LMFDB - Embedded newform 1840.3.c.b.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.53

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.16732i q^{3} +2.23607 q^{5} +5.17862i q^{7} -17.7012 q^{9} +O(q^{10})$	$q-5.16732i q^{3} +2.23607 q^{5} +5.17862i q^{7} -17.7012 q^{9} +3.16157i q^{11} +11.6522 q^{13} -11.5545i q^{15} -29.8317 q^{17} +7.91406i q^{19} +26.7596 q^{21} -4.79583i q^{23} +5.00000 q^{25} +44.9621i q^{27} -53.6667 q^{29} +17.7204i q^{31} +16.3369 q^{33} +11.5797i q^{35} +38.9312 q^{37} -60.2109i q^{39} +37.3023 q^{41} +47.3531i q^{43} -39.5812 q^{45} +4.76342i q^{47} +22.1819 q^{49} +154.150i q^{51} -38.4537 q^{53} +7.06949i q^{55} +40.8945 q^{57} +31.4262i q^{59} +23.2413 q^{61} -91.6679i q^{63} +26.0552 q^{65} +104.707i q^{67} -24.7816 q^{69} +94.6766i q^{71} +67.9495 q^{73} -25.8366i q^{75} -16.3726 q^{77} -33.4259i q^{79} +73.0225 q^{81} +163.930i q^{83} -66.7057 q^{85} +277.313i q^{87} -127.551 q^{89} +60.3425i q^{91} +91.5671 q^{93} +17.6964i q^{95} +36.2808 q^{97} -55.9637i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 5.16732i	− 1.72244i	−0.508231	−	0.861221i	$-0.669701\pi$
$3$	− 5.16732i	− 1.72244i	0.508231	−	0.861221i	$-0.330299\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	5.17862i	0.739802i	0.929071	+	0.369901i	$0.120608\pi$
$7$	5.17862i	0.739802i	−0.929071	+	0.369901i	$0.879392\pi$
$8$	0	0
$9$	−17.7012	−1.96680
$10$	0	0
$11$	3.16157i	0.287415i	0.989620	+	0.143708i	$0.0459025\pi$
$11$	3.16157i	0.287415i	−0.989620	+	0.143708i	$0.954097\pi$
$12$	0	0
$13$	11.6522	0.896327	0.448163	−	0.893952i	$-0.352078\pi$
$13$	11.6522	0.896327	0.448163	+	0.893952i	$0.352078\pi$
$14$	0	0
$15$	− 11.5545i	− 0.770299i
$16$	0	0
$17$	−29.8317	−1.75480	−0.877402	−	0.479755i	$-0.840726\pi$
$17$	−29.8317	−1.75480	−0.877402	+	0.479755i	$0.840726\pi$
$18$	0	0
$19$	7.91406i	0.416529i	0.978072	+	0.208265i	$0.0667816\pi$
$19$	7.91406i	0.416529i	−0.978072	+	0.208265i	$0.933218\pi$
$20$	0	0
$21$	26.7596	1.27427
$22$	0	0
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	44.9621i	1.66526i
$28$	0	0
$29$	−53.6667	−1.85057	−0.925287	−	0.379267i	$-0.876176\pi$
$29$	−53.6667	−1.85057	−0.925287	+	0.379267i	$0.876176\pi$
$30$	0	0
$31$	17.7204i	0.571626i	0.958285	+	0.285813i	$0.0922636\pi$
$31$	17.7204i	0.571626i	−0.958285	+	0.285813i	$0.907736\pi$
$32$	0	0
$33$	16.3369	0.495056
$34$	0	0
$35$	11.5797i	0.330850i
$36$	0	0
$37$	38.9312	1.05220	0.526098	−	0.850424i	$-0.323655\pi$
$37$	38.9312	1.05220	0.526098	+	0.850424i	$0.323655\pi$
$38$	0	0
$39$	− 60.2109i	− 1.54387i
$40$	0	0
$41$	37.3023	0.909811	0.454906	−	0.890540i	$-0.349673\pi$
$41$	37.3023	0.909811	0.454906	+	0.890540i	$0.349673\pi$
$42$	0	0
$43$	47.3531i	1.10123i	0.834758	+	0.550617i	$0.185608\pi$
$43$	47.3531i	1.10123i	−0.834758	+	0.550617i	$0.814392\pi$
$44$	0	0
$45$	−39.5812	−0.879581
$46$	0	0
$47$	4.76342i	0.101349i	0.998715	+	0.0506747i	$0.0161372\pi$
$47$	4.76342i	0.101349i	−0.998715	+	0.0506747i	$0.983863\pi$
$48$	0	0
$49$	22.1819	0.452693
$50$	0	0
$51$	154.150i	3.02255i
$52$	0	0
$53$	−38.4537	−0.725542	−0.362771	−	0.931878i	$-0.618169\pi$
$53$	−38.4537	−0.725542	−0.362771	+	0.931878i	$0.618169\pi$
$54$	0	0
$55$	7.06949i	0.128536i
$56$	0	0
$57$	40.8945	0.717447
$58$	0	0
$59$	31.4262i	0.532648i	0.963884	+	0.266324i	$0.0858092\pi$
$59$	31.4262i	0.532648i	−0.963884	+	0.266324i	$0.914191\pi$
$60$	0	0
$61$	23.2413	0.381005	0.190503	−	0.981687i	$-0.438988\pi$
$61$	23.2413	0.381005	0.190503	+	0.981687i	$0.438988\pi$
$62$	0	0
$63$	− 91.6679i	− 1.45505i
$64$	0	0
$65$	26.0552	0.400850
$66$	0	0
$67$	104.707i	1.56278i	0.624040	+	0.781392i	$0.285490\pi$
$67$	104.707i	1.56278i	−0.624040	+	0.781392i	$0.714510\pi$
$68$	0	0
$69$	−24.7816	−0.359154
$70$	0	0
$71$	94.6766i	1.33347i	0.745294	+	0.666736i	$0.232309\pi$
$71$	94.6766i	1.33347i	−0.745294	+	0.666736i	$0.767691\pi$
$72$	0	0
$73$	67.9495	0.930816	0.465408	−	0.885096i	$-0.345908\pi$
$73$	67.9495	0.930816	0.465408	+	0.885096i	$0.345908\pi$
$74$	0	0
$75$	− 25.8366i	− 0.344488i
$76$	0	0
$77$	−16.3726	−0.212631
$78$	0	0
$79$	− 33.4259i	− 0.423112i	−0.977366	−	0.211556i	$-0.932147\pi$
$79$	− 33.4259i	− 0.423112i	0.977366	−	0.211556i	$-0.0678531\pi$
$80$	0	0
$81$	73.0225	0.901512
$82$	0	0
$83$	163.930i	1.97506i	0.157421	+	0.987532i	$0.449682\pi$
$83$	163.930i	1.97506i	−0.157421	+	0.987532i	$0.550318\pi$
$84$	0	0
$85$	−66.7057	−0.784773
$86$	0	0
$87$	277.313i	3.18751i
$88$	0	0
$89$	−127.551	−1.43315	−0.716577	−	0.697508i	$-0.754292\pi$
$89$	−127.551	−1.43315	−0.716577	+	0.697508i	$0.754292\pi$
$90$	0	0
$91$	60.3425i	0.663105i
$92$	0	0
$93$	91.5671	0.984592
$94$	0	0
$95$	17.6964i	0.186278i
$96$	0	0
$97$	36.2808	0.374029	0.187014	−	0.982357i	$-0.440119\pi$
$97$	36.2808	0.374029	0.187014	+	0.982357i	$0.440119\pi$
$98$	0	0
$99$	− 55.9637i	− 0.565290i
$100$	0	0
$101$	−51.8114	−0.512985	−0.256492	−	0.966546i	$-0.582567\pi$
$101$	−51.8114	−0.512985	−0.256492	+	0.966546i	$0.582567\pi$
$102$	0	0
$103$	59.3336i	0.576054i	0.957622	+	0.288027i	$0.0929993\pi$
$103$	59.3336i	0.576054i	−0.957622	+	0.288027i	$0.907001\pi$
$104$	0	0
$105$	59.8362	0.569869
$106$	0	0
$107$	− 55.9834i	− 0.523209i	−0.965175	−	0.261604i	$-0.915748\pi$
$107$	− 55.9834i	− 0.523209i	0.965175	−	0.261604i	$-0.0842516\pi$
$108$	0	0
$109$	146.243	1.34168	0.670840	−	0.741602i	$-0.265934\pi$
$109$	146.243	1.34168	0.670840	+	0.741602i	$0.265934\pi$
$110$	0	0
$111$	− 201.170i	− 1.81234i
$112$	0	0
$113$	−48.5044	−0.429242	−0.214621	−	0.976697i	$-0.568852\pi$
$113$	−48.5044	−0.429242	−0.214621	+	0.976697i	$0.568852\pi$
$114$	0	0
$115$	− 10.7238i	− 0.0932505i
$116$	0	0
$117$	−206.259	−1.76290
$118$	0	0
$119$	− 154.487i	− 1.29821i
$120$	0	0
$121$	111.004	0.917392
$122$	0	0
$123$	− 192.753i	− 1.56710i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	56.7151i	0.446576i	0.974753	+	0.223288i	$0.0716790\pi$
$127$	56.7151i	0.446576i	−0.974753	+	0.223288i	$0.928321\pi$
$128$	0	0
$129$	244.689	1.89681
$130$	0	0
$131$	− 110.355i	− 0.842402i	−0.906967	−	0.421201i	$-0.861609\pi$
$131$	− 110.355i	− 0.842402i	0.906967	−	0.421201i	$-0.138391\pi$
$132$	0	0
$133$	−40.9839	−0.308149
$134$	0	0
$135$	100.538i	0.744728i
$136$	0	0
$137$	43.9502	0.320805	0.160402	−	0.987052i	$-0.448721\pi$
$137$	43.9502	0.320805	0.160402	+	0.987052i	$0.448721\pi$
$138$	0	0
$139$	40.2475i	0.289550i	0.989465	+	0.144775i	$0.0462459\pi$
$139$	40.2475i	0.289550i	−0.989465	+	0.144775i	$0.953754\pi$
$140$	0	0
$141$	24.6141	0.174568
$142$	0	0
$143$	36.8394i	0.257618i
$144$	0	0
$145$	−120.002	−0.827602
$146$	0	0
$147$	− 114.621i	− 0.779737i
$148$	0	0
$149$	−88.7552	−0.595673	−0.297836	−	0.954617i	$-0.596265\pi$
$149$	−88.7552	−0.595673	−0.297836	+	0.954617i	$0.596265\pi$
$150$	0	0
$151$	− 66.7713i	− 0.442194i	−0.975252	−	0.221097i	$-0.929036\pi$
$151$	− 66.7713i	− 0.442194i	0.975252	−	0.221097i	$-0.0709637\pi$
$152$	0	0
$153$	528.058	3.45136
$154$	0	0
$155$	39.6240i	0.255639i
$156$	0	0
$157$	−125.101	−0.796819	−0.398409	−	0.917208i	$-0.630438\pi$
$157$	−125.101	−0.796819	−0.398409	+	0.917208i	$0.630438\pi$
$158$	0	0
$159$	198.703i	1.24970i
$160$	0	0
$161$	24.8358	0.154259
$162$	0	0
$163$	13.0126i	0.0798317i	0.999203	+	0.0399158i	$0.0127090\pi$
$163$	13.0126i	0.0798317i	−0.999203	+	0.0399158i	$0.987291\pi$
$164$	0	0
$165$	36.5303	0.221396
$166$	0	0
$167$	47.1211i	0.282162i	0.989998	+	0.141081i	$0.0450578\pi$
$167$	47.1211i	0.282162i	−0.989998	+	0.141081i	$0.954942\pi$
$168$	0	0
$169$	−33.2251	−0.196598
$170$	0	0
$171$	− 140.089i	− 0.819232i
$172$	0	0
$173$	309.799	1.79075	0.895374	−	0.445315i	$-0.146908\pi$
$173$	309.799	1.79075	0.895374	+	0.445315i	$0.146908\pi$
$174$	0	0
$175$	25.8931i	0.147960i
$176$	0	0
$177$	162.390	0.917455
$178$	0	0
$179$	− 244.369i	− 1.36519i	−0.730796	−	0.682596i	$-0.760851\pi$
$179$	− 244.369i	− 1.36519i	0.730796	−	0.682596i	$-0.239149\pi$
$180$	0	0
$181$	−252.510	−1.39508	−0.697541	−	0.716545i	$-0.745722\pi$
$181$	−252.510	−1.39508	−0.697541	+	0.716545i	$0.745722\pi$
$182$	0	0
$183$	− 120.095i	− 0.656259i
$184$	0	0
$185$	87.0529	0.470556
$186$	0	0
$187$	− 94.3150i	− 0.504358i
$188$	0	0
$189$	−232.841	−1.23196
$190$	0	0
$191$	315.994i	1.65442i	0.561894	+	0.827209i	$0.310073\pi$
$191$	315.994i	1.65442i	−0.561894	+	0.827209i	$0.689927\pi$
$192$	0	0
$193$	−86.9324	−0.450427	−0.225213	−	0.974309i	$-0.572308\pi$
$193$	−86.9324	−0.450427	−0.225213	+	0.974309i	$0.572308\pi$
$194$	0	0
$195$	− 134.636i	− 0.690440i
$196$	0	0
$197$	25.7909	0.130918	0.0654592	−	0.997855i	$-0.479149\pi$
$197$	25.7909	0.130918	0.0654592	+	0.997855i	$0.479149\pi$
$198$	0	0
$199$	− 332.349i	− 1.67010i	−0.550176	−	0.835049i	$-0.685439\pi$
$199$	− 332.349i	− 1.67010i	0.550176	−	0.835049i	$-0.314561\pi$
$200$	0	0
$201$	541.053	2.69180
$202$	0	0
$203$	− 277.919i	− 1.36906i
$204$	0	0
$205$	83.4104	0.406880
$206$	0	0
$207$	84.8921i	0.410107i
$208$	0	0
$209$	−25.0209	−0.119717
$210$	0	0
$211$	− 64.2911i	− 0.304697i	−0.988327	−	0.152349i	$-0.951316\pi$
$211$	− 64.2911i	− 0.304697i	0.988327	−	0.152349i	$-0.0486836\pi$
$212$	0	0
$213$	489.224	2.29683
$214$	0	0
$215$	105.885i	0.492487i
$216$	0	0
$217$	−91.7672	−0.422890
$218$	0	0
$219$	− 351.117i	− 1.60328i
$220$	0	0
$221$	−347.606	−1.57288
$222$	0	0
$223$	− 180.537i	− 0.809583i	−0.914409	−	0.404792i	$-0.867344\pi$
$223$	− 180.537i	− 0.809583i	0.914409	−	0.404792i	$-0.132656\pi$
$224$	0	0
$225$	−88.5062	−0.393361
$226$	0	0
$227$	234.132i	1.03142i	0.856763	+	0.515710i	$0.172472\pi$
$227$	234.132i	1.03142i	−0.856763	+	0.515710i	$0.827528\pi$
$228$	0	0
$229$	371.141	1.62070	0.810351	−	0.585944i	$-0.199276\pi$
$229$	371.141	1.62070	0.810351	+	0.585944i	$0.199276\pi$
$230$	0	0
$231$	84.6023i	0.366244i
$232$	0	0
$233$	74.7781	0.320936	0.160468	−	0.987041i	$-0.448700\pi$
$233$	74.7781	0.320936	0.160468	+	0.987041i	$0.448700\pi$
$234$	0	0
$235$	10.6513i	0.0453248i
$236$	0	0
$237$	−172.722	−0.728786
$238$	0	0
$239$	− 12.0676i	− 0.0504919i	−0.999681	−	0.0252459i	$-0.991963\pi$
$239$	− 12.0676i	− 0.0504919i	0.999681	−	0.0252459i	$-0.00803689\pi$
$240$	0	0
$241$	19.7435	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$241$	19.7435	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$242$	0	0
$243$	27.3278i	0.112460i
$244$	0	0
$245$	49.6003	0.202450
$246$	0	0
$247$	92.2166i	0.373347i
$248$	0	0
$249$	847.081	3.40193
$250$	0	0
$251$	298.018i	1.18732i	0.804716	+	0.593661i	$0.202318\pi$
$251$	298.018i	1.18732i	−0.804716	+	0.593661i	$0.797682\pi$
$252$	0	0
$253$	15.1624	0.0599303
$254$	0	0
$255$	344.690i	1.35172i
$256$	0	0
$257$	−324.783	−1.26375	−0.631873	−	0.775072i	$-0.717714\pi$
$257$	−324.783	−1.26375	−0.631873	+	0.775072i	$0.717714\pi$
$258$	0	0
$259$	201.610i	0.778416i
$260$	0	0
$261$	949.966	3.63972
$262$	0	0
$263$	97.1969i	0.369570i	0.982779	+	0.184785i	$0.0591589\pi$
$263$	97.1969i	0.369570i	−0.982779	+	0.184785i	$0.940841\pi$
$264$	0	0
$265$	−85.9851	−0.324472
$266$	0	0
$267$	659.095i	2.46852i
$268$	0	0
$269$	220.635	0.820203	0.410102	−	0.912040i	$-0.365493\pi$
$269$	220.635	0.820203	0.410102	+	0.912040i	$0.365493\pi$
$270$	0	0
$271$	390.373i	1.44049i	0.693719	+	0.720246i	$0.255971\pi$
$271$	390.373i	1.44049i	−0.693719	+	0.720246i	$0.744029\pi$
$272$	0	0
$273$	311.809	1.14216
$274$	0	0
$275$	15.8079i	0.0574831i
$276$	0	0
$277$	−312.957	−1.12981	−0.564904	−	0.825157i	$-0.691087\pi$
$277$	−312.957	−1.12981	−0.564904	+	0.825157i	$0.691087\pi$
$278$	0	0
$279$	− 313.673i	− 1.12428i
$280$	0	0
$281$	375.065	1.33475	0.667375	−	0.744722i	$-0.267418\pi$
$281$	375.065	1.33475	0.667375	+	0.744722i	$0.267418\pi$
$282$	0	0
$283$	− 236.307i	− 0.835008i	−0.908675	−	0.417504i	$-0.862905\pi$
$283$	− 236.307i	− 0.835008i	0.908675	−	0.417504i	$-0.137095\pi$
$284$	0	0
$285$	91.4429	0.320852
$286$	0	0
$287$	193.174i	0.673080i
$288$	0	0
$289$	600.929	2.07934
$290$	0	0
$291$	− 187.474i	− 0.644242i
$292$	0	0
$293$	−505.487	−1.72521	−0.862606	−	0.505877i	$-0.831169\pi$
$293$	−505.487	−1.72521	−0.862606	+	0.505877i	$0.831169\pi$
$294$	0	0
$295$	70.2712i	0.238208i
$296$	0	0
$297$	−142.151	−0.478622
$298$	0	0
$299$	− 55.8822i	− 0.186897i
$300$	0	0
$301$	−245.223	−0.814695
$302$	0	0
$303$	267.726i	0.883586i
$304$	0	0
$305$	51.9692	0.170391
$306$	0	0
$307$	− 44.3239i	− 0.144378i	−0.997391	−	0.0721888i	$-0.977002\pi$
$307$	− 44.3239i	− 0.144378i	0.997391	−	0.0721888i	$-0.0229984\pi$
$308$	0	0
$309$	306.596	0.992219
$310$	0	0
$311$	− 46.9079i	− 0.150829i	−0.997152	−	0.0754146i	$-0.975972\pi$
$311$	− 46.9079i	− 0.150829i	0.997152	−	0.0754146i	$-0.0240280\pi$
$312$	0	0
$313$	−496.937	−1.58766	−0.793829	−	0.608141i	$-0.791915\pi$
$313$	−496.937	−1.58766	−0.793829	+	0.608141i	$0.791915\pi$
$314$	0	0
$315$	− 204.976i	− 0.650716i
$316$	0	0
$317$	310.408	0.979204	0.489602	−	0.871946i	$-0.337142\pi$
$317$	310.408	0.979204	0.489602	+	0.871946i	$0.337142\pi$
$318$	0	0
$319$	− 169.671i	− 0.531884i
$320$	0	0
$321$	−289.284	−0.901197
$322$	0	0
$323$	− 236.090i	− 0.730928i
$324$	0	0
$325$	58.2612	0.179265
$326$	0	0
$327$	− 755.686i	− 2.31097i
$328$	0	0
$329$	−24.6679	−0.0749785
$330$	0	0
$331$	− 532.798i	− 1.60966i	−0.593503	−	0.804832i	$-0.702256\pi$
$331$	− 532.798i	− 1.60966i	0.593503	−	0.804832i	$-0.297744\pi$
$332$	0	0
$333$	−689.131	−2.06946
$334$	0	0
$335$	234.131i	0.698898i
$336$	0	0
$337$	−404.912	−1.20152	−0.600760	−	0.799430i	$-0.705135\pi$
$337$	−404.912	−1.20152	−0.600760	+	0.799430i	$0.705135\pi$
$338$	0	0
$339$	250.638i	0.739345i
$340$	0	0
$341$	−56.0243	−0.164294
$342$	0	0
$343$	368.624i	1.07471i
$344$	0	0
$345$	−55.4134	−0.160618
$346$	0	0
$347$	671.288i	1.93455i	0.253735	+	0.967274i	$0.418341\pi$
$347$	671.288i	1.93455i	−0.253735	+	0.967274i	$0.581659\pi$
$348$	0	0
$349$	−508.284	−1.45640	−0.728200	−	0.685364i	$-0.759643\pi$
$349$	−508.284	−1.45640	−0.728200	+	0.685364i	$0.759643\pi$
$350$	0	0
$351$	523.909i	1.49262i
$352$	0	0
$353$	−521.749	−1.47804	−0.739021	−	0.673683i	$-0.764711\pi$
$353$	−521.749	−1.47804	−0.739021	+	0.673683i	$0.764711\pi$
$354$	0	0
$355$	211.703i	0.596347i
$356$	0	0
$357$	−798.283	−2.23609
$358$	0	0
$359$	− 148.011i	− 0.412288i	−0.978522	−	0.206144i	$-0.933908\pi$
$359$	− 148.011i	− 0.412288i	0.978522	−	0.206144i	$-0.0660915\pi$
$360$	0	0
$361$	298.368	0.826503
$362$	0	0
$363$	− 573.596i	− 1.58015i
$364$	0	0
$365$	151.940	0.416273
$366$	0	0
$367$	− 38.0028i	− 0.103550i	−0.998659	−	0.0517749i	$-0.983512\pi$
$367$	− 38.0028i	− 0.103550i	0.998659	−	0.0517749i	$-0.0164878\pi$
$368$	0	0
$369$	−660.296	−1.78942
$370$	0	0
$371$	− 199.137i	− 0.536757i
$372$	0	0
$373$	−468.343	−1.25561	−0.627806	−	0.778370i	$-0.716047\pi$
$373$	−468.343	−1.25561	−0.627806	+	0.778370i	$0.716047\pi$
$374$	0	0
$375$	− 57.7724i	− 0.154060i
$376$	0	0
$377$	−625.337	−1.65872
$378$	0	0
$379$	229.758i	0.606223i	0.952955	+	0.303111i	$0.0980254\pi$
$379$	229.758i	0.606223i	−0.952955	+	0.303111i	$0.901975\pi$
$380$	0	0
$381$	293.065	0.769200
$382$	0	0
$383$	− 514.384i	− 1.34304i	−0.740987	−	0.671519i	$-0.765642\pi$
$383$	− 514.384i	− 1.34304i	0.740987	−	0.671519i	$-0.234358\pi$
$384$	0	0
$385$	−36.6101	−0.0950913
$386$	0	0
$387$	− 838.208i	− 2.16591i
$388$	0	0
$389$	−137.598	−0.353722	−0.176861	−	0.984236i	$-0.556594\pi$
$389$	−137.598	−0.353722	−0.176861	+	0.984236i	$0.556594\pi$
$390$	0	0
$391$	143.068i	0.365902i
$392$	0	0
$393$	−570.238	−1.45099
$394$	0	0
$395$	− 74.7425i	− 0.189222i
$396$	0	0
$397$	−311.642	−0.784993	−0.392497	−	0.919753i	$-0.628389\pi$
$397$	−311.642	−0.784993	−0.392497	+	0.919753i	$0.628389\pi$
$398$	0	0
$399$	211.777i	0.530769i
$400$	0	0
$401$	196.425	0.489837	0.244919	−	0.969544i	$-0.421239\pi$
$401$	196.425	0.489837	0.244919	+	0.969544i	$0.421239\pi$
$402$	0	0
$403$	206.483i	0.512364i
$404$	0	0
$405$	163.283	0.403169
$406$	0	0
$407$	123.084i	0.302417i
$408$	0	0
$409$	519.140	1.26929	0.634646	−	0.772803i	$-0.281146\pi$
$409$	519.140	1.26929	0.634646	+	0.772803i	$0.281146\pi$
$410$	0	0
$411$	− 227.105i	− 0.552567i
$412$	0	0
$413$	−162.744	−0.394054
$414$	0	0
$415$	366.559i	0.883275i
$416$	0	0
$417$	207.972	0.498734
$418$	0	0
$419$	293.730i	0.701027i	0.936558	+	0.350513i	$0.113993\pi$
$419$	293.730i	0.701027i	−0.936558	+	0.350513i	$0.886007\pi$
$420$	0	0
$421$	−92.9317	−0.220740	−0.110370	−	0.993891i	$-0.535204\pi$
$421$	−92.9317	−0.220740	−0.110370	+	0.993891i	$0.535204\pi$
$422$	0	0
$423$	− 84.3184i	− 0.199334i
$424$	0	0
$425$	−149.158	−0.350961
$426$	0	0
$427$	120.358i	0.281868i
$428$	0	0
$429$	190.361	0.443732
$430$	0	0
$431$	354.579i	0.822689i	0.911480	+	0.411344i	$0.134941\pi$
$431$	354.579i	0.822689i	−0.911480	+	0.411344i	$0.865059\pi$
$432$	0	0
$433$	−500.118	−1.15501	−0.577504	−	0.816388i	$-0.695973\pi$
$433$	−500.118	−1.15501	−0.577504	+	0.816388i	$0.695973\pi$
$434$	0	0
$435$	620.091i	1.42550i
$436$	0	0
$437$	37.9545	0.0868524
$438$	0	0
$439$	579.910i	1.32098i	0.750835	+	0.660490i	$0.229651\pi$
$439$	579.910i	1.32098i	−0.750835	+	0.660490i	$0.770349\pi$
$440$	0	0
$441$	−392.648	−0.890358
$442$	0	0
$443$	638.775i	1.44193i	0.692972	+	0.720965i	$0.256301\pi$
$443$	638.775i	1.44193i	−0.692972	+	0.720965i	$0.743699\pi$
$444$	0	0
$445$	−285.212	−0.640926
$446$	0	0
$447$	458.627i	1.02601i
$448$	0	0
$449$	30.8170	0.0686348	0.0343174	−	0.999411i	$-0.489074\pi$
$449$	30.8170	0.0686348	0.0343174	+	0.999411i	$0.489074\pi$
$450$	0	0
$451$	117.934i	0.261494i
$452$	0	0
$453$	−345.029	−0.761653
$454$	0	0
$455$	134.930i	0.296549i
$456$	0	0
$457$	622.095	1.36126	0.680629	−	0.732628i	$-0.261707\pi$
$457$	622.095	1.36126	0.680629	+	0.732628i	$0.261707\pi$
$458$	0	0
$459$	− 1341.29i	− 2.92221i
$460$	0	0
$461$	−45.1133	−0.0978596	−0.0489298	−	0.998802i	$-0.515581\pi$
$461$	−45.1133	−0.0978596	−0.0489298	+	0.998802i	$0.515581\pi$
$462$	0	0
$463$	− 60.8672i	− 0.131463i	−0.997837	−	0.0657313i	$-0.979062\pi$
$463$	− 60.8672i	− 0.131463i	0.997837	−	0.0657313i	$-0.0209380\pi$
$464$	0	0
$465$	204.750	0.440323
$466$	0	0
$467$	372.833i	0.798357i	0.916873	+	0.399179i	$0.130705\pi$
$467$	372.833i	0.798357i	−0.916873	+	0.399179i	$0.869295\pi$
$468$	0	0
$469$	−542.235	−1.15615
$470$	0	0
$471$	646.435i	1.37247i
$472$	0	0
$473$	−149.710	−0.316512
$474$	0	0
$475$	39.5703i	0.0833059i
$476$	0	0
$477$	680.678	1.42700
$478$	0	0
$479$	515.754i	1.07673i	0.842712	+	0.538365i	$0.180958\pi$
$479$	515.754i	1.07673i	−0.842712	+	0.538365i	$0.819042\pi$
$480$	0	0
$481$	453.636	0.943111
$482$	0	0
$483$	− 128.334i	− 0.265703i
$484$	0	0
$485$	81.1263	0.167271
$486$	0	0
$487$	160.652i	0.329882i	0.986303	+	0.164941i	$0.0527433\pi$
$487$	160.652i	0.329882i	−0.986303	+	0.164941i	$0.947257\pi$
$488$	0	0
$489$	67.2401	0.137505
$490$	0	0
$491$	− 804.440i	− 1.63837i	−0.573529	−	0.819186i	$-0.694426\pi$
$491$	− 804.440i	− 1.63837i	0.573529	−	0.819186i	$-0.305574\pi$
$492$	0	0
$493$	1600.97	3.24740
$494$	0	0
$495$	− 125.139i	− 0.252805i
$496$	0	0
$497$	−490.293	−0.986506
$498$	0	0
$499$	− 577.731i	− 1.15778i	−0.815406	−	0.578889i	$-0.803486\pi$
$499$	− 577.731i	− 1.15778i	0.815406	−	0.578889i	$-0.196514\pi$
$500$	0	0
$501$	243.490	0.486008
$502$	0	0
$503$	414.732i	0.824517i	0.911067	+	0.412259i	$0.135260\pi$
$503$	414.732i	0.824517i	−0.911067	+	0.412259i	$0.864740\pi$
$504$	0	0
$505$	−115.854	−0.229414
$506$	0	0
$507$	171.685i	0.338629i
$508$	0	0
$509$	−884.588	−1.73789	−0.868946	−	0.494906i	$-0.835202\pi$
$509$	−884.588	−1.73789	−0.868946	+	0.494906i	$0.835202\pi$
$510$	0	0
$511$	351.885i	0.688619i
$512$	0	0
$513$	−355.833	−0.693631
$514$	0	0
$515$	132.674i	0.257619i
$516$	0	0
$517$	−15.0599	−0.0291294
$518$	0	0
$519$	− 1600.83i	− 3.08446i
$520$	0	0
$521$	555.582	1.06638	0.533188	−	0.845997i	$-0.320994\pi$
$521$	555.582	1.06638	0.533188	+	0.845997i	$0.320994\pi$
$522$	0	0
$523$	− 762.487i	− 1.45791i	−0.684562	−	0.728955i	$-0.740006\pi$
$523$	− 762.487i	− 1.45791i	0.684562	−	0.728955i	$-0.259994\pi$
$524$	0	0
$525$	133.798	0.254853
$526$	0	0
$527$	− 528.630i	− 1.00309i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	− 556.283i	− 1.04761i
$532$	0	0
$533$	434.655	0.815488
$534$	0	0
$535$	− 125.183i	− 0.233986i
$536$	0	0
$537$	−1262.74	−2.35146
$538$	0	0
$539$	70.1298i	0.130111i
$540$	0	0
$541$	−274.240	−0.506913	−0.253457	−	0.967347i	$-0.581568\pi$
$541$	−274.240	−0.506913	−0.253457	+	0.967347i	$0.581568\pi$
$542$	0	0
$543$	1304.80i	2.40294i
$544$	0	0
$545$	327.010	0.600018
$546$	0	0
$547$	237.473i	0.434138i	0.976156	+	0.217069i	$0.0696496\pi$
$547$	237.473i	0.434138i	−0.976156	+	0.217069i	$0.930350\pi$
$548$	0	0
$549$	−411.400	−0.749362
$550$	0	0
$551$	− 424.721i	− 0.770819i
$552$	0	0
$553$	173.100	0.313019
$554$	0	0
$555$	− 449.830i	− 0.810505i
$556$	0	0
$557$	331.415	0.595000	0.297500	−	0.954722i	$-0.403847\pi$
$557$	331.415	0.595000	0.297500	+	0.954722i	$0.403847\pi$
$558$	0	0
$559$	551.770i	0.987066i
$560$	0	0
$561$	−487.356	−0.868727
$562$	0	0
$563$	499.944i	0.888000i	0.896027	+	0.444000i	$0.146441\pi$
$563$	499.944i	0.888000i	−0.896027	+	0.444000i	$0.853559\pi$
$564$	0	0
$565$	−108.459	−0.191963
$566$	0	0
$567$	378.155i	0.666941i
$568$	0	0
$569$	98.3777	0.172896	0.0864479	−	0.996256i	$-0.472448\pi$
$569$	98.3777	0.172896	0.0864479	+	0.996256i	$0.472448\pi$
$570$	0	0
$571$	− 9.53563i	− 0.0166999i	−0.999965	−	0.00834994i	$-0.997342\pi$
$571$	− 9.53563i	− 0.0166999i	0.999965	−	0.00834994i	$-0.00265790\pi$
$572$	0	0
$573$	1632.84	2.84964
$574$	0	0
$575$	− 23.9792i	− 0.0417029i
$576$	0	0
$577$	−108.150	−0.187435	−0.0937177	−	0.995599i	$-0.529875\pi$
$577$	−108.150	−0.187435	−0.0937177	+	0.995599i	$0.529875\pi$
$578$	0	0
$579$	449.208i	0.775834i
$580$	0	0
$581$	−848.932	−1.46116
$582$	0	0
$583$	− 121.574i	− 0.208532i
$584$	0	0
$585$	−461.209	−0.788392
$586$	0	0
$587$	− 492.786i	− 0.839499i	−0.907640	−	0.419749i	$-0.862118\pi$
$587$	− 492.786i	− 0.839499i	0.907640	−	0.419749i	$-0.137882\pi$
$588$	0	0
$589$	−140.240	−0.238099
$590$	0	0
$591$	− 133.270i	− 0.225499i
$592$	0	0
$593$	−515.766	−0.869757	−0.434878	−	0.900489i	$-0.643209\pi$
$593$	−515.766	−0.869757	−0.434878	+	0.900489i	$0.643209\pi$
$594$	0	0
$595$	− 345.443i	− 0.580577i
$596$	0	0
$597$	−1717.36	−2.87665
$598$	0	0
$599$	− 1031.81i	− 1.72255i	−0.508137	−	0.861276i	$-0.669666\pi$
$599$	− 1031.81i	− 1.72255i	0.508137	−	0.861276i	$-0.330334\pi$
$600$	0	0
$601$	1027.64	1.70988	0.854939	−	0.518729i	$-0.173595\pi$
$601$	1027.64	1.70988	0.854939	+	0.518729i	$0.173595\pi$
$602$	0	0
$603$	− 1853.43i	− 3.07369i
$604$	0	0
$605$	248.214	0.410270
$606$	0	0
$607$	− 415.867i	− 0.685119i	−0.939496	−	0.342559i	$-0.888706\pi$
$607$	− 415.867i	− 0.685119i	0.939496	−	0.342559i	$-0.111294\pi$
$608$	0	0
$609$	−1436.10	−2.35812
$610$	0	0
$611$	55.5046i	0.0908422i
$612$	0	0
$613$	−37.3898	−0.0609948	−0.0304974	−	0.999535i	$-0.509709\pi$
$613$	−37.3898	−0.0609948	−0.0304974	+	0.999535i	$0.509709\pi$
$614$	0	0
$615$	− 431.008i	− 0.700827i
$616$	0	0
$617$	738.418	1.19679	0.598394	−	0.801202i	$-0.295806\pi$
$617$	738.418	1.19679	0.598394	+	0.801202i	$0.295806\pi$
$618$	0	0
$619$	355.422i	0.574188i	0.957902	+	0.287094i	$0.0926892\pi$
$619$	355.422i	0.574188i	−0.957902	+	0.287094i	$0.907311\pi$
$620$	0	0
$621$	215.631	0.347231
$622$	0	0
$623$	− 660.536i	− 1.06025i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	129.291i	0.206206i
$628$	0	0
$629$	−1161.38	−1.84640
$630$	0	0
$631$	705.901i	1.11870i	0.828931	+	0.559351i	$0.188950\pi$
$631$	705.901i	1.11870i	−0.828931	+	0.559351i	$0.811050\pi$
$632$	0	0
$633$	−332.213	−0.524823
$634$	0	0
$635$	126.819i	0.199715i
$636$	0	0
$637$	258.470	0.405761
$638$	0	0
$639$	− 1675.89i	− 2.62268i
$640$	0	0
$641$	61.7900	0.0963963	0.0481981	−	0.998838i	$-0.484652\pi$
$641$	61.7900	0.0963963	0.0481981	+	0.998838i	$0.484652\pi$
$642$	0	0
$643$	237.659i	0.369609i	0.982775	+	0.184804i	$0.0591652\pi$
$643$	237.659i	0.369609i	−0.982775	+	0.184804i	$0.940835\pi$
$644$	0	0
$645$	547.140	0.848280
$646$	0	0
$647$	− 1243.39i	− 1.92177i	−0.276942	−	0.960887i	$-0.589321\pi$
$647$	− 1243.39i	− 1.92177i	0.276942	−	0.960887i	$-0.410679\pi$
$648$	0	0
$649$	−99.3563	−0.153091
$650$	0	0
$651$	474.191i	0.728404i
$652$	0	0
$653$	−596.394	−0.913314	−0.456657	−	0.889643i	$-0.650953\pi$
$653$	−596.394	−0.913314	−0.456657	+	0.889643i	$0.650953\pi$
$654$	0	0
$655$	− 246.761i	− 0.376734i
$656$	0	0
$657$	−1202.79	−1.83073
$658$	0	0
$659$	1262.74i	1.91615i	0.286518	+	0.958075i	$0.407502\pi$
$659$	1262.74i	1.91615i	−0.286518	+	0.958075i	$0.592498\pi$
$660$	0	0
$661$	231.900	0.350832	0.175416	−	0.984494i	$-0.443873\pi$
$661$	231.900	0.350832	0.175416	+	0.984494i	$0.443873\pi$
$662$	0	0
$663$	1796.19i	2.70919i
$664$	0	0
$665$	−91.6427	−0.137809
$666$	0	0
$667$	257.376i	0.385871i
$668$	0	0
$669$	−932.894	−1.39446
$670$	0	0
$671$	73.4790i	0.109507i
$672$	0	0
$673$	−316.183	−0.469812	−0.234906	−	0.972018i	$-0.575478\pi$
$673$	−316.183	−0.469812	−0.234906	+	0.972018i	$0.575478\pi$
$674$	0	0
$675$	224.810i	0.333052i
$676$	0	0
$677$	402.761	0.594921	0.297460	−	0.954734i	$-0.403860\pi$
$677$	402.761	0.594921	0.297460	+	0.954734i	$0.403860\pi$
$678$	0	0
$679$	187.884i	0.276707i
$680$	0	0
$681$	1209.84	1.77656
$682$	0	0
$683$	− 153.709i	− 0.225050i	−0.993649	−	0.112525i	$-0.964106\pi$
$683$	− 153.709i	− 0.225050i	0.993649	−	0.112525i	$-0.0358938\pi$
$684$	0	0
$685$	98.2757	0.143468
$686$	0	0
$687$	− 1917.80i	− 2.79156i
$688$	0	0
$689$	−448.072	−0.650323
$690$	0	0
$691$	− 218.085i	− 0.315608i	−0.987470	−	0.157804i	$-0.949559\pi$
$691$	− 218.085i	− 0.315608i	0.987470	−	0.157804i	$-0.0504414\pi$
$692$	0	0
$693$	289.814	0.418203
$694$	0	0
$695$	89.9962i	0.129491i
$696$	0	0
$697$	−1112.79	−1.59654
$698$	0	0
$699$	− 386.403i	− 0.552793i
$700$	0	0
$701$	603.827	0.861379	0.430690	−	0.902500i	$-0.358270\pi$
$701$	603.827	0.861379	0.430690	+	0.902500i	$0.358270\pi$
$702$	0	0
$703$	308.104i	0.438270i
$704$	0	0
$705$	55.0389	0.0780694
$706$	0	0
$707$	− 268.312i	− 0.379507i
$708$	0	0
$709$	−436.716	−0.615961	−0.307981	−	0.951393i	$-0.599653\pi$
$709$	−436.716	−0.615961	−0.307981	+	0.951393i	$0.599653\pi$
$710$	0	0
$711$	591.679i	0.832179i
$712$	0	0
$713$	84.9841	0.119192
$714$	0	0
$715$	82.3754i	0.115210i
$716$	0	0
$717$	−62.3570	−0.0869693
$718$	0	0
$719$	1126.01i	1.56607i	0.621977	+	0.783036i	$0.286330\pi$
$719$	1126.01i	1.56607i	−0.621977	+	0.783036i	$0.713670\pi$
$720$	0	0
$721$	−307.266	−0.426166
$722$	0	0
$723$	− 102.021i	− 0.141108i
$724$	0	0
$725$	−268.333	−0.370115
$726$	0	0
$727$	608.262i	0.836675i	0.908292	+	0.418337i	$0.137387\pi$
$727$	608.262i	0.836675i	−0.908292	+	0.418337i	$0.862613\pi$
$728$	0	0
$729$	798.414	1.09522
$730$	0	0
$731$	− 1412.62i	− 1.93245i
$732$	0	0
$733$	135.455	0.184795	0.0923975	−	0.995722i	$-0.470547\pi$
$733$	135.455	0.184795	0.0923975	+	0.995722i	$0.470547\pi$
$734$	0	0
$735$	− 256.301i	− 0.348709i
$736$	0	0
$737$	−331.037	−0.449168
$738$	0	0
$739$	− 1091.70i	− 1.47727i	−0.674106	−	0.738635i	$-0.735471\pi$
$739$	− 1091.70i	− 1.47727i	0.674106	−	0.738635i	$-0.264529\pi$
$740$	0	0
$741$	476.513	0.643067
$742$	0	0
$743$	− 477.809i	− 0.643081i	−0.946896	−	0.321540i	$-0.895799\pi$
$743$	− 477.809i	− 0.643081i	0.946896	−	0.321540i	$-0.104201\pi$
$744$	0	0
$745$	−198.463	−0.266393
$746$	0	0
$747$	− 2901.77i	− 3.88456i
$748$	0	0
$749$	289.916	0.387071
$750$	0	0
$751$	− 34.8894i	− 0.0464572i	−0.999730	−	0.0232286i	$-0.992605\pi$
$751$	− 34.8894i	− 0.0464572i	0.999730	−	0.0232286i	$-0.00739457\pi$
$752$	0	0
$753$	1539.95	2.04509
$754$	0	0
$755$	− 149.305i	− 0.197755i
$756$	0	0
$757$	933.790	1.23354	0.616770	−	0.787143i	$-0.288441\pi$
$757$	933.790	1.23354	0.616770	+	0.787143i	$0.288441\pi$
$758$	0	0
$759$	− 78.3488i	− 0.103226i
$760$	0	0
$761$	−1362.21	−1.79003	−0.895015	−	0.446035i	$-0.852836\pi$
$761$	−1362.21	−1.79003	−0.895015	+	0.446035i	$0.852836\pi$
$762$	0	0
$763$	757.337i	0.992578i
$764$	0	0
$765$	1180.77	1.54349
$766$	0	0
$767$	366.186i	0.477427i
$768$	0	0
$769$	−818.414	−1.06426	−0.532129	−	0.846663i	$-0.678608\pi$
$769$	−818.414	−1.06426	−0.532129	+	0.846663i	$0.678608\pi$
$770$	0	0
$771$	1678.26i	2.17673i
$772$	0	0
$773$	−208.216	−0.269361	−0.134680	−	0.990889i	$-0.543001\pi$
$773$	−208.216	−0.269361	−0.134680	+	0.990889i	$0.543001\pi$
$774$	0	0
$775$	88.6020i	0.114325i
$776$	0	0
$777$	1041.78	1.34078
$778$	0	0
$779$	295.212i	0.378963i
$780$	0	0
$781$	−299.327	−0.383261
$782$	0	0
$783$	− 2412.96i	− 3.08169i
$784$	0	0
$785$	−279.733	−0.356348
$786$	0	0
$787$	− 247.005i	− 0.313856i	−0.987610	−	0.156928i	$-0.949841\pi$
$787$	− 247.005i	− 0.313856i	0.987610	−	0.156928i	$-0.0501591\pi$
$788$	0	0
$789$	502.248	0.636563
$790$	0	0
$791$	− 251.186i	− 0.317554i
$792$	0	0
$793$	270.814	0.341505
$794$	0	0
$795$	444.313i	0.558884i
$796$	0	0
$797$	−1417.09	−1.77804	−0.889018	−	0.457872i	$-0.848612\pi$
$797$	−1417.09	−1.77804	−0.889018	+	0.457872i	$0.848612\pi$
$798$	0	0
$799$	− 142.101i	− 0.177848i
$800$	0	0
$801$	2257.80	2.81873
$802$	0	0
$803$	214.827i	0.267531i
$804$	0	0
$805$	55.5345	0.0689869
$806$	0	0
$807$	− 1140.09i	− 1.41275i
$808$	0	0
$809$	−1007.85	−1.24580	−0.622901	−	0.782301i	$-0.714046\pi$
$809$	−1007.85	−1.24580	−0.622901	+	0.782301i	$0.714046\pi$
$810$	0	0
$811$	376.346i	0.464052i	0.972710	+	0.232026i	$0.0745354\pi$
$811$	376.346i	0.464052i	−0.972710	+	0.232026i	$0.925465\pi$
$812$	0	0
$813$	2017.19	2.48116
$814$	0	0
$815$	29.0970i	0.0357018i
$816$	0	0
$817$	−374.755	−0.458697
$818$	0	0
$819$	− 1068.14i	− 1.30420i
$820$	0	0
$821$	−652.187	−0.794382	−0.397191	−	0.917736i	$-0.630015\pi$
$821$	−652.187	−0.794382	−0.397191	+	0.917736i	$0.630015\pi$
$822$	0	0
$823$	1105.25i	1.34295i	0.741027	+	0.671476i	$0.234339\pi$
$823$	1105.25i	1.34295i	−0.741027	+	0.671476i	$0.765661\pi$
$824$	0	0
$825$	81.6843	0.0990112
$826$	0	0
$827$	− 893.934i	− 1.08094i	−0.841365	−	0.540468i	$-0.818247\pi$
$827$	− 893.934i	− 1.08094i	0.841365	−	0.540468i	$-0.181753\pi$
$828$	0	0
$829$	950.405	1.14645	0.573224	−	0.819399i	$-0.305693\pi$
$829$	950.405	1.14645	0.573224	+	0.819399i	$0.305693\pi$
$830$	0	0
$831$	1617.15i	1.94603i
$832$	0	0
$833$	−661.725	−0.794387
$834$	0	0
$835$	105.366i	0.126187i
$836$	0	0
$837$	−796.746	−0.951907
$838$	0	0
$839$	− 567.229i	− 0.676078i	−0.941132	−	0.338039i	$-0.890236\pi$
$839$	− 567.229i	− 0.676078i	0.941132	−	0.338039i	$-0.109764\pi$
$840$	0	0
$841$	2039.11	2.42463
$842$	0	0
$843$	− 1938.08i	− 2.29903i
$844$	0	0
$845$	−74.2936	−0.0879214
$846$	0	0
$847$	574.849i	0.678689i
$848$	0	0
$849$	−1221.08	−1.43825
$850$	0	0
$851$	− 186.708i	− 0.219398i
$852$	0	0
$853$	−1461.09	−1.71289	−0.856444	−	0.516241i	$-0.827331\pi$
$853$	−1461.09	−1.71289	−0.856444	+	0.516241i	$0.827331\pi$
$854$	0	0
$855$	− 313.248i	− 0.366372i
$856$	0	0
$857$	831.399	0.970127	0.485064	−	0.874479i	$-0.338796\pi$
$857$	831.399	0.970127	0.485064	+	0.874479i	$0.338796\pi$
$858$	0	0
$859$	− 536.680i	− 0.624773i	−0.949955	−	0.312386i	$-0.898872\pi$
$859$	− 536.680i	− 0.624773i	0.949955	−	0.312386i	$-0.101128\pi$
$860$	0	0
$861$	998.193	1.15934
$862$	0	0
$863$	85.6990i	0.0993036i	0.998767	+	0.0496518i	$0.0158112\pi$
$863$	85.6990i	0.0993036i	−0.998767	+	0.0496518i	$0.984189\pi$
$864$	0	0
$865$	692.733	0.800847
$866$	0	0
$867$	− 3105.20i	− 3.58154i
$868$	0	0
$869$	105.678	0.121609
$870$	0	0
$871$	1220.07i	1.40077i
$872$	0	0
$873$	−642.214	−0.735641
$874$	0	0
$875$	57.8987i	0.0661699i
$876$	0	0
$877$	−727.697	−0.829757	−0.414878	−	0.909877i	$-0.636176\pi$
$877$	−727.697	−0.829757	−0.414878	+	0.909877i	$0.636176\pi$
$878$	0	0
$879$	2612.01i	2.97157i
$880$	0	0
$881$	−908.233	−1.03091	−0.515456	−	0.856916i	$-0.672377\pi$
$881$	−908.233	−1.03091	−0.515456	+	0.856916i	$0.672377\pi$
$882$	0	0
$883$	339.481i	0.384464i	0.981350	+	0.192232i	$0.0615726\pi$
$883$	339.481i	0.384464i	−0.981350	+	0.192232i	$0.938427\pi$
$884$	0	0
$885$	363.114	0.410299
$886$	0	0
$887$	492.257i	0.554969i	0.960730	+	0.277484i	$0.0895007\pi$
$887$	492.257i	0.554969i	−0.960730	+	0.277484i	$0.910499\pi$
$888$	0	0
$889$	−293.706	−0.330378
$890$	0	0
$891$	230.866i	0.259109i
$892$	0	0
$893$	−37.6980	−0.0422150
$894$	0	0
$895$	− 546.427i	− 0.610533i
$896$	0	0
$897$	−288.762	−0.321919
$898$	0	0
$899$	− 950.995i	− 1.05784i
$900$	0	0
$901$	1147.14	1.27318
$902$	0	0
$903$	1267.15i	1.40327i
$904$	0	0
$905$	−564.629	−0.623899
$906$	0	0
$907$	1057.96i	1.16644i	0.812313	+	0.583222i	$0.198208\pi$
$907$	1057.96i	1.16644i	−0.812313	+	0.583222i	$0.801792\pi$
$908$	0	0
$909$	917.126	1.00894
$910$	0	0
$911$	752.266i	0.825759i	0.910786	+	0.412879i	$0.135477\pi$
$911$	752.266i	0.825759i	−0.910786	+	0.412879i	$0.864523\pi$
$912$	0	0
$913$	−518.277	−0.567664
$914$	0	0
$915$	− 268.541i	− 0.293488i
$916$	0	0
$917$	571.485	0.623211
$918$	0	0
$919$	721.246i	0.784816i	0.919791	+	0.392408i	$0.128358\pi$
$919$	721.246i	0.784816i	−0.919791	+	0.392408i	$0.871642\pi$
$920$	0	0
$921$	−229.036	−0.248682
$922$	0	0
$923$	1103.19i	1.19523i
$924$	0	0
$925$	194.656	0.210439
$926$	0	0
$927$	− 1050.28i	− 1.13298i
$928$	0	0
$929$	−1132.09	−1.21861	−0.609304	−	0.792937i	$-0.708551\pi$
$929$	−1132.09	−1.21861	−0.609304	+	0.792937i	$0.708551\pi$
$930$	0	0
$931$	175.549i	0.188560i
$932$	0	0
$933$	−242.388	−0.259794
$934$	0	0
$935$	− 210.895i	− 0.225556i
$936$	0	0
$937$	235.415	0.251244	0.125622	−	0.992078i	$-0.459907\pi$
$937$	235.415	0.251244	0.125622	+	0.992078i	$0.459907\pi$
$938$	0	0
$939$	2567.83i	2.73465i
$940$	0	0
$941$	1134.89	1.20605	0.603024	−	0.797723i	$-0.293962\pi$
$941$	1134.89	1.20605	0.603024	+	0.797723i	$0.293962\pi$
$942$	0	0
$943$	− 178.895i	− 0.189709i
$944$	0	0
$945$	−520.649	−0.550951
$946$	0	0
$947$	− 1830.66i	− 1.93311i	−0.256454	−	0.966556i	$-0.582554\pi$
$947$	− 1830.66i	− 1.93311i	0.256454	−	0.966556i	$-0.417446\pi$
$948$	0	0
$949$	791.765	0.834315
$950$	0	0
$951$	− 1603.98i	− 1.68662i
$952$	0	0
$953$	−874.902	−0.918050	−0.459025	−	0.888423i	$-0.651801\pi$
$953$	−874.902	−0.918050	−0.459025	+	0.888423i	$0.651801\pi$
$954$	0	0
$955$	706.584i	0.739878i
$956$	0	0
$957$	−876.744	−0.916138
$958$	0	0
$959$	227.601i	0.237332i
$960$	0	0
$961$	646.987	0.673244
$962$	0	0
$963$	990.974i	1.02905i
$964$	0	0
$965$	−194.387	−0.201437
$966$	0	0
$967$	1100.44i	1.13799i	0.822341	+	0.568995i	$0.192668\pi$
$967$	1100.44i	1.13799i	−0.822341	+	0.568995i	$0.807332\pi$
$968$	0	0
$969$	−1219.95	−1.25898
$970$	0	0
$971$	− 958.235i	− 0.986853i	−0.869787	−	0.493427i	$-0.835744\pi$
$971$	− 958.235i	− 0.986853i	0.869787	−	0.493427i	$-0.164256\pi$
$972$	0	0
$973$	−208.426	−0.214210
$974$	0	0
$975$	− 301.055i	− 0.308774i
$976$	0	0
$977$	960.906	0.983527	0.491764	−	0.870729i	$-0.336352\pi$
$977$	960.906	0.983527	0.491764	+	0.870729i	$0.336352\pi$
$978$	0	0
$979$	− 403.260i	− 0.411910i
$980$	0	0
$981$	−2588.68	−2.63882
$982$	0	0
$983$	1011.91i	1.02941i	0.857367	+	0.514705i	$0.172099\pi$
$983$	1011.91i	1.02941i	−0.857367	+	0.514705i	$0.827901\pi$
$984$	0	0
$985$	57.6703	0.0585485
$986$	0	0
$987$	127.467i	0.129146i
$988$	0	0
$989$	227.097	0.229623
$990$	0	0
$991$	− 389.053i	− 0.392586i	−0.980545	−	0.196293i	$-0.937110\pi$
$991$	− 389.053i	− 0.392586i	0.980545	−	0.196293i	$-0.0628904\pi$
$992$	0	0
$993$	−2753.14	−2.77255
$994$	0	0
$995$	− 743.156i	− 0.746890i
$996$	0	0
$997$	1649.34	1.65431	0.827153	−	0.561976i	$-0.189959\pi$
$997$	1649.34	1.65431	0.827153	+	0.561976i	$0.189959\pi$
$998$	0	0
$999$	1750.43i	1.75218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.4	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.53	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.4	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.53	yes	56	4.3	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.16732i q^{3} +2.23607 q^{5} +5.17862i q^{7} -17.7012 q^{9} +O(q^{10})\)	\(q-5.16732i q^{3} +2.23607 q^{5} +5.17862i q^{7} -17.7012 q^{9} +3.16157i q^{11} +11.6522 q^{13} -11.5545i q^{15} -29.8317 q^{17} +7.91406i q^{19} +26.7596 q^{21} -4.79583i q^{23} +5.00000 q^{25} +44.9621i q^{27} -53.6667 q^{29} +17.7204i q^{31} +16.3369 q^{33} +11.5797i q^{35} +38.9312 q^{37} -60.2109i q^{39} +37.3023 q^{41} +47.3531i q^{43} -39.5812 q^{45} +4.76342i q^{47} +22.1819 q^{49} +154.150i q^{51} -38.4537 q^{53} +7.06949i q^{55} +40.8945 q^{57} +31.4262i q^{59} +23.2413 q^{61} -91.6679i q^{63} +26.0552 q^{65} +104.707i q^{67} -24.7816 q^{69} +94.6766i q^{71} +67.9495 q^{73} -25.8366i q^{75} -16.3726 q^{77} -33.4259i q^{79} +73.0225 q^{81} +163.930i q^{83} -66.7057 q^{85} +277.313i q^{87} -127.551 q^{89} +60.3425i q^{91} +91.5671 q^{93} +17.6964i q^{95} +36.2808 q^{97} -55.9637i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more