LMFDB - Embedded newform 2475.4.a.bv.1.1

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 49 x^{8} + 180 x^{7} + 753 x^{6} - 2420 x^{5} - 4059 x^{4} + 9796 x^{3} + 7226 x^{2} + \cdots - 2112 $ x^10 - 4x^9 - 49x^8 + 180x^7 + 753x^6 - 2420x^5 - 4059x^4 + 9796x^3 + 7226x^2 - 9120*x - 2112
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.47062$ 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)

$f(q)$	$=$	$q-5.47062 q^{2} +21.9277 q^{4} -16.1408 q^{7} -76.1933 q^{8} +O(q^{10})$	$q-5.47062 q^{2} +21.9277 q^{4} -16.1408 q^{7} -76.1933 q^{8} -11.0000 q^{11} +4.25817 q^{13} +88.3003 q^{14} +241.403 q^{16} -88.3099 q^{17} +40.8637 q^{19} +60.1769 q^{22} -37.9141 q^{23} -23.2948 q^{26} -353.931 q^{28} -56.6204 q^{29} -5.95951 q^{31} -711.079 q^{32} +483.110 q^{34} +323.020 q^{37} -223.550 q^{38} +161.020 q^{41} -437.141 q^{43} -241.205 q^{44} +207.414 q^{46} +397.406 q^{47} -82.4739 q^{49} +93.3719 q^{52} +211.063 q^{53} +1229.82 q^{56} +309.749 q^{58} -338.406 q^{59} +777.456 q^{61} +32.6022 q^{62} +1958.82 q^{64} +428.951 q^{67} -1936.44 q^{68} +848.538 q^{71} +789.116 q^{73} -1767.12 q^{74} +896.048 q^{76} +177.549 q^{77} +903.745 q^{79} -880.877 q^{82} -1269.17 q^{83} +2391.43 q^{86} +838.126 q^{88} +247.212 q^{89} -68.7303 q^{91} -831.370 q^{92} -2174.06 q^{94} -202.067 q^{97} +451.184 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8}+O(q^{10}) $ 10 * q - 4 * q^2 + 34 * q^4 + 2 * q^7 - 48 * q^8	$ 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8} - 110 q^{11} + 26 q^{13} + 72 q^{14} + 206 q^{16} - 148 q^{17} - 114 q^{19} + 44 q^{22} + 34 q^{23} + 100 q^{26} - 86 q^{28} - 38 q^{29} + 232 q^{31} - 448 q^{32} - 20 q^{34} + 754 q^{37} - 780 q^{38} + 160 q^{41} - 66 q^{43} - 374 q^{44} + 682 q^{46} - 450 q^{47} + 590 q^{49} + 200 q^{52} - 1068 q^{53} + 268 q^{56} - 138 q^{58} - 838 q^{59} - 566 q^{61} - 1230 q^{62} + 462 q^{64} + 430 q^{67} - 2234 q^{68} + 518 q^{71} - 184 q^{73} + 402 q^{74} + 386 q^{76} - 22 q^{77} + 956 q^{79} - 2180 q^{82} - 2094 q^{83} + 892 q^{86} + 528 q^{88} - 512 q^{89} - 858 q^{91} - 4476 q^{92} - 294 q^{94} - 1006 q^{97} - 2226 q^{98}+O(q^{100}) $ 10 * q - 4 * q^2 + 34 * q^4 + 2 * q^7 - 48 * q^8 - 110 * q^11 + 26 * q^13 + 72 * q^14 + 206 * q^16 - 148 * q^17 - 114 * q^19 + 44 * q^22 + 34 * q^23 + 100 * q^26 - 86 * q^28 - 38 * q^29 + 232 * q^31 - 448 * q^32 - 20 * q^34 + 754 * q^37 - 780 * q^38 + 160 * q^41 - 66 * q^43 - 374 * q^44 + 682 * q^46 - 450 * q^47 + 590 * q^49 + 200 * q^52 - 1068 * q^53 + 268 * q^56 - 138 * q^58 - 838 * q^59 - 566 * q^61 - 1230 * q^62 + 462 * q^64 + 430 * q^67 - 2234 * q^68 + 518 * q^71 - 184 * q^73 + 402 * q^74 + 386 * q^76 - 22 * q^77 + 956 * q^79 - 2180 * q^82 - 2094 * q^83 + 892 * q^86 + 528 * q^88 - 512 * q^89 - 858 * q^91 - 4476 * q^92 - 294 * q^94 - 1006 * q^97 - 2226 * 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 $

$2$	−5.47062	−1.93416	−0.967079	−	0.254478i	$-0.918097\pi$
$2$	−5.47062	−1.93416	−0.967079	+	0.254478i	$0.918097\pi$
$3$	0	0
$3$	0	0
$4$	21.9277	2.74096
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−16.1408	−0.871522	−0.435761	−	0.900062i	$-0.643521\pi$
$7$	−16.1408	−0.871522	−0.435761	+	0.900062i	$0.643521\pi$
$8$	−76.1933	−3.36730
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	4.25817	0.0908464	0.0454232	−	0.998968i	$-0.485536\pi$
$13$	4.25817	0.0908464	0.0454232	+	0.998968i	$0.485536\pi$
$14$	88.3003	1.68566
$15$	0	0
$16$	241.403	3.77192
$17$	−88.3099	−1.25990	−0.629950	−	0.776636i	$-0.716925\pi$
$17$	−88.3099	−1.25990	−0.629950	+	0.776636i	$0.716925\pi$
$18$	0	0
$19$	40.8637	0.493409	0.246705	−	0.969091i	$-0.420652\pi$
$19$	40.8637	0.493409	0.246705	+	0.969091i	$0.420652\pi$
$20$	0	0
$21$	0	0
$22$	60.1769	0.583170
$23$	−37.9141	−0.343724	−0.171862	−	0.985121i	$-0.554978\pi$
$23$	−37.9141	−0.343724	−0.171862	+	0.985121i	$0.554978\pi$
$24$	0	0
$25$	0	0
$26$	−23.2948	−0.175711
$27$	0	0
$28$	−353.931	−2.38881
$29$	−56.6204	−0.362556	−0.181278	−	0.983432i	$-0.558023\pi$
$29$	−56.6204	−0.362556	−0.181278	+	0.983432i	$0.558023\pi$
$30$	0	0
$31$	−5.95951	−0.0345277	−0.0172639	−	0.999851i	$-0.505496\pi$
$31$	−5.95951	−0.0345277	−0.0172639	+	0.999851i	$0.505496\pi$
$32$	−711.079	−3.92819
$33$	0	0
$34$	483.110	2.43685
$35$	0	0
$36$	0	0
$37$	323.020	1.43525	0.717625	−	0.696430i	$-0.245229\pi$
$37$	323.020	1.43525	0.717625	+	0.696430i	$0.245229\pi$
$38$	−223.550	−0.954331
$39$	0	0
$40$	0	0
$41$	161.020	0.613342	0.306671	−	0.951816i	$-0.400785\pi$
$41$	161.020	0.613342	0.306671	+	0.951816i	$0.400785\pi$
$42$	0	0
$43$	−437.141	−1.55031	−0.775155	−	0.631771i	$-0.782328\pi$
$43$	−437.141	−1.55031	−0.775155	+	0.631771i	$0.782328\pi$
$44$	−241.205	−0.826432
$45$	0	0
$46$	207.414	0.664816
$47$	397.406	1.23335	0.616677	−	0.787217i	$-0.288479\pi$
$47$	397.406	1.23335	0.616677	+	0.787217i	$0.288479\pi$
$48$	0	0
$49$	−82.4739	−0.240449
$50$	0	0
$51$	0	0
$52$	93.3719	0.249007
$53$	211.063	0.547014	0.273507	−	0.961870i	$-0.411816\pi$
$53$	211.063	0.547014	0.273507	+	0.961870i	$0.411816\pi$
$54$	0	0
$55$	0	0
$56$	1229.82	2.93468
$57$	0	0
$58$	309.749	0.701241
$59$	−338.406	−0.746723	−0.373361	−	0.927686i	$-0.621795\pi$
$59$	−338.406	−0.746723	−0.373361	+	0.927686i	$0.621795\pi$
$60$	0	0
$61$	777.456	1.63185	0.815926	−	0.578156i	$-0.196228\pi$
$61$	777.456	1.63185	0.815926	+	0.578156i	$0.196228\pi$
$62$	32.6022	0.0667821
$63$	0	0
$64$	1958.82	3.82582
$65$	0	0
$66$	0	0
$67$	428.951	0.782160	0.391080	−	0.920357i	$-0.372102\pi$
$67$	428.951	0.782160	0.391080	+	0.920357i	$0.372102\pi$
$68$	−1936.44	−3.45334
$69$	0	0
$70$	0	0
$71$	848.538	1.41835	0.709175	−	0.705032i	$-0.249067\pi$
$71$	848.538	1.41835	0.709175	+	0.705032i	$0.249067\pi$
$72$	0	0
$73$	789.116	1.26519	0.632596	−	0.774482i	$-0.281989\pi$
$73$	789.116	1.26519	0.632596	+	0.774482i	$0.281989\pi$
$74$	−1767.12	−2.77600
$75$	0	0
$76$	896.048	1.35242
$77$	177.549	0.262774
$78$	0	0
$79$	903.745	1.28708	0.643540	−	0.765413i	$-0.277465\pi$
$79$	903.745	1.28708	0.643540	+	0.765413i	$0.277465\pi$
$80$	0	0
$81$	0	0
$82$	−880.877	−1.18630
$83$	−1269.17	−1.67842	−0.839212	−	0.543805i	$-0.816983\pi$
$83$	−1269.17	−1.67842	−0.839212	+	0.543805i	$0.816983\pi$
$84$	0	0
$85$	0	0
$86$	2391.43	2.99854
$87$	0	0
$88$	838.126	1.01528
$89$	247.212	0.294432	0.147216	−	0.989104i	$-0.452969\pi$
$89$	247.212	0.294432	0.147216	+	0.989104i	$0.452969\pi$
$90$	0	0
$91$	−68.7303	−0.0791747
$92$	−831.370	−0.942134
$93$	0	0
$94$	−2174.06	−2.38550
$95$	0	0
$96$	0	0
$97$	−202.067	−0.211513	−0.105756	−	0.994392i	$-0.533726\pi$
$97$	−202.067	−0.211513	−0.105756	+	0.994392i	$0.533726\pi$
$98$	451.184	0.465066
$99$	0	0
$100$	0	0
$101$	−1929.59	−1.90100	−0.950502	−	0.310718i	$-0.899431\pi$
$101$	−1929.59	−1.90100	−0.950502	+	0.310718i	$0.899431\pi$
$102$	0	0
$103$	867.936	0.830294	0.415147	−	0.909754i	$-0.363730\pi$
$103$	867.936	0.830294	0.415147	+	0.909754i	$0.363730\pi$
$104$	−324.444	−0.305907
$105$	0	0
$106$	−1154.65	−1.05801
$107$	−300.052	−0.271095	−0.135547	−	0.990771i	$-0.543279\pi$
$107$	−300.052	−0.271095	−0.135547	+	0.990771i	$0.543279\pi$
$108$	0	0
$109$	−332.072	−0.291804	−0.145902	−	0.989299i	$-0.546608\pi$
$109$	−332.072	−0.291804	−0.145902	+	0.989299i	$0.546608\pi$
$110$	0	0
$111$	0	0
$112$	−3896.44	−3.28732
$113$	−1433.96	−1.19377	−0.596885	−	0.802327i	$-0.703595\pi$
$113$	−1433.96	−1.19377	−0.596885	+	0.802327i	$0.703595\pi$
$114$	0	0
$115$	0	0
$116$	−1241.56	−0.993754
$117$	0	0
$118$	1851.29	1.44428
$119$	1425.39	1.09803
$120$	0	0
$121$	121.000	0.0909091
$122$	−4253.17	−3.15626
$123$	0	0
$124$	−130.678	−0.0946393
$125$	0	0
$126$	0	0
$127$	1415.45	0.988984	0.494492	−	0.869182i	$-0.335354\pi$
$127$	1415.45	0.988984	0.494492	+	0.869182i	$0.335354\pi$
$128$	−5027.34	−3.47155
$129$	0	0
$130$	0	0
$131$	1447.77	0.965590	0.482795	−	0.875733i	$-0.339622\pi$
$131$	1447.77	0.965590	0.482795	+	0.875733i	$0.339622\pi$
$132$	0	0
$133$	−659.574	−0.430017
$134$	−2346.63	−1.51282
$135$	0	0
$136$	6728.62	4.24246
$137$	−1913.57	−1.19334	−0.596668	−	0.802488i	$-0.703509\pi$
$137$	−1913.57	−1.19334	−0.596668	+	0.802488i	$0.703509\pi$
$138$	0	0
$139$	−1832.40	−1.11814	−0.559072	−	0.829119i	$-0.688843\pi$
$139$	−1832.40	−1.11814	−0.559072	+	0.829119i	$0.688843\pi$
$140$	0	0
$141$	0	0
$142$	−4642.03	−2.74331
$143$	−46.8399	−0.0273912
$144$	0	0
$145$	0	0
$146$	−4316.96	−2.44708
$147$	0	0
$148$	7083.10	3.93397
$149$	−1888.28	−1.03822	−0.519108	−	0.854709i	$-0.673736\pi$
$149$	−1888.28	−1.03822	−0.519108	+	0.854709i	$0.673736\pi$
$150$	0	0
$151$	95.3097	0.0513655	0.0256828	−	0.999670i	$-0.491824\pi$
$151$	95.3097	0.0513655	0.0256828	+	0.999670i	$0.491824\pi$
$152$	−3113.54	−1.66146
$153$	0	0
$154$	−971.304	−0.508246
$155$	0	0
$156$	0	0
$157$	−3591.59	−1.82573	−0.912866	−	0.408260i	$-0.866136\pi$
$157$	−3591.59	−1.82573	−0.912866	+	0.408260i	$0.866136\pi$
$158$	−4944.05	−2.48941
$159$	0	0
$160$	0	0
$161$	611.965	0.299563
$162$	0	0
$163$	424.954	0.204202	0.102101	−	0.994774i	$-0.467443\pi$
$163$	424.954	0.204202	0.102101	+	0.994774i	$0.467443\pi$
$164$	3530.79	1.68115
$165$	0	0
$166$	6943.13	3.24634
$167$	800.995	0.371155	0.185577	−	0.982630i	$-0.440584\pi$
$167$	800.995	0.371155	0.185577	+	0.982630i	$0.440584\pi$
$168$	0	0
$169$	−2178.87	−0.991747
$170$	0	0
$171$	0	0
$172$	−9585.50	−4.24934
$173$	527.097	0.231644	0.115822	−	0.993270i	$-0.463050\pi$
$173$	527.097	0.231644	0.115822	+	0.993270i	$0.463050\pi$
$174$	0	0
$175$	0	0
$176$	−2655.43	−1.13728
$177$	0	0
$178$	−1352.41	−0.569478
$179$	3148.11	1.31453	0.657265	−	0.753660i	$-0.271713\pi$
$179$	3148.11	1.31453	0.657265	+	0.753660i	$0.271713\pi$
$180$	0	0
$181$	946.985	0.388889	0.194444	−	0.980914i	$-0.437710\pi$
$181$	946.985	0.388889	0.194444	+	0.980914i	$0.437710\pi$
$182$	375.998	0.153136
$183$	0	0
$184$	2888.80	1.15742
$185$	0	0
$186$	0	0
$187$	971.409	0.379874
$188$	8714.20	3.38058
$189$	0	0
$190$	0	0
$191$	3946.28	1.49499	0.747494	−	0.664268i	$-0.231257\pi$
$191$	3946.28	1.49499	0.747494	+	0.664268i	$0.231257\pi$
$192$	0	0
$193$	5026.15	1.87456	0.937281	−	0.348575i	$-0.113334\pi$
$193$	5026.15	1.87456	0.937281	+	0.348575i	$0.113334\pi$
$194$	1105.43	0.409099
$195$	0	0
$196$	−1808.46	−0.659062
$197$	4033.73	1.45884	0.729420	−	0.684066i	$-0.239790\pi$
$197$	4033.73	1.45884	0.729420	+	0.684066i	$0.239790\pi$
$198$	0	0
$199$	−2616.06	−0.931896	−0.465948	−	0.884812i	$-0.654287\pi$
$199$	−2616.06	−0.931896	−0.465948	+	0.884812i	$0.654287\pi$
$200$	0	0
$201$	0	0
$202$	10556.1	3.67684
$203$	913.899	0.315976
$204$	0	0
$205$	0	0
$206$	−4748.15	−1.60592
$207$	0	0
$208$	1027.94	0.342666
$209$	−449.501	−0.148769
$210$	0	0
$211$	−2606.47	−0.850411	−0.425205	−	0.905097i	$-0.639798\pi$
$211$	−2606.47	−0.850411	−0.425205	+	0.905097i	$0.639798\pi$
$212$	4628.13	1.49934
$213$	0	0
$214$	1641.47	0.524340
$215$	0	0
$216$	0	0
$217$	96.1914	0.0300917
$218$	1816.64	0.564396
$219$	0	0
$220$	0	0
$221$	−376.039	−0.114457
$222$	0	0
$223$	−1606.87	−0.482529	−0.241265	−	0.970459i	$-0.577562\pi$
$223$	−1606.87	−0.482529	−0.241265	+	0.970459i	$0.577562\pi$
$224$	11477.4	3.42351
$225$	0	0
$226$	7844.68	2.30894
$227$	1624.37	0.474949	0.237474	−	0.971394i	$-0.423680\pi$
$227$	1624.37	0.474949	0.237474	+	0.971394i	$0.423680\pi$
$228$	0	0
$229$	6380.22	1.84112	0.920561	−	0.390599i	$-0.127732\pi$
$229$	6380.22	1.84112	0.920561	+	0.390599i	$0.127732\pi$
$230$	0	0
$231$	0	0
$232$	4314.09	1.22084
$233$	3697.07	1.03950	0.519749	−	0.854319i	$-0.326026\pi$
$233$	3697.07	1.03950	0.519749	+	0.854319i	$0.326026\pi$
$234$	0	0
$235$	0	0
$236$	−7420.46	−2.04674
$237$	0	0
$238$	−7797.80	−2.12377
$239$	−5744.82	−1.55482	−0.777409	−	0.628995i	$-0.783467\pi$
$239$	−5744.82	−1.55482	−0.777409	+	0.628995i	$0.783467\pi$
$240$	0	0
$241$	−4276.62	−1.14308	−0.571538	−	0.820576i	$-0.693653\pi$
$241$	−4276.62	−1.14308	−0.571538	+	0.820576i	$0.693653\pi$
$242$	−661.945	−0.175832
$243$	0	0
$244$	17047.8	4.47285
$245$	0	0
$246$	0	0
$247$	174.005	0.0448245
$248$	454.075	0.116265
$249$	0	0
$250$	0	0
$251$	−3566.03	−0.896755	−0.448378	−	0.893844i	$-0.647998\pi$
$251$	−3566.03	−0.896755	−0.448378	+	0.893844i	$0.647998\pi$
$252$	0	0
$253$	417.055	0.103637
$254$	−7743.40	−1.91285
$255$	0	0
$256$	11832.1	2.88870
$257$	4672.89	1.13419	0.567095	−	0.823652i	$-0.308067\pi$
$257$	4672.89	1.13419	0.567095	+	0.823652i	$0.308067\pi$
$258$	0	0
$259$	−5213.81	−1.25085
$260$	0	0
$261$	0	0
$262$	−7920.21	−1.86760
$263$	5768.59	1.35250	0.676248	−	0.736674i	$-0.263605\pi$
$263$	5768.59	1.35250	0.676248	+	0.736674i	$0.263605\pi$
$264$	0	0
$265$	0	0
$266$	3608.28	0.831721
$267$	0	0
$268$	9405.92	2.14387
$269$	2852.10	0.646452	0.323226	−	0.946322i	$-0.395233\pi$
$269$	2852.10	0.646452	0.323226	+	0.946322i	$0.395233\pi$
$270$	0	0
$271$	2155.32	0.483123	0.241562	−	0.970385i	$-0.422340\pi$
$271$	2155.32	0.483123	0.241562	+	0.970385i	$0.422340\pi$
$272$	−21318.3	−4.75225
$273$	0	0
$274$	10468.4	2.30810
$275$	0	0
$276$	0	0
$277$	−5882.13	−1.27590	−0.637948	−	0.770080i	$-0.720216\pi$
$277$	−5882.13	−1.27590	−0.637948	+	0.770080i	$0.720216\pi$
$278$	10024.4	2.16267
$279$	0	0
$280$	0	0
$281$	1267.11	0.269001	0.134501	−	0.990914i	$-0.457057\pi$
$281$	1267.11	0.269001	0.134501	+	0.990914i	$0.457057\pi$
$282$	0	0
$283$	−3496.02	−0.734334	−0.367167	−	0.930155i	$-0.619672\pi$
$283$	−3496.02	−0.734334	−0.367167	+	0.930155i	$0.619672\pi$
$284$	18606.5	3.88765
$285$	0	0
$286$	256.243	0.0529790
$287$	−2598.99	−0.534541
$288$	0	0
$289$	2885.64	0.587348
$290$	0	0
$291$	0	0
$292$	17303.5	3.46785
$293$	−5566.81	−1.10995	−0.554977	−	0.831866i	$-0.687273\pi$
$293$	−5566.81	−1.10995	−0.554977	+	0.831866i	$0.687273\pi$
$294$	0	0
$295$	0	0
$296$	−24612.0	−4.83291
$297$	0	0
$298$	10330.1	2.00807
$299$	−161.445	−0.0312261
$300$	0	0
$301$	7055.81	1.35113
$302$	−521.404	−0.0993490
$303$	0	0
$304$	9864.62	1.86110
$305$	0	0
$306$	0	0
$307$	−9918.64	−1.84393	−0.921965	−	0.387274i	$-0.873417\pi$
$307$	−9918.64	−1.84393	−0.921965	+	0.387274i	$0.873417\pi$
$308$	3893.24	0.720254
$309$	0	0
$310$	0	0
$311$	2159.49	0.393741	0.196870	−	0.980430i	$-0.436922\pi$
$311$	2159.49	0.393741	0.196870	+	0.980430i	$0.436922\pi$
$312$	0	0
$313$	−1346.72	−0.243199	−0.121600	−	0.992579i	$-0.538802\pi$
$313$	−1346.72	−0.243199	−0.121600	+	0.992579i	$0.538802\pi$
$314$	19648.2	3.53125
$315$	0	0
$316$	19817.1	3.52784
$317$	6805.64	1.20581	0.602907	−	0.797812i	$-0.294009\pi$
$317$	6805.64	1.20581	0.602907	+	0.797812i	$0.294009\pi$
$318$	0	0
$319$	622.824	0.109315
$320$	0	0
$321$	0	0
$322$	−3347.83	−0.579402
$323$	−3608.67	−0.621647
$324$	0	0
$325$	0	0
$326$	−2324.76	−0.394959
$327$	0	0
$328$	−12268.6	−2.06531
$329$	−6414.46	−1.07489
$330$	0	0
$331$	−8942.71	−1.48500	−0.742501	−	0.669845i	$-0.766361\pi$
$331$	−8942.71	−1.48500	−0.742501	+	0.669845i	$0.766361\pi$
$332$	−27829.9	−4.60050
$333$	0	0
$334$	−4381.94	−0.717872
$335$	0	0
$336$	0	0
$337$	−248.202	−0.0401200	−0.0200600	−	0.999799i	$-0.506386\pi$
$337$	−248.202	−0.0401200	−0.0200600	+	0.999799i	$0.506386\pi$
$338$	11919.8	1.91819
$339$	0	0
$340$	0	0
$341$	65.5546	0.0104105
$342$	0	0
$343$	6867.50	1.08108
$344$	33307.2	5.22036
$345$	0	0
$346$	−2883.55	−0.448036
$347$	−1349.14	−0.208719	−0.104359	−	0.994540i	$-0.533279\pi$
$347$	−1349.14	−0.208719	−0.104359	+	0.994540i	$0.533279\pi$
$348$	0	0
$349$	−12041.5	−1.84689	−0.923447	−	0.383727i	$-0.874640\pi$
$349$	−12041.5	−1.84689	−0.923447	+	0.383727i	$0.874640\pi$
$350$	0	0
$351$	0	0
$352$	7821.87	1.18439
$353$	−4275.93	−0.644716	−0.322358	−	0.946618i	$-0.604476\pi$
$353$	−4275.93	−0.644716	−0.322358	+	0.946618i	$0.604476\pi$
$354$	0	0
$355$	0	0
$356$	5420.80	0.807028
$357$	0	0
$358$	−17222.1	−2.54251
$359$	8599.42	1.26423	0.632117	−	0.774873i	$-0.282186\pi$
$359$	8599.42	1.26423	0.632117	+	0.774873i	$0.282186\pi$
$360$	0	0
$361$	−5189.16	−0.756547
$362$	−5180.60	−0.752172
$363$	0	0
$364$	−1507.10	−0.217015
$365$	0	0
$366$	0	0
$367$	7366.68	1.04779	0.523893	−	0.851784i	$-0.324479\pi$
$367$	7366.68	1.04779	0.523893	+	0.851784i	$0.324479\pi$
$368$	−9152.59	−1.29650
$369$	0	0
$370$	0	0
$371$	−3406.73	−0.476735
$372$	0	0
$373$	1589.29	0.220618	0.110309	−	0.993897i	$-0.464816\pi$
$373$	1589.29	0.220618	0.110309	+	0.993897i	$0.464816\pi$
$374$	−5314.21	−0.734736
$375$	0	0
$376$	−30279.7	−4.15307
$377$	−241.099	−0.0329370
$378$	0	0
$379$	−436.884	−0.0592117	−0.0296058	−	0.999562i	$-0.509425\pi$
$379$	−436.884	−0.0592117	−0.0296058	+	0.999562i	$0.509425\pi$
$380$	0	0
$381$	0	0
$382$	−21588.6	−2.89154
$383$	−2874.52	−0.383501	−0.191751	−	0.981444i	$-0.561416\pi$
$383$	−2874.52	−0.383501	−0.191751	+	0.981444i	$0.561416\pi$
$384$	0	0
$385$	0	0
$386$	−27496.2	−3.62570
$387$	0	0
$388$	−4430.86	−0.579749
$389$	14560.8	1.89785	0.948923	−	0.315507i	$-0.102174\pi$
$389$	14560.8	1.89785	0.948923	+	0.315507i	$0.102174\pi$
$390$	0	0
$391$	3348.19	0.433057
$392$	6283.96	0.809663
$393$	0	0
$394$	−22067.0	−2.82163
$395$	0	0
$396$	0	0
$397$	242.624	0.0306724	0.0153362	−	0.999882i	$-0.495118\pi$
$397$	242.624	0.0306724	0.0153362	+	0.999882i	$0.495118\pi$
$398$	14311.5	1.80243
$399$	0	0
$400$	0	0
$401$	2836.58	0.353246	0.176623	−	0.984279i	$-0.443483\pi$
$401$	2836.58	0.353246	0.176623	+	0.984279i	$0.443483\pi$
$402$	0	0
$403$	−25.3766	−0.00313672
$404$	−42311.5	−5.21059
$405$	0	0
$406$	−4999.60	−0.611147
$407$	−3553.22	−0.432744
$408$	0	0
$409$	−9001.94	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$409$	−9001.94	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$410$	0	0
$411$	0	0
$412$	19031.9	2.27581
$413$	5462.14	0.650786
$414$	0	0
$415$	0	0
$416$	−3027.89	−0.356862
$417$	0	0
$418$	2459.05	0.287742
$419$	−10374.8	−1.20965	−0.604823	−	0.796360i	$-0.706756\pi$
$419$	−10374.8	−1.20965	−0.604823	+	0.796360i	$0.706756\pi$
$420$	0	0
$421$	−3936.91	−0.455756	−0.227878	−	0.973690i	$-0.573179\pi$
$421$	−3936.91	−0.455756	−0.227878	+	0.973690i	$0.573179\pi$
$422$	14259.0	1.64483
$423$	0	0
$424$	−16081.6	−1.84196
$425$	0	0
$426$	0	0
$427$	−12548.8	−1.42220
$428$	−6579.47	−0.743062
$429$	0	0
$430$	0	0
$431$	14452.5	1.61520	0.807601	−	0.589729i	$-0.200765\pi$
$431$	14452.5	1.61520	0.807601	+	0.589729i	$0.200765\pi$
$432$	0	0
$433$	13717.0	1.52240	0.761200	−	0.648518i	$-0.224611\pi$
$433$	13717.0	1.52240	0.761200	+	0.648518i	$0.224611\pi$
$434$	−526.227	−0.0582021
$435$	0	0
$436$	−7281.57	−0.799826
$437$	−1549.31	−0.169596
$438$	0	0
$439$	−811.243	−0.0881971	−0.0440985	−	0.999027i	$-0.514042\pi$
$439$	−811.243	−0.0881971	−0.0440985	+	0.999027i	$0.514042\pi$
$440$	0	0
$441$	0	0
$442$	2057.17	0.221379
$443$	−8100.18	−0.868739	−0.434369	−	0.900735i	$-0.643029\pi$
$443$	−8100.18	−0.868739	−0.434369	+	0.900735i	$0.643029\pi$
$444$	0	0
$445$	0	0
$446$	8790.59	0.933288
$447$	0	0
$448$	−31617.0	−3.33429
$449$	−9345.18	−0.982242	−0.491121	−	0.871091i	$-0.663413\pi$
$449$	−9345.18	−0.982242	−0.491121	+	0.871091i	$0.663413\pi$
$450$	0	0
$451$	−1771.21	−0.184930
$452$	−31443.6	−3.27208
$453$	0	0
$454$	−8886.33	−0.918626
$455$	0	0
$456$	0	0
$457$	−12315.1	−1.26056	−0.630282	−	0.776366i	$-0.717061\pi$
$457$	−12315.1	−1.26056	−0.630282	+	0.776366i	$0.717061\pi$
$458$	−34903.8	−3.56102
$459$	0	0
$460$	0	0
$461$	−7568.03	−0.764595	−0.382298	−	0.924039i	$-0.624867\pi$
$461$	−7568.03	−0.764595	−0.382298	+	0.924039i	$0.624867\pi$
$462$	0	0
$463$	4014.54	0.402963	0.201481	−	0.979492i	$-0.435424\pi$
$463$	4014.54	0.402963	0.201481	+	0.979492i	$0.435424\pi$
$464$	−13668.3	−1.36754
$465$	0	0
$466$	−20225.3	−2.01055
$467$	6585.39	0.652539	0.326270	−	0.945277i	$-0.394208\pi$
$467$	6585.39	0.652539	0.326270	+	0.945277i	$0.394208\pi$
$468$	0	0
$469$	−6923.63	−0.681670
$470$	0	0
$471$	0	0
$472$	25784.2	2.51444
$473$	4808.55	0.467436
$474$	0	0
$475$	0	0
$476$	31255.7	3.00966
$477$	0	0
$478$	31427.8	3.00726
$479$	−11795.1	−1.12512	−0.562558	−	0.826758i	$-0.690183\pi$
$479$	−11795.1	−1.12512	−0.562558	+	0.826758i	$0.690183\pi$
$480$	0	0
$481$	1375.48	0.130387
$482$	23395.8	2.21089
$483$	0	0
$484$	2653.25	0.249179
$485$	0	0
$486$	0	0
$487$	−13069.9	−1.21613	−0.608063	−	0.793889i	$-0.708053\pi$
$487$	−13069.9	−1.21613	−0.608063	+	0.793889i	$0.708053\pi$
$488$	−59236.9	−5.49494
$489$	0	0
$490$	0	0
$491$	4966.91	0.456524	0.228262	−	0.973600i	$-0.426696\pi$
$491$	4966.91	0.456524	0.228262	+	0.973600i	$0.426696\pi$
$492$	0	0
$493$	5000.14	0.456785
$494$	−951.913	−0.0866976
$495$	0	0
$496$	−1438.64	−0.130236
$497$	−13696.1	−1.23612
$498$	0	0
$499$	−12123.2	−1.08760	−0.543799	−	0.839216i	$-0.683015\pi$
$499$	−12123.2	−1.08760	−0.543799	+	0.839216i	$0.683015\pi$
$500$	0	0
$501$	0	0
$502$	19508.4	1.73447
$503$	6863.58	0.608414	0.304207	−	0.952606i	$-0.401609\pi$
$503$	6863.58	0.608414	0.304207	+	0.952606i	$0.401609\pi$
$504$	0	0
$505$	0	0
$506$	−2281.55	−0.200449
$507$	0	0
$508$	31037.6	2.71077
$509$	6183.49	0.538464	0.269232	−	0.963075i	$-0.413230\pi$
$509$	6183.49	0.538464	0.269232	+	0.963075i	$0.413230\pi$
$510$	0	0
$511$	−12737.0	−1.10264
$512$	−24510.3	−2.11565
$513$	0	0
$514$	−25563.6	−2.19370
$515$	0	0
$516$	0	0
$517$	−4371.46	−0.371870
$518$	28522.8	2.41934
$519$	0	0
$520$	0	0
$521$	2204.01	0.185335	0.0926674	−	0.995697i	$-0.470461\pi$
$521$	2204.01	0.185335	0.0926674	+	0.995697i	$0.470461\pi$
$522$	0	0
$523$	−4508.45	−0.376942	−0.188471	−	0.982079i	$-0.560353\pi$
$523$	−4508.45	−0.376942	−0.188471	+	0.982079i	$0.560353\pi$
$524$	31746.3	2.64665
$525$	0	0
$526$	−31557.8	−2.61594
$527$	526.284	0.0435015
$528$	0	0
$529$	−10729.5	−0.881854
$530$	0	0
$531$	0	0
$532$	−14462.9	−1.17866
$533$	685.648	0.0557199
$534$	0	0
$535$	0	0
$536$	−32683.2	−2.63377
$537$	0	0
$538$	−15602.8	−1.25034
$539$	907.213	0.0724980
$540$	0	0
$541$	−9335.59	−0.741901	−0.370950	−	0.928653i	$-0.620968\pi$
$541$	−9335.59	−0.741901	−0.370950	+	0.928653i	$0.620968\pi$
$542$	−11790.9	−0.934436
$543$	0	0
$544$	62795.3	4.94913
$545$	0	0
$546$	0	0
$547$	21554.8	1.68486	0.842430	−	0.538805i	$-0.181124\pi$
$547$	21554.8	1.68486	0.842430	+	0.538805i	$0.181124\pi$
$548$	−41960.2	−3.27089
$549$	0	0
$550$	0	0
$551$	−2313.72	−0.178889
$552$	0	0
$553$	−14587.2	−1.12172
$554$	32178.9	2.46778
$555$	0	0
$556$	−40180.3	−3.06480
$557$	890.535	0.0677436	0.0338718	−	0.999426i	$-0.489216\pi$
$557$	890.535	0.0677436	0.0338718	+	0.999426i	$0.489216\pi$
$558$	0	0
$559$	−1861.42	−0.140840
$560$	0	0
$561$	0	0
$562$	−6931.87	−0.520291
$563$	−12347.7	−0.924325	−0.462163	−	0.886795i	$-0.652926\pi$
$563$	−12347.7	−0.924325	−0.462163	+	0.886795i	$0.652926\pi$
$564$	0	0
$565$	0	0
$566$	19125.4	1.42032
$567$	0	0
$568$	−64652.9	−4.77601
$569$	−15315.4	−1.12839	−0.564196	−	0.825641i	$-0.690814\pi$
$569$	−15315.4	−1.12839	−0.564196	+	0.825641i	$0.690814\pi$
$570$	0	0
$571$	901.566	0.0660759	0.0330380	−	0.999454i	$-0.489482\pi$
$571$	901.566	0.0660759	0.0330380	+	0.999454i	$0.489482\pi$
$572$	−1027.09	−0.0750784
$573$	0	0
$574$	14218.1	1.03389
$575$	0	0
$576$	0	0
$577$	4109.31	0.296486	0.148243	−	0.988951i	$-0.452638\pi$
$577$	4109.31	0.296486	0.148243	+	0.988951i	$0.452638\pi$
$578$	−15786.3	−1.13602
$579$	0	0
$580$	0	0
$581$	20485.4	1.46278
$582$	0	0
$583$	−2321.69	−0.164931
$584$	−60125.4	−4.26028
$585$	0	0
$586$	30453.9	2.14683
$587$	4246.38	0.298581	0.149290	−	0.988793i	$-0.452301\pi$
$587$	4246.38	0.298581	0.149290	+	0.988793i	$0.452301\pi$
$588$	0	0
$589$	−243.528	−0.0170363
$590$	0	0
$591$	0	0
$592$	77978.1	5.41365
$593$	−20200.4	−1.39887	−0.699435	−	0.714696i	$-0.746565\pi$
$593$	−20200.4	−1.39887	−0.699435	+	0.714696i	$0.746565\pi$
$594$	0	0
$595$	0	0
$596$	−41405.7	−2.84571
$597$	0	0
$598$	883.204	0.0603961
$599$	−18031.3	−1.22995	−0.614976	−	0.788546i	$-0.710834\pi$
$599$	−18031.3	−1.22995	−0.614976	+	0.788546i	$0.710834\pi$
$600$	0	0
$601$	17708.1	1.20188	0.600938	−	0.799295i	$-0.294794\pi$
$601$	17708.1	1.20188	0.600938	+	0.799295i	$0.294794\pi$
$602$	−38599.7	−2.61330
$603$	0	0
$604$	2089.92	0.140791
$605$	0	0
$606$	0	0
$607$	24991.9	1.67115	0.835575	−	0.549376i	$-0.185135\pi$
$607$	24991.9	1.67115	0.835575	+	0.549376i	$0.185135\pi$
$608$	−29057.3	−1.93821
$609$	0	0
$610$	0	0
$611$	1692.22	0.112046
$612$	0	0
$613$	−6784.54	−0.447023	−0.223511	−	0.974701i	$-0.571752\pi$
$613$	−6784.54	−0.447023	−0.223511	+	0.974701i	$0.571752\pi$
$614$	54261.1	3.56645
$615$	0	0
$616$	−13528.0	−0.884838
$617$	−16032.0	−1.04607	−0.523034	−	0.852312i	$-0.675200\pi$
$617$	−16032.0	−1.04607	−0.523034	+	0.852312i	$0.675200\pi$
$618$	0	0
$619$	−23414.7	−1.52038	−0.760192	−	0.649698i	$-0.774895\pi$
$619$	−23414.7	−1.52038	−0.760192	+	0.649698i	$0.774895\pi$
$620$	0	0
$621$	0	0
$622$	−11813.7	−0.761556
$623$	−3990.21	−0.256604
$624$	0	0
$625$	0	0
$626$	7367.42	0.470386
$627$	0	0
$628$	−78755.3	−5.00426
$629$	−28525.9	−1.80827
$630$	0	0
$631$	−15251.2	−0.962189	−0.481094	−	0.876669i	$-0.659761\pi$
$631$	−15251.2	−0.962189	−0.481094	+	0.876669i	$0.659761\pi$
$632$	−68859.3	−4.33398
$633$	0	0
$634$	−37231.1	−2.33223
$635$	0	0
$636$	0	0
$637$	−351.188	−0.0218439
$638$	−3407.24	−0.211432
$639$	0	0
$640$	0	0
$641$	−6697.04	−0.412664	−0.206332	−	0.978482i	$-0.566153\pi$
$641$	−6697.04	−0.412664	−0.206332	+	0.978482i	$0.566153\pi$
$642$	0	0
$643$	25649.1	1.57310	0.786549	−	0.617528i	$-0.211866\pi$
$643$	25649.1	1.57310	0.786549	+	0.617528i	$0.211866\pi$
$644$	13419.0	0.821091
$645$	0	0
$646$	19741.7	1.20236
$647$	−26760.5	−1.62607	−0.813033	−	0.582218i	$-0.802185\pi$
$647$	−26760.5	−1.62607	−0.813033	+	0.582218i	$0.802185\pi$
$648$	0	0
$649$	3722.46	0.225145
$650$	0	0
$651$	0	0
$652$	9318.27	0.559711
$653$	−9094.20	−0.544998	−0.272499	−	0.962156i	$-0.587850\pi$
$653$	−9094.20	−0.544998	−0.272499	+	0.962156i	$0.587850\pi$
$654$	0	0
$655$	0	0
$656$	38870.6	2.31348
$657$	0	0
$658$	35091.1	2.07902
$659$	−5205.47	−0.307703	−0.153852	−	0.988094i	$-0.549168\pi$
$659$	−5205.47	−0.307703	−0.153852	+	0.988094i	$0.549168\pi$
$660$	0	0
$661$	7479.50	0.440119	0.220060	−	0.975486i	$-0.429375\pi$
$661$	7479.50	0.440119	0.220060	+	0.975486i	$0.429375\pi$
$662$	48922.2	2.87223
$663$	0	0
$664$	96702.0	5.65176
$665$	0	0
$666$	0	0
$667$	2146.71	0.124619
$668$	17564.0	1.01732
$669$	0	0
$670$	0	0
$671$	−8552.01	−0.492022
$672$	0	0
$673$	26740.4	1.53160	0.765799	−	0.643080i	$-0.222344\pi$
$673$	26740.4	1.53160	0.765799	+	0.643080i	$0.222344\pi$
$674$	1357.82	0.0775984
$675$	0	0
$676$	−47777.6	−2.71834
$677$	13179.9	0.748221	0.374110	−	0.927384i	$-0.377948\pi$
$677$	13179.9	0.748221	0.374110	+	0.927384i	$0.377948\pi$
$678$	0	0
$679$	3261.52	0.184338
$680$	0	0
$681$	0	0
$682$	−358.625	−0.0201355
$683$	23658.6	1.32543	0.662717	−	0.748870i	$-0.269403\pi$
$683$	23658.6	1.32543	0.662717	+	0.748870i	$0.269403\pi$
$684$	0	0
$685$	0	0
$686$	−37569.5	−2.09098
$687$	0	0
$688$	−105527.	−5.84765
$689$	898.742	0.0496942
$690$	0	0
$691$	13920.4	0.766362	0.383181	−	0.923673i	$-0.374828\pi$
$691$	13920.4	0.766362	0.383181	+	0.923673i	$0.374828\pi$
$692$	11558.0	0.634928
$693$	0	0
$694$	7380.61	0.403695
$695$	0	0
$696$	0	0
$697$	−14219.6	−0.772750
$698$	65874.4	3.57218
$699$	0	0
$700$	0	0
$701$	−6114.52	−0.329447	−0.164723	−	0.986340i	$-0.552673\pi$
$701$	−6114.52	−0.329447	−0.164723	+	0.986340i	$0.552673\pi$
$702$	0	0
$703$	13199.8	0.708165
$704$	−21547.0	−1.15353
$705$	0	0
$706$	23392.0	1.24698
$707$	31145.2	1.65677
$708$	0	0
$709$	6233.43	0.330185	0.165093	−	0.986278i	$-0.447208\pi$
$709$	6233.43	0.330185	0.165093	+	0.986278i	$0.447208\pi$
$710$	0	0
$711$	0	0
$712$	−18835.9	−0.991441
$713$	225.950	0.0118680
$714$	0	0
$715$	0	0
$716$	69030.9	3.60308
$717$	0	0
$718$	−47044.2	−2.44523
$719$	8718.05	0.452195	0.226098	−	0.974105i	$-0.427403\pi$
$719$	8718.05	0.452195	0.226098	+	0.974105i	$0.427403\pi$
$720$	0	0
$721$	−14009.2	−0.723620
$722$	28387.9	1.46328
$723$	0	0
$724$	20765.2	1.06593
$725$	0	0
$726$	0	0
$727$	−24431.1	−1.24635	−0.623177	−	0.782081i	$-0.714158\pi$
$727$	−24431.1	−1.24635	−0.623177	+	0.782081i	$0.714158\pi$
$728$	5236.79	0.266605
$729$	0	0
$730$	0	0
$731$	38603.9	1.95324
$732$	0	0
$733$	14558.6	0.733609	0.366804	−	0.930298i	$-0.380452\pi$
$733$	14558.6	0.733609	0.366804	+	0.930298i	$0.380452\pi$
$734$	−40300.3	−2.02658
$735$	0	0
$736$	26959.9	1.35021
$737$	−4718.46	−0.235830
$738$	0	0
$739$	−31004.4	−1.54332	−0.771662	−	0.636033i	$-0.780574\pi$
$739$	−31004.4	−1.54332	−0.771662	+	0.636033i	$0.780574\pi$
$740$	0	0
$741$	0	0
$742$	18636.9	0.922080
$743$	14739.8	0.727794	0.363897	−	0.931439i	$-0.381446\pi$
$743$	14739.8	0.727794	0.363897	+	0.931439i	$0.381446\pi$
$744$	0	0
$745$	0	0
$746$	−8694.42	−0.426710
$747$	0	0
$748$	21300.8	1.04122
$749$	4843.09	0.236265
$750$	0	0
$751$	−19553.8	−0.950105	−0.475053	−	0.879957i	$-0.657571\pi$
$751$	−19553.8	−0.950105	−0.475053	+	0.879957i	$0.657571\pi$
$752$	95935.0	4.65211
$753$	0	0
$754$	1318.96	0.0637053
$755$	0	0
$756$	0	0
$757$	19785.5	0.949954	0.474977	−	0.879998i	$-0.342456\pi$
$757$	19785.5	0.949954	0.474977	+	0.879998i	$0.342456\pi$
$758$	2390.03	0.114525
$759$	0	0
$760$	0	0
$761$	−25370.2	−1.20850	−0.604250	−	0.796795i	$-0.706527\pi$
$761$	−25370.2	−1.20850	−0.604250	+	0.796795i	$0.706527\pi$
$762$	0	0
$763$	5359.91	0.254314
$764$	86532.9	4.09771
$765$	0	0
$766$	15725.4	0.741752
$767$	−1440.99	−0.0678371
$768$	0	0
$769$	−40974.5	−1.92143	−0.960715	−	0.277537i	$-0.910482\pi$
$769$	−40974.5	−1.92143	−0.960715	+	0.277537i	$0.910482\pi$
$770$	0	0
$771$	0	0
$772$	110212.	5.13811
$773$	−36869.1	−1.71551	−0.857756	−	0.514058i	$-0.828142\pi$
$773$	−36869.1	−1.71551	−0.857756	+	0.514058i	$0.828142\pi$
$774$	0	0
$775$	0	0
$776$	15396.1	0.712227
$777$	0	0
$778$	−79656.7	−3.67073
$779$	6579.85	0.302629
$780$	0	0
$781$	−9333.92	−0.427649
$782$	−18316.7	−0.837601
$783$	0	0
$784$	−19909.5	−0.906954
$785$	0	0
$786$	0	0
$787$	−39992.9	−1.81143	−0.905714	−	0.423890i	$-0.860664\pi$
$787$	−39992.9	−1.81143	−0.905714	+	0.423890i	$0.860664\pi$
$788$	88450.6	3.99863
$789$	0	0
$790$	0	0
$791$	23145.4	1.04040
$792$	0	0
$793$	3310.54	0.148248
$794$	−1327.30	−0.0593253
$795$	0	0
$796$	−57364.1	−2.55429
$797$	23016.9	1.02296	0.511480	−	0.859295i	$-0.329097\pi$
$797$	23016.9	1.02296	0.511480	+	0.859295i	$0.329097\pi$
$798$	0	0
$799$	−35094.9	−1.55390
$800$	0	0
$801$	0	0
$802$	−15517.8	−0.683234
$803$	−8680.28	−0.381470
$804$	0	0
$805$	0	0
$806$	138.826	0.00606691
$807$	0	0
$808$	147022.	6.40125
$809$	−17595.7	−0.764687	−0.382343	−	0.924020i	$-0.624883\pi$
$809$	−17595.7	−0.764687	−0.382343	+	0.924020i	$0.624883\pi$
$810$	0	0
$811$	7677.42	0.332418	0.166209	−	0.986091i	$-0.446847\pi$
$811$	7677.42	0.332418	0.166209	+	0.986091i	$0.446847\pi$
$812$	20039.7	0.866079
$813$	0	0
$814$	19438.3	0.836995
$815$	0	0
$816$	0	0
$817$	−17863.2	−0.764937
$818$	49246.2	2.10496
$819$	0	0
$820$	0	0
$821$	−27596.1	−1.17309	−0.586547	−	0.809915i	$-0.699513\pi$
$821$	−27596.1	−1.17309	−0.586547	+	0.809915i	$0.699513\pi$
$822$	0	0
$823$	−1542.68	−0.0653394	−0.0326697	−	0.999466i	$-0.510401\pi$
$823$	−1542.68	−0.0653394	−0.0326697	+	0.999466i	$0.510401\pi$
$824$	−66130.9	−2.79585
$825$	0	0
$826$	−29881.3	−1.25872
$827$	−13108.6	−0.551187	−0.275593	−	0.961274i	$-0.588874\pi$
$827$	−13108.6	−0.551187	−0.275593	+	0.961274i	$0.588874\pi$
$828$	0	0
$829$	2179.55	0.0913137	0.0456568	−	0.998957i	$-0.485462\pi$
$829$	2179.55	0.0913137	0.0456568	+	0.998957i	$0.485462\pi$
$830$	0	0
$831$	0	0
$832$	8340.99	0.347562
$833$	7283.27	0.302941
$834$	0	0
$835$	0	0
$836$	−9856.53	−0.407769
$837$	0	0
$838$	56756.6	2.33965
$839$	29066.9	1.19607	0.598034	−	0.801470i	$-0.295949\pi$
$839$	29066.9	1.19607	0.598034	+	0.801470i	$0.295949\pi$
$840$	0	0
$841$	−21183.1	−0.868553
$842$	21537.4	0.881504
$843$	0	0
$844$	−57153.9	−2.33095
$845$	0	0
$846$	0	0
$847$	−1953.04	−0.0792293
$848$	50951.2	2.06329
$849$	0	0
$850$	0	0
$851$	−12247.0	−0.493329
$852$	0	0
$853$	−24515.0	−0.984030	−0.492015	−	0.870587i	$-0.663740\pi$
$853$	−24515.0	−0.984030	−0.492015	+	0.870587i	$0.663740\pi$
$854$	68649.6	2.75075
$855$	0	0
$856$	22862.0	0.912858
$857$	−7776.28	−0.309956	−0.154978	−	0.987918i	$-0.549531\pi$
$857$	−7776.28	−0.309956	−0.154978	+	0.987918i	$0.549531\pi$
$858$	0	0
$859$	18927.4	0.751797	0.375898	−	0.926661i	$-0.377334\pi$
$859$	18927.4	0.751797	0.375898	+	0.926661i	$0.377334\pi$
$860$	0	0
$861$	0	0
$862$	−79064.2	−3.12406
$863$	−10971.1	−0.432746	−0.216373	−	0.976311i	$-0.569423\pi$
$863$	−10971.1	−0.432746	−0.216373	+	0.976311i	$0.569423\pi$
$864$	0	0
$865$	0	0
$866$	−75040.8	−2.94456
$867$	0	0
$868$	2109.26	0.0824802
$869$	−9941.19	−0.388069
$870$	0	0
$871$	1826.55	0.0710565
$872$	25301.6	0.982593
$873$	0	0
$874$	8475.70	0.328026
$875$	0	0
$876$	0	0
$877$	−47950.7	−1.84627	−0.923136	−	0.384474i	$-0.874383\pi$
$877$	−47950.7	−1.84627	−0.923136	+	0.384474i	$0.874383\pi$
$878$	4438.01	0.170587
$879$	0	0
$880$	0	0
$881$	−17065.5	−0.652612	−0.326306	−	0.945264i	$-0.605804\pi$
$881$	−17065.5	−0.652612	−0.326306	+	0.945264i	$0.605804\pi$
$882$	0	0
$883$	50399.3	1.92080	0.960402	−	0.278618i	$-0.0898763\pi$
$883$	50399.3	1.92080	0.960402	+	0.278618i	$0.0898763\pi$
$884$	−8245.67	−0.313724
$885$	0	0
$886$	44313.1	1.68028
$887$	−46843.9	−1.77324	−0.886620	−	0.462498i	$-0.846953\pi$
$887$	−46843.9	−1.77324	−0.886620	+	0.462498i	$0.846953\pi$
$888$	0	0
$889$	−22846.5	−0.861921
$890$	0	0
$891$	0	0
$892$	−35235.0	−1.32260
$893$	16239.5	0.608548
$894$	0	0
$895$	0	0
$896$	81145.3	3.02553
$897$	0	0
$898$	51124.0	1.89981
$899$	337.430	0.0125183
$900$	0	0
$901$	−18638.9	−0.689182
$902$	9689.65	0.357683
$903$	0	0
$904$	109258.	4.01978
$905$	0	0
$906$	0	0
$907$	−30936.3	−1.13255	−0.566275	−	0.824217i	$-0.691616\pi$
$907$	−30936.3	−1.13255	−0.566275	+	0.824217i	$0.691616\pi$
$908$	35618.8	1.30182
$909$	0	0
$910$	0	0
$911$	−8721.86	−0.317199	−0.158599	−	0.987343i	$-0.550698\pi$
$911$	−8721.86	−0.317199	−0.158599	+	0.987343i	$0.550698\pi$
$912$	0	0
$913$	13960.8	0.506064
$914$	67371.5	2.43813
$915$	0	0
$916$	139904.	5.04645
$917$	−23368.2	−0.841534
$918$	0	0
$919$	−44433.1	−1.59490	−0.797450	−	0.603386i	$-0.793818\pi$
$919$	−44433.1	−1.59490	−0.797450	+	0.603386i	$0.793818\pi$
$920$	0	0
$921$	0	0
$922$	41401.8	1.47885
$923$	3613.22	0.128852
$924$	0	0
$925$	0	0
$926$	−21962.1	−0.779393
$927$	0	0
$928$	40261.5	1.42419
$929$	11331.5	0.400187	0.200093	−	0.979777i	$-0.435875\pi$
$929$	11331.5	0.400187	0.200093	+	0.979777i	$0.435875\pi$
$930$	0	0
$931$	−3370.19	−0.118640
$932$	81068.2	2.84923
$933$	0	0
$934$	−36026.2	−1.26211
$935$	0	0
$936$	0	0
$937$	−16721.3	−0.582989	−0.291495	−	0.956572i	$-0.594153\pi$
$937$	−16721.3	−0.582989	−0.291495	+	0.956572i	$0.594153\pi$
$938$	37876.6	1.31846
$939$	0	0
$940$	0	0
$941$	28480.6	0.986655	0.493327	−	0.869844i	$-0.335780\pi$
$941$	28480.6	0.986655	0.493327	+	0.869844i	$0.335780\pi$
$942$	0	0
$943$	−6104.92	−0.210820
$944$	−81692.1	−2.81658
$945$	0	0
$946$	−26305.7	−0.904095
$947$	−47080.3	−1.61553	−0.807763	−	0.589507i	$-0.799322\pi$
$947$	−47080.3	−1.61553	−0.807763	+	0.589507i	$0.799322\pi$
$948$	0	0
$949$	3360.19	0.114938
$950$	0	0
$951$	0	0
$952$	−108606.	−3.69740
$953$	−15510.5	−0.527213	−0.263607	−	0.964630i	$-0.584912\pi$
$953$	−15510.5	−0.527213	−0.263607	+	0.964630i	$0.584912\pi$
$954$	0	0
$955$	0	0
$956$	−125971.	−4.26170
$957$	0	0
$958$	64526.4	2.17615
$959$	30886.6	1.04002
$960$	0	0
$961$	−29755.5	−0.998808
$962$	−7524.71	−0.252189
$963$	0	0
$964$	−93776.4	−3.13313
$965$	0	0
$966$	0	0
$967$	20926.2	0.695906	0.347953	−	0.937512i	$-0.386877\pi$
$967$	20926.2	0.695906	0.347953	+	0.937512i	$0.386877\pi$
$968$	−9219.39	−0.306118
$969$	0	0
$970$	0	0
$971$	15375.7	0.508166	0.254083	−	0.967182i	$-0.418226\pi$
$971$	15375.7	0.508166	0.254083	+	0.967182i	$0.418226\pi$
$972$	0	0
$973$	29576.4	0.974488
$974$	71500.5	2.35218
$975$	0	0
$976$	187680.	6.15522
$977$	43839.1	1.43555	0.717777	−	0.696273i	$-0.245160\pi$
$977$	43839.1	1.43555	0.717777	+	0.696273i	$0.245160\pi$
$978$	0	0
$979$	−2719.34	−0.0887746
$980$	0	0
$981$	0	0
$982$	−27172.1	−0.882990
$983$	−32179.2	−1.04411	−0.522054	−	0.852912i	$-0.674834\pi$
$983$	−32179.2	−1.04411	−0.522054	+	0.852912i	$0.674834\pi$
$984$	0	0
$985$	0	0
$986$	−27353.9	−0.883494
$987$	0	0
$988$	3815.52	0.122862
$989$	16573.8	0.532878
$990$	0	0
$991$	5334.79	0.171004	0.0855021	−	0.996338i	$-0.472751\pi$
$991$	5334.79	0.171004	0.0855021	+	0.996338i	$0.472751\pi$
$992$	4237.68	0.135632
$993$	0	0
$994$	74926.2	2.39086
$995$	0	0
$996$	0	0
$997$	−10593.9	−0.336520	−0.168260	−	0.985743i	$-0.553815\pi$
$997$	−10593.9	−0.336520	−0.168260	+	0.985743i	$0.553815\pi$
$998$	66321.7	2.10358
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bv.1.1	yes	10
3.2	odd	2		2475.4.a.by.1.10	yes	10
5.4	even	2		2475.4.a.bx.1.10	yes	10
15.14	odd	2		2475.4.a.bu.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.4.a.bu.1.1	✓	10	15.14	odd	2
2475.4.a.bv.1.1	yes	10	1.1	even	1	trivial
2475.4.a.bx.1.10	yes	10	5.4	even	2
2475.4.a.by.1.10	yes	10	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-5.47062 q^{2} +21.9277 q^{4} -16.1408 q^{7} -76.1933 q^{8} +O(q^{10})\)	\(q-5.47062 q^{2} +21.9277 q^{4} -16.1408 q^{7} -76.1933 q^{8} -11.0000 q^{11} +4.25817 q^{13} +88.3003 q^{14} +241.403 q^{16} -88.3099 q^{17} +40.8637 q^{19} +60.1769 q^{22} -37.9141 q^{23} -23.2948 q^{26} -353.931 q^{28} -56.6204 q^{29} -5.95951 q^{31} -711.079 q^{32} +483.110 q^{34} +323.020 q^{37} -223.550 q^{38} +161.020 q^{41} -437.141 q^{43} -241.205 q^{44} +207.414 q^{46} +397.406 q^{47} -82.4739 q^{49} +93.3719 q^{52} +211.063 q^{53} +1229.82 q^{56} +309.749 q^{58} -338.406 q^{59} +777.456 q^{61} +32.6022 q^{62} +1958.82 q^{64} +428.951 q^{67} -1936.44 q^{68} +848.538 q^{71} +789.116 q^{73} -1767.12 q^{74} +896.048 q^{76} +177.549 q^{77} +903.745 q^{79} -880.877 q^{82} -1269.17 q^{83} +2391.43 q^{86} +838.126 q^{88} +247.212 q^{89} -68.7303 q^{91} -831.370 q^{92} -2174.06 q^{94} -202.067 q^{97} +451.184 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8}+O(q^{10}) \) 10 * q - 4 * q^2 + 34 * q^4 + 2 * q^7 - 48 * q^8	\( 10 q - 4 q^{2} + 34 q^{4} + 2 q^{7} - 48 q^{8} - 110 q^{11} + 26 q^{13} + 72 q^{14} + 206 q^{16} - 148 q^{17} - 114 q^{19} + 44 q^{22} + 34 q^{23} + 100 q^{26} - 86 q^{28} - 38 q^{29} + 232 q^{31} - 448 q^{32} - 20 q^{34} + 754 q^{37} - 780 q^{38} + 160 q^{41} - 66 q^{43} - 374 q^{44} + 682 q^{46} - 450 q^{47} + 590 q^{49} + 200 q^{52} - 1068 q^{53} + 268 q^{56} - 138 q^{58} - 838 q^{59} - 566 q^{61} - 1230 q^{62} + 462 q^{64} + 430 q^{67} - 2234 q^{68} + 518 q^{71} - 184 q^{73} + 402 q^{74} + 386 q^{76} - 22 q^{77} + 956 q^{79} - 2180 q^{82} - 2094 q^{83} + 892 q^{86} + 528 q^{88} - 512 q^{89} - 858 q^{91} - 4476 q^{92} - 294 q^{94} - 1006 q^{97} - 2226 q^{98}+O(q^{100}) \) 10 * q - 4 * q^2 + 34 * q^4 + 2 * q^7 - 48 * q^8 - 110 * q^11 + 26 * q^13 + 72 * q^14 + 206 * q^16 - 148 * q^17 - 114 * q^19 + 44 * q^22 + 34 * q^23 + 100 * q^26 - 86 * q^28 - 38 * q^29 + 232 * q^31 - 448 * q^32 - 20 * q^34 + 754 * q^37 - 780 * q^38 + 160 * q^41 - 66 * q^43 - 374 * q^44 + 682 * q^46 - 450 * q^47 + 590 * q^49 + 200 * q^52 - 1068 * q^53 + 268 * q^56 - 138 * q^58 - 838 * q^59 - 566 * q^61 - 1230 * q^62 + 462 * q^64 + 430 * q^67 - 2234 * q^68 + 518 * q^71 - 184 * q^73 + 402 * q^74 + 386 * q^76 - 22 * q^77 + 956 * q^79 - 2180 * q^82 - 2094 * q^83 + 892 * q^86 + 528 * q^88 - 512 * q^89 - 858 * q^91 - 4476 * q^92 - 294 * q^94 - 1006 * q^97 - 2226 * q^98

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