LMFDB - Embedded newform 2700.3.g.s.701.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(701,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2700.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.g (of order $2$, degree $1$, 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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.488455618816.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{5} + 105x^{4} - 238x^{3} - 426x^{2} + 548x + 3140 $ x^8 - 4x^7 + 14x^5 + 105x^4 - 238x^3 - 426x^2 + 548x + 3140
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.7
Root			$-1.67945 - 2.15831i$ of defining polynomial
Character	$\chi$	$=$	2700.701
Dual form			2700.3.g.s.701.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.15439 q^{7} +O(q^{10})$	$q+7.15439 q^{7} -5.06984i q^{11} -3.12682 q^{13} -8.72842i q^{17} +20.1852 q^{19} -14.7284i q^{23} -39.7995i q^{29} -39.3705 q^{31} +34.8712 q^{37} +13.2665i q^{41} -66.6156 q^{43} +16.9137i q^{47} +2.18525 q^{49} -4.62950i q^{53} -25.7738i q^{59} -12.1852 q^{61} +106.415 q^{67} -101.487i q^{71} -23.2646 q^{73} -36.2716i q^{77} +66.5557 q^{79} -144.284i q^{83} +154.553i q^{89} -22.3705 q^{91} -175.733 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 12 q^{19} + 32 q^{31} - 156 q^{49} + 76 q^{61} + 12 q^{79} + 168 q^{91}+O(q^{100}) $ 8 * q - 12 * q^19 + 32 * q^31 - 156 * q^49 + 76 * q^61 + 12 * q^79 + 168 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.15439	1.02206	0.511028	−	0.859564i	$-0.329265\pi$
$7$	7.15439	1.02206	0.511028	+	0.859564i	$0.329265\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.06984i	− 0.460894i	−0.973085	−	0.230447i	$-0.925981\pi$
$11$	− 5.06984i	− 0.460894i	0.973085	−	0.230447i	$-0.0740189\pi$
$12$	0	0
$13$	−3.12682	−0.240525	−0.120262	−	0.992742i	$-0.538374\pi$
$13$	−3.12682	−0.240525	−0.120262	+	0.992742i	$0.538374\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 8.72842i	− 0.513436i	−0.966486	−	0.256718i	$-0.917359\pi$
$17$	− 8.72842i	− 0.513436i	0.966486	−	0.256718i	$-0.0826412\pi$
$18$	0	0
$19$	20.1852	1.06238	0.531191	−	0.847252i	$-0.321745\pi$
$19$	20.1852	1.06238	0.531191	+	0.847252i	$0.321745\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.7284i	− 0.640366i	−0.947356	−	0.320183i	$-0.896256\pi$
$23$	− 14.7284i	− 0.640366i	0.947356	−	0.320183i	$-0.103744\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 39.7995i	− 1.37240i	−0.727415	−	0.686198i	$-0.759278\pi$
$29$	− 39.7995i	− 1.37240i	0.727415	−	0.686198i	$-0.240722\pi$
$30$	0	0
$31$	−39.3705	−1.27002	−0.635008	−	0.772506i	$-0.719003\pi$
$31$	−39.3705	−1.27002	−0.635008	+	0.772506i	$0.719003\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	34.8712	0.942465	0.471232	−	0.882009i	$-0.343809\pi$
$37$	34.8712	0.942465	0.471232	+	0.882009i	$0.343809\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	13.2665i	0.323573i	0.986826	+	0.161787i	$0.0517256\pi$
$41$	13.2665i	0.323573i	−0.986826	+	0.161787i	$0.948274\pi$
$42$	0	0
$43$	−66.6156	−1.54920	−0.774600	−	0.632452i	$-0.782049\pi$
$43$	−66.6156	−1.54920	−0.774600	+	0.632452i	$0.782049\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16.9137i	0.359865i	0.983679	+	0.179933i	$0.0575880\pi$
$47$	16.9137i	0.359865i	−0.983679	+	0.179933i	$0.942412\pi$
$48$	0	0
$49$	2.18525	0.0445969
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.62950i	− 0.0873491i	−0.999046	−	0.0436746i	$-0.986094\pi$
$53$	− 4.62950i	− 0.0873491i	0.999046	−	0.0436746i	$-0.0139065\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 25.7738i	− 0.436844i	−0.975854	−	0.218422i	$-0.929909\pi$
$59$	− 25.7738i	− 0.436844i	0.975854	−	0.218422i	$-0.0700909\pi$
$60$	0	0
$61$	−12.1852	−0.199758	−0.0998791	−	0.995000i	$-0.531846\pi$
$61$	−12.1852	−0.199758	−0.0998791	+	0.995000i	$0.531846\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	106.415	1.58828	0.794142	−	0.607732i	$-0.207920\pi$
$67$	106.415	1.58828	0.794142	+	0.607732i	$0.207920\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 101.487i	− 1.42939i	−0.699436	−	0.714695i	$-0.746565\pi$
$71$	− 101.487i	− 1.42939i	0.699436	−	0.714695i	$-0.253435\pi$
$72$	0	0
$73$	−23.2646	−0.318694	−0.159347	−	0.987223i	$-0.550939\pi$
$73$	−23.2646	−0.318694	−0.159347	+	0.987223i	$0.550939\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 36.2716i	− 0.471060i
$78$	0	0
$79$	66.5557	0.842478	0.421239	−	0.906950i	$-0.361595\pi$
$79$	66.5557	0.842478	0.421239	+	0.906950i	$0.361595\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 144.284i	− 1.73836i	−0.494493	−	0.869182i	$-0.664646\pi$
$83$	− 144.284i	− 1.73836i	0.494493	−	0.869182i	$-0.335354\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	154.553i	1.73655i	0.496085	+	0.868274i	$0.334770\pi$
$89$	154.553i	1.73655i	−0.496085	+	0.868274i	$0.665230\pi$
$90$	0	0
$91$	−22.3705	−0.245830
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−175.733	−1.81168	−0.905839	−	0.423621i	$-0.860759\pi$
$97$	−175.733	−1.81168	−0.905839	+	0.423621i	$0.860759\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	149.483i	1.48003i	0.672591	+	0.740014i	$0.265181\pi$
$101$	149.483i	1.48003i	−0.672591	+	0.740014i	$0.734819\pi$
$102$	0	0
$103$	−28.6175	−0.277840	−0.138920	−	0.990304i	$-0.544363\pi$
$103$	−28.6175	−0.277840	−0.138920	+	0.990304i	$0.544363\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 156.297i	− 1.46072i	−0.683064	−	0.730359i	$-0.739353\pi$
$107$	− 156.297i	− 1.46072i	0.683064	−	0.730359i	$-0.260647\pi$
$108$	0	0
$109$	−8.55575	−0.0784931	−0.0392465	−	0.999230i	$-0.512496\pi$
$109$	−8.55575	−0.0784931	−0.0392465	+	0.999230i	$0.512496\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 184.370i	− 1.63160i	−0.578336	−	0.815799i	$-0.696298\pi$
$113$	− 184.370i	− 1.63160i	0.578336	−	0.815799i	$-0.303702\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 62.4465i	− 0.524760i
$120$	0	0
$121$	95.2967	0.787576
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	40.7002	0.320474	0.160237	−	0.987079i	$-0.448774\pi$
$127$	40.7002	0.320474	0.160237	+	0.987079i	$0.448774\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	80.0236i	0.610867i	0.952213	+	0.305434i	$0.0988014\pi$
$131$	80.0236i	0.610867i	−0.952213	+	0.305434i	$0.901199\pi$
$132$	0	0
$133$	144.413	1.08581
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.0738i	− 0.0954289i	−0.998861	−	0.0477144i	$-0.984806\pi$
$137$	− 13.0738i	− 0.0954289i	0.998861	−	0.0477144i	$-0.0151937\pi$
$138$	0	0
$139$	−144.370	−1.03864	−0.519318	−	0.854581i	$-0.673814\pi$
$139$	−144.370	−1.03864	−0.519318	+	0.854581i	$0.673814\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.8525i	0.110857i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	54.2498i	0.364093i	0.983290	+	0.182046i	$0.0582721\pi$
$149$	54.2498i	0.364093i	−0.983290	+	0.182046i	$0.941728\pi$
$150$	0	0
$151$	21.1852	0.140300	0.0701498	−	0.997536i	$-0.477652\pi$
$151$	21.1852	0.140300	0.0701498	+	0.997536i	$0.477652\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−105.939	−0.674770	−0.337385	−	0.941367i	$-0.609542\pi$
$157$	−105.939	−0.674770	−0.337385	+	0.941367i	$0.609542\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 105.373i	− 0.654489i
$162$	0	0
$163$	154.746	0.949361	0.474680	−	0.880158i	$-0.342564\pi$
$163$	154.746	0.949361	0.474680	+	0.880158i	$0.342564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	151.667i	0.908187i	0.890954	+	0.454094i	$0.150037\pi$
$167$	151.667i	0.908187i	−0.890954	+	0.454094i	$0.849963\pi$
$168$	0	0
$169$	−159.223	−0.942148
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 311.198i	− 1.79883i	−0.437094	−	0.899416i	$-0.643992\pi$
$173$	− 311.198i	− 1.79883i	0.437094	−	0.899416i	$-0.356008\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	211.170i	1.17972i	0.807505	+	0.589861i	$0.200817\pi$
$179$	211.170i	1.17972i	−0.807505	+	0.589861i	$0.799183\pi$
$180$	0	0
$181$	149.297	0.824844	0.412422	−	0.910993i	$-0.364683\pi$
$181$	149.297	0.824844	0.412422	+	0.910993i	$0.364683\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−44.2517	−0.236640
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 24.1654i	− 0.126520i	−0.997997	−	0.0632602i	$-0.979850\pi$
$191$	− 24.1654i	− 0.126520i	0.997997	−	0.0632602i	$-0.0201498\pi$
$192$	0	0
$193$	332.653	1.72359	0.861796	−	0.507255i	$-0.169340\pi$
$193$	332.653	1.72359	0.861796	+	0.507255i	$0.169340\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 279.568i	− 1.41913i	−0.704641	−	0.709564i	$-0.748892\pi$
$197$	− 279.568i	− 1.41913i	0.704641	−	0.709564i	$-0.251108\pi$
$198$	0	0
$199$	161.185	0.809976	0.404988	−	0.914322i	$-0.367276\pi$
$199$	161.185	0.809976	0.404988	+	0.914322i	$0.367276\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 284.741i	− 1.40266i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 102.336i	− 0.489646i
$210$	0	0
$211$	−92.1852	−0.436897	−0.218448	−	0.975848i	$-0.570100\pi$
$211$	−92.1852	−0.436897	−0.218448	+	0.975848i	$0.570100\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−281.672	−1.29803
$218$	0	0
$219$	0	0
$220$	0	0
$221$	27.2922i	0.123494i
$222$	0	0
$223$	−197.672	−0.886422	−0.443211	−	0.896417i	$-0.646161\pi$
$223$	−197.672	−0.886422	−0.443211	+	0.896417i	$0.646161\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 32.7284i	− 0.144178i	−0.997398	−	0.0720890i	$-0.977033\pi$
$227$	− 32.7284i	− 0.144178i	0.997398	−	0.0720890i	$-0.0229666\pi$
$228$	0	0
$229$	−350.926	−1.53243	−0.766215	−	0.642585i	$-0.777862\pi$
$229$	−350.926	−1.53243	−0.766215	+	0.642585i	$0.777862\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 399.827i	− 1.71600i	−0.513652	−	0.857999i	$-0.671708\pi$
$233$	− 399.827i	− 1.71600i	0.513652	−	0.857999i	$-0.328292\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 363.690i	− 1.52172i	−0.648919	−	0.760858i	$-0.724779\pi$
$239$	− 363.690i	− 1.52172i	0.648919	−	0.760858i	$-0.275221\pi$
$240$	0	0
$241$	227.667	0.944677	0.472339	−	0.881417i	$-0.343410\pi$
$241$	227.667	0.944677	0.472339	+	0.881417i	$0.343410\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−63.1157	−0.255529
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 227.988i	− 0.908319i	−0.890920	−	0.454160i	$-0.849940\pi$
$251$	− 227.988i	− 0.908319i	0.890920	−	0.454160i	$-0.150060\pi$
$252$	0	0
$253$	−74.6707	−0.295141
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 456.038i	− 1.77447i	−0.461322	−	0.887233i	$-0.652625\pi$
$257$	− 456.038i	− 1.77447i	0.461322	−	0.887233i	$-0.347375\pi$
$258$	0	0
$259$	249.482	0.963251
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 186.543i	− 0.709290i	−0.935001	−	0.354645i	$-0.884602\pi$
$263$	− 186.543i	− 0.709290i	0.935001	−	0.354645i	$-0.115398\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 337.157i	− 1.25337i	−0.779272	−	0.626686i	$-0.784411\pi$
$269$	− 337.157i	− 1.25337i	0.779272	−	0.626686i	$-0.215589\pi$
$270$	0	0
$271$	58.2590	0.214978	0.107489	−	0.994206i	$-0.465719\pi$
$271$	58.2590	0.214978	0.107489	+	0.994206i	$0.465719\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	332.229	1.19938	0.599691	−	0.800232i	$-0.295290\pi$
$277$	332.229	1.19938	0.599691	+	0.800232i	$0.295290\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	112.296i	0.399629i	0.979834	+	0.199814i	$0.0640339\pi$
$281$	112.296i	0.399629i	−0.979834	+	0.199814i	$0.935966\pi$
$282$	0	0
$283$	−325.975	−1.15185	−0.575927	−	0.817501i	$-0.695359\pi$
$283$	−325.975	−1.15185	−0.575927	+	0.817501i	$0.695359\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	94.9137i	0.330710i
$288$	0	0
$289$	212.815	0.736383
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 71.9623i	− 0.245605i	−0.992431	−	0.122803i	$-0.960812\pi$
$293$	− 71.9623i	− 0.245605i	0.992431	−	0.122803i	$-0.0391882\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	46.0531i	0.154024i
$300$	0	0
$301$	−476.593	−1.58337
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	245.900	0.800977	0.400488	−	0.916302i	$-0.368841\pi$
$307$	245.900	0.800977	0.400488	+	0.916302i	$0.368841\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 69.3693i	− 0.223053i	−0.993761	−	0.111526i	$-0.964426\pi$
$311$	− 69.3693i	− 0.223053i	0.993761	−	0.111526i	$-0.0355739\pi$
$312$	0	0
$313$	218.659	0.698592	0.349296	−	0.937013i	$-0.386421\pi$
$313$	218.659	0.698592	0.349296	+	0.937013i	$0.386421\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 74.4317i	− 0.234800i	−0.993085	−	0.117400i	$-0.962544\pi$
$317$	− 74.4317i	− 0.234800i	0.993085	−	0.117400i	$-0.0374560\pi$
$318$	0	0
$319$	−201.777	−0.632530
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 176.185i	− 0.545465i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	121.007i	0.367802i
$330$	0	0
$331$	−220.223	−0.665326	−0.332663	−	0.943046i	$-0.607947\pi$
$331$	−220.223	−0.665326	−0.332663	+	0.943046i	$0.607947\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.6415	0.0731203	0.0365601	−	0.999331i	$-0.488360\pi$
$337$	24.6415	0.0731203	0.0365601	+	0.999331i	$0.488360\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	199.602i	0.585343i
$342$	0	0
$343$	−334.931	−0.976475
$344$	0	0
$345$	0	0
$346$	0	0
$347$	147.556i	0.425233i	0.977136	+	0.212616i	$0.0681985\pi$
$347$	147.556i	0.425233i	−0.977136	+	0.212616i	$0.931802\pi$
$348$	0	0
$349$	503.149	1.44169	0.720844	−	0.693097i	$-0.243754\pi$
$349$	503.149	1.44169	0.720844	+	0.693097i	$0.243754\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 236.716i	− 0.670583i	−0.942114	−	0.335292i	$-0.891165\pi$
$353$	− 236.716i	− 0.670583i	0.942114	−	0.335292i	$-0.108835\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	195.871i	0.545601i	0.962071	+	0.272800i	$0.0879499\pi$
$359$	195.871i	0.545601i	−0.962071	+	0.272800i	$0.912050\pi$
$360$	0	0
$361$	46.4443	0.128654
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	339.756	0.925766	0.462883	−	0.886419i	$-0.346815\pi$
$367$	339.756	0.925766	0.462883	+	0.886419i	$0.346815\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 33.1213i	− 0.0892756i
$372$	0	0
$373$	−540.980	−1.45035	−0.725174	−	0.688566i	$-0.758241\pi$
$373$	−540.980	−1.45035	−0.725174	+	0.688566i	$0.758241\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	124.446i	0.330095i
$378$	0	0
$379$	432.926	1.14229	0.571143	−	0.820851i	$-0.306500\pi$
$379$	432.926	1.14229	0.571143	+	0.820851i	$0.306500\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 478.977i	− 1.25059i	−0.780388	−	0.625296i	$-0.784978\pi$
$383$	− 478.977i	− 1.25059i	0.780388	−	0.625296i	$-0.215022\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	339.190i	0.871954i	0.899958	+	0.435977i	$0.143597\pi$
$389$	339.190i	0.871954i	−0.899958	+	0.435977i	$0.856403\pi$
$390$	0	0
$391$	−128.556	−0.328787
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	113.042	0.284740	0.142370	−	0.989814i	$-0.454528\pi$
$397$	113.042	0.284740	0.142370	+	0.989814i	$0.454528\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 477.774i	− 1.19146i	−0.803186	−	0.595728i	$-0.796864\pi$
$401$	− 477.774i	− 1.19146i	0.803186	−	0.595728i	$-0.203136\pi$
$402$	0	0
$403$	123.105	0.305470
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 176.791i	− 0.434377i
$408$	0	0
$409$	−677.370	−1.65616	−0.828081	−	0.560608i	$-0.810567\pi$
$409$	−677.370	−1.65616	−0.828081	+	0.560608i	$0.810567\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 184.396i	− 0.446479i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	344.170i	0.821408i	0.911769	+	0.410704i	$0.134717\pi$
$419$	344.170i	0.821408i	−0.911769	+	0.410704i	$0.865283\pi$
$420$	0	0
$421$	29.5918	0.0702892	0.0351446	−	0.999382i	$-0.488811\pi$
$421$	29.5918	0.0702892	0.0351446	+	0.999382i	$0.488811\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−87.1780	−0.204164
$428$	0	0
$429$	0	0
$430$	0	0
$431$	724.253i	1.68040i	0.542276	+	0.840201i	$0.317563\pi$
$431$	724.253i	1.68040i	−0.542276	+	0.840201i	$0.682437\pi$
$432$	0	0
$433$	−96.6100	−0.223118	−0.111559	−	0.993758i	$-0.535584\pi$
$433$	−96.6100	−0.223118	−0.111559	+	0.993758i	$0.535584\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 297.297i	− 0.680313i
$438$	0	0
$439$	−101.223	−0.230576	−0.115288	−	0.993332i	$-0.536779\pi$
$439$	−101.223	−0.230576	−0.115288	+	0.993332i	$0.536779\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	377.520i	0.852189i	0.904679	+	0.426094i	$0.140111\pi$
$443$	377.520i	0.852189i	−0.904679	+	0.426094i	$0.859889\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 654.974i	− 1.45874i	−0.684120	−	0.729369i	$-0.739814\pi$
$449$	− 654.974i	− 1.45874i	0.684120	−	0.729369i	$-0.260186\pi$
$450$	0	0
$451$	67.2590	0.149133
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	321.574	0.703664	0.351832	−	0.936063i	$-0.385559\pi$
$457$	321.574	0.703664	0.351832	+	0.936063i	$0.385559\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 465.511i	− 1.00979i	−0.863182	−	0.504893i	$-0.831532\pi$
$461$	− 465.511i	− 1.00979i	0.863182	−	0.504893i	$-0.168468\pi$
$462$	0	0
$463$	250.880	0.541857	0.270928	−	0.962600i	$-0.412669\pi$
$463$	250.880	0.541857	0.270928	+	0.962600i	$0.412669\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	246.259i	0.527321i	0.964616	+	0.263661i	$0.0849299\pi$
$467$	246.259i	0.527321i	−0.964616	+	0.263661i	$0.915070\pi$
$468$	0	0
$469$	761.334	1.62331
$470$	0	0
$471$	0	0
$472$	0	0
$473$	337.730i	0.714017i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	175.012i	0.365370i	0.983172	+	0.182685i	$0.0584788\pi$
$479$	175.012i	0.365370i	−0.983172	+	0.182685i	$0.941521\pi$
$480$	0	0
$481$	−109.036	−0.226686
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−28.9906	−0.0595289	−0.0297645	−	0.999557i	$-0.509476\pi$
$487$	−28.9906	−0.0595289	−0.0297645	+	0.999557i	$0.509476\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	974.373i	1.98447i	0.124388	+	0.992234i	$0.460303\pi$
$491$	974.373i	1.98447i	−0.124388	+	0.992234i	$0.539697\pi$
$492$	0	0
$493$	−347.387	−0.704638
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 726.075i	− 1.46092i
$498$	0	0
$499$	−626.185	−1.25488	−0.627440	−	0.778665i	$-0.715897\pi$
$499$	−626.185	−1.25488	−0.627440	+	0.778665i	$0.715897\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 71.5197i	− 0.142186i	−0.997470	−	0.0710932i	$-0.977351\pi$
$503$	− 71.5197i	− 0.142186i	0.997470	−	0.0710932i	$-0.0226488\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	932.296i	1.83162i	0.401608	+	0.915812i	$0.368451\pi$
$509$	932.296i	1.83162i	−0.401608	+	0.915812i	$0.631549\pi$
$510$	0	0
$511$	−166.444	−0.325723
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	85.7495	0.165860
$518$	0	0
$519$	0	0
$520$	0	0
$521$	373.070i	0.716066i	0.933709	+	0.358033i	$0.116552\pi$
$521$	373.070i	0.716066i	−0.933709	+	0.358033i	$0.883448\pi$
$522$	0	0
$523$	828.480	1.58409	0.792046	−	0.610461i	$-0.209016\pi$
$523$	828.480	1.58409	0.792046	+	0.610461i	$0.209016\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	343.642i	0.652072i
$528$	0	0
$529$	312.074	0.589931
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 41.4820i	− 0.0778274i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 11.0789i	− 0.0205545i
$540$	0	0
$541$	−50.5935	−0.0935184	−0.0467592	−	0.998906i	$-0.514889\pi$
$541$	−50.5935	−0.0935184	−0.0467592	+	0.998906i	$0.514889\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−485.070	−0.886782	−0.443391	−	0.896328i	$-0.646225\pi$
$547$	−485.070	−0.886782	−0.443391	+	0.896328i	$0.646225\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 803.363i	− 1.45801i
$552$	0	0
$553$	476.166	0.861059
$554$	0	0
$555$	0	0
$556$	0	0
$557$	198.372i	0.356144i	0.984017	+	0.178072i	$0.0569860\pi$
$557$	198.372i	0.356144i	−0.984017	+	0.178072i	$0.943014\pi$
$558$	0	0
$559$	208.295	0.372621
$560$	0	0
$561$	0	0
$562$	0	0
$563$	660.840i	1.17378i	0.809666	+	0.586892i	$0.199649\pi$
$563$	660.840i	1.17378i	−0.809666	+	0.586892i	$0.800351\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 162.994i	− 0.286457i	−0.989690	−	0.143229i	$-0.954252\pi$
$569$	− 162.994i	− 0.286457i	0.989690	−	0.143229i	$-0.0457484\pi$
$570$	0	0
$571$	262.631	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$571$	262.631	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−532.551	−0.922966	−0.461483	−	0.887149i	$-0.652682\pi$
$577$	−532.551	−0.922966	−0.461483	+	0.887149i	$0.652682\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1032.26i	− 1.77670i
$582$	0	0
$583$	−23.4708	−0.0402587
$584$	0	0
$585$	0	0
$586$	0	0
$587$	927.593i	1.58023i	0.612960	+	0.790114i	$0.289979\pi$
$587$	927.593i	1.58023i	−0.612960	+	0.790114i	$0.710021\pi$
$588$	0	0
$589$	−794.703	−1.34924
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 890.336i	− 1.50141i	−0.660638	−	0.750705i	$-0.729714\pi$
$593$	− 890.336i	− 1.50141i	0.660638	−	0.750705i	$-0.270286\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1036.76i	− 1.73081i	−0.501073	−	0.865405i	$-0.667061\pi$
$599$	− 1036.76i	− 1.73081i	0.501073	−	0.865405i	$-0.332939\pi$
$600$	0	0
$601$	865.816	1.44063	0.720313	−	0.693649i	$-0.243998\pi$
$601$	865.816	1.44063	0.720313	+	0.693649i	$0.243998\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−708.606	−1.16739	−0.583695	−	0.811973i	$-0.698394\pi$
$607$	−708.606	−1.16739	−0.583695	+	0.811973i	$0.698394\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 52.8860i	− 0.0865565i
$612$	0	0
$613$	−677.814	−1.10573	−0.552866	−	0.833270i	$-0.686466\pi$
$613$	−677.814	−1.10573	−0.552866	+	0.833270i	$0.686466\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 474.063i	− 0.768335i	−0.923263	−	0.384168i	$-0.874488\pi$
$617$	− 474.063i	− 0.768335i	0.923263	−	0.384168i	$-0.125512\pi$
$618$	0	0
$619$	−323.187	−0.522111	−0.261056	−	0.965324i	$-0.584071\pi$
$619$	−323.187	−0.522111	−0.261056	+	0.965324i	$0.584071\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1105.73i	1.77485i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 304.370i	− 0.483896i
$630$	0	0
$631$	977.669	1.54940	0.774698	−	0.632331i	$-0.217902\pi$
$631$	977.669	1.54940	0.774698	+	0.632331i	$0.217902\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.83288	−0.0107267
$638$	0	0
$639$	0	0
$640$	0	0
$641$	245.231i	0.382575i	0.981534	+	0.191288i	$0.0612663\pi$
$641$	245.231i	0.382575i	−0.981534	+	0.191288i	$0.938734\pi$
$642$	0	0
$643$	244.471	0.380204	0.190102	−	0.981764i	$-0.439118\pi$
$643$	244.471	0.380204	0.190102	+	0.981764i	$0.439118\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	866.779i	1.33969i	0.742501	+	0.669844i	$0.233639\pi$
$647$	866.779i	1.33969i	−0.742501	+	0.669844i	$0.766361\pi$
$648$	0	0
$649$	−130.669	−0.201339
$650$	0	0
$651$	0	0
$652$	0	0
$653$	641.198i	0.981926i	0.871180	+	0.490963i	$0.163355\pi$
$653$	641.198i	0.981926i	−0.871180	+	0.490963i	$0.836645\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	771.490i	1.17070i	0.810781	+	0.585349i	$0.199042\pi$
$659$	771.490i	1.17070i	−0.810781	+	0.585349i	$0.800958\pi$
$660$	0	0
$661$	−796.075	−1.20435	−0.602175	−	0.798364i	$-0.705699\pi$
$661$	−796.075	−1.20435	−0.602175	+	0.798364i	$0.705699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−586.184	−0.878836
$668$	0	0
$669$	0	0
$670$	0	0
$671$	61.7772i	0.0920674i
$672$	0	0
$673$	698.325	1.03763	0.518815	−	0.854887i	$-0.326373\pi$
$673$	698.325	1.03763	0.518815	+	0.854887i	$0.326373\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 433.655i	− 0.640553i	−0.947324	−	0.320277i	$-0.896224\pi$
$677$	− 433.655i	− 0.640553i	0.947324	−	0.320277i	$-0.103776\pi$
$678$	0	0
$679$	−1257.26	−1.85164
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1112.83i	1.62933i	0.579935	+	0.814663i	$0.303078\pi$
$683$	1112.83i	1.62933i	−0.579935	+	0.814663i	$0.696922\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.4756i	0.0210096i
$690$	0	0
$691$	497.743	0.720322	0.360161	−	0.932890i	$-0.382722\pi$
$691$	497.743	0.720322	0.360161	+	0.932890i	$0.382722\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	115.796	0.166134
$698$	0	0
$699$	0	0
$700$	0	0
$701$	660.713i	0.942529i	0.881992	+	0.471264i	$0.156202\pi$
$701$	660.713i	0.942529i	−0.881992	+	0.471264i	$0.843798\pi$
$702$	0	0
$703$	703.884	1.00126
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1069.46i	1.51267i
$708$	0	0
$709$	1013.56	1.42956	0.714780	−	0.699350i	$-0.246527\pi$
$709$	1013.56	1.42956	0.714780	+	0.699350i	$0.246527\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	579.865i	0.813275i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1067.57i	− 1.48480i	−0.669955	−	0.742402i	$-0.733687\pi$
$719$	− 1067.57i	− 1.48480i	0.669955	−	0.742402i	$-0.266313\pi$
$720$	0	0
$721$	−204.741	−0.283968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−878.561	−1.20847	−0.604237	−	0.796804i	$-0.706522\pi$
$727$	−878.561	−1.20847	−0.604237	+	0.796804i	$0.706522\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	581.448i	0.795415i
$732$	0	0
$733$	−76.9483	−0.104977	−0.0524886	−	0.998622i	$-0.516715\pi$
$733$	−76.9483	−0.104977	−0.0524886	+	0.998622i	$0.516715\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 539.507i	− 0.732031i
$738$	0	0
$739$	613.815	0.830602	0.415301	−	0.909684i	$-0.363676\pi$
$739$	613.815	0.830602	0.415301	+	0.909684i	$0.363676\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	331.036i	0.445540i	0.974871	+	0.222770i	$0.0715098\pi$
$743$	331.036i	0.445540i	−0.974871	+	0.222770i	$0.928490\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1118.21i	− 1.49293i
$750$	0	0
$751$	−496.852	−0.661588	−0.330794	−	0.943703i	$-0.607317\pi$
$751$	−496.852	−0.661588	−0.330794	+	0.943703i	$0.607317\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1234.26	1.63046	0.815232	−	0.579135i	$-0.196610\pi$
$757$	1234.26	1.63046	0.815232	+	0.579135i	$0.196610\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 69.3693i	− 0.0911555i	−0.998961	−	0.0455777i	$-0.985487\pi$
$761$	− 69.3693i	− 0.0911555i	0.998961	−	0.0455777i	$-0.0145129\pi$
$762$	0	0
$763$	−61.2111	−0.0802243
$764$	0	0
$765$	0	0
$766$	0	0
$767$	80.5901i	0.105072i
$768$	0	0
$769$	−1110.85	−1.44454	−0.722271	−	0.691610i	$-0.756902\pi$
$769$	−1110.85	−1.44454	−0.722271	+	0.691610i	$0.756902\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	733.074i	0.948349i	0.880431	+	0.474174i	$0.157253\pi$
$773$	733.074i	0.948349i	−0.880431	+	0.474174i	$0.842747\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	267.788i	0.343758i
$780$	0	0
$781$	−514.521	−0.658798
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1004.84	1.27680	0.638402	−	0.769703i	$-0.279596\pi$
$787$	1004.84	1.27680	0.638402	+	0.769703i	$0.279596\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1319.06i	− 1.66758i
$792$	0	0
$793$	38.1011	0.0480468
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 621.101i	− 0.779298i	−0.920963	−	0.389649i	$-0.872596\pi$
$797$	− 621.101i	− 0.779298i	0.920963	−	0.389649i	$-0.127404\pi$
$798$	0	0
$799$	147.630	0.184768
$800$	0	0
$801$	0	0
$802$	0	0
$803$	117.948i	0.146884i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	276.049i	0.341222i	0.985338	+	0.170611i	$0.0545742\pi$
$809$	276.049i	0.341222i	−0.985338	+	0.170611i	$0.945426\pi$
$810$	0	0
$811$	1024.08	1.26273	0.631366	−	0.775485i	$-0.282495\pi$
$811$	1024.08	1.26273	0.631366	+	0.775485i	$0.282495\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1344.65	−1.64584
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 6.43362i	− 0.00783632i	−0.999992	−	0.00391816i	$-0.998753\pi$
$821$	− 6.43362i	− 0.00783632i	0.999992	−	0.00391816i	$-0.00124719\pi$
$822$	0	0
$823$	282.727	0.343532	0.171766	−	0.985138i	$-0.445053\pi$
$823$	282.727	0.343532	0.171766	+	0.985138i	$0.445053\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	223.567i	0.270334i	0.990823	+	0.135167i	$0.0431572\pi$
$827$	223.567i	0.270334i	−0.990823	+	0.135167i	$0.956843\pi$
$828$	0	0
$829$	1355.63	1.63526	0.817631	−	0.575742i	$-0.195287\pi$
$829$	1355.63	1.63526	0.817631	+	0.575742i	$0.195287\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 19.0738i	− 0.0228977i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	643.715i	0.767241i	0.923491	+	0.383620i	$0.125323\pi$
$839$	643.715i	0.767241i	−0.923491	+	0.383620i	$0.874677\pi$
$840$	0	0
$841$	−743.000	−0.883472
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	681.790	0.804947
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 513.597i	− 0.603522i
$852$	0	0
$853$	−964.735	−1.13099	−0.565495	−	0.824751i	$-0.691315\pi$
$853$	−964.735	−1.13099	−0.565495	+	0.824751i	$0.691315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	455.077i	0.531012i	0.964109	+	0.265506i	$0.0855390\pi$
$857$	455.077i	0.531012i	−0.964109	+	0.265506i	$0.914461\pi$
$858$	0	0
$859$	−1086.78	−1.26517	−0.632584	−	0.774492i	$-0.718006\pi$
$859$	−1086.78	−1.26517	−0.632584	+	0.774492i	$0.718006\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1273.67i	1.47586i	0.674877	+	0.737930i	$0.264197\pi$
$863$	1273.67i	1.47586i	−0.674877	+	0.737930i	$0.735803\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 337.427i	− 0.388293i
$870$	0	0
$871$	−332.741	−0.382022
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1169.03	1.33299	0.666496	−	0.745509i	$-0.267793\pi$
$877$	1169.03	1.33299	0.666496	+	0.745509i	$0.267793\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	547.812i	0.621808i	0.950441	+	0.310904i	$0.100632\pi$
$881$	547.812i	0.621808i	−0.950441	+	0.310904i	$0.899368\pi$
$882$	0	0
$883$	−494.077	−0.559544	−0.279772	−	0.960066i	$-0.590259\pi$
$883$	−494.077	−0.559544	−0.279772	+	0.960066i	$0.590259\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	919.667i	1.03683i	0.855129	+	0.518414i	$0.173478\pi$
$887$	919.667i	1.03683i	−0.855129	+	0.518414i	$0.826522\pi$
$888$	0	0
$889$	291.185	0.327542
$890$	0	0
$891$	0	0
$892$	0	0
$893$	341.407i	0.382314i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1566.93i	1.74297i
$900$	0	0
$901$	−40.4082	−0.0448482
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	142.135	0.156709	0.0783547	−	0.996926i	$-0.475033\pi$
$907$	142.135	0.156709	0.0783547	+	0.996926i	$0.475033\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	15.6341i	0.0171615i	0.999963	+	0.00858074i	$0.00273137\pi$
$911$	15.6341i	0.0171615i	−0.999963	+	0.00858074i	$0.997269\pi$
$912$	0	0
$913$	−731.497	−0.801202
$914$	0	0
$915$	0	0
$916$	0	0
$917$	572.520i	0.624340i
$918$	0	0
$919$	1549.48	1.68605	0.843027	−	0.537871i	$-0.180771\pi$
$919$	1549.48	1.68605	0.843027	+	0.537871i	$0.180771\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	317.331i	0.343804i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	982.480i	1.05757i	0.848757	+	0.528784i	$0.177352\pi$
$929$	982.480i	1.05757i	−0.848757	+	0.528784i	$0.822648\pi$
$930$	0	0
$931$	44.1098	0.0473789
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−329.526	−0.351682	−0.175841	−	0.984419i	$-0.556265\pi$
$937$	−329.526	−0.351682	−0.175841	+	0.984419i	$0.556265\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	939.888i	0.998819i	0.866366	+	0.499409i	$0.166450\pi$
$941$	939.888i	0.998819i	−0.866366	+	0.499409i	$0.833550\pi$
$942$	0	0
$943$	195.395	0.207205
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1456.78i	1.53831i	0.639062	+	0.769155i	$0.279323\pi$
$947$	1456.78i	1.53831i	−0.639062	+	0.769155i	$0.720677\pi$
$948$	0	0
$949$	72.7444	0.0766538
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1391.14i	1.45975i	0.683582	+	0.729874i	$0.260421\pi$
$953$	1391.14i	1.45975i	−0.683582	+	0.729874i	$0.739579\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 93.5347i	− 0.0975336i
$960$	0	0
$961$	589.036	0.612941
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.9796	−0.0123884	−0.00619421	−	0.999981i	$-0.501972\pi$
$967$	−11.9796	−0.0123884	−0.00619421	+	0.999981i	$0.501972\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 504.732i	− 0.519806i	−0.965635	−	0.259903i	$-0.916309\pi$
$971$	− 504.732i	− 0.519806i	0.965635	−	0.259903i	$-0.0836906\pi$
$972$	0	0
$973$	−1032.88	−1.06154
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 529.554i	− 0.542021i	−0.962576	−	0.271010i	$-0.912642\pi$
$977$	− 529.554i	− 0.542021i	0.962576	−	0.271010i	$-0.0873577\pi$
$978$	0	0
$979$	783.557	0.800365
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1110.66i	1.12986i	0.825137	+	0.564932i	$0.191098\pi$
$983$	1110.66i	1.12986i	−0.825137	+	0.564932i	$0.808902\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	981.142i	0.992054i
$990$	0	0
$991$	−1048.11	−1.05763	−0.528817	−	0.848736i	$-0.677364\pi$
$991$	−1048.11	−1.05763	−0.528817	+	0.848736i	$0.677364\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1596.81	1.60161	0.800805	−	0.598925i	$-0.204405\pi$
$997$	1596.81	1.60161	0.800805	+	0.598925i	$0.204405\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.g.s.701.7		8
3.2	odd	2	inner	2700.3.g.s.701.8		8
5.2	odd	4		540.3.b.b.269.1	yes	4
5.3	odd	4		540.3.b.a.269.3	✓	4
5.4	even	2	inner	2700.3.g.s.701.1		8
15.2	even	4		540.3.b.a.269.4	yes	4
15.8	even	4		540.3.b.b.269.2	yes	4
15.14	odd	2	inner	2700.3.g.s.701.2		8
20.3	even	4		2160.3.c.h.1889.3		4
20.7	even	4		2160.3.c.l.1889.1		4
45.2	even	12		1620.3.t.d.1349.1		8
45.7	odd	12		1620.3.t.a.1349.4		8
45.13	odd	12		1620.3.t.d.269.1		8
45.22	odd	12		1620.3.t.a.269.2		8
45.23	even	12		1620.3.t.a.269.4		8
45.32	even	12		1620.3.t.d.269.3		8
45.38	even	12		1620.3.t.a.1349.2		8
45.43	odd	12		1620.3.t.d.1349.3		8
60.23	odd	4		2160.3.c.l.1889.2		4
60.47	odd	4		2160.3.c.h.1889.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.b.a.269.3	✓	4	5.3	odd	4
540.3.b.a.269.4	yes	4	15.2	even	4
540.3.b.b.269.1	yes	4	5.2	odd	4
540.3.b.b.269.2	yes	4	15.8	even	4
1620.3.t.a.269.2		8	45.22	odd	12
1620.3.t.a.269.4		8	45.23	even	12
1620.3.t.a.1349.2		8	45.38	even	12
1620.3.t.a.1349.4		8	45.7	odd	12
1620.3.t.d.269.1		8	45.13	odd	12
1620.3.t.d.269.3		8	45.32	even	12
1620.3.t.d.1349.1		8	45.2	even	12
1620.3.t.d.1349.3		8	45.43	odd	12
2160.3.c.h.1889.3		4	20.3	even	4
2160.3.c.h.1889.4		4	60.47	odd	4
2160.3.c.l.1889.1		4	20.7	even	4
2160.3.c.l.1889.2		4	60.23	odd	4
2700.3.g.s.701.1		8	5.4	even	2	inner
2700.3.g.s.701.2		8	15.14	odd	2	inner
2700.3.g.s.701.7		8	1.1	even	1	trivial
2700.3.g.s.701.8		8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2700.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.15439 q^{7} +O(q^{10})\)	\(q+7.15439 q^{7} -5.06984i q^{11} -3.12682 q^{13} -8.72842i q^{17} +20.1852 q^{19} -14.7284i q^{23} -39.7995i q^{29} -39.3705 q^{31} +34.8712 q^{37} +13.2665i q^{41} -66.6156 q^{43} +16.9137i q^{47} +2.18525 q^{49} -4.62950i q^{53} -25.7738i q^{59} -12.1852 q^{61} +106.415 q^{67} -101.487i q^{71} -23.2646 q^{73} -36.2716i q^{77} +66.5557 q^{79} -144.284i q^{83} +154.553i q^{89} -22.3705 q^{91} -175.733 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 12 q^{19} + 32 q^{31} - 156 q^{49} + 76 q^{61} + 12 q^{79} + 168 q^{91}+O(q^{100}) \) 8 * q - 12 * q^19 + 32 * q^31 - 156 * q^49 + 76 * q^61 + 12 * q^79 + 168 * q^91

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