LMFDB - Embedded newform 1764.4.k.v.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, 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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.1
Root			$-1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.4.k.v.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.93725 + 13.7477i) q^{5} +(-7.93725 - 13.7477i) q^{11} +26.0000 q^{13} +(39.6863 + 68.7386i) q^{17} +(-34.0000 + 58.8897i) q^{19} +(-23.8118 + 41.2432i) q^{23} +(-63.5000 - 109.985i) q^{25} -253.992 q^{29} +(-106.000 - 183.597i) q^{31} +(-109.000 + 188.794i) q^{37} +396.863 q^{41} +260.000 q^{43} +(-206.369 + 357.441i) q^{47} +(238.118 + 412.432i) q^{53} +252.000 q^{55} +(142.871 + 247.459i) q^{59} +(161.000 - 278.860i) q^{61} +(-206.369 + 357.441i) q^{65} +(-178.000 - 308.305i) q^{67} -1127.09 q^{71} +(113.000 + 195.722i) q^{73} +(-220.000 + 381.051i) q^{79} -253.992 q^{83} -1260.00 q^{85} +(-103.184 + 178.720i) q^{89} +(-539.733 - 934.845i) q^{95} -1330.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 104 q^{13} - 136 q^{19} - 254 q^{25} - 424 q^{31} - 436 q^{37} + 1040 q^{43} + 1008 q^{55} + 644 q^{61} - 712 q^{67} + 452 q^{73} - 880 q^{79} - 5040 q^{85} - 5320 q^{97}+O(q^{100}) $ 4 * q + 104 * q^13 - 136 * q^19 - 254 * q^25 - 424 * q^31 - 436 * q^37 + 1040 * q^43 + 1008 * q^55 + 644 * q^61 - 712 * q^67 + 452 * q^73 - 880 * q^79 - 5040 * q^85 - 5320 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−7.93725	+	13.7477i	−0.709930	+	1.22963i	0.254953	+	0.966953i	$0.417940\pi$
$5$	−7.93725	+	13.7477i	−0.709930	+	1.22963i	−0.964883	+	0.262681i	$0.915393\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.93725	−	13.7477i	−0.217561	−	0.376827i	0.736501	−	0.676437i	$-0.236477\pi$
$11$	−7.93725	−	13.7477i	−0.217561	−	0.376827i	−0.954062	+	0.299610i	$0.903143\pi$
$12$		0			0
$13$	26.0000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	39.6863	+	68.7386i	0.566196	+	0.980680i	0.996937	+	0.0782050i	$0.0249189\pi$
$17$	39.6863	+	68.7386i	0.566196	+	0.980680i	−0.430741	+	0.902476i	$0.641748\pi$
$18$		0			0
$19$	−34.0000	+	58.8897i	−0.410533	+	0.711065i	−0.994948	−	0.100390i	$-0.967991\pi$
$19$	−34.0000	+	58.8897i	−0.410533	+	0.711065i	0.584415	+	0.811455i	$0.301324\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−23.8118	+	41.2432i	−0.215874	+	0.373904i	−0.953543	−	0.301259i	$-0.902593\pi$
$23$	−23.8118	+	41.2432i	−0.215874	+	0.373904i	0.737669	+	0.675163i	$0.235927\pi$
$24$		0			0
$25$	−63.5000	−	109.985i	−0.508000	−	0.879882i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−253.992			−1.62638			−0.813192	−	0.581995i	$-0.802272\pi$
$29$	−253.992			−1.62638			−0.813192	+	0.581995i	$0.802272\pi$
$30$		0			0
$31$	−106.000	−	183.597i	−0.614134	−	1.06371i	−0.990536	−	0.137255i	$-0.956172\pi$
$31$	−106.000	−	183.597i	−0.614134	−	1.06371i	0.376401	−	0.926457i	$-0.377161\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−109.000	+	188.794i	−0.484311	+	0.838850i	−0.999838	−	0.0180229i	$-0.994263\pi$
$37$	−109.000	+	188.794i	−0.484311	+	0.838850i	0.515527	+	0.856873i	$0.327596\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	396.863			1.51170			0.755848	−	0.654747i	$-0.227225\pi$
$41$	396.863			1.51170			0.755848	+	0.654747i	$0.227225\pi$
$42$		0			0
$43$	260.000			0.922084			0.461042	−	0.887378i	$-0.347476\pi$
$43$	260.000			0.922084			0.461042	+	0.887378i	$0.347476\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−206.369	+	357.441i	−0.640467	+	1.10932i	0.344861	+	0.938654i	$0.387926\pi$
$47$	−206.369	+	357.441i	−0.640467	+	1.10932i	−0.985329	+	0.170668i	$0.945407\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	238.118	+	412.432i	0.617132	+	1.06890i	0.990007	+	0.141022i	$0.0450387\pi$
$53$	238.118	+	412.432i	0.617132	+	1.06890i	−0.372875	+	0.927882i	$0.621628\pi$
$54$		0			0
$55$	252.000			0.617812
$56$		0			0
$57$		0			0
$58$		0			0
$59$	142.871	+	247.459i	0.315257	+	0.546041i	0.979492	−	0.201483i	$-0.0645759\pi$
$59$	142.871	+	247.459i	0.315257	+	0.546041i	−0.664235	+	0.747524i	$0.731243\pi$
$60$		0			0
$61$	161.000	−	278.860i	0.337933	−	0.585318i	−0.646110	−	0.763244i	$-0.723605\pi$
$61$	161.000	−	278.860i	0.337933	−	0.585318i	0.984044	+	0.177926i	$0.0569388\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−206.369	+	357.441i	−0.393798	+	0.682078i
$66$		0			0
$67$	−178.000	−	308.305i	−0.324570	−	0.562171i	0.656856	−	0.754016i	$-0.271886\pi$
$67$	−178.000	−	308.305i	−0.324570	−	0.562171i	−0.981425	+	0.191845i	$0.938553\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1127.09			−1.88396			−0.941979	−	0.335673i	$-0.891036\pi$
$71$	−1127.09			−1.88396			−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$	113.000	+	195.722i	0.181173	+	0.313801i	0.942280	−	0.334825i	$-0.108677\pi$
$73$	113.000	+	195.722i	0.181173	+	0.313801i	−0.761107	+	0.648626i	$0.775344\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−220.000	+	381.051i	−0.313316	+	0.542679i	−0.979078	−	0.203485i	$-0.934773\pi$
$79$	−220.000	+	381.051i	−0.313316	+	0.542679i	0.665762	+	0.746164i	$0.268106\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−253.992			−0.335895			−0.167947	−	0.985796i	$-0.553714\pi$
$83$	−253.992			−0.335895			−0.167947	+	0.985796i	$0.553714\pi$
$84$		0			0
$85$	−1260.00			−1.60784
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−103.184	+	178.720i	−0.122893	+	0.212858i	−0.920908	−	0.389781i	$-0.872551\pi$
$89$	−103.184	+	178.720i	−0.122893	+	0.212858i	0.798014	+	0.602639i	$0.205884\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−539.733	−	934.845i	−0.582900	−	1.00961i
$96$		0			0
$97$	−1330.00			−1.39218			−0.696088	−	0.717957i	$-0.745078\pi$
$97$	−1330.00			−1.39218			−0.696088	+	0.717957i	$0.745078\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−119.059	−	206.216i	−0.117295	−	0.203161i	0.801400	−	0.598129i	$-0.204089\pi$
$101$	−119.059	−	206.216i	−0.117295	−	0.203161i	−0.918695	+	0.394968i	$0.870756\pi$
$102$		0			0
$103$	902.000	−	1562.31i	0.862881	−	1.49455i	−0.00625573	−	0.999980i	$-0.501991\pi$
$103$	902.000	−	1562.31i	0.862881	−	1.49455i	0.869136	−	0.494573i	$-0.164675\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	87.3098	−	151.225i	0.0788837	−	0.136631i	−0.823885	−	0.566757i	$-0.808198\pi$
$107$	87.3098	−	151.225i	0.0788837	−	0.136631i	0.902769	+	0.430127i	$0.141531\pi$
$108$		0			0
$109$	227.000	+	393.176i	0.199474	+	0.345499i	0.948358	−	0.317202i	$-0.102743\pi$
$109$	227.000	+	393.176i	0.199474	+	0.345499i	−0.748884	+	0.662701i	$0.769410\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1777.94			1.48013			0.740066	−	0.672534i	$-0.234794\pi$
$113$	1777.94			1.48013			0.740066	+	0.672534i	$0.234794\pi$
$114$		0			0
$115$	−378.000	−	654.715i	−0.306510	−	0.530891i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	539.500	−	934.441i	0.405334	−	0.702060i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	31.7490			0.0227177
$126$		0			0
$127$	−904.000			−0.631630			−0.315815	−	0.948821i	$-0.602278\pi$
$127$	−904.000			−0.631630			−0.315815	+	0.948821i	$0.602278\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	666.729	−	1154.81i	0.444675	−	0.770200i	−0.553355	−	0.832946i	$-0.686653\pi$
$131$	666.729	−	1154.81i	0.444675	−	0.770200i	0.998030	+	0.0627463i	$0.0199859\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	809.600	+	1402.27i	0.504882	+	0.874481i	0.999984	+	0.00564606i	$0.00179721\pi$
$137$	809.600	+	1402.27i	0.504882	+	0.874481i	−0.495102	+	0.868835i	$0.664869\pi$
$138$		0			0
$139$	−880.000			−0.536983			−0.268491	−	0.963282i	$-0.586525\pi$
$139$	−880.000			−0.536983			−0.268491	+	0.963282i	$0.586525\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−206.369	−	357.441i	−0.120681	−	0.209026i
$144$		0			0
$145$	2016.00	−	3491.81i	1.15462	−	1.99986i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1285.84	−	2227.13i	0.706978	−	1.22452i	−0.258995	−	0.965879i	$-0.583391\pi$
$149$	1285.84	−	2227.13i	0.706978	−	1.22452i	0.965973	−	0.258643i	$-0.0832754\pi$
$150$		0			0
$151$	−1756.00	−	3041.48i	−0.946366	−	1.63915i	−0.752993	−	0.658029i	$-0.771391\pi$
$151$	−1756.00	−	3041.48i	−0.946366	−	1.63915i	−0.193373	−	0.981125i	$-0.561943\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3365.40			1.74397
$156$		0			0
$157$	161.000	+	278.860i	0.0818420	+	0.141755i	0.904041	−	0.427445i	$-0.140586\pi$
$157$	161.000	+	278.860i	0.0818420	+	0.141755i	−0.822199	+	0.569200i	$0.807253\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1702.00	+	2947.95i	−0.817858	+	1.41657i	0.0893986	+	0.995996i	$0.471505\pi$
$163$	−1702.00	+	2947.95i	−0.817858	+	1.41657i	−0.907257	+	0.420576i	$0.861828\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2762.16			−1.27990			−0.639948	−	0.768418i	$-0.721044\pi$
$167$	−2762.16			−1.27990			−0.639948	+	0.768418i	$0.721044\pi$
$168$		0			0
$169$	−1521.00			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1754.13	−	3038.25i	0.770892	−	1.33522i	−0.166183	−	0.986095i	$-0.553144\pi$
$173$	1754.13	−	3038.25i	0.770892	−	1.33522i	0.937075	−	0.349129i	$-0.113522\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1801.76	−	3120.73i	−0.752344	−	1.30310i	−0.946684	−	0.322164i	$-0.895590\pi$
$179$	−1801.76	−	3120.73i	−0.752344	−	1.30310i	0.194340	−	0.980934i	$-0.437744\pi$
$180$		0			0
$181$	−502.000			−0.206151			−0.103076	−	0.994674i	$-0.532868\pi$
$181$	−502.000			−0.206151			−0.103076	+	0.994674i	$0.532868\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1730.32	−	2997.00i	−0.687653	−	1.19105i
$186$		0			0
$187$	630.000	−	1091.19i	0.246365	−	0.426716i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1246.15	+	2158.39i	−0.472085	+	0.817675i	−0.999490	−	0.0319393i	$-0.989832\pi$
$191$	−1246.15	+	2158.39i	−0.472085	+	0.817675i	0.527405	+	0.849614i	$0.323165\pi$
$192$		0			0
$193$	−655.000	−	1134.49i	−0.244290	−	0.423122i	0.717642	−	0.696412i	$-0.245221\pi$
$193$	−655.000	−	1134.49i	−0.244290	−	0.423122i	−0.961932	+	0.273290i	$0.911888\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1174.71			0.424847			0.212424	−	0.977178i	$-0.431864\pi$
$197$	1174.71			0.424847			0.212424	+	0.977178i	$0.431864\pi$
$198$		0			0
$199$	−2296.00	−	3976.79i	−0.817885	−	1.41662i	−0.907238	−	0.420618i	$-0.861813\pi$
$199$	−2296.00	−	3976.79i	−0.817885	−	1.41662i	0.0893526	−	0.996000i	$-0.471520\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−3150.00	+	5455.96i	−1.07320	+	1.85883i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1079.47			0.357264
$210$		0			0
$211$	−1780.00			−0.580759			−0.290380	−	0.956911i	$-0.593782\pi$
$211$	−1780.00			−0.580759			−0.290380	+	0.956911i	$0.593782\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2063.69	+	3574.41i	−0.654615	+	1.13383i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1031.84	+	1787.20i	0.314069	+	0.543984i
$222$		0			0
$223$	4088.00			1.22759			0.613795	−	0.789465i	$-0.289642\pi$
$223$	4088.00			1.22759			0.613795	+	0.789465i	$0.289642\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2428.80	+	4206.80i	0.710155	+	1.23002i	0.964799	+	0.262989i	$0.0847083\pi$
$227$	2428.80	+	4206.80i	0.710155	+	1.23002i	−0.254644	+	0.967035i	$0.581958\pi$
$228$		0			0
$229$	3191.00	−	5526.97i	0.920818	−	1.59490i	0.122664	−	0.992448i	$-0.460856\pi$
$229$	3191.00	−	5526.97i	0.920818	−	1.59490i	0.798153	−	0.602455i	$-0.205811\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1095.34	−	1897.19i	0.307975	−	0.533428i	−0.669944	−	0.742411i	$-0.733682\pi$
$233$	1095.34	−	1897.19i	0.307975	−	0.533428i	0.977919	+	0.208983i	$0.0670153\pi$
$234$		0			0
$235$	−3276.00	−	5674.20i	−0.909373	−	1.57508i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5349.71			−1.44788			−0.723941	−	0.689862i	$-0.757671\pi$
$239$	−5349.71			−1.44788			−0.723941	+	0.689862i	$0.757671\pi$
$240$		0			0
$241$	−2107.00	−	3649.43i	−0.563169	−	0.975438i	−0.997217	−	0.0745482i	$-0.976249\pi$
$241$	−2107.00	−	3649.43i	−0.563169	−	0.975438i	0.434048	−	0.900890i	$-0.357085\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−884.000	+	1531.13i	−0.227723	+	0.394428i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1428.71			−0.359279			−0.179640	−	0.983732i	$-0.557493\pi$
$251$	−1428.71			−0.359279			−0.179640	+	0.983732i	$0.557493\pi$
$252$		0			0
$253$	756.000			0.187863
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−420.674	+	728.630i	−0.102105	+	0.176851i	−0.912552	−	0.408961i	$-0.865891\pi$
$257$	−420.674	+	728.630i	−0.102105	+	0.176851i	0.810447	+	0.585812i	$0.199224\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	134.933	+	233.711i	0.0316363	+	0.0547957i	0.881410	−	0.472352i	$-0.156595\pi$
$263$	134.933	+	233.711i	0.0316363	+	0.0547957i	−0.849774	+	0.527148i	$0.823262\pi$
$264$		0			0
$265$	−7560.00			−1.75248
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1373.14	−	2378.36i	−0.311235	−	0.539074i	0.667395	−	0.744704i	$-0.267409\pi$
$269$	−1373.14	−	2378.36i	−0.311235	−	0.539074i	−0.978630	+	0.205629i	$0.934076\pi$
$270$		0			0
$271$	3122.00	−	5407.46i	0.699808	−	1.21210i	−0.268725	−	0.963217i	$-0.586602\pi$
$271$	3122.00	−	5407.46i	0.699808	−	1.21210i	0.968533	−	0.248886i	$-0.0800645\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1008.03	+	1745.96i	−0.221042	+	0.382856i
$276$		0			0
$277$	−4063.00	−	7037.32i	−0.881307	−	1.52647i	−0.849889	−	0.526962i	$-0.823331\pi$
$277$	−4063.00	−	7037.32i	−0.881307	−	1.52647i	−0.0314180	−	0.999506i	$-0.510002\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3079.65			−0.653796			−0.326898	−	0.945060i	$-0.606003\pi$
$281$	−3079.65			−0.653796			−0.326898	+	0.945060i	$0.606003\pi$
$282$		0			0
$283$	302.000	+	523.079i	0.0634348	+	0.109872i	0.895999	−	0.444057i	$-0.146461\pi$
$283$	302.000	+	523.079i	0.0634348	+	0.109872i	−0.832564	+	0.553929i	$0.813128\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−693.500	+	1201.18i	−0.141156	+	0.244490i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7667.39			1.52878			0.764392	−	0.644752i	$-0.223039\pi$
$293$	7667.39			1.52878			0.764392	+	0.644752i	$0.223039\pi$
$294$		0			0
$295$	−4536.00			−0.895241
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−619.106	+	1072.32i	−0.119745	+	0.207405i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2555.80	+	4426.77i	0.479818	+	0.831069i
$306$		0			0
$307$	−1636.00			−0.304142			−0.152071	−	0.988370i	$-0.548594\pi$
$307$	−1636.00			−0.304142			−0.152071	+	0.988370i	$0.548594\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1635.07	+	2832.03i	0.298124	+	0.516366i	0.975707	−	0.219081i	$-0.0703058\pi$
$311$	1635.07	+	2832.03i	0.298124	+	0.516366i	−0.677583	+	0.735447i	$0.736972\pi$
$312$		0			0
$313$	2519.00	−	4363.04i	0.454896	−	0.787902i	−0.543787	−	0.839223i	$-0.683010\pi$
$313$	2519.00	−	4363.04i	0.454896	−	0.787902i	0.998682	+	0.0513213i	$0.0163433\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4936.97	+	8551.09i	−0.874725	+	1.51507i	−0.0176705	+	0.999844i	$0.505625\pi$
$317$	−4936.97	+	8551.09i	−0.874725	+	1.51507i	−0.857055	+	0.515225i	$0.827708\pi$
$318$		0			0
$319$	2016.00	+	3491.81i	0.353838	+	0.612865i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5397.33			−0.929770
$324$		0			0
$325$	−1651.00	−	2859.62i	−0.281788	−	0.488071i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−214.000	+	370.659i	−0.0355363	+	0.0615506i	−0.883247	−	0.468909i	$-0.844647\pi$
$331$	−214.000	+	370.659i	−0.0355363	+	0.0615506i	0.847710	+	0.530460i	$0.177981\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5651.32			0.921686
$336$		0			0
$337$	−9106.00			−1.47192			−0.735958	−	0.677028i	$-0.763268\pi$
$337$	−9106.00			−1.47192			−0.735958	+	0.677028i	$0.763268\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1682.70	+	2914.52i	−0.267223	+	0.462845i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−976.282	−	1690.97i	−0.151036	−	0.261602i	0.780572	−	0.625065i	$-0.214928\pi$
$347$	−976.282	−	1690.97i	−0.151036	−	0.261602i	−0.931609	+	0.363463i	$0.881594\pi$
$348$		0			0
$349$	−1834.00			−0.281294			−0.140647	−	0.990060i	$-0.544918\pi$
$349$	−1834.00			−0.281294			−0.140647	+	0.990060i	$0.544918\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5246.52	+	9087.25i	0.791060	+	1.37016i	0.925311	+	0.379209i	$0.123804\pi$
$353$	5246.52	+	9087.25i	0.791060	+	1.37016i	−0.134250	+	0.990947i	$0.542863\pi$
$354$		0			0
$355$	8946.00	−	15494.9i	1.33748	−	2.31658i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6691.11	−	11589.3i	0.983685	−	1.70379i	0.336046	−	0.941845i	$-0.390910\pi$
$359$	6691.11	−	11589.3i	0.983685	−	1.70379i	0.647639	−	0.761948i	$-0.275757\pi$
$360$		0			0
$361$	1117.50	+	1935.57i	0.162925	+	0.282194i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3587.64			−0.514481
$366$		0			0
$367$	104.000	+	180.133i	0.0147923	+	0.0256209i	0.873327	−	0.487135i	$-0.161958\pi$
$367$	104.000	+	180.133i	0.0147923	+	0.0256209i	−0.858535	+	0.512756i	$0.828625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2887.00	+	5000.43i	−0.400759	+	0.694135i	−0.993818	−	0.111024i	$-0.964587\pi$
$373$	−2887.00	+	5000.43i	−0.400759	+	0.694135i	0.593058	+	0.805159i	$0.297920\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6603.80			−0.902156
$378$		0			0
$379$	−12004.0			−1.62692			−0.813462	−	0.581618i	$-0.802420\pi$
$379$	−12004.0			−1.62692			−0.813462	+	0.581618i	$0.802420\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3524.14	+	6103.99i	−0.470170	+	0.814359i	−0.999418	−	0.0341086i	$-0.989141\pi$
$383$	−3524.14	+	6103.99i	−0.470170	+	0.814359i	0.529248	+	0.848467i	$0.322474\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−603.231	−	1044.83i	−0.0786248	−	0.136182i	0.824032	−	0.566543i	$-0.191720\pi$
$389$	−603.231	−	1044.83i	−0.0786248	−	0.136182i	−0.902657	+	0.430361i	$0.858386\pi$
$390$		0			0
$391$	−3780.00			−0.488907
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3492.39	−	6049.00i	−0.444864	−	0.770527i
$396$		0			0
$397$	−1735.00	+	3005.11i	−0.219338	+	0.379904i	−0.954606	−	0.297872i	$-0.903723\pi$
$397$	−1735.00	+	3005.11i	−0.219338	+	0.379904i	0.735268	+	0.677777i	$0.237056\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−682.604	+	1182.30i	−0.0850065	+	0.147236i	−0.905394	−	0.424572i	$-0.860424\pi$
$401$	−682.604	+	1182.30i	−0.0850065	+	0.147236i	0.820387	+	0.571808i	$0.193758\pi$
$402$		0			0
$403$	−2756.00	−	4773.53i	−0.340660	−	0.590041i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3460.64			0.421469
$408$		0			0
$409$	5921.00	+	10255.5i	0.715830	+	1.23985i	0.962638	+	0.270790i	$0.0872849\pi$
$409$	5921.00	+	10255.5i	0.715830	+	1.23985i	−0.246808	+	0.969064i	$0.579382\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2016.00	−	3491.81i	0.238462	−	0.413028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3460.64			−0.403493			−0.201746	−	0.979438i	$-0.564662\pi$
$419$	−3460.64			−0.403493			−0.201746	+	0.979438i	$0.564662\pi$
$420$		0			0
$421$	15518.0			1.79644			0.898220	−	0.439547i	$-0.144861\pi$
$421$	15518.0			1.79644			0.898220	+	0.439547i	$0.144861\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5040.16	−	8729.81i	0.575255	−	0.996371i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3468.58	−	6007.76i	−0.387646	−	0.671423i	0.604486	−	0.796616i	$-0.293379\pi$
$431$	−3468.58	−	6007.76i	−0.387646	−	0.671423i	−0.992132	+	0.125192i	$0.960045\pi$
$432$		0			0
$433$	−4006.00			−0.444610			−0.222305	−	0.974977i	$-0.571358\pi$
$433$	−4006.00			−0.444610			−0.222305	+	0.974977i	$0.571358\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1619.20	−	2804.54i	−0.177247	−	0.307000i
$438$		0			0
$439$	3752.00	−	6498.65i	0.407912	−	0.706524i	−0.586744	−	0.809773i	$-0.699590\pi$
$439$	3752.00	−	6498.65i	0.407912	−	0.706524i	0.994656	+	0.103249i	$0.0329238\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9135.78	+	15823.6i	−0.979806	+	1.69707i	−0.316740	+	0.948512i	$0.602588\pi$
$443$	−9135.78	+	15823.6i	−0.979806	+	1.69707i	−0.663066	+	0.748561i	$0.730745\pi$
$444$		0			0
$445$	−1638.00	−	2837.10i	−0.174491	−	0.302228i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8445.24			0.887651			0.443826	−	0.896113i	$-0.353621\pi$
$449$	8445.24			0.887651			0.443826	+	0.896113i	$0.353621\pi$
$450$		0			0
$451$	−3150.00	−	5455.96i	−0.328886	−	0.569648i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5107.00	+	8845.58i	−0.522747	+	0.905424i	0.476903	+	0.878956i	$0.341759\pi$
$457$	−5107.00	+	8845.58i	−0.522747	+	0.905424i	−0.999650	+	0.0264683i	$0.991574\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1285.84			−0.129907			−0.0649537	−	0.997888i	$-0.520690\pi$
$461$	−1285.84			−0.129907			−0.0649537	+	0.997888i	$0.520690\pi$
$462$		0			0
$463$	−1408.00			−0.141329			−0.0706645	−	0.997500i	$-0.522512\pi$
$463$	−1408.00			−0.141329			−0.0706645	+	0.997500i	$0.522512\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5698.95	−	9870.87i	0.564702	−	0.978093i	−0.432375	−	0.901694i	$-0.642324\pi$
$467$	5698.95	−	9870.87i	0.564702	−	0.978093i	0.997077	−	0.0763989i	$-0.0243423\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2063.69	−	3574.41i	−0.200610	−	0.347466i
$474$		0			0
$475$	8636.00			0.834204
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5381.46	−	9320.96i	−0.513330	−	0.889114i	−0.999880	−	0.0154612i	$-0.995078\pi$
$479$	−5381.46	−	9320.96i	−0.513330	−	0.889114i	0.486550	−	0.873653i	$-0.338255\pi$
$480$		0			0
$481$	−2834.00	+	4908.63i	−0.268647	+	0.465311i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10556.5	−	18284.5i	0.988347	−	1.71187i
$486$		0			0
$487$	−2356.00	−	4080.71i	−0.219221	−	0.379702i	0.735349	−	0.677688i	$-0.237018\pi$
$487$	−2356.00	−	4080.71i	−0.219221	−	0.379702i	−0.954570	+	0.297987i	$0.903685\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15795.1			1.45178			0.725891	−	0.687810i	$-0.241428\pi$
$491$	15795.1			1.45178			0.725891	+	0.687810i	$0.241428\pi$
$492$		0			0
$493$	−10080.0	−	17459.1i	−0.920853	−	1.59496i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	8270.00	−	14324.1i	0.741916	−	1.28504i	−0.209706	−	0.977764i	$-0.567251\pi$
$499$	8270.00	−	14324.1i	0.741916	−	1.28504i	0.951622	−	0.307272i	$-0.0994160\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10921.7			−0.968137			−0.484068	−	0.875030i	$-0.660841\pi$
$503$	−10921.7			−0.968137			−0.484068	+	0.875030i	$0.660841\pi$
$504$		0			0
$505$	3780.00			0.333085
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1087.40	+	1883.44i	−0.0946922	+	0.164012i	−0.909480	−	0.415747i	$-0.863520\pi$
$509$	−1087.40	+	1883.44i	−0.0946922	+	0.164012i	0.814788	+	0.579759i	$0.196853\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14318.8	+	24800.9i	1.22517	+	2.12205i
$516$		0			0
$517$	6552.00			0.557363
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5865.63	−	10159.6i	−0.493240	−	0.854316i	0.506730	−	0.862105i	$-0.330854\pi$
$521$	−5865.63	−	10159.6i	−0.493240	−	0.854316i	−0.999970	+	0.00778848i	$0.997521\pi$
$522$		0			0
$523$	−7720.00	+	13371.4i	−0.645453	+	1.11796i	0.338743	+	0.940879i	$0.389998\pi$
$523$	−7720.00	+	13371.4i	−0.645453	+	1.11796i	−0.984197	+	0.177079i	$0.943335\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8413.49	−	14572.6i	0.695441	−	1.20454i
$528$		0			0
$529$	4949.50	+	8572.79i	0.406797	+	0.704593i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10318.4			0.838538
$534$		0			0
$535$	1386.00	+	2400.62i	0.112004	+	0.193996i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9523.00	+	16494.3i	−0.756794	+	1.31081i	0.187683	+	0.982230i	$0.439902\pi$
$541$	−9523.00	+	16494.3i	−0.756794	+	1.31081i	−0.944477	+	0.328577i	$0.893431\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7207.03			−0.566450
$546$		0			0
$547$	1892.00			0.147890			0.0739452	−	0.997262i	$-0.476441\pi$
$547$	1892.00			0.147890			0.0739452	+	0.997262i	$0.476441\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8635.73	−	14957.5i	0.667685	−	1.15646i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11826.5	−	20484.1i	−0.899650	−	1.55824i	−0.827941	−	0.560814i	$-0.810488\pi$
$557$	−11826.5	−	20484.1i	−0.899650	−	1.55824i	−0.0717089	−	0.997426i	$-0.522845\pi$
$558$		0			0
$559$	6760.00			0.511480
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3476.52	+	6021.50i	0.260245	+	0.450757i	0.966307	−	0.257393i	$-0.0828635\pi$
$563$	3476.52	+	6021.50i	0.260245	+	0.450757i	−0.706062	+	0.708150i	$0.749530\pi$
$564$		0			0
$565$	−14112.0	+	24442.7i	−1.05079	+	1.82002i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5508.45	+	9540.92i	−0.405846	+	0.702946i	−0.994420	−	0.105498i	$-0.966356\pi$
$569$	−5508.45	+	9540.92i	−0.405846	+	0.702946i	0.588573	+	0.808444i	$0.299690\pi$
$570$		0			0
$571$	−3790.00	−	6564.47i	−0.277770	−	0.481111i	0.693060	−	0.720880i	$-0.256262\pi$
$571$	−3790.00	−	6564.47i	−0.277770	−	0.481111i	−0.970830	+	0.239768i	$0.922929\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6048.19			0.438655
$576$		0			0
$577$	3359.00	+	5817.96i	0.242352	+	0.419766i	0.961384	−	0.275211i	$-0.0887478\pi$
$577$	3359.00	+	5817.96i	0.242352	+	0.419766i	−0.719032	+	0.694977i	$0.755415\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3780.00	−	6547.15i	0.268528	−	0.465103i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24288.0			1.70779			0.853895	−	0.520445i	$-0.174234\pi$
$587$	24288.0			1.70779			0.853895	+	0.520445i	$0.174234\pi$
$588$		0			0
$589$	14416.0			1.00849
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10056.5	+	17418.4i	−0.696410	+	1.20622i	0.273293	+	0.961931i	$0.411887\pi$
$593$	−10056.5	+	17418.4i	−0.696410	+	1.20622i	−0.969703	+	0.244286i	$0.921446\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2992.34	−	5182.89i	−0.204113	−	0.353535i	0.745737	−	0.666241i	$-0.232098\pi$
$599$	−2992.34	−	5182.89i	−0.204113	−	0.353535i	−0.949850	+	0.312706i	$0.898764\pi$
$600$		0			0
$601$	−8230.00			−0.558584			−0.279292	−	0.960206i	$-0.590100\pi$
$601$	−8230.00			−0.558584			−0.279292	+	0.960206i	$0.590100\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8564.30	+	14833.8i	0.575518	+	0.996826i
$606$		0			0
$607$	−4348.00	+	7530.96i	−0.290741	+	0.503578i	−0.973985	−	0.226612i	$-0.927235\pi$
$607$	−4348.00	+	7530.96i	−0.290741	+	0.503578i	0.683244	+	0.730190i	$0.260569\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5365.58	+	9293.46i	−0.355267	+	0.615341i
$612$		0			0
$613$	2897.00	+	5017.75i	0.190879	+	0.330612i	0.945542	−	0.325501i	$-0.105533\pi$
$613$	2897.00	+	5017.75i	0.190879	+	0.330612i	−0.754663	+	0.656113i	$0.772200\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13493.3			−0.880423			−0.440212	−	0.897894i	$-0.645097\pi$
$617$	−13493.3			−0.880423			−0.440212	+	0.897894i	$0.645097\pi$
$618$		0			0
$619$	9668.00	+	16745.5i	0.627770	+	1.08733i	0.987998	+	0.154466i	$0.0493656\pi$
$619$	9668.00	+	16745.5i	0.627770	+	1.08733i	−0.360228	+	0.932864i	$0.617301\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7685.50	−	13311.7i	0.491872	−	0.851947i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−17303.2			−1.09686
$630$		0			0
$631$	632.000			0.0398725			0.0199362	−	0.999801i	$-0.493654\pi$
$631$	632.000			0.0398725			0.0199362	+	0.999801i	$0.493654\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7175.28	−	12427.9i	0.448413	−	0.776674i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9445.33	−	16359.8i	−0.582010	−	1.00807i	−0.995241	−	0.0974442i	$-0.968933\pi$
$641$	−9445.33	−	16359.8i	−0.582010	−	1.00807i	0.413231	−	0.910626i	$-0.364400\pi$
$642$		0			0
$643$	28460.0			1.74549			0.872747	−	0.488173i	$-0.162336\pi$
$643$	28460.0			1.74549			0.872747	+	0.488173i	$0.162336\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5127.47	−	8881.03i	−0.311563	−	0.539644i	0.667138	−	0.744934i	$-0.267519\pi$
$647$	−5127.47	−	8881.03i	−0.311563	−	0.539644i	−0.978701	+	0.205291i	$0.934186\pi$
$648$		0			0
$649$	2268.00	−	3928.29i	0.137175	−	0.237595i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2984.41	+	5169.15i	−0.178850	+	0.309777i	−0.941487	−	0.337050i	$-0.890571\pi$
$653$	−2984.41	+	5169.15i	−0.178850	+	0.309777i	0.762637	+	0.646827i	$0.223904\pi$
$654$		0			0
$655$	10584.0	+	18332.0i	0.631376	+	1.09357i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8048.38			−0.475751			−0.237876	−	0.971296i	$-0.576451\pi$
$659$	−8048.38			−0.475751			−0.237876	+	0.971296i	$0.576451\pi$
$660$		0			0
$661$	6089.00	+	10546.5i	0.358298	+	0.620589i	0.987677	−	0.156509i	$-0.0500240\pi$
$661$	6089.00	+	10546.5i	0.358298	+	0.620589i	−0.629379	+	0.777098i	$0.716691\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6048.00	−	10475.4i	0.351094	−	0.608112i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5111.59			−0.294085
$672$		0			0
$673$	12902.0			0.738983			0.369491	−	0.929234i	$-0.379532\pi$
$673$	12902.0			0.738983			0.369491	+	0.929234i	$0.379532\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7040.34	+	12194.2i	−0.399679	+	0.692264i	−0.993686	−	0.112196i	$-0.964212\pi$
$677$	−7040.34	+	12194.2i	−0.399679	+	0.692264i	0.594008	+	0.804459i	$0.297545\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9977.13	+	17280.9i	0.558952	+	0.968133i	0.997584	+	0.0694666i	$0.0221297\pi$
$683$	9977.13	+	17280.9i	0.558952	+	0.968133i	−0.438632	+	0.898667i	$0.644537\pi$
$684$		0			0
$685$	−25704.0			−1.43372
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6191.06	+	10723.2i	0.342323	+	0.592921i
$690$		0			0
$691$	−1084.00	+	1877.54i	−0.0596777	+	0.103365i	−0.894321	−	0.447426i	$-0.852341\pi$
$691$	−1084.00	+	1877.54i	−0.0596777	+	0.103365i	0.834643	+	0.550791i	$0.185674\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6984.78	−	12098.0i	0.381220	−	0.660293i
$696$		0			0
$697$	15750.0	+	27279.8i	0.855916	+	1.48249i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30987.0			−1.66956			−0.834782	−	0.550581i	$-0.814406\pi$
$701$	−30987.0			−1.66956			−0.834782	+	0.550581i	$0.814406\pi$
$702$		0			0
$703$	−7412.00	−	12838.0i	−0.397651	−	0.688752i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9647.00	−	16709.1i	0.511002	−	0.885082i	−0.488916	−	0.872331i	$-0.662608\pi$
$709$	9647.00	−	16709.1i	0.511002	−	0.885082i	0.999919	−	0.0127514i	$-0.00405901\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10096.2			0.530302
$714$		0			0
$715$	6552.00			0.342701
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14160.1	+	24525.9i	−0.734466	+	1.27213i	0.220491	+	0.975389i	$0.429234\pi$
$719$	−14160.1	+	24525.9i	−0.734466	+	1.27213i	−0.954957	+	0.296744i	$0.904099\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16128.5	+	27935.4i	0.826203	+	1.43103i
$726$		0			0
$727$	15860.0			0.809099			0.404549	−	0.914516i	$-0.367428\pi$
$727$	15860.0			0.809099			0.404549	+	0.914516i	$0.367428\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10318.4	+	17872.0i	0.522081	+	0.904270i
$732$		0			0
$733$	−19249.0	+	33340.2i	−0.969956	+	1.68001i	−0.274293	+	0.961646i	$0.588444\pi$
$733$	−19249.0	+	33340.2i	−0.969956	+	1.68001i	−0.695664	+	0.718367i	$0.744890\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2825.66	+	4894.19i	−0.141227	+	0.244613i
$738$		0			0
$739$	13178.0	+	22825.0i	0.655968	+	1.13617i	0.981650	+	0.190691i	$0.0610728\pi$
$739$	13178.0	+	22825.0i	0.655968	+	1.13617i	−0.325682	+	0.945479i	$0.605594\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9000.85			−0.444427			−0.222213	−	0.974998i	$-0.571328\pi$
$743$	−9000.85			−0.444427			−0.222213	+	0.974998i	$0.571328\pi$
$744$		0			0
$745$	20412.0	+	35354.6i	1.00381	+	1.73865i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	8900.00	−	15415.3i	0.432444	−	0.749015i	−0.564639	−	0.825338i	$-0.690985\pi$
$751$	8900.00	−	15415.3i	0.432444	−	0.749015i	0.997083	+	0.0763227i	$0.0243179\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	55751.3			2.68741
$756$		0			0
$757$	2882.00			0.138373			0.0691863	−	0.997604i	$-0.477960\pi$
$757$	2882.00			0.138373			0.0691863	+	0.997604i	$0.477960\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4198.81	+	7272.55i	−0.200009	+	0.346425i	−0.948531	−	0.316684i	$-0.897430\pi$
$761$	−4198.81	+	7272.55i	−0.200009	+	0.346425i	0.748522	+	0.663110i	$0.230764\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3714.63	+	6433.94i	0.174873	+	0.302889i
$768$		0			0
$769$	−40630.0			−1.90527			−0.952637	−	0.304111i	$-0.901641\pi$
$769$	−40630.0			−1.90527			−0.952637	+	0.304111i	$0.901641\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2262.12	+	3918.10i	0.105256	+	0.182308i	0.913843	−	0.406068i	$-0.133101\pi$
$773$	2262.12	+	3918.10i	0.105256	+	0.182308i	−0.808587	+	0.588377i	$0.799767\pi$
$774$		0			0
$775$	−13462.0	+	23316.9i	−0.623960	+	1.08073i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13493.3	+	23371.1i	−0.620602	+	1.07491i
$780$		0			0
$781$	8946.00	+	15494.9i	0.409876	+	0.709926i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5111.59			−0.232408
$786$		0			0
$787$	3248.00	+	5625.70i	0.147114	+	0.254809i	0.930160	−	0.367155i	$-0.119668\pi$
$787$	3248.00	+	5625.70i	0.147114	+	0.254809i	−0.783046	+	0.621964i	$0.786335\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4186.00	−	7250.36i	0.187452	−	0.324676i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2301.80			−0.102301			−0.0511506	−	0.998691i	$-0.516289\pi$
$797$	−2301.80			−0.102301			−0.0511506	+	0.998691i	$0.516289\pi$
$798$		0			0
$799$	−32760.0			−1.45052
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1793.82	−	3106.99i	0.0788325	−	0.136542i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3746.38	−	6488.93i	−0.162813	−	0.282001i	0.773063	−	0.634329i	$-0.218723\pi$
$809$	−3746.38	−	6488.93i	−0.162813	−	0.282001i	−0.935877	+	0.352328i	$0.885390\pi$
$810$		0			0
$811$	38864.0			1.68274			0.841368	−	0.540462i	$-0.181751\pi$
$811$	38864.0			1.68274			0.841368	+	0.540462i	$0.181751\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27018.4	−	46797.3i	−1.16124	−	2.01133i
$816$		0			0
$817$	−8840.00	+	15311.3i	−0.378546	+	0.655662i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7857.88	+	13610.2i	−0.334034	+	0.578564i	−0.983299	−	0.181999i	$-0.941743\pi$
$821$	−7857.88	+	13610.2i	−0.334034	+	0.578564i	0.649265	+	0.760562i	$0.275077\pi$
$822$		0			0
$823$	−18208.0	−	31537.2i	−0.771192	−	1.33574i	−0.936910	−	0.349570i	$-0.886328\pi$
$823$	−18208.0	−	31537.2i	−0.771192	−	1.33574i	0.165718	−	0.986173i	$-0.447006\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24684.9			−1.03794			−0.518970	−	0.854792i	$-0.673684\pi$
$827$	−24684.9			−1.03794			−0.518970	+	0.854792i	$0.673684\pi$
$828$		0			0
$829$	8651.00	+	14984.0i	0.362439	+	0.627762i	0.988362	−	0.152123i	$-0.0486109\pi$
$829$	8651.00	+	14984.0i	0.362439	+	0.627762i	−0.625923	+	0.779885i	$0.715278\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21924.0	−	37973.5i	0.908636	−	1.57380i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−29240.8			−1.20323			−0.601613	−	0.798788i	$-0.705475\pi$
$839$	−29240.8			−1.20323			−0.601613	+	0.798788i	$0.705475\pi$
$840$		0			0
$841$	40123.0			1.64513
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12072.6	−	20910.3i	0.491490	−	0.851285i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5190.96	−	8991.01i	−0.209100	−	0.362172i
$852$		0			0
$853$	−82.0000			−0.00329147			−0.00164574	−	0.999999i	$-0.500524\pi$
$853$	−82.0000			−0.00329147			−0.00164574	+	0.999999i	$0.500524\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5325.90	+	9224.72i	0.212286	+	0.367690i	0.952430	−	0.304759i	$-0.0985758\pi$
$857$	5325.90	+	9224.72i	0.212286	+	0.367690i	−0.740144	+	0.672449i	$0.765242\pi$
$858$		0			0
$859$	−12754.0	+	22090.6i	−0.506590	+	0.877440i	0.493381	+	0.869813i	$0.335761\pi$
$859$	−12754.0	+	22090.6i	−0.506590	+	0.877440i	−0.999971	+	0.00762634i	$0.997572\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17088.9	−	29598.9i	0.674059	−	1.16750i	−0.302684	−	0.953091i	$-0.597883\pi$
$863$	17088.9	−	29598.9i	0.674059	−	1.16750i	0.976743	−	0.214414i	$-0.0687841\pi$
$864$		0			0
$865$	27846.0	+	48230.7i	1.09456	+	1.89583i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6984.78			0.272661
$870$		0			0
$871$	−4628.00	−	8015.93i	−0.180039	−	0.311836i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5419.00	+	9385.98i	−0.208651	+	0.361394i	−0.951290	−	0.308298i	$-0.900240\pi$
$877$	−5419.00	+	9385.98i	−0.208651	+	0.361394i	0.742639	+	0.669692i	$0.233574\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6175.18			−0.236149			−0.118074	−	0.993005i	$-0.537672\pi$
$881$	−6175.18			−0.236149			−0.118074	+	0.993005i	$0.537672\pi$
$882$		0			0
$883$	22100.0			0.842270			0.421135	−	0.906998i	$-0.361632\pi$
$883$	22100.0			0.842270			0.421135	+	0.906998i	$0.361632\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10366.1	+	17954.5i	−0.392399	+	0.679655i	−0.992765	−	0.120070i	$-0.961688\pi$
$887$	−10366.1	+	17954.5i	−0.392399	+	0.679655i	0.600366	+	0.799725i	$0.295021\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14033.1	−	24306.0i	−0.525866	−	0.910827i
$894$		0			0
$895$	57204.0			2.13645
$896$		0			0
$897$		0			0
$898$		0			0
$899$	26923.2	+	46632.3i	0.998819	+	1.73000i
$900$		0			0
$901$	−18900.0	+	32735.8i	−0.698835	+	1.21042i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3984.50	−	6901.36i	0.146353	−	0.253491i
$906$		0			0
$907$	−3406.00	−	5899.37i	−0.124691	−	0.215970i	0.796921	−	0.604083i	$-0.206460\pi$
$907$	−3406.00	−	5899.37i	−0.124691	−	0.215970i	−0.921612	+	0.388113i	$0.873127\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11636.0			0.423182			0.211591	−	0.977358i	$-0.432136\pi$
$911$	11636.0			0.423182			0.211591	+	0.977358i	$0.432136\pi$
$912$		0			0
$913$	2016.00	+	3491.81i	0.0730776	+	0.126574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	584.000	−	1011.52i	0.0209623	−	0.0363078i	−0.855354	−	0.518044i	$-0.826660\pi$
$919$	584.000	−	1011.52i	0.0209623	−	0.0363078i	0.876316	+	0.481736i	$0.159994\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−29304.3			−1.04503
$924$		0			0
$925$	27686.0			0.984119
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4817.91	−	8344.87i	0.170151	−	0.294711i	−0.768321	−	0.640064i	$-0.778908\pi$
$929$	4817.91	−	8344.87i	0.170151	−	0.294711i	0.938473	+	0.345354i	$0.112241\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10000.9	+	17322.1i	0.349803	+	0.605876i
$936$		0			0
$937$	−32254.0			−1.12454			−0.562269	−	0.826954i	$-0.690071\pi$
$937$	−32254.0			−1.12454			−0.562269	+	0.826954i	$0.690071\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2055.75	+	3560.66i	0.0712173	+	0.123352i	0.899435	−	0.437054i	$-0.143978\pi$
$941$	2055.75	+	3560.66i	0.0712173	+	0.123352i	−0.828218	+	0.560406i	$0.810645\pi$
$942$		0			0
$943$	−9450.00	+	16367.9i	−0.326335	+	0.565230i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3436.83	+	5952.77i	−0.117932	+	0.204265i	−0.918948	−	0.394378i	$-0.870960\pi$
$947$	−3436.83	+	5952.77i	−0.117932	+	0.204265i	0.801016	+	0.598643i	$0.204293\pi$
$948$		0			0
$949$	2938.00	+	5088.77i	0.100497	+	0.174066i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	33971.4			1.15471			0.577357	−	0.816492i	$-0.304084\pi$
$953$	33971.4			1.15471			0.577357	+	0.816492i	$0.304084\pi$
$954$		0			0
$955$	−19782.0	−	34263.4i	−0.670294	−	1.16098i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−7576.50	+	13122.9i	−0.254322	+	0.440498i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	20795.6			0.693714
$966$		0			0
$967$	43952.0			1.46163			0.730817	−	0.682573i	$-0.239139\pi$
$967$	43952.0			1.46163			0.730817	+	0.682573i	$0.239139\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1968.44	−	3409.44i	0.0650569	−	0.112682i	−0.831662	−	0.555282i	$-0.812610\pi$
$971$	1968.44	−	3409.44i	0.0650569	−	0.112682i	0.896719	+	0.442600i	$0.145944\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6079.94	−	10530.8i	−0.199094	−	0.344840i	0.749141	−	0.662410i	$-0.230466\pi$
$977$	−6079.94	−	10530.8i	−0.199094	−	0.344840i	−0.948235	+	0.317570i	$0.897133\pi$
$978$		0			0
$979$	3276.00			0.106947
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19716.1	−	34149.4i	−0.639722	−	1.10803i	−0.985494	−	0.169713i	$-0.945716\pi$
$983$	−19716.1	−	34149.4i	−0.639722	−	1.10803i	0.345771	−	0.938319i	$-0.387617\pi$
$984$		0			0
$985$	−9324.00	+	16149.6i	−0.301612	+	0.522406i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−6191.06	+	10723.2i	−0.199054	+	0.344771i
$990$		0			0
$991$	−5380.00	−	9318.43i	−0.172453	−	0.298698i	0.766824	−	0.641858i	$-0.221836\pi$
$991$	−5380.00	−	9318.43i	−0.172453	−	0.298698i	−0.939277	+	0.343160i	$0.888503\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	72895.7			2.32256
$996$		0			0
$997$	4985.00	+	8634.27i	0.158352	+	0.274273i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	4985.00	+	8634.27i	0.158352	+	0.274273i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.v.1549.1		4
3.2	odd	2	inner	1764.4.k.v.1549.2		4
7.2	even	3		252.4.a.f.1.2	yes	2
7.3	odd	6		1764.4.k.u.361.2		4
7.4	even	3	inner	1764.4.k.v.361.1		4
7.5	odd	6		1764.4.a.s.1.1		2
7.6	odd	2		1764.4.k.u.1549.2		4
21.2	odd	6		252.4.a.f.1.1	✓	2
21.5	even	6		1764.4.a.s.1.2		2
21.11	odd	6	inner	1764.4.k.v.361.2		4
21.17	even	6		1764.4.k.u.361.1		4
21.20	even	2		1764.4.k.u.1549.1		4
28.23	odd	6		1008.4.a.bc.1.2		2
84.23	even	6		1008.4.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.a.f.1.1	✓	2	21.2	odd	6
252.4.a.f.1.2	yes	2	7.2	even	3
1008.4.a.bc.1.1		2	84.23	even	6
1008.4.a.bc.1.2		2	28.23	odd	6
1764.4.a.s.1.1		2	7.5	odd	6
1764.4.a.s.1.2		2	21.5	even	6
1764.4.k.u.361.1		4	21.17	even	6
1764.4.k.u.361.2		4	7.3	odd	6
1764.4.k.u.1549.1		4	21.20	even	2
1764.4.k.u.1549.2		4	7.6	odd	2
1764.4.k.v.361.1		4	7.4	even	3	inner
1764.4.k.v.361.2		4	21.11	odd	6	inner
1764.4.k.v.1549.1		4	1.1	even	1	trivial
1764.4.k.v.1549.2		4	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-7.93725 + 13.7477i) q^{5} +(-7.93725 - 13.7477i) q^{11} +26.0000 q^{13} +(39.6863 + 68.7386i) q^{17} +(-34.0000 + 58.8897i) q^{19} +(-23.8118 + 41.2432i) q^{23} +(-63.5000 - 109.985i) q^{25} -253.992 q^{29} +(-106.000 - 183.597i) q^{31} +(-109.000 + 188.794i) q^{37} +396.863 q^{41} +260.000 q^{43} +(-206.369 + 357.441i) q^{47} +(238.118 + 412.432i) q^{53} +252.000 q^{55} +(142.871 + 247.459i) q^{59} +(161.000 - 278.860i) q^{61} +(-206.369 + 357.441i) q^{65} +(-178.000 - 308.305i) q^{67} -1127.09 q^{71} +(113.000 + 195.722i) q^{73} +(-220.000 + 381.051i) q^{79} -253.992 q^{83} -1260.00 q^{85} +(-103.184 + 178.720i) q^{89} +(-539.733 - 934.845i) q^{95} -1330.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 104 q^{13} - 136 q^{19} - 254 q^{25} - 424 q^{31} - 436 q^{37} + 1040 q^{43} + 1008 q^{55} + 644 q^{61} - 712 q^{67} + 452 q^{73} - 880 q^{79} - 5040 q^{85} - 5320 q^{97}+O(q^{100}) \) 4 * q + 104 * q^13 - 136 * q^19 - 254 * q^25 - 424 * q^31 - 436 * q^37 + 1040 * q^43 + 1008 * q^55 + 644 * q^61 - 712 * q^67 + 452 * q^73 - 880 * q^79 - 5040 * q^85 - 5320 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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