LMFDB - Embedded newform 2475.4.a.bs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$146.029727264$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 31x^{5} + 50x^{4} + 272x^{3} - 322x^{2} - 704x + 512 $ x^7 - 2x^6 - 31x^5 + 50x^4 + 272x^3 - 322x^2 - 704x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.647712$ 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)

$f(q)$	$=$	$q+0.352288 q^{2} -7.87589 q^{4} -19.8486 q^{7} -5.59288 q^{8} +O(q^{10})$	$q+0.352288 q^{2} -7.87589 q^{4} -19.8486 q^{7} -5.59288 q^{8} -11.0000 q^{11} -25.5286 q^{13} -6.99242 q^{14} +61.0368 q^{16} -77.3644 q^{17} +96.1609 q^{19} -3.87517 q^{22} -173.860 q^{23} -8.99340 q^{26} +156.326 q^{28} +189.619 q^{29} -275.829 q^{31} +66.2456 q^{32} -27.2545 q^{34} -269.805 q^{37} +33.8763 q^{38} -306.118 q^{41} -477.907 q^{43} +86.6348 q^{44} -61.2487 q^{46} -257.405 q^{47} +50.9676 q^{49} +201.060 q^{52} +495.761 q^{53} +111.011 q^{56} +66.8003 q^{58} -102.064 q^{59} -585.738 q^{61} -97.1710 q^{62} -464.957 q^{64} -474.381 q^{67} +609.314 q^{68} -453.954 q^{71} -655.838 q^{73} -95.0491 q^{74} -757.353 q^{76} +218.335 q^{77} +482.235 q^{79} -107.842 q^{82} +523.492 q^{83} -168.361 q^{86} +61.5217 q^{88} +750.979 q^{89} +506.707 q^{91} +1369.30 q^{92} -90.6806 q^{94} +572.084 q^{97} +17.9553 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 5 q^{2} + 13 q^{4} - 34 q^{7} + 75 q^{8}+O(q^{10}) $ 7 * q + 5 * q^2 + 13 * q^4 - 34 * q^7 + 75 * q^8	$ 7 q + 5 q^{2} + 13 q^{4} - 34 q^{7} + 75 q^{8} - 77 q^{11} - 80 q^{13} - 42 q^{14} - 43 q^{16} + 162 q^{17} + 58 q^{19} - 55 q^{22} + 324 q^{23} + 200 q^{26} + 168 q^{28} - 64 q^{29} - 348 q^{31} - 75 q^{32} + 206 q^{34} - 664 q^{37} + 334 q^{38} + 332 q^{41} - 774 q^{43} - 143 q^{44} - 328 q^{46} + 872 q^{47} - 417 q^{49} - 134 q^{52} + 1628 q^{53} + 1618 q^{56} - 1568 q^{58} + 332 q^{59} + 22 q^{61} - 260 q^{62} + 561 q^{64} - 1524 q^{67} + 2324 q^{68} + 516 q^{71} - 1700 q^{73} - 1628 q^{74} + 2794 q^{76} + 374 q^{77} + 1746 q^{79} - 364 q^{82} + 2344 q^{83} - 1270 q^{86} - 825 q^{88} + 2226 q^{89} + 1072 q^{91} + 4184 q^{92} + 4736 q^{94} - 1048 q^{97} + 3057 q^{98}+O(q^{100}) $ 7 * q + 5 * q^2 + 13 * q^4 - 34 * q^7 + 75 * q^8 - 77 * q^11 - 80 * q^13 - 42 * q^14 - 43 * q^16 + 162 * q^17 + 58 * q^19 - 55 * q^22 + 324 * q^23 + 200 * q^26 + 168 * q^28 - 64 * q^29 - 348 * q^31 - 75 * q^32 + 206 * q^34 - 664 * q^37 + 334 * q^38 + 332 * q^41 - 774 * q^43 - 143 * q^44 - 328 * q^46 + 872 * q^47 - 417 * q^49 - 134 * q^52 + 1628 * q^53 + 1618 * q^56 - 1568 * q^58 + 332 * q^59 + 22 * q^61 - 260 * q^62 + 561 * q^64 - 1524 * q^67 + 2324 * q^68 + 516 * q^71 - 1700 * q^73 - 1628 * q^74 + 2794 * q^76 + 374 * q^77 + 1746 * q^79 - 364 * q^82 + 2344 * q^83 - 1270 * q^86 - 825 * q^88 + 2226 * q^89 + 1072 * q^91 + 4184 * q^92 + 4736 * q^94 - 1048 * q^97 + 3057 * 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 $

$2$	0.352288	0.124553	0.0622763	−	0.998059i	$-0.480164\pi$
$2$	0.352288	0.124553	0.0622763	+	0.998059i	$0.480164\pi$
$3$	0	0
$3$	0	0
$4$	−7.87589	−0.984487
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−19.8486	−1.07172	−0.535862	−	0.844305i	$-0.680013\pi$
$7$	−19.8486	−1.07172	−0.535862	+	0.844305i	$0.680013\pi$
$8$	−5.59288	−0.247173
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−25.5286	−0.544643	−0.272321	−	0.962206i	$-0.587791\pi$
$13$	−25.5286	−0.544643	−0.272321	+	0.962206i	$0.587791\pi$
$14$	−6.99242	−0.133486
$15$	0	0
$16$	61.0368	0.953701
$17$	−77.3644	−1.10374	−0.551871	−	0.833929i	$-0.686086\pi$
$17$	−77.3644	−1.10374	−0.551871	+	0.833929i	$0.686086\pi$
$18$	0	0
$19$	96.1609	1.16110	0.580548	−	0.814226i	$-0.302838\pi$
$19$	96.1609	1.16110	0.580548	+	0.814226i	$0.302838\pi$
$20$	0	0
$21$	0	0
$22$	−3.87517	−0.0375540
$23$	−173.860	−1.57619	−0.788094	−	0.615556i	$-0.788932\pi$
$23$	−173.860	−1.57619	−0.788094	+	0.615556i	$0.788932\pi$
$24$	0	0
$25$	0	0
$26$	−8.99340	−0.0678366
$27$	0	0
$28$	156.326	1.05510
$29$	189.619	1.21418	0.607091	−	0.794632i	$-0.292336\pi$
$29$	189.619	1.21418	0.607091	+	0.794632i	$0.292336\pi$
$30$	0	0
$31$	−275.829	−1.59807	−0.799037	−	0.601282i	$-0.794657\pi$
$31$	−275.829	−1.59807	−0.799037	+	0.601282i	$0.794657\pi$
$32$	66.2456	0.365959
$33$	0	0
$34$	−27.2545	−0.137474
$35$	0	0
$36$	0	0
$37$	−269.805	−1.19880	−0.599401	−	0.800449i	$-0.704595\pi$
$37$	−269.805	−1.19880	−0.599401	+	0.800449i	$0.704595\pi$
$38$	33.8763	0.144618
$39$	0	0
$40$	0	0
$41$	−306.118	−1.16604	−0.583020	−	0.812458i	$-0.698129\pi$
$41$	−306.118	−1.16604	−0.583020	+	0.812458i	$0.698129\pi$
$42$	0	0
$43$	−477.907	−1.69489	−0.847443	−	0.530887i	$-0.821859\pi$
$43$	−477.907	−1.69489	−0.847443	+	0.530887i	$0.821859\pi$
$44$	86.6348	0.296834
$45$	0	0
$46$	−61.2487	−0.196318
$47$	−257.405	−0.798859	−0.399429	−	0.916764i	$-0.630792\pi$
$47$	−257.405	−0.798859	−0.399429	+	0.916764i	$0.630792\pi$
$48$	0	0
$49$	50.9676	0.148594
$50$	0	0
$51$	0	0
$52$	201.060	0.536193
$53$	495.761	1.28487	0.642434	−	0.766341i	$-0.277925\pi$
$53$	495.761	1.28487	0.642434	+	0.766341i	$0.277925\pi$
$54$	0	0
$55$	0	0
$56$	111.011	0.264901
$57$	0	0
$58$	66.8003	0.151229
$59$	−102.064	−0.225214	−0.112607	−	0.993640i	$-0.535920\pi$
$59$	−102.064	−0.225214	−0.112607	+	0.993640i	$0.535920\pi$
$60$	0	0
$61$	−585.738	−1.22944	−0.614721	−	0.788744i	$-0.710732\pi$
$61$	−585.738	−1.22944	−0.614721	+	0.788744i	$0.710732\pi$
$62$	−97.1710	−0.199044
$63$	0	0
$64$	−464.957	−0.908120
$65$	0	0
$66$	0	0
$67$	−474.381	−0.864997	−0.432499	−	0.901635i	$-0.642368\pi$
$67$	−474.381	−0.864997	−0.432499	+	0.901635i	$0.642368\pi$
$68$	609.314	1.08662
$69$	0	0
$70$	0	0
$71$	−453.954	−0.758794	−0.379397	−	0.925234i	$-0.623869\pi$
$71$	−453.954	−0.758794	−0.379397	+	0.925234i	$0.623869\pi$
$72$	0	0
$73$	−655.838	−1.05151	−0.525753	−	0.850637i	$-0.676216\pi$
$73$	−655.838	−1.05151	−0.525753	+	0.850637i	$0.676216\pi$
$74$	−95.0491	−0.149314
$75$	0	0
$76$	−757.353	−1.14308
$77$	218.335	0.323137
$78$	0	0
$79$	482.235	0.686781	0.343390	−	0.939193i	$-0.388425\pi$
$79$	482.235	0.686781	0.343390	+	0.939193i	$0.388425\pi$
$80$	0	0
$81$	0	0
$82$	−107.842	−0.145233
$83$	523.492	0.692297	0.346149	−	0.938180i	$-0.387489\pi$
$83$	523.492	0.692297	0.346149	+	0.938180i	$0.387489\pi$
$84$	0	0
$85$	0	0
$86$	−168.361	−0.211102
$87$	0	0
$88$	61.5217	0.0745254
$89$	750.979	0.894423	0.447211	−	0.894428i	$-0.352417\pi$
$89$	750.979	0.894423	0.447211	+	0.894428i	$0.352417\pi$
$90$	0	0
$91$	506.707	0.583707
$92$	1369.30	1.55174
$93$	0	0
$94$	−90.6806	−0.0994999
$95$	0	0
$96$	0	0
$97$	572.084	0.598829	0.299414	−	0.954123i	$-0.403209\pi$
$97$	572.084	0.598829	0.299414	+	0.954123i	$0.403209\pi$
$98$	17.9553	0.0185077
$99$	0	0
$100$	0	0
$101$	−79.5187	−0.0783407	−0.0391703	−	0.999233i	$-0.512471\pi$
$101$	−79.5187	−0.0783407	−0.0391703	+	0.999233i	$0.512471\pi$
$102$	0	0
$103$	−1026.44	−0.981925	−0.490963	−	0.871181i	$-0.663355\pi$
$103$	−1026.44	−0.981925	−0.490963	+	0.871181i	$0.663355\pi$
$104$	142.778	0.134621
$105$	0	0
$106$	174.650	0.160034
$107$	−1087.33	−0.982397	−0.491199	−	0.871048i	$-0.663441\pi$
$107$	−1087.33	−0.982397	−0.491199	+	0.871048i	$0.663441\pi$
$108$	0	0
$109$	1330.39	1.16907	0.584534	−	0.811369i	$-0.301277\pi$
$109$	1330.39	1.16907	0.584534	+	0.811369i	$0.301277\pi$
$110$	0	0
$111$	0	0
$112$	−1211.50	−1.02210
$113$	1460.20	1.21561	0.607805	−	0.794086i	$-0.292050\pi$
$113$	1460.20	1.21561	0.607805	+	0.794086i	$0.292050\pi$
$114$	0	0
$115$	0	0
$116$	−1493.42	−1.19535
$117$	0	0
$118$	−35.9560	−0.0280510
$119$	1535.58	1.18291
$120$	0	0
$121$	121.000	0.0909091
$122$	−206.348	−0.153130
$123$	0	0
$124$	2172.40	1.57328
$125$	0	0
$126$	0	0
$127$	−1022.85	−0.714674	−0.357337	−	0.933975i	$-0.616315\pi$
$127$	−1022.85	−0.714674	−0.357337	+	0.933975i	$0.616315\pi$
$128$	−693.763	−0.479067
$129$	0	0
$130$	0	0
$131$	−1308.85	−0.872936	−0.436468	−	0.899720i	$-0.643771\pi$
$131$	−1308.85	−0.872936	−0.436468	+	0.899720i	$0.643771\pi$
$132$	0	0
$133$	−1908.66	−1.24438
$134$	−167.118	−0.107738
$135$	0	0
$136$	432.690	0.272815
$137$	1405.87	0.876725	0.438363	−	0.898798i	$-0.355559\pi$
$137$	1405.87	0.876725	0.438363	+	0.898798i	$0.355559\pi$
$138$	0	0
$139$	−725.138	−0.442485	−0.221242	−	0.975219i	$-0.571011\pi$
$139$	−725.138	−0.442485	−0.221242	+	0.975219i	$0.571011\pi$
$140$	0	0
$141$	0	0
$142$	−159.922	−0.0945098
$143$	280.814	0.164216
$144$	0	0
$145$	0	0
$146$	−231.044	−0.130968
$147$	0	0
$148$	2124.96	1.18021
$149$	−2812.26	−1.54624	−0.773118	−	0.634262i	$-0.781304\pi$
$149$	−2812.26	−1.54624	−0.773118	+	0.634262i	$0.781304\pi$
$150$	0	0
$151$	−469.633	−0.253100	−0.126550	−	0.991960i	$-0.540390\pi$
$151$	−469.633	−0.253100	−0.126550	+	0.991960i	$0.540390\pi$
$152$	−537.817	−0.286992
$153$	0	0
$154$	76.9167	0.0402475
$155$	0	0
$156$	0	0
$157$	−276.945	−0.140781	−0.0703906	−	0.997520i	$-0.522425\pi$
$157$	−276.945	−0.140781	−0.0703906	+	0.997520i	$0.522425\pi$
$158$	169.885	0.0855402
$159$	0	0
$160$	0	0
$161$	3450.88	1.68924
$162$	0	0
$163$	2654.73	1.27567	0.637836	−	0.770172i	$-0.279830\pi$
$163$	2654.73	1.27567	0.637836	+	0.770172i	$0.279830\pi$
$164$	2410.95	1.14795
$165$	0	0
$166$	184.420	0.0862274
$167$	3219.81	1.49196	0.745978	−	0.665970i	$-0.231982\pi$
$167$	3219.81	1.49196	0.745978	+	0.665970i	$0.231982\pi$
$168$	0	0
$169$	−1545.29	−0.703365
$170$	0	0
$171$	0	0
$172$	3763.94	1.66859
$173$	2540.33	1.11640	0.558201	−	0.829706i	$-0.311492\pi$
$173$	2540.33	1.11640	0.558201	+	0.829706i	$0.311492\pi$
$174$	0	0
$175$	0	0
$176$	−671.405	−0.287552
$177$	0	0
$178$	264.561	0.111403
$179$	86.1324	0.0359656	0.0179828	−	0.999838i	$-0.494276\pi$
$179$	86.1324	0.0359656	0.0179828	+	0.999838i	$0.494276\pi$
$180$	0	0
$181$	1745.36	0.716748	0.358374	−	0.933578i	$-0.383331\pi$
$181$	1745.36	0.716748	0.358374	+	0.933578i	$0.383331\pi$
$182$	178.507	0.0727021
$183$	0	0
$184$	972.378	0.389591
$185$	0	0
$186$	0	0
$187$	851.008	0.332791
$188$	2027.29	0.786466
$189$	0	0
$190$	0	0
$191$	5019.02	1.90138	0.950690	−	0.310144i	$-0.100377\pi$
$191$	5019.02	1.90138	0.950690	+	0.310144i	$0.100377\pi$
$192$	0	0
$193$	−4275.03	−1.59442	−0.797211	−	0.603700i	$-0.793692\pi$
$193$	−4275.03	−1.59442	−0.797211	+	0.603700i	$0.793692\pi$
$194$	201.538	0.0745856
$195$	0	0
$196$	−401.415	−0.146288
$197$	2006.68	0.725735	0.362867	−	0.931841i	$-0.381798\pi$
$197$	2006.68	0.725735	0.362867	+	0.931841i	$0.381798\pi$
$198$	0	0
$199$	−3668.78	−1.30690	−0.653450	−	0.756970i	$-0.726679\pi$
$199$	−3668.78	−1.30690	−0.653450	+	0.756970i	$0.726679\pi$
$200$	0	0
$201$	0	0
$202$	−28.0135	−0.00975753
$203$	−3763.67	−1.30127
$204$	0	0
$205$	0	0
$206$	−361.603	−0.122301
$207$	0	0
$208$	−1558.18	−0.519426
$209$	−1057.77	−0.350084
$210$	0	0
$211$	1600.30	0.522129	0.261064	−	0.965321i	$-0.415927\pi$
$211$	1600.30	0.522129	0.261064	+	0.965321i	$0.415927\pi$
$212$	−3904.56	−1.26494
$213$	0	0
$214$	−383.054	−0.122360
$215$	0	0
$216$	0	0
$217$	5474.81	1.71269
$218$	468.681	0.145610
$219$	0	0
$220$	0	0
$221$	1975.00	0.601145
$222$	0	0
$223$	−3904.42	−1.17246	−0.586232	−	0.810143i	$-0.699389\pi$
$223$	−3904.42	−1.17246	−0.586232	+	0.810143i	$0.699389\pi$
$224$	−1314.88	−0.392207
$225$	0	0
$226$	514.410	0.151407
$227$	1364.98	0.399107	0.199553	−	0.979887i	$-0.436051\pi$
$227$	1364.98	0.399107	0.199553	+	0.979887i	$0.436051\pi$
$228$	0	0
$229$	5045.66	1.45601	0.728005	−	0.685571i	$-0.240448\pi$
$229$	5045.66	1.45601	0.728005	+	0.685571i	$0.240448\pi$
$230$	0	0
$231$	0	0
$232$	−1060.51	−0.300113
$233$	−6083.69	−1.71054	−0.855270	−	0.518182i	$-0.826609\pi$
$233$	−6083.69	−1.71054	−0.855270	+	0.518182i	$0.826609\pi$
$234$	0	0
$235$	0	0
$236$	803.848	0.221720
$237$	0	0
$238$	540.965	0.147334
$239$	−3413.30	−0.923799	−0.461900	−	0.886932i	$-0.652832\pi$
$239$	−3413.30	−0.923799	−0.461900	+	0.886932i	$0.652832\pi$
$240$	0	0
$241$	979.432	0.261787	0.130894	−	0.991396i	$-0.458215\pi$
$241$	979.432	0.261787	0.130894	+	0.991396i	$0.458215\pi$
$242$	42.6268	0.0113230
$243$	0	0
$244$	4613.21	1.21037
$245$	0	0
$246$	0	0
$247$	−2454.85	−0.632383
$248$	1542.68	0.395000
$249$	0	0
$250$	0	0
$251$	2414.09	0.607075	0.303537	−	0.952820i	$-0.401832\pi$
$251$	2414.09	0.607075	0.303537	+	0.952820i	$0.401832\pi$
$252$	0	0
$253$	1912.46	0.475238
$254$	−360.339	−0.0890145
$255$	0	0
$256$	3475.25	0.848451
$257$	−2888.26	−0.701029	−0.350514	−	0.936557i	$-0.613993\pi$
$257$	−2888.26	−0.701029	−0.350514	+	0.936557i	$0.613993\pi$
$258$	0	0
$259$	5355.26	1.28479
$260$	0	0
$261$	0	0
$262$	−461.091	−0.108726
$263$	3916.08	0.918158	0.459079	−	0.888395i	$-0.348179\pi$
$263$	3916.08	0.918158	0.459079	+	0.888395i	$0.348179\pi$
$264$	0	0
$265$	0	0
$266$	−672.398	−0.154990
$267$	0	0
$268$	3736.17	0.851578
$269$	3495.41	0.792263	0.396131	−	0.918194i	$-0.370353\pi$
$269$	3495.41	0.792263	0.396131	+	0.918194i	$0.370353\pi$
$270$	0	0
$271$	774.974	0.173713	0.0868567	−	0.996221i	$-0.472318\pi$
$271$	774.974	0.173713	0.0868567	+	0.996221i	$0.472318\pi$
$272$	−4722.08	−1.05264
$273$	0	0
$274$	495.270	0.109198
$275$	0	0
$276$	0	0
$277$	−7491.71	−1.62503	−0.812515	−	0.582940i	$-0.801902\pi$
$277$	−7491.71	−1.62503	−0.812515	+	0.582940i	$0.801902\pi$
$278$	−255.457	−0.0551126
$279$	0	0
$280$	0	0
$281$	125.514	0.0266461	0.0133230	−	0.999911i	$-0.495759\pi$
$281$	125.514	0.0266461	0.0133230	+	0.999911i	$0.495759\pi$
$282$	0	0
$283$	−8.28147	−0.00173951	−0.000869757	−	1.00000i	$-0.500277\pi$
$283$	−8.28147	−0.00173951	−0.000869757		1.00000i	$0.500277\pi$
$284$	3575.29	0.747023
$285$	0	0
$286$	98.9274	0.0204535
$287$	6076.02	1.24967
$288$	0	0
$289$	1072.25	0.218247
$290$	0	0
$291$	0	0
$292$	5165.31	1.03519
$293$	3055.93	0.609316	0.304658	−	0.952462i	$-0.401458\pi$
$293$	3055.93	0.609316	0.304658	+	0.952462i	$0.401458\pi$
$294$	0	0
$295$	0	0
$296$	1508.99	0.296312
$297$	0	0
$298$	−990.724	−0.192588
$299$	4438.40	0.858458
$300$	0	0
$301$	9485.78	1.81645
$302$	−165.446	−0.0315243
$303$	0	0
$304$	5869.36	1.10734
$305$	0	0
$306$	0	0
$307$	−8063.04	−1.49896	−0.749482	−	0.662024i	$-0.769698\pi$
$307$	−8063.04	−1.49896	−0.749482	+	0.662024i	$0.769698\pi$
$308$	−1719.58	−0.318124
$309$	0	0
$310$	0	0
$311$	6633.87	1.20956	0.604778	−	0.796394i	$-0.293262\pi$
$311$	6633.87	1.20956	0.604778	+	0.796394i	$0.293262\pi$
$312$	0	0
$313$	4751.78	0.858104	0.429052	−	0.903280i	$-0.358848\pi$
$313$	4751.78	0.858104	0.429052	+	0.903280i	$0.358848\pi$
$314$	−97.5644	−0.0175346
$315$	0	0
$316$	−3798.03	−0.676126
$317$	−1207.20	−0.213890	−0.106945	−	0.994265i	$-0.534107\pi$
$317$	−1207.20	−0.213890	−0.106945	+	0.994265i	$0.534107\pi$
$318$	0	0
$319$	−2085.80	−0.366090
$320$	0	0
$321$	0	0
$322$	1215.70	0.210399
$323$	−7439.43	−1.28155
$324$	0	0
$325$	0	0
$326$	935.230	0.158888
$327$	0	0
$328$	1712.08	0.288213
$329$	5109.13	0.856157
$330$	0	0
$331$	2437.37	0.404744	0.202372	−	0.979309i	$-0.435135\pi$
$331$	2437.37	0.404744	0.202372	+	0.979309i	$0.435135\pi$
$332$	−4122.97	−0.681558
$333$	0	0
$334$	1134.30	0.185827
$335$	0	0
$336$	0	0
$337$	4057.48	0.655861	0.327930	−	0.944702i	$-0.393649\pi$
$337$	4057.48	0.655861	0.327930	+	0.944702i	$0.393649\pi$
$338$	−544.387	−0.0876058
$339$	0	0
$340$	0	0
$341$	3034.11	0.481837
$342$	0	0
$343$	5796.44	0.912473
$344$	2672.88	0.418930
$345$	0	0
$346$	894.926	0.139051
$347$	7263.85	1.12376	0.561879	−	0.827219i	$-0.310079\pi$
$347$	7263.85	1.12376	0.561879	+	0.827219i	$0.310079\pi$
$348$	0	0
$349$	5331.58	0.817745	0.408872	−	0.912592i	$-0.365922\pi$
$349$	5331.58	0.817745	0.408872	+	0.912592i	$0.365922\pi$
$350$	0	0
$351$	0	0
$352$	−728.702	−0.110341
$353$	−6448.09	−0.972231	−0.486115	−	0.873895i	$-0.661586\pi$
$353$	−6448.09	−0.972231	−0.486115	+	0.873895i	$0.661586\pi$
$354$	0	0
$355$	0	0
$356$	−5914.63	−0.880547
$357$	0	0
$358$	30.3434	0.00447960
$359$	1845.68	0.271341	0.135670	−	0.990754i	$-0.456681\pi$
$359$	1845.68	0.271341	0.135670	+	0.990754i	$0.456681\pi$
$360$	0	0
$361$	2387.93	0.348145
$362$	614.868	0.0892727
$363$	0	0
$364$	−3990.77	−0.574652
$365$	0	0
$366$	0	0
$367$	−11384.6	−1.61926	−0.809632	−	0.586938i	$-0.800333\pi$
$367$	−11384.6	−1.61926	−0.809632	+	0.586938i	$0.800333\pi$
$368$	−10611.9	−1.50321
$369$	0	0
$370$	0	0
$371$	−9840.17	−1.37702
$372$	0	0
$373$	3543.87	0.491943	0.245972	−	0.969277i	$-0.420893\pi$
$373$	3543.87	0.491943	0.245972	+	0.969277i	$0.420893\pi$
$374$	299.800	0.0414499
$375$	0	0
$376$	1439.64	0.197456
$377$	−4840.69	−0.661295
$378$	0	0
$379$	9256.66	1.25457	0.627286	−	0.778789i	$-0.284166\pi$
$379$	9256.66	1.25457	0.627286	+	0.778789i	$0.284166\pi$
$380$	0	0
$381$	0	0
$382$	1768.14	0.236822
$383$	−3260.36	−0.434978	−0.217489	−	0.976063i	$-0.569787\pi$
$383$	−3260.36	−0.434978	−0.217489	+	0.976063i	$0.569787\pi$
$384$	0	0
$385$	0	0
$386$	−1506.04	−0.198589
$387$	0	0
$388$	−4505.68	−0.589539
$389$	−139.121	−0.0181329	−0.00906646	−	0.999959i	$-0.502886\pi$
$389$	−139.121	−0.0181329	−0.00906646	+	0.999959i	$0.502886\pi$
$390$	0	0
$391$	13450.6	1.73970
$392$	−285.056	−0.0367283
$393$	0	0
$394$	706.927	0.0903921
$395$	0	0
$396$	0	0
$397$	1324.39	0.167428	0.0837141	−	0.996490i	$-0.473322\pi$
$397$	1324.39	0.167428	0.0837141	+	0.996490i	$0.473322\pi$
$398$	−1292.47	−0.162778
$399$	0	0
$400$	0	0
$401$	638.767	0.0795473	0.0397737	−	0.999209i	$-0.487336\pi$
$401$	638.767	0.0795473	0.0397737	+	0.999209i	$0.487336\pi$
$402$	0	0
$403$	7041.51	0.870379
$404$	626.281	0.0771253
$405$	0	0
$406$	−1325.89	−0.162076
$407$	2967.86	0.361453
$408$	0	0
$409$	7828.03	0.946384	0.473192	−	0.880959i	$-0.343102\pi$
$409$	7828.03	0.946384	0.473192	+	0.880959i	$0.343102\pi$
$410$	0	0
$411$	0	0
$412$	8084.15	0.966692
$413$	2025.84	0.241368
$414$	0	0
$415$	0	0
$416$	−1691.16	−0.199317
$417$	0	0
$418$	−372.640	−0.0436038
$419$	−3829.62	−0.446513	−0.223257	−	0.974760i	$-0.571669\pi$
$419$	−3829.62	−0.446513	−0.223257	+	0.974760i	$0.571669\pi$
$420$	0	0
$421$	−16051.8	−1.85824	−0.929120	−	0.369779i	$-0.879433\pi$
$421$	−16051.8	−1.85824	−0.929120	+	0.369779i	$0.879433\pi$
$422$	563.766	0.0650324
$423$	0	0
$424$	−2772.73	−0.317584
$425$	0	0
$426$	0	0
$427$	11626.1	1.31762
$428$	8563.72	0.967157
$429$	0	0
$430$	0	0
$431$	−1613.07	−0.180276	−0.0901379	−	0.995929i	$-0.528731\pi$
$431$	−1613.07	−0.180276	−0.0901379	+	0.995929i	$0.528731\pi$
$432$	0	0
$433$	−1137.16	−0.126209	−0.0631044	−	0.998007i	$-0.520100\pi$
$433$	−1137.16	−0.126209	−0.0631044	+	0.998007i	$0.520100\pi$
$434$	1928.71	0.213320
$435$	0	0
$436$	−10478.0	−1.15093
$437$	−16718.5	−1.83011
$438$	0	0
$439$	15605.6	1.69662	0.848309	−	0.529502i	$-0.177621\pi$
$439$	15605.6	1.69662	0.848309	+	0.529502i	$0.177621\pi$
$440$	0	0
$441$	0	0
$442$	695.769	0.0748741
$443$	−417.501	−0.0447767	−0.0223884	−	0.999749i	$-0.507127\pi$
$443$	−417.501	−0.0447767	−0.0223884	+	0.999749i	$0.507127\pi$
$444$	0	0
$445$	0	0
$446$	−1375.48	−0.146033
$447$	0	0
$448$	9228.76	0.973254
$449$	2601.70	0.273456	0.136728	−	0.990609i	$-0.456341\pi$
$449$	2601.70	0.273456	0.136728	+	0.990609i	$0.456341\pi$
$450$	0	0
$451$	3367.30	0.351574
$452$	−11500.4	−1.19675
$453$	0	0
$454$	480.867	0.0497097
$455$	0	0
$456$	0	0
$457$	6957.44	0.712156	0.356078	−	0.934456i	$-0.384114\pi$
$457$	6957.44	0.712156	0.356078	+	0.934456i	$0.384114\pi$
$458$	1777.52	0.181350
$459$	0	0
$460$	0	0
$461$	−3120.45	−0.315258	−0.157629	−	0.987498i	$-0.550385\pi$
$461$	−3120.45	−0.315258	−0.157629	+	0.987498i	$0.550385\pi$
$462$	0	0
$463$	−76.4453	−0.00767325	−0.00383662	−	0.999993i	$-0.501221\pi$
$463$	−76.4453	−0.00767325	−0.00383662	+	0.999993i	$0.501221\pi$
$464$	11573.7	1.15797
$465$	0	0
$466$	−2143.21	−0.213052
$467$	−9038.75	−0.895639	−0.447819	−	0.894124i	$-0.647799\pi$
$467$	−9038.75	−0.895639	−0.447819	+	0.894124i	$0.647799\pi$
$468$	0	0
$469$	9415.80	0.927039
$470$	0	0
$471$	0	0
$472$	570.834	0.0556668
$473$	5256.97	0.511027
$474$	0	0
$475$	0	0
$476$	−12094.0	−1.16456
$477$	0	0
$478$	−1202.46	−0.115062
$479$	7767.93	0.740972	0.370486	−	0.928838i	$-0.379191\pi$
$479$	7767.93	0.740972	0.370486	+	0.928838i	$0.379191\pi$
$480$	0	0
$481$	6887.74	0.652919
$482$	345.042	0.0326063
$483$	0	0
$484$	−952.983	−0.0894988
$485$	0	0
$486$	0	0
$487$	−311.944	−0.0290258	−0.0145129	−	0.999895i	$-0.504620\pi$
$487$	−311.944	−0.0290258	−0.0145129	+	0.999895i	$0.504620\pi$
$488$	3275.96	0.303885
$489$	0	0
$490$	0	0
$491$	−12908.1	−1.18642	−0.593211	−	0.805047i	$-0.702140\pi$
$491$	−12908.1	−1.18642	−0.593211	+	0.805047i	$0.702140\pi$
$492$	0	0
$493$	−14669.7	−1.34014
$494$	−864.814	−0.0787648
$495$	0	0
$496$	−16835.7	−1.52408
$497$	9010.35	0.813219
$498$	0	0
$499$	−14175.0	−1.27166	−0.635831	−	0.771828i	$-0.719343\pi$
$499$	−14175.0	−1.27166	−0.635831	+	0.771828i	$0.719343\pi$
$500$	0	0
$501$	0	0
$502$	850.453	0.0756127
$503$	−7144.07	−0.633277	−0.316639	−	0.948546i	$-0.602554\pi$
$503$	−7144.07	−0.633277	−0.316639	+	0.948546i	$0.602554\pi$
$504$	0	0
$505$	0	0
$506$	673.736	0.0591921
$507$	0	0
$508$	8055.89	0.703587
$509$	−8333.89	−0.725724	−0.362862	−	0.931843i	$-0.618200\pi$
$509$	−8333.89	−0.725724	−0.362862	+	0.931843i	$0.618200\pi$
$510$	0	0
$511$	13017.5	1.12693
$512$	6774.40	0.584744
$513$	0	0
$514$	−1017.50	−0.0873149
$515$	0	0
$516$	0	0
$517$	2831.45	0.240865
$518$	1886.59	0.160023
$519$	0	0
$520$	0	0
$521$	11728.3	0.986232	0.493116	−	0.869963i	$-0.335858\pi$
$521$	11728.3	0.986232	0.493116	+	0.869963i	$0.335858\pi$
$522$	0	0
$523$	15253.6	1.27532	0.637662	−	0.770316i	$-0.279901\pi$
$523$	15253.6	1.27532	0.637662	+	0.770316i	$0.279901\pi$
$524$	10308.3	0.859393
$525$	0	0
$526$	1379.59	0.114359
$527$	21339.3	1.76386
$528$	0	0
$529$	18060.3	1.48437
$530$	0	0
$531$	0	0
$532$	15032.4	1.22507
$533$	7814.76	0.635075
$534$	0	0
$535$	0	0
$536$	2653.15	0.213804
$537$	0	0
$538$	1231.39	0.0986783
$539$	−560.643	−0.0448026
$540$	0	0
$541$	−22147.6	−1.76008	−0.880039	−	0.474902i	$-0.842483\pi$
$541$	−22147.6	−1.76008	−0.880039	+	0.474902i	$0.842483\pi$
$542$	273.014	0.0216364
$543$	0	0
$544$	−5125.05	−0.403924
$545$	0	0
$546$	0	0
$547$	−2806.38	−0.219364	−0.109682	−	0.993967i	$-0.534983\pi$
$547$	−2806.38	−0.219364	−0.109682	+	0.993967i	$0.534983\pi$
$548$	−11072.5	−0.863124
$549$	0	0
$550$	0	0
$551$	18233.9	1.40978
$552$	0	0
$553$	−9571.69	−0.736040
$554$	−2639.24	−0.202402
$555$	0	0
$556$	5711.11	0.435621
$557$	−8190.23	−0.623036	−0.311518	−	0.950240i	$-0.600837\pi$
$557$	−8190.23	−0.623036	−0.311518	+	0.950240i	$0.600837\pi$
$558$	0	0
$559$	12200.3	0.923107
$560$	0	0
$561$	0	0
$562$	44.2171	0.00331883
$563$	21508.4	1.61007	0.805036	−	0.593226i	$-0.202146\pi$
$563$	21508.4	1.61007	0.805036	+	0.593226i	$0.202146\pi$
$564$	0	0
$565$	0	0
$566$	−2.91746	−0.000216661 0
$567$	0	0
$568$	2538.91	0.187553
$569$	7128.25	0.525188	0.262594	−	0.964906i	$-0.415422\pi$
$569$	7128.25	0.525188	0.262594	+	0.964906i	$0.415422\pi$
$570$	0	0
$571$	14565.4	1.06750	0.533750	−	0.845642i	$-0.320782\pi$
$571$	14565.4	1.06750	0.533750	+	0.845642i	$0.320782\pi$
$572$	−2211.66	−0.161668
$573$	0	0
$574$	2140.51	0.155650
$575$	0	0
$576$	0	0
$577$	−15188.8	−1.09587	−0.547937	−	0.836520i	$-0.684587\pi$
$577$	−15188.8	−1.09587	−0.547937	+	0.836520i	$0.684587\pi$
$578$	377.739	0.0271832
$579$	0	0
$580$	0	0
$581$	−10390.6	−0.741952
$582$	0	0
$583$	−5453.37	−0.387402
$584$	3668.02	0.259904
$585$	0	0
$586$	1076.57	0.0758918
$587$	−10096.8	−0.709948	−0.354974	−	0.934876i	$-0.615510\pi$
$587$	−10096.8	−0.709948	−0.354974	+	0.934876i	$0.615510\pi$
$588$	0	0
$589$	−26523.9	−1.85552
$590$	0	0
$591$	0	0
$592$	−16468.1	−1.14330
$593$	566.942	0.0392605	0.0196303	−	0.999807i	$-0.493751\pi$
$593$	566.942	0.0392605	0.0196303	+	0.999807i	$0.493751\pi$
$594$	0	0
$595$	0	0
$596$	22149.0	1.52225
$597$	0	0
$598$	1563.59	0.106923
$599$	−3326.86	−0.226931	−0.113466	−	0.993542i	$-0.536195\pi$
$599$	−3326.86	−0.226931	−0.113466	+	0.993542i	$0.536195\pi$
$600$	0	0
$601$	−27089.1	−1.83858	−0.919290	−	0.393581i	$-0.871236\pi$
$601$	−27089.1	−1.83858	−0.919290	+	0.393581i	$0.871236\pi$
$602$	3341.73	0.226243
$603$	0	0
$604$	3698.78	0.249174
$605$	0	0
$606$	0	0
$607$	−732.349	−0.0489706	−0.0244853	−	0.999700i	$-0.507795\pi$
$607$	−732.349	−0.0489706	−0.0244853	+	0.999700i	$0.507795\pi$
$608$	6370.24	0.424913
$609$	0	0
$610$	0	0
$611$	6571.18	0.435092
$612$	0	0
$613$	−7678.49	−0.505924	−0.252962	−	0.967476i	$-0.581405\pi$
$613$	−7678.49	−0.505924	−0.252962	+	0.967476i	$0.581405\pi$
$614$	−2840.51	−0.186700
$615$	0	0
$616$	−1221.12	−0.0798707
$617$	21512.8	1.40368	0.701841	−	0.712334i	$-0.252362\pi$
$617$	21512.8	1.40368	0.701841	+	0.712334i	$0.252362\pi$
$618$	0	0
$619$	−11921.8	−0.774112	−0.387056	−	0.922056i	$-0.626508\pi$
$619$	−11921.8	−0.774112	−0.387056	+	0.922056i	$0.626508\pi$
$620$	0	0
$621$	0	0
$622$	2337.03	0.150653
$623$	−14905.9	−0.958575
$624$	0	0
$625$	0	0
$626$	1673.99	0.106879
$627$	0	0
$628$	2181.19	0.138597
$629$	20873.3	1.32317
$630$	0	0
$631$	6522.97	0.411530	0.205765	−	0.978601i	$-0.434032\pi$
$631$	6522.97	0.411530	0.205765	+	0.978601i	$0.434032\pi$
$632$	−2697.08	−0.169753
$633$	0	0
$634$	−425.282	−0.0266405
$635$	0	0
$636$	0	0
$637$	−1301.13	−0.0809304
$638$	−734.803	−0.0455974
$639$	0	0
$640$	0	0
$641$	−2973.80	−0.183242	−0.0916209	−	0.995794i	$-0.529205\pi$
$641$	−2973.80	−0.183242	−0.0916209	+	0.995794i	$0.529205\pi$
$642$	0	0
$643$	−25884.2	−1.58752	−0.793758	−	0.608234i	$-0.791878\pi$
$643$	−25884.2	−1.58752	−0.793758	+	0.608234i	$0.791878\pi$
$644$	−27178.8	−1.66303
$645$	0	0
$646$	−2620.82	−0.159620
$647$	13929.9	0.846433	0.423217	−	0.906028i	$-0.360901\pi$
$647$	13929.9	0.846433	0.423217	+	0.906028i	$0.360901\pi$
$648$	0	0
$649$	1122.71	0.0679047
$650$	0	0
$651$	0	0
$652$	−20908.4	−1.25588
$653$	−18254.9	−1.09398	−0.546989	−	0.837140i	$-0.684226\pi$
$653$	−18254.9	−1.09398	−0.546989	+	0.837140i	$0.684226\pi$
$654$	0	0
$655$	0	0
$656$	−18684.5	−1.11205
$657$	0	0
$658$	1799.88	0.106636
$659$	−24637.9	−1.45638	−0.728192	−	0.685373i	$-0.759639\pi$
$659$	−24637.9	−1.45638	−0.728192	+	0.685373i	$0.759639\pi$
$660$	0	0
$661$	22721.6	1.33702	0.668508	−	0.743705i	$-0.266933\pi$
$661$	22721.6	1.33702	0.668508	+	0.743705i	$0.266933\pi$
$662$	858.657	0.0504118
$663$	0	0
$664$	−2927.83	−0.171117
$665$	0	0
$666$	0	0
$667$	−32967.1	−1.91378
$668$	−25358.9	−1.46881
$669$	0	0
$670$	0	0
$671$	6443.11	0.370691
$672$	0	0
$673$	10281.6	0.588898	0.294449	−	0.955667i	$-0.404864\pi$
$673$	10281.6	0.588898	0.294449	+	0.955667i	$0.404864\pi$
$674$	1429.40	0.0816891
$675$	0	0
$676$	12170.6	0.692453
$677$	18937.3	1.07507	0.537533	−	0.843243i	$-0.319356\pi$
$677$	18937.3	1.07507	0.537533	+	0.843243i	$0.319356\pi$
$678$	0	0
$679$	−11355.1	−0.641779
$680$	0	0
$681$	0	0
$682$	1068.88	0.0600140
$683$	9590.55	0.537294	0.268647	−	0.963239i	$-0.413423\pi$
$683$	9590.55	0.537294	0.268647	+	0.963239i	$0.413423\pi$
$684$	0	0
$685$	0	0
$686$	2042.01	0.113651
$687$	0	0
$688$	−29169.9	−1.61641
$689$	−12656.1	−0.699794
$690$	0	0
$691$	−29379.5	−1.61744	−0.808718	−	0.588197i	$-0.799838\pi$
$691$	−29379.5	−1.61744	−0.808718	+	0.588197i	$0.799838\pi$
$692$	−20007.3	−1.09908
$693$	0	0
$694$	2558.97	0.139967
$695$	0	0
$696$	0	0
$697$	23682.6	1.28701
$698$	1878.25	0.101852
$699$	0	0
$700$	0	0
$701$	−9839.37	−0.530140	−0.265070	−	0.964229i	$-0.585395\pi$
$701$	−9839.37	−0.530140	−0.265070	+	0.964229i	$0.585395\pi$
$702$	0	0
$703$	−25944.7	−1.39193
$704$	5114.53	0.273808
$705$	0	0
$706$	−2271.58	−0.121094
$707$	1578.34	0.0839596
$708$	0	0
$709$	2732.03	0.144716	0.0723579	−	0.997379i	$-0.476948\pi$
$709$	2732.03	0.144716	0.0723579	+	0.997379i	$0.476948\pi$
$710$	0	0
$711$	0	0
$712$	−4200.14	−0.221077
$713$	47955.5	2.51886
$714$	0	0
$715$	0	0
$716$	−678.370	−0.0354076
$717$	0	0
$718$	650.211	0.0337962
$719$	12352.7	0.640719	0.320359	−	0.947296i	$-0.396196\pi$
$719$	12352.7	0.640719	0.320359	+	0.947296i	$0.396196\pi$
$720$	0	0
$721$	20373.4	1.05235
$722$	841.238	0.0433624
$723$	0	0
$724$	−13746.2	−0.705628
$725$	0	0
$726$	0	0
$727$	−4292.36	−0.218975	−0.109488	−	0.993988i	$-0.534921\pi$
$727$	−4292.36	−0.218975	−0.109488	+	0.993988i	$0.534921\pi$
$728$	−2833.95	−0.144276
$729$	0	0
$730$	0	0
$731$	36972.9	1.87072
$732$	0	0
$733$	28377.0	1.42991	0.714957	−	0.699168i	$-0.246446\pi$
$733$	28377.0	1.42991	0.714957	+	0.699168i	$0.246446\pi$
$734$	−4010.65	−0.201683
$735$	0	0
$736$	−11517.5	−0.576819
$737$	5218.19	0.260806
$738$	0	0
$739$	15807.4	0.786853	0.393426	−	0.919356i	$-0.371290\pi$
$739$	15807.4	0.786853	0.393426	+	0.919356i	$0.371290\pi$
$740$	0	0
$741$	0	0
$742$	−3466.57	−0.171512
$743$	12809.5	0.632481	0.316241	−	0.948679i	$-0.397579\pi$
$743$	12809.5	0.632481	0.316241	+	0.948679i	$0.397579\pi$
$744$	0	0
$745$	0	0
$746$	1248.46	0.0612728
$747$	0	0
$748$	−6702.45	−0.327628
$749$	21582.1	1.05286
$750$	0	0
$751$	2792.75	0.135697	0.0678487	−	0.997696i	$-0.478386\pi$
$751$	2792.75	0.135697	0.0678487	+	0.997696i	$0.478386\pi$
$752$	−15711.2	−0.761872
$753$	0	0
$754$	−1705.32	−0.0823660
$755$	0	0
$756$	0	0
$757$	19648.3	0.943366	0.471683	−	0.881768i	$-0.343647\pi$
$757$	19648.3	0.943366	0.471683	+	0.881768i	$0.343647\pi$
$758$	3261.01	0.156260
$759$	0	0
$760$	0	0
$761$	−3048.76	−0.145226	−0.0726132	−	0.997360i	$-0.523134\pi$
$761$	−3048.76	−0.145226	−0.0726132	+	0.997360i	$0.523134\pi$
$762$	0	0
$763$	−26406.4	−1.25292
$764$	−39529.3	−1.87188
$765$	0	0
$766$	−1148.59	−0.0541776
$767$	2605.56	0.122661
$768$	0	0
$769$	1792.84	0.0840722	0.0420361	−	0.999116i	$-0.486616\pi$
$769$	1792.84	0.0840722	0.0420361	+	0.999116i	$0.486616\pi$
$770$	0	0
$771$	0	0
$772$	33669.7	1.56969
$773$	17731.5	0.825044	0.412522	−	0.910948i	$-0.364648\pi$
$773$	17731.5	0.825044	0.412522	+	0.910948i	$0.364648\pi$
$774$	0	0
$775$	0	0
$776$	−3199.60	−0.148014
$777$	0	0
$778$	−49.0106	−0.00225850
$779$	−29436.6	−1.35388
$780$	0	0
$781$	4993.49	0.228785
$782$	4738.47	0.216685
$783$	0	0
$784$	3110.90	0.141714
$785$	0	0
$786$	0	0
$787$	−7735.83	−0.350385	−0.175192	−	0.984534i	$-0.556055\pi$
$787$	−7735.83	−0.350385	−0.175192	+	0.984534i	$0.556055\pi$
$788$	−15804.4	−0.714476
$789$	0	0
$790$	0	0
$791$	−28982.9	−1.30280
$792$	0	0
$793$	14953.0	0.669607
$794$	466.565	0.0208536
$795$	0	0
$796$	28894.9	1.28663
$797$	−25823.4	−1.14769	−0.573846	−	0.818963i	$-0.694549\pi$
$797$	−25823.4	−1.14769	−0.573846	+	0.818963i	$0.694549\pi$
$798$	0	0
$799$	19914.0	0.881734
$800$	0	0
$801$	0	0
$802$	225.030	0.00990782
$803$	7214.21	0.317041
$804$	0	0
$805$	0	0
$806$	2480.64	0.108408
$807$	0	0
$808$	444.739	0.0193637
$809$	−3169.71	−0.137752	−0.0688759	−	0.997625i	$-0.521941\pi$
$809$	−3169.71	−0.137752	−0.0688759	+	0.997625i	$0.521941\pi$
$810$	0	0
$811$	23051.5	0.998085	0.499043	−	0.866577i	$-0.333685\pi$
$811$	23051.5	0.998085	0.499043	+	0.866577i	$0.333685\pi$
$812$	29642.2	1.28108
$813$	0	0
$814$	1045.54	0.0450198
$815$	0	0
$816$	0	0
$817$	−45955.9	−1.96793
$818$	2757.72	0.117875
$819$	0	0
$820$	0	0
$821$	−6922.35	−0.294265	−0.147133	−	0.989117i	$-0.547004\pi$
$821$	−6922.35	−0.294265	−0.147133	+	0.989117i	$0.547004\pi$
$822$	0	0
$823$	−6344.57	−0.268722	−0.134361	−	0.990932i	$-0.542898\pi$
$823$	−6344.57	−0.268722	−0.134361	+	0.990932i	$0.542898\pi$
$824$	5740.77	0.242705
$825$	0	0
$826$	713.677	0.0300630
$827$	−40602.9	−1.70726	−0.853629	−	0.520882i	$-0.825603\pi$
$827$	−40602.9	−1.70726	−0.853629	+	0.520882i	$0.825603\pi$
$828$	0	0
$829$	−20652.2	−0.865238	−0.432619	−	0.901577i	$-0.642410\pi$
$829$	−20652.2	−0.865238	−0.432619	+	0.901577i	$0.642410\pi$
$830$	0	0
$831$	0	0
$832$	11869.7	0.494601
$833$	−3943.08	−0.164009
$834$	0	0
$835$	0	0
$836$	8330.89	0.344653
$837$	0	0
$838$	−1349.13	−0.0556143
$839$	30274.0	1.24574	0.622870	−	0.782326i	$-0.285967\pi$
$839$	30274.0	1.24574	0.622870	+	0.782326i	$0.285967\pi$
$840$	0	0
$841$	11566.2	0.474238
$842$	−5654.87	−0.231448
$843$	0	0
$844$	−12603.8	−0.514029
$845$	0	0
$846$	0	0
$847$	−2401.68	−0.0974295
$848$	30259.7	1.22538
$849$	0	0
$850$	0	0
$851$	46908.3	1.88954
$852$	0	0
$853$	19771.8	0.793639	0.396820	−	0.917897i	$-0.370114\pi$
$853$	19771.8	0.793639	0.396820	+	0.917897i	$0.370114\pi$
$854$	4095.73	0.164113
$855$	0	0
$856$	6081.33	0.242822
$857$	−26489.4	−1.05585	−0.527923	−	0.849292i	$-0.677029\pi$
$857$	−26489.4	−1.05585	−0.527923	+	0.849292i	$0.677029\pi$
$858$	0	0
$859$	−24876.2	−0.988085	−0.494043	−	0.869438i	$-0.664481\pi$
$859$	−24876.2	−0.988085	−0.494043	+	0.869438i	$0.664481\pi$
$860$	0	0
$861$	0	0
$862$	−568.265	−0.0224538
$863$	13871.8	0.547162	0.273581	−	0.961849i	$-0.411792\pi$
$863$	13871.8	0.547162	0.273581	+	0.961849i	$0.411792\pi$
$864$	0	0
$865$	0	0
$866$	−400.608	−0.0157196
$867$	0	0
$868$	−43119.1	−1.68612
$869$	−5304.58	−0.207072
$870$	0	0
$871$	12110.3	0.471114
$872$	−7440.72	−0.288962
$873$	0	0
$874$	−5889.74	−0.227944
$875$	0	0
$876$	0	0
$877$	−34035.3	−1.31048	−0.655240	−	0.755421i	$-0.727432\pi$
$877$	−34035.3	−1.31048	−0.655240	+	0.755421i	$0.727432\pi$
$878$	5497.66	0.211318
$879$	0	0
$880$	0	0
$881$	642.501	0.0245703	0.0122851	−	0.999925i	$-0.496089\pi$
$881$	642.501	0.0245703	0.0122851	+	0.999925i	$0.496089\pi$
$882$	0	0
$883$	−37886.4	−1.44392	−0.721958	−	0.691937i	$-0.756758\pi$
$883$	−37886.4	−1.44392	−0.721958	+	0.691937i	$0.756758\pi$
$884$	−15554.9	−0.591819
$885$	0	0
$886$	−147.081	−0.00557705
$887$	17674.7	0.669063	0.334532	−	0.942385i	$-0.391422\pi$
$887$	17674.7	0.669063	0.334532	+	0.942385i	$0.391422\pi$
$888$	0	0
$889$	20302.2	0.765934
$890$	0	0
$891$	0	0
$892$	30750.8	1.15428
$893$	−24752.3	−0.927552
$894$	0	0
$895$	0	0
$896$	13770.2	0.513428
$897$	0	0
$898$	916.547	0.0340597
$899$	−52302.2	−1.94035
$900$	0	0
$901$	−38354.2	−1.41816
$902$	1186.26	0.0437894
$903$	0	0
$904$	−8166.72	−0.300466
$905$	0	0
$906$	0	0
$907$	46463.4	1.70098	0.850491	−	0.525989i	$-0.176305\pi$
$907$	46463.4	1.70098	0.850491	+	0.525989i	$0.176305\pi$
$908$	−10750.5	−0.392915
$909$	0	0
$910$	0	0
$911$	−15455.9	−0.562103	−0.281051	−	0.959693i	$-0.590683\pi$
$911$	−15455.9	−0.562103	−0.281051	+	0.959693i	$0.590683\pi$
$912$	0	0
$913$	−5758.41	−0.208736
$914$	2451.02	0.0887008
$915$	0	0
$916$	−39739.1	−1.43342
$917$	25978.8	0.935546
$918$	0	0
$919$	−35769.7	−1.28393	−0.641965	−	0.766734i	$-0.721881\pi$
$919$	−35769.7	−1.28393	−0.641965	+	0.766734i	$0.721881\pi$
$920$	0	0
$921$	0	0
$922$	−1099.30	−0.0392661
$923$	11588.8	0.413272
$924$	0	0
$925$	0	0
$926$	−26.9307	−0.000955722 0
$927$	0	0
$928$	12561.4	0.444340
$929$	50926.4	1.79854	0.899268	−	0.437398i	$-0.144100\pi$
$929$	50926.4	1.79854	0.899268	+	0.437398i	$0.144100\pi$
$930$	0	0
$931$	4901.09	0.172531
$932$	47914.5	1.68400
$933$	0	0
$934$	−3184.24	−0.111554
$935$	0	0
$936$	0	0
$937$	15370.9	0.535906	0.267953	−	0.963432i	$-0.413653\pi$
$937$	15370.9	0.535906	0.267953	+	0.963432i	$0.413653\pi$
$938$	3317.07	0.115465
$939$	0	0
$940$	0	0
$941$	−35629.1	−1.23430	−0.617149	−	0.786846i	$-0.711713\pi$
$941$	−35629.1	−1.23430	−0.617149	+	0.786846i	$0.711713\pi$
$942$	0	0
$943$	53221.7	1.83790
$944$	−6229.68	−0.214787
$945$	0	0
$946$	1851.97	0.0636497
$947$	−22204.2	−0.761920	−0.380960	−	0.924591i	$-0.624407\pi$
$947$	−22204.2	−0.761920	−0.380960	+	0.924591i	$0.624407\pi$
$948$	0	0
$949$	16742.6	0.572695
$950$	0	0
$951$	0	0
$952$	−8588.30	−0.292383
$953$	−50757.0	−1.72527	−0.862635	−	0.505827i	$-0.831187\pi$
$953$	−50757.0	−1.72527	−0.862635	+	0.505827i	$0.831187\pi$
$954$	0	0
$955$	0	0
$956$	26882.8	0.909468
$957$	0	0
$958$	2736.55	0.0922900
$959$	−27904.5	−0.939608
$960$	0	0
$961$	46290.4	1.55384
$962$	2426.47	0.0813227
$963$	0	0
$964$	−7713.90	−0.257726
$965$	0	0
$966$	0	0
$967$	−33405.1	−1.11089	−0.555447	−	0.831552i	$-0.687453\pi$
$967$	−33405.1	−1.11089	−0.555447	+	0.831552i	$0.687453\pi$
$968$	−676.739	−0.0224703
$969$	0	0
$970$	0	0
$971$	−48382.3	−1.59903	−0.799516	−	0.600644i	$-0.794911\pi$
$971$	−48382.3	−1.59903	−0.799516	+	0.600644i	$0.794911\pi$
$972$	0	0
$973$	14393.0	0.474222
$974$	−109.894	−0.00361523
$975$	0	0
$976$	−35751.6	−1.17252
$977$	14604.6	0.478241	0.239121	−	0.970990i	$-0.423141\pi$
$977$	14604.6	0.478241	0.239121	+	0.970990i	$0.423141\pi$
$978$	0	0
$979$	−8260.77	−0.269679
$980$	0	0
$981$	0	0
$982$	−4547.35	−0.147772
$983$	−22492.6	−0.729809	−0.364904	−	0.931045i	$-0.618898\pi$
$983$	−22492.6	−0.729809	−0.364904	+	0.931045i	$0.618898\pi$
$984$	0	0
$985$	0	0
$986$	−5167.96	−0.166918
$987$	0	0
$988$	19334.2	0.622572
$989$	83088.8	2.67146
$990$	0	0
$991$	22276.1	0.714049	0.357025	−	0.934095i	$-0.383791\pi$
$991$	22276.1	0.714049	0.357025	+	0.934095i	$0.383791\pi$
$992$	−18272.4	−0.584829
$993$	0	0
$994$	3174.24	0.101288
$995$	0	0
$996$	0	0
$997$	25955.4	0.824490	0.412245	−	0.911073i	$-0.364745\pi$
$997$	25955.4	0.824490	0.412245	+	0.911073i	$0.364745\pi$
$998$	−4993.68	−0.158389
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bs.1.4		7
3.2	odd	2		825.4.a.ba.1.4		7
5.2	odd	4		495.4.c.d.199.8		14
5.3	odd	4		495.4.c.d.199.7		14
5.4	even	2		2475.4.a.bo.1.4		7
15.2	even	4		165.4.c.b.34.7	✓	14
15.8	even	4		165.4.c.b.34.8	yes	14
15.14	odd	2		825.4.a.bd.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.c.b.34.7	✓	14	15.2	even	4
165.4.c.b.34.8	yes	14	15.8	even	4
495.4.c.d.199.7		14	5.3	odd	4
495.4.c.d.199.8		14	5.2	odd	4
825.4.a.ba.1.4		7	3.2	odd	2
825.4.a.bd.1.4		7	15.14	odd	2
2475.4.a.bo.1.4		7	5.4	even	2
2475.4.a.bs.1.4		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.352288 q^{2} -7.87589 q^{4} -19.8486 q^{7} -5.59288 q^{8} +O(q^{10})\)	\(q+0.352288 q^{2} -7.87589 q^{4} -19.8486 q^{7} -5.59288 q^{8} -11.0000 q^{11} -25.5286 q^{13} -6.99242 q^{14} +61.0368 q^{16} -77.3644 q^{17} +96.1609 q^{19} -3.87517 q^{22} -173.860 q^{23} -8.99340 q^{26} +156.326 q^{28} +189.619 q^{29} -275.829 q^{31} +66.2456 q^{32} -27.2545 q^{34} -269.805 q^{37} +33.8763 q^{38} -306.118 q^{41} -477.907 q^{43} +86.6348 q^{44} -61.2487 q^{46} -257.405 q^{47} +50.9676 q^{49} +201.060 q^{52} +495.761 q^{53} +111.011 q^{56} +66.8003 q^{58} -102.064 q^{59} -585.738 q^{61} -97.1710 q^{62} -464.957 q^{64} -474.381 q^{67} +609.314 q^{68} -453.954 q^{71} -655.838 q^{73} -95.0491 q^{74} -757.353 q^{76} +218.335 q^{77} +482.235 q^{79} -107.842 q^{82} +523.492 q^{83} -168.361 q^{86} +61.5217 q^{88} +750.979 q^{89} +506.707 q^{91} +1369.30 q^{92} -90.6806 q^{94} +572.084 q^{97} +17.9553 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 13 q^{4} - 34 q^{7} + 75 q^{8}+O(q^{10}) \) 7 * q + 5 * q^2 + 13 * q^4 - 34 * q^7 + 75 * q^8	\( 7 q + 5 q^{2} + 13 q^{4} - 34 q^{7} + 75 q^{8} - 77 q^{11} - 80 q^{13} - 42 q^{14} - 43 q^{16} + 162 q^{17} + 58 q^{19} - 55 q^{22} + 324 q^{23} + 200 q^{26} + 168 q^{28} - 64 q^{29} - 348 q^{31} - 75 q^{32} + 206 q^{34} - 664 q^{37} + 334 q^{38} + 332 q^{41} - 774 q^{43} - 143 q^{44} - 328 q^{46} + 872 q^{47} - 417 q^{49} - 134 q^{52} + 1628 q^{53} + 1618 q^{56} - 1568 q^{58} + 332 q^{59} + 22 q^{61} - 260 q^{62} + 561 q^{64} - 1524 q^{67} + 2324 q^{68} + 516 q^{71} - 1700 q^{73} - 1628 q^{74} + 2794 q^{76} + 374 q^{77} + 1746 q^{79} - 364 q^{82} + 2344 q^{83} - 1270 q^{86} - 825 q^{88} + 2226 q^{89} + 1072 q^{91} + 4184 q^{92} + 4736 q^{94} - 1048 q^{97} + 3057 q^{98}+O(q^{100}) \) 7 * q + 5 * q^2 + 13 * q^4 - 34 * q^7 + 75 * q^8 - 77 * q^11 - 80 * q^13 - 42 * q^14 - 43 * q^16 + 162 * q^17 + 58 * q^19 - 55 * q^22 + 324 * q^23 + 200 * q^26 + 168 * q^28 - 64 * q^29 - 348 * q^31 - 75 * q^32 + 206 * q^34 - 664 * q^37 + 334 * q^38 + 332 * q^41 - 774 * q^43 - 143 * q^44 - 328 * q^46 + 872 * q^47 - 417 * q^49 - 134 * q^52 + 1628 * q^53 + 1618 * q^56 - 1568 * q^58 + 332 * q^59 + 22 * q^61 - 260 * q^62 + 561 * q^64 - 1524 * q^67 + 2324 * q^68 + 516 * q^71 - 1700 * q^73 - 1628 * q^74 + 2794 * q^76 + 374 * q^77 + 1746 * q^79 - 364 * q^82 + 2344 * q^83 - 1270 * q^86 - 825 * q^88 + 2226 * q^89 + 1072 * q^91 + 4184 * q^92 + 4736 * q^94 - 1048 * q^97 + 3057 * q^98

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