LMFDB - Embedded newform 2475.4.a.l.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-5.56155 q^{2} +22.9309 q^{4} -6.05398 q^{7} -83.0388 q^{8} +O(q^{10})$	$q-5.56155 q^{2} +22.9309 q^{4} -6.05398 q^{7} -83.0388 q^{8} +11.0000 q^{11} +4.38447 q^{13} +33.6695 q^{14} +278.378 q^{16} -110.546 q^{17} -94.2699 q^{19} -61.1771 q^{22} +15.7538 q^{23} -24.3845 q^{26} -138.823 q^{28} +256.870 q^{29} -170.702 q^{31} -883.902 q^{32} +614.810 q^{34} +190.853 q^{37} +524.287 q^{38} -249.602 q^{41} -291.602 q^{43} +252.240 q^{44} -87.6155 q^{46} +182.155 q^{47} -306.349 q^{49} +100.540 q^{52} -289.902 q^{53} +502.715 q^{56} -1428.60 q^{58} -282.725 q^{59} +167.825 q^{61} +949.366 q^{62} +2688.85 q^{64} +176.233 q^{67} -2534.93 q^{68} -919.255 q^{71} -154.570 q^{73} -1061.44 q^{74} -2161.69 q^{76} -66.5937 q^{77} -882.017 q^{79} +1388.18 q^{82} +277.619 q^{83} +1621.76 q^{86} -913.427 q^{88} +977.147 q^{89} -26.5435 q^{91} +361.248 q^{92} -1013.07 q^{94} +1102.94 q^{97} +1703.78 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7 q^{2} + 17 q^{4} + 25 q^{7} - 63 q^{8}+O(q^{10}) $ 2 * q - 7 * q^2 + 17 * q^4 + 25 * q^7 - 63 * q^8	$ 2 q - 7 q^{2} + 17 q^{4} + 25 q^{7} - 63 q^{8} + 22 q^{11} + 50 q^{13} - 11 q^{14} + 297 q^{16} - 151 q^{17} - 3 q^{19} - 77 q^{22} + 48 q^{23} - 90 q^{26} - 323 q^{28} + 221 q^{29} + 141 q^{31} - 1071 q^{32} + 673 q^{34} + 559 q^{37} + 393 q^{38} + 144 q^{41} + 60 q^{43} + 187 q^{44} - 134 q^{46} - 48 q^{47} + 315 q^{49} - 170 q^{52} + 117 q^{53} + 1125 q^{56} - 1377 q^{58} + 86 q^{59} - 155 q^{61} + 501 q^{62} + 2809 q^{64} - 266 q^{67} - 2295 q^{68} - 1587 q^{71} - 70 q^{73} - 1591 q^{74} - 2703 q^{76} + 275 q^{77} - 1294 q^{79} + 822 q^{82} - 558 q^{83} + 1116 q^{86} - 693 q^{88} + 1777 q^{89} + 1390 q^{91} + 170 q^{92} - 682 q^{94} + 334 q^{97} + 810 q^{98}+O(q^{100}) $ 2 * q - 7 * q^2 + 17 * q^4 + 25 * q^7 - 63 * q^8 + 22 * q^11 + 50 * q^13 - 11 * q^14 + 297 * q^16 - 151 * q^17 - 3 * q^19 - 77 * q^22 + 48 * q^23 - 90 * q^26 - 323 * q^28 + 221 * q^29 + 141 * q^31 - 1071 * q^32 + 673 * q^34 + 559 * q^37 + 393 * q^38 + 144 * q^41 + 60 * q^43 + 187 * q^44 - 134 * q^46 - 48 * q^47 + 315 * q^49 - 170 * q^52 + 117 * q^53 + 1125 * q^56 - 1377 * q^58 + 86 * q^59 - 155 * q^61 + 501 * q^62 + 2809 * q^64 - 266 * q^67 - 2295 * q^68 - 1587 * q^71 - 70 * q^73 - 1591 * q^74 - 2703 * q^76 + 275 * q^77 - 1294 * q^79 + 822 * q^82 - 558 * q^83 + 1116 * q^86 - 693 * q^88 + 1777 * q^89 + 1390 * q^91 + 170 * q^92 - 682 * q^94 + 334 * q^97 + 810 * 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 $

$2$	−5.56155	−1.96631	−0.983153	−	0.182785i	$-0.941489\pi$
$2$	−5.56155	−1.96631	−0.983153	+	0.182785i	$0.941489\pi$
$3$	0	0
$3$	0	0
$4$	22.9309	2.86636
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.05398	−0.326884	−0.163442	−	0.986553i	$-0.552260\pi$
$7$	−6.05398	−0.326884	−0.163442	+	0.986553i	$0.552260\pi$
$8$	−83.0388	−3.66983
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	4.38447	0.0935411	0.0467705	−	0.998906i	$-0.485107\pi$
$13$	4.38447	0.0935411	0.0467705	+	0.998906i	$0.485107\pi$
$14$	33.6695	0.642754
$15$	0	0
$16$	278.378	4.34965
$17$	−110.546	−1.57714	−0.788572	−	0.614943i	$-0.789179\pi$
$17$	−110.546	−1.57714	−0.788572	+	0.614943i	$0.789179\pi$
$18$	0	0
$19$	−94.2699	−1.13826	−0.569131	−	0.822247i	$-0.692720\pi$
$19$	−94.2699	−1.13826	−0.569131	+	0.822247i	$0.692720\pi$
$20$	0	0
$21$	0	0
$22$	−61.1771	−0.592864
$23$	15.7538	0.142821	0.0714107	−	0.997447i	$-0.477250\pi$
$23$	15.7538	0.142821	0.0714107	+	0.997447i	$0.477250\pi$
$24$	0	0
$25$	0	0
$26$	−24.3845	−0.183930
$27$	0	0
$28$	−138.823	−0.936967
$29$	256.870	1.64481	0.822407	−	0.568900i	$-0.192631\pi$
$29$	256.870	1.64481	0.822407	+	0.568900i	$0.192631\pi$
$30$	0	0
$31$	−170.702	−0.988998	−0.494499	−	0.869178i	$-0.664648\pi$
$31$	−170.702	−0.988998	−0.494499	+	0.869178i	$0.664648\pi$
$32$	−883.902	−4.88292
$33$	0	0
$34$	614.810	3.10115
$35$	0	0
$36$	0	0
$37$	190.853	0.848002	0.424001	−	0.905662i	$-0.360625\pi$
$37$	190.853	0.848002	0.424001	+	0.905662i	$0.360625\pi$
$38$	524.287	2.23817
$39$	0	0
$40$	0	0
$41$	−249.602	−0.950764	−0.475382	−	0.879780i	$-0.657690\pi$
$41$	−249.602	−0.950764	−0.475382	+	0.879780i	$0.657690\pi$
$42$	0	0
$43$	−291.602	−1.03416	−0.517081	−	0.855937i	$-0.672981\pi$
$43$	−291.602	−1.03416	−0.517081	+	0.855937i	$0.672981\pi$
$44$	252.240	0.864240
$45$	0	0
$46$	−87.6155	−0.280831
$47$	182.155	0.565321	0.282660	−	0.959220i	$-0.408783\pi$
$47$	182.155	0.565321	0.282660	+	0.959220i	$0.408783\pi$
$48$	0	0
$49$	−306.349	−0.893147
$50$	0	0
$51$	0	0
$52$	100.540	0.268122
$53$	−289.902	−0.751343	−0.375671	−	0.926753i	$-0.622588\pi$
$53$	−289.902	−0.751343	−0.375671	+	0.926753i	$0.622588\pi$
$54$	0	0
$55$	0	0
$56$	502.715	1.19961
$57$	0	0
$58$	−1428.60	−3.23421
$59$	−282.725	−0.623859	−0.311930	−	0.950105i	$-0.600975\pi$
$59$	−282.725	−0.623859	−0.311930	+	0.950105i	$0.600975\pi$
$60$	0	0
$61$	167.825	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$61$	167.825	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$62$	949.366	1.94467
$63$	0	0
$64$	2688.85	5.25166
$65$	0	0
$66$	0	0
$67$	176.233	0.321347	0.160674	−	0.987008i	$-0.448633\pi$
$67$	176.233	0.321347	0.160674	+	0.987008i	$0.448633\pi$
$68$	−2534.93	−4.52066
$69$	0	0
$70$	0	0
$71$	−919.255	−1.53656	−0.768278	−	0.640116i	$-0.778886\pi$
$71$	−919.255	−1.53656	−0.768278	+	0.640116i	$0.778886\pi$
$72$	0	0
$73$	−154.570	−0.247823	−0.123911	−	0.992293i	$-0.539544\pi$
$73$	−154.570	−0.247823	−0.123911	+	0.992293i	$0.539544\pi$
$74$	−1061.44	−1.66743
$75$	0	0
$76$	−2161.69	−3.26267
$77$	−66.5937	−0.0985592
$78$	0	0
$79$	−882.017	−1.25614	−0.628068	−	0.778159i	$-0.716154\pi$
$79$	−882.017	−1.25614	−0.628068	+	0.778159i	$0.716154\pi$
$80$	0	0
$81$	0	0
$82$	1388.18	1.86949
$83$	277.619	0.367141	0.183570	−	0.983007i	$-0.441234\pi$
$83$	277.619	0.367141	0.183570	+	0.983007i	$0.441234\pi$
$84$	0	0
$85$	0	0
$86$	1621.76	2.03348
$87$	0	0
$88$	−913.427	−1.10650
$89$	977.147	1.16379	0.581895	−	0.813264i	$-0.302311\pi$
$89$	977.147	1.16379	0.581895	+	0.813264i	$0.302311\pi$
$90$	0	0
$91$	−26.5435	−0.0305771
$92$	361.248	0.409377
$93$	0	0
$94$	−1013.07	−1.11159
$95$	0	0
$96$	0	0
$97$	1102.94	1.15451	0.577253	−	0.816565i	$-0.304125\pi$
$97$	1102.94	1.15451	0.577253	+	0.816565i	$0.304125\pi$
$98$	1703.78	1.75620
$99$	0	0
$100$	0	0
$101$	−484.314	−0.477139	−0.238570	−	0.971125i	$-0.576679\pi$
$101$	−484.314	−0.477139	−0.238570	+	0.971125i	$0.576679\pi$
$102$	0	0
$103$	874.419	0.836495	0.418248	−	0.908333i	$-0.362644\pi$
$103$	874.419	0.836495	0.418248	+	0.908333i	$0.362644\pi$
$104$	−364.081	−0.343280
$105$	0	0
$106$	1612.31	1.47737
$107$	119.845	0.108279	0.0541394	−	0.998533i	$-0.482758\pi$
$107$	119.845	0.108279	0.0541394	+	0.998533i	$0.482758\pi$
$108$	0	0
$109$	414.621	0.364344	0.182172	−	0.983267i	$-0.441687\pi$
$109$	414.621	0.364344	0.182172	+	0.983267i	$0.441687\pi$
$110$	0	0
$111$	0	0
$112$	−1685.29	−1.42183
$113$	534.453	0.444930	0.222465	−	0.974941i	$-0.428590\pi$
$113$	534.453	0.444930	0.222465	+	0.974941i	$0.428590\pi$
$114$	0	0
$115$	0	0
$116$	5890.26	4.71463
$117$	0	0
$118$	1572.39	1.22670
$119$	669.245	0.515543
$120$	0	0
$121$	121.000	0.0909091
$122$	−933.366	−0.692648
$123$	0	0
$124$	−3914.34	−2.83482
$125$	0	0
$126$	0	0
$127$	−640.121	−0.447256	−0.223628	−	0.974675i	$-0.571790\pi$
$127$	−640.121	−0.447256	−0.223628	+	0.974675i	$0.571790\pi$
$128$	−7882.95	−5.44344
$129$	0	0
$130$	0	0
$131$	−1051.05	−0.700999	−0.350499	−	0.936563i	$-0.613988\pi$
$131$	−1051.05	−0.700999	−0.350499	+	0.936563i	$0.613988\pi$
$132$	0	0
$133$	570.708	0.372080
$134$	−980.129	−0.631867
$135$	0	0
$136$	9179.64	5.78785
$137$	1690.68	1.05434	0.527169	−	0.849761i	$-0.323254\pi$
$137$	1690.68	1.05434	0.527169	+	0.849761i	$0.323254\pi$
$138$	0	0
$139$	2789.43	1.70213	0.851067	−	0.525058i	$-0.175956\pi$
$139$	2789.43	1.70213	0.851067	+	0.525058i	$0.175956\pi$
$140$	0	0
$141$	0	0
$142$	5112.48	3.02134
$143$	48.2292	0.0282037
$144$	0	0
$145$	0	0
$146$	859.650	0.487295
$147$	0	0
$148$	4376.43	2.43068
$149$	−1090.62	−0.599647	−0.299823	−	0.953995i	$-0.596928\pi$
$149$	−1090.62	−0.599647	−0.299823	+	0.953995i	$0.596928\pi$
$150$	0	0
$151$	623.574	0.336064	0.168032	−	0.985782i	$-0.446259\pi$
$151$	623.574	0.336064	0.168032	+	0.985782i	$0.446259\pi$
$152$	7828.06	4.17723
$153$	0	0
$154$	370.365	0.193798
$155$	0	0
$156$	0	0
$157$	2114.96	1.07511	0.537555	−	0.843228i	$-0.319348\pi$
$157$	2114.96	1.07511	0.537555	+	0.843228i	$0.319348\pi$
$158$	4905.38	2.46995
$159$	0	0
$160$	0	0
$161$	−95.3730	−0.0466860
$162$	0	0
$163$	−1153.53	−0.554301	−0.277151	−	0.960827i	$-0.589390\pi$
$163$	−1153.53	−0.554301	−0.277151	+	0.960827i	$0.589390\pi$
$164$	−5723.60	−2.72523
$165$	0	0
$166$	−1543.99	−0.721911
$167$	−1100.93	−0.510137	−0.255068	−	0.966923i	$-0.582098\pi$
$167$	−1100.93	−0.510137	−0.255068	+	0.966923i	$0.582098\pi$
$168$	0	0
$169$	−2177.78	−0.991250
$170$	0	0
$171$	0	0
$172$	−6686.69	−2.96428
$173$	−2369.25	−1.04122	−0.520609	−	0.853795i	$-0.674295\pi$
$173$	−2369.25	−1.04122	−0.520609	+	0.853795i	$0.674295\pi$
$174$	0	0
$175$	0	0
$176$	3062.16	1.31147
$177$	0	0
$178$	−5434.45	−2.28837
$179$	−1226.77	−0.512250	−0.256125	−	0.966644i	$-0.582446\pi$
$179$	−1226.77	−0.512250	−0.256125	+	0.966644i	$0.582446\pi$
$180$	0	0
$181$	−439.606	−0.180528	−0.0902642	−	0.995918i	$-0.528771\pi$
$181$	−439.606	−0.180528	−0.0902642	+	0.995918i	$0.528771\pi$
$182$	147.623	0.0601239
$183$	0	0
$184$	−1308.18	−0.524131
$185$	0	0
$186$	0	0
$187$	−1216.01	−0.475527
$188$	4176.98	1.62041
$189$	0	0
$190$	0	0
$191$	4968.96	1.88241	0.941207	−	0.337829i	$-0.109693\pi$
$191$	4968.96	1.88241	0.941207	+	0.337829i	$0.109693\pi$
$192$	0	0
$193$	1362.22	0.508054	0.254027	−	0.967197i	$-0.418245\pi$
$193$	1362.22	0.508054	0.254027	+	0.967197i	$0.418245\pi$
$194$	−6134.09	−2.27011
$195$	0	0
$196$	−7024.86	−2.56008
$197$	2195.91	0.794174	0.397087	−	0.917781i	$-0.370021\pi$
$197$	2195.91	0.794174	0.397087	+	0.917781i	$0.370021\pi$
$198$	0	0
$199$	−558.189	−0.198839	−0.0994194	−	0.995046i	$-0.531699\pi$
$199$	−558.189	−0.198839	−0.0994194	+	0.995046i	$0.531699\pi$
$200$	0	0
$201$	0	0
$202$	2693.54	0.938202
$203$	−1555.09	−0.537663
$204$	0	0
$205$	0	0
$206$	−4863.12	−1.64481
$207$	0	0
$208$	1220.54	0.406871
$209$	−1036.97	−0.343199
$210$	0	0
$211$	−3002.01	−0.979463	−0.489732	−	0.871873i	$-0.662905\pi$
$211$	−3002.01	−0.979463	−0.489732	+	0.871873i	$0.662905\pi$
$212$	−6647.71	−2.15362
$213$	0	0
$214$	−666.523	−0.212909
$215$	0	0
$216$	0	0
$217$	1033.42	0.323287
$218$	−2305.94	−0.716412
$219$	0	0
$220$	0	0
$221$	−484.688	−0.147528
$222$	0	0
$223$	−854.595	−0.256627	−0.128314	−	0.991734i	$-0.540956\pi$
$223$	−854.595	−0.256627	−0.128314	+	0.991734i	$0.540956\pi$
$224$	5351.12	1.59615
$225$	0	0
$226$	−2972.39	−0.874868
$227$	394.002	0.115202	0.0576010	−	0.998340i	$-0.481655\pi$
$227$	394.002	0.115202	0.0576010	+	0.998340i	$0.481655\pi$
$228$	0	0
$229$	−491.822	−0.141924	−0.0709618	−	0.997479i	$-0.522607\pi$
$229$	−491.822	−0.141924	−0.0709618	+	0.997479i	$0.522607\pi$
$230$	0	0
$231$	0	0
$232$	−21330.2	−6.03619
$233$	−6884.63	−1.93574	−0.967869	−	0.251456i	$-0.919091\pi$
$233$	−6884.63	−1.93574	−0.967869	+	0.251456i	$0.919091\pi$
$234$	0	0
$235$	0	0
$236$	−6483.14	−1.78820
$237$	0	0
$238$	−3722.04	−1.01371
$239$	−3012.77	−0.815397	−0.407699	−	0.913117i	$-0.633669\pi$
$239$	−3012.77	−0.815397	−0.407699	+	0.913117i	$0.633669\pi$
$240$	0	0
$241$	3106.98	0.830448	0.415224	−	0.909719i	$-0.363703\pi$
$241$	3106.98	0.830448	0.415224	+	0.909719i	$0.363703\pi$
$242$	−672.948	−0.178755
$243$	0	0
$244$	3848.37	1.00970
$245$	0	0
$246$	0	0
$247$	−413.324	−0.106474
$248$	14174.9	3.62946
$249$	0	0
$250$	0	0
$251$	834.313	0.209806	0.104903	−	0.994482i	$-0.466547\pi$
$251$	834.313	0.209806	0.104903	+	0.994482i	$0.466547\pi$
$252$	0	0
$253$	173.292	0.0430623
$254$	3560.07	0.879443
$255$	0	0
$256$	22330.6	5.45182
$257$	−7536.63	−1.82927	−0.914635	−	0.404281i	$-0.867522\pi$
$257$	−7536.63	−1.82927	−0.914635	+	0.404281i	$0.867522\pi$
$258$	0	0
$259$	−1155.42	−0.277198
$260$	0	0
$261$	0	0
$262$	5845.48	1.37838
$263$	−6242.10	−1.46351	−0.731757	−	0.681565i	$-0.761300\pi$
$263$	−6242.10	−1.46351	−0.731757	+	0.681565i	$0.761300\pi$
$264$	0	0
$265$	0	0
$266$	−3174.02	−0.731623
$267$	0	0
$268$	4041.17	0.921097
$269$	−1636.95	−0.371027	−0.185514	−	0.982642i	$-0.559395\pi$
$269$	−1636.95	−0.371027	−0.185514	+	0.982642i	$0.559395\pi$
$270$	0	0
$271$	787.212	0.176457	0.0882283	−	0.996100i	$-0.471880\pi$
$271$	787.212	0.176457	0.0882283	+	0.996100i	$0.471880\pi$
$272$	−30773.7	−6.86003
$273$	0	0
$274$	−9402.78	−2.07315
$275$	0	0
$276$	0	0
$277$	−1954.96	−0.424052	−0.212026	−	0.977264i	$-0.568006\pi$
$277$	−1954.96	−0.424052	−0.212026	+	0.977264i	$0.568006\pi$
$278$	−15513.6	−3.34691
$279$	0	0
$280$	0	0
$281$	5097.58	1.08219	0.541097	−	0.840960i	$-0.318009\pi$
$281$	5097.58	1.08219	0.541097	+	0.840960i	$0.318009\pi$
$282$	0	0
$283$	7187.71	1.50977	0.754885	−	0.655857i	$-0.227693\pi$
$283$	7187.71	1.50977	0.754885	+	0.655857i	$0.227693\pi$
$284$	−21079.3	−4.40432
$285$	0	0
$286$	−268.229	−0.0554571
$287$	1511.09	0.310789
$288$	0	0
$289$	7307.51	1.48738
$290$	0	0
$291$	0	0
$292$	−3544.43	−0.710349
$293$	6388.82	1.27385	0.636926	−	0.770925i	$-0.280206\pi$
$293$	6388.82	1.27385	0.636926	+	0.770925i	$0.280206\pi$
$294$	0	0
$295$	0	0
$296$	−15848.2	−3.11203
$297$	0	0
$298$	6065.56	1.17909
$299$	69.0720	0.0133597
$300$	0	0
$301$	1765.35	0.338051
$302$	−3468.04	−0.660806
$303$	0	0
$304$	−26242.6	−4.95105
$305$	0	0
$306$	0	0
$307$	−4882.07	−0.907603	−0.453802	−	0.891103i	$-0.649933\pi$
$307$	−4882.07	−0.907603	−0.453802	+	0.891103i	$0.649933\pi$
$308$	−1527.05	−0.282506
$309$	0	0
$310$	0	0
$311$	−2846.01	−0.518914	−0.259457	−	0.965755i	$-0.583544\pi$
$311$	−2846.01	−0.518914	−0.259457	+	0.965755i	$0.583544\pi$
$312$	0	0
$313$	−8009.49	−1.44640	−0.723200	−	0.690639i	$-0.757330\pi$
$313$	−8009.49	−1.44640	−0.723200	+	0.690639i	$0.757330\pi$
$314$	−11762.5	−2.11400
$315$	0	0
$316$	−20225.4	−3.60053
$317$	4668.89	0.827227	0.413613	−	0.910453i	$-0.364267\pi$
$317$	4668.89	0.827227	0.413613	+	0.910453i	$0.364267\pi$
$318$	0	0
$319$	2825.57	0.495930
$320$	0	0
$321$	0	0
$322$	530.422	0.0917990
$323$	10421.2	1.79520
$324$	0	0
$325$	0	0
$326$	6415.39	1.08993
$327$	0	0
$328$	20726.7	3.48914
$329$	−1102.76	−0.184794
$330$	0	0
$331$	2581.31	0.428645	0.214323	−	0.976763i	$-0.431246\pi$
$331$	2581.31	0.428645	0.214323	+	0.976763i	$0.431246\pi$
$332$	6366.05	1.05236
$333$	0	0
$334$	6122.91	1.00309
$335$	0	0
$336$	0	0
$337$	8152.45	1.31778	0.658890	−	0.752239i	$-0.271026\pi$
$337$	8152.45	1.31778	0.658890	+	0.752239i	$0.271026\pi$
$338$	12111.8	1.94910
$339$	0	0
$340$	0	0
$341$	−1877.72	−0.298194
$342$	0	0
$343$	3931.15	0.618839
$344$	24214.3	3.79520
$345$	0	0
$346$	13176.7	2.04735
$347$	−3426.21	−0.530054	−0.265027	−	0.964241i	$-0.585381\pi$
$347$	−3426.21	−0.530054	−0.265027	+	0.964241i	$0.585381\pi$
$348$	0	0
$349$	1334.33	0.204656	0.102328	−	0.994751i	$-0.467371\pi$
$349$	1334.33	0.204656	0.102328	+	0.994751i	$0.467371\pi$
$350$	0	0
$351$	0	0
$352$	−9722.93	−1.47225
$353$	4406.21	0.664360	0.332180	−	0.943216i	$-0.392216\pi$
$353$	4406.21	0.664360	0.332180	+	0.943216i	$0.392216\pi$
$354$	0	0
$355$	0	0
$356$	22406.8	3.33584
$357$	0	0
$358$	6822.72	1.00724
$359$	8623.04	1.26771	0.633853	−	0.773453i	$-0.281472\pi$
$359$	8623.04	1.26771	0.633853	+	0.773453i	$0.281472\pi$
$360$	0	0
$361$	2027.81	0.295642
$362$	2444.89	0.354974
$363$	0	0
$364$	−608.665	−0.0876448
$365$	0	0
$366$	0	0
$367$	3585.58	0.509989	0.254995	−	0.966942i	$-0.417926\pi$
$367$	3585.58	0.509989	0.254995	+	0.966942i	$0.417926\pi$
$368$	4385.51	0.621224
$369$	0	0
$370$	0	0
$371$	1755.06	0.245602
$372$	0	0
$373$	−9855.90	−1.36815	−0.684074	−	0.729413i	$-0.739793\pi$
$373$	−9855.90	−1.36815	−0.684074	+	0.729413i	$0.739793\pi$
$374$	6762.91	0.935031
$375$	0	0
$376$	−15126.0	−2.07463
$377$	1126.24	0.153858
$378$	0	0
$379$	−10837.8	−1.46887	−0.734435	−	0.678679i	$-0.762553\pi$
$379$	−10837.8	−1.46887	−0.734435	+	0.678679i	$0.762553\pi$
$380$	0	0
$381$	0	0
$382$	−27635.1	−3.70140
$383$	−2025.55	−0.270237	−0.135119	−	0.990829i	$-0.543142\pi$
$383$	−2025.55	−0.270237	−0.135119	+	0.990829i	$0.543142\pi$
$384$	0	0
$385$	0	0
$386$	−7576.04	−0.998990
$387$	0	0
$388$	25291.5	3.30923
$389$	978.894	0.127588	0.0637942	−	0.997963i	$-0.479680\pi$
$389$	978.894	0.127588	0.0637942	+	0.997963i	$0.479680\pi$
$390$	0	0
$391$	−1741.52	−0.225250
$392$	25438.9	3.27770
$393$	0	0
$394$	−12212.7	−1.56159
$395$	0	0
$396$	0	0
$397$	5008.28	0.633144	0.316572	−	0.948568i	$-0.397468\pi$
$397$	5008.28	0.633144	0.316572	+	0.948568i	$0.397468\pi$
$398$	3104.39	0.390978
$399$	0	0
$400$	0	0
$401$	15584.1	1.94073	0.970366	−	0.241639i	$-0.0776850\pi$
$401$	15584.1	1.94073	0.970366	+	0.241639i	$0.0776850\pi$
$402$	0	0
$403$	−748.437	−0.0925119
$404$	−11105.7	−1.36765
$405$	0	0
$406$	8648.69	1.05721
$407$	2099.39	0.255682
$408$	0	0
$409$	15106.6	1.82634	0.913171	−	0.407576i	$-0.133626\pi$
$409$	15106.6	1.82634	0.913171	+	0.407576i	$0.133626\pi$
$410$	0	0
$411$	0	0
$412$	20051.2	2.39770
$413$	1711.61	0.203930
$414$	0	0
$415$	0	0
$416$	−3875.45	−0.456753
$417$	0	0
$418$	5767.16	0.674834
$419$	1518.17	0.177011	0.0885056	−	0.996076i	$-0.471791\pi$
$419$	1518.17	0.177011	0.0885056	+	0.996076i	$0.471791\pi$
$420$	0	0
$421$	4637.05	0.536808	0.268404	−	0.963306i	$-0.413504\pi$
$421$	4637.05	0.536808	0.268404	+	0.963306i	$0.413504\pi$
$422$	16695.8	1.92592
$423$	0	0
$424$	24073.2	2.75730
$425$	0	0
$426$	0	0
$427$	−1016.01	−0.115148
$428$	2748.14	0.310366
$429$	0	0
$430$	0	0
$431$	11477.9	1.28276	0.641380	−	0.767223i	$-0.278362\pi$
$431$	11477.9	1.28276	0.641380	+	0.767223i	$0.278362\pi$
$432$	0	0
$433$	−10204.5	−1.13256	−0.566280	−	0.824213i	$-0.691618\pi$
$433$	−10204.5	−1.13256	−0.566280	+	0.824213i	$0.691618\pi$
$434$	−5747.44	−0.635682
$435$	0	0
$436$	9507.62	1.04434
$437$	−1485.11	−0.162568
$438$	0	0
$439$	−6919.06	−0.752229	−0.376115	−	0.926573i	$-0.622740\pi$
$439$	−6919.06	−0.752229	−0.376115	+	0.926573i	$0.622740\pi$
$440$	0	0
$441$	0	0
$442$	2695.62	0.290085
$443$	−2912.53	−0.312366	−0.156183	−	0.987728i	$-0.549919\pi$
$443$	−2912.53	−0.312366	−0.156183	+	0.987728i	$0.549919\pi$
$444$	0	0
$445$	0	0
$446$	4752.87	0.504608
$447$	0	0
$448$	−16278.2	−1.71668
$449$	1155.53	0.121454	0.0607270	−	0.998154i	$-0.480658\pi$
$449$	1155.53	0.121454	0.0607270	+	0.998154i	$0.480658\pi$
$450$	0	0
$451$	−2745.62	−0.286666
$452$	12255.5	1.27533
$453$	0	0
$454$	−2191.26	−0.226522
$455$	0	0
$456$	0	0
$457$	−2745.62	−0.281039	−0.140519	−	0.990078i	$-0.544877\pi$
$457$	−2745.62	−0.281039	−0.140519	+	0.990078i	$0.544877\pi$
$458$	2735.29	0.279065
$459$	0	0
$460$	0	0
$461$	−11224.1	−1.13397	−0.566984	−	0.823729i	$-0.691890\pi$
$461$	−11224.1	−1.13397	−0.566984	+	0.823729i	$0.691890\pi$
$462$	0	0
$463$	−15994.8	−1.60549	−0.802743	−	0.596325i	$-0.796627\pi$
$463$	−15994.8	−1.60549	−0.802743	+	0.596325i	$0.796627\pi$
$464$	71507.0	7.15437
$465$	0	0
$466$	38289.2	3.80625
$467$	6674.60	0.661378	0.330689	−	0.943740i	$-0.392719\pi$
$467$	6674.60	0.661378	0.330689	+	0.943740i	$0.392719\pi$
$468$	0	0
$469$	−1066.91	−0.105043
$470$	0	0
$471$	0	0
$472$	23477.2	2.28946
$473$	−3207.62	−0.311811
$474$	0	0
$475$	0	0
$476$	15346.4	1.47773
$477$	0	0
$478$	16755.7	1.60332
$479$	11582.0	1.10479	0.552396	−	0.833582i	$-0.313714\pi$
$479$	11582.0	1.10479	0.552396	+	0.833582i	$0.313714\pi$
$480$	0	0
$481$	836.791	0.0793230
$482$	−17279.6	−1.63292
$483$	0	0
$484$	2774.64	0.260578
$485$	0	0
$486$	0	0
$487$	10618.7	0.988047	0.494024	−	0.869448i	$-0.335526\pi$
$487$	10618.7	0.988047	0.494024	+	0.869448i	$0.335526\pi$
$488$	−13936.0	−1.29273
$489$	0	0
$490$	0	0
$491$	17948.0	1.64966	0.824829	−	0.565382i	$-0.191271\pi$
$491$	17948.0	1.64966	0.824829	+	0.565382i	$0.191271\pi$
$492$	0	0
$493$	−28396.1	−2.59411
$494$	2298.72	0.209361
$495$	0	0
$496$	−47519.6	−4.30180
$497$	5565.15	0.502275
$498$	0	0
$499$	−10409.6	−0.933865	−0.466932	−	0.884293i	$-0.654641\pi$
$499$	−10409.6	−0.933865	−0.466932	+	0.884293i	$0.654641\pi$
$500$	0	0
$501$	0	0
$502$	−4640.07	−0.412543
$503$	7319.98	0.648870	0.324435	−	0.945908i	$-0.394826\pi$
$503$	7319.98	0.648870	0.324435	+	0.945908i	$0.394826\pi$
$504$	0	0
$505$	0	0
$506$	−963.771	−0.0846736
$507$	0	0
$508$	−14678.5	−1.28200
$509$	−7619.94	−0.663552	−0.331776	−	0.943358i	$-0.607648\pi$
$509$	−7619.94	−0.663552	−0.331776	+	0.943358i	$0.607648\pi$
$510$	0	0
$511$	935.763	0.0810093
$512$	−61129.5	−5.27650
$513$	0	0
$514$	41915.4	3.59690
$515$	0	0
$516$	0	0
$517$	2003.71	0.170451
$518$	6425.93	0.545057
$519$	0	0
$520$	0	0
$521$	12413.4	1.04384	0.521921	−	0.852994i	$-0.325215\pi$
$521$	12413.4	1.04384	0.521921	+	0.852994i	$0.325215\pi$
$522$	0	0
$523$	2524.30	0.211051	0.105526	−	0.994417i	$-0.466347\pi$
$523$	2524.30	0.211051	0.105526	+	0.994417i	$0.466347\pi$
$524$	−24101.5	−2.00931
$525$	0	0
$526$	34715.8	2.87772
$527$	18870.5	1.55979
$528$	0	0
$529$	−11918.8	−0.979602
$530$	0	0
$531$	0	0
$532$	13086.8	1.06651
$533$	−1094.37	−0.0889355
$534$	0	0
$535$	0	0
$536$	−14634.2	−1.17929
$537$	0	0
$538$	9103.96	0.729553
$539$	−3369.84	−0.269294
$540$	0	0
$541$	10271.4	0.816269	0.408135	−	0.912922i	$-0.366179\pi$
$541$	10271.4	0.816269	0.408135	+	0.912922i	$0.366179\pi$
$542$	−4378.12	−0.346968
$543$	0	0
$544$	97712.2	7.70106
$545$	0	0
$546$	0	0
$547$	−7810.11	−0.610487	−0.305243	−	0.952274i	$-0.598738\pi$
$547$	−7810.11	−0.610487	−0.305243	+	0.952274i	$0.598738\pi$
$548$	38768.7	3.02211
$549$	0	0
$550$	0	0
$551$	−24215.1	−1.87223
$552$	0	0
$553$	5339.71	0.410610
$554$	10872.6	0.833815
$555$	0	0
$556$	63964.1	4.87892
$557$	−18348.8	−1.39580	−0.697902	−	0.716193i	$-0.745883\pi$
$557$	−18348.8	−1.39580	−0.697902	+	0.716193i	$0.745883\pi$
$558$	0	0
$559$	−1278.52	−0.0967365
$560$	0	0
$561$	0	0
$562$	−28350.5	−2.12792
$563$	−174.680	−0.0130761	−0.00653807	−	0.999979i	$-0.502081\pi$
$563$	−174.680	−0.0130761	−0.00653807	+	0.999979i	$0.502081\pi$
$564$	0	0
$565$	0	0
$566$	−39974.8	−2.96867
$567$	0	0
$568$	76333.8	5.63890
$569$	3208.08	0.236362	0.118181	−	0.992992i	$-0.462294\pi$
$569$	3208.08	0.236362	0.118181	+	0.992992i	$0.462294\pi$
$570$	0	0
$571$	−11660.4	−0.854592	−0.427296	−	0.904112i	$-0.640534\pi$
$571$	−11660.4	−0.854592	−0.427296	+	0.904112i	$0.640534\pi$
$572$	1105.94	0.0808419
$573$	0	0
$574$	−8403.98	−0.611107
$575$	0	0
$576$	0	0
$577$	12906.9	0.931233	0.465617	−	0.884987i	$-0.345833\pi$
$577$	12906.9	0.931233	0.465617	+	0.884987i	$0.345833\pi$
$578$	−40641.1	−2.92465
$579$	0	0
$580$	0	0
$581$	−1680.70	−0.120012
$582$	0	0
$583$	−3188.93	−0.226538
$584$	12835.3	0.909468
$585$	0	0
$586$	−35531.8	−2.50478
$587$	27427.0	1.92850	0.964252	−	0.264986i	$-0.0853673\pi$
$587$	27427.0	1.92850	0.964252	+	0.264986i	$0.0853673\pi$
$588$	0	0
$589$	16092.0	1.12574
$590$	0	0
$591$	0	0
$592$	53129.3	3.68852
$593$	5332.11	0.369247	0.184623	−	0.982809i	$-0.440893\pi$
$593$	5332.11	0.369247	0.184623	+	0.982809i	$0.440893\pi$
$594$	0	0
$595$	0	0
$596$	−25009.0	−1.71880
$597$	0	0
$598$	−384.148	−0.0262692
$599$	22329.2	1.52312	0.761558	−	0.648097i	$-0.224435\pi$
$599$	22329.2	1.52312	0.761558	+	0.648097i	$0.224435\pi$
$600$	0	0
$601$	15511.8	1.05281	0.526405	−	0.850234i	$-0.323540\pi$
$601$	15511.8	1.05281	0.526405	+	0.850234i	$0.323540\pi$
$602$	−9818.10	−0.664711
$603$	0	0
$604$	14299.1	0.963281
$605$	0	0
$606$	0	0
$607$	−7205.25	−0.481799	−0.240900	−	0.970550i	$-0.577442\pi$
$607$	−7205.25	−0.481799	−0.240900	+	0.970550i	$0.577442\pi$
$608$	83325.4	5.55804
$609$	0	0
$610$	0	0
$611$	798.655	0.0528807
$612$	0	0
$613$	2837.16	0.186936	0.0934682	−	0.995622i	$-0.470205\pi$
$613$	2837.16	0.186936	0.0934682	+	0.995622i	$0.470205\pi$
$614$	27151.9	1.78463
$615$	0	0
$616$	5529.86	0.361696
$617$	−7423.58	−0.484379	−0.242190	−	0.970229i	$-0.577866\pi$
$617$	−7423.58	−0.484379	−0.242190	+	0.970229i	$0.577866\pi$
$618$	0	0
$619$	9747.62	0.632940	0.316470	−	0.948603i	$-0.397502\pi$
$619$	9747.62	0.632940	0.316470	+	0.948603i	$0.397502\pi$
$620$	0	0
$621$	0	0
$622$	15828.2	1.02034
$623$	−5915.62	−0.380424
$624$	0	0
$625$	0	0
$626$	44545.2	2.84407
$627$	0	0
$628$	48497.9	3.08165
$629$	−21098.1	−1.33742
$630$	0	0
$631$	5914.75	0.373157	0.186579	−	0.982440i	$-0.440260\pi$
$631$	5914.75	0.373157	0.186579	+	0.982440i	$0.440260\pi$
$632$	73241.7	4.60980
$633$	0	0
$634$	−25966.3	−1.62658
$635$	0	0
$636$	0	0
$637$	−1343.18	−0.0835459
$638$	−15714.6	−0.975150
$639$	0	0
$640$	0	0
$641$	25438.0	1.56746	0.783728	−	0.621104i	$-0.213316\pi$
$641$	25438.0	1.56746	0.783728	+	0.621104i	$0.213316\pi$
$642$	0	0
$643$	−769.253	−0.0471794	−0.0235897	−	0.999722i	$-0.507510\pi$
$643$	−769.253	−0.0471794	−0.0235897	+	0.999722i	$0.507510\pi$
$644$	−2186.99	−0.133819
$645$	0	0
$646$	−57958.0	−3.52992
$647$	25813.2	1.56850	0.784250	−	0.620445i	$-0.213048\pi$
$647$	25813.2	1.56850	0.784250	+	0.620445i	$0.213048\pi$
$648$	0	0
$649$	−3109.98	−0.188101
$650$	0	0
$651$	0	0
$652$	−26451.3	−1.58883
$653$	−13138.6	−0.787372	−0.393686	−	0.919245i	$-0.628800\pi$
$653$	−13138.6	−0.787372	−0.393686	+	0.919245i	$0.628800\pi$
$654$	0	0
$655$	0	0
$656$	−69483.7	−4.13549
$657$	0	0
$658$	6133.08	0.363362
$659$	19105.0	1.12933	0.564663	−	0.825322i	$-0.309006\pi$
$659$	19105.0	1.12933	0.564663	+	0.825322i	$0.309006\pi$
$660$	0	0
$661$	−31694.3	−1.86500	−0.932502	−	0.361166i	$-0.882379\pi$
$661$	−31694.3	−1.86500	−0.932502	+	0.361166i	$0.882379\pi$
$662$	−14356.1	−0.842848
$663$	0	0
$664$	−23053.2	−1.34734
$665$	0	0
$666$	0	0
$667$	4046.68	0.234915
$668$	−25245.4	−1.46224
$669$	0	0
$670$	0	0
$671$	1846.07	0.106210
$672$	0	0
$673$	23110.6	1.32370	0.661848	−	0.749638i	$-0.269772\pi$
$673$	23110.6	1.32370	0.661848	+	0.749638i	$0.269772\pi$
$674$	−45340.3	−2.59116
$675$	0	0
$676$	−49938.3	−2.84128
$677$	−17052.4	−0.968062	−0.484031	−	0.875051i	$-0.660828\pi$
$677$	−17052.4	−0.968062	−0.484031	+	0.875051i	$0.660828\pi$
$678$	0	0
$679$	−6677.20	−0.377390
$680$	0	0
$681$	0	0
$682$	10443.0	0.586341
$683$	28542.8	1.59907	0.799533	−	0.600623i	$-0.205081\pi$
$683$	28542.8	1.59907	0.799533	+	0.600623i	$0.205081\pi$
$684$	0	0
$685$	0	0
$686$	−21863.3	−1.21683
$687$	0	0
$688$	−81175.6	−4.49824
$689$	−1271.07	−0.0702814
$690$	0	0
$691$	6479.20	0.356701	0.178351	−	0.983967i	$-0.442924\pi$
$691$	6479.20	0.356701	0.178351	+	0.983967i	$0.442924\pi$
$692$	−54328.9	−2.98450
$693$	0	0
$694$	19055.1	1.04225
$695$	0	0
$696$	0	0
$697$	27592.6	1.49949
$698$	−7420.94	−0.402417
$699$	0	0
$700$	0	0
$701$	−21118.6	−1.13786	−0.568929	−	0.822386i	$-0.692642\pi$
$701$	−21118.6	−1.13786	−0.568929	+	0.822386i	$0.692642\pi$
$702$	0	0
$703$	−17991.7	−0.965249
$704$	29577.3	1.58343
$705$	0	0
$706$	−24505.4	−1.30633
$707$	2932.03	0.155969
$708$	0	0
$709$	7072.53	0.374632	0.187316	−	0.982300i	$-0.440021\pi$
$709$	7072.53	0.374632	0.187316	+	0.982300i	$0.440021\pi$
$710$	0	0
$711$	0	0
$712$	−81141.1	−4.27092
$713$	−2689.20	−0.141250
$714$	0	0
$715$	0	0
$716$	−28130.8	−1.46829
$717$	0	0
$718$	−47957.5	−2.49270
$719$	−22177.9	−1.15034	−0.575170	−	0.818034i	$-0.695064\pi$
$719$	−22177.9	−1.15034	−0.575170	+	0.818034i	$0.695064\pi$
$720$	0	0
$721$	−5293.71	−0.273437
$722$	−11277.8	−0.581323
$723$	0	0
$724$	−10080.5	−0.517459
$725$	0	0
$726$	0	0
$727$	17390.7	0.887186	0.443593	−	0.896228i	$-0.353704\pi$
$727$	17390.7	0.887186	0.443593	+	0.896228i	$0.353704\pi$
$728$	2204.14	0.112213
$729$	0	0
$730$	0	0
$731$	32235.6	1.63102
$732$	0	0
$733$	21877.0	1.10238	0.551191	−	0.834379i	$-0.314174\pi$
$733$	21877.0	1.10238	0.551191	+	0.834379i	$0.314174\pi$
$734$	−19941.4	−1.00279
$735$	0	0
$736$	−13924.8	−0.697385
$737$	1938.56	0.0968899
$738$	0	0
$739$	14203.1	0.706994	0.353497	−	0.935436i	$-0.384992\pi$
$739$	14203.1	0.706994	0.353497	+	0.935436i	$0.384992\pi$
$740$	0	0
$741$	0	0
$742$	−9760.87	−0.482928
$743$	−3933.68	−0.194230	−0.0971148	−	0.995273i	$-0.530961\pi$
$743$	−3933.68	−0.194230	−0.0971148	+	0.995273i	$0.530961\pi$
$744$	0	0
$745$	0	0
$746$	54814.1	2.69020
$747$	0	0
$748$	−27884.2	−1.36303
$749$	−725.537	−0.0353946
$750$	0	0
$751$	22554.3	1.09590	0.547949	−	0.836512i	$-0.315409\pi$
$751$	22554.3	1.09590	0.547949	+	0.836512i	$0.315409\pi$
$752$	50708.0	2.45895
$753$	0	0
$754$	−6263.65	−0.302531
$755$	0	0
$756$	0	0
$757$	−11432.0	−0.548883	−0.274441	−	0.961604i	$-0.588493\pi$
$757$	−11432.0	−0.548883	−0.274441	+	0.961604i	$0.588493\pi$
$758$	60275.2	2.88825
$759$	0	0
$760$	0	0
$761$	32660.1	1.55575	0.777877	−	0.628416i	$-0.216296\pi$
$761$	32660.1	1.55575	0.777877	+	0.628416i	$0.216296\pi$
$762$	0	0
$763$	−2510.11	−0.119098
$764$	113943.	5.39568
$765$	0	0
$766$	11265.2	0.531369
$767$	−1239.60	−0.0583565
$768$	0	0
$769$	−8569.93	−0.401872	−0.200936	−	0.979604i	$-0.564398\pi$
$769$	−8569.93	−0.401872	−0.200936	+	0.979604i	$0.564398\pi$
$770$	0	0
$771$	0	0
$772$	31236.8	1.45627
$773$	−29158.0	−1.35671	−0.678357	−	0.734733i	$-0.737307\pi$
$773$	−29158.0	−1.35671	−0.678357	+	0.734733i	$0.737307\pi$
$774$	0	0
$775$	0	0
$776$	−91587.2	−4.23684
$777$	0	0
$778$	−5444.17	−0.250878
$779$	23530.0	1.08222
$780$	0	0
$781$	−10111.8	−0.463289
$782$	9685.58	0.442910
$783$	0	0
$784$	−85280.9	−3.88488
$785$	0	0
$786$	0	0
$787$	8501.30	0.385055	0.192528	−	0.981292i	$-0.438332\pi$
$787$	8501.30	0.385055	0.192528	+	0.981292i	$0.438332\pi$
$788$	50354.2	2.27639
$789$	0	0
$790$	0	0
$791$	−3235.56	−0.145440
$792$	0	0
$793$	735.823	0.0329506
$794$	−27853.8	−1.24496
$795$	0	0
$796$	−12799.7	−0.569944
$797$	37459.5	1.66485	0.832424	−	0.554139i	$-0.186953\pi$
$797$	37459.5	1.66485	0.832424	+	0.554139i	$0.186953\pi$
$798$	0	0
$799$	−20136.6	−0.891592
$800$	0	0
$801$	0	0
$802$	−86671.9	−3.81607
$803$	−1700.27	−0.0747214
$804$	0	0
$805$	0	0
$806$	4162.47	0.181907
$807$	0	0
$808$	40216.9	1.75102
$809$	−18045.8	−0.784246	−0.392123	−	0.919913i	$-0.628259\pi$
$809$	−18045.8	−0.784246	−0.392123	+	0.919913i	$0.628259\pi$
$810$	0	0
$811$	914.961	0.0396161	0.0198080	−	0.999804i	$-0.493694\pi$
$811$	914.961	0.0396161	0.0198080	+	0.999804i	$0.493694\pi$
$812$	−35659.5	−1.54114
$813$	0	0
$814$	−11675.8	−0.502750
$815$	0	0
$816$	0	0
$817$	27489.3	1.17715
$818$	−84016.3	−3.59115
$819$	0	0
$820$	0	0
$821$	−15188.9	−0.645672	−0.322836	−	0.946455i	$-0.604636\pi$
$821$	−15188.9	−0.645672	−0.322836	+	0.946455i	$0.604636\pi$
$822$	0	0
$823$	37930.3	1.60652	0.803261	−	0.595628i	$-0.203097\pi$
$823$	37930.3	1.60652	0.803261	+	0.595628i	$0.203097\pi$
$824$	−72610.7	−3.06980
$825$	0	0
$826$	−9519.22	−0.400988
$827$	29679.9	1.24797	0.623985	−	0.781437i	$-0.285513\pi$
$827$	29679.9	1.24797	0.623985	+	0.781437i	$0.285513\pi$
$828$	0	0
$829$	29094.2	1.21892	0.609458	−	0.792818i	$-0.291387\pi$
$829$	29094.2	1.21892	0.609458	+	0.792818i	$0.291387\pi$
$830$	0	0
$831$	0	0
$832$	11789.2	0.491245
$833$	33865.8	1.40862
$834$	0	0
$835$	0	0
$836$	−23778.6	−0.983732
$837$	0	0
$838$	−8443.41	−0.348058
$839$	−38791.2	−1.59621	−0.798105	−	0.602519i	$-0.794164\pi$
$839$	−38791.2	−1.59621	−0.798105	+	0.602519i	$0.794164\pi$
$840$	0	0
$841$	41593.3	1.70541
$842$	−25789.2	−1.05553
$843$	0	0
$844$	−68838.7	−2.80749
$845$	0	0
$846$	0	0
$847$	−732.531	−0.0297167
$848$	−80702.4	−3.26808
$849$	0	0
$850$	0	0
$851$	3006.66	0.121113
$852$	0	0
$853$	42933.7	1.72335	0.861677	−	0.507458i	$-0.169415\pi$
$853$	42933.7	1.72335	0.861677	+	0.507458i	$0.169415\pi$
$854$	5650.58	0.226415
$855$	0	0
$856$	−9951.76	−0.397365
$857$	−2664.36	−0.106199	−0.0530997	−	0.998589i	$-0.516910\pi$
$857$	−2664.36	−0.106199	−0.0530997	+	0.998589i	$0.516910\pi$
$858$	0	0
$859$	−25002.7	−0.993111	−0.496556	−	0.868005i	$-0.665402\pi$
$859$	−25002.7	−0.993111	−0.496556	+	0.868005i	$0.665402\pi$
$860$	0	0
$861$	0	0
$862$	−63834.8	−2.52230
$863$	27509.8	1.08510	0.542551	−	0.840023i	$-0.317459\pi$
$863$	27509.8	1.08510	0.542551	+	0.840023i	$0.317459\pi$
$864$	0	0
$865$	0	0
$866$	56753.1	2.22696
$867$	0	0
$868$	23697.3	0.926658
$869$	−9702.19	−0.378739
$870$	0	0
$871$	772.688	0.0300592
$872$	−34429.6	−1.33708
$873$	0	0
$874$	8259.51	0.319659
$875$	0	0
$876$	0	0
$877$	25993.8	1.00085	0.500426	−	0.865779i	$-0.333177\pi$
$877$	25993.8	1.00085	0.500426	+	0.865779i	$0.333177\pi$
$878$	38480.7	1.47911
$879$	0	0
$880$	0	0
$881$	29528.7	1.12922	0.564612	−	0.825357i	$-0.309026\pi$
$881$	29528.7	1.12922	0.564612	+	0.825357i	$0.309026\pi$
$882$	0	0
$883$	50497.5	1.92455	0.962274	−	0.272081i	$-0.0877119\pi$
$883$	50497.5	1.92455	0.962274	+	0.272081i	$0.0877119\pi$
$884$	−11114.3	−0.422867
$885$	0	0
$886$	16198.2	0.614208
$887$	36471.9	1.38062	0.690309	−	0.723515i	$-0.257475\pi$
$887$	36471.9	1.38062	0.690309	+	0.723515i	$0.257475\pi$
$888$	0	0
$889$	3875.28	0.146201
$890$	0	0
$891$	0	0
$892$	−19596.6	−0.735586
$893$	−17171.8	−0.643484
$894$	0	0
$895$	0	0
$896$	47723.2	1.77937
$897$	0	0
$898$	−6426.54	−0.238816
$899$	−43848.2	−1.62672
$900$	0	0
$901$	32047.7	1.18498
$902$	15269.9	0.563673
$903$	0	0
$904$	−44380.3	−1.63282
$905$	0	0
$906$	0	0
$907$	−15130.2	−0.553902	−0.276951	−	0.960884i	$-0.589324\pi$
$907$	−15130.2	−0.553902	−0.276951	+	0.960884i	$0.589324\pi$
$908$	9034.81	0.330210
$909$	0	0
$910$	0	0
$911$	−13937.0	−0.506864	−0.253432	−	0.967353i	$-0.581559\pi$
$911$	−13937.0	−0.506864	−0.253432	+	0.967353i	$0.581559\pi$
$912$	0	0
$913$	3053.81	0.110697
$914$	15269.9	0.552608
$915$	0	0
$916$	−11277.9	−0.406804
$917$	6363.04	0.229145
$918$	0	0
$919$	−40897.5	−1.46799	−0.733996	−	0.679153i	$-0.762347\pi$
$919$	−40897.5	−1.46799	−0.733996	+	0.679153i	$0.762347\pi$
$920$	0	0
$921$	0	0
$922$	62423.5	2.22973
$923$	−4030.45	−0.143731
$924$	0	0
$925$	0	0
$926$	88955.7	3.15688
$927$	0	0
$928$	−227048.	−8.03149
$929$	12154.1	0.429239	0.214620	−	0.976698i	$-0.431149\pi$
$929$	12154.1	0.429239	0.214620	+	0.976698i	$0.431149\pi$
$930$	0	0
$931$	28879.5	1.01664
$932$	−157870.	−5.54852
$933$	0	0
$934$	−37121.1	−1.30047
$935$	0	0
$936$	0	0
$937$	15754.1	0.549267	0.274634	−	0.961549i	$-0.411443\pi$
$937$	15754.1	0.549267	0.274634	+	0.961549i	$0.411443\pi$
$938$	5933.67	0.206547
$939$	0	0
$940$	0	0
$941$	4217.53	0.146108	0.0730539	−	0.997328i	$-0.476725\pi$
$941$	4217.53	0.146108	0.0730539	+	0.997328i	$0.476725\pi$
$942$	0	0
$943$	−3932.18	−0.135789
$944$	−78704.5	−2.71357
$945$	0	0
$946$	17839.4	0.613116
$947$	49839.0	1.71019	0.855095	−	0.518471i	$-0.173498\pi$
$947$	49839.0	1.71019	0.855095	+	0.518471i	$0.173498\pi$
$948$	0	0
$949$	−677.708	−0.0231816
$950$	0	0
$951$	0	0
$952$	−55573.3	−1.89196
$953$	−12845.7	−0.436635	−0.218318	−	0.975878i	$-0.570057\pi$
$953$	−12845.7	−0.436635	−0.218318	+	0.975878i	$0.570057\pi$
$954$	0	0
$955$	0	0
$956$	−69085.4	−2.33722
$957$	0	0
$958$	−64413.9	−2.17236
$959$	−10235.3	−0.344646
$960$	0	0
$961$	−651.937	−0.0218837
$962$	−4653.86	−0.155973
$963$	0	0
$964$	71245.7	2.38036
$965$	0	0
$966$	0	0
$967$	38829.6	1.29129	0.645645	−	0.763638i	$-0.276589\pi$
$967$	38829.6	1.29129	0.645645	+	0.763638i	$0.276589\pi$
$968$	−10047.7	−0.333621
$969$	0	0
$970$	0	0
$971$	11438.8	0.378053	0.189026	−	0.981972i	$-0.439467\pi$
$971$	11438.8	0.378053	0.189026	+	0.981972i	$0.439467\pi$
$972$	0	0
$973$	−16887.1	−0.556400
$974$	−59056.4	−1.94280
$975$	0	0
$976$	46718.7	1.53220
$977$	−12084.1	−0.395706	−0.197853	−	0.980232i	$-0.563397\pi$
$977$	−12084.1	−0.395706	−0.197853	+	0.980232i	$0.563397\pi$
$978$	0	0
$979$	10748.6	0.350896
$980$	0	0
$981$	0	0
$982$	−99818.8	−3.24373
$983$	−23502.2	−0.762569	−0.381284	−	0.924458i	$-0.624518\pi$
$983$	−23502.2	−0.762569	−0.381284	+	0.924458i	$0.624518\pi$
$984$	0	0
$985$	0	0
$986$	157926.	5.10081
$987$	0	0
$988$	−9477.87	−0.305194
$989$	−4593.84	−0.147700
$990$	0	0
$991$	−18664.1	−0.598268	−0.299134	−	0.954211i	$-0.596698\pi$
$991$	−18664.1	−0.598268	−0.299134	+	0.954211i	$0.596698\pi$
$992$	150884.	4.82919
$993$	0	0
$994$	−30950.9	−0.987627
$995$	0	0
$996$	0	0
$997$	−24528.4	−0.779158	−0.389579	−	0.920993i	$-0.627380\pi$
$997$	−24528.4	−0.779158	−0.389579	+	0.920993i	$0.627380\pi$
$998$	57893.7	1.83626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.l.1.1		2
3.2	odd	2		275.4.a.c.1.2		2
5.4	even	2		495.4.a.e.1.2		2
15.2	even	4		275.4.b.b.199.4		4
15.8	even	4		275.4.b.b.199.1		4
15.14	odd	2		55.4.a.b.1.1	✓	2
60.59	even	2		880.4.a.r.1.2		2
165.164	even	2		605.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.b.1.1	✓	2	15.14	odd	2
275.4.a.c.1.2		2	3.2	odd	2
275.4.b.b.199.1		4	15.8	even	4
275.4.b.b.199.4		4	15.2	even	4
495.4.a.e.1.2		2	5.4	even	2
605.4.a.g.1.2		2	165.164	even	2
880.4.a.r.1.2		2	60.59	even	2
2475.4.a.l.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-5.56155 q^{2} +22.9309 q^{4} -6.05398 q^{7} -83.0388 q^{8} +O(q^{10})\)	\(q-5.56155 q^{2} +22.9309 q^{4} -6.05398 q^{7} -83.0388 q^{8} +11.0000 q^{11} +4.38447 q^{13} +33.6695 q^{14} +278.378 q^{16} -110.546 q^{17} -94.2699 q^{19} -61.1771 q^{22} +15.7538 q^{23} -24.3845 q^{26} -138.823 q^{28} +256.870 q^{29} -170.702 q^{31} -883.902 q^{32} +614.810 q^{34} +190.853 q^{37} +524.287 q^{38} -249.602 q^{41} -291.602 q^{43} +252.240 q^{44} -87.6155 q^{46} +182.155 q^{47} -306.349 q^{49} +100.540 q^{52} -289.902 q^{53} +502.715 q^{56} -1428.60 q^{58} -282.725 q^{59} +167.825 q^{61} +949.366 q^{62} +2688.85 q^{64} +176.233 q^{67} -2534.93 q^{68} -919.255 q^{71} -154.570 q^{73} -1061.44 q^{74} -2161.69 q^{76} -66.5937 q^{77} -882.017 q^{79} +1388.18 q^{82} +277.619 q^{83} +1621.76 q^{86} -913.427 q^{88} +977.147 q^{89} -26.5435 q^{91} +361.248 q^{92} -1013.07 q^{94} +1102.94 q^{97} +1703.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7 q^{2} + 17 q^{4} + 25 q^{7} - 63 q^{8}+O(q^{10}) \) 2 * q - 7 * q^2 + 17 * q^4 + 25 * q^7 - 63 * q^8	\( 2 q - 7 q^{2} + 17 q^{4} + 25 q^{7} - 63 q^{8} + 22 q^{11} + 50 q^{13} - 11 q^{14} + 297 q^{16} - 151 q^{17} - 3 q^{19} - 77 q^{22} + 48 q^{23} - 90 q^{26} - 323 q^{28} + 221 q^{29} + 141 q^{31} - 1071 q^{32} + 673 q^{34} + 559 q^{37} + 393 q^{38} + 144 q^{41} + 60 q^{43} + 187 q^{44} - 134 q^{46} - 48 q^{47} + 315 q^{49} - 170 q^{52} + 117 q^{53} + 1125 q^{56} - 1377 q^{58} + 86 q^{59} - 155 q^{61} + 501 q^{62} + 2809 q^{64} - 266 q^{67} - 2295 q^{68} - 1587 q^{71} - 70 q^{73} - 1591 q^{74} - 2703 q^{76} + 275 q^{77} - 1294 q^{79} + 822 q^{82} - 558 q^{83} + 1116 q^{86} - 693 q^{88} + 1777 q^{89} + 1390 q^{91} + 170 q^{92} - 682 q^{94} + 334 q^{97} + 810 q^{98}+O(q^{100}) \) 2 * q - 7 * q^2 + 17 * q^4 + 25 * q^7 - 63 * q^8 + 22 * q^11 + 50 * q^13 - 11 * q^14 + 297 * q^16 - 151 * q^17 - 3 * q^19 - 77 * q^22 + 48 * q^23 - 90 * q^26 - 323 * q^28 + 221 * q^29 + 141 * q^31 - 1071 * q^32 + 673 * q^34 + 559 * q^37 + 393 * q^38 + 144 * q^41 + 60 * q^43 + 187 * q^44 - 134 * q^46 - 48 * q^47 + 315 * q^49 - 170 * q^52 + 117 * q^53 + 1125 * q^56 - 1377 * q^58 + 86 * q^59 - 155 * q^61 + 501 * q^62 + 2809 * q^64 - 266 * q^67 - 2295 * q^68 - 1587 * q^71 - 70 * q^73 - 1591 * q^74 - 2703 * q^76 + 275 * q^77 - 1294 * q^79 + 822 * q^82 - 558 * q^83 + 1116 * q^86 - 693 * q^88 + 1777 * q^89 + 1390 * q^91 + 170 * q^92 - 682 * q^94 + 334 * q^97 + 810 * q^98

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