.999954i	$0.503058\pi$
$338$	−8061.53	−1.29731
$339$	0	0
$340$	0	0
$341$	3200.67	0.508287
$342$	0	0
$343$	−3013.65	−0.474407
$344$	−535.255	−0.0838925
$345$	0	0
$346$	−3938.87	−0.612009
$347$	−10434.7	−1.61431	−0.807154	−	0.590341i	$-0.798993\pi$
$347$	−10434.7	−1.61431	−0.807154	+	0.590341i	$0.798993\pi$
$348$	0	0
$349$	7528.91	1.15477	0.577383	−	0.816473i	$-0.304074\pi$
$349$	7528.91	1.15477	0.577383	+	0.816473i	$0.304074\pi$
$350$	0	0
$351$	0	0
$352$	2509.04	0.379922
$353$	−215.454	−0.0324857	−0.0162428	−	0.999868i	$-0.505170\pi$
$353$	−215.454	−0.0324857	−0.0162428	+	0.999868i	$0.505170\pi$
$354$	0	0
$355$	0	0
$356$	−4811.64	−0.716338
$357$	0	0
$358$	−7077.45	−1.04485
$359$	−9105.02	−1.33856	−0.669282	−	0.743008i	$-0.733398\pi$
$359$	−9105.02	−1.33856	−0.669282	+	0.743008i	$0.733398\pi$
$360$	0	0
$361$	−2000.88	−0.291715
$362$	−7231.83	−1.04999
$363$	0	0
$364$	−1076.57	−0.155021
$365$	0	0
$366$	0	0
$367$	−3763.06	−0.535232	−0.267616	−	0.963526i	$-0.586236\pi$
$367$	−3763.06	−0.535232	−0.267616	+	0.963526i	$0.586236\pi$
$368$	765.717	0.108467
$369$	0	0
$370$	0	0
$371$	4855.90	0.679530
$372$	0	0
$373$	−375.989	−0.0521929	−0.0260965	−	0.999659i	$-0.508308\pi$
$373$	−375.989	−0.0521929	−0.0260965	+	0.999659i	$0.508308\pi$
$374$	−359.634	−0.0497226
$375$	0	0
$376$	3274.54	0.449127
$377$	−541.916	−0.0740321
$378$	0	0
$379$	2967.48	0.402187	0.201094	−	0.979572i	$-0.435550\pi$
$379$	2967.48	0.402187	0.201094	+	0.979572i	$0.435550\pi$
$380$	0	0
$381$	0	0
$382$	−766.584	−0.102675
$383$	−1550.93	−0.206916	−0.103458	−	0.994634i	$-0.532991\pi$
$383$	−1550.93	−0.206916	−0.103458	+	0.994634i	$0.532991\pi$
$384$	0	0
$385$	0	0
$386$	−10756.0	−1.41831
$387$	0	0
$388$	3275.76	0.428612
$389$	7511.63	0.979061	0.489530	−	0.871986i	$-0.337168\pi$
$389$	7511.63	0.979061	0.489530	+	0.871986i	$0.337168\pi$
$390$	0	0
$391$	−88.3107	−0.0114222
$392$	−1491.79	−0.192211
$393$	0	0
$394$	−4291.97	−0.548798
$395$	0	0
$396$	0	0
$397$	−11125.1	−1.40643	−0.703217	−	0.710975i	$-0.748254\pi$
$397$	−11125.1	−1.40643	−0.703217	+	0.710975i	$0.748254\pi$
$398$	−12130.3	−1.52773
$399$	0	0
$400$	0	0
$401$	13478.0	1.67845	0.839225	−	0.543784i	$-0.183009\pi$
$401$	13478.0	1.67845	0.839225	+	0.543784i	$0.183009\pi$
$402$	0	0
$403$	2151.75	0.265971
$404$	−11693.1	−1.43998
$405$	0	0
$406$	6516.65	0.796591
$407$	−1161.80	−0.141494
$408$	0	0
$409$	12911.4	1.56095	0.780474	−	0.625188i	$-0.214978\pi$
$409$	12911.4	1.56095	0.780474	+	0.625188i	$0.214978\pi$
$410$	0	0
$411$	0	0
$412$	−8061.61	−0.963997
$413$	−7804.76	−0.929897
$414$	0	0
$415$	0	0
$416$	1686.79	0.198802
$417$	0	0
$418$	2885.10	0.337596
$419$	8906.41	1.03844	0.519220	−	0.854640i	$-0.326222\pi$
$419$	8906.41	1.03844	0.519220	+	0.854640i	$0.326222\pi$
$420$	0	0
$421$	1852.36	0.214439	0.107219	−	0.994235i	$-0.465805\pi$
$421$	1852.36	0.214439	0.107219	+	0.994235i	$0.465805\pi$
$422$	6319.51	0.728978
$423$	0	0
$424$	−1422.57	−0.162939
$425$	0	0
$426$	0	0
$427$	−22024.9	−2.49616
$428$	1273.03	0.143772
$429$	0	0
$430$	0	0
$431$	4090.74	0.457178	0.228589	−	0.973523i	$-0.426589\pi$
$431$	4090.74	0.457178	0.228589	+	0.973523i	$0.426589\pi$
$432$	0	0
$433$	−8253.94	−0.916071	−0.458036	−	0.888934i	$-0.651447\pi$
$433$	−8253.94	−0.916071	−0.458036	+	0.888934i	$0.651447\pi$
$434$	−25875.3	−2.86187
$435$	0	0
$436$	3669.63	0.403082
$437$	708.458	0.0775518
$438$	0	0
$439$	979.240	0.106461	0.0532307	−	0.998582i	$-0.483048\pi$
$439$	979.240	0.106461	0.0532307	+	0.998582i	$0.483048\pi$
$440$	0	0
$441$	0	0
$442$	−241.776	−0.0260183
$443$	89.5194	0.00960089	0.00480045	−	0.999988i	$-0.498472\pi$
$443$	89.5194	0.00960089	0.00480045	+	0.999988i	$0.498472\pi$
$444$	0	0
$445$	0	0
$446$	23036.2	2.44573
$447$	0	0
$448$	−6041.63	−0.637143
$449$	17119.1	1.79933	0.899666	−	0.436579i	$-0.143810\pi$
$449$	17119.1	1.79933	0.899666	+	0.436579i	$0.143810\pi$
$450$	0	0
$451$	445.619	0.0465263
$452$	9320.40	0.969899
$453$	0	0
$454$	−6623.74	−0.684730
$455$	0	0
$456$	0	0
$457$	1780.97	0.182298	0.0911492	−	0.995837i	$-0.470946\pi$
$457$	1780.97	0.182298	0.0911492	+	0.995837i	$0.470946\pi$
$458$	−21752.3	−2.21926
$459$	0	0
$460$	0	0
$461$	7216.14	0.729043	0.364522	−	0.931195i	$-0.381233\pi$
$461$	7216.14	0.729043	0.364522	+	0.931195i	$0.381233\pi$
$462$	0	0
$463$	−9451.28	−0.948679	−0.474339	−	0.880342i	$-0.657313\pi$
$463$	−9451.28	−0.948679	−0.474339	+	0.880342i	$0.657313\pi$
$464$	−5520.47	−0.552330
$465$	0	0
$466$	20447.9	2.03269
$467$	7188.38	0.712289	0.356144	−	0.934431i	$-0.384091\pi$
$467$	7188.38	0.712289	0.356144	+	0.934431i	$0.384091\pi$
$468$	0	0
$469$	9884.19	0.973154
$470$	0	0
$471$	0	0
$472$	2286.46	0.222972
$473$	−850.446	−0.0826714
$474$	0	0
$475$	0	0
$476$	1264.83	0.121792
$477$	0	0
$478$	−2196.94	−0.210221
$479$	8224.85	0.784558	0.392279	−	0.919846i	$-0.371687\pi$
$479$	8224.85	0.784558	0.392279	+	0.919846i	$0.371687\pi$
$480$	0	0
$481$	−781.058	−0.0740399
$482$	15871.8	1.49988
$483$	0	0
$484$	745.383	0.0700022
$485$	0	0
$486$	0	0
$487$	4817.49	0.448257	0.224129	−	0.974560i	$-0.428046\pi$
$487$	4817.49	0.448257	0.224129	+	0.974560i	$0.428046\pi$
$488$	6452.36	0.598534
$489$	0	0
$490$	0	0
$491$	4944.04	0.454422	0.227211	−	0.973846i	$-0.427039\pi$
$491$	4944.04	0.454422	0.227211	+	0.973846i	$0.427039\pi$
$492$	0	0
$493$	636.680	0.0581635
$494$	1939.61	0.176654
$495$	0	0
$496$	21919.8	1.98433
$497$	11965.6	1.07994
$498$	0	0
$499$	−3616.97	−0.324485	−0.162242	−	0.986751i	$-0.551873\pi$
$499$	−3616.97	−0.324485	−0.162242	+	0.986751i	$0.551873\pi$
$500$	0	0
$501$	0	0
$502$	18200.7	1.61820
$503$	−1176.71	−0.104308	−0.0521540	−	0.998639i	$-0.516609\pi$
$503$	−1176.71	−0.104308	−0.0521540	+	0.998639i	$0.516609\pi$
$504$	0	0
$505$	0	0
$506$	420.733	0.0369642
$507$	0	0
$508$	−480.062	−0.0419278
$509$	14606.6	1.27196	0.635981	−	0.771705i	$-0.280596\pi$
$509$	14606.6	1.27196	0.635981	+	0.771705i	$0.280596\pi$
$510$	0	0
$511$	−14474.6	−1.25307
$512$	13010.8	1.12305
$513$	0	0
$514$	−17513.5	−1.50290
$515$	0	0
$516$	0	0
$517$	5202.79	0.442589
$518$	9392.38	0.796674
$519$	0	0
$520$	0	0
$521$	−4175.52	−0.351119	−0.175559	−	0.984469i	$-0.556173\pi$
$521$	−4175.52	−0.351119	−0.175559	+	0.984469i	$0.556173\pi$
$522$	0	0
$523$	−7136.72	−0.596686	−0.298343	−	0.954459i	$-0.596434\pi$
$523$	−7136.72	−0.596686	−0.298343	+	0.954459i	$0.596434\pi$
$524$	52.9908	0.00441777
$525$	0	0
$526$	19199.3	1.59150
$527$	−2528.03	−0.208961
$528$	0	0
$529$	−12063.7	−0.991509
$530$	0	0
$531$	0	0
$532$	−10146.8	−0.826920
$533$	299.582	0.0243458
$534$	0	0
$535$	0	0
$536$	−2895.64	−0.233345
$537$	0	0
$538$	−16840.0	−1.34949
$539$	−2370.24	−0.189413
$540$	0	0
$541$	18984.0	1.50866	0.754329	−	0.656496i	$-0.227962\pi$
$541$	18984.0	1.50866	0.754329	+	0.656496i	$0.227962\pi$
$542$	−29462.1	−2.33488
$543$	0	0
$544$	−1981.75	−0.156189
$545$	0	0
$546$	0	0
$547$	−13084.1	−1.02273	−0.511366	−	0.859363i	$-0.670860\pi$
$547$	−13084.1	−1.02273	−0.511366	+	0.859363i	$0.670860\pi$
$548$	−4097.52	−0.319411
$549$	0	0
$550$	0	0
$551$	−5107.66	−0.394907
$552$	0	0
$553$	1291.87	0.0993413
$554$	−13894.9	−1.06559
$555$	0	0
$556$	4593.40	0.350366
$557$	−11677.1	−0.888284	−0.444142	−	0.895956i	$-0.646491\pi$
$557$	−11677.1	−0.888284	−0.444142	+	0.895956i	$0.646491\pi$
$558$	0	0
$559$	−571.740	−0.0432595
$560$	0	0
$561$	0	0
$562$	4877.63	0.366104
$563$	13214.1	0.989180	0.494590	−	0.869127i	$-0.335318\pi$
$563$	13214.1	0.989180	0.494590	+	0.869127i	$0.335318\pi$
$564$	0	0
$565$	0	0
$566$	−4397.06	−0.326541
$567$	0	0
$568$	−3505.41	−0.258951
$569$	22207.0	1.63614	0.818070	−	0.575118i	$-0.195044\pi$
$569$	22207.0	1.63614	0.818070	+	0.575118i	$0.195044\pi$
$570$	0	0
$571$	−7251.49	−0.531463	−0.265731	−	0.964047i	$-0.585613\pi$
$571$	−7251.49	−0.531463	−0.265731	+	0.964047i	$0.585613\pi$
$572$	501.109	0.0366301
$573$	0	0
$574$	−3602.53	−0.261963
$575$	0	0
$576$	0	0
$577$	8190.57	0.590949	0.295475	−	0.955351i	$-0.404522\pi$
$577$	8190.57	0.590949	0.295475	+	0.955351i	$0.404522\pi$
$578$	−18203.6	−1.30998
$579$	0	0
$580$	0	0
$581$	12736.9	0.909490
$582$	0	0
$583$	−2260.27	−0.160567
$584$	4240.44	0.300463
$585$	0	0
$586$	−33413.3	−2.35545
$587$	9678.56	0.680540	0.340270	−	0.940328i	$-0.389482\pi$
$587$	9678.56	0.680540	0.340270	+	0.940328i	$0.389482\pi$
$588$	0	0
$589$	20280.7	1.41876
$590$	0	0
$591$	0	0
$592$	−7956.59	−0.552388
$593$	−12938.6	−0.895994	−0.447997	−	0.894035i	$-0.647863\pi$
$593$	−12938.6	−0.895994	−0.447997	+	0.894035i	$0.647863\pi$
$594$	0	0
$595$	0	0
$596$	14775.2	1.01546
$597$	0	0
$598$	282.852	0.0193423
$599$	11678.3	0.796601	0.398300	−	0.917255i	$-0.369600\pi$
$599$	11678.3	0.796601	0.398300	+	0.917255i	$0.369600\pi$
$600$	0	0
$601$	17335.0	1.17655	0.588277	−	0.808660i	$-0.299807\pi$
$601$	17335.0	1.17655	0.588277	+	0.808660i	$0.299807\pi$
$602$	6875.29	0.465475
$603$	0	0
$604$	−20047.0	−1.35050
$605$	0	0
$606$	0	0
$607$	−10259.9	−0.686057	−0.343028	−	0.939325i	$-0.611453\pi$
$607$	−10259.9	−0.686057	−0.343028	+	0.939325i	$0.611453\pi$
$608$	15898.3	1.06046
$609$	0	0
$610$	0	0
$611$	3497.75	0.231594
$612$	0	0
$613$	−8691.75	−0.572686	−0.286343	−	0.958127i	$-0.592440\pi$
$613$	−8691.75	−0.572686	−0.286343	+	0.958127i	$0.592440\pi$
$614$	17777.0	1.16844
$615$	0	0
$616$	1799.71	0.117715
$617$	7438.86	0.485377	0.242688	−	0.970104i	$-0.421971\pi$
$617$	7438.86	0.485377	0.242688	+	0.970104i	$0.421971\pi$
$618$	0	0
$619$	−8091.72	−0.525418	−0.262709	−	0.964875i	$-0.584616\pi$
$619$	−8091.72	−0.525418	−0.262709	+	0.964875i	$0.584616\pi$
$620$	0	0
$621$	0	0
$622$	−202.132	−0.0130301
$623$	−18458.7	−1.18705
$624$	0	0
$625$	0	0
$626$	17549.8	1.12050
$627$	0	0
$628$	−15352.1	−0.975503
$629$	917.639	0.0581696
$630$	0	0
$631$	10374.5	0.654518	0.327259	−	0.944935i	$-0.393875\pi$
$631$	10374.5	0.654518	0.327259	+	0.944935i	$0.393875\pi$
$632$	−378.462	−0.0238202
$633$	0	0
$634$	32606.3	2.04253
$635$	0	0
$636$	0	0
$637$	−1593.47	−0.0991142
$638$	−3033.29	−0.188228
$639$	0	0
$640$	0	0
$641$	−25560.8	−1.57502	−0.787511	−	0.616300i	$-0.788631\pi$
$641$	−25560.8	−1.57502	−0.787511	+	0.616300i	$0.788631\pi$
$642$	0	0
$643$	22746.6	1.39508	0.697541	−	0.716544i	$-0.254277\pi$
$643$	22746.6	1.39508	0.697541	+	0.716544i	$0.254277\pi$
$644$	−1479.71	−0.0905415
$645$	0	0
$646$	−2278.78	−0.138789
$647$	24638.3	1.49711	0.748555	−	0.663073i	$-0.230748\pi$
$647$	24638.3	1.49711	0.748555	+	0.663073i	$0.230748\pi$
$648$	0	0
$649$	3632.87	0.219727
$650$	0	0
$651$	0	0
$652$	4974.66	0.298808
$653$	−17926.1	−1.07428	−0.537139	−	0.843493i	$-0.680495\pi$
$653$	−17926.1	−1.07428	−0.537139	+	0.843493i	$0.680495\pi$
$654$	0	0
$655$	0	0
$656$	3051.82	0.181637
$657$	0	0
$658$	−42061.1	−2.49197
$659$	28662.2	1.69426	0.847132	−	0.531382i	$-0.178327\pi$
$659$	28662.2	1.69426	0.847132	+	0.531382i	$0.178327\pi$
$660$	0	0
$661$	13447.8	0.791312	0.395656	−	0.918399i	$-0.370517\pi$
$661$	13447.8	0.791312	0.395656	+	0.918399i	$0.370517\pi$
$662$	−12879.0	−0.756129
$663$	0	0
$664$	−3731.35	−0.218079
$665$	0	0
$666$	0	0
$667$	−744.847	−0.0432392
$668$	11293.4	0.654123
$669$	0	0
$670$	0	0
$671$	10251.9	0.589822
$672$	0	0
$673$	9621.01	0.551059	0.275529	−	0.961293i	$-0.411147\pi$
$673$	9621.01	0.551059	0.275529	+	0.961293i	$0.411147\pi$
$674$	−447.283	−0.0255619
$675$	0	0
$676$	−13197.1	−0.750857
$677$	15901.5	0.902723	0.451361	−	0.892341i	$-0.350939\pi$
$677$	15901.5	0.902723	0.451361	+	0.892341i	$0.350939\pi$
$678$	0	0
$679$	12566.7	0.710257
$680$	0	0
$681$	0	0
$682$	12044.1	0.676236
$683$	611.084	0.0342350	0.0171175	−	0.999853i	$-0.494551\pi$
$683$	611.084	0.0342350	0.0171175	+	0.999853i	$0.494551\pi$
$684$	0	0
$685$	0	0
$686$	−11340.4	−0.631162
$687$	0	0
$688$	−5824.29	−0.322745
$689$	−1519.54	−0.0840201
$690$	0	0
$691$	24067.1	1.32497	0.662487	−	0.749074i	$-0.269501\pi$
$691$	24067.1	1.32497	0.662487	+	0.749074i	$0.269501\pi$
$692$	−6448.09	−0.354219
$693$	0	0
$694$	−39265.9	−2.14771
$695$	0	0
$696$	0	0
$697$	−351.969	−0.0191274
$698$	28331.3	1.53633
$699$	0	0
$700$	0	0
$701$	−10093.5	−0.543834	−0.271917	−	0.962321i	$-0.587658\pi$
$701$	−10093.5	−0.543834	−0.271917	+	0.962321i	$0.587658\pi$
$702$	0	0
$703$	−7361.61	−0.394948
$704$	2812.19	0.150551
$705$	0	0
$706$	−810.753	−0.0432197
$707$	−44857.6	−2.38620
$708$	0	0
$709$	−4970.98	−0.263313	−0.131657	−	0.991295i	$-0.542030\pi$
$709$	−4970.98	−0.263313	−0.131657	+	0.991295i	$0.542030\pi$
$710$	0	0
$711$	0	0
$712$	5407.61	0.284633
$713$	2957.52	0.155343
$714$	0	0
$715$	0	0
$716$	−11586.1	−0.604737
$717$	0	0
$718$	−34262.2	−1.78086
$719$	35411.6	1.83676	0.918381	−	0.395698i	$-0.129497\pi$
$719$	35411.6	1.83676	0.918381	+	0.395698i	$0.129497\pi$
$720$	0	0
$721$	−30926.4	−1.59745
$722$	−7529.30	−0.388105
$723$	0	0
$724$	−11838.8	−0.607715
$725$	0	0
$726$	0	0
$727$	−34598.6	−1.76505	−0.882525	−	0.470266i	$-0.844158\pi$
$727$	−34598.6	−1.76505	−0.882525	+	0.470266i	$0.844158\pi$
$728$	1209.91	0.0615966
$729$	0	0
$730$	0	0
$731$	671.719	0.0339869
$732$	0	0
$733$	32108.2	1.61793	0.808966	−	0.587856i	$-0.200028\pi$
$733$	32108.2	1.61793	0.808966	+	0.587856i	$0.200028\pi$
$734$	−14160.4	−0.712085
$735$	0	0
$736$	2318.44	0.116112
$737$	−4600.78	−0.229948
$738$	0	0
$739$	−27536.9	−1.37072	−0.685359	−	0.728206i	$-0.740355\pi$
$739$	−27536.9	−1.37072	−0.685359	+	0.728206i	$0.740355\pi$
$740$	0	0
$741$	0	0
$742$	18272.8	0.904062
$743$	2437.44	0.120351	0.0601757	−	0.998188i	$-0.480834\pi$
$743$	2437.44	0.120351	0.0601757	+	0.998188i	$0.480834\pi$
$744$	0	0
$745$	0	0
$746$	−1414.85	−0.0694386
$747$	0	0
$748$	−588.736	−0.0287785
$749$	4883.69	0.238246
$750$	0	0
$751$	−29534.7	−1.43507	−0.717534	−	0.696523i	$-0.754729\pi$
$751$	−29534.7	−1.43507	−0.717534	+	0.696523i	$0.754729\pi$
$752$	35631.4	1.72785
$753$	0	0
$754$	−2039.23	−0.0984940
$755$	0	0
$756$	0	0
$757$	−747.476	−0.0358883	−0.0179442	−	0.999839i	$-0.505712\pi$
$757$	−747.476	−0.0358883	−0.0179442	+	0.999839i	$0.505712\pi$
$758$	11166.6	0.535079
$759$	0	0
$760$	0	0
$761$	21031.2	1.00181	0.500906	−	0.865502i	$-0.333000\pi$
$761$	21031.2	1.00181	0.500906	+	0.865502i	$0.333000\pi$
$762$	0	0
$763$	14077.7	0.667950
$764$	−1254.93	−0.0594264
$765$	0	0
$766$	−5836.15	−0.275286
$767$	2442.32	0.114976
$768$	0	0
$769$	23107.4	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$769$	23107.4	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$770$	0	0
$771$	0	0
$772$	−17608.1	−0.820892
$773$	−24367.7	−1.13382	−0.566912	−	0.823778i	$-0.691862\pi$
$773$	−24367.7	−1.13382	−0.566912	+	0.823778i	$0.691862\pi$
$774$	0	0
$775$	0	0
$776$	−3681.49	−0.170307
$777$	0	0
$778$	28266.3	1.30256
$779$	2823.61	0.129867
$780$	0	0
$781$	−5569.62	−0.255181
$782$	−332.313	−0.0151963
$783$	0	0
$784$	−16232.6	−0.739460
$785$	0	0
$786$	0	0
$787$	11308.1	0.512188	0.256094	−	0.966652i	$-0.417564\pi$
$787$	11308.1	0.512188	0.256094	+	0.966652i	$0.417564\pi$
$788$	−7026.14	−0.317634
$789$	0	0
$790$	0	0
$791$	35755.5	1.60723
$792$	0	0
$793$	6892.18	0.308636
$794$	−41863.9	−1.87115
$795$	0	0
$796$	−19857.8	−0.884221
$797$	−43620.5	−1.93867	−0.969333	−	0.245752i	$-0.920965\pi$
$797$	−43620.5	−1.93867	−0.969333	+	0.245752i	$0.920965\pi$
$798$	0	0
$799$	−4109.39	−0.181952
$800$	0	0
$801$	0	0
$802$	50717.7	2.23305
$803$	6737.47	0.296090
$804$	0	0
$805$	0	0
$806$	8097.06	0.353854
$807$	0	0
$808$	13141.4	0.572168
$809$	7491.05	0.325551	0.162776	−	0.986663i	$-0.447955\pi$
$809$	7491.05	0.325551	0.162776	+	0.986663i	$0.447955\pi$
$810$	0	0
$811$	−7335.75	−0.317624	−0.158812	−	0.987309i	$-0.550766\pi$
$811$	−7335.75	−0.317624	−0.158812	+	0.987309i	$0.550766\pi$
$812$	10668.0	0.461052
$813$	0	0
$814$	−4371.85	−0.188247
$815$	0	0
$816$	0	0
$817$	−5388.76	−0.230757
$818$	48585.7	2.07672
$819$	0	0
$820$	0	0
$821$	37482.9	1.59338	0.796689	−	0.604390i	$-0.206583\pi$
$821$	37482.9	1.59338	0.796689	+	0.604390i	$0.206583\pi$
$822$	0	0
$823$	1213.87	0.0514128	0.0257064	−	0.999670i	$-0.491816\pi$
$823$	1213.87	0.0514128	0.0257064	+	0.999670i	$0.491816\pi$
$824$	9060.12	0.383039
$825$	0	0
$826$	−29369.4	−1.23716
$827$	7999.10	0.336343	0.168172	−	0.985758i	$-0.446214\pi$
$827$	7999.10	0.336343	0.168172	+	0.985758i	$0.446214\pi$
$828$	0	0
$829$	−14809.5	−0.620451	−0.310225	−	0.950663i	$-0.600405\pi$
$829$	−14809.5	−0.620451	−0.310225	+	0.950663i	$0.600405\pi$
$830$	0	0
$831$	0	0
$832$	1890.58	0.0787791
$833$	1872.12	0.0778693
$834$	0	0
$835$	0	0
$836$	4723.04	0.195394
$837$	0	0
$838$	33514.8	1.38156
$839$	44598.1	1.83516	0.917578	−	0.397555i	$-0.130141\pi$
$839$	44598.1	1.83516	0.917578	+	0.397555i	$0.130141\pi$
$840$	0	0
$841$	−19019.0	−0.779819
$842$	6970.45	0.285294
$843$	0	0
$844$	10345.3	0.421919
$845$	0	0
$846$	0	0
$847$	2859.49	0.116001
$848$	−15479.5	−0.626848
$849$	0	0
$850$	0	0
$851$	−1073.54	−0.0432438
$852$	0	0
$853$	28250.2	1.13396	0.566980	−	0.823732i	$-0.308112\pi$
$853$	28250.2	1.13396	0.566980	+	0.823732i	$0.308112\pi$
$854$	−82879.9	−3.32095
$855$	0	0
$856$	−1430.71	−0.0571270
$857$	−28018.4	−1.11679	−0.558395	−	0.829575i	$-0.688583\pi$
$857$	−28018.4	−1.11679	−0.558395	+	0.829575i	$0.688583\pi$
$858$	0	0
$859$	17367.5	0.689837	0.344919	−	0.938633i	$-0.387906\pi$
$859$	17367.5	0.689837	0.344919	+	0.938633i	$0.387906\pi$
$860$	0	0
$861$	0	0
$862$	15393.4	0.608240
$863$	49064.8	1.93532	0.967662	−	0.252252i	$-0.0811712\pi$
$863$	49064.8	1.93532	0.967662	+	0.252252i	$0.0811712\pi$
$864$	0	0
$865$	0	0
$866$	−31059.6	−1.21876
$867$	0	0
$868$	−42358.9	−1.65640
$869$	−601.323	−0.0234735
$870$	0	0
$871$	−3093.02	−0.120325
$872$	−4124.15	−0.160162
$873$	0	0
$874$	2665.93	0.103177
$875$	0	0
$876$	0	0
$877$	42355.0	1.63082	0.815409	−	0.578885i	$-0.196512\pi$
$877$	42355.0	1.63082	0.815409	+	0.578885i	$0.196512\pi$
$878$	3684.89	0.141639
$879$	0	0
$880$	0	0
$881$	36424.8	1.39294	0.696472	−	0.717584i	$-0.254752\pi$
$881$	36424.8	1.39294	0.696472	+	0.717584i	$0.254752\pi$
$882$	0	0
$883$	−27158.6	−1.03506	−0.517532	−	0.855664i	$-0.673149\pi$
$883$	−27158.6	−1.03506	−0.517532	+	0.855664i	$0.673149\pi$
$884$	−395.797	−0.0150589
$885$	0	0
$886$	336.862	0.0127732
$887$	−45832.2	−1.73494	−0.867472	−	0.497486i	$-0.834257\pi$
$887$	−45832.2	−1.73494	−0.867472	+	0.497486i	$0.834257\pi$
$888$	0	0
$889$	−1841.64	−0.0694789
$890$	0	0
$891$	0	0
$892$	37711.2	1.41554
$893$	32966.9	1.23538
$894$	0	0
$895$	0	0
$896$	20388.2	0.760183
$897$	0	0
$898$	64419.2	2.39387
$899$	−21322.4	−0.791035
$900$	0	0
$901$	1785.26	0.0660106
$902$	1676.86	0.0618996
$903$	0	0
$904$	−10474.8	−0.385384
$905$	0	0
$906$	0	0
$907$	42596.1	1.55940	0.779702	−	0.626151i	$-0.215371\pi$
$907$	42596.1	1.55940	0.779702	+	0.626151i	$0.215371\pi$
$908$	−10843.3	−0.396309
$909$	0	0
$910$	0	0
$911$	−45531.0	−1.65588	−0.827941	−	0.560816i	$-0.810488\pi$
$911$	−45531.0	−1.65588	−0.827941	+	0.560816i	$0.810488\pi$
$912$	0	0
$913$	−5928.60	−0.214905
$914$	6701.80	0.242534
$915$	0	0
$916$	−35609.5	−1.28447
$917$	203.286	0.00732073
$918$	0	0
$919$	14353.6	0.515215	0.257607	−	0.966250i	$-0.417066\pi$
$919$	14353.6	0.515215	0.257607	+	0.966250i	$0.417066\pi$
$920$	0	0
$921$	0	0
$922$	27154.3	0.969936
$923$	−3744.36	−0.133529
$924$	0	0
$925$	0	0
$926$	−35565.2	−1.26214
$927$	0	0
$928$	−16714.9	−0.591263
$929$	−38477.7	−1.35890	−0.679448	−	0.733724i	$-0.737781\pi$
$929$	−38477.7	−1.35890	−0.679448	+	0.733724i	$0.737781\pi$
$930$	0	0
$931$	−15018.8	−0.528701
$932$	33474.1	1.17648
$933$	0	0
$934$	27049.9	0.947645
$935$	0	0
$936$	0	0
$937$	−6917.67	−0.241185	−0.120592	−	0.992702i	$-0.538479\pi$
$937$	−6917.67	−0.241185	−0.120592	+	0.992702i	$0.538479\pi$
$938$	37194.2	1.29471
$939$	0	0
$940$	0	0
$941$	−41270.2	−1.42972	−0.714862	−	0.699265i	$-0.753511\pi$
$941$	−41270.2	−1.42972	−0.714862	+	0.699265i	$0.753511\pi$
$942$	0	0
$943$	411.766	0.0142194
$944$	24879.7	0.857803
$945$	0	0
$946$	−3200.23	−0.109988
$947$	7099.57	0.243617	0.121808	−	0.992554i	$-0.461131\pi$
$947$	7099.57	0.243617	0.121808	+	0.992554i	$0.461131\pi$
$948$	0	0
$949$	4529.48	0.154935
$950$	0	0
$951$	0	0
$952$	−1421.49	−0.0483935
$953$	−27983.8	−0.951191	−0.475595	−	0.879664i	$-0.657767\pi$
$953$	−27983.8	−0.951191	−0.475595	+	0.879664i	$0.657767\pi$
$954$	0	0
$955$	0	0
$956$	−3596.49	−0.121672
$957$	0	0
$958$	30950.2	1.04379
$959$	−15719.2	−0.529299
$960$	0	0
$961$	54872.3	1.84191
$962$	−2939.12	−0.0985043
$963$	0	0
$964$	25982.9	0.868103
$965$	0	0
$966$	0	0
$967$	22702.4	0.754972	0.377486	−	0.926015i	$-0.376789\pi$
$967$	22702.4	0.754972	0.377486	+	0.926015i	$0.376789\pi$
$968$	−837.707	−0.0278150
$969$	0	0
$970$	0	0
$971$	−11642.7	−0.384791	−0.192396	−	0.981317i	$-0.561626\pi$
$971$	−11642.7	−0.384791	−0.192396	+	0.981317i	$0.561626\pi$
$972$	0	0
$973$	17621.5	0.580595
$974$	18128.2	0.596372
$975$	0	0
$976$	70210.2	2.30264
$977$	6166.32	0.201922	0.100961	−	0.994890i	$-0.467808\pi$
$977$	6166.32	0.201922	0.100961	+	0.994890i	$0.467808\pi$
$978$	0	0
$979$	8591.94	0.280490
$980$	0	0
$981$	0	0
$982$	18604.4	0.604573
$983$	−18059.0	−0.585953	−0.292976	−	0.956120i	$-0.594646\pi$
$983$	−18059.0	−0.585953	−0.292976	+	0.956120i	$0.594646\pi$
$984$	0	0
$985$	0	0
$986$	2395.83	0.0773821
$987$	0	0
$988$	3175.22	0.102244
$989$	−785.839	−0.0252662
$990$	0	0
$991$	−62330.7	−1.99798	−0.998991	−	0.0449065i	$-0.985701\pi$
$991$	−62330.7	−1.99798	−0.998991	+	0.0449065i	$0.985701\pi$
$992$	66368.7	2.12420
$993$	0	0
$994$	45026.7	1.43678
$995$	0	0
$996$	0	0
$997$	13560.3	0.430752	0.215376	−	0.976531i	$-0.430902\pi$
$997$	13560.3	0.430752	0.215376	+	0.976531i	$0.430902\pi$
$998$	−13610.7	−0.431702
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.v.1.3		3
3.2	odd	2		825.4.a.q.1.1	yes	3
5.4	even	2		2475.4.a.y.1.1		3
15.2	even	4		825.4.c.n.199.2		6
15.8	even	4		825.4.c.n.199.5		6
15.14	odd	2		825.4.a.o.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.3	✓	3	15.14	odd	2
825.4.a.q.1.1	yes	3	3.2	odd	2
825.4.c.n.199.2		6	15.2	even	4
825.4.c.n.199.5		6	15.8	even	4
2475.4.a.v.1.3		3	1.1	even	1	trivial
2475.4.a.y.1.1		3	5.4	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+3.76300 q^{2} +6.16019 q^{4} +23.6321 q^{7} -6.92320 q^{8} +O(q^{10})\)	\(q+3.76300 q^{2} +6.16019 q^{4} +23.6321 q^{7} -6.92320 q^{8} -11.0000 q^{11} -7.39511 q^{13} +88.9277 q^{14} -75.3336 q^{16} +8.68828 q^{17} -69.7002 q^{19} -41.3930 q^{22} -10.1643 q^{23} -27.8278 q^{26} +145.578 q^{28} +73.2803 q^{29} -290.970 q^{31} -228.095 q^{32} +32.6940 q^{34} +105.618 q^{37} -262.282 q^{38} -40.5108 q^{41} +77.3133 q^{43} -67.7621 q^{44} -38.2485 q^{46} -472.981 q^{47} +215.477 q^{49} -45.5553 q^{52} +205.479 q^{53} -163.610 q^{56} +275.754 q^{58} -330.261 q^{59} -931.991 q^{61} -1094.92 q^{62} -255.653 q^{64} +418.252 q^{67} +53.5215 q^{68} +506.329 q^{71} -612.497 q^{73} +397.441 q^{74} -429.367 q^{76} -259.953 q^{77} +54.6657 q^{79} -152.442 q^{82} +538.964 q^{83} +290.930 q^{86} +76.1552 q^{88} -781.086 q^{89} -174.762 q^{91} -62.6144 q^{92} -1779.83 q^{94} +531.762 q^{97} +810.839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8	\( 3 q - q^{2} + 7 q^{4} + 16 q^{7} + 3 q^{8} - 33 q^{11} + 45 q^{13} + 116 q^{14} - 85 q^{16} + 58 q^{17} - 169 q^{19} + 11 q^{22} - 155 q^{23} - 167 q^{26} + 100 q^{28} + 277 q^{29} - 173 q^{31} - 97 q^{32} - 146 q^{34} + 60 q^{37} - 169 q^{38} - 44 q^{41} - 109 q^{43} - 77 q^{44} + 425 q^{46} - 270 q^{47} - 427 q^{49} + 45 q^{52} + 148 q^{53} - 168 q^{56} - 783 q^{58} + 684 q^{59} - 1038 q^{61} - 953 q^{62} - 1129 q^{64} + 314 q^{67} + 366 q^{68} + 1459 q^{71} - 1170 q^{73} + 1764 q^{74} + 211 q^{76} - 176 q^{77} - 506 q^{79} - 1040 q^{82} + 347 q^{83} + 527 q^{86} - 33 q^{88} + 607 q^{89} - 398 q^{91} - 687 q^{92} - 1822 q^{94} + 1263 q^{97} + 2273 q^{98}+O(q^{100}) \) 3 * q - q^2 + 7 * q^4 + 16 * q^7 + 3 * q^8 - 33 * q^11 + 45 * q^13 + 116 * q^14 - 85 * q^16 + 58 * q^17 - 169 * q^19 + 11 * q^22 - 155 * q^23 - 167 * q^26 + 100 * q^28 + 277 * q^29 - 173 * q^31 - 97 * q^32 - 146 * q^34 + 60 * q^37 - 169 * q^38 - 44 * q^41 - 109 * q^43 - 77 * q^44 + 425 * q^46 - 270 * q^47 - 427 * q^49 + 45 * q^52 + 148 * q^53 - 168 * q^56 - 783 * q^58 + 684 * q^59 - 1038 * q^61 - 953 * q^62 - 1129 * q^64 + 314 * q^67 + 366 * q^68 + 1459 * q^71 - 1170 * q^73 + 1764 * q^74 + 211 * q^76 - 176 * q^77 - 506 * q^79 - 1040 * q^82 + 347 * q^83 + 527 * q^86 - 33 * q^88 + 607 * q^89 - 398 * q^91 - 687 * q^92 - 1822 * q^94 + 1263 * q^97 + 2273 * q^98

Introduction

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