LMFDB - Embedded newform 75.4.b.c.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.4.b.c.49.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.35890i q^{2} +3.00000i q^{3} -3.28220 q^{4} +10.0767 q^{6} -30.4356i q^{7} -15.8466i q^{8} -9.00000 q^{9} +31.4356 q^{11} -9.84661i q^{12} -60.7424i q^{13} -102.230 q^{14} -79.4848 q^{16} +121.178i q^{17} +30.2301i q^{18} +14.4356 q^{19} +91.3068 q^{21} -105.589i q^{22} +13.6932i q^{23} +47.5398 q^{24} -204.028 q^{26} -27.0000i q^{27} +99.8958i q^{28} +76.0492 q^{29} +183.049 q^{31} +140.208i q^{32} +94.3068i q^{33} +407.025 q^{34} +29.5398 q^{36} +37.3864i q^{37} -48.4877i q^{38} +182.227 q^{39} -30.6627 q^{41} -306.690i q^{42} +327.564i q^{43} -103.178 q^{44} +45.9941 q^{46} +449.485i q^{47} -238.454i q^{48} -583.325 q^{49} -363.534 q^{51} +199.369i q^{52} -301.951i q^{53} -90.6903 q^{54} -482.301 q^{56} +43.3068i q^{57} -255.441i q^{58} -340.970 q^{59} +619.098 q^{61} -614.844i q^{62} +273.920i q^{63} -164.932 q^{64} +316.767 q^{66} -256.890i q^{67} -397.731i q^{68} -41.0796 q^{69} +499.178 q^{71} +142.619i q^{72} -19.1288i q^{73} +125.577 q^{74} -47.3805 q^{76} -956.761i q^{77} -612.083i q^{78} -257.424 q^{79} +81.0000 q^{81} +102.993i q^{82} +914.909i q^{83} -299.687 q^{84} +1100.26 q^{86} +228.148i q^{87} -498.148i q^{88} +1059.68 q^{89} -1848.73 q^{91} -44.9439i q^{92} +549.148i q^{93} +1509.77 q^{94} -420.625 q^{96} -521.000i q^{97} +1959.33i q^{98} -282.920 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41}+ \cdots - 504 q^{99}+O(q^{100}) $ 4 * q - 48 * q^4 - 12 * q^6 - 36 * q^9 + 56 * q^11 - 252 * q^14 + 240 * q^16 - 12 * q^19 + 156 * q^21 + 504 * q^24 - 1252 * q^26 - 184 * q^29 + 244 * q^31 + 1384 * q^34 + 432 * q^36 - 108 * q^39 + 784 * q^41 - 64 * q^44 - 1176 * q^46 - 520 * q^49 - 408 * q^51 + 108 * q^54 - 360 * q^56 - 248 * q^59 + 1500 * q^61 - 2752 * q^64 + 744 * q^66 - 792 * q^69 + 1648 * q^71 - 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 324 * q^81 - 48 * q^84 - 1684 * q^86 + 1728 * q^89 - 4396 * q^91 + 1192 * q^94 - 3984 * q^96 - 504 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.35890i	− 1.18755i	−0.804631	−	0.593775i	$-0.797637\pi$
$2$	− 3.35890i	− 1.18755i	0.804631	−	0.593775i	$-0.202363\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−3.28220	−0.410275
$5$	0	0
$5$	0	0
$6$	10.0767	0.685632
$7$	− 30.4356i	− 1.64337i	−0.569944	−	0.821684i	$-0.693035\pi$
$7$	− 30.4356i	− 1.64337i	0.569944	−	0.821684i	$-0.306965\pi$
$8$	− 15.8466i	− 0.700328i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	31.4356	0.861654	0.430827	−	0.902435i	$-0.358222\pi$
$11$	31.4356	0.861654	0.430827	+	0.902435i	$0.358222\pi$
$12$	− 9.84661i	− 0.236873i
$13$	− 60.7424i	− 1.29592i	−0.761676	−	0.647958i	$-0.775623\pi$
$13$	− 60.7424i	− 1.29592i	0.761676	−	0.647958i	$-0.224377\pi$
$14$	−102.230	−1.95158
$15$	0	0
$16$	−79.4848	−1.24195
$17$	121.178i	1.72882i	0.502786	+	0.864411i	$0.332308\pi$
$17$	121.178i	1.72882i	−0.502786	+	0.864411i	$0.667692\pi$
$18$	30.2301i	0.395850i
$19$	14.4356	0.174303	0.0871514	−	0.996195i	$-0.472224\pi$
$19$	14.4356	0.174303	0.0871514	+	0.996195i	$0.472224\pi$
$20$	0	0
$21$	91.3068	0.948799
$22$	− 105.589i	− 1.02326i
$23$	13.6932i	0.124141i	0.998072	+	0.0620703i	$0.0197703\pi$
$23$	13.6932i	0.124141i	−0.998072	+	0.0620703i	$0.980230\pi$
$24$	47.5398	0.404334
$25$	0	0
$26$	−204.028	−1.53896
$27$	− 27.0000i	− 0.192450i
$28$	99.8958i	0.674233i
$29$	76.0492	0.486965	0.243482	−	0.969905i	$-0.421710\pi$
$29$	76.0492	0.486965	0.243482	+	0.969905i	$0.421710\pi$
$30$	0	0
$31$	183.049	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$31$	183.049	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$32$	140.208i	0.774550i
$33$	94.3068i	0.497476i
$34$	407.025	2.05306
$35$	0	0
$36$	29.5398	0.136758
$37$	37.3864i	0.166116i	0.996545	+	0.0830580i	$0.0264687\pi$
$37$	37.3864i	0.166116i	−0.996545	+	0.0830580i	$0.973531\pi$
$38$	− 48.4877i	− 0.206993i
$39$	182.227	0.748197
$40$	0	0
$41$	−30.6627	−0.116798	−0.0583990	−	0.998293i	$-0.518600\pi$
$41$	−30.6627	−0.116798	−0.0583990	+	0.998293i	$0.518600\pi$
$42$	− 306.690i	− 1.12675i
$43$	327.564i	1.16170i	0.814011	+	0.580850i	$0.197280\pi$
$43$	327.564i	1.16170i	−0.814011	+	0.580850i	$0.802720\pi$
$44$	−103.178	−0.353515
$45$	0	0
$46$	45.9941	0.147423
$47$	449.485i	1.39498i	0.716594	+	0.697490i	$0.245700\pi$
$47$	449.485i	1.39498i	−0.716594	+	0.697490i	$0.754300\pi$
$48$	− 238.454i	− 0.717040i
$49$	−583.325	−1.70066
$50$	0	0
$51$	−363.534	−0.998136
$52$	199.369i	0.531682i
$53$	− 301.951i	− 0.782569i	−0.920270	−	0.391284i	$-0.872031\pi$
$53$	− 301.951i	− 0.782569i	0.920270	−	0.391284i	$-0.127969\pi$
$54$	−90.6903	−0.228544
$55$	0	0
$56$	−482.301	−1.15090
$57$	43.3068i	0.100634i
$58$	− 255.441i	− 0.578295i
$59$	−340.970	−0.752381	−0.376190	−	0.926542i	$-0.622766\pi$
$59$	−340.970	−0.752381	−0.376190	+	0.926542i	$0.622766\pi$
$60$	0	0
$61$	619.098	1.29947	0.649733	−	0.760163i	$-0.274881\pi$
$61$	619.098	1.29947	0.649733	+	0.760163i	$0.274881\pi$
$62$	− 614.844i	− 1.25944i
$63$	273.920i	0.547789i
$64$	−164.932	−0.322133
$65$	0	0
$66$	316.767	0.590778
$67$	− 256.890i	− 0.468419i	−0.972186	−	0.234210i	$-0.924750\pi$
$67$	− 256.890i	− 0.468419i	0.972186	−	0.234210i	$-0.0752502\pi$
$68$	− 397.731i	− 0.709293i
$69$	−41.0796	−0.0716726
$70$	0	0
$71$	499.178	0.834388	0.417194	−	0.908818i	$-0.363014\pi$
$71$	499.178	0.834388	0.417194	+	0.908818i	$0.363014\pi$
$72$	142.619i	0.233443i
$73$	− 19.1288i	− 0.0306693i	−0.999882	−	0.0153346i	$-0.995119\pi$
$73$	− 19.1288i	− 0.0306693i	0.999882	−	0.0153346i	$-0.00488136\pi$
$74$	125.577	0.197271
$75$	0	0
$76$	−47.3805	−0.0715121
$77$	− 956.761i	− 1.41601i
$78$	− 612.083i	− 0.888522i
$79$	−257.424	−0.366613	−0.183307	−	0.983056i	$-0.558680\pi$
$79$	−257.424	−0.366613	−0.183307	+	0.983056i	$0.558680\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	102.993i	0.138703i
$83$	914.909i	1.20993i	0.796252	+	0.604965i	$0.206813\pi$
$83$	914.909i	1.20993i	−0.796252	+	0.604965i	$0.793187\pi$
$84$	−299.687	−0.389269
$85$	0	0
$86$	1100.26	1.37958
$87$	228.148i	0.281149i
$88$	− 498.148i	− 0.603440i
$89$	1059.68	1.26209	0.631045	−	0.775746i	$-0.282626\pi$
$89$	1059.68	1.26209	0.631045	+	0.775746i	$0.282626\pi$
$90$	0	0
$91$	−1848.73	−2.12967
$92$	− 44.9439i	− 0.0509318i
$93$	549.148i	0.612300i
$94$	1509.77	1.65661
$95$	0	0
$96$	−420.625	−0.447186
$97$	− 521.000i	− 0.545356i	−0.962105	−	0.272678i	$-0.912091\pi$
$97$	− 521.000i	− 0.545356i	0.962105	−	0.272678i	$-0.0879094\pi$
$98$	1959.33i	2.01962i
$99$	−282.920	−0.287218
$100$	0	0
$101$	347.080	0.341938	0.170969	−	0.985276i	$-0.445310\pi$
$101$	347.080	0.341938	0.170969	+	0.985276i	$0.445310\pi$
$102$	1221.07i	1.18534i
$103$	− 770.749i	− 0.737322i	−0.929564	−	0.368661i	$-0.879816\pi$
$103$	− 770.749i	− 0.737322i	0.929564	−	0.368661i	$-0.120184\pi$
$104$	−962.561	−0.907566
$105$	0	0
$106$	−1014.22	−0.929339
$107$	1415.37i	1.27878i	0.768883	+	0.639390i	$0.220813\pi$
$107$	1415.37i	1.27878i	−0.768883	+	0.639390i	$0.779187\pi$
$108$	88.6195i	0.0789575i
$109$	−908.386	−0.798235	−0.399118	−	0.916900i	$-0.630683\pi$
$109$	−908.386	−0.798235	−0.399118	+	0.916900i	$0.630683\pi$
$110$	0	0
$111$	−112.159	−0.0959071
$112$	2419.17i	2.04098i
$113$	− 2049.94i	− 1.70657i	−0.521447	−	0.853283i	$-0.674608\pi$
$113$	− 2049.94i	− 1.70657i	0.521447	−	0.853283i	$-0.325392\pi$
$114$	145.463	0.119508
$115$	0	0
$116$	−249.609	−0.199790
$117$	546.681i	0.431972i
$118$	1145.28i	0.893490i
$119$	3688.12	2.84109
$120$	0	0
$121$	−342.803	−0.257553
$122$	− 2079.49i	− 1.54318i
$123$	− 91.9882i	− 0.0674333i
$124$	−600.804	−0.435111
$125$	0	0
$126$	920.071	0.650527
$127$	− 281.644i	− 0.196786i	−0.995148	−	0.0983932i	$-0.968630\pi$
$127$	− 281.644i	− 0.196786i	0.995148	−	0.0983932i	$-0.0313703\pi$
$128$	1675.66i	1.15710i
$129$	−982.693	−0.670708
$130$	0	0
$131$	243.056	0.162106	0.0810531	−	0.996710i	$-0.474172\pi$
$131$	243.056	0.162106	0.0810531	+	0.996710i	$0.474172\pi$
$132$	− 309.534i	− 0.204102i
$133$	− 439.356i	− 0.286444i
$134$	−862.867	−0.556271
$135$	0	0
$136$	1920.26	1.21074
$137$	− 909.386i	− 0.567110i	−0.958956	−	0.283555i	$-0.908486\pi$
$137$	− 909.386i	− 0.567110i	0.958956	−	0.283555i	$-0.0915139\pi$
$138$	137.982i	0.0851148i
$139$	2049.52	1.25063	0.625317	−	0.780371i	$-0.284970\pi$
$139$	2049.52	1.25063	0.625317	+	0.780371i	$0.284970\pi$
$140$	0	0
$141$	−1348.45	−0.805392
$142$	− 1676.69i	− 0.990877i
$143$	− 1909.47i	− 1.11663i
$144$	715.363	0.413983
$145$	0	0
$146$	−64.2517	−0.0364213
$147$	− 1749.98i	− 0.981875i
$148$	− 122.710i	− 0.0681533i
$149$	−3601.14	−1.97998	−0.989990	−	0.141136i	$-0.954925\pi$
$149$	−3601.14	−1.97998	−0.989990	+	0.141136i	$0.954925\pi$
$150$	0	0
$151$	1383.38	0.745550	0.372775	−	0.927922i	$-0.378406\pi$
$151$	1383.38	0.745550	0.372775	+	0.927922i	$0.378406\pi$
$152$	− 228.755i	− 0.122069i
$153$	− 1090.60i	− 0.576274i
$154$	−3213.66	−1.68159
$155$	0	0
$156$	−598.106	−0.306967
$157$	131.749i	0.0669729i	0.999439	+	0.0334864i	$0.0106611\pi$
$157$	131.749i	0.0669729i	−0.999439	+	0.0334864i	$0.989339\pi$
$158$	864.661i	0.435372i
$159$	905.852	0.451816
$160$	0	0
$161$	416.761	0.204009
$162$	− 272.071i	− 0.131950i
$163$	2897.74i	1.39244i	0.717827	+	0.696222i	$0.245137\pi$
$163$	2897.74i	1.39244i	−0.717827	+	0.696222i	$0.754863\pi$
$164$	100.641	0.0479193
$165$	0	0
$166$	3073.09	1.43685
$167$	− 260.283i	− 0.120607i	−0.998180	−	0.0603034i	$-0.980793\pi$
$167$	− 260.283i	− 0.120607i	0.998180	−	0.0603034i	$-0.0192068\pi$
$168$	− 1446.90i	− 0.664470i
$169$	−1492.64	−0.679398
$170$	0	0
$171$	−129.920	−0.0581009
$172$	− 1075.13i	− 0.476617i
$173$	− 1935.83i	− 0.850742i	−0.905019	−	0.425371i	$-0.860144\pi$
$173$	− 1935.83i	− 0.850742i	0.905019	−	0.425371i	$-0.139856\pi$
$174$	766.324	0.333879
$175$	0	0
$176$	−2498.65	−1.07013
$177$	− 1022.91i	− 0.434387i
$178$	− 3559.36i	− 1.49880i
$179$	−576.627	−0.240777	−0.120389	−	0.992727i	$-0.538414\pi$
$179$	−576.627	−0.240777	−0.120389	+	0.992727i	$0.538414\pi$
$180$	0	0
$181$	−1962.04	−0.805733	−0.402866	−	0.915259i	$-0.631986\pi$
$181$	−1962.04	−0.805733	−0.402866	+	0.915259i	$0.631986\pi$
$182$	6209.70i	2.52909i
$183$	1857.30i	0.750247i
$184$	216.991	0.0869390
$185$	0	0
$186$	1844.53	0.727137
$187$	3809.30i	1.48965i
$188$	− 1475.30i	− 0.572326i
$189$	−821.761	−0.316266
$190$	0	0
$191$	−4318.75	−1.63609	−0.818047	−	0.575152i	$-0.804943\pi$
$191$	−4318.75	−1.63609	−0.818047	+	0.575152i	$0.804943\pi$
$192$	− 494.796i	− 0.185984i
$193$	2.97647i	0.00111011i	1.00000		0.000555054i	$0.000176679\pi$
$193$	2.97647i	0.00111011i	−1.00000		0.000555054i	$0.999823\pi$
$194$	−1749.99	−0.647638
$195$	0	0
$196$	1914.59	0.697738
$197$	− 569.705i	− 0.206040i	−0.994679	−	0.103020i	$-0.967149\pi$
$197$	− 569.705i	− 0.206040i	0.994679	−	0.103020i	$-0.0328505\pi$
$198$	950.301i	0.341086i
$199$	3050.73	1.08674	0.543368	−	0.839494i	$-0.317149\pi$
$199$	3050.73	1.08674	0.543368	+	0.839494i	$0.317149\pi$
$200$	0	0
$201$	770.670	0.270442
$202$	− 1165.81i	− 0.406068i
$203$	− 2314.60i	− 0.800262i
$204$	1193.19	0.409510
$205$	0	0
$206$	−2588.87	−0.875607
$207$	− 123.239i	− 0.0413802i
$208$	4828.09i	1.60946i
$209$	453.792	0.150189
$210$	0	0
$211$	−50.5104	−0.0164800	−0.00824000	−	0.999966i	$-0.502623\pi$
$211$	−50.5104	−0.0164800	−0.00824000	+	0.999966i	$0.502623\pi$
$212$	991.064i	0.321069i
$213$	1497.53i	0.481734i
$214$	4754.10	1.51862
$215$	0	0
$216$	−427.858	−0.134778
$217$	− 5571.21i	− 1.74285i
$218$	3051.18i	0.947944i
$219$	57.3864	0.0177069
$220$	0	0
$221$	7360.64	2.24041
$222$	376.732i	0.113894i
$223$	5453.55i	1.63765i	0.574040	+	0.818827i	$0.305375\pi$
$223$	5453.55i	1.63765i	−0.574040	+	0.818827i	$0.694625\pi$
$224$	4267.33	1.27287
$225$	0	0
$226$	−6885.54	−2.02663
$227$	4777.14i	1.39678i	0.715715	+	0.698392i	$0.246101\pi$
$227$	4777.14i	1.39678i	−0.715715	+	0.698392i	$0.753899\pi$
$228$	− 142.142i	− 0.0412875i
$229$	2085.51	0.601808	0.300904	−	0.953654i	$-0.402712\pi$
$229$	2085.51	0.601808	0.300904	+	0.953654i	$0.402712\pi$
$230$	0	0
$231$	2870.28	0.817536
$232$	− 1205.12i	− 0.341035i
$233$	6484.53i	1.82324i	0.411030	+	0.911622i	$0.365169\pi$
$233$	6484.53i	1.82324i	−0.411030	+	0.911622i	$0.634831\pi$
$234$	1836.25	0.512988
$235$	0	0
$236$	1119.13	0.308683
$237$	− 772.271i	− 0.211664i
$238$	− 12388.0i	− 3.37394i
$239$	−2234.62	−0.604792	−0.302396	−	0.953182i	$-0.597786\pi$
$239$	−2234.62	−0.604792	−0.302396	+	0.953182i	$0.597786\pi$
$240$	0	0
$241$	−2393.01	−0.639616	−0.319808	−	0.947482i	$-0.603618\pi$
$241$	−2393.01	−0.639616	−0.319808	+	0.947482i	$0.603618\pi$
$242$	1151.44i	0.305857i
$243$	243.000i	0.0641500i
$244$	−2032.01	−0.533139
$245$	0	0
$246$	−308.979	−0.0800805
$247$	− 876.852i	− 0.225882i
$248$	− 2900.71i	− 0.742722i
$249$	−2744.73	−0.698554
$250$	0	0
$251$	−612.661	−0.154067	−0.0770335	−	0.997029i	$-0.524545\pi$
$251$	−612.661	−0.154067	−0.0770335	+	0.997029i	$0.524545\pi$
$252$	− 899.062i	− 0.224744i
$253$	430.454i	0.106966i
$254$	−946.014	−0.233694
$255$	0	0
$256$	4308.91	1.05198
$257$	306.112i	0.0742987i	0.999310	+	0.0371493i	$0.0118277\pi$
$257$	306.112i	0.0742987i	−0.999310	+	0.0371493i	$0.988172\pi$
$258$	3300.77i	0.796499i
$259$	1137.88	0.272990
$260$	0	0
$261$	−684.443	−0.162322
$262$	− 816.401i	− 0.192509i
$263$	283.839i	0.0665484i	0.999446	+	0.0332742i	$0.0105935\pi$
$263$	283.839i	0.0665484i	−0.999446	+	0.0332742i	$0.989407\pi$
$264$	1494.44	0.348396
$265$	0	0
$266$	−1475.75	−0.340166
$267$	3179.04i	0.728668i
$268$	843.165i	0.192181i
$269$	−2426.21	−0.549920	−0.274960	−	0.961456i	$-0.588665\pi$
$269$	−2426.21	−0.549920	−0.274960	+	0.961456i	$0.588665\pi$
$270$	0	0
$271$	−174.946	−0.0392148	−0.0196074	−	0.999808i	$-0.506242\pi$
$271$	−174.946	−0.0392148	−0.0196074	+	0.999808i	$0.506242\pi$
$272$	− 9631.80i	− 2.14711i
$273$	− 5546.19i	− 1.22956i
$274$	−3054.54	−0.673472
$275$	0	0
$276$	134.832	0.0294055
$277$	7807.07i	1.69344i	0.532042	+	0.846718i	$0.321425\pi$
$277$	7807.07i	1.69344i	−0.532042	+	0.846718i	$0.678575\pi$
$278$	− 6884.14i	− 1.48519i
$279$	−1647.44	−0.353512
$280$	0	0
$281$	584.171	0.124017	0.0620084	−	0.998076i	$-0.480249\pi$
$281$	584.171	0.124017	0.0620084	+	0.998076i	$0.480249\pi$
$282$	4529.32i	0.956444i
$283$	− 5897.31i	− 1.23872i	−0.785106	−	0.619362i	$-0.787391\pi$
$283$	− 5897.31i	− 1.23872i	0.785106	−	0.619362i	$-0.212609\pi$
$284$	−1638.40	−0.342329
$285$	0	0
$286$	−6413.73	−1.32605
$287$	933.239i	0.191942i
$288$	− 1261.88i	− 0.258183i
$289$	−9771.10	−1.98883
$290$	0	0
$291$	1563.00	0.314861
$292$	62.7846i	0.0125828i
$293$	1609.73i	0.320960i	0.987039	+	0.160480i	$0.0513042\pi$
$293$	1609.73i	0.320960i	−0.987039	+	0.160480i	$0.948696\pi$
$294$	−5877.99	−1.16603
$295$	0	0
$296$	592.448	0.116336
$297$	− 848.761i	− 0.165825i
$298$	12095.9i	2.35133i
$299$	831.758	0.160876
$300$	0	0
$301$	9969.62	1.90910
$302$	− 4646.64i	− 0.885378i
$303$	1041.24i	0.197418i
$304$	−1147.41	−0.216475
$305$	0	0
$306$	−3663.22	−0.684354
$307$	− 234.473i	− 0.0435898i	−0.999762	−	0.0217949i	$-0.993062\pi$
$307$	− 234.473i	− 0.0435898i	0.999762	−	0.0217949i	$-0.00693809\pi$
$308$	3140.28i	0.580955i
$309$	2312.25	0.425693
$310$	0	0
$311$	−1795.25	−0.327329	−0.163665	−	0.986516i	$-0.552331\pi$
$311$	−1795.25	−0.327329	−0.163665	+	0.986516i	$0.552331\pi$
$312$	− 2887.68i	− 0.523983i
$313$	− 8440.61i	− 1.52425i	−0.647427	−	0.762127i	$-0.724155\pi$
$313$	− 8440.61i	− 1.52425i	0.647427	−	0.762127i	$-0.275845\pi$
$314$	442.533	0.0795336
$315$	0	0
$316$	844.917	0.150412
$317$	− 10551.7i	− 1.86953i	−0.355264	−	0.934766i	$-0.615609\pi$
$317$	− 10551.7i	− 1.86953i	0.355264	−	0.934766i	$-0.384391\pi$
$318$	− 3042.67i	− 0.536554i
$319$	2390.65	0.419595
$320$	0	0
$321$	−4246.12	−0.738304
$322$	− 1399.86i	− 0.242270i
$323$	1749.28i	0.301339i
$324$	−265.858	−0.0455861
$325$	0	0
$326$	9733.21	1.65360
$327$	− 2725.16i	− 0.460861i
$328$	485.900i	0.0817968i
$329$	13680.3	2.29247
$330$	0	0
$331$	6743.17	1.11975	0.559876	−	0.828576i	$-0.310849\pi$
$331$	6743.17	1.11975	0.559876	+	0.828576i	$0.310849\pi$
$332$	− 3002.91i	− 0.496405i
$333$	− 336.478i	− 0.0553720i
$334$	−874.265	−0.143227
$335$	0	0
$336$	−7257.50	−1.17836
$337$	8437.26i	1.36382i	0.731437	+	0.681909i	$0.238850\pi$
$337$	8437.26i	1.36382i	−0.731437	+	0.681909i	$0.761150\pi$
$338$	5013.62i	0.806819i
$339$	6149.82	0.985287
$340$	0	0
$341$	5754.26	0.913814
$342$	436.389i	0.0689978i
$343$	7314.45i	1.15144i
$344$	5190.78	0.813571
$345$	0	0
$346$	−6502.25	−1.01030
$347$	1848.85i	0.286027i	0.989721	+	0.143013i	$0.0456792\pi$
$347$	1848.85i	0.286027i	−0.989721	+	0.143013i	$0.954321\pi$
$348$	− 748.826i	− 0.115349i
$349$	1148.38	0.176136	0.0880678	−	0.996114i	$-0.471931\pi$
$349$	1148.38	0.176136	0.0880678	+	0.996114i	$0.471931\pi$
$350$	0	0
$351$	−1640.04	−0.249399
$352$	4407.54i	0.667393i
$353$	− 5753.60i	− 0.867516i	−0.901029	−	0.433758i	$-0.857187\pi$
$353$	− 5753.60i	− 0.867516i	0.901029	−	0.433758i	$-0.142813\pi$
$354$	−3435.85	−0.515856
$355$	0	0
$356$	−3478.09	−0.517804
$357$	11064.4i	1.64030i
$358$	1936.83i	0.285935i
$359$	−5452.01	−0.801521	−0.400761	−	0.916183i	$-0.631254\pi$
$359$	−5452.01	−0.801521	−0.400761	+	0.916183i	$0.631254\pi$
$360$	0	0
$361$	−6650.61	−0.969619
$362$	6590.31i	0.956848i
$363$	− 1028.41i	− 0.148698i
$364$	6067.91	0.873749
$365$	0	0
$366$	6238.47	0.890956
$367$	− 8385.93i	− 1.19276i	−0.802703	−	0.596379i	$-0.796606\pi$
$367$	− 8385.93i	− 1.19276i	0.802703	−	0.596379i	$-0.203394\pi$
$368$	− 1088.40i	− 0.154176i
$369$	275.965	0.0389327
$370$	0	0
$371$	−9190.05	−1.28605
$372$	− 1802.41i	− 0.251212i
$373$	2728.30i	0.378730i	0.981907	+	0.189365i	$0.0606429\pi$
$373$	2728.30i	0.378730i	−0.981907	+	0.189365i	$0.939357\pi$
$374$	12795.1	1.76903
$375$	0	0
$376$	7122.81	0.976944
$377$	− 4619.41i	− 0.631065i
$378$	2760.21i	0.375582i
$379$	−3348.99	−0.453895	−0.226947	−	0.973907i	$-0.572875\pi$
$379$	−3348.99	−0.453895	−0.226947	+	0.973907i	$0.572875\pi$
$380$	0	0
$381$	844.932	0.113615
$382$	14506.2i	1.94294i
$383$	10430.3i	1.39155i	0.718261	+	0.695774i	$0.244938\pi$
$383$	10430.3i	1.39155i	−0.718261	+	0.695774i	$0.755062\pi$
$384$	−5026.97	−0.668051
$385$	0	0
$386$	9.99766	0.00131831
$387$	− 2948.08i	− 0.387233i
$388$	1710.03i	0.223746i
$389$	−9827.23	−1.28088	−0.640438	−	0.768010i	$-0.721247\pi$
$389$	−9827.23	−1.28088	−0.640438	+	0.768010i	$0.721247\pi$
$390$	0	0
$391$	−1659.32	−0.214617
$392$	9243.73i	1.19102i
$393$	729.168i	0.0935921i
$394$	−1913.58	−0.244682
$395$	0	0
$396$	928.602	0.117838
$397$	436.382i	0.0551672i	0.999619	+	0.0275836i	$0.00878125\pi$
$397$	436.382i	0.0551672i	−0.999619	+	0.0275836i	$0.991219\pi$
$398$	− 10247.1i	− 1.29055i
$399$	1318.07	0.165378
$400$	0	0
$401$	−14501.5	−1.80591	−0.902955	−	0.429736i	$-0.858607\pi$
$401$	−14501.5	−1.80591	−0.902955	+	0.429736i	$0.858607\pi$
$402$	− 2588.60i	− 0.321163i
$403$	− 11118.8i	− 1.37436i
$404$	−1139.19	−0.140289
$405$	0	0
$406$	−7774.51	−0.950351
$407$	1175.26i	0.143134i
$408$	5760.78i	0.699022i
$409$	12058.4	1.45782	0.728911	−	0.684609i	$-0.240027\pi$
$409$	12058.4	1.45782	0.728911	+	0.684609i	$0.240027\pi$
$410$	0	0
$411$	2728.16	0.327421
$412$	2529.76i	0.302505i
$413$	10377.6i	1.23644i
$414$	−413.947	−0.0491410
$415$	0	0
$416$	8516.60	1.00375
$417$	6148.57i	0.722054i
$418$	− 1524.24i	− 0.178356i
$419$	−6042.95	−0.704577	−0.352288	−	0.935892i	$-0.614596\pi$
$419$	−6042.95	−0.704577	−0.352288	+	0.935892i	$0.614596\pi$
$420$	0	0
$421$	−9994.67	−1.15703	−0.578516	−	0.815671i	$-0.696368\pi$
$421$	−9994.67	−1.15703	−0.578516	+	0.815671i	$0.696368\pi$
$422$	169.659i	0.0195708i
$423$	− 4045.36i	− 0.464994i
$424$	−4784.90	−0.548054
$425$	0	0
$426$	5030.07	0.572083
$427$	− 18842.6i	− 2.13550i
$428$	− 4645.55i	− 0.524652i
$429$	5728.42	0.644687
$430$	0	0
$431$	9327.32	1.04242	0.521208	−	0.853430i	$-0.325482\pi$
$431$	9327.32	1.04242	0.521208	+	0.853430i	$0.325482\pi$
$432$	2146.09i	0.239013i
$433$	− 7861.22i	− 0.872485i	−0.899829	−	0.436243i	$-0.856309\pi$
$433$	− 7861.22i	− 0.872485i	0.899829	−	0.436243i	$-0.143691\pi$
$434$	−18713.1	−2.06972
$435$	0	0
$436$	2981.51	0.327496
$437$	197.670i	0.0216380i
$438$	− 192.755i	− 0.0210279i
$439$	−7412.06	−0.805828	−0.402914	−	0.915238i	$-0.632003\pi$
$439$	−7412.06	−0.805828	−0.402914	+	0.915238i	$0.632003\pi$
$440$	0	0
$441$	5249.93	0.566886
$442$	− 24723.6i	− 2.66060i
$443$	− 3043.66i	− 0.326430i	−0.986591	−	0.163215i	$-0.947814\pi$
$443$	− 3043.66i	− 0.326430i	0.986591	−	0.163215i	$-0.0521864\pi$
$444$	368.129	0.0393483
$445$	0	0
$446$	18317.9	1.94480
$447$	− 10803.4i	− 1.14314i
$448$	5019.81i	0.529383i
$449$	9547.87	1.00355	0.501773	−	0.864999i	$-0.332681\pi$
$449$	9547.87	1.00355	0.501773	+	0.864999i	$0.332681\pi$
$450$	0	0
$451$	−963.902	−0.100639
$452$	6728.31i	0.700162i
$453$	4150.14i	0.430443i
$454$	16045.9	1.65875
$455$	0	0
$456$	686.266	0.0704766
$457$	− 13401.9i	− 1.37180i	−0.727695	−	0.685901i	$-0.759408\pi$
$457$	− 13401.9i	− 1.37180i	0.727695	−	0.685901i	$-0.240592\pi$
$458$	− 7005.00i	− 0.714677i
$459$	3271.81	0.332712
$460$	0	0
$461$	−4137.03	−0.417962	−0.208981	−	0.977920i	$-0.567015\pi$
$461$	−4137.03	−0.417962	−0.208981	+	0.977920i	$0.567015\pi$
$462$	− 9640.99i	− 0.970865i
$463$	− 13976.3i	− 1.40288i	−0.712729	−	0.701439i	$-0.752541\pi$
$463$	− 13976.3i	− 1.40288i	0.712729	−	0.701439i	$-0.247459\pi$
$464$	−6044.75	−0.604786
$465$	0	0
$466$	21780.9	2.16519
$467$	10796.5i	1.06982i	0.844910	+	0.534908i	$0.179654\pi$
$467$	10796.5i	1.06982i	−0.844910	+	0.534908i	$0.820346\pi$
$468$	− 1794.32i	− 0.177227i
$469$	−7818.60	−0.769785
$470$	0	0
$471$	−395.248	−0.0386668
$472$	5403.21i	0.526913i
$473$	10297.2i	1.00098i
$474$	−2593.98	−0.251362
$475$	0	0
$476$	−12105.2	−1.16563
$477$	2717.56i	0.260856i
$478$	7505.85i	0.718221i
$479$	−14568.4	−1.38966	−0.694830	−	0.719174i	$-0.744521\pi$
$479$	−14568.4	−1.38966	−0.694830	+	0.719174i	$0.744521\pi$
$480$	0	0
$481$	2270.94	0.215272
$482$	8037.89i	0.759577i
$483$	1250.28i	0.117784i
$484$	1125.15	0.105668
$485$	0	0
$486$	816.212	0.0761814
$487$	11456.6i	1.06601i	0.846113	+	0.533004i	$0.178937\pi$
$487$	11456.6i	1.06601i	−0.846113	+	0.533004i	$0.821063\pi$
$488$	− 9810.61i	− 0.910052i
$489$	−8693.21	−0.803928
$490$	0	0
$491$	−19666.5	−1.80761	−0.903804	−	0.427948i	$-0.859237\pi$
$491$	−19666.5	−1.80761	−0.903804	+	0.427948i	$0.859237\pi$
$492$	301.924i	0.0276662i
$493$	9215.48i	0.841875i
$494$	−2945.26	−0.268246
$495$	0	0
$496$	−14549.6	−1.31713
$497$	− 15192.8i	− 1.37121i
$498$	9219.26i	0.829568i
$499$	−8379.31	−0.751722	−0.375861	−	0.926676i	$-0.622653\pi$
$499$	−8379.31	−0.751722	−0.375861	+	0.926676i	$0.622653\pi$
$500$	0	0
$501$	780.850	0.0696323
$502$	2057.87i	0.182962i
$503$	− 15678.1i	− 1.38976i	−0.719124	−	0.694881i	$-0.755457\pi$
$503$	− 15678.1i	− 1.38976i	0.719124	−	0.694881i	$-0.244543\pi$
$504$	4340.71	0.383632
$505$	0	0
$506$	1445.85	0.127028
$507$	− 4477.91i	− 0.392251i
$508$	924.413i	0.0807366i
$509$	17037.3	1.48363	0.741813	−	0.670606i	$-0.233966\pi$
$509$	17037.3	1.48363	0.741813	+	0.670606i	$0.233966\pi$
$510$	0	0
$511$	−582.197	−0.0504009
$512$	− 1067.93i	− 0.0921798i
$513$	− 389.761i	− 0.0335446i
$514$	1028.20	0.0882334
$515$	0	0
$516$	3225.40	0.275175
$517$	14129.8i	1.20199i
$518$	− 3822.02i	− 0.324189i
$519$	5807.49	0.491176
$520$	0	0
$521$	−8776.12	−0.737982	−0.368991	−	0.929433i	$-0.620297\pi$
$521$	−8776.12	−0.737982	−0.368991	+	0.929433i	$0.620297\pi$
$522$	2298.97i	0.192765i
$523$	11120.4i	0.929753i	0.885375	+	0.464877i	$0.153901\pi$
$523$	11120.4i	0.929753i	−0.885375	+	0.464877i	$0.846099\pi$
$524$	−797.759	−0.0665082
$525$	0	0
$526$	953.385	0.0790296
$527$	22181.5i	1.83348i
$528$	− 7495.95i	− 0.617840i
$529$	11979.5	0.984589
$530$	0	0
$531$	3068.73	0.250794
$532$	1442.06i	0.117521i
$533$	1862.53i	0.151360i
$534$	10678.1	0.865330
$535$	0	0
$536$	−4070.83	−0.328047
$537$	− 1729.88i	− 0.139013i
$538$	8149.39i	0.653058i
$539$	−18337.2	−1.46538
$540$	0	0
$541$	21730.6	1.72693	0.863467	−	0.504405i	$-0.168288\pi$
$541$	21730.6	1.72693	0.863467	+	0.504405i	$0.168288\pi$
$542$	587.626i	0.0465695i
$543$	− 5886.13i	− 0.465190i
$544$	−16990.2	−1.33906
$545$	0	0
$546$	−18629.1	−1.46017
$547$	6926.17i	0.541392i	0.962665	+	0.270696i	$0.0872539\pi$
$547$	6926.17i	0.541392i	−0.962665	+	0.270696i	$0.912746\pi$
$548$	2984.79i	0.232671i
$549$	−5571.89	−0.433155
$550$	0	0
$551$	1097.82	0.0848793
$552$	650.973i	0.0501943i
$553$	7834.85i	0.602480i
$554$	26223.2	2.01104
$555$	0	0
$556$	−6726.95	−0.513104
$557$	6589.22i	0.501246i	0.968085	+	0.250623i	$0.0806355\pi$
$557$	6589.22i	0.501246i	−0.968085	+	0.250623i	$0.919364\pi$
$558$	5533.59i	0.419813i
$559$	19897.0	1.50547
$560$	0	0
$561$	−11427.9	−0.860047
$562$	− 1962.17i	− 0.147276i
$563$	3839.63i	0.287426i	0.989619	+	0.143713i	$0.0459043\pi$
$563$	3839.63i	0.287426i	−0.989619	+	0.143713i	$0.954096\pi$
$564$	4425.90	0.330433
$565$	0	0
$566$	−19808.5	−1.47105
$567$	− 2465.28i	− 0.182596i
$568$	− 7910.28i	− 0.584345i
$569$	20874.0	1.53794	0.768968	−	0.639287i	$-0.220771\pi$
$569$	20874.0	1.53794	0.768968	+	0.639287i	$0.220771\pi$
$570$	0	0
$571$	21175.9	1.55199	0.775994	−	0.630740i	$-0.217248\pi$
$571$	21175.9	1.55199	0.775994	+	0.630740i	$0.217248\pi$
$572$	6267.28i	0.458126i
$573$	− 12956.3i	− 0.944599i
$574$	3134.66	0.227941
$575$	0	0
$576$	1484.39	0.107378
$577$	14924.2i	1.07678i	0.842695	+	0.538391i	$0.180968\pi$
$577$	14924.2i	1.07678i	−0.842695	+	0.538391i	$0.819032\pi$
$578$	32820.1i	2.36183i
$579$	−8.92941	−0.000640922 0
$580$	0	0
$581$	27845.8	1.98836
$582$	− 5249.96i	− 0.373914i
$583$	− 9492.00i	− 0.674303i
$584$	−303.127	−0.0214785
$585$	0	0
$586$	5406.92	0.381156
$587$	− 25218.0i	− 1.77318i	−0.462552	−	0.886592i	$-0.653066\pi$
$587$	− 25218.0i	− 1.77318i	0.462552	−	0.886592i	$-0.346934\pi$
$588$	5743.78i	0.402839i
$589$	2642.42	0.184854
$590$	0	0
$591$	1709.11	0.118957
$592$	− 2971.65i	− 0.206308i
$593$	5011.77i	0.347063i	0.984828	+	0.173532i	$0.0555179\pi$
$593$	5011.77i	0.347063i	−0.984828	+	0.173532i	$0.944482\pi$
$594$	−2850.90	−0.196926
$595$	0	0
$596$	11819.7	0.812337
$597$	9152.19i	0.627428i
$598$	− 2793.79i	− 0.191048i
$599$	−4943.41	−0.337199	−0.168600	−	0.985685i	$-0.553925\pi$
$599$	−4943.41	−0.337199	−0.168600	+	0.985685i	$0.553925\pi$
$600$	0	0
$601$	−24334.8	−1.65164	−0.825821	−	0.563932i	$-0.809288\pi$
$601$	−24334.8	−1.65164	−0.825821	+	0.563932i	$0.809288\pi$
$602$	− 33486.9i	− 2.26715i
$603$	2312.01i	0.156140i
$604$	−4540.54	−0.305881
$605$	0	0
$606$	3497.42	0.234444
$607$	− 28973.8i	− 1.93741i	−0.248207	−	0.968707i	$-0.579841\pi$
$607$	− 28973.8i	− 1.93741i	0.248207	−	0.968707i	$-0.420159\pi$
$608$	2023.99i	0.135006i
$609$	6943.81	0.462032
$610$	0	0
$611$	27302.8	1.80778
$612$	3579.58i	0.236431i
$613$	15139.1i	0.997490i	0.866749	+	0.498745i	$0.166206\pi$
$613$	15139.1i	0.997490i	−0.866749	+	0.498745i	$0.833794\pi$
$614$	−787.571	−0.0517651
$615$	0	0
$616$	−15161.4	−0.991673
$617$	− 13894.8i	− 0.906617i	−0.891354	−	0.453309i	$-0.850244\pi$
$617$	− 13894.8i	− 0.906617i	0.891354	−	0.453309i	$-0.149756\pi$
$618$	− 7766.61i	− 0.505532i
$619$	4589.69	0.298021	0.149011	−	0.988836i	$-0.452391\pi$
$619$	4589.69	0.298021	0.149011	+	0.988836i	$0.452391\pi$
$620$	0	0
$621$	369.717	0.0238909
$622$	6030.07i	0.388720i
$623$	− 32252.0i	− 2.07408i
$624$	−14484.3	−0.929223
$625$	0	0
$626$	−28351.2	−1.81013
$627$	1361.37i	0.0867114i
$628$	− 432.428i	− 0.0274773i
$629$	−4530.41	−0.287185
$630$	0	0
$631$	3005.77	0.189632	0.0948160	−	0.995495i	$-0.469774\pi$
$631$	3005.77	0.189632	0.0948160	+	0.995495i	$0.469774\pi$
$632$	4079.29i	0.256749i
$633$	− 151.531i	− 0.00951473i
$634$	−35442.0	−2.22016
$635$	0	0
$636$	−2973.19	−0.185369
$637$	35432.6i	2.20391i
$638$	− 8029.96i	− 0.498290i
$639$	−4492.60	−0.278129
$640$	0	0
$641$	−5631.47	−0.347004	−0.173502	−	0.984834i	$-0.555508\pi$
$641$	−5631.47	−0.347004	−0.173502	+	0.984834i	$0.555508\pi$
$642$	14262.3i	0.876773i
$643$	11305.1i	0.693358i	0.937984	+	0.346679i	$0.112691\pi$
$643$	11305.1i	0.693358i	−0.937984	+	0.346679i	$0.887309\pi$
$644$	−1367.89	−0.0836997
$645$	0	0
$646$	5875.64	0.357855
$647$	8614.30i	0.523436i	0.965144	+	0.261718i	$0.0842891\pi$
$647$	8614.30i	0.523436i	−0.965144	+	0.261718i	$0.915711\pi$
$648$	− 1283.58i	− 0.0778142i
$649$	−10718.6	−0.648291
$650$	0	0
$651$	16713.6	1.00623
$652$	− 9510.96i	− 0.571285i
$653$	12639.8i	0.757479i	0.925503	+	0.378739i	$0.123642\pi$
$653$	12639.8i	0.757479i	−0.925503	+	0.378739i	$0.876358\pi$
$654$	−9153.53	−0.547296
$655$	0	0
$656$	2437.22	0.145057
$657$	172.159i	0.0102231i
$658$	− 45950.9i	− 2.72242i
$659$	−13640.4	−0.806306	−0.403153	−	0.915133i	$-0.632086\pi$
$659$	−13640.4	−0.806306	−0.403153	+	0.915133i	$0.632086\pi$
$660$	0	0
$661$	−17052.0	−1.00340	−0.501699	−	0.865042i	$-0.667291\pi$
$661$	−17052.0	−1.00340	−0.501699	+	0.865042i	$0.667291\pi$
$662$	− 22649.6i	− 1.32976i
$663$	22081.9i	1.29350i
$664$	14498.2	0.847348
$665$	0	0
$666$	−1130.20	−0.0657570
$667$	1041.36i	0.0604521i
$668$	854.302i	0.0494820i
$669$	−16360.7	−0.945500
$670$	0	0
$671$	19461.7	1.11969
$672$	12802.0i	0.734892i
$673$	− 16419.7i	− 0.940467i	−0.882542	−	0.470234i	$-0.844170\pi$
$673$	− 16419.7i	− 0.940467i	0.882542	−	0.470234i	$-0.155830\pi$
$674$	28339.9	1.61960
$675$	0	0
$676$	4899.14	0.278740
$677$	8670.47i	0.492221i	0.969242	+	0.246110i	$0.0791525\pi$
$677$	8670.47i	0.492221i	−0.969242	+	0.246110i	$0.920847\pi$
$678$	− 20656.6i	− 1.17008i
$679$	−15856.9	−0.896220
$680$	0	0
$681$	−14331.4	−0.806434
$682$	− 19328.0i	− 1.08520i
$683$	5973.36i	0.334647i	0.985902	+	0.167324i	$0.0535125\pi$
$683$	5973.36i	0.334647i	−0.985902	+	0.167324i	$0.946488\pi$
$684$	426.425	0.0238374
$685$	0	0
$686$	24568.5	1.36739
$687$	6256.52i	0.347454i
$688$	− 26036.4i	− 1.44277i
$689$	−18341.2	−1.01414
$690$	0	0
$691$	−15316.3	−0.843212	−0.421606	−	0.906779i	$-0.638533\pi$
$691$	−15316.3	−0.843212	−0.421606	+	0.906779i	$0.638533\pi$
$692$	6353.78i	0.349038i
$693$	8610.85i	0.472005i
$694$	6210.09	0.339671
$695$	0	0
$696$	3615.36	0.196897
$697$	− 3715.65i	− 0.201923i
$698$	− 3857.29i	− 0.209170i
$699$	−19453.6	−1.05265
$700$	0	0
$701$	34583.1	1.86332	0.931660	−	0.363333i	$-0.118361\pi$
$701$	34583.1	1.86332	0.931660	+	0.363333i	$0.118361\pi$
$702$	5508.74i	0.296174i
$703$	539.695i	0.0289545i
$704$	−5184.74	−0.277567
$705$	0	0
$706$	−19325.8	−1.03022
$707$	− 10563.6i	− 0.561929i
$708$	3357.39i	0.178218i
$709$	−11194.1	−0.592955	−0.296477	−	0.955040i	$-0.595812\pi$
$709$	−11194.1	−0.592955	−0.296477	+	0.955040i	$0.595812\pi$
$710$	0	0
$711$	2316.81	0.122204
$712$	− 16792.4i	− 0.883877i
$713$	2506.53i	0.131655i
$714$	37164.1	1.94794
$715$	0	0
$716$	1892.61	0.0987850
$717$	− 6703.85i	− 0.349177i
$718$	18312.8i	0.951847i
$719$	15491.9	0.803549	0.401774	−	0.915739i	$-0.368394\pi$
$719$	15491.9	0.803549	0.401774	+	0.915739i	$0.368394\pi$
$720$	0	0
$721$	−23458.2	−1.21169
$722$	22338.7i	1.15147i
$723$	− 7179.04i	− 0.369283i
$724$	6439.83	0.330572
$725$	0	0
$726$	−3454.33	−0.176587
$727$	6272.72i	0.320003i	0.987117	+	0.160002i	$0.0511500\pi$
$727$	6272.72i	0.320003i	−0.987117	+	0.160002i	$0.948850\pi$
$728$	29296.1i	1.49146i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−39693.6	−2.00837
$732$	− 6096.02i	− 0.307808i
$733$	24980.5i	1.25877i	0.777095	+	0.629383i	$0.216692\pi$
$733$	24980.5i	1.25877i	−0.777095	+	0.629383i	$0.783308\pi$
$734$	−28167.5	−1.41646
$735$	0	0
$736$	−1919.90	−0.0961530
$737$	− 8075.49i	− 0.403615i
$738$	− 926.938i	− 0.0462345i
$739$	30660.7	1.52621	0.763107	−	0.646272i	$-0.223673\pi$
$739$	30660.7	1.52621	0.763107	+	0.646272i	$0.223673\pi$
$740$	0	0
$741$	2630.56	0.130413
$742$	30868.5i	1.52725i
$743$	− 17205.7i	− 0.849551i	−0.905299	−	0.424776i	$-0.860353\pi$
$743$	− 17205.7i	− 0.849551i	0.905299	−	0.424776i	$-0.139647\pi$
$744$	8702.12	0.428811
$745$	0	0
$746$	9164.09	0.449760
$747$	− 8234.18i	− 0.403310i
$748$	− 12502.9i	− 0.611165i
$749$	43077.8	2.10151
$750$	0	0
$751$	18397.1	0.893901	0.446950	−	0.894559i	$-0.352510\pi$
$751$	18397.1	0.893901	0.446950	+	0.894559i	$0.352510\pi$
$752$	− 35727.2i	− 1.73250i
$753$	− 1837.98i	− 0.0889506i
$754$	−15516.1	−0.749422
$755$	0	0
$756$	2697.19	0.129756
$757$	− 22305.1i	− 1.07093i	−0.844557	−	0.535465i	$-0.820136\pi$
$757$	− 22305.1i	− 1.07093i	0.844557	−	0.535465i	$-0.179864\pi$
$758$	11248.9i	0.539023i
$759$	−1291.36	−0.0617569
$760$	0	0
$761$	14458.5	0.688727	0.344364	−	0.938836i	$-0.388095\pi$
$761$	14458.5	0.688727	0.344364	+	0.938836i	$0.388095\pi$
$762$	− 2838.04i	− 0.134923i
$763$	27647.3i	1.31179i
$764$	14175.0	0.671249
$765$	0	0
$766$	35034.3	1.65253
$767$	20711.3i	0.975022i
$768$	12926.7i	0.607361i
$769$	39897.6	1.87093	0.935463	−	0.353424i	$-0.114983\pi$
$769$	39897.6	1.87093	0.935463	+	0.353424i	$0.114983\pi$
$770$	0	0
$771$	−918.337	−0.0428964
$772$	− 9.76938i	0 0.000455450i
$773$	20070.2i	0.933863i	0.884294	+	0.466931i	$0.154641\pi$
$773$	20070.2i	0.933863i	−0.884294	+	0.466931i	$0.845359\pi$
$774$	−9902.30	−0.459859
$775$	0	0
$776$	−8256.08	−0.381928
$777$	3413.63i	0.157611i
$778$	33008.7i	1.52110i
$779$	−442.635	−0.0203582
$780$	0	0
$781$	15692.0	0.718953
$782$	5573.47i	0.254868i
$783$	− 2053.33i	− 0.0937164i
$784$	46365.5	2.11213
$785$	0	0
$786$	2449.20	0.111145
$787$	− 10733.0i	− 0.486137i	−0.970009	−	0.243069i	$-0.921846\pi$
$787$	− 10733.0i	− 0.486137i	0.970009	−	0.243069i	$-0.0781540\pi$
$788$	1869.89i	0.0845329i
$789$	−851.516	−0.0384218
$790$	0	0
$791$	−62391.1	−2.80452
$792$	4483.33i	0.201147i
$793$	− 37605.5i	− 1.68400i
$794$	1465.76	0.0655139
$795$	0	0
$796$	−10013.1	−0.445861
$797$	14335.8i	0.637140i	0.947899	+	0.318570i	$0.103203\pi$
$797$	14335.8i	0.637140i	−0.947899	+	0.318570i	$0.896797\pi$
$798$	− 4427.26i	− 0.196395i
$799$	−54467.7	−2.41167
$800$	0	0
$801$	−9537.13	−0.420697
$802$	48709.0i	2.14461i
$803$	− 601.325i	− 0.0264263i
$804$	−2529.49	−0.110956
$805$	0	0
$806$	−37347.1	−1.63213
$807$	− 7278.63i	− 0.317497i
$808$	− 5500.03i	− 0.239468i
$809$	−20920.7	−0.909187	−0.454593	−	0.890699i	$-0.650215\pi$
$809$	−20920.7	−0.909187	−0.454593	+	0.890699i	$0.650215\pi$
$810$	0	0
$811$	12816.5	0.554931	0.277465	−	0.960736i	$-0.410506\pi$
$811$	12816.5	0.554931	0.277465	+	0.960736i	$0.410506\pi$
$812$	7596.99i	0.328328i
$813$	− 524.838i	− 0.0226407i
$814$	3947.59	0.169979
$815$	0	0
$816$	28895.4	1.23963
$817$	4728.59i	0.202488i
$818$	− 40502.9i	− 1.73124i
$819$	16638.6	0.709889
$820$	0	0
$821$	7253.55	0.308344	0.154172	−	0.988044i	$-0.450729\pi$
$821$	7253.55	0.308344	0.154172	+	0.988044i	$0.450729\pi$
$822$	− 9163.61i	− 0.388829i
$823$	35288.1i	1.49461i	0.664479	+	0.747307i	$0.268654\pi$
$823$	35288.1i	1.49461i	−0.664479	+	0.747307i	$0.731346\pi$
$824$	−12213.8	−0.516367
$825$	0	0
$826$	34857.3	1.46833
$827$	32205.9i	1.35418i	0.735899	+	0.677092i	$0.236760\pi$
$827$	32205.9i	1.35418i	−0.735899	+	0.677092i	$0.763240\pi$
$828$	404.495i	0.0169773i
$829$	−29993.3	−1.25659	−0.628294	−	0.777976i	$-0.716246\pi$
$829$	−29993.3	−1.25659	−0.628294	+	0.777976i	$0.716246\pi$
$830$	0	0
$831$	−23421.2	−0.977706
$832$	10018.4i	0.417457i
$833$	− 70686.2i	− 2.94013i
$834$	20652.4	0.857476
$835$	0	0
$836$	−1489.44	−0.0616187
$837$	− 4942.33i	− 0.204100i
$838$	20297.7i	0.836720i
$839$	35608.7	1.46526	0.732628	−	0.680629i	$-0.238294\pi$
$839$	35608.7	1.46526	0.732628	+	0.680629i	$0.238294\pi$
$840$	0	0
$841$	−18605.5	−0.762865
$842$	33571.1i	1.37403i
$843$	1752.51i	0.0716011i
$844$	165.785	0.00676134
$845$	0	0
$846$	−13588.0	−0.552203
$847$	10433.4i	0.423255i
$848$	24000.5i	0.971911i
$849$	17691.9	0.715177
$850$	0	0
$851$	−511.940	−0.0206217
$852$	− 4915.21i	− 0.197644i
$853$	− 11229.3i	− 0.450744i	−0.974273	−	0.225372i	$-0.927640\pi$
$853$	− 11229.3i	− 0.450744i	0.974273	−	0.225372i	$-0.0723598\pi$
$854$	−63290.5	−2.53601
$855$	0	0
$856$	22428.9	0.895565
$857$	− 22136.1i	− 0.882327i	−0.897427	−	0.441164i	$-0.854566\pi$
$857$	− 22136.1i	− 0.882327i	0.897427	−	0.441164i	$-0.145434\pi$
$858$	− 19241.2i	− 0.765598i
$859$	820.727	0.0325994	0.0162997	−	0.999867i	$-0.494811\pi$
$859$	820.727	0.0325994	0.0162997	+	0.999867i	$0.494811\pi$
$860$	0	0
$861$	−2799.72	−0.110818
$862$	− 31329.5i	− 1.23792i
$863$	− 245.223i	− 0.00967264i	−0.999988	−	0.00483632i	$-0.998461\pi$
$863$	− 245.223i	− 0.00967264i	0.999988	−	0.00483632i	$-0.00153945\pi$
$864$	3785.63	0.149062
$865$	0	0
$866$	−26405.0	−1.03612
$867$	− 29313.3i	− 1.14825i
$868$	18285.8i	0.715048i
$869$	−8092.27	−0.315894
$870$	0	0
$871$	−15604.1	−0.607032
$872$	14394.8i	0.559026i
$873$	4689.00i	0.181785i
$874$	663.952	0.0256963
$875$	0	0
$876$	−188.354	−0.00726471
$877$	− 37727.8i	− 1.45265i	−0.687350	−	0.726326i	$-0.741226\pi$
$877$	− 37727.8i	− 1.45265i	0.687350	−	0.726326i	$-0.258774\pi$
$878$	24896.4i	0.956961i
$879$	−4829.19	−0.185306
$880$	0	0
$881$	21738.8	0.831326	0.415663	−	0.909519i	$-0.363550\pi$
$881$	21738.8	0.831326	0.415663	+	0.909519i	$0.363550\pi$
$882$	− 17634.0i	− 0.673205i
$883$	− 44340.4i	− 1.68989i	−0.534852	−	0.844946i	$-0.679632\pi$
$883$	− 44340.4i	− 1.68989i	0.534852	−	0.844946i	$-0.320368\pi$
$884$	−24159.1	−0.919184
$885$	0	0
$886$	−10223.3	−0.387652
$887$	681.008i	0.0257790i	0.999917	+	0.0128895i	$0.00410298\pi$
$887$	681.008i	0.0257790i	−0.999917	+	0.0128895i	$0.995897\pi$
$888$	1777.34i	0.0671664i
$889$	−8572.00	−0.323392
$890$	0	0
$891$	2546.28	0.0957393
$892$	− 17899.7i	− 0.671889i
$893$	6488.58i	0.243149i
$894$	−36287.6	−1.35754
$895$	0	0
$896$	50999.6	1.90154
$897$	2495.28i	0.0928816i
$898$	− 32070.3i	− 1.19176i
$899$	13920.7	0.516443
$900$	0	0
$901$	36589.8	1.35292
$902$	3237.65i	0.119514i
$903$	29908.9i	1.10222i
$904$	−32484.6	−1.19516
$905$	0	0
$906$	13939.9	0.511173
$907$	− 5348.01i	− 0.195786i	−0.995197	−	0.0978929i	$-0.968790\pi$
$907$	− 5348.01i	− 0.195786i	0.995197	−	0.0978929i	$-0.0312103\pi$
$908$	− 15679.5i	− 0.573066i
$909$	−3123.72	−0.113979
$910$	0	0
$911$	14488.7	0.526930	0.263465	−	0.964669i	$-0.415135\pi$
$911$	14488.7	0.526930	0.263465	+	0.964669i	$0.415135\pi$
$912$	− 3442.23i	− 0.124982i
$913$	28760.7i	1.04254i
$914$	−45015.6	−1.62908
$915$	0	0
$916$	−6845.05	−0.246907
$917$	− 7397.56i	− 0.266400i
$918$	− 10989.7i	− 0.395112i
$919$	22546.9	0.809308	0.404654	−	0.914470i	$-0.367392\pi$
$919$	22546.9	0.809308	0.404654	+	0.914470i	$0.367392\pi$
$920$	0	0
$921$	703.419	0.0251666
$922$	13895.9i	0.496351i
$923$	− 30321.3i	− 1.08130i
$924$	−9420.85	−0.335415
$925$	0	0
$926$	−46944.9	−1.66599
$927$	6936.74i	0.245774i
$928$	10662.7i	0.377178i
$929$	13980.7	0.493747	0.246874	−	0.969048i	$-0.420597\pi$
$929$	13980.7	0.493747	0.246874	+	0.969048i	$0.420597\pi$
$930$	0	0
$931$	−8420.65	−0.296429
$932$	− 21283.5i	− 0.748032i
$933$	− 5385.75i	− 0.188984i
$934$	36264.5	1.27046
$935$	0	0
$936$	8663.05	0.302522
$937$	− 26362.6i	− 0.919133i	−0.888143	−	0.459567i	$-0.848005\pi$
$937$	− 26362.6i	− 0.919133i	0.888143	−	0.459567i	$-0.151995\pi$
$938$	26261.9i	0.914159i
$939$	25321.8	0.880029
$940$	0	0
$941$	−14715.6	−0.509792	−0.254896	−	0.966968i	$-0.582041\pi$
$941$	−14715.6	−0.509792	−0.254896	+	0.966968i	$0.582041\pi$
$942$	1327.60i	0.0459188i
$943$	− 419.871i	− 0.0144994i
$944$	27101.9	0.934419
$945$	0	0
$946$	34587.2	1.18872
$947$	11818.2i	0.405532i	0.979227	+	0.202766i	$0.0649931\pi$
$947$	11818.2i	0.405532i	−0.979227	+	0.202766i	$0.935007\pi$
$948$	2534.75i	0.0868406i
$949$	−1161.93	−0.0397448
$950$	0	0
$951$	31655.1	1.07937
$952$	− 58444.2i	− 1.98969i
$953$	− 43832.3i	− 1.48989i	−0.667125	−	0.744945i	$-0.732476\pi$
$953$	− 43832.3i	− 1.48989i	0.667125	−	0.744945i	$-0.267524\pi$
$954$	9128.00	0.309780
$955$	0	0
$956$	7334.46	0.248131
$957$	7171.95i	0.242253i
$958$	48933.8i	1.65029i
$959$	−27677.7	−0.931971
$960$	0	0
$961$	3716.00	0.124736
$962$	− 7627.86i	− 0.255647i
$963$	− 12738.4i	− 0.426260i
$964$	7854.36	0.262419
$965$	0	0
$966$	4199.58	0.139875
$967$	− 10696.2i	− 0.355706i	−0.984057	−	0.177853i	$-0.943085\pi$
$967$	− 10696.2i	− 0.355706i	0.984057	−	0.177853i	$-0.0569152\pi$
$968$	5432.27i	0.180372i
$969$	−5247.83	−0.173978
$970$	0	0
$971$	27933.0	0.923187	0.461593	−	0.887092i	$-0.347278\pi$
$971$	27933.0	0.923187	0.461593	+	0.887092i	$0.347278\pi$
$972$	− 797.575i	− 0.0263192i
$973$	− 62378.4i	− 2.05525i
$974$	38481.4	1.26594
$975$	0	0
$976$	−49208.9	−1.61387
$977$	− 24341.7i	− 0.797094i	−0.917148	−	0.398547i	$-0.869515\pi$
$977$	− 24341.7i	− 0.797094i	0.917148	−	0.398547i	$-0.130485\pi$
$978$	29199.6i	0.954704i
$979$	33311.7	1.08748
$980$	0	0
$981$	8175.48	0.266078
$982$	66057.7i	2.14662i
$983$	12553.0i	0.407301i	0.979044	+	0.203651i	$0.0652807\pi$
$983$	12553.0i	0.407301i	−0.979044	+	0.203651i	$0.934719\pi$
$984$	−1457.70	−0.0472254
$985$	0	0
$986$	30953.9	0.999769
$987$	41041.0i	1.32356i
$988$	2878.01i	0.0926737i
$989$	−4485.41	−0.144214
$990$	0	0
$991$	−45631.8	−1.46271	−0.731353	−	0.681999i	$-0.761111\pi$
$991$	−45631.8	−1.46271	−0.731353	+	0.681999i	$0.761111\pi$
$992$	25665.0i	0.821437i
$993$	20229.5i	0.646490i
$994$	−51031.0	−1.62838
$995$	0	0
$996$	9008.74	0.286599
$997$	34499.0i	1.09588i	0.836517	+	0.547941i	$0.184588\pi$
$997$	34499.0i	1.09588i	−0.836517	+	0.547941i	$0.815412\pi$
$998$	28145.3i	0.892708i
$999$	1009.43	0.0319690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.b.c.49.2		4
3.2	odd	2		225.4.b.h.199.3		4
4.3	odd	2		1200.4.f.v.49.2		4
5.2	odd	4		75.4.a.d.1.2	✓	2
5.3	odd	4		75.4.a.e.1.1	yes	2
5.4	even	2	inner	75.4.b.c.49.3		4
15.2	even	4		225.4.a.n.1.1		2
15.8	even	4		225.4.a.j.1.2		2
15.14	odd	2		225.4.b.h.199.2		4
20.3	even	4		1200.4.a.bu.1.2		2
20.7	even	4		1200.4.a.bl.1.1		2
20.19	odd	2		1200.4.f.v.49.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.2	✓	2	5.2	odd	4
75.4.a.e.1.1	yes	2	5.3	odd	4
75.4.b.c.49.2		4	1.1	even	1	trivial
75.4.b.c.49.3		4	5.4	even	2	inner
225.4.a.j.1.2		2	15.8	even	4
225.4.a.n.1.1		2	15.2	even	4
225.4.b.h.199.2		4	15.14	odd	2
225.4.b.h.199.3		4	3.2	odd	2
1200.4.a.bl.1.1		2	20.7	even	4
1200.4.a.bu.1.2		2	20.3	even	4
1200.4.f.v.49.2		4	4.3	odd	2
1200.4.f.v.49.3		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.35890i q^{2} +3.00000i q^{3} -3.28220 q^{4} +10.0767 q^{6} -30.4356i q^{7} -15.8466i q^{8} -9.00000 q^{9} +31.4356 q^{11} -9.84661i q^{12} -60.7424i q^{13} -102.230 q^{14} -79.4848 q^{16} +121.178i q^{17} +30.2301i q^{18} +14.4356 q^{19} +91.3068 q^{21} -105.589i q^{22} +13.6932i q^{23} +47.5398 q^{24} -204.028 q^{26} -27.0000i q^{27} +99.8958i q^{28} +76.0492 q^{29} +183.049 q^{31} +140.208i q^{32} +94.3068i q^{33} +407.025 q^{34} +29.5398 q^{36} +37.3864i q^{37} -48.4877i q^{38} +182.227 q^{39} -30.6627 q^{41} -306.690i q^{42} +327.564i q^{43} -103.178 q^{44} +45.9941 q^{46} +449.485i q^{47} -238.454i q^{48} -583.325 q^{49} -363.534 q^{51} +199.369i q^{52} -301.951i q^{53} -90.6903 q^{54} -482.301 q^{56} +43.3068i q^{57} -255.441i q^{58} -340.970 q^{59} +619.098 q^{61} -614.844i q^{62} +273.920i q^{63} -164.932 q^{64} +316.767 q^{66} -256.890i q^{67} -397.731i q^{68} -41.0796 q^{69} +499.178 q^{71} +142.619i q^{72} -19.1288i q^{73} +125.577 q^{74} -47.3805 q^{76} -956.761i q^{77} -612.083i q^{78} -257.424 q^{79} +81.0000 q^{81} +102.993i q^{82} +914.909i q^{83} -299.687 q^{84} +1100.26 q^{86} +228.148i q^{87} -498.148i q^{88} +1059.68 q^{89} -1848.73 q^{91} -44.9439i q^{92} +549.148i q^{93} +1509.77 q^{94} -420.625 q^{96} -521.000i q^{97} +1959.33i q^{98} -282.920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41}+ \cdots - 504 q^{99}+O(q^{100}) \) 4 * q - 48 * q^4 - 12 * q^6 - 36 * q^9 + 56 * q^11 - 252 * q^14 + 240 * q^16 - 12 * q^19 + 156 * q^21 + 504 * q^24 - 1252 * q^26 - 184 * q^29 + 244 * q^31 + 1384 * q^34 + 432 * q^36 - 108 * q^39 + 784 * q^41 - 64 * q^44 - 1176 * q^46 - 520 * q^49 - 408 * q^51 + 108 * q^54 - 360 * q^56 - 248 * q^59 + 1500 * q^61 - 2752 * q^64 + 744 * q^66 - 792 * q^69 + 1648 * q^71 - 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 324 * q^81 - 48 * q^84 - 1684 * q^86 + 1728 * q^89 - 4396 * q^91 + 1192 * q^94 - 3984 * q^96 - 504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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