LMFDB - Embedded newform 2475.4.a.bi.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 21x^{3} + 17x^{2} + 78x - 30 $ x^5 - x^4 - 21x^3 + 17x^2 + 78*x - 30
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			$0.368634$ 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)

$f(q)$	$=$	$q-0.368634 q^{2} -7.86411 q^{4} -26.5824 q^{7} +5.84806 q^{8} +O(q^{10})$	$q-0.368634 q^{2} -7.86411 q^{4} -26.5824 q^{7} +5.84806 q^{8} +11.0000 q^{11} +50.4402 q^{13} +9.79918 q^{14} +60.7571 q^{16} -108.843 q^{17} +19.1278 q^{19} -4.05498 q^{22} -60.4434 q^{23} -18.5940 q^{26} +209.047 q^{28} +39.2097 q^{29} -22.4715 q^{31} -69.1816 q^{32} +40.1234 q^{34} -345.619 q^{37} -7.05117 q^{38} +96.3091 q^{41} +335.088 q^{43} -86.5052 q^{44} +22.2815 q^{46} +514.880 q^{47} +363.623 q^{49} -396.667 q^{52} -131.589 q^{53} -155.455 q^{56} -14.4541 q^{58} +210.728 q^{59} -68.9791 q^{61} +8.28378 q^{62} -460.554 q^{64} +202.618 q^{67} +855.955 q^{68} +645.234 q^{71} +1021.75 q^{73} +127.407 q^{74} -150.423 q^{76} -292.406 q^{77} +321.381 q^{79} -35.5029 q^{82} -840.583 q^{83} -123.525 q^{86} +64.3286 q^{88} +1449.66 q^{89} -1340.82 q^{91} +475.333 q^{92} -189.803 q^{94} +1602.70 q^{97} -134.044 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 3 q^{4} + 18 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q - q^2 + 3 * q^4 + 18 * q^7 + 3 * q^8	$ 5 q - q^{2} + 3 q^{4} + 18 q^{7} + 3 q^{8} + 55 q^{11} + 31 q^{13} - 8 q^{14} - 125 q^{16} - 38 q^{17} - 57 q^{19} - 11 q^{22} - 161 q^{23} + 125 q^{26} + 324 q^{28} + 107 q^{29} - 295 q^{31} + 23 q^{32} + 34 q^{34} - 260 q^{37} - 619 q^{38} + 128 q^{41} + 377 q^{43} + 33 q^{44} - 577 q^{46} + 114 q^{47} - 415 q^{49} - 395 q^{52} - 812 q^{53} + 132 q^{56} + 339 q^{58} + 1152 q^{59} - 344 q^{61} + 235 q^{62} - 545 q^{64} - 928 q^{67} - 654 q^{68} + 707 q^{71} + 322 q^{73} + 1176 q^{74} - 1699 q^{76} + 198 q^{77} - 2494 q^{79} - 2776 q^{82} - 1657 q^{83} + 799 q^{86} + 33 q^{88} + 2435 q^{89} - 804 q^{91} + 2775 q^{92} - 502 q^{94} - 1901 q^{97} - 343 q^{98}+O(q^{100}) $ 5 * q - q^2 + 3 * q^4 + 18 * q^7 + 3 * q^8 + 55 * q^11 + 31 * q^13 - 8 * q^14 - 125 * q^16 - 38 * q^17 - 57 * q^19 - 11 * q^22 - 161 * q^23 + 125 * q^26 + 324 * q^28 + 107 * q^29 - 295 * q^31 + 23 * q^32 + 34 * q^34 - 260 * q^37 - 619 * q^38 + 128 * q^41 + 377 * q^43 + 33 * q^44 - 577 * q^46 + 114 * q^47 - 415 * q^49 - 395 * q^52 - 812 * q^53 + 132 * q^56 + 339 * q^58 + 1152 * q^59 - 344 * q^61 + 235 * q^62 - 545 * q^64 - 928 * q^67 - 654 * q^68 + 707 * q^71 + 322 * q^73 + 1176 * q^74 - 1699 * q^76 + 198 * q^77 - 2494 * q^79 - 2776 * q^82 - 1657 * q^83 + 799 * q^86 + 33 * q^88 + 2435 * q^89 - 804 * q^91 + 2775 * q^92 - 502 * q^94 - 1901 * q^97 - 343 * 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 $

$2$	−0.368634	−0.130332	−0.0651660	−	0.997874i	$-0.520758\pi$
$2$	−0.368634	−0.130332	−0.0651660	+	0.997874i	$0.520758\pi$
$3$	0	0
$3$	0	0
$4$	−7.86411	−0.983014
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−26.5824	−1.43531	−0.717657	−	0.696397i	$-0.754785\pi$
$7$	−26.5824	−1.43531	−0.717657	+	0.696397i	$0.754785\pi$
$8$	5.84806	0.258450
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	50.4402	1.07612	0.538061	−	0.842906i	$-0.319157\pi$
$13$	50.4402	1.07612	0.538061	+	0.842906i	$0.319157\pi$
$14$	9.79918	0.187067
$15$	0	0
$16$	60.7571	0.949329
$17$	−108.843	−1.55284	−0.776422	−	0.630213i	$-0.782968\pi$
$17$	−108.843	−1.55284	−0.776422	+	0.630213i	$0.782968\pi$
$18$	0	0
$19$	19.1278	0.230959	0.115479	−	0.993310i	$-0.463160\pi$
$19$	19.1278	0.230959	0.115479	+	0.993310i	$0.463160\pi$
$20$	0	0
$21$	0	0
$22$	−4.05498	−0.0392966
$23$	−60.4434	−0.547970	−0.273985	−	0.961734i	$-0.588342\pi$
$23$	−60.4434	−0.547970	−0.273985	+	0.961734i	$0.588342\pi$
$24$	0	0
$25$	0	0
$26$	−18.5940	−0.140253
$27$	0	0
$28$	209.047	1.41093
$29$	39.2097	0.251071	0.125536	−	0.992089i	$-0.459935\pi$
$29$	39.2097	0.251071	0.125536	+	0.992089i	$0.459935\pi$
$30$	0	0
$31$	−22.4715	−0.130194	−0.0650969	−	0.997879i	$-0.520736\pi$
$31$	−22.4715	−0.130194	−0.0650969	+	0.997879i	$0.520736\pi$
$32$	−69.1816	−0.382178
$33$	0	0
$34$	40.1234	0.202385
$35$	0	0
$36$	0	0
$37$	−345.619	−1.53566	−0.767830	−	0.640654i	$-0.778663\pi$
$37$	−345.619	−1.53566	−0.767830	+	0.640654i	$0.778663\pi$
$38$	−7.05117	−0.0301013
$39$	0	0
$40$	0	0
$41$	96.3091	0.366853	0.183426	−	0.983033i	$-0.441281\pi$
$41$	96.3091	0.366853	0.183426	+	0.983033i	$0.441281\pi$
$42$	0	0
$43$	335.088	1.18838	0.594191	−	0.804324i	$-0.297472\pi$
$43$	335.088	1.18838	0.594191	+	0.804324i	$0.297472\pi$
$44$	−86.5052	−0.296390
$45$	0	0
$46$	22.2815	0.0714180
$47$	514.880	1.59794	0.798968	−	0.601373i	$-0.205379\pi$
$47$	514.880	1.59794	0.798968	+	0.601373i	$0.205379\pi$
$48$	0	0
$49$	363.623	1.06012
$50$	0	0
$51$	0	0
$52$	−396.667	−1.05784
$53$	−131.589	−0.341041	−0.170521	−	0.985354i	$-0.554545\pi$
$53$	−131.589	−0.341041	−0.170521	+	0.985354i	$0.554545\pi$
$54$	0	0
$55$	0	0
$56$	−155.455	−0.370957
$57$	0	0
$58$	−14.4541	−0.0327226
$59$	210.728	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$59$	210.728	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$60$	0	0
$61$	−68.9791	−0.144785	−0.0723924	−	0.997376i	$-0.523063\pi$
$61$	−68.9791	−0.144785	−0.0723924	+	0.997376i	$0.523063\pi$
$62$	8.28378	0.0169684
$63$	0	0
$64$	−460.554	−0.899519
$65$	0	0
$66$	0	0
$67$	202.618	0.369459	0.184729	−	0.982789i	$-0.440859\pi$
$67$	202.618	0.369459	0.184729	+	0.982789i	$0.440859\pi$
$68$	855.955	1.52647
$69$	0	0
$70$	0	0
$71$	645.234	1.07852	0.539262	−	0.842138i	$-0.318703\pi$
$71$	645.234	1.07852	0.539262	+	0.842138i	$0.318703\pi$
$72$	0	0
$73$	1021.75	1.63818	0.819089	−	0.573666i	$-0.194479\pi$
$73$	1021.75	1.63818	0.819089	+	0.573666i	$0.194479\pi$
$74$	127.407	0.200145
$75$	0	0
$76$	−150.423	−0.227036
$77$	−292.406	−0.432763
$78$	0	0
$79$	321.381	0.457699	0.228849	−	0.973462i	$-0.426504\pi$
$79$	321.381	0.457699	0.228849	+	0.973462i	$0.426504\pi$
$80$	0	0
$81$	0	0
$82$	−35.5029	−0.0478126
$83$	−840.583	−1.11164	−0.555819	−	0.831303i	$-0.687595\pi$
$83$	−840.583	−1.11164	−0.555819	+	0.831303i	$0.687595\pi$
$84$	0	0
$85$	0	0
$86$	−123.525	−0.154884
$87$	0	0
$88$	64.3286	0.0779256
$89$	1449.66	1.72655	0.863277	−	0.504731i	$-0.168408\pi$
$89$	1449.66	1.72655	0.863277	+	0.504731i	$0.168408\pi$
$90$	0	0
$91$	−1340.82	−1.54457
$92$	475.333	0.538662
$93$	0	0
$94$	−189.803	−0.208262
$95$	0	0
$96$	0	0
$97$	1602.70	1.67762	0.838811	−	0.544422i	$-0.183251\pi$
$97$	1602.70	1.67762	0.838811	+	0.544422i	$0.183251\pi$
$98$	−134.044	−0.138168
$99$	0	0
$100$	0	0
$101$	970.653	0.956273	0.478136	−	0.878286i	$-0.341312\pi$
$101$	970.653	0.956273	0.478136	+	0.878286i	$0.341312\pi$
$102$	0	0
$103$	−185.474	−0.177430	−0.0887152	−	0.996057i	$-0.528276\pi$
$103$	−185.474	−0.177430	−0.0887152	+	0.996057i	$0.528276\pi$
$104$	294.977	0.278124
$105$	0	0
$106$	48.5084	0.0444486
$107$	652.518	0.589545	0.294773	−	0.955567i	$-0.404756\pi$
$107$	652.518	0.589545	0.294773	+	0.955567i	$0.404756\pi$
$108$	0	0
$109$	−915.109	−0.804143	−0.402071	−	0.915608i	$-0.631710\pi$
$109$	−915.109	−0.804143	−0.402071	+	0.915608i	$0.631710\pi$
$110$	0	0
$111$	0	0
$112$	−1615.07	−1.36258
$113$	−327.063	−0.272279	−0.136139	−	0.990690i	$-0.543470\pi$
$113$	−327.063	−0.272279	−0.136139	+	0.990690i	$0.543470\pi$
$114$	0	0
$115$	0	0
$116$	−308.349	−0.246806
$117$	0	0
$118$	−77.6816	−0.0606031
$119$	2893.31	2.22882
$120$	0	0
$121$	121.000	0.0909091
$122$	25.4281	0.0188701
$123$	0	0
$124$	176.719	0.127982
$125$	0	0
$126$	0	0
$127$	−1739.39	−1.21532	−0.607661	−	0.794196i	$-0.707892\pi$
$127$	−1739.39	−1.21532	−0.607661	+	0.794196i	$0.707892\pi$
$128$	723.229	0.499414
$129$	0	0
$130$	0	0
$131$	−1537.93	−1.02572	−0.512860	−	0.858472i	$-0.671414\pi$
$131$	−1537.93	−1.02572	−0.512860	+	0.858472i	$0.671414\pi$
$132$	0	0
$133$	−508.463	−0.331498
$134$	−74.6920	−0.0481523
$135$	0	0
$136$	−636.521	−0.401333
$137$	−1885.08	−1.17557	−0.587787	−	0.809016i	$-0.700001\pi$
$137$	−1885.08	−1.17557	−0.587787	+	0.809016i	$0.700001\pi$
$138$	0	0
$139$	−482.323	−0.294317	−0.147159	−	0.989113i	$-0.547013\pi$
$139$	−482.323	−0.294317	−0.147159	+	0.989113i	$0.547013\pi$
$140$	0	0
$141$	0	0
$142$	−237.855	−0.140566
$143$	554.842	0.324463
$144$	0	0
$145$	0	0
$146$	−376.653	−0.213507
$147$	0	0
$148$	2717.98	1.50957
$149$	757.043	0.416237	0.208119	−	0.978104i	$-0.433266\pi$
$149$	757.043	0.416237	0.208119	+	0.978104i	$0.433266\pi$
$150$	0	0
$151$	−2919.01	−1.57315	−0.786576	−	0.617494i	$-0.788148\pi$
$151$	−2919.01	−1.57315	−0.786576	+	0.617494i	$0.788148\pi$
$152$	111.861	0.0596914
$153$	0	0
$154$	107.791	0.0564029
$155$	0	0
$156$	0	0
$157$	−1352.65	−0.687599	−0.343799	−	0.939043i	$-0.611714\pi$
$157$	−1352.65	−0.687599	−0.343799	+	0.939043i	$0.611714\pi$
$158$	−118.472	−0.0596527
$159$	0	0
$160$	0	0
$161$	1606.73	0.786509
$162$	0	0
$163$	−3027.63	−1.45486	−0.727431	−	0.686181i	$-0.759286\pi$
$163$	−3027.63	−1.45486	−0.727431	+	0.686181i	$0.759286\pi$
$164$	−757.385	−0.360621
$165$	0	0
$166$	309.868	0.144882
$167$	4225.69	1.95805	0.979023	−	0.203749i	$-0.0653126\pi$
$167$	4225.69	1.95805	0.979023	+	0.203749i	$0.0653126\pi$
$168$	0	0
$169$	347.214	0.158040
$170$	0	0
$171$	0	0
$172$	−2635.17	−1.16820
$173$	−1515.20	−0.665886	−0.332943	−	0.942947i	$-0.608042\pi$
$173$	−1515.20	−0.665886	−0.332943	+	0.942947i	$0.608042\pi$
$174$	0	0
$175$	0	0
$176$	668.328	0.286234
$177$	0	0
$178$	−534.393	−0.225025
$179$	−1087.42	−0.454066	−0.227033	−	0.973887i	$-0.572902\pi$
$179$	−1087.42	−0.454066	−0.227033	+	0.973887i	$0.572902\pi$
$180$	0	0
$181$	−145.994	−0.0599538	−0.0299769	−	0.999551i	$-0.509543\pi$
$181$	−145.994	−0.0599538	−0.0299769	+	0.999551i	$0.509543\pi$
$182$	494.273	0.201307
$183$	0	0
$184$	−353.476	−0.141623
$185$	0	0
$186$	0	0
$187$	−1197.28	−0.468200
$188$	−4049.08	−1.57079
$189$	0	0
$190$	0	0
$191$	−3586.76	−1.35879	−0.679395	−	0.733773i	$-0.737757\pi$
$191$	−3586.76	−1.35879	−0.679395	+	0.733773i	$0.737757\pi$
$192$	0	0
$193$	1015.55	0.378762	0.189381	−	0.981904i	$-0.439352\pi$
$193$	1015.55	0.378762	0.189381	+	0.981904i	$0.439352\pi$
$194$	−590.810	−0.218648
$195$	0	0
$196$	−2859.57	−1.04212
$197$	−2992.63	−1.08231	−0.541157	−	0.840921i	$-0.682014\pi$
$197$	−2992.63	−1.08231	−0.541157	+	0.840921i	$0.682014\pi$
$198$	0	0
$199$	−3267.57	−1.16398	−0.581989	−	0.813196i	$-0.697725\pi$
$199$	−3267.57	−1.16398	−0.581989	+	0.813196i	$0.697725\pi$
$200$	0	0
$201$	0	0
$202$	−357.816	−0.124633
$203$	−1042.29	−0.360366
$204$	0	0
$205$	0	0
$206$	68.3722	0.0231248
$207$	0	0
$208$	3064.60	1.02159
$209$	210.406	0.0696368
$210$	0	0
$211$	−4275.02	−1.39481	−0.697403	−	0.716679i	$-0.745661\pi$
$211$	−4275.02	−1.39481	−0.697403	+	0.716679i	$0.745661\pi$
$212$	1034.83	0.335248
$213$	0	0
$214$	−240.541	−0.0768366
$215$	0	0
$216$	0	0
$217$	597.347	0.186869
$218$	337.341	0.104805
$219$	0	0
$220$	0	0
$221$	−5490.07	−1.67105
$222$	0	0
$223$	−3771.24	−1.13247	−0.566235	−	0.824244i	$-0.691601\pi$
$223$	−3771.24	−1.13247	−0.566235	+	0.824244i	$0.691601\pi$
$224$	1839.01	0.548545
$225$	0	0
$226$	120.567	0.0354866
$227$	−2891.58	−0.845467	−0.422733	−	0.906254i	$-0.638929\pi$
$227$	−2891.58	−0.845467	−0.422733	+	0.906254i	$0.638929\pi$
$228$	0	0
$229$	921.309	0.265859	0.132930	−	0.991125i	$-0.457562\pi$
$229$	921.309	0.265859	0.132930	+	0.991125i	$0.457562\pi$
$230$	0	0
$231$	0	0
$232$	229.301	0.0648893
$233$	−2818.65	−0.792516	−0.396258	−	0.918139i	$-0.629691\pi$
$233$	−2818.65	−0.792516	−0.396258	+	0.918139i	$0.629691\pi$
$234$	0	0
$235$	0	0
$236$	−1657.19	−0.457092
$237$	0	0
$238$	−1066.57	−0.290486
$239$	−783.622	−0.212085	−0.106042	−	0.994362i	$-0.533818\pi$
$239$	−783.622	−0.212085	−0.106042	+	0.994362i	$0.533818\pi$
$240$	0	0
$241$	5891.99	1.57484	0.787420	−	0.616417i	$-0.211417\pi$
$241$	5891.99	1.57484	0.787420	+	0.616417i	$0.211417\pi$
$242$	−44.6048	−0.0118484
$243$	0	0
$244$	542.459	0.142325
$245$	0	0
$246$	0	0
$247$	964.811	0.248540
$248$	−131.415	−0.0336486
$249$	0	0
$250$	0	0
$251$	7892.69	1.98479	0.992394	−	0.123101i	$-0.0392838\pi$
$251$	7892.69	1.98479	0.992394	+	0.123101i	$0.0392838\pi$
$252$	0	0
$253$	−664.877	−0.165219
$254$	641.199	0.158395
$255$	0	0
$256$	3417.82	0.834430
$257$	−3375.73	−0.819348	−0.409674	−	0.912232i	$-0.634357\pi$
$257$	−3375.73	−0.819348	−0.409674	+	0.912232i	$0.634357\pi$
$258$	0	0
$259$	9187.37	2.20415
$260$	0	0
$261$	0	0
$262$	566.933	0.133684
$263$	1823.12	0.427447	0.213724	−	0.976894i	$-0.431441\pi$
$263$	1823.12	0.427447	0.213724	+	0.976894i	$0.431441\pi$
$264$	0	0
$265$	0	0
$266$	187.437	0.0432048
$267$	0	0
$268$	−1593.41	−0.363183
$269$	7312.95	1.65754	0.828770	−	0.559590i	$-0.189041\pi$
$269$	7312.95	1.65754	0.828770	+	0.559590i	$0.189041\pi$
$270$	0	0
$271$	6168.14	1.38261	0.691306	−	0.722562i	$-0.257036\pi$
$271$	6168.14	1.38261	0.691306	+	0.722562i	$0.257036\pi$
$272$	−6612.99	−1.47416
$273$	0	0
$274$	694.907	0.153215
$275$	0	0
$276$	0	0
$277$	−1858.13	−0.403048	−0.201524	−	0.979484i	$-0.564589\pi$
$277$	−1858.13	−0.403048	−0.201524	+	0.979484i	$0.564589\pi$
$278$	177.801	0.0383589
$279$	0	0
$280$	0	0
$281$	7716.88	1.63826	0.819129	−	0.573609i	$-0.194457\pi$
$281$	7716.88	1.63826	0.819129	+	0.573609i	$0.194457\pi$
$282$	0	0
$283$	−6582.51	−1.38265	−0.691324	−	0.722545i	$-0.742972\pi$
$283$	−6582.51	−1.38265	−0.691324	+	0.722545i	$0.742972\pi$
$284$	−5074.19	−1.06020
$285$	0	0
$286$	−204.534	−0.0422879
$287$	−2560.12	−0.526548
$288$	0	0
$289$	6933.84	1.41133
$290$	0	0
$291$	0	0
$292$	−8035.17	−1.61035
$293$	4788.12	0.954692	0.477346	−	0.878715i	$-0.341599\pi$
$293$	4788.12	0.954692	0.477346	+	0.878715i	$0.341599\pi$
$294$	0	0
$295$	0	0
$296$	−2021.20	−0.396891
$297$	0	0
$298$	−279.072	−0.0542490
$299$	−3048.78	−0.589683
$300$	0	0
$301$	−8907.43	−1.70570
$302$	1076.05	0.205032
$303$	0	0
$304$	1162.15	0.219256
$305$	0	0
$306$	0	0
$307$	−6612.72	−1.22934	−0.614671	−	0.788784i	$-0.710711\pi$
$307$	−6612.72	−1.22934	−0.614671	+	0.788784i	$0.710711\pi$
$308$	2299.51	0.425412
$309$	0	0
$310$	0	0
$311$	1915.55	0.349264	0.174632	−	0.984634i	$-0.444126\pi$
$311$	1915.55	0.349264	0.174632	+	0.984634i	$0.444126\pi$
$312$	0	0
$313$	−6383.65	−1.15280	−0.576398	−	0.817169i	$-0.695542\pi$
$313$	−6383.65	−1.15280	−0.576398	+	0.817169i	$0.695542\pi$
$314$	498.632	0.0896161
$315$	0	0
$316$	−2527.38	−0.449924
$317$	−10128.7	−1.79459	−0.897293	−	0.441434i	$-0.854470\pi$
$317$	−10128.7	−1.79459	−0.897293	+	0.441434i	$0.854470\pi$
$318$	0	0
$319$	431.307	0.0757008
$320$	0	0
$321$	0	0
$322$	−592.295	−0.102507
$323$	−2081.93	−0.358643
$324$	0	0
$325$	0	0
$326$	1116.09	0.189615
$327$	0	0
$328$	563.221	0.0948131
$329$	−13686.7	−2.29354
$330$	0	0
$331$	3453.52	0.573483	0.286741	−	0.958008i	$-0.407428\pi$
$331$	3453.52	0.573483	0.286741	+	0.958008i	$0.407428\pi$
$332$	6610.44	1.09276
$333$	0	0
$334$	−1557.73	−0.255196
$335$	0	0
$336$	0	0
$337$	−4246.90	−0.686480	−0.343240	−	0.939248i	$-0.611524\pi$
$337$	−4246.90	−0.686480	−0.343240	+	0.939248i	$0.611524\pi$
$338$	−127.995	−0.0205977
$339$	0	0
$340$	0	0
$341$	−247.187	−0.0392549
$342$	0	0
$343$	−548.198	−0.0862971
$344$	1959.61	0.307137
$345$	0	0
$346$	558.554	0.0867863
$347$	−10332.8	−1.59854	−0.799268	−	0.600975i	$-0.794779\pi$
$347$	−10332.8	−1.59854	−0.799268	+	0.600975i	$0.794779\pi$
$348$	0	0
$349$	−3899.90	−0.598156	−0.299078	−	0.954229i	$-0.596679\pi$
$349$	−3899.90	−0.598156	−0.299078	+	0.954229i	$0.596679\pi$
$350$	0	0
$351$	0	0
$352$	−760.998	−0.115231
$353$	7564.03	1.14049	0.570245	−	0.821475i	$-0.306848\pi$
$353$	7564.03	1.14049	0.570245	+	0.821475i	$0.306848\pi$
$354$	0	0
$355$	0	0
$356$	−11400.3	−1.69723
$357$	0	0
$358$	400.861	0.0591793
$359$	11198.7	1.64637	0.823185	−	0.567773i	$-0.192195\pi$
$359$	11198.7	1.64637	0.823185	+	0.567773i	$0.192195\pi$
$360$	0	0
$361$	−6493.13	−0.946658
$362$	53.8183	0.00781389
$363$	0	0
$364$	10544.4	1.51834
$365$	0	0
$366$	0	0
$367$	−5939.65	−0.844816	−0.422408	−	0.906406i	$-0.638815\pi$
$367$	−5939.65	−0.844816	−0.422408	+	0.906406i	$0.638815\pi$
$368$	−3672.36	−0.520204
$369$	0	0
$370$	0	0
$371$	3497.96	0.489501
$372$	0	0
$373$	−20.1820	−0.00280157	−0.00140078	−	0.999999i	$-0.500446\pi$
$373$	−20.1820	−0.00280157	−0.00140078	+	0.999999i	$0.500446\pi$
$374$	441.357	0.0610214
$375$	0	0
$376$	3011.05	0.412987
$377$	1977.75	0.270183
$378$	0	0
$379$	8241.81	1.11703	0.558514	−	0.829495i	$-0.311372\pi$
$379$	8241.81	1.11703	0.558514	+	0.829495i	$0.311372\pi$
$380$	0	0
$381$	0	0
$382$	1322.20	0.177094
$383$	9971.48	1.33034	0.665168	−	0.746694i	$-0.268360\pi$
$383$	9971.48	1.33034	0.665168	+	0.746694i	$0.268360\pi$
$384$	0	0
$385$	0	0
$386$	−374.367	−0.0493647
$387$	0	0
$388$	−12603.8	−1.64913
$389$	−6897.31	−0.898991	−0.449496	−	0.893283i	$-0.648396\pi$
$389$	−6897.31	−0.898991	−0.449496	+	0.893283i	$0.648396\pi$
$390$	0	0
$391$	6578.85	0.850913
$392$	2126.49	0.273989
$393$	0	0
$394$	1103.19	0.141060
$395$	0	0
$396$	0	0
$397$	−10587.7	−1.33849	−0.669243	−	0.743043i	$-0.733382\pi$
$397$	−10587.7	−1.33849	−0.669243	+	0.743043i	$0.733382\pi$
$398$	1204.54	0.151704
$399$	0	0
$400$	0	0
$401$	−5700.87	−0.709944	−0.354972	−	0.934877i	$-0.615510\pi$
$401$	−5700.87	−0.709944	−0.354972	+	0.934877i	$0.615510\pi$
$402$	0	0
$403$	−1133.47	−0.140105
$404$	−7633.32	−0.940029
$405$	0	0
$406$	384.223	0.0469672
$407$	−3801.81	−0.463019
$408$	0	0
$409$	2389.92	0.288934	0.144467	−	0.989510i	$-0.453853\pi$
$409$	2389.92	0.288934	0.144467	+	0.989510i	$0.453853\pi$
$410$	0	0
$411$	0	0
$412$	1458.59	0.174416
$413$	−5601.65	−0.667407
$414$	0	0
$415$	0	0
$416$	−3489.53	−0.411270
$417$	0	0
$418$	−77.5629	−0.00907589
$419$	−2338.30	−0.272633	−0.136316	−	0.990665i	$-0.543526\pi$
$419$	−2338.30	−0.272633	−0.136316	+	0.990665i	$0.543526\pi$
$420$	0	0
$421$	−9848.22	−1.14008	−0.570039	−	0.821618i	$-0.693072\pi$
$421$	−9848.22	−1.14008	−0.570039	+	0.821618i	$0.693072\pi$
$422$	1575.92	0.181788
$423$	0	0
$424$	−769.542	−0.0881422
$425$	0	0
$426$	0	0
$427$	1833.63	0.207811
$428$	−5131.48	−0.579531
$429$	0	0
$430$	0	0
$431$	−13734.9	−1.53501	−0.767504	−	0.641045i	$-0.778501\pi$
$431$	−13734.9	−1.53501	−0.767504	+	0.641045i	$0.778501\pi$
$432$	0	0
$433$	−300.831	−0.0333881	−0.0166940	−	0.999861i	$-0.505314\pi$
$433$	−300.831	−0.0333881	−0.0166940	+	0.999861i	$0.505314\pi$
$434$	−220.203	−0.0243550
$435$	0	0
$436$	7196.52	0.790483
$437$	−1156.15	−0.126559
$438$	0	0
$439$	12002.9	1.30494	0.652468	−	0.757816i	$-0.273734\pi$
$439$	12002.9	1.30494	0.652468	+	0.757816i	$0.273734\pi$
$440$	0	0
$441$	0	0
$442$	2023.83	0.217791
$443$	−9448.03	−1.01329	−0.506647	−	0.862153i	$-0.669115\pi$
$443$	−9448.03	−1.01329	−0.506647	+	0.862153i	$0.669115\pi$
$444$	0	0
$445$	0	0
$446$	1390.21	0.147597
$447$	0	0
$448$	12242.6	1.29109
$449$	5412.07	0.568845	0.284422	−	0.958699i	$-0.408198\pi$
$449$	5412.07	0.568845	0.284422	+	0.958699i	$0.408198\pi$
$450$	0	0
$451$	1059.40	0.110610
$452$	2572.06	0.267654
$453$	0	0
$454$	1065.94	0.110191
$455$	0	0
$456$	0	0
$457$	10009.7	1.02458	0.512291	−	0.858812i	$-0.328797\pi$
$457$	10009.7	1.02458	0.512291	+	0.858812i	$0.328797\pi$
$458$	−339.626	−0.0346500
$459$	0	0
$460$	0	0
$461$	−9058.72	−0.915199	−0.457599	−	0.889159i	$-0.651291\pi$
$461$	−9058.72	−0.915199	−0.457599	+	0.889159i	$0.651291\pi$
$462$	0	0
$463$	−16017.2	−1.60774	−0.803871	−	0.594804i	$-0.797230\pi$
$463$	−16017.2	−1.60774	−0.803871	+	0.594804i	$0.797230\pi$
$464$	2382.27	0.238349
$465$	0	0
$466$	1039.05	0.103290
$467$	10318.0	1.02240	0.511201	−	0.859461i	$-0.329201\pi$
$467$	10318.0	1.02240	0.511201	+	0.859461i	$0.329201\pi$
$468$	0	0
$469$	−5386.07	−0.530289
$470$	0	0
$471$	0	0
$472$	1232.35	0.120177
$473$	3685.97	0.358311
$474$	0	0
$475$	0	0
$476$	−22753.3	−2.19096
$477$	0	0
$478$	288.870	0.0276414
$479$	9810.96	0.935854	0.467927	−	0.883767i	$-0.345001\pi$
$479$	9810.96	0.935854	0.467927	+	0.883767i	$0.345001\pi$
$480$	0	0
$481$	−17433.1	−1.65256
$482$	−2171.99	−0.205252
$483$	0	0
$484$	−951.557	−0.0893649
$485$	0	0
$486$	0	0
$487$	8081.87	0.752001	0.376001	−	0.926619i	$-0.377299\pi$
$487$	8081.87	0.752001	0.376001	+	0.926619i	$0.377299\pi$
$488$	−403.394	−0.0374196
$489$	0	0
$490$	0	0
$491$	5645.54	0.518900	0.259450	−	0.965757i	$-0.416459\pi$
$491$	5645.54	0.518900	0.259450	+	0.965757i	$0.416459\pi$
$492$	0	0
$493$	−4267.71	−0.389874
$494$	−355.662	−0.0323927
$495$	0	0
$496$	−1365.31	−0.123597
$497$	−17151.8	−1.54802
$498$	0	0
$499$	4176.04	0.374640	0.187320	−	0.982299i	$-0.440020\pi$
$499$	4176.04	0.374640	0.187320	+	0.982299i	$0.440020\pi$
$500$	0	0
$501$	0	0
$502$	−2909.52	−0.258681
$503$	−9131.76	−0.809473	−0.404737	−	0.914433i	$-0.632637\pi$
$503$	−9131.76	−0.809473	−0.404737	+	0.914433i	$0.632637\pi$
$504$	0	0
$505$	0	0
$506$	245.097	0.0215333
$507$	0	0
$508$	13678.8	1.19468
$509$	−21062.6	−1.83415	−0.917076	−	0.398712i	$-0.869457\pi$
$509$	−21062.6	−1.83415	−0.917076	+	0.398712i	$0.869457\pi$
$510$	0	0
$511$	−27160.6	−2.35130
$512$	−7045.76	−0.608167
$513$	0	0
$514$	1244.41	0.106787
$515$	0	0
$516$	0	0
$517$	5663.68	0.481796
$518$	−3386.78	−0.287271
$519$	0	0
$520$	0	0
$521$	6858.67	0.576744	0.288372	−	0.957518i	$-0.406886\pi$
$521$	6858.67	0.576744	0.288372	+	0.957518i	$0.406886\pi$
$522$	0	0
$523$	5752.51	0.480956	0.240478	−	0.970655i	$-0.422696\pi$
$523$	5752.51	0.480956	0.240478	+	0.970655i	$0.422696\pi$
$524$	12094.4	1.00830
$525$	0	0
$526$	−672.067	−0.0557101
$527$	2445.87	0.202171
$528$	0	0
$529$	−8513.60	−0.699729
$530$	0	0
$531$	0	0
$532$	3998.61	0.325868
$533$	4857.85	0.394778
$534$	0	0
$535$	0	0
$536$	1184.92	0.0954866
$537$	0	0
$538$	−2695.80	−0.216030
$539$	3999.85	0.319639
$540$	0	0
$541$	4957.66	0.393986	0.196993	−	0.980405i	$-0.436882\pi$
$541$	4957.66	0.393986	0.196993	+	0.980405i	$0.436882\pi$
$542$	−2273.79	−0.180199
$543$	0	0
$544$	7529.95	0.593463
$545$	0	0
$546$	0	0
$547$	−19515.0	−1.52542	−0.762709	−	0.646742i	$-0.776131\pi$
$547$	−19515.0	−1.52542	−0.762709	+	0.646742i	$0.776131\pi$
$548$	14824.5	1.15560
$549$	0	0
$550$	0	0
$551$	749.996	0.0579871
$552$	0	0
$553$	−8543.07	−0.656941
$554$	684.971	0.0525301
$555$	0	0
$556$	3793.04	0.289318
$557$	227.163	0.0172804	0.00864022	−	0.999963i	$-0.497250\pi$
$557$	227.163	0.0172804	0.00864022	+	0.999963i	$0.497250\pi$
$558$	0	0
$559$	16901.9	1.27884
$560$	0	0
$561$	0	0
$562$	−2844.71	−0.213517
$563$	18313.6	1.37092	0.685460	−	0.728110i	$-0.259601\pi$
$563$	18313.6	1.37092	0.685460	+	0.728110i	$0.259601\pi$
$564$	0	0
$565$	0	0
$566$	2426.54	0.180203
$567$	0	0
$568$	3773.36	0.278744
$569$	5703.35	0.420206	0.210103	−	0.977679i	$-0.432620\pi$
$569$	5703.35	0.420206	0.210103	+	0.977679i	$0.432620\pi$
$570$	0	0
$571$	−14797.1	−1.08448	−0.542241	−	0.840223i	$-0.682424\pi$
$571$	−14797.1	−1.08448	−0.542241	+	0.840223i	$0.682424\pi$
$572$	−4363.34	−0.318952
$573$	0	0
$574$	943.750	0.0686261
$575$	0	0
$576$	0	0
$577$	2045.44	0.147578	0.0737891	−	0.997274i	$-0.476491\pi$
$577$	2045.44	0.147578	0.0737891	+	0.997274i	$0.476491\pi$
$578$	−2556.05	−0.183941
$579$	0	0
$580$	0	0
$581$	22344.7	1.59555
$582$	0	0
$583$	−1447.48	−0.102828
$584$	5975.27	0.423387
$585$	0	0
$586$	−1765.06	−0.124427
$587$	10482.6	0.737075	0.368537	−	0.929613i	$-0.379859\pi$
$587$	10482.6	0.737075	0.368537	+	0.929613i	$0.379859\pi$
$588$	0	0
$589$	−429.831	−0.0300694
$590$	0	0
$591$	0	0
$592$	−20998.8	−1.45785
$593$	938.019	0.0649575	0.0324788	−	0.999472i	$-0.489660\pi$
$593$	938.019	0.0649575	0.0324788	+	0.999472i	$0.489660\pi$
$594$	0	0
$595$	0	0
$596$	−5953.47	−0.409167
$597$	0	0
$598$	1123.88	0.0768546
$599$	−2885.83	−0.196848	−0.0984239	−	0.995145i	$-0.531380\pi$
$599$	−2885.83	−0.196848	−0.0984239	+	0.995145i	$0.531380\pi$
$600$	0	0
$601$	−25636.7	−1.74000	−0.870001	−	0.493049i	$-0.835882\pi$
$601$	−25636.7	−1.74000	−0.870001	+	0.493049i	$0.835882\pi$
$602$	3283.59	0.222307
$603$	0	0
$604$	22955.4	1.54643
$605$	0	0
$606$	0	0
$607$	−9924.91	−0.663657	−0.331828	−	0.943340i	$-0.607665\pi$
$607$	−9924.91	−0.663657	−0.331828	+	0.943340i	$0.607665\pi$
$608$	−1323.29	−0.0882674
$609$	0	0
$610$	0	0
$611$	25970.7	1.71958
$612$	0	0
$613$	−9514.05	−0.626866	−0.313433	−	0.949610i	$-0.601479\pi$
$613$	−9514.05	−0.626866	−0.313433	+	0.949610i	$0.601479\pi$
$614$	2437.68	0.160222
$615$	0	0
$616$	−1710.01	−0.111848
$617$	25714.5	1.67784	0.838920	−	0.544255i	$-0.183188\pi$
$617$	25714.5	1.67784	0.838920	+	0.544255i	$0.183188\pi$
$618$	0	0
$619$	−17933.5	−1.16447	−0.582235	−	0.813020i	$-0.697822\pi$
$619$	−17933.5	−1.16447	−0.582235	+	0.813020i	$0.697822\pi$
$620$	0	0
$621$	0	0
$622$	−706.139	−0.0455203
$623$	−38535.3	−2.47815
$624$	0	0
$625$	0	0
$626$	2353.23	0.150246
$627$	0	0
$628$	10637.4	0.675919
$629$	37618.3	2.38464
$630$	0	0
$631$	−3836.00	−0.242011	−0.121005	−	0.992652i	$-0.538612\pi$
$631$	−3836.00	−0.242011	−0.121005	+	0.992652i	$0.538612\pi$
$632$	1879.45	0.118292
$633$	0	0
$634$	3733.78	0.233892
$635$	0	0
$636$	0	0
$637$	18341.2	1.14082
$638$	−158.995	−0.00986623
$639$	0	0
$640$	0	0
$641$	−20430.4	−1.25890	−0.629449	−	0.777042i	$-0.716719\pi$
$641$	−20430.4	−1.25890	−0.629449	+	0.777042i	$0.716719\pi$
$642$	0	0
$643$	14252.9	0.874152	0.437076	−	0.899424i	$-0.356014\pi$
$643$	14252.9	0.874152	0.437076	+	0.899424i	$0.356014\pi$
$644$	−12635.5	−0.773149
$645$	0	0
$646$	767.472	0.0467427
$647$	−18729.4	−1.13807	−0.569033	−	0.822315i	$-0.692682\pi$
$647$	−18729.4	−1.13807	−0.569033	+	0.822315i	$0.692682\pi$
$648$	0	0
$649$	2318.01	0.140200
$650$	0	0
$651$	0	0
$652$	23809.6	1.43015
$653$	4415.88	0.264635	0.132318	−	0.991207i	$-0.457758\pi$
$653$	4415.88	0.264635	0.132318	+	0.991207i	$0.457758\pi$
$654$	0	0
$655$	0	0
$656$	5851.46	0.348264
$657$	0	0
$658$	5045.40	0.298922
$659$	−7391.89	−0.436946	−0.218473	−	0.975843i	$-0.570107\pi$
$659$	−7391.89	−0.436946	−0.218473	+	0.975843i	$0.570107\pi$
$660$	0	0
$661$	−640.842	−0.0377093	−0.0188547	−	0.999822i	$-0.506002\pi$
$661$	−640.842	−0.0377093	−0.0188547	+	0.999822i	$0.506002\pi$
$662$	−1273.09	−0.0747431
$663$	0	0
$664$	−4915.78	−0.287303
$665$	0	0
$666$	0	0
$667$	−2369.97	−0.137579
$668$	−33231.3	−1.92479
$669$	0	0
$670$	0	0
$671$	−758.770	−0.0436542
$672$	0	0
$673$	19129.0	1.09564	0.547822	−	0.836595i	$-0.315457\pi$
$673$	19129.0	1.09564	0.547822	+	0.836595i	$0.315457\pi$
$674$	1565.56	0.0894702
$675$	0	0
$676$	−2730.53	−0.155355
$677$	−4043.27	−0.229536	−0.114768	−	0.993392i	$-0.536612\pi$
$677$	−4043.27	−0.229536	−0.114768	+	0.993392i	$0.536612\pi$
$678$	0	0
$679$	−42603.5	−2.40791
$680$	0	0
$681$	0	0
$682$	91.1216	0.00511617
$683$	26633.3	1.49209	0.746044	−	0.665897i	$-0.231951\pi$
$683$	26633.3	1.49209	0.746044	+	0.665897i	$0.231951\pi$
$684$	0	0
$685$	0	0
$686$	202.085	0.0112473
$687$	0	0
$688$	20359.0	1.12817
$689$	−6637.40	−0.367002
$690$	0	0
$691$	30662.4	1.68806	0.844032	−	0.536293i	$-0.180176\pi$
$691$	30662.4	1.68806	0.844032	+	0.536293i	$0.180176\pi$
$692$	11915.7	0.654575
$693$	0	0
$694$	3809.01	0.208340
$695$	0	0
$696$	0	0
$697$	−10482.6	−0.569665
$698$	1437.64	0.0779589
$699$	0	0
$700$	0	0
$701$	−18443.3	−0.993717	−0.496858	−	0.867832i	$-0.665513\pi$
$701$	−18443.3	−0.993717	−0.496858	+	0.867832i	$0.665513\pi$
$702$	0	0
$703$	−6610.93	−0.354674
$704$	−5066.09	−0.271215
$705$	0	0
$706$	−2788.36	−0.148642
$707$	−25802.3	−1.37255
$708$	0	0
$709$	−21693.3	−1.14910	−0.574548	−	0.818471i	$-0.694822\pi$
$709$	−21693.3	−1.14910	−0.574548	+	0.818471i	$0.694822\pi$
$710$	0	0
$711$	0	0
$712$	8477.67	0.446228
$713$	1358.26	0.0713423
$714$	0	0
$715$	0	0
$716$	8551.61	0.446353
$717$	0	0
$718$	−4128.24	−0.214575
$719$	28555.1	1.48112	0.740560	−	0.671990i	$-0.234560\pi$
$719$	28555.1	1.48112	0.740560	+	0.671990i	$0.234560\pi$
$720$	0	0
$721$	4930.35	0.254668
$722$	2393.59	0.123380
$723$	0	0
$724$	1148.11	0.0589354
$725$	0	0
$726$	0	0
$727$	−21629.1	−1.10341	−0.551704	−	0.834040i	$-0.686022\pi$
$727$	−21629.1	−1.10341	−0.551704	+	0.834040i	$0.686022\pi$
$728$	−7841.19	−0.399195
$729$	0	0
$730$	0	0
$731$	−36472.0	−1.84537
$732$	0	0
$733$	−13081.1	−0.659157	−0.329579	−	0.944128i	$-0.606907\pi$
$733$	−13081.1	−0.659157	−0.329579	+	0.944128i	$0.606907\pi$
$734$	2189.56	0.110106
$735$	0	0
$736$	4181.57	0.209422
$737$	2228.80	0.111396
$738$	0	0
$739$	−24296.5	−1.20942	−0.604710	−	0.796446i	$-0.706711\pi$
$739$	−24296.5	−1.20942	−0.604710	+	0.796446i	$0.706711\pi$
$740$	0	0
$741$	0	0
$742$	−1289.47	−0.0637977
$743$	7572.81	0.373916	0.186958	−	0.982368i	$-0.440137\pi$
$743$	7572.81	0.373916	0.186958	+	0.982368i	$0.440137\pi$
$744$	0	0
$745$	0	0
$746$	7.43978	0.000365134 0
$747$	0	0
$748$	9415.50	0.460247
$749$	−17345.5	−0.846182
$750$	0	0
$751$	38506.1	1.87098	0.935492	−	0.353347i	$-0.114957\pi$
$751$	38506.1	1.87098	0.935492	+	0.353347i	$0.114957\pi$
$752$	31282.6	1.51697
$753$	0	0
$754$	−729.065	−0.0352135
$755$	0	0
$756$	0	0
$757$	−22513.0	−1.08091	−0.540455	−	0.841373i	$-0.681748\pi$
$757$	−22513.0	−1.08091	−0.540455	+	0.841373i	$0.681748\pi$
$758$	−3038.21	−0.145584
$759$	0	0
$760$	0	0
$761$	29062.0	1.38436	0.692179	−	0.721726i	$-0.256651\pi$
$761$	29062.0	1.38436	0.692179	+	0.721726i	$0.256651\pi$
$762$	0	0
$763$	24325.8	1.15420
$764$	28206.7	1.33571
$765$	0	0
$766$	−3675.83	−0.173385
$767$	10629.2	0.500387
$768$	0	0
$769$	−17769.2	−0.833257	−0.416628	−	0.909077i	$-0.636788\pi$
$769$	−17769.2	−0.833257	−0.416628	+	0.909077i	$0.636788\pi$
$770$	0	0
$771$	0	0
$772$	−7986.41	−0.372328
$773$	−15074.0	−0.701391	−0.350695	−	0.936490i	$-0.614055\pi$
$773$	−15074.0	−0.701391	−0.350695	+	0.936490i	$0.614055\pi$
$774$	0	0
$775$	0	0
$776$	9372.67	0.433582
$777$	0	0
$778$	2542.59	0.117167
$779$	1842.18	0.0847279
$780$	0	0
$781$	7097.57	0.325187
$782$	−2425.19	−0.110901
$783$	0	0
$784$	22092.6	1.00641
$785$	0	0
$786$	0	0
$787$	−15445.3	−0.699573	−0.349787	−	0.936829i	$-0.613746\pi$
$787$	−15445.3	−0.699573	−0.349787	+	0.936829i	$0.613746\pi$
$788$	23534.4	1.06393
$789$	0	0
$790$	0	0
$791$	8694.12	0.390806
$792$	0	0
$793$	−3479.32	−0.155806
$794$	3902.97	0.174448
$795$	0	0
$796$	25696.5	1.14421
$797$	14902.8	0.662337	0.331169	−	0.943572i	$-0.392557\pi$
$797$	14902.8	0.662337	0.331169	+	0.943572i	$0.392557\pi$
$798$	0	0
$799$	−56041.2	−2.48135
$800$	0	0
$801$	0	0
$802$	2101.54	0.0925284
$803$	11239.3	0.493930
$804$	0	0
$805$	0	0
$806$	417.836	0.0182601
$807$	0	0
$808$	5676.43	0.247149
$809$	−19642.4	−0.853633	−0.426816	−	0.904338i	$-0.640365\pi$
$809$	−19642.4	−0.853633	−0.426816	+	0.904338i	$0.640365\pi$
$810$	0	0
$811$	16729.7	0.724365	0.362182	−	0.932107i	$-0.382032\pi$
$811$	16729.7	0.724365	0.362182	+	0.932107i	$0.382032\pi$
$812$	8196.66	0.354244
$813$	0	0
$814$	1401.48	0.0603461
$815$	0	0
$816$	0	0
$817$	6409.50	0.274467
$818$	−881.007	−0.0376573
$819$	0	0
$820$	0	0
$821$	10838.2	0.460726	0.230363	−	0.973105i	$-0.426009\pi$
$821$	10838.2	0.460726	0.230363	+	0.973105i	$0.426009\pi$
$822$	0	0
$823$	12375.6	0.524164	0.262082	−	0.965046i	$-0.415591\pi$
$823$	12375.6	0.524164	0.262082	+	0.965046i	$0.415591\pi$
$824$	−1084.66	−0.0458569
$825$	0	0
$826$	2064.96	0.0869845
$827$	30336.4	1.27558	0.637788	−	0.770212i	$-0.279850\pi$
$827$	30336.4	1.27558	0.637788	+	0.770212i	$0.279850\pi$
$828$	0	0
$829$	−26159.0	−1.09595	−0.547973	−	0.836496i	$-0.684600\pi$
$829$	−26159.0	−1.09595	−0.547973	+	0.836496i	$0.684600\pi$
$830$	0	0
$831$	0	0
$832$	−23230.4	−0.967993
$833$	−39577.9	−1.64621
$834$	0	0
$835$	0	0
$836$	−1654.65	−0.0684539
$837$	0	0
$838$	861.976	0.0355328
$839$	−22058.7	−0.907690	−0.453845	−	0.891081i	$-0.649948\pi$
$839$	−22058.7	−0.907690	−0.453845	+	0.891081i	$0.649948\pi$
$840$	0	0
$841$	−22851.6	−0.936963
$842$	3630.39	0.148589
$843$	0	0
$844$	33619.2	1.37111
$845$	0	0
$846$	0	0
$847$	−3216.47	−0.130483
$848$	−7994.99	−0.323761
$849$	0	0
$850$	0	0
$851$	20890.4	0.841496
$852$	0	0
$853$	−41626.5	−1.67088	−0.835442	−	0.549578i	$-0.814788\pi$
$853$	−41626.5	−1.67088	−0.835442	+	0.549578i	$0.814788\pi$
$854$	−675.938	−0.0270845
$855$	0	0
$856$	3815.96	0.152368
$857$	−44478.1	−1.77286	−0.886431	−	0.462860i	$-0.846823\pi$
$857$	−44478.1	−1.77286	−0.886431	+	0.462860i	$0.846823\pi$
$858$	0	0
$859$	−5250.15	−0.208537	−0.104268	−	0.994549i	$-0.533250\pi$
$859$	−5250.15	−0.208537	−0.104268	+	0.994549i	$0.533250\pi$
$860$	0	0
$861$	0	0
$862$	5063.17	0.200061
$863$	8110.49	0.319912	0.159956	−	0.987124i	$-0.448865\pi$
$863$	8110.49	0.319912	0.159956	+	0.987124i	$0.448865\pi$
$864$	0	0
$865$	0	0
$866$	110.897	0.00435153
$867$	0	0
$868$	−4697.60	−0.183695
$869$	3535.19	0.138001
$870$	0	0
$871$	10220.1	0.397583
$872$	−5351.61	−0.207831
$873$	0	0
$874$	426.196	0.0164946
$875$	0	0
$876$	0	0
$877$	−11508.7	−0.443126	−0.221563	−	0.975146i	$-0.571116\pi$
$877$	−11508.7	−0.443126	−0.221563	+	0.975146i	$0.571116\pi$
$878$	−4424.68	−0.170075
$879$	0	0
$880$	0	0
$881$	15689.1	0.599978	0.299989	−	0.953943i	$-0.403017\pi$
$881$	15689.1	0.599978	0.299989	+	0.953943i	$0.403017\pi$
$882$	0	0
$883$	−12353.4	−0.470809	−0.235405	−	0.971897i	$-0.575642\pi$
$883$	−12353.4	−0.470809	−0.235405	+	0.971897i	$0.575642\pi$
$884$	43174.5	1.64267
$885$	0	0
$886$	3482.87	0.132065
$887$	5146.08	0.194801	0.0974005	−	0.995245i	$-0.468947\pi$
$887$	5146.08	0.194801	0.0974005	+	0.995245i	$0.468947\pi$
$888$	0	0
$889$	46237.1	1.74437
$890$	0	0
$891$	0	0
$892$	29657.4	1.11323
$893$	9848.53	0.369058
$894$	0	0
$895$	0	0
$896$	−19225.1	−0.716816
$897$	0	0
$898$	−1995.07	−0.0741386
$899$	−881.103	−0.0326879
$900$	0	0
$901$	14322.6	0.529584
$902$	−390.531	−0.0144160
$903$	0	0
$904$	−1912.68	−0.0703705
$905$	0	0
$906$	0	0
$907$	19902.0	0.728594	0.364297	−	0.931283i	$-0.381309\pi$
$907$	19902.0	0.728594	0.364297	+	0.931283i	$0.381309\pi$
$908$	22739.7	0.831105
$909$	0	0
$910$	0	0
$911$	6980.14	0.253856	0.126928	−	0.991912i	$-0.459488\pi$
$911$	6980.14	0.253856	0.126928	+	0.991912i	$0.459488\pi$
$912$	0	0
$913$	−9246.42	−0.335172
$914$	−3689.92	−0.133536
$915$	0	0
$916$	−7245.27	−0.261343
$917$	40881.8	1.47223
$918$	0	0
$919$	−13632.2	−0.489320	−0.244660	−	0.969609i	$-0.578676\pi$
$919$	−13632.2	−0.489320	−0.244660	+	0.969609i	$0.578676\pi$
$920$	0	0
$921$	0	0
$922$	3339.36	0.119280
$923$	32545.7	1.16062
$924$	0	0
$925$	0	0
$926$	5904.50	0.209540
$927$	0	0
$928$	−2712.59	−0.0959538
$929$	−48008.6	−1.69549	−0.847746	−	0.530403i	$-0.822041\pi$
$929$	−48008.6	−1.69549	−0.847746	+	0.530403i	$0.822041\pi$
$930$	0	0
$931$	6955.30	0.244845
$932$	22166.2	0.779054
$933$	0	0
$934$	−3803.58	−0.133252
$935$	0	0
$936$	0	0
$937$	19708.7	0.687144	0.343572	−	0.939126i	$-0.388363\pi$
$937$	19708.7	0.687144	0.343572	+	0.939126i	$0.388363\pi$
$938$	1985.49	0.0691136
$939$	0	0
$940$	0	0
$941$	40078.3	1.38843	0.694216	−	0.719767i	$-0.255751\pi$
$941$	40078.3	1.38843	0.694216	+	0.719767i	$0.255751\pi$
$942$	0	0
$943$	−5821.25	−0.201024
$944$	12803.2	0.441429
$945$	0	0
$946$	−1358.77	−0.0466993
$947$	−34849.6	−1.19584	−0.597919	−	0.801556i	$-0.704006\pi$
$947$	−34849.6	−1.19584	−0.597919	+	0.801556i	$0.704006\pi$
$948$	0	0
$949$	51537.4	1.76288
$950$	0	0
$951$	0	0
$952$	16920.2	0.576038
$953$	−40133.0	−1.36415	−0.682076	−	0.731282i	$-0.738923\pi$
$953$	−40133.0	−1.36415	−0.682076	+	0.731282i	$0.738923\pi$
$954$	0	0
$955$	0	0
$956$	6162.49	0.208482
$957$	0	0
$958$	−3616.66	−0.121972
$959$	50110.0	1.68732
$960$	0	0
$961$	−29286.0	−0.983050
$962$	6426.43	0.215381
$963$	0	0
$964$	−46335.2	−1.54809
$965$	0	0
$966$	0	0
$967$	36825.4	1.22464	0.612320	−	0.790610i	$-0.290237\pi$
$967$	36825.4	1.22464	0.612320	+	0.790610i	$0.290237\pi$
$968$	707.615	0.0234955
$969$	0	0
$970$	0	0
$971$	−72.3675	−0.00239174	−0.00119587	−	0.999999i	$-0.500381\pi$
$971$	−72.3675	−0.00239174	−0.00119587	+	0.999999i	$0.500381\pi$
$972$	0	0
$973$	12821.3	0.422437
$974$	−2979.26	−0.0980098
$975$	0	0
$976$	−4190.97	−0.137448
$977$	−25134.8	−0.823065	−0.411533	−	0.911395i	$-0.635006\pi$
$977$	−25134.8	−0.823065	−0.411533	+	0.911395i	$0.635006\pi$
$978$	0	0
$979$	15946.2	0.520576
$980$	0	0
$981$	0	0
$982$	−2081.14	−0.0676292
$983$	−43144.8	−1.39990	−0.699952	−	0.714190i	$-0.746795\pi$
$983$	−43144.8	−1.39990	−0.699952	+	0.714190i	$0.746795\pi$
$984$	0	0
$985$	0	0
$986$	1573.23	0.0508131
$987$	0	0
$988$	−7587.38	−0.244318
$989$	−20253.8	−0.651198
$990$	0	0
$991$	28318.0	0.907721	0.453860	−	0.891073i	$-0.350046\pi$
$991$	28318.0	0.907721	0.453860	+	0.891073i	$0.350046\pi$
$992$	1554.62	0.0497572
$993$	0	0
$994$	6322.76	0.201756
$995$	0	0
$996$	0	0
$997$	17969.5	0.570811	0.285405	−	0.958407i	$-0.407872\pi$
$997$	17969.5	0.570811	0.285405	+	0.958407i	$0.407872\pi$
$998$	−1539.43	−0.0488275
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bi.1.3		5
3.2	odd	2		825.4.a.y.1.3	yes	5
5.4	even	2		2475.4.a.bj.1.3		5
15.2	even	4		825.4.c.s.199.6		10
15.8	even	4		825.4.c.s.199.5		10
15.14	odd	2		825.4.a.x.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.x.1.3	✓	5	15.14	odd	2
825.4.a.y.1.3	yes	5	3.2	odd	2
825.4.c.s.199.5		10	15.8	even	4
825.4.c.s.199.6		10	15.2	even	4
2475.4.a.bi.1.3		5	1.1	even	1	trivial
2475.4.a.bj.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.368634 q^{2} -7.86411 q^{4} -26.5824 q^{7} +5.84806 q^{8} +O(q^{10})\)	\(q-0.368634 q^{2} -7.86411 q^{4} -26.5824 q^{7} +5.84806 q^{8} +11.0000 q^{11} +50.4402 q^{13} +9.79918 q^{14} +60.7571 q^{16} -108.843 q^{17} +19.1278 q^{19} -4.05498 q^{22} -60.4434 q^{23} -18.5940 q^{26} +209.047 q^{28} +39.2097 q^{29} -22.4715 q^{31} -69.1816 q^{32} +40.1234 q^{34} -345.619 q^{37} -7.05117 q^{38} +96.3091 q^{41} +335.088 q^{43} -86.5052 q^{44} +22.2815 q^{46} +514.880 q^{47} +363.623 q^{49} -396.667 q^{52} -131.589 q^{53} -155.455 q^{56} -14.4541 q^{58} +210.728 q^{59} -68.9791 q^{61} +8.28378 q^{62} -460.554 q^{64} +202.618 q^{67} +855.955 q^{68} +645.234 q^{71} +1021.75 q^{73} +127.407 q^{74} -150.423 q^{76} -292.406 q^{77} +321.381 q^{79} -35.5029 q^{82} -840.583 q^{83} -123.525 q^{86} +64.3286 q^{88} +1449.66 q^{89} -1340.82 q^{91} +475.333 q^{92} -189.803 q^{94} +1602.70 q^{97} -134.044 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 3 q^{4} + 18 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q - q^2 + 3 * q^4 + 18 * q^7 + 3 * q^8	\( 5 q - q^{2} + 3 q^{4} + 18 q^{7} + 3 q^{8} + 55 q^{11} + 31 q^{13} - 8 q^{14} - 125 q^{16} - 38 q^{17} - 57 q^{19} - 11 q^{22} - 161 q^{23} + 125 q^{26} + 324 q^{28} + 107 q^{29} - 295 q^{31} + 23 q^{32} + 34 q^{34} - 260 q^{37} - 619 q^{38} + 128 q^{41} + 377 q^{43} + 33 q^{44} - 577 q^{46} + 114 q^{47} - 415 q^{49} - 395 q^{52} - 812 q^{53} + 132 q^{56} + 339 q^{58} + 1152 q^{59} - 344 q^{61} + 235 q^{62} - 545 q^{64} - 928 q^{67} - 654 q^{68} + 707 q^{71} + 322 q^{73} + 1176 q^{74} - 1699 q^{76} + 198 q^{77} - 2494 q^{79} - 2776 q^{82} - 1657 q^{83} + 799 q^{86} + 33 q^{88} + 2435 q^{89} - 804 q^{91} + 2775 q^{92} - 502 q^{94} - 1901 q^{97} - 343 q^{98}+O(q^{100}) \) 5 * q - q^2 + 3 * q^4 + 18 * q^7 + 3 * q^8 + 55 * q^11 + 31 * q^13 - 8 * q^14 - 125 * q^16 - 38 * q^17 - 57 * q^19 - 11 * q^22 - 161 * q^23 + 125 * q^26 + 324 * q^28 + 107 * q^29 - 295 * q^31 + 23 * q^32 + 34 * q^34 - 260 * q^37 - 619 * q^38 + 128 * q^41 + 377 * q^43 + 33 * q^44 - 577 * q^46 + 114 * q^47 - 415 * q^49 - 395 * q^52 - 812 * q^53 + 132 * q^56 + 339 * q^58 + 1152 * q^59 - 344 * q^61 + 235 * q^62 - 545 * q^64 - 928 * q^67 - 654 * q^68 + 707 * q^71 + 322 * q^73 + 1176 * q^74 - 1699 * q^76 + 198 * q^77 - 2494 * q^79 - 2776 * q^82 - 1657 * q^83 + 799 * q^86 + 33 * q^88 + 2435 * q^89 - 804 * q^91 + 2775 * q^92 - 502 * q^94 - 1901 * q^97 - 343 * q^98

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