LMFDB - Embedded newform 2475.4.a.br.1.2

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:	$7$
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} - 3x^{6} - 44x^{5} + 118x^{4} + 515x^{3} - 1279x^{2} - 892x + 1840 $ x^7 - 3x^6 - 44x^5 + 118x^4 + 515x^3 - 1279x^2 - 892x + 1840
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.31207$ 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)

$f(q)$	$=$	$q-4.31207 q^{2} +10.5939 q^{4} +9.06309 q^{7} -11.1853 q^{8} +O(q^{10})$	$q-4.31207 q^{2} +10.5939 q^{4} +9.06309 q^{7} -11.1853 q^{8} +11.0000 q^{11} -40.4128 q^{13} -39.0807 q^{14} -36.5198 q^{16} +76.4137 q^{17} -44.7264 q^{19} -47.4328 q^{22} -50.4845 q^{23} +174.263 q^{26} +96.0139 q^{28} +81.1094 q^{29} +100.171 q^{31} +246.958 q^{32} -329.501 q^{34} -38.7941 q^{37} +192.863 q^{38} -18.6954 q^{41} -40.8344 q^{43} +116.533 q^{44} +217.693 q^{46} +36.4307 q^{47} -260.860 q^{49} -428.131 q^{52} +652.838 q^{53} -101.373 q^{56} -349.749 q^{58} -744.649 q^{59} -516.203 q^{61} -431.945 q^{62} -772.743 q^{64} +430.849 q^{67} +809.523 q^{68} +272.484 q^{71} -692.483 q^{73} +167.283 q^{74} -473.829 q^{76} +99.6940 q^{77} -1086.01 q^{79} +80.6157 q^{82} +595.328 q^{83} +176.081 q^{86} -123.038 q^{88} +706.539 q^{89} -366.265 q^{91} -534.830 q^{92} -157.092 q^{94} +1311.14 q^{97} +1124.85 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8	$ 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8} + 77 q^{11} - 24 q^{13} - 142 q^{14} + 181 q^{16} + 38 q^{17} + 26 q^{19} + 33 q^{22} + 228 q^{23} - 476 q^{26} - 840 q^{28} - 572 q^{29} - 140 q^{31} + 991 q^{32} - 806 q^{34} + 104 q^{37} + 498 q^{38} - 896 q^{41} - 614 q^{43} + 451 q^{44} - 344 q^{46} + 520 q^{47} + 295 q^{49} + 26 q^{52} + 380 q^{53} - 1522 q^{56} - 1600 q^{58} - 1316 q^{59} - 386 q^{61} + 440 q^{62} + 869 q^{64} - 348 q^{67} + 332 q^{68} - 804 q^{71} - 468 q^{73} + 748 q^{74} - 1698 q^{76} - 550 q^{77} - 374 q^{79} + 620 q^{82} + 3128 q^{83} + 2534 q^{86} + 231 q^{88} - 694 q^{89} - 3376 q^{91} - 1184 q^{92} - 2920 q^{94} + 8 q^{97} + 4211 q^{98}+O(q^{100}) $ 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8 + 77 * q^11 - 24 * q^13 - 142 * q^14 + 181 * q^16 + 38 * q^17 + 26 * q^19 + 33 * q^22 + 228 * q^23 - 476 * q^26 - 840 * q^28 - 572 * q^29 - 140 * q^31 + 991 * q^32 - 806 * q^34 + 104 * q^37 + 498 * q^38 - 896 * q^41 - 614 * q^43 + 451 * q^44 - 344 * q^46 + 520 * q^47 + 295 * q^49 + 26 * q^52 + 380 * q^53 - 1522 * q^56 - 1600 * q^58 - 1316 * q^59 - 386 * q^61 + 440 * q^62 + 869 * q^64 - 348 * q^67 + 332 * q^68 - 804 * q^71 - 468 * q^73 + 748 * q^74 - 1698 * q^76 - 550 * q^77 - 374 * q^79 + 620 * q^82 + 3128 * q^83 + 2534 * q^86 + 231 * q^88 - 694 * q^89 - 3376 * q^91 - 1184 * q^92 - 2920 * q^94 + 8 * q^97 + 4211 * 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 $

$2$	−4.31207	−1.52455	−0.762274	−	0.647255i	$-0.775917\pi$
$2$	−4.31207	−1.52455	−0.762274	+	0.647255i	$0.775917\pi$
$3$	0	0
$3$	0	0
$4$	10.5939	1.32424
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.06309	0.489361	0.244680	−	0.969604i	$-0.421317\pi$
$7$	9.06309	0.489361	0.244680	+	0.969604i	$0.421317\pi$
$8$	−11.1853	−0.494325
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−40.4128	−0.862192	−0.431096	−	0.902306i	$-0.641873\pi$
$13$	−40.4128	−0.862192	−0.431096	+	0.902306i	$0.641873\pi$
$14$	−39.0807	−0.746054
$15$	0	0
$16$	−36.5198	−0.570622
$17$	76.4137	1.09018	0.545089	−	0.838378i	$-0.316496\pi$
$17$	76.4137	1.09018	0.545089	+	0.838378i	$0.316496\pi$
$18$	0	0
$19$	−44.7264	−0.540050	−0.270025	−	0.962853i	$-0.587032\pi$
$19$	−44.7264	−0.540050	−0.270025	+	0.962853i	$0.587032\pi$
$20$	0	0
$21$	0	0
$22$	−47.4328	−0.459668
$23$	−50.4845	−0.457685	−0.228842	−	0.973464i	$-0.573494\pi$
$23$	−50.4845	−0.457685	−0.228842	+	0.973464i	$0.573494\pi$
$24$	0	0
$25$	0	0
$26$	174.263	1.31445
$27$	0	0
$28$	96.0139	0.648033
$29$	81.1094	0.519367	0.259683	−	0.965694i	$-0.416382\pi$
$29$	81.1094	0.519367	0.259683	+	0.965694i	$0.416382\pi$
$30$	0	0
$31$	100.171	0.580363	0.290182	−	0.956972i	$-0.406284\pi$
$31$	100.171	0.580363	0.290182	+	0.956972i	$0.406284\pi$
$32$	246.958	1.36427
$33$	0	0
$34$	−329.501	−1.66203
$35$	0	0
$36$	0	0
$37$	−38.7941	−0.172371	−0.0861853	−	0.996279i	$-0.527468\pi$
$37$	−38.7941	−0.172371	−0.0861853	+	0.996279i	$0.527468\pi$
$38$	192.863	0.823331
$39$	0	0
$40$	0	0
$41$	−18.6954	−0.0712128	−0.0356064	−	0.999366i	$-0.511336\pi$
$41$	−18.6954	−0.0712128	−0.0356064	+	0.999366i	$0.511336\pi$
$42$	0	0
$43$	−40.8344	−0.144818	−0.0724091	−	0.997375i	$-0.523069\pi$
$43$	−40.8344	−0.144818	−0.0724091	+	0.997375i	$0.523069\pi$
$44$	116.533	0.399274
$45$	0	0
$46$	217.693	0.697762
$47$	36.4307	0.113063	0.0565315	−	0.998401i	$-0.481996\pi$
$47$	36.4307	0.113063	0.0565315	+	0.998401i	$0.481996\pi$
$48$	0	0
$49$	−260.860	−0.760526
$50$	0	0
$51$	0	0
$52$	−428.131	−1.14175
$53$	652.838	1.69197	0.845983	−	0.533209i	$-0.179014\pi$
$53$	652.838	1.69197	0.845983	+	0.533209i	$0.179014\pi$
$54$	0	0
$55$	0	0
$56$	−101.373	−0.241903
$57$	0	0
$58$	−349.749	−0.791799
$59$	−744.649	−1.64314	−0.821568	−	0.570110i	$-0.806900\pi$
$59$	−744.649	−1.64314	−0.821568	+	0.570110i	$0.806900\pi$
$60$	0	0
$61$	−516.203	−1.08349	−0.541746	−	0.840542i	$-0.682237\pi$
$61$	−516.203	−1.08349	−0.541746	+	0.840542i	$0.682237\pi$
$62$	−431.945	−0.884791
$63$	0	0
$64$	−772.743	−1.50926
$65$	0	0
$66$	0	0
$67$	430.849	0.785621	0.392811	−	0.919619i	$-0.371503\pi$
$67$	430.849	0.785621	0.392811	+	0.919619i	$0.371503\pi$
$68$	809.523	1.44366
$69$	0	0
$70$	0	0
$71$	272.484	0.455464	0.227732	−	0.973724i	$-0.426869\pi$
$71$	272.484	0.455464	0.227732	+	0.973724i	$0.426869\pi$
$72$	0	0
$73$	−692.483	−1.11026	−0.555130	−	0.831763i	$-0.687332\pi$
$73$	−692.483	−1.11026	−0.555130	+	0.831763i	$0.687332\pi$
$74$	167.283	0.262787
$75$	0	0
$76$	−473.829	−0.715157
$77$	99.6940	0.147548
$78$	0	0
$79$	−1086.01	−1.54665	−0.773324	−	0.634011i	$-0.781407\pi$
$79$	−1086.01	−1.54665	−0.773324	+	0.634011i	$0.781407\pi$
$80$	0	0
$81$	0	0
$82$	80.6157	0.108567
$83$	595.328	0.787299	0.393649	−	0.919261i	$-0.371212\pi$
$83$	595.328	0.787299	0.393649	+	0.919261i	$0.371212\pi$
$84$	0	0
$85$	0	0
$86$	176.081	0.220782
$87$	0	0
$88$	−123.038	−0.149045
$89$	706.539	0.841494	0.420747	−	0.907178i	$-0.361768\pi$
$89$	706.539	0.841494	0.420747	+	0.907178i	$0.361768\pi$
$90$	0	0
$91$	−366.265	−0.421923
$92$	−534.830	−0.606086
$93$	0	0
$94$	−157.092	−0.172370
$95$	0	0
$96$	0	0
$97$	1311.14	1.37244	0.686218	−	0.727396i	$-0.259270\pi$
$97$	1311.14	1.37244	0.686218	+	0.727396i	$0.259270\pi$
$98$	1124.85	1.15946
$99$	0	0
$100$	0	0
$101$	−1377.31	−1.35690	−0.678452	−	0.734644i	$-0.737349\pi$
$101$	−1377.31	−1.35690	−0.678452	+	0.734644i	$0.737349\pi$
$102$	0	0
$103$	−714.721	−0.683724	−0.341862	−	0.939750i	$-0.611058\pi$
$103$	−714.721	−0.683724	−0.341862	+	0.939750i	$0.611058\pi$
$104$	452.029	0.426203
$105$	0	0
$106$	−2815.09	−2.57948
$107$	1495.34	1.35102	0.675512	−	0.737349i	$-0.263922\pi$
$107$	1495.34	1.35102	0.675512	+	0.737349i	$0.263922\pi$
$108$	0	0
$109$	−743.908	−0.653702	−0.326851	−	0.945076i	$-0.605987\pi$
$109$	−743.908	−0.653702	−0.326851	+	0.945076i	$0.605987\pi$
$110$	0	0
$111$	0	0
$112$	−330.983	−0.279240
$113$	−1639.90	−1.36521	−0.682604	−	0.730789i	$-0.739153\pi$
$113$	−1639.90	−1.36521	−0.682604	+	0.730789i	$0.739153\pi$
$114$	0	0
$115$	0	0
$116$	859.269	0.687768
$117$	0	0
$118$	3210.98	2.50504
$119$	692.544	0.533491
$120$	0	0
$121$	121.000	0.0909091
$122$	2225.90	1.65183
$123$	0	0
$124$	1061.21	0.768543
$125$	0	0
$126$	0	0
$127$	−2329.48	−1.62762	−0.813810	−	0.581131i	$-0.802610\pi$
$127$	−2329.48	−1.62762	−0.813810	+	0.581131i	$0.802610\pi$
$128$	1356.46	0.936679
$129$	0	0
$130$	0	0
$131$	−1842.66	−1.22896	−0.614480	−	0.788932i	$-0.710634\pi$
$131$	−1842.66	−1.22896	−0.614480	+	0.788932i	$0.710634\pi$
$132$	0	0
$133$	−405.360	−0.264279
$134$	−1857.85	−1.19772
$135$	0	0
$136$	−854.709	−0.538902
$137$	−893.224	−0.557031	−0.278516	−	0.960432i	$-0.589842\pi$
$137$	−893.224	−0.557031	−0.278516	+	0.960432i	$0.589842\pi$
$138$	0	0
$139$	2777.79	1.69503	0.847514	−	0.530773i	$-0.178098\pi$
$139$	2777.79	1.69503	0.847514	+	0.530773i	$0.178098\pi$
$140$	0	0
$141$	0	0
$142$	−1174.97	−0.694376
$143$	−444.541	−0.259961
$144$	0	0
$145$	0	0
$146$	2986.04	1.69264
$147$	0	0
$148$	−410.983	−0.228261
$149$	−969.362	−0.532975	−0.266487	−	0.963838i	$-0.585863\pi$
$149$	−969.362	−0.532975	−0.266487	+	0.963838i	$0.585863\pi$
$150$	0	0
$151$	3477.75	1.87428	0.937138	−	0.348959i	$-0.113465\pi$
$151$	3477.75	1.87428	0.937138	+	0.348959i	$0.113465\pi$
$152$	500.278	0.266960
$153$	0	0
$154$	−429.888	−0.224944
$155$	0	0
$156$	0	0
$157$	2374.99	1.20729	0.603646	−	0.797252i	$-0.293714\pi$
$157$	2374.99	1.20729	0.603646	+	0.797252i	$0.293714\pi$
$158$	4682.93	2.35794
$159$	0	0
$160$	0	0
$161$	−457.546	−0.223973
$162$	0	0
$163$	363.778	0.174805	0.0874027	−	0.996173i	$-0.472143\pi$
$163$	363.778	0.174805	0.0874027	+	0.996173i	$0.472143\pi$
$164$	−198.058	−0.0943031
$165$	0	0
$166$	−2567.10	−1.20027
$167$	3804.91	1.76307	0.881535	−	0.472119i	$-0.156511\pi$
$167$	3804.91	1.76307	0.881535	+	0.472119i	$0.156511\pi$
$168$	0	0
$169$	−563.806	−0.256625
$170$	0	0
$171$	0	0
$172$	−432.597	−0.191775
$173$	−1008.59	−0.443248	−0.221624	−	0.975132i	$-0.571136\pi$
$173$	−1008.59	−0.443248	−0.221624	+	0.975132i	$0.571136\pi$
$174$	0	0
$175$	0	0
$176$	−401.718	−0.172049
$177$	0	0
$178$	−3046.65	−1.28290
$179$	−3347.66	−1.39785	−0.698926	−	0.715194i	$-0.746339\pi$
$179$	−3347.66	−1.39785	−0.698926	+	0.715194i	$0.746339\pi$
$180$	0	0
$181$	1470.36	0.603819	0.301909	−	0.953337i	$-0.402376\pi$
$181$	1470.36	0.603819	0.301909	+	0.953337i	$0.402376\pi$
$182$	1579.36	0.643241
$183$	0	0
$184$	564.684	0.226245
$185$	0	0
$186$	0	0
$187$	840.550	0.328701
$188$	385.945	0.149723
$189$	0	0
$190$	0	0
$191$	−3232.07	−1.22442	−0.612210	−	0.790695i	$-0.709719\pi$
$191$	−3232.07	−1.22442	−0.612210	+	0.790695i	$0.709719\pi$
$192$	0	0
$193$	−3983.00	−1.48551	−0.742753	−	0.669566i	$-0.766480\pi$
$193$	−3983.00	−1.48551	−0.742753	+	0.669566i	$0.766480\pi$
$194$	−5653.74	−2.09234
$195$	0	0
$196$	−2763.54	−1.00712
$197$	4893.00	1.76960	0.884802	−	0.465967i	$-0.154293\pi$
$197$	4893.00	1.76960	0.884802	+	0.465967i	$0.154293\pi$
$198$	0	0
$199$	254.519	0.0906651	0.0453326	−	0.998972i	$-0.485565\pi$
$199$	254.519	0.0906651	0.0453326	+	0.998972i	$0.485565\pi$
$200$	0	0
$201$	0	0
$202$	5939.05	2.06867
$203$	735.102	0.254158
$204$	0	0
$205$	0	0
$206$	3081.93	1.04237
$207$	0	0
$208$	1475.87	0.491986
$209$	−491.990	−0.162831
$210$	0	0
$211$	−1289.71	−0.420794	−0.210397	−	0.977616i	$-0.567476\pi$
$211$	−1289.71	−0.420794	−0.210397	+	0.977616i	$0.567476\pi$
$212$	6916.14	2.24058
$213$	0	0
$214$	−6448.00	−2.05970
$215$	0	0
$216$	0	0
$217$	907.860	0.284007
$218$	3207.78	0.996599
$219$	0	0
$220$	0	0
$221$	−3088.09	−0.939943
$222$	0	0
$223$	−1519.82	−0.456388	−0.228194	−	0.973616i	$-0.573282\pi$
$223$	−1519.82	−0.456388	−0.228194	+	0.973616i	$0.573282\pi$
$224$	2238.21	0.667618
$225$	0	0
$226$	7071.35	2.08132
$227$	2735.43	0.799809	0.399904	−	0.916557i	$-0.369043\pi$
$227$	2735.43	0.799809	0.399904	+	0.916557i	$0.369043\pi$
$228$	0	0
$229$	1529.19	0.441274	0.220637	−	0.975356i	$-0.429186\pi$
$229$	1529.19	0.441274	0.220637	+	0.975356i	$0.429186\pi$
$230$	0	0
$231$	0	0
$232$	−907.232	−0.256736
$233$	−1660.31	−0.466826	−0.233413	−	0.972378i	$-0.574989\pi$
$233$	−1660.31	−0.466826	−0.233413	+	0.972378i	$0.574989\pi$
$234$	0	0
$235$	0	0
$236$	−7888.77	−2.17591
$237$	0	0
$238$	−2986.30	−0.813332
$239$	−558.538	−0.151167	−0.0755834	−	0.997139i	$-0.524082\pi$
$239$	−558.538	−0.151167	−0.0755834	+	0.997139i	$0.524082\pi$
$240$	0	0
$241$	6132.52	1.63913	0.819565	−	0.572986i	$-0.194215\pi$
$241$	6132.52	1.63913	0.819565	+	0.572986i	$0.194215\pi$
$242$	−521.760	−0.138595
$243$	0	0
$244$	−5468.63	−1.43481
$245$	0	0
$246$	0	0
$247$	1807.52	0.465626
$248$	−1120.44	−0.286888
$249$	0	0
$250$	0	0
$251$	−2130.42	−0.535741	−0.267870	−	0.963455i	$-0.586320\pi$
$251$	−2130.42	−0.535741	−0.267870	+	0.963455i	$0.586320\pi$
$252$	0	0
$253$	−555.329	−0.137997
$254$	10044.9	2.48138
$255$	0	0
$256$	332.812	0.0812530
$257$	2559.71	0.621285	0.310643	−	0.950527i	$-0.399456\pi$
$257$	2559.71	0.621285	0.310643	+	0.950527i	$0.399456\pi$
$258$	0	0
$259$	−351.595	−0.0843515
$260$	0	0
$261$	0	0
$262$	7945.67	1.87361
$263$	69.1920	0.0162227	0.00811133	−	0.999967i	$-0.497418\pi$
$263$	69.1920	0.0162227	0.00811133	+	0.999967i	$0.497418\pi$
$264$	0	0
$265$	0	0
$266$	1747.94	0.402906
$267$	0	0
$268$	4564.40	1.04035
$269$	−2985.84	−0.676765	−0.338382	−	0.941009i	$-0.609880\pi$
$269$	−2985.84	−0.676765	−0.338382	+	0.941009i	$0.609880\pi$
$270$	0	0
$271$	337.747	0.0757073	0.0378536	−	0.999283i	$-0.487948\pi$
$271$	337.747	0.0757073	0.0378536	+	0.999283i	$0.487948\pi$
$272$	−2790.61	−0.622080
$273$	0	0
$274$	3851.65	0.849221
$275$	0	0
$276$	0	0
$277$	5468.83	1.18625	0.593123	−	0.805112i	$-0.297895\pi$
$277$	5468.83	1.18625	0.593123	+	0.805112i	$0.297895\pi$
$278$	−11978.0	−2.58415
$279$	0	0
$280$	0	0
$281$	3846.36	0.816563	0.408282	−	0.912856i	$-0.366128\pi$
$281$	3846.36	0.816563	0.408282	+	0.912856i	$0.366128\pi$
$282$	0	0
$283$	6289.97	1.32120	0.660601	−	0.750737i	$-0.270302\pi$
$283$	6289.97	1.32120	0.660601	+	0.750737i	$0.270302\pi$
$284$	2886.68	0.603145
$285$	0	0
$286$	1916.89	0.396322
$287$	−169.438	−0.0348488
$288$	0	0
$289$	926.050	0.188490
$290$	0	0
$291$	0	0
$292$	−7336.13	−1.47026
$293$	−1660.81	−0.331146	−0.165573	−	0.986198i	$-0.552947\pi$
$293$	−1660.81	−0.331146	−0.165573	+	0.986198i	$0.552947\pi$
$294$	0	0
$295$	0	0
$296$	433.923	0.0852071
$297$	0	0
$298$	4179.96	0.812545
$299$	2040.22	0.394612
$300$	0	0
$301$	−370.086	−0.0708684
$302$	−14996.3	−2.85742
$303$	0	0
$304$	1633.40	0.308164
$305$	0	0
$306$	0	0
$307$	−4067.69	−0.756207	−0.378103	−	0.925763i	$-0.623424\pi$
$307$	−4067.69	−0.756207	−0.378103	+	0.925763i	$0.623424\pi$
$308$	1056.15	0.195389
$309$	0	0
$310$	0	0
$311$	3923.20	0.715320	0.357660	−	0.933852i	$-0.383575\pi$
$311$	3923.20	0.715320	0.357660	+	0.933852i	$0.383575\pi$
$312$	0	0
$313$	806.771	0.145691	0.0728457	−	0.997343i	$-0.476792\pi$
$313$	806.771	0.145691	0.0728457	+	0.997343i	$0.476792\pi$
$314$	−10241.1	−1.84057
$315$	0	0
$316$	−11505.1	−2.04814
$317$	−8976.60	−1.59046	−0.795231	−	0.606307i	$-0.792650\pi$
$317$	−8976.60	−1.59046	−0.795231	+	0.606307i	$0.792650\pi$
$318$	0	0
$319$	892.203	0.156595
$320$	0	0
$321$	0	0
$322$	1972.97	0.341457
$323$	−3417.71	−0.588751
$324$	0	0
$325$	0	0
$326$	−1568.64	−0.266499
$327$	0	0
$328$	209.113	0.0352023
$329$	330.175	0.0553286
$330$	0	0
$331$	−7482.31	−1.24249	−0.621246	−	0.783615i	$-0.713373\pi$
$331$	−7482.31	−1.24249	−0.621246	+	0.783615i	$0.713373\pi$
$332$	6306.88	1.04258
$333$	0	0
$334$	−16407.0	−2.68788
$335$	0	0
$336$	0	0
$337$	−6904.38	−1.11604	−0.558020	−	0.829827i	$-0.688439\pi$
$337$	−6904.38	−1.11604	−0.558020	+	0.829827i	$0.688439\pi$
$338$	2431.17	0.391238
$339$	0	0
$340$	0	0
$341$	1101.88	0.174986
$342$	0	0
$343$	−5472.84	−0.861533
$344$	456.744	0.0715872
$345$	0	0
$346$	4349.12	0.675752
$347$	6050.06	0.935977	0.467989	−	0.883735i	$-0.344979\pi$
$347$	6050.06	0.935977	0.467989	+	0.883735i	$0.344979\pi$
$348$	0	0
$349$	−1697.12	−0.260299	−0.130150	−	0.991494i	$-0.541546\pi$
$349$	−1697.12	−0.260299	−0.130150	+	0.991494i	$0.541546\pi$
$350$	0	0
$351$	0	0
$352$	2716.54	0.411341
$353$	8505.96	1.28251	0.641256	−	0.767327i	$-0.278414\pi$
$353$	8505.96	1.28251	0.641256	+	0.767327i	$0.278414\pi$
$354$	0	0
$355$	0	0
$356$	7485.04	1.11434
$357$	0	0
$358$	14435.3	2.13109
$359$	4937.28	0.725848	0.362924	−	0.931819i	$-0.381778\pi$
$359$	4937.28	0.725848	0.362924	+	0.931819i	$0.381778\pi$
$360$	0	0
$361$	−4858.55	−0.708347
$362$	−6340.31	−0.920550
$363$	0	0
$364$	−3880.19	−0.558729
$365$	0	0
$366$	0	0
$367$	−12698.5	−1.80614	−0.903072	−	0.429490i	$-0.858693\pi$
$367$	−12698.5	−1.80614	−0.903072	+	0.429490i	$0.858693\pi$
$368$	1843.69	0.261165
$369$	0	0
$370$	0	0
$371$	5916.73	0.827983
$372$	0	0
$373$	1441.18	0.200058	0.100029	−	0.994985i	$-0.468106\pi$
$373$	1441.18	0.200058	0.100029	+	0.994985i	$0.468106\pi$
$374$	−3624.51	−0.501121
$375$	0	0
$376$	−407.488	−0.0558899
$377$	−3277.86	−0.447794
$378$	0	0
$379$	12908.9	1.74957	0.874785	−	0.484511i	$-0.161003\pi$
$379$	12908.9	1.74957	0.874785	+	0.484511i	$0.161003\pi$
$380$	0	0
$381$	0	0
$382$	13936.9	1.86669
$383$	−10281.2	−1.37166	−0.685829	−	0.727762i	$-0.740560\pi$
$383$	−10281.2	−1.37166	−0.685829	+	0.727762i	$0.740560\pi$
$384$	0	0
$385$	0	0
$386$	17175.0	2.26472
$387$	0	0
$388$	13890.2	1.81744
$389$	11485.2	1.49698	0.748490	−	0.663146i	$-0.230779\pi$
$389$	11485.2	1.49698	0.748490	+	0.663146i	$0.230779\pi$
$390$	0	0
$391$	−3857.71	−0.498958
$392$	2917.80	0.375947
$393$	0	0
$394$	−21099.0	−2.69785
$395$	0	0
$396$	0	0
$397$	−12163.2	−1.53766	−0.768831	−	0.639452i	$-0.779161\pi$
$397$	−12163.2	−1.53766	−0.768831	+	0.639452i	$0.779161\pi$
$398$	−1097.50	−0.138223
$399$	0	0
$400$	0	0
$401$	−15307.9	−1.90634	−0.953169	−	0.302437i	$-0.902200\pi$
$401$	−15307.9	−1.90634	−0.953169	+	0.302437i	$0.902200\pi$
$402$	0	0
$403$	−4048.20	−0.500385
$404$	−14591.1	−1.79687
$405$	0	0
$406$	−3169.81	−0.387476
$407$	−426.735	−0.0519717
$408$	0	0
$409$	−10396.2	−1.25686	−0.628431	−	0.777865i	$-0.716303\pi$
$409$	−10396.2	−1.25686	−0.628431	+	0.777865i	$0.716303\pi$
$410$	0	0
$411$	0	0
$412$	−7571.72	−0.905417
$413$	−6748.82	−0.804087
$414$	0	0
$415$	0	0
$416$	−9980.28	−1.17626
$417$	0	0
$418$	2121.50	0.248244
$419$	−1588.33	−0.185191	−0.0925953	−	0.995704i	$-0.529516\pi$
$419$	−1588.33	−0.185191	−0.0925953	+	0.995704i	$0.529516\pi$
$420$	0	0
$421$	3793.62	0.439168	0.219584	−	0.975594i	$-0.429530\pi$
$421$	3793.62	0.439168	0.219584	+	0.975594i	$0.429530\pi$
$422$	5561.34	0.641521
$423$	0	0
$424$	−7302.19	−0.836381
$425$	0	0
$426$	0	0
$427$	−4678.39	−0.530219
$428$	15841.5	1.78909
$429$	0	0
$430$	0	0
$431$	6013.25	0.672037	0.336018	−	0.941855i	$-0.390920\pi$
$431$	6013.25	0.672037	0.336018	+	0.941855i	$0.390920\pi$
$432$	0	0
$433$	2624.08	0.291236	0.145618	−	0.989341i	$-0.453483\pi$
$433$	2624.08	0.291236	0.145618	+	0.989341i	$0.453483\pi$
$434$	−3914.76	−0.432982
$435$	0	0
$436$	−7880.93	−0.865660
$437$	2257.99	0.247172
$438$	0	0
$439$	1085.10	0.117971	0.0589854	−	0.998259i	$-0.481213\pi$
$439$	1085.10	0.117971	0.0589854	+	0.998259i	$0.481213\pi$
$440$	0	0
$441$	0	0
$442$	13316.1	1.43299
$443$	−5173.59	−0.554864	−0.277432	−	0.960745i	$-0.589483\pi$
$443$	−5173.59	−0.554864	−0.277432	+	0.960745i	$0.589483\pi$
$444$	0	0
$445$	0	0
$446$	6553.55	0.695784
$447$	0	0
$448$	−7003.44	−0.738575
$449$	16047.9	1.68674	0.843370	−	0.537333i	$-0.180568\pi$
$449$	16047.9	1.68674	0.843370	+	0.537333i	$0.180568\pi$
$450$	0	0
$451$	−205.649	−0.0214715
$452$	−17373.0	−1.80787
$453$	0	0
$454$	−11795.4	−1.21935
$455$	0	0
$456$	0	0
$457$	−4243.55	−0.434365	−0.217183	−	0.976131i	$-0.569687\pi$
$457$	−4243.55	−0.434365	−0.217183	+	0.976131i	$0.569687\pi$
$458$	−6593.98	−0.672743
$459$	0	0
$460$	0	0
$461$	−15118.3	−1.52740	−0.763698	−	0.645573i	$-0.776619\pi$
$461$	−15118.3	−1.52740	−0.763698	+	0.645573i	$0.776619\pi$
$462$	0	0
$463$	−3300.13	−0.331252	−0.165626	−	0.986189i	$-0.552965\pi$
$463$	−3300.13	−0.331252	−0.165626	+	0.986189i	$0.552965\pi$
$464$	−2962.10	−0.296362
$465$	0	0
$466$	7159.38	0.711699
$467$	−1267.20	−0.125566	−0.0627829	−	0.998027i	$-0.519998\pi$
$467$	−1267.20	−0.125566	−0.0627829	+	0.998027i	$0.519998\pi$
$468$	0	0
$469$	3904.83	0.384452
$470$	0	0
$471$	0	0
$472$	8329.11	0.812243
$473$	−449.178	−0.0436643
$474$	0	0
$475$	0	0
$476$	7336.78	0.706472
$477$	0	0
$478$	2408.46	0.230461
$479$	2938.23	0.280274	0.140137	−	0.990132i	$-0.455246\pi$
$479$	2938.23	0.280274	0.140137	+	0.990132i	$0.455246\pi$
$480$	0	0
$481$	1567.78	0.148617
$482$	−26443.9	−2.49893
$483$	0	0
$484$	1281.87	0.120386
$485$	0	0
$486$	0	0
$487$	−5071.53	−0.471895	−0.235948	−	0.971766i	$-0.575819\pi$
$487$	−5071.53	−0.471895	−0.235948	+	0.971766i	$0.575819\pi$
$488$	5773.88	0.535597
$489$	0	0
$490$	0	0
$491$	−1890.86	−0.173795	−0.0868976	−	0.996217i	$-0.527695\pi$
$491$	−1890.86	−0.173795	−0.0868976	+	0.996217i	$0.527695\pi$
$492$	0	0
$493$	6197.87	0.566203
$494$	−7794.15	−0.709869
$495$	0	0
$496$	−3658.23	−0.331168
$497$	2469.55	0.222886
$498$	0	0
$499$	−16537.6	−1.48362	−0.741808	−	0.670613i	$-0.766031\pi$
$499$	−16537.6	−1.48362	−0.741808	+	0.670613i	$0.766031\pi$
$500$	0	0
$501$	0	0
$502$	9186.52	0.816762
$503$	−8330.23	−0.738422	−0.369211	−	0.929346i	$-0.620372\pi$
$503$	−8330.23	−0.738422	−0.369211	+	0.929346i	$0.620372\pi$
$504$	0	0
$505$	0	0
$506$	2394.62	0.210383
$507$	0	0
$508$	−24678.4	−2.15537
$509$	−20701.7	−1.80273	−0.901363	−	0.433064i	$-0.857432\pi$
$509$	−20701.7	−1.80273	−0.901363	+	0.433064i	$0.857432\pi$
$510$	0	0
$511$	−6276.04	−0.543318
$512$	−12286.8	−1.06055
$513$	0	0
$514$	−11037.6	−0.947178
$515$	0	0
$516$	0	0
$517$	400.738	0.0340898
$518$	1516.10	0.128598
$519$	0	0
$520$	0	0
$521$	−9311.54	−0.783005	−0.391503	−	0.920177i	$-0.628045\pi$
$521$	−9311.54	−0.783005	−0.391503	+	0.920177i	$0.628045\pi$
$522$	0	0
$523$	−18192.0	−1.52099	−0.760497	−	0.649341i	$-0.775045\pi$
$523$	−18192.0	−1.52099	−0.760497	+	0.649341i	$0.775045\pi$
$524$	−19521.0	−1.62744
$525$	0	0
$526$	−298.361	−0.0247322
$527$	7654.45	0.632700
$528$	0	0
$529$	−9618.32	−0.790525
$530$	0	0
$531$	0	0
$532$	−4294.36	−0.349970
$533$	755.532	0.0613991
$534$	0	0
$535$	0	0
$536$	−4819.18	−0.388352
$537$	0	0
$538$	12875.1	1.03176
$539$	−2869.46	−0.229307
$540$	0	0
$541$	−12799.7	−1.01720	−0.508598	−	0.861004i	$-0.669836\pi$
$541$	−12799.7	−1.01720	−0.508598	+	0.861004i	$0.669836\pi$
$542$	−1456.39	−0.115419
$543$	0	0
$544$	18871.0	1.48729
$545$	0	0
$546$	0	0
$547$	4564.37	0.356779	0.178390	−	0.983960i	$-0.442911\pi$
$547$	4564.37	0.356779	0.178390	+	0.983960i	$0.442911\pi$
$548$	−9462.77	−0.737645
$549$	0	0
$550$	0	0
$551$	−3627.73	−0.280484
$552$	0	0
$553$	−9842.57	−0.756869
$554$	−23582.0	−1.80849
$555$	0	0
$556$	29427.7	2.24463
$557$	−14228.8	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$557$	−14228.8	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$558$	0	0
$559$	1650.23	0.124861
$560$	0	0
$561$	0	0
$562$	−16585.8	−1.24489
$563$	−18321.3	−1.37149	−0.685747	−	0.727840i	$-0.740524\pi$
$563$	−18321.3	−1.37149	−0.685747	+	0.727840i	$0.740524\pi$
$564$	0	0
$565$	0	0
$566$	−27122.8	−2.01423
$567$	0	0
$568$	−3047.81	−0.225147
$569$	1949.83	0.143658	0.0718288	−	0.997417i	$-0.477116\pi$
$569$	1949.83	0.143658	0.0718288	+	0.997417i	$0.477116\pi$
$570$	0	0
$571$	−20121.3	−1.47469	−0.737347	−	0.675514i	$-0.763922\pi$
$571$	−20121.3	−1.47469	−0.737347	+	0.675514i	$0.763922\pi$
$572$	−4709.44	−0.344251
$573$	0	0
$574$	730.628	0.0531286
$575$	0	0
$576$	0	0
$577$	4171.22	0.300953	0.150477	−	0.988614i	$-0.451919\pi$
$577$	4171.22	0.300953	0.150477	+	0.988614i	$0.451919\pi$
$578$	−3993.19	−0.287362
$579$	0	0
$580$	0	0
$581$	5395.52	0.385273
$582$	0	0
$583$	7181.22	0.510147
$584$	7745.63	0.548829
$585$	0	0
$586$	7161.54	0.504847
$587$	−3593.98	−0.252708	−0.126354	−	0.991985i	$-0.540327\pi$
$587$	−3593.98	−0.252708	−0.126354	+	0.991985i	$0.540327\pi$
$588$	0	0
$589$	−4480.29	−0.313425
$590$	0	0
$591$	0	0
$592$	1416.75	0.0983585
$593$	−2893.64	−0.200384	−0.100192	−	0.994968i	$-0.531946\pi$
$593$	−2893.64	−0.200384	−0.100192	+	0.994968i	$0.531946\pi$
$594$	0	0
$595$	0	0
$596$	−10269.4	−0.705789
$597$	0	0
$598$	−8797.57	−0.601604
$599$	−26534.5	−1.80997	−0.904984	−	0.425445i	$-0.860118\pi$
$599$	−26534.5	−1.80997	−0.904984	+	0.425445i	$0.860118\pi$
$600$	0	0
$601$	2770.19	0.188017	0.0940085	−	0.995571i	$-0.470032\pi$
$601$	2770.19	0.188017	0.0940085	+	0.995571i	$0.470032\pi$
$602$	1595.84	0.108042
$603$	0	0
$604$	36843.2	2.48200
$605$	0	0
$606$	0	0
$607$	−5653.45	−0.378034	−0.189017	−	0.981974i	$-0.560530\pi$
$607$	−5653.45	−0.378034	−0.189017	+	0.981974i	$0.560530\pi$
$608$	−11045.6	−0.736771
$609$	0	0
$610$	0	0
$611$	−1472.27	−0.0974820
$612$	0	0
$613$	2464.19	0.162362	0.0811808	−	0.996699i	$-0.474131\pi$
$613$	2464.19	0.162362	0.0811808	+	0.996699i	$0.474131\pi$
$614$	17540.2	1.15287
$615$	0	0
$616$	−1115.11	−0.0729366
$617$	−24848.9	−1.62136	−0.810678	−	0.585492i	$-0.800902\pi$
$617$	−24848.9	−1.62136	−0.810678	+	0.585492i	$0.800902\pi$
$618$	0	0
$619$	−12904.2	−0.837905	−0.418953	−	0.908008i	$-0.637603\pi$
$619$	−12904.2	−0.837905	−0.418953	+	0.908008i	$0.637603\pi$
$620$	0	0
$621$	0	0
$622$	−16917.1	−1.09054
$623$	6403.43	0.411794
$624$	0	0
$625$	0	0
$626$	−3478.85	−0.222113
$627$	0	0
$628$	25160.5	1.59875
$629$	−2964.40	−0.187915
$630$	0	0
$631$	−5744.27	−0.362402	−0.181201	−	0.983446i	$-0.557998\pi$
$631$	−5744.27	−0.362402	−0.181201	+	0.983446i	$0.557998\pi$
$632$	12147.3	0.764546
$633$	0	0
$634$	38707.7	2.42473
$635$	0	0
$636$	0	0
$637$	10542.1	0.655719
$638$	−3847.24	−0.238736
$639$	0	0
$640$	0	0
$641$	18325.2	1.12917	0.564587	−	0.825374i	$-0.309036\pi$
$641$	18325.2	1.12917	0.564587	+	0.825374i	$0.309036\pi$
$642$	0	0
$643$	−25589.3	−1.56943	−0.784716	−	0.619855i	$-0.787191\pi$
$643$	−25589.3	−1.56943	−0.784716	+	0.619855i	$0.787191\pi$
$644$	−4847.21	−0.296595
$645$	0	0
$646$	14737.4	0.897578
$647$	10244.3	0.622479	0.311240	−	0.950331i	$-0.399256\pi$
$647$	10244.3	0.622479	0.311240	+	0.950331i	$0.399256\pi$
$648$	0	0
$649$	−8191.14	−0.495424
$650$	0	0
$651$	0	0
$652$	3853.85	0.231485
$653$	9356.38	0.560710	0.280355	−	0.959896i	$-0.409548\pi$
$653$	9356.38	0.560710	0.280355	+	0.959896i	$0.409548\pi$
$654$	0	0
$655$	0	0
$656$	682.752	0.0406356
$657$	0	0
$658$	−1423.74	−0.0843511
$659$	8188.52	0.484036	0.242018	−	0.970272i	$-0.422191\pi$
$659$	8188.52	0.484036	0.242018	+	0.970272i	$0.422191\pi$
$660$	0	0
$661$	6886.53	0.405227	0.202613	−	0.979259i	$-0.435057\pi$
$661$	6886.53	0.405227	0.202613	+	0.979259i	$0.435057\pi$
$662$	32264.3	1.89424
$663$	0	0
$664$	−6658.92	−0.389181
$665$	0	0
$666$	0	0
$667$	−4094.77	−0.237706
$668$	40309.0	2.33473
$669$	0	0
$670$	0	0
$671$	−5678.23	−0.326685
$672$	0	0
$673$	8450.15	0.483996	0.241998	−	0.970277i	$-0.422197\pi$
$673$	8450.15	0.483996	0.241998	+	0.970277i	$0.422197\pi$
$674$	29772.2	1.70146
$675$	0	0
$676$	−5972.93	−0.339835
$677$	−23308.6	−1.32322	−0.661612	−	0.749846i	$-0.730128\pi$
$677$	−23308.6	−1.32322	−0.661612	+	0.749846i	$0.730128\pi$
$678$	0	0
$679$	11883.0	0.671617
$680$	0	0
$681$	0	0
$682$	−4751.39	−0.266775
$683$	19580.0	1.09693	0.548467	−	0.836172i	$-0.315212\pi$
$683$	19580.0	1.09693	0.548467	+	0.836172i	$0.315212\pi$
$684$	0	0
$685$	0	0
$686$	23599.3	1.31345
$687$	0	0
$688$	1491.26	0.0826365
$689$	−26383.0	−1.45880
$690$	0	0
$691$	−33164.9	−1.82584	−0.912918	−	0.408142i	$-0.866177\pi$
$691$	−33164.9	−1.82584	−0.912918	+	0.408142i	$0.866177\pi$
$692$	−10685.0	−0.586968
$693$	0	0
$694$	−26088.3	−1.42694
$695$	0	0
$696$	0	0
$697$	−1428.58	−0.0776347
$698$	7318.08	0.396839
$699$	0	0
$700$	0	0
$701$	20819.6	1.12175	0.560874	−	0.827901i	$-0.310465\pi$
$701$	20819.6	1.12175	0.560874	+	0.827901i	$0.310465\pi$
$702$	0	0
$703$	1735.12	0.0930887
$704$	−8500.18	−0.455060
$705$	0	0
$706$	−36678.3	−1.95525
$707$	−12482.7	−0.664016
$708$	0	0
$709$	14367.7	0.761058	0.380529	−	0.924769i	$-0.375742\pi$
$709$	14367.7	0.761058	0.380529	+	0.924769i	$0.375742\pi$
$710$	0	0
$711$	0	0
$712$	−7902.84	−0.415971
$713$	−5057.09	−0.265623
$714$	0	0
$715$	0	0
$716$	−35464.9	−1.85110
$717$	0	0
$718$	−21289.9	−1.10659
$719$	27403.4	1.42138	0.710691	−	0.703504i	$-0.248382\pi$
$719$	27403.4	1.42138	0.710691	+	0.703504i	$0.248382\pi$
$720$	0	0
$721$	−6477.58	−0.334588
$722$	20950.4	1.07991
$723$	0	0
$724$	15576.9	0.799603
$725$	0	0
$726$	0	0
$727$	−6652.03	−0.339354	−0.169677	−	0.985500i	$-0.554272\pi$
$727$	−6652.03	−0.339354	−0.169677	+	0.985500i	$0.554272\pi$
$728$	4096.78	0.208567
$729$	0	0
$730$	0	0
$731$	−3120.30	−0.157878
$732$	0	0
$733$	−11674.6	−0.588285	−0.294142	−	0.955762i	$-0.595034\pi$
$733$	−11674.6	−0.588285	−0.294142	+	0.955762i	$0.595034\pi$
$734$	54756.7	2.75355
$735$	0	0
$736$	−12467.6	−0.624403
$737$	4739.34	0.236874
$738$	0	0
$739$	6253.35	0.311276	0.155638	−	0.987814i	$-0.450257\pi$
$739$	6253.35	0.311276	0.155638	+	0.987814i	$0.450257\pi$
$740$	0	0
$741$	0	0
$742$	−25513.4	−1.26230
$743$	−5832.24	−0.287973	−0.143987	−	0.989580i	$-0.545992\pi$
$743$	−5832.24	−0.287973	−0.143987	+	0.989580i	$0.545992\pi$
$744$	0	0
$745$	0	0
$746$	−6214.47	−0.304997
$747$	0	0
$748$	8904.75	0.435281
$749$	13552.4	0.661139
$750$	0	0
$751$	−9600.53	−0.466483	−0.233241	−	0.972419i	$-0.574933\pi$
$751$	−9600.53	−0.466483	−0.233241	+	0.972419i	$0.574933\pi$
$752$	−1330.44	−0.0645163
$753$	0	0
$754$	14134.4	0.682683
$755$	0	0
$756$	0	0
$757$	−18731.0	−0.899324	−0.449662	−	0.893199i	$-0.648456\pi$
$757$	−18731.0	−0.899324	−0.449662	+	0.893199i	$0.648456\pi$
$758$	−55664.2	−2.66730
$759$	0	0
$760$	0	0
$761$	−23588.9	−1.12365	−0.561823	−	0.827257i	$-0.689900\pi$
$761$	−23588.9	−1.12365	−0.561823	+	0.827257i	$0.689900\pi$
$762$	0	0
$763$	−6742.11	−0.319896
$764$	−34240.4	−1.62143
$765$	0	0
$766$	44333.3	2.09116
$767$	30093.3	1.41670
$768$	0	0
$769$	−84.7916	−0.00397616	−0.00198808	−	0.999998i	$-0.500633\pi$
$769$	−84.7916	−0.00397616	−0.00198808	+	0.999998i	$0.500633\pi$
$770$	0	0
$771$	0	0
$772$	−42195.7	−1.96717
$773$	17416.9	0.810403	0.405202	−	0.914227i	$-0.367201\pi$
$773$	17416.9	0.810403	0.405202	+	0.914227i	$0.367201\pi$
$774$	0	0
$775$	0	0
$776$	−14665.5	−0.678429
$777$	0	0
$778$	−49525.2	−2.28222
$779$	836.177	0.0384585
$780$	0	0
$781$	2997.32	0.137327
$782$	16634.7	0.760685
$783$	0	0
$784$	9526.58	0.433973
$785$	0	0
$786$	0	0
$787$	8941.25	0.404983	0.202491	−	0.979284i	$-0.435096\pi$
$787$	8941.25	0.404983	0.202491	+	0.979284i	$0.435096\pi$
$788$	51836.2	2.34339
$789$	0	0
$790$	0	0
$791$	−14862.5	−0.668079
$792$	0	0
$793$	20861.2	0.934178
$794$	52448.4	2.34424
$795$	0	0
$796$	2696.36	0.120063
$797$	24769.1	1.10084	0.550419	−	0.834889i	$-0.314468\pi$
$797$	24769.1	1.10084	0.550419	+	0.834889i	$0.314468\pi$
$798$	0	0
$799$	2783.80	0.123259
$800$	0	0
$801$	0	0
$802$	66008.9	2.90630
$803$	−7617.32	−0.334756
$804$	0	0
$805$	0	0
$806$	17456.1	0.762860
$807$	0	0
$808$	15405.6	0.670752
$809$	882.312	0.0383442	0.0191721	−	0.999816i	$-0.493897\pi$
$809$	882.312	0.0383442	0.0191721	+	0.999816i	$0.493897\pi$
$810$	0	0
$811$	41973.3	1.81736	0.908682	−	0.417490i	$-0.137090\pi$
$811$	41973.3	1.81736	0.908682	+	0.417490i	$0.137090\pi$
$812$	7787.63	0.336567
$813$	0	0
$814$	1840.11	0.0792333
$815$	0	0
$816$	0	0
$817$	1826.37	0.0782090
$818$	44829.0	1.91615
$819$	0	0
$820$	0	0
$821$	32888.4	1.39807	0.699033	−	0.715089i	$-0.253614\pi$
$821$	32888.4	1.39807	0.699033	+	0.715089i	$0.253614\pi$
$822$	0	0
$823$	−36607.4	−1.55049	−0.775245	−	0.631661i	$-0.782373\pi$
$823$	−36607.4	−1.55049	−0.775245	+	0.631661i	$0.782373\pi$
$824$	7994.36	0.337982
$825$	0	0
$826$	29101.4	1.22587
$827$	−41798.2	−1.75752	−0.878759	−	0.477266i	$-0.841628\pi$
$827$	−41798.2	−1.75752	−0.878759	+	0.477266i	$0.841628\pi$
$828$	0	0
$829$	−18948.8	−0.793872	−0.396936	−	0.917846i	$-0.629926\pi$
$829$	−18948.8	−0.793872	−0.396936	+	0.917846i	$0.629926\pi$
$830$	0	0
$831$	0	0
$832$	31228.7	1.30128
$833$	−19933.3	−0.829109
$834$	0	0
$835$	0	0
$836$	−5212.12	−0.215628
$837$	0	0
$838$	6848.98	0.282332
$839$	25373.5	1.04409	0.522045	−	0.852918i	$-0.325169\pi$
$839$	25373.5	1.04409	0.522045	+	0.852918i	$0.325169\pi$
$840$	0	0
$841$	−17810.3	−0.730258
$842$	−16358.4	−0.669533
$843$	0	0
$844$	−13663.2	−0.557234
$845$	0	0
$846$	0	0
$847$	1096.63	0.0444874
$848$	−23841.5	−0.965474
$849$	0	0
$850$	0	0
$851$	1958.50	0.0788914
$852$	0	0
$853$	−3778.34	−0.151662	−0.0758310	−	0.997121i	$-0.524161\pi$
$853$	−3778.34	−0.151662	−0.0758310	+	0.997121i	$0.524161\pi$
$854$	20173.6	0.808343
$855$	0	0
$856$	−16725.8	−0.667845
$857$	22069.1	0.879655	0.439828	−	0.898082i	$-0.355040\pi$
$857$	22069.1	0.879655	0.439828	+	0.898082i	$0.355040\pi$
$858$	0	0
$859$	−42408.4	−1.68447	−0.842234	−	0.539113i	$-0.818760\pi$
$859$	−42408.4	−1.68447	−0.842234	+	0.539113i	$0.818760\pi$
$860$	0	0
$861$	0	0
$862$	−25929.5	−1.02455
$863$	−19295.8	−0.761110	−0.380555	−	0.924758i	$-0.624267\pi$
$863$	−19295.8	−0.761110	−0.380555	+	0.924758i	$0.624267\pi$
$864$	0	0
$865$	0	0
$866$	−11315.2	−0.444003
$867$	0	0
$868$	9617.83	0.376095
$869$	−11946.1	−0.466332
$870$	0	0
$871$	−17411.8	−0.677356
$872$	8320.83	0.323141
$873$	0	0
$874$	−9736.61	−0.376826
$875$	0	0
$876$	0	0
$877$	−32873.7	−1.26575	−0.632877	−	0.774253i	$-0.718126\pi$
$877$	−32873.7	−1.26575	−0.632877	+	0.774253i	$0.718126\pi$
$878$	−4679.05	−0.179852
$879$	0	0
$880$	0	0
$881$	44772.8	1.71218	0.856092	−	0.516823i	$-0.172886\pi$
$881$	44772.8	1.71218	0.856092	+	0.516823i	$0.172886\pi$
$882$	0	0
$883$	30710.6	1.17043	0.585217	−	0.810877i	$-0.301009\pi$
$883$	30710.6	1.17043	0.585217	+	0.810877i	$0.301009\pi$
$884$	−32715.1	−1.24471
$885$	0	0
$886$	22308.9	0.845916
$887$	65.5662	0.00248196	0.00124098	−	0.999999i	$-0.499605\pi$
$887$	65.5662	0.00248196	0.00124098	+	0.999999i	$0.499605\pi$
$888$	0	0
$889$	−21112.3	−0.796494
$890$	0	0
$891$	0	0
$892$	−16100.9	−0.604368
$893$	−1629.41	−0.0610596
$894$	0	0
$895$	0	0
$896$	12293.7	0.458374
$897$	0	0
$898$	−69199.6	−2.57152
$899$	8124.82	0.301422
$900$	0	0
$901$	49885.8	1.84455
$902$	886.773	0.0327343
$903$	0	0
$904$	18342.7	0.674856
$905$	0	0
$906$	0	0
$907$	−30133.2	−1.10315	−0.551575	−	0.834125i	$-0.685973\pi$
$907$	−30133.2	−1.10315	−0.551575	+	0.834125i	$0.685973\pi$
$908$	28979.0	1.05914
$909$	0	0
$910$	0	0
$911$	52126.0	1.89573	0.947865	−	0.318671i	$-0.103237\pi$
$911$	52126.0	1.89573	0.947865	+	0.318671i	$0.103237\pi$
$912$	0	0
$913$	6548.61	0.237379
$914$	18298.5	0.662210
$915$	0	0
$916$	16200.2	0.584354
$917$	−16700.2	−0.601405
$918$	0	0
$919$	−16211.2	−0.581891	−0.290945	−	0.956740i	$-0.593970\pi$
$919$	−16211.2	−0.581891	−0.290945	+	0.956740i	$0.593970\pi$
$920$	0	0
$921$	0	0
$922$	65191.2	2.32859
$923$	−11011.8	−0.392697
$924$	0	0
$925$	0	0
$926$	14230.4	0.505010
$927$	0	0
$928$	20030.6	0.708554
$929$	−46797.6	−1.65272	−0.826362	−	0.563139i	$-0.809594\pi$
$929$	−46797.6	−1.65272	−0.826362	+	0.563139i	$0.809594\pi$
$930$	0	0
$931$	11667.3	0.410722
$932$	−17589.2	−0.618192
$933$	0	0
$934$	5464.27	0.191431
$935$	0	0
$936$	0	0
$937$	31594.6	1.10155	0.550775	−	0.834654i	$-0.314332\pi$
$937$	31594.6	1.10155	0.550775	+	0.834654i	$0.314332\pi$
$938$	−16837.9	−0.586116
$939$	0	0
$940$	0	0
$941$	54758.6	1.89700	0.948501	−	0.316775i	$-0.102600\pi$
$941$	54758.6	1.89700	0.948501	+	0.316775i	$0.102600\pi$
$942$	0	0
$943$	943.826	0.0325930
$944$	27194.5	0.937610
$945$	0	0
$946$	1936.89	0.0665683
$947$	32471.6	1.11424	0.557120	−	0.830432i	$-0.311906\pi$
$947$	32471.6	1.11424	0.557120	+	0.830432i	$0.311906\pi$
$948$	0	0
$949$	27985.2	0.957258
$950$	0	0
$951$	0	0
$952$	−7746.31	−0.263718
$953$	−721.649	−0.0245294	−0.0122647	−	0.999925i	$-0.503904\pi$
$953$	−721.649	−0.0245294	−0.0122647	+	0.999925i	$0.503904\pi$
$954$	0	0
$955$	0	0
$956$	−5917.13	−0.200182
$957$	0	0
$958$	−12669.9	−0.427291
$959$	−8095.37	−0.272589
$960$	0	0
$961$	−19756.7	−0.663178
$962$	−6760.37	−0.226573
$963$	0	0
$964$	64967.6	2.17061
$965$	0	0
$966$	0	0
$967$	45604.2	1.51658	0.758290	−	0.651917i	$-0.226035\pi$
$967$	45604.2	1.51658	0.758290	+	0.651917i	$0.226035\pi$
$968$	−1353.42	−0.0449386
$969$	0	0
$970$	0	0
$971$	−35321.3	−1.16737	−0.583685	−	0.811980i	$-0.698390\pi$
$971$	−35321.3	−1.16737	−0.583685	+	0.811980i	$0.698390\pi$
$972$	0	0
$973$	25175.3	0.829480
$974$	21868.8	0.719426
$975$	0	0
$976$	18851.6	0.618265
$977$	29255.2	0.957991	0.478996	−	0.877817i	$-0.341001\pi$
$977$	29255.2	0.957991	0.478996	+	0.877817i	$0.341001\pi$
$978$	0	0
$979$	7771.93	0.253720
$980$	0	0
$981$	0	0
$982$	8153.54	0.264959
$983$	−4054.06	−0.131541	−0.0657703	−	0.997835i	$-0.520950\pi$
$983$	−4054.06	−0.131541	−0.0657703	+	0.997835i	$0.520950\pi$
$984$	0	0
$985$	0	0
$986$	−26725.6	−0.863203
$987$	0	0
$988$	19148.8	0.616603
$989$	2061.50	0.0662811
$990$	0	0
$991$	−9972.08	−0.319650	−0.159825	−	0.987145i	$-0.551093\pi$
$991$	−9972.08	−0.319650	−0.159825	+	0.987145i	$0.551093\pi$
$992$	24738.1	0.791770
$993$	0	0
$994$	−10648.9	−0.339800
$995$	0	0
$996$	0	0
$997$	7920.62	0.251603	0.125802	−	0.992055i	$-0.459850\pi$
$997$	7920.62	0.251603	0.125802	+	0.992055i	$0.459850\pi$
$998$	71311.2	2.26184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.br.1.2		7
3.2	odd	2		825.4.a.bb.1.6		7
5.2	odd	4		495.4.c.c.199.3		14
5.3	odd	4		495.4.c.c.199.12		14
5.4	even	2		2475.4.a.bq.1.6		7
15.2	even	4		165.4.c.a.34.12	yes	14
15.8	even	4		165.4.c.a.34.3	✓	14
15.14	odd	2		825.4.a.bc.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.c.a.34.3	✓	14	15.8	even	4
165.4.c.a.34.12	yes	14	15.2	even	4
495.4.c.c.199.3		14	5.2	odd	4
495.4.c.c.199.12		14	5.3	odd	4
825.4.a.bb.1.6		7	3.2	odd	2
825.4.a.bc.1.2		7	15.14	odd	2
2475.4.a.bq.1.6		7	5.4	even	2
2475.4.a.br.1.2		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)

\(f(q)\)	\(=\)	\(q-4.31207 q^{2} +10.5939 q^{4} +9.06309 q^{7} -11.1853 q^{8} +O(q^{10})\)	\(q-4.31207 q^{2} +10.5939 q^{4} +9.06309 q^{7} -11.1853 q^{8} +11.0000 q^{11} -40.4128 q^{13} -39.0807 q^{14} -36.5198 q^{16} +76.4137 q^{17} -44.7264 q^{19} -47.4328 q^{22} -50.4845 q^{23} +174.263 q^{26} +96.0139 q^{28} +81.1094 q^{29} +100.171 q^{31} +246.958 q^{32} -329.501 q^{34} -38.7941 q^{37} +192.863 q^{38} -18.6954 q^{41} -40.8344 q^{43} +116.533 q^{44} +217.693 q^{46} +36.4307 q^{47} -260.860 q^{49} -428.131 q^{52} +652.838 q^{53} -101.373 q^{56} -349.749 q^{58} -744.649 q^{59} -516.203 q^{61} -431.945 q^{62} -772.743 q^{64} +430.849 q^{67} +809.523 q^{68} +272.484 q^{71} -692.483 q^{73} +167.283 q^{74} -473.829 q^{76} +99.6940 q^{77} -1086.01 q^{79} +80.6157 q^{82} +595.328 q^{83} +176.081 q^{86} -123.038 q^{88} +706.539 q^{89} -366.265 q^{91} -534.830 q^{92} -157.092 q^{94} +1311.14 q^{97} +1124.85 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8	\( 7 q + 3 q^{2} + 41 q^{4} - 50 q^{7} + 21 q^{8} + 77 q^{11} - 24 q^{13} - 142 q^{14} + 181 q^{16} + 38 q^{17} + 26 q^{19} + 33 q^{22} + 228 q^{23} - 476 q^{26} - 840 q^{28} - 572 q^{29} - 140 q^{31} + 991 q^{32} - 806 q^{34} + 104 q^{37} + 498 q^{38} - 896 q^{41} - 614 q^{43} + 451 q^{44} - 344 q^{46} + 520 q^{47} + 295 q^{49} + 26 q^{52} + 380 q^{53} - 1522 q^{56} - 1600 q^{58} - 1316 q^{59} - 386 q^{61} + 440 q^{62} + 869 q^{64} - 348 q^{67} + 332 q^{68} - 804 q^{71} - 468 q^{73} + 748 q^{74} - 1698 q^{76} - 550 q^{77} - 374 q^{79} + 620 q^{82} + 3128 q^{83} + 2534 q^{86} + 231 q^{88} - 694 q^{89} - 3376 q^{91} - 1184 q^{92} - 2920 q^{94} + 8 q^{97} + 4211 q^{98}+O(q^{100}) \) 7 * q + 3 * q^2 + 41 * q^4 - 50 * q^7 + 21 * q^8 + 77 * q^11 - 24 * q^13 - 142 * q^14 + 181 * q^16 + 38 * q^17 + 26 * q^19 + 33 * q^22 + 228 * q^23 - 476 * q^26 - 840 * q^28 - 572 * q^29 - 140 * q^31 + 991 * q^32 - 806 * q^34 + 104 * q^37 + 498 * q^38 - 896 * q^41 - 614 * q^43 + 451 * q^44 - 344 * q^46 + 520 * q^47 + 295 * q^49 + 26 * q^52 + 380 * q^53 - 1522 * q^56 - 1600 * q^58 - 1316 * q^59 - 386 * q^61 + 440 * q^62 + 869 * q^64 - 348 * q^67 + 332 * q^68 - 804 * q^71 - 468 * q^73 + 748 * q^74 - 1698 * q^76 - 550 * q^77 - 374 * q^79 + 620 * q^82 + 3128 * q^83 + 2534 * q^86 + 231 * q^88 - 694 * q^89 - 3376 * q^91 - 1184 * q^92 - 2920 * q^94 + 8 * q^97 + 4211 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more