LMFDB - Embedded newform 2100.4.k.d.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,4,Mod(1849,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1849");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.k (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:	$123.904011012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.d.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} -26.0000 q^{11} +39.0000i q^{13} -101.000i q^{17} +48.0000 q^{19} -21.0000 q^{21} -57.0000i q^{23} -27.0000i q^{27} +291.000 q^{29} -79.0000 q^{31} -78.0000i q^{33} +322.000i q^{37} -117.000 q^{39} +455.000 q^{41} +307.000i q^{43} +508.000i q^{47} -49.0000 q^{49} +303.000 q^{51} +31.0000i q^{53} +144.000i q^{57} -211.000 q^{59} -797.000 q^{61} -63.0000i q^{63} -924.000i q^{67} +171.000 q^{69} +212.000 q^{71} -22.0000i q^{73} -182.000i q^{77} -874.000 q^{79} +81.0000 q^{81} +587.000i q^{83} +873.000i q^{87} +266.000 q^{89} -273.000 q^{91} -237.000i q^{93} +1822.00i q^{97} +234.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} - 52 q^{11} + 96 q^{19} - 42 q^{21} + 582 q^{29} - 158 q^{31} - 234 q^{39} + 910 q^{41} - 98 q^{49} + 606 q^{51} - 422 q^{59} - 1594 q^{61} + 342 q^{69} + 424 q^{71} - 1748 q^{79} + 162 q^{81}+ \cdots + 468 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 52 * q^11 + 96 * q^19 - 42 * q^21 + 582 * q^29 - 158 * q^31 - 234 * q^39 + 910 * q^41 - 98 * q^49 + 606 * q^51 - 422 * q^59 - 1594 * q^61 + 342 * q^69 + 424 * q^71 - 1748 * q^79 + 162 * q^81 + 532 * q^89 - 546 * q^91 + 468 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	0	0
$13$	39.0000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	39.0000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 101.000i	− 1.44095i	−0.693482	−	0.720473i	$-0.743925\pi$
$17$	− 101.000i	− 1.44095i	0.693482	−	0.720473i	$-0.256075\pi$
$18$	0	0
$19$	48.0000	0.579577	0.289788	−	0.957091i	$-0.406415\pi$
$19$	48.0000	0.579577	0.289788	+	0.957091i	$0.406415\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	− 57.0000i	− 0.516753i	−0.966044	−	0.258377i	$-0.916812\pi$
$23$	− 57.0000i	− 0.516753i	0.966044	−	0.258377i	$-0.0831875\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	291.000	1.86336	0.931678	−	0.363284i	$-0.118345\pi$
$29$	291.000	1.86336	0.931678	+	0.363284i	$0.118345\pi$
$30$	0	0
$31$	−79.0000	−0.457704	−0.228852	−	0.973461i	$-0.573497\pi$
$31$	−79.0000	−0.457704	−0.228852	+	0.973461i	$0.573497\pi$
$32$	0	0
$33$	− 78.0000i	− 0.411456i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	322.000i	1.43072i	0.698758	+	0.715358i	$0.253736\pi$
$37$	322.000i	1.43072i	−0.698758	+	0.715358i	$0.746264\pi$
$38$	0	0
$39$	−117.000	−0.480384
$40$	0	0
$41$	455.000	1.73315	0.866574	−	0.499049i	$-0.166317\pi$
$41$	455.000	1.73315	0.866574	+	0.499049i	$0.166317\pi$
$42$	0	0
$43$	307.000i	1.08877i	0.838836	+	0.544384i	$0.183237\pi$
$43$	307.000i	1.08877i	−0.838836	+	0.544384i	$0.816763\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	508.000i	1.57658i	0.615302	+	0.788292i	$0.289034\pi$
$47$	508.000i	1.57658i	−0.615302	+	0.788292i	$0.710966\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	303.000	0.831931
$52$	0	0
$53$	31.0000i	0.0803430i	0.999193	+	0.0401715i	$0.0127904\pi$
$53$	31.0000i	0.0803430i	−0.999193	+	0.0401715i	$0.987210\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	144.000i	0.334619i
$58$	0	0
$59$	−211.000	−0.465591	−0.232795	−	0.972526i	$-0.574787\pi$
$59$	−211.000	−0.465591	−0.232795	+	0.972526i	$0.574787\pi$
$60$	0	0
$61$	−797.000	−1.67288	−0.836438	−	0.548062i	$-0.815366\pi$
$61$	−797.000	−1.67288	−0.836438	+	0.548062i	$0.815366\pi$
$62$	0	0
$63$	− 63.0000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 924.000i	− 1.68484i	−0.538818	−	0.842422i	$-0.681129\pi$
$67$	− 924.000i	− 1.68484i	0.538818	−	0.842422i	$-0.318871\pi$
$68$	0	0
$69$	171.000	0.298348
$70$	0	0
$71$	212.000	0.354363	0.177181	−	0.984178i	$-0.443302\pi$
$71$	212.000	0.354363	0.177181	+	0.984178i	$0.443302\pi$
$72$	0	0
$73$	− 22.0000i	− 0.0352727i	−0.999844	−	0.0176363i	$-0.994386\pi$
$73$	− 22.0000i	− 0.0352727i	0.999844	−	0.0176363i	$-0.00561411\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 182.000i	− 0.269361i
$78$	0	0
$79$	−874.000	−1.24472	−0.622359	−	0.782732i	$-0.713825\pi$
$79$	−874.000	−1.24472	−0.622359	+	0.782732i	$0.713825\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	587.000i	0.776285i	0.921599	+	0.388142i	$0.126883\pi$
$83$	587.000i	0.776285i	−0.921599	+	0.388142i	$0.873117\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	873.000i	1.07581i
$88$	0	0
$89$	266.000	0.316808	0.158404	−	0.987374i	$-0.449365\pi$
$89$	266.000	0.316808	0.158404	+	0.987374i	$0.449365\pi$
$90$	0	0
$91$	−273.000	−0.314485
$92$	0	0
$93$	− 237.000i	− 0.264255i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1822.00i	1.90718i	0.301114	+	0.953588i	$0.402641\pi$
$97$	1822.00i	1.90718i	−0.301114	+	0.953588i	$0.597359\pi$
$98$	0	0
$99$	234.000	0.237554
$100$	0	0
$101$	−1872.00	−1.84427	−0.922133	−	0.386872i	$-0.873556\pi$
$101$	−1872.00	−1.84427	−0.922133	+	0.386872i	$0.873556\pi$
$102$	0	0
$103$	− 1595.00i	− 1.52583i	−0.646501	−	0.762913i	$-0.723769\pi$
$103$	− 1595.00i	− 1.52583i	0.646501	−	0.762913i	$-0.276231\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1026.00i	0.926983i	0.886101	+	0.463491i	$0.153404\pi$
$107$	1026.00i	0.926983i	−0.886101	+	0.463491i	$0.846596\pi$
$108$	0	0
$109$	588.000	0.516699	0.258349	−	0.966052i	$-0.416821\pi$
$109$	588.000	0.516699	0.258349	+	0.966052i	$0.416821\pi$
$110$	0	0
$111$	−966.000	−0.826024
$112$	0	0
$113$	− 22.0000i	− 0.0183149i	−0.999958	−	0.00915746i	$-0.997085\pi$
$113$	− 22.0000i	− 0.0183149i	0.999958	−	0.00915746i	$-0.00291495\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 351.000i	− 0.277350i
$118$	0	0
$119$	707.000	0.544627
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	1365.00i	1.00063i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1750.00i	− 1.22274i	−0.791347	−	0.611368i	$-0.790620\pi$
$127$	− 1750.00i	− 1.22274i	0.791347	−	0.611368i	$-0.209380\pi$
$128$	0	0
$129$	−921.000	−0.628601
$130$	0	0
$131$	1664.00	1.10980	0.554902	−	0.831916i	$-0.312756\pi$
$131$	1664.00	1.10980	0.554902	+	0.831916i	$0.312756\pi$
$132$	0	0
$133$	336.000i	0.219059i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1080.00i	0.673508i	0.941593	+	0.336754i	$0.109329\pi$
$137$	1080.00i	0.673508i	−0.941593	+	0.336754i	$0.890671\pi$
$138$	0	0
$139$	896.000	0.546746	0.273373	−	0.961908i	$-0.411861\pi$
$139$	896.000	0.546746	0.273373	+	0.961908i	$0.411861\pi$
$140$	0	0
$141$	−1524.00	−0.910241
$142$	0	0
$143$	− 1014.00i	− 0.592972i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	591.000	0.324944	0.162472	−	0.986713i	$-0.448053\pi$
$149$	591.000	0.324944	0.162472	+	0.986713i	$0.448053\pi$
$150$	0	0
$151$	−1606.00	−0.865526	−0.432763	−	0.901508i	$-0.642461\pi$
$151$	−1606.00	−0.865526	−0.432763	+	0.901508i	$0.642461\pi$
$152$	0	0
$153$	909.000i	0.480316i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3554.00i	1.80662i	0.428983	+	0.903312i	$0.358872\pi$
$157$	3554.00i	1.80662i	−0.428983	+	0.903312i	$0.641128\pi$
$158$	0	0
$159$	−93.0000	−0.0463860
$160$	0	0
$161$	399.000	0.195314
$162$	0	0
$163$	2897.00i	1.39209i	0.717999	+	0.696045i	$0.245058\pi$
$163$	2897.00i	1.39209i	−0.717999	+	0.696045i	$0.754942\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 66.0000i	− 0.0305822i	−0.999883	−	0.0152911i	$-0.995132\pi$
$167$	− 66.0000i	− 0.0305822i	0.999883	−	0.0152911i	$-0.00486750\pi$
$168$	0	0
$169$	676.000	0.307692
$170$	0	0
$171$	−432.000	−0.193192
$172$	0	0
$173$	− 1672.00i	− 0.734797i	−0.930064	−	0.367398i	$-0.880249\pi$
$173$	− 1672.00i	− 0.734797i	0.930064	−	0.367398i	$-0.119751\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 633.000i	− 0.268809i
$178$	0	0
$179$	−4710.00	−1.96671	−0.983357	−	0.181682i	$-0.941846\pi$
$179$	−4710.00	−1.96671	−0.983357	+	0.181682i	$0.941846\pi$
$180$	0	0
$181$	2842.00	1.16710	0.583548	−	0.812079i	$-0.301664\pi$
$181$	2842.00	1.16710	0.583548	+	0.812079i	$0.301664\pi$
$182$	0	0
$183$	− 2391.00i	− 0.965835i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2626.00i	1.02691i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−4197.00	−1.58997	−0.794985	−	0.606629i	$-0.792521\pi$
$191$	−4197.00	−1.58997	−0.794985	+	0.606629i	$0.792521\pi$
$192$	0	0
$193$	− 2662.00i	− 0.992824i	−0.868087	−	0.496412i	$-0.834651\pi$
$193$	− 2662.00i	− 0.992824i	0.868087	−	0.496412i	$-0.165349\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3601.00i	1.30234i	0.758933	+	0.651169i	$0.225721\pi$
$197$	3601.00i	1.30234i	−0.758933	+	0.651169i	$0.774279\pi$
$198$	0	0
$199$	1840.00	0.655448	0.327724	−	0.944774i	$-0.393718\pi$
$199$	1840.00	0.655448	0.327724	+	0.944774i	$0.393718\pi$
$200$	0	0
$201$	2772.00	0.972745
$202$	0	0
$203$	2037.00i	0.704283i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	513.000i	0.172251i
$208$	0	0
$209$	−1248.00	−0.413043
$210$	0	0
$211$	−1733.00	−0.565425	−0.282712	−	0.959205i	$-0.591234\pi$
$211$	−1733.00	−0.565425	−0.282712	+	0.959205i	$0.591234\pi$
$212$	0	0
$213$	636.000i	0.204592i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 553.000i	− 0.172996i
$218$	0	0
$219$	66.0000	0.0203647
$220$	0	0
$221$	3939.00	1.19894
$222$	0	0
$223$	− 4727.00i	− 1.41948i	−0.704465	−	0.709738i	$-0.748813\pi$
$223$	− 4727.00i	− 1.41948i	0.704465	−	0.709738i	$-0.251187\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2863.00i	0.837110i	0.908191	+	0.418555i	$0.137463\pi$
$227$	2863.00i	0.837110i	−0.908191	+	0.418555i	$0.862537\pi$
$228$	0	0
$229$	−5578.00	−1.60963	−0.804814	−	0.593528i	$-0.797735\pi$
$229$	−5578.00	−1.60963	−0.804814	+	0.593528i	$0.797735\pi$
$230$	0	0
$231$	546.000	0.155516
$232$	0	0
$233$	− 5016.00i	− 1.41034i	−0.709039	−	0.705170i	$-0.750871\pi$
$233$	− 5016.00i	− 1.41034i	0.709039	−	0.705170i	$-0.249129\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2622.00i	− 0.718638i
$238$	0	0
$239$	−2992.00	−0.809776	−0.404888	−	0.914366i	$-0.632689\pi$
$239$	−2992.00	−0.809776	−0.404888	+	0.914366i	$0.632689\pi$
$240$	0	0
$241$	−1574.00	−0.420706	−0.210353	−	0.977625i	$-0.567461\pi$
$241$	−1574.00	−0.420706	−0.210353	+	0.977625i	$0.567461\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1872.00i	0.482237i
$248$	0	0
$249$	−1761.00	−0.448188
$250$	0	0
$251$	−5697.00	−1.43264	−0.716318	−	0.697774i	$-0.754174\pi$
$251$	−5697.00	−1.43264	−0.716318	+	0.697774i	$0.754174\pi$
$252$	0	0
$253$	1482.00i	0.368271i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2599.00i	0.630822i	0.948955	+	0.315411i	$0.102142\pi$
$257$	2599.00i	0.630822i	−0.948955	+	0.315411i	$0.897858\pi$
$258$	0	0
$259$	−2254.00	−0.540760
$260$	0	0
$261$	−2619.00	−0.621119
$262$	0	0
$263$	1115.00i	0.261421i	0.991421	+	0.130711i	$0.0417259\pi$
$263$	1115.00i	0.261421i	−0.991421	+	0.130711i	$0.958274\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	798.000i	0.182909i
$268$	0	0
$269$	−4194.00	−0.950605	−0.475302	−	0.879822i	$-0.657661\pi$
$269$	−4194.00	−0.950605	−0.475302	+	0.879822i	$0.657661\pi$
$270$	0	0
$271$	−3972.00	−0.890339	−0.445169	−	0.895446i	$-0.646857\pi$
$271$	−3972.00	−0.890339	−0.445169	+	0.895446i	$0.646857\pi$
$272$	0	0
$273$	− 819.000i	− 0.181568i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1842.00i	− 0.399549i	−0.979842	−	0.199774i	$-0.935979\pi$
$277$	− 1842.00i	− 0.399549i	0.979842	−	0.199774i	$-0.0640209\pi$
$278$	0	0
$279$	711.000	0.152568
$280$	0	0
$281$	−2680.00	−0.568952	−0.284476	−	0.958683i	$-0.591820\pi$
$281$	−2680.00	−0.568952	−0.284476	+	0.958683i	$0.591820\pi$
$282$	0	0
$283$	− 2680.00i	− 0.562931i	−0.959571	−	0.281465i	$-0.909180\pi$
$283$	− 2680.00i	− 0.562931i	0.959571	−	0.281465i	$-0.0908205\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3185.00i	0.655068i
$288$	0	0
$289$	−5288.00	−1.07633
$290$	0	0
$291$	−5466.00	−1.10111
$292$	0	0
$293$	5656.00i	1.12774i	0.825864	+	0.563869i	$0.190688\pi$
$293$	5656.00i	1.12774i	−0.825864	+	0.563869i	$0.809312\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	702.000i	0.137152i
$298$	0	0
$299$	2223.00	0.429965
$300$	0	0
$301$	−2149.00	−0.411516
$302$	0	0
$303$	− 5616.00i	− 1.06479i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6998.00i	1.30097i	0.759520	+	0.650484i	$0.225434\pi$
$307$	6998.00i	1.30097i	−0.759520	+	0.650484i	$0.774566\pi$
$308$	0	0
$309$	4785.00	0.880936
$310$	0	0
$311$	5610.00	1.02287	0.511437	−	0.859321i	$-0.329113\pi$
$311$	5610.00	1.02287	0.511437	+	0.859321i	$0.329113\pi$
$312$	0	0
$313$	− 166.000i	− 0.0299772i	−0.999888	−	0.0149886i	$-0.995229\pi$
$313$	− 166.000i	− 0.0299772i	0.999888	−	0.0149886i	$-0.00477120\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5333.00i	0.944893i	0.881359	+	0.472447i	$0.156629\pi$
$317$	5333.00i	0.944893i	−0.881359	+	0.472447i	$0.843371\pi$
$318$	0	0
$319$	−7566.00	−1.32795
$320$	0	0
$321$	−3078.00	−0.535194
$322$	0	0
$323$	− 4848.00i	− 0.835139i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1764.00i	0.298316i
$328$	0	0
$329$	−3556.00	−0.595892
$330$	0	0
$331$	6481.00	1.07622	0.538109	−	0.842875i	$-0.319139\pi$
$331$	6481.00	1.07622	0.538109	+	0.842875i	$0.319139\pi$
$332$	0	0
$333$	− 2898.00i	− 0.476905i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9221.00i	1.49050i	0.666783	+	0.745252i	$0.267671\pi$
$337$	9221.00i	1.49050i	−0.666783	+	0.745252i	$0.732329\pi$
$338$	0	0
$339$	66.0000	0.0105741
$340$	0	0
$341$	2054.00	0.326189
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1910.00i	0.295488i	0.989026	+	0.147744i	$0.0472011\pi$
$347$	1910.00i	0.295488i	−0.989026	+	0.147744i	$0.952799\pi$
$348$	0	0
$349$	−3409.00	−0.522864	−0.261432	−	0.965222i	$-0.584195\pi$
$349$	−3409.00	−0.522864	−0.261432	+	0.965222i	$0.584195\pi$
$350$	0	0
$351$	1053.00	0.160128
$352$	0	0
$353$	− 2130.00i	− 0.321157i	−0.987023	−	0.160579i	$-0.948664\pi$
$353$	− 2130.00i	− 0.321157i	0.987023	−	0.160579i	$-0.0513360\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2121.00i	0.314440i
$358$	0	0
$359$	−647.000	−0.0951180	−0.0475590	−	0.998868i	$-0.515144\pi$
$359$	−647.000	−0.0951180	−0.0475590	+	0.998868i	$0.515144\pi$
$360$	0	0
$361$	−4555.00	−0.664091
$362$	0	0
$363$	− 1965.00i	− 0.284121i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 11503.0i	− 1.63611i	−0.575141	−	0.818054i	$-0.695053\pi$
$367$	− 11503.0i	− 1.63611i	0.575141	−	0.818054i	$-0.304947\pi$
$368$	0	0
$369$	−4095.00	−0.577716
$370$	0	0
$371$	−217.000	−0.0303668
$372$	0	0
$373$	3692.00i	0.512505i	0.966610	+	0.256253i	$0.0824879\pi$
$373$	3692.00i	0.512505i	−0.966610	+	0.256253i	$0.917512\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11349.0i	1.55041i
$378$	0	0
$379$	−8103.00	−1.09821	−0.549107	−	0.835752i	$-0.685032\pi$
$379$	−8103.00	−1.09821	−0.549107	+	0.835752i	$0.685032\pi$
$380$	0	0
$381$	5250.00	0.705947
$382$	0	0
$383$	2664.00i	0.355415i	0.984083	+	0.177708i	$0.0568681\pi$
$383$	2664.00i	0.355415i	−0.984083	+	0.177708i	$0.943132\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2763.00i	− 0.362923i
$388$	0	0
$389$	2050.00	0.267196	0.133598	−	0.991036i	$-0.457347\pi$
$389$	2050.00	0.267196	0.133598	+	0.991036i	$0.457347\pi$
$390$	0	0
$391$	−5757.00	−0.744614
$392$	0	0
$393$	4992.00i	0.640746i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1213.00i	− 0.153347i	−0.997056	−	0.0766735i	$-0.975570\pi$
$397$	− 1213.00i	− 0.153347i	0.997056	−	0.0766735i	$-0.0244299\pi$
$398$	0	0
$399$	−1008.00	−0.126474
$400$	0	0
$401$	10876.0	1.35442	0.677209	−	0.735791i	$-0.263189\pi$
$401$	10876.0	1.35442	0.677209	+	0.735791i	$0.263189\pi$
$402$	0	0
$403$	− 3081.00i	− 0.380833i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8372.00i	− 1.01962i
$408$	0	0
$409$	12534.0	1.51532	0.757661	−	0.652649i	$-0.226342\pi$
$409$	12534.0	1.51532	0.757661	+	0.652649i	$0.226342\pi$
$410$	0	0
$411$	−3240.00	−0.388850
$412$	0	0
$413$	− 1477.00i	− 0.175977i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2688.00i	0.315664i
$418$	0	0
$419$	−4569.00	−0.532721	−0.266361	−	0.963873i	$-0.585821\pi$
$419$	−4569.00	−0.532721	−0.266361	+	0.963873i	$0.585821\pi$
$420$	0	0
$421$	−9262.00	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−9262.00	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	− 4572.00i	− 0.525528i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5579.00i	− 0.632287i
$428$	0	0
$429$	3042.00	0.342352
$430$	0	0
$431$	−9429.00	−1.05378	−0.526890	−	0.849934i	$-0.676642\pi$
$431$	−9429.00	−1.05378	−0.526890	+	0.849934i	$0.676642\pi$
$432$	0	0
$433$	− 418.000i	− 0.0463921i	−0.999731	−	0.0231961i	$-0.992616\pi$
$433$	− 418.000i	− 0.0463921i	0.999731	−	0.0231961i	$-0.00738420\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2736.00i	− 0.299498i
$438$	0	0
$439$	419.000	0.0455530	0.0227765	−	0.999741i	$-0.492749\pi$
$439$	419.000	0.0455530	0.0227765	+	0.999741i	$0.492749\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	9628.00i	1.03260i	0.856409	+	0.516298i	$0.172690\pi$
$443$	9628.00i	1.03260i	−0.856409	+	0.516298i	$0.827310\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1773.00i	0.187606i
$448$	0	0
$449$	−14052.0	−1.47696	−0.738480	−	0.674276i	$-0.764456\pi$
$449$	−14052.0	−1.47696	−0.738480	+	0.674276i	$0.764456\pi$
$450$	0	0
$451$	−11830.0	−1.23515
$452$	0	0
$453$	− 4818.00i	− 0.499712i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12089.0i	1.23742i	0.785621	+	0.618708i	$0.212344\pi$
$457$	12089.0i	1.23742i	−0.785621	+	0.618708i	$0.787656\pi$
$458$	0	0
$459$	−2727.00	−0.277310
$460$	0	0
$461$	−2744.00	−0.277225	−0.138613	−	0.990347i	$-0.544264\pi$
$461$	−2744.00	−0.277225	−0.138613	+	0.990347i	$0.544264\pi$
$462$	0	0
$463$	− 592.000i	− 0.0594224i	−0.999559	−	0.0297112i	$-0.990541\pi$
$463$	− 592.000i	− 0.0594224i	0.999559	−	0.0297112i	$-0.00945876\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3361.00i	0.333038i	0.986038	+	0.166519i	$0.0532526\pi$
$467$	3361.00i	0.333038i	−0.986038	+	0.166519i	$0.946747\pi$
$468$	0	0
$469$	6468.00	0.636811
$470$	0	0
$471$	−10662.0	−1.04306
$472$	0	0
$473$	− 7982.00i	− 0.775925i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 279.000i	− 0.0267810i
$478$	0	0
$479$	−8604.00	−0.820724	−0.410362	−	0.911923i	$-0.634598\pi$
$479$	−8604.00	−0.820724	−0.410362	+	0.911923i	$0.634598\pi$
$480$	0	0
$481$	−12558.0	−1.19043
$482$	0	0
$483$	1197.00i	0.112765i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9826.00i	0.914289i	0.889393	+	0.457144i	$0.151128\pi$
$487$	9826.00i	0.914289i	−0.889393	+	0.457144i	$0.848872\pi$
$488$	0	0
$489$	−8691.00	−0.803723
$490$	0	0
$491$	4758.00	0.437323	0.218661	−	0.975801i	$-0.429831\pi$
$491$	4758.00	0.437323	0.218661	+	0.975801i	$0.429831\pi$
$492$	0	0
$493$	− 29391.0i	− 2.68500i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1484.00i	0.133937i
$498$	0	0
$499$	20631.0	1.85084	0.925421	−	0.378940i	$-0.123711\pi$
$499$	20631.0	1.85084	0.925421	+	0.378940i	$0.123711\pi$
$500$	0	0
$501$	198.000	0.0176567
$502$	0	0
$503$	7566.00i	0.670678i	0.942097	+	0.335339i	$0.108851\pi$
$503$	7566.00i	0.670678i	−0.942097	+	0.335339i	$0.891149\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2028.00i	0.177646i
$508$	0	0
$509$	−12028.0	−1.04741	−0.523705	−	0.851900i	$-0.675451\pi$
$509$	−12028.0	−1.04741	−0.523705	+	0.851900i	$0.675451\pi$
$510$	0	0
$511$	154.000	0.0133318
$512$	0	0
$513$	− 1296.00i	− 0.111540i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13208.0i	− 1.12357i
$518$	0	0
$519$	5016.00	0.424235
$520$	0	0
$521$	21317.0	1.79254	0.896271	−	0.443506i	$-0.146266\pi$
$521$	21317.0	1.79254	0.896271	+	0.443506i	$0.146266\pi$
$522$	0	0
$523$	− 15416.0i	− 1.28890i	−0.764647	−	0.644450i	$-0.777086\pi$
$523$	− 15416.0i	− 1.28890i	0.764647	−	0.644450i	$-0.222914\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7979.00i	0.659527i
$528$	0	0
$529$	8918.00	0.732966
$530$	0	0
$531$	1899.00	0.155197
$532$	0	0
$533$	17745.0i	1.44207i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 14130.0i	− 1.13548i
$538$	0	0
$539$	1274.00	0.101809
$540$	0	0
$541$	17520.0	1.39232	0.696159	−	0.717888i	$-0.254891\pi$
$541$	17520.0	1.39232	0.696159	+	0.717888i	$0.254891\pi$
$542$	0	0
$543$	8526.00i	0.673823i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17379.0i	1.35845i	0.733930	+	0.679225i	$0.237684\pi$
$547$	17379.0i	1.35845i	−0.733930	+	0.679225i	$0.762316\pi$
$548$	0	0
$549$	7173.00	0.557625
$550$	0	0
$551$	13968.0	1.07996
$552$	0	0
$553$	− 6118.00i	− 0.470459i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15830.0i	1.20420i	0.798421	+	0.602099i	$0.205669\pi$
$557$	15830.0i	1.20420i	−0.798421	+	0.602099i	$0.794331\pi$
$558$	0	0
$559$	−11973.0	−0.905910
$560$	0	0
$561$	−7878.00	−0.592887
$562$	0	0
$563$	7343.00i	0.549681i	0.961490	+	0.274841i	$0.0886251\pi$
$563$	7343.00i	0.549681i	−0.961490	+	0.274841i	$0.911375\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	567.000i	0.0419961i
$568$	0	0
$569$	10806.0	0.796153	0.398077	−	0.917352i	$-0.369678\pi$
$569$	10806.0	0.796153	0.398077	+	0.917352i	$0.369678\pi$
$570$	0	0
$571$	−1593.00	−0.116751	−0.0583756	−	0.998295i	$-0.518592\pi$
$571$	−1593.00	−0.116751	−0.0583756	+	0.998295i	$0.518592\pi$
$572$	0	0
$573$	− 12591.0i	− 0.917970i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9314.00i	0.672005i	0.941861	+	0.336003i	$0.109075\pi$
$577$	9314.00i	0.672005i	−0.941861	+	0.336003i	$0.890925\pi$
$578$	0	0
$579$	7986.00	0.573207
$580$	0	0
$581$	−4109.00	−0.293408
$582$	0	0
$583$	− 806.000i	− 0.0572575i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3083.00i	− 0.216779i	−0.994109	−	0.108389i	$-0.965431\pi$
$587$	− 3083.00i	− 0.216779i	0.994109	−	0.108389i	$-0.0345693\pi$
$588$	0	0
$589$	−3792.00	−0.265274
$590$	0	0
$591$	−10803.0	−0.751905
$592$	0	0
$593$	18882.0i	1.30757i	0.756679	+	0.653787i	$0.226821\pi$
$593$	18882.0i	1.30757i	−0.756679	+	0.653787i	$0.773179\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5520.00i	0.378423i
$598$	0	0
$599$	−1707.00	−0.116438	−0.0582188	−	0.998304i	$-0.518542\pi$
$599$	−1707.00	−0.116438	−0.0582188	+	0.998304i	$0.518542\pi$
$600$	0	0
$601$	1440.00	0.0977352	0.0488676	−	0.998805i	$-0.484439\pi$
$601$	1440.00	0.0977352	0.0488676	+	0.998805i	$0.484439\pi$
$602$	0	0
$603$	8316.00i	0.561615i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8284.00i	− 0.553933i	−0.960880	−	0.276966i	$-0.910671\pi$
$607$	− 8284.00i	− 0.553933i	0.960880	−	0.276966i	$-0.0893291\pi$
$608$	0	0
$609$	−6111.00	−0.406618
$610$	0	0
$611$	−19812.0	−1.31180
$612$	0	0
$613$	24042.0i	1.58409i	0.610463	+	0.792045i	$0.290984\pi$
$613$	24042.0i	1.58409i	−0.610463	+	0.792045i	$0.709016\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	862.000i	0.0562444i	0.999604	+	0.0281222i	$0.00895276\pi$
$617$	862.000i	0.0562444i	−0.999604	+	0.0281222i	$0.991047\pi$
$618$	0	0
$619$	−9890.00	−0.642185	−0.321093	−	0.947048i	$-0.604050\pi$
$619$	−9890.00	−0.642185	−0.321093	+	0.947048i	$0.604050\pi$
$620$	0	0
$621$	−1539.00	−0.0994492
$622$	0	0
$623$	1862.00i	0.119742i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3744.00i	− 0.238470i
$628$	0	0
$629$	32522.0	2.06159
$630$	0	0
$631$	−21646.0	−1.36563	−0.682816	−	0.730590i	$-0.739245\pi$
$631$	−21646.0	−1.36563	−0.682816	+	0.730590i	$0.739245\pi$
$632$	0	0
$633$	− 5199.00i	− 0.326448i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1911.00i	− 0.118864i
$638$	0	0
$639$	−1908.00	−0.118121
$640$	0	0
$641$	19658.0	1.21130	0.605651	−	0.795731i	$-0.292913\pi$
$641$	19658.0	1.21130	0.605651	+	0.795731i	$0.292913\pi$
$642$	0	0
$643$	23818.0i	1.46079i	0.683023	+	0.730397i	$0.260665\pi$
$643$	23818.0i	1.46079i	−0.683023	+	0.730397i	$0.739335\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28490.0i	1.73116i	0.500775	+	0.865578i	$0.333049\pi$
$647$	28490.0i	1.73116i	−0.500775	+	0.865578i	$0.666951\pi$
$648$	0	0
$649$	5486.00	0.331809
$650$	0	0
$651$	1659.00	0.0998792
$652$	0	0
$653$	− 11002.0i	− 0.659329i	−0.944098	−	0.329664i	$-0.893064\pi$
$653$	− 11002.0i	− 0.659329i	0.944098	−	0.329664i	$-0.106936\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	198.000i	0.0117576i
$658$	0	0
$659$	−11506.0	−0.680137	−0.340068	−	0.940401i	$-0.610450\pi$
$659$	−11506.0	−0.680137	−0.340068	+	0.940401i	$0.610450\pi$
$660$	0	0
$661$	8398.00	0.494167	0.247083	−	0.968994i	$-0.420528\pi$
$661$	8398.00	0.494167	0.247083	+	0.968994i	$0.420528\pi$
$662$	0	0
$663$	11817.0i	0.692209i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16587.0i	− 0.962895i
$668$	0	0
$669$	14181.0	0.819535
$670$	0	0
$671$	20722.0	1.19220
$672$	0	0
$673$	− 1901.00i	− 0.108883i	−0.998517	−	0.0544414i	$-0.982662\pi$
$673$	− 1901.00i	− 0.108883i	0.998517	−	0.0544414i	$-0.0173378\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19896.0i	1.12949i	0.825265	+	0.564745i	$0.191026\pi$
$677$	19896.0i	1.12949i	−0.825265	+	0.564745i	$0.808974\pi$
$678$	0	0
$679$	−12754.0	−0.720845
$680$	0	0
$681$	−8589.00	−0.483306
$682$	0	0
$683$	15662.0i	0.877437i	0.898624	+	0.438719i	$0.144568\pi$
$683$	15662.0i	0.877437i	−0.898624	+	0.438719i	$0.855432\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16734.0i	− 0.929319i
$688$	0	0
$689$	−1209.00	−0.0668494
$690$	0	0
$691$	1162.00	0.0639719	0.0319859	−	0.999488i	$-0.489817\pi$
$691$	1162.00	0.0639719	0.0319859	+	0.999488i	$0.489817\pi$
$692$	0	0
$693$	1638.00i	0.0897871i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 45955.0i	− 2.49737i
$698$	0	0
$699$	15048.0	0.814260
$700$	0	0
$701$	13501.0	0.727426	0.363713	−	0.931511i	$-0.381509\pi$
$701$	13501.0	0.727426	0.363713	+	0.931511i	$0.381509\pi$
$702$	0	0
$703$	15456.0i	0.829209i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 13104.0i	− 0.697067i
$708$	0	0
$709$	20672.0	1.09500	0.547499	−	0.836806i	$-0.315580\pi$
$709$	20672.0	1.09500	0.547499	+	0.836806i	$0.315580\pi$
$710$	0	0
$711$	7866.00	0.414906
$712$	0	0
$713$	4503.00i	0.236520i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8976.00i	− 0.467524i
$718$	0	0
$719$	7098.00	0.368165	0.184083	−	0.982911i	$-0.441069\pi$
$719$	7098.00	0.368165	0.184083	+	0.982911i	$0.441069\pi$
$720$	0	0
$721$	11165.0	0.576708
$722$	0	0
$723$	− 4722.00i	− 0.242895i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 4053.00i	− 0.206764i	−0.994642	−	0.103382i	$-0.967034\pi$
$727$	− 4053.00i	− 0.206764i	0.994642	−	0.103382i	$-0.0329664\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	31007.0	1.56886
$732$	0	0
$733$	− 10331.0i	− 0.520579i	−0.965531	−	0.260289i	$-0.916182\pi$
$733$	− 10331.0i	− 0.520579i	0.965531	−	0.260289i	$-0.0838180\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24024.0i	1.20073i
$738$	0	0
$739$	8969.00	0.446455	0.223227	−	0.974766i	$-0.428341\pi$
$739$	8969.00	0.446455	0.223227	+	0.974766i	$0.428341\pi$
$740$	0	0
$741$	−5616.00	−0.278420
$742$	0	0
$743$	11613.0i	0.573405i	0.958020	+	0.286702i	$0.0925591\pi$
$743$	11613.0i	0.573405i	−0.958020	+	0.286702i	$0.907441\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5283.00i	− 0.258762i
$748$	0	0
$749$	−7182.00	−0.350367
$750$	0	0
$751$	−32476.0	−1.57798	−0.788992	−	0.614403i	$-0.789397\pi$
$751$	−32476.0	−1.57798	−0.788992	+	0.614403i	$0.789397\pi$
$752$	0	0
$753$	− 17091.0i	− 0.827132i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 29914.0i	− 1.43625i	−0.695913	−	0.718126i	$-0.745000\pi$
$757$	− 29914.0i	− 1.43625i	0.695913	−	0.718126i	$-0.255000\pi$
$758$	0	0
$759$	−4446.00	−0.212621
$760$	0	0
$761$	−3090.00	−0.147191	−0.0735955	−	0.997288i	$-0.523447\pi$
$761$	−3090.00	−0.147191	−0.0735955	+	0.997288i	$0.523447\pi$
$762$	0	0
$763$	4116.00i	0.195294i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8229.00i	− 0.387395i
$768$	0	0
$769$	7760.00	0.363892	0.181946	−	0.983309i	$-0.441760\pi$
$769$	7760.00	0.363892	0.181946	+	0.983309i	$0.441760\pi$
$770$	0	0
$771$	−7797.00	−0.364205
$772$	0	0
$773$	− 34020.0i	− 1.58294i	−0.611207	−	0.791471i	$-0.709316\pi$
$773$	− 34020.0i	− 1.58294i	0.611207	−	0.791471i	$-0.290684\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6762.00i	− 0.312208i
$778$	0	0
$779$	21840.0	1.00449
$780$	0	0
$781$	−5512.00	−0.252541
$782$	0	0
$783$	− 7857.00i	− 0.358603i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 15154.0i	− 0.686381i	−0.939266	−	0.343190i	$-0.888492\pi$
$787$	− 15154.0i	− 0.686381i	0.939266	−	0.343190i	$-0.111508\pi$
$788$	0	0
$789$	−3345.00	−0.150932
$790$	0	0
$791$	154.000	0.00692239
$792$	0	0
$793$	− 31083.0i	− 1.39192i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13386.0i	− 0.594927i	−0.954733	−	0.297463i	$-0.903859\pi$
$797$	− 13386.0i	− 0.594927i	0.954733	−	0.297463i	$-0.0961406\pi$
$798$	0	0
$799$	51308.0	2.27177
$800$	0	0
$801$	−2394.00	−0.105603
$802$	0	0
$803$	572.000i	0.0251375i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 12582.0i	− 0.548832i
$808$	0	0
$809$	−36468.0	−1.58485	−0.792427	−	0.609967i	$-0.791183\pi$
$809$	−36468.0	−1.58485	−0.792427	+	0.609967i	$0.791183\pi$
$810$	0	0
$811$	−38422.0	−1.66360	−0.831800	−	0.555076i	$-0.812689\pi$
$811$	−38422.0	−1.66360	−0.831800	+	0.555076i	$0.812689\pi$
$812$	0	0
$813$	− 11916.0i	− 0.514037i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14736.0i	0.631025i
$818$	0	0
$819$	2457.00	0.104828
$820$	0	0
$821$	19194.0	0.815926	0.407963	−	0.912998i	$-0.366239\pi$
$821$	19194.0	0.815926	0.407963	+	0.912998i	$0.366239\pi$
$822$	0	0
$823$	7426.00i	0.314525i	0.987557	+	0.157263i	$0.0502669\pi$
$823$	7426.00i	0.314525i	−0.987557	+	0.157263i	$0.949733\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 33170.0i	− 1.39472i	−0.716721	−	0.697360i	$-0.754358\pi$
$827$	− 33170.0i	− 1.39472i	0.716721	−	0.697360i	$-0.245642\pi$
$828$	0	0
$829$	−16397.0	−0.686962	−0.343481	−	0.939160i	$-0.611606\pi$
$829$	−16397.0	−0.686962	−0.343481	+	0.939160i	$0.611606\pi$
$830$	0	0
$831$	5526.00	0.230680
$832$	0	0
$833$	4949.00i	0.205850i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2133.00i	0.0880851i
$838$	0	0
$839$	−14996.0	−0.617067	−0.308534	−	0.951213i	$-0.599838\pi$
$839$	−14996.0	−0.617067	−0.308534	+	0.951213i	$0.599838\pi$
$840$	0	0
$841$	60292.0	2.47210
$842$	0	0
$843$	− 8040.00i	− 0.328484i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4585.00i	− 0.186001i
$848$	0	0
$849$	8040.00	0.325008
$850$	0	0
$851$	18354.0	0.739327
$852$	0	0
$853$	21053.0i	0.845066i	0.906348	+	0.422533i	$0.138859\pi$
$853$	21053.0i	0.845066i	−0.906348	+	0.422533i	$0.861141\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6390.00i	− 0.254700i	−0.991858	−	0.127350i	$-0.959353\pi$
$857$	− 6390.00i	− 0.254700i	0.991858	−	0.127350i	$-0.0406472\pi$
$858$	0	0
$859$	−22176.0	−0.880833	−0.440416	−	0.897794i	$-0.645169\pi$
$859$	−22176.0	−0.880833	−0.440416	+	0.897794i	$0.645169\pi$
$860$	0	0
$861$	−9555.00	−0.378204
$862$	0	0
$863$	42472.0i	1.67528i	0.546225	+	0.837638i	$0.316064\pi$
$863$	42472.0i	1.67528i	−0.546225	+	0.837638i	$0.683936\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 15864.0i	− 0.621418i
$868$	0	0
$869$	22724.0	0.887064
$870$	0	0
$871$	36036.0	1.40188
$872$	0	0
$873$	− 16398.0i	− 0.635725i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3472.00i	0.133684i	0.997764	+	0.0668421i	$0.0212924\pi$
$877$	3472.00i	0.133684i	−0.997764	+	0.0668421i	$0.978708\pi$
$878$	0	0
$879$	−16968.0	−0.651099
$880$	0	0
$881$	32639.0	1.24817	0.624084	−	0.781357i	$-0.285472\pi$
$881$	32639.0	1.24817	0.624084	+	0.781357i	$0.285472\pi$
$882$	0	0
$883$	45513.0i	1.73458i	0.497803	+	0.867290i	$0.334140\pi$
$883$	45513.0i	1.73458i	−0.497803	+	0.867290i	$0.665860\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 27626.0i	− 1.04576i	−0.852406	−	0.522881i	$-0.824857\pi$
$887$	− 27626.0i	− 1.04576i	0.852406	−	0.522881i	$-0.175143\pi$
$888$	0	0
$889$	12250.0	0.462151
$890$	0	0
$891$	−2106.00	−0.0791848
$892$	0	0
$893$	24384.0i	0.913751i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6669.00i	0.248240i
$898$	0	0
$899$	−22989.0	−0.852865
$900$	0	0
$901$	3131.00	0.115770
$902$	0	0
$903$	− 6447.00i	− 0.237589i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 11561.0i	− 0.423238i	−0.977352	−	0.211619i	$-0.932126\pi$
$907$	− 11561.0i	− 0.423238i	0.977352	−	0.211619i	$-0.0678736\pi$
$908$	0	0
$909$	16848.0	0.614756
$910$	0	0
$911$	25773.0	0.937319	0.468659	−	0.883379i	$-0.344737\pi$
$911$	25773.0	0.937319	0.468659	+	0.883379i	$0.344737\pi$
$912$	0	0
$913$	− 15262.0i	− 0.553229i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11648.0i	0.419467i
$918$	0	0
$919$	13240.0	0.475242	0.237621	−	0.971358i	$-0.423632\pi$
$919$	13240.0	0.475242	0.237621	+	0.971358i	$0.423632\pi$
$920$	0	0
$921$	−20994.0	−0.751114
$922$	0	0
$923$	8268.00i	0.294848i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14355.0i	0.508608i
$928$	0	0
$929$	30653.0	1.08255	0.541277	−	0.840844i	$-0.317941\pi$
$929$	30653.0	1.08255	0.541277	+	0.840844i	$0.317941\pi$
$930$	0	0
$931$	−2352.00	−0.0827967
$932$	0	0
$933$	16830.0i	0.590557i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19030.0i	0.663482i	0.943370	+	0.331741i	$0.107636\pi$
$937$	19030.0i	0.663482i	−0.943370	+	0.331741i	$0.892364\pi$
$938$	0	0
$939$	498.000	0.0173074
$940$	0	0
$941$	−5320.00	−0.184301	−0.0921504	−	0.995745i	$-0.529374\pi$
$941$	−5320.00	−0.184301	−0.0921504	+	0.995745i	$0.529374\pi$
$942$	0	0
$943$	− 25935.0i	− 0.895610i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14524.0i	0.498381i	0.968455	+	0.249190i	$0.0801645\pi$
$947$	14524.0i	0.498381i	−0.968455	+	0.249190i	$0.919836\pi$
$948$	0	0
$949$	858.000	0.0293486
$950$	0	0
$951$	−15999.0	−0.545534
$952$	0	0
$953$	8688.00i	0.295312i	0.989039	+	0.147656i	$0.0471728\pi$
$953$	8688.00i	0.295312i	−0.989039	+	0.147656i	$0.952827\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 22698.0i	− 0.766690i
$958$	0	0
$959$	−7560.00	−0.254562
$960$	0	0
$961$	−23550.0	−0.790507
$962$	0	0
$963$	− 9234.00i	− 0.308994i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 19672.0i	− 0.654197i	−0.944990	−	0.327099i	$-0.893929\pi$
$967$	− 19672.0i	− 0.654197i	0.944990	−	0.327099i	$-0.106071\pi$
$968$	0	0
$969$	14544.0	0.482168
$970$	0	0
$971$	45996.0	1.52017	0.760083	−	0.649826i	$-0.225158\pi$
$971$	45996.0	1.52017	0.760083	+	0.649826i	$0.225158\pi$
$972$	0	0
$973$	6272.00i	0.206651i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32454.0i	− 1.06274i	−0.847140	−	0.531369i	$-0.821678\pi$
$977$	− 32454.0i	− 1.06274i	0.847140	−	0.531369i	$-0.178322\pi$
$978$	0	0
$979$	−6916.00	−0.225778
$980$	0	0
$981$	−5292.00	−0.172233
$982$	0	0
$983$	− 34048.0i	− 1.10474i	−0.833598	−	0.552372i	$-0.813723\pi$
$983$	− 34048.0i	− 1.10474i	0.833598	−	0.552372i	$-0.186277\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 10668.0i	− 0.344039i
$988$	0	0
$989$	17499.0	0.562625
$990$	0	0
$991$	−14804.0	−0.474535	−0.237268	−	0.971444i	$-0.576252\pi$
$991$	−14804.0	−0.474535	−0.237268	+	0.971444i	$0.576252\pi$
$992$	0	0
$993$	19443.0i	0.621354i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 35546.0i	− 1.12914i	−0.825385	−	0.564570i	$-0.809042\pi$
$997$	− 35546.0i	− 1.12914i	0.825385	−	0.564570i	$-0.190958\pi$
$998$	0	0
$999$	8694.00	0.275341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.d.1849.2		2
5.2	odd	4		2100.4.a.j.1.1	yes	1
5.3	odd	4		2100.4.a.e.1.1	✓	1
5.4	even	2	inner	2100.4.k.d.1849.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.4.a.e.1.1	✓	1	5.3	odd	4
2100.4.a.j.1.1	yes	1	5.2	odd	4
2100.4.k.d.1849.1		2	5.4	even	2	inner
2100.4.k.d.1849.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} -26.0000 q^{11} +39.0000i q^{13} -101.000i q^{17} +48.0000 q^{19} -21.0000 q^{21} -57.0000i q^{23} -27.0000i q^{27} +291.000 q^{29} -79.0000 q^{31} -78.0000i q^{33} +322.000i q^{37} -117.000 q^{39} +455.000 q^{41} +307.000i q^{43} +508.000i q^{47} -49.0000 q^{49} +303.000 q^{51} +31.0000i q^{53} +144.000i q^{57} -211.000 q^{59} -797.000 q^{61} -63.0000i q^{63} -924.000i q^{67} +171.000 q^{69} +212.000 q^{71} -22.0000i q^{73} -182.000i q^{77} -874.000 q^{79} +81.0000 q^{81} +587.000i q^{83} +873.000i q^{87} +266.000 q^{89} -273.000 q^{91} -237.000i q^{93} +1822.00i q^{97} +234.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} - 52 q^{11} + 96 q^{19} - 42 q^{21} + 582 q^{29} - 158 q^{31} - 234 q^{39} + 910 q^{41} - 98 q^{49} + 606 q^{51} - 422 q^{59} - 1594 q^{61} + 342 q^{69} + 424 q^{71} - 1748 q^{79} + 162 q^{81}+ \cdots + 468 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 52 * q^11 + 96 * q^19 - 42 * q^21 + 582 * q^29 - 158 * q^31 - 234 * q^39 + 910 * q^41 - 98 * q^49 + 606 * q^51 - 422 * q^59 - 1594 * q^61 + 342 * q^69 + 424 * q^71 - 1748 * q^79 + 162 * q^81 + 532 * q^89 - 546 * q^91 + 468 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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