LMFDB - Embedded newform 2240.2.b.e.1121.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3534400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 $ x^6 - 2x^5 - 3x^4 + 16x^3 + x^2 - 12x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.6
Root			$2.19082 + 1.44755i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.e.1121.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.89511i q^{3} +1.00000i q^{5} -1.00000 q^{7} -5.38164 q^{9} +O(q^{10})$	$q+2.89511i q^{3} +1.00000i q^{5} -1.00000 q^{7} -5.38164 q^{9} +2.38164i q^{11} +3.40857i q^{13} -2.89511 q^{15} -1.92204 q^{17} +5.79021i q^{19} -2.89511i q^{21} +8.76328 q^{23} -1.00000 q^{25} -6.89511i q^{27} +5.86818i q^{29} +1.02693 q^{31} -6.89511 q^{33} -1.00000i q^{35} -2.81714i q^{37} -9.86818 q^{39} -11.7902 q^{41} -2.00000i q^{43} -5.38164i q^{45} -5.40857 q^{47} +1.00000 q^{49} -5.56450i q^{51} -6.97307i q^{53} -2.38164 q^{55} -16.7633 q^{57} -6.97307i q^{59} -14.7633i q^{61} +5.38164 q^{63} -3.40857 q^{65} -0.209787i q^{67} +25.3706i q^{69} +13.7902 q^{71} -14.7633 q^{73} -2.89511i q^{75} -2.38164i q^{77} +13.4486 q^{79} +3.81714 q^{81} +5.79021i q^{83} -1.92204i q^{85} -16.9890 q^{87} -1.23672 q^{89} -3.40857i q^{91} +2.97307i q^{93} -5.79021 q^{95} +5.92204 q^{97} -12.8171i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 10 * q^9	$ 6 q - 6 q^{7} - 10 q^{9} - 4 q^{17} + 8 q^{23} - 6 q^{25} + 16 q^{31} - 24 q^{33} - 32 q^{39} - 36 q^{41} - 20 q^{47} + 6 q^{49} + 8 q^{55} - 56 q^{57} + 10 q^{63} - 8 q^{65} + 48 q^{71} - 44 q^{73} - 16 q^{79} - 2 q^{81} - 20 q^{87} - 52 q^{89} + 28 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 10 * q^9 - 4 * q^17 + 8 * q^23 - 6 * q^25 + 16 * q^31 - 24 * q^33 - 32 * q^39 - 36 * q^41 - 20 * q^47 + 6 * q^49 + 8 * q^55 - 56 * q^57 + 10 * q^63 - 8 * q^65 + 48 * q^71 - 44 * q^73 - 16 * q^79 - 2 * q^81 - 20 * q^87 - 52 * q^89 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	2.89511i	1.67149i	0.549117	+	0.835745i	$0.314964\pi$
$3$	2.89511i	1.67149i	−0.549117	+	0.835745i	$0.685036\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−5.38164	−1.79388
$10$	0	0
$11$	2.38164i	0.718092i	0.933320	+	0.359046i	$0.116898\pi$
$11$	2.38164i	0.718092i	−0.933320	+	0.359046i	$0.883102\pi$
$12$	0	0
$13$	3.40857i	0.945368i	0.881232	+	0.472684i	$0.156715\pi$
$13$	3.40857i	0.945368i	−0.881232	+	0.472684i	$0.843285\pi$
$14$	0	0
$15$	−2.89511	−0.747513
$16$	0	0
$17$	−1.92204	−0.466162	−0.233081	−	0.972457i	$-0.574881\pi$
$17$	−1.92204	−0.466162	−0.233081	+	0.972457i	$0.574881\pi$
$18$	0	0
$19$	5.79021i	1.32837i	0.747570	+	0.664183i	$0.231220\pi$
$19$	5.79021i	1.32837i	−0.747570	+	0.664183i	$0.768780\pi$
$20$	0	0
$21$	− 2.89511i	− 0.631764i
$22$	0	0
$23$	8.76328	1.82727	0.913635	−	0.406534i	$-0.133263\pi$
$23$	8.76328	1.82727	0.913635	+	0.406534i	$0.133263\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 6.89511i	− 1.32696i
$28$	0	0
$29$	5.86818i	1.08969i	0.838536	+	0.544847i	$0.183412\pi$
$29$	5.86818i	1.08969i	−0.838536	+	0.544847i	$0.816588\pi$
$30$	0	0
$31$	1.02693	0.184442	0.0922210	−	0.995739i	$-0.470603\pi$
$31$	1.02693	0.184442	0.0922210	+	0.995739i	$0.470603\pi$
$32$	0	0
$33$	−6.89511	−1.20028
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	− 2.81714i	− 0.463135i	−0.972819	−	0.231568i	$-0.925615\pi$
$37$	− 2.81714i	− 0.463135i	0.972819	−	0.231568i	$-0.0743855\pi$
$38$	0	0
$39$	−9.86818	−1.58017
$40$	0	0
$41$	−11.7902	−1.84132	−0.920661	−	0.390363i	$-0.872349\pi$
$41$	−11.7902	−1.84132	−0.920661	+	0.390363i	$0.872349\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	0	0
$45$	− 5.38164i	− 0.802248i
$46$	0	0
$47$	−5.40857	−0.788921	−0.394461	−	0.918913i	$-0.629069\pi$
$47$	−5.40857	−0.788921	−0.394461	+	0.918913i	$0.629069\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 5.56450i	− 0.779186i
$52$	0	0
$53$	− 6.97307i	− 0.957825i	−0.877863	−	0.478912i	$-0.841031\pi$
$53$	− 6.97307i	− 0.957825i	0.877863	−	0.478912i	$-0.158969\pi$
$54$	0	0
$55$	−2.38164	−0.321141
$56$	0	0
$57$	−16.7633	−2.22035
$58$	0	0
$59$	− 6.97307i	− 0.907816i	−0.891048	−	0.453908i	$-0.850029\pi$
$59$	− 6.97307i	− 0.907816i	0.891048	−	0.453908i	$-0.149971\pi$
$60$	0	0
$61$	− 14.7633i	− 1.89024i	−0.326717	−	0.945122i	$-0.605942\pi$
$61$	− 14.7633i	− 1.89024i	0.326717	−	0.945122i	$-0.394058\pi$
$62$	0	0
$63$	5.38164	0.678023
$64$	0	0
$65$	−3.40857	−0.422781
$66$	0	0
$67$	− 0.209787i	− 0.0256296i	−0.999918	−	0.0128148i	$-0.995921\pi$
$67$	− 0.209787i	− 0.0256296i	0.999918	−	0.0128148i	$-0.00407918\pi$
$68$	0	0
$69$	25.3706i	3.05427i
$70$	0	0
$71$	13.7902	1.63660	0.818299	−	0.574793i	$-0.194918\pi$
$71$	13.7902	1.63660	0.818299	+	0.574793i	$0.194918\pi$
$72$	0	0
$73$	−14.7633	−1.72791	−0.863956	−	0.503568i	$-0.832020\pi$
$73$	−14.7633	−1.72791	−0.863956	+	0.503568i	$0.832020\pi$
$74$	0	0
$75$	− 2.89511i	− 0.334298i
$76$	0	0
$77$	− 2.38164i	− 0.271413i
$78$	0	0
$79$	13.4486	1.51309	0.756543	−	0.653944i	$-0.226887\pi$
$79$	13.4486	1.51309	0.756543	+	0.653944i	$0.226887\pi$
$80$	0	0
$81$	3.81714	0.424127
$82$	0	0
$83$	5.79021i	0.635558i	0.948165	+	0.317779i	$0.102937\pi$
$83$	5.79021i	0.635558i	−0.948165	+	0.317779i	$0.897063\pi$
$84$	0	0
$85$	− 1.92204i	− 0.208474i
$86$	0	0
$87$	−16.9890	−1.82141
$88$	0	0
$89$	−1.23672	−0.131092	−0.0655458	−	0.997850i	$-0.520879\pi$
$89$	−1.23672	−0.131092	−0.0655458	+	0.997850i	$0.520879\pi$
$90$	0	0
$91$	− 3.40857i	− 0.357315i
$92$	0	0
$93$	2.97307i	0.308293i
$94$	0	0
$95$	−5.79021	−0.594063
$96$	0	0
$97$	5.92204	0.601292	0.300646	−	0.953736i	$-0.402798\pi$
$97$	5.92204	0.601292	0.300646	+	0.953736i	$0.402798\pi$
$98$	0	0
$99$	− 12.8171i	− 1.28817i
$100$	0	0
$101$	9.58043i	0.953288i	0.879096	+	0.476644i	$0.158147\pi$
$101$	9.58043i	0.953288i	−0.879096	+	0.476644i	$0.841853\pi$
$102$	0	0
$103$	−6.59143	−0.649473	−0.324736	−	0.945805i	$-0.605276\pi$
$103$	−6.59143	−0.649473	−0.324736	+	0.945805i	$0.605276\pi$
$104$	0	0
$105$	2.89511	0.282533
$106$	0	0
$107$	− 0.0538591i	− 0.00520676i	−0.999997	−	0.00260338i	$-0.999171\pi$
$107$	− 0.0538591i	− 0.00520676i	0.999997	−	0.00260338i	$-0.000828682\pi$
$108$	0	0
$109$	9.44860i	0.905012i	0.891761	+	0.452506i	$0.149470\pi$
$109$	9.44860i	0.905012i	−0.891761	+	0.452506i	$0.850530\pi$
$110$	0	0
$111$	8.15593	0.774126
$112$	0	0
$113$	12.5535	1.18093	0.590467	−	0.807062i	$-0.298944\pi$
$113$	12.5535	1.18093	0.590467	+	0.807062i	$0.298944\pi$
$114$	0	0
$115$	8.76328i	0.817180i
$116$	0	0
$117$	− 18.3437i	− 1.69588i
$118$	0	0
$119$	1.92204	0.176193
$120$	0	0
$121$	5.32778	0.484344
$122$	0	0
$123$	− 34.1339i	− 3.07775i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−2.97307	−0.263817	−0.131909	−	0.991262i	$-0.542111\pi$
$127$	−2.97307	−0.263817	−0.131909	+	0.991262i	$0.542111\pi$
$128$	0	0
$129$	5.79021	0.509800
$130$	0	0
$131$	− 10.8171i	− 0.945098i	−0.881304	−	0.472549i	$-0.843334\pi$
$131$	− 10.8171i	− 0.945098i	0.881304	−	0.472549i	$-0.156666\pi$
$132$	0	0
$133$	− 5.79021i	− 0.502075i
$134$	0	0
$135$	6.89511	0.593436
$136$	0	0
$137$	2.76328	0.236083	0.118042	−	0.993009i	$-0.462338\pi$
$137$	2.76328	0.236083	0.118042	+	0.993009i	$0.462338\pi$
$138$	0	0
$139$	− 2.81714i	− 0.238947i	−0.992837	−	0.119473i	$-0.961879\pi$
$139$	− 2.81714i	− 0.238947i	0.992837	−	0.119473i	$-0.0381206\pi$
$140$	0	0
$141$	− 15.6584i	− 1.31867i
$142$	0	0
$143$	−8.11800	−0.678861
$144$	0	0
$145$	−5.86818	−0.487326
$146$	0	0
$147$	2.89511i	0.238784i
$148$	0	0
$149$	2.05386i	0.168259i	0.996455	+	0.0841293i	$0.0268109\pi$
$149$	2.05386i	0.168259i	−0.996455	+	0.0841293i	$0.973189\pi$
$150$	0	0
$151$	−9.71225	−0.790372	−0.395186	−	0.918601i	$-0.629320\pi$
$151$	−9.71225	−0.790372	−0.395186	+	0.918601i	$0.629320\pi$
$152$	0	0
$153$	10.3437	0.836239
$154$	0	0
$155$	1.02693i	0.0824850i
$156$	0	0
$157$	2.41957i	0.193103i	0.995328	+	0.0965515i	$0.0307813\pi$
$157$	2.41957i	0.193103i	−0.995328	+	0.0965515i	$0.969219\pi$
$158$	0	0
$159$	20.1878	1.60100
$160$	0	0
$161$	−8.76328	−0.690643
$162$	0	0
$163$	12.5535i	0.983266i	0.870803	+	0.491633i	$0.163600\pi$
$163$	12.5535i	0.983266i	−0.870803	+	0.491633i	$0.836400\pi$
$164$	0	0
$165$	− 6.89511i	− 0.536783i
$166$	0	0
$167$	−24.5694	−1.90124	−0.950620	−	0.310359i	$-0.899551\pi$
$167$	−24.5694	−1.90124	−0.950620	+	0.310359i	$0.899551\pi$
$168$	0	0
$169$	1.38164	0.106280
$170$	0	0
$171$	− 31.1609i	− 2.38293i
$172$	0	0
$173$	− 1.46243i	− 0.111187i	−0.998453	−	0.0555933i	$-0.982295\pi$
$173$	− 1.46243i	− 0.111187i	0.998453	−	0.0555933i	$-0.0177050\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	20.1878	1.51741
$178$	0	0
$179$	8.81714i	0.659024i	0.944151	+	0.329512i	$0.106884\pi$
$179$	8.81714i	0.659024i	−0.944151	+	0.329512i	$0.893116\pi$
$180$	0	0
$181$	17.3168i	1.28715i	0.765385	+	0.643573i	$0.222549\pi$
$181$	17.3168i	1.28715i	−0.765385	+	0.643573i	$0.777451\pi$
$182$	0	0
$183$	42.7413	3.15953
$184$	0	0
$185$	2.81714	0.207120
$186$	0	0
$187$	− 4.57760i	− 0.334747i
$188$	0	0
$189$	6.89511i	0.501545i
$190$	0	0
$191$	15.5025	1.12172	0.560859	−	0.827911i	$-0.310471\pi$
$191$	15.5025	1.12172	0.560859	+	0.827911i	$0.310471\pi$
$192$	0	0
$193$	−8.39757	−0.604470	−0.302235	−	0.953233i	$-0.597733\pi$
$193$	−8.39757	−0.604470	−0.302235	+	0.953233i	$0.597733\pi$
$194$	0	0
$195$	− 9.86818i	− 0.706675i
$196$	0	0
$197$	− 14.1339i	− 1.00700i	−0.863995	−	0.503500i	$-0.832045\pi$
$197$	− 14.1339i	− 1.00700i	0.863995	−	0.503500i	$-0.167955\pi$
$198$	0	0
$199$	20.4996	1.45318	0.726590	−	0.687071i	$-0.241104\pi$
$199$	20.4996	1.45318	0.726590	+	0.687071i	$0.241104\pi$
$200$	0	0
$201$	0.607356	0.0428396
$202$	0	0
$203$	− 5.86818i	− 0.411865i
$204$	0	0
$205$	− 11.7902i	− 0.823464i
$206$	0	0
$207$	−47.1609	−3.27791
$208$	0	0
$209$	−13.7902	−0.953889
$210$	0	0
$211$	25.9621i	1.78730i	0.448762	+	0.893651i	$0.351865\pi$
$211$	25.9621i	1.78730i	−0.448762	+	0.893651i	$0.648135\pi$
$212$	0	0
$213$	39.9241i	2.73556i
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−1.02693	−0.0697125
$218$	0	0
$219$	− 42.7413i	− 2.88819i
$220$	0	0
$221$	− 6.55140i	− 0.440695i
$222$	0	0
$223$	−24.5694	−1.64529	−0.822645	−	0.568555i	$-0.807503\pi$
$223$	−24.5694	−1.64529	−0.822645	+	0.568555i	$0.807503\pi$
$224$	0	0
$225$	5.38164	0.358776
$226$	0	0
$227$	− 20.8412i	− 1.38328i	−0.722241	−	0.691641i	$-0.756888\pi$
$227$	− 20.8412i	− 1.38328i	0.722241	−	0.691641i	$-0.243112\pi$
$228$	0	0
$229$	1.84407i	0.121860i	0.998142	+	0.0609299i	$0.0194066\pi$
$229$	1.84407i	0.121860i	−0.998142	+	0.0609299i	$0.980593\pi$
$230$	0	0
$231$	6.89511	0.453665
$232$	0	0
$233$	−1.12900	−0.0739631	−0.0369816	−	0.999316i	$-0.511774\pi$
$233$	−1.12900	−0.0739631	−0.0369816	+	0.999316i	$0.511774\pi$
$234$	0	0
$235$	− 5.40857i	− 0.352816i
$236$	0	0
$237$	38.9351i	2.52911i
$238$	0	0
$239$	−6.13182	−0.396635	−0.198317	−	0.980138i	$-0.563548\pi$
$239$	−6.13182	−0.396635	−0.198317	+	0.980138i	$0.563548\pi$
$240$	0	0
$241$	25.5804	1.64778	0.823890	−	0.566750i	$-0.191799\pi$
$241$	25.5804	1.64778	0.823890	+	0.566750i	$0.191799\pi$
$242$	0	0
$243$	− 9.63429i	− 0.618040i
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	−19.7364	−1.25579
$248$	0	0
$249$	−16.7633	−1.06233
$250$	0	0
$251$	2.81714i	0.177816i	0.996040	+	0.0889082i	$0.0283378\pi$
$251$	2.81714i	0.177816i	−0.996040	+	0.0889082i	$0.971662\pi$
$252$	0	0
$253$	20.8710i	1.31215i
$254$	0	0
$255$	5.56450	0.348462
$256$	0	0
$257$	−14.3437	−0.894736	−0.447368	−	0.894350i	$-0.647639\pi$
$257$	−14.3437	−0.894736	−0.447368	+	0.894350i	$0.647639\pi$
$258$	0	0
$259$	2.81714i	0.175049i
$260$	0	0
$261$	− 31.5804i	− 1.95478i
$262$	0	0
$263$	−10.2098	−0.629562	−0.314781	−	0.949164i	$-0.601931\pi$
$263$	−10.2098	−0.629562	−0.314781	+	0.949164i	$0.601931\pi$
$264$	0	0
$265$	6.97307	0.428352
$266$	0	0
$267$	− 3.58043i	− 0.219119i
$268$	0	0
$269$	− 8.55350i	− 0.521516i	−0.965404	−	0.260758i	$-0.916028\pi$
$269$	− 8.55350i	− 0.521516i	0.965404	−	0.260758i	$-0.0839724\pi$
$270$	0	0
$271$	19.5804	1.18943	0.594713	−	0.803938i	$-0.297266\pi$
$271$	19.5804	1.18943	0.594713	+	0.803938i	$0.297266\pi$
$272$	0	0
$273$	9.86818	0.597249
$274$	0	0
$275$	− 2.38164i	− 0.143618i
$276$	0	0
$277$	− 12.7633i	− 0.766871i	−0.923568	−	0.383436i	$-0.874741\pi$
$277$	− 12.7633i	− 0.766871i	0.923568	−	0.383436i	$-0.125259\pi$
$278$	0	0
$279$	−5.52657	−0.330867
$280$	0	0
$281$	−25.9621	−1.54877	−0.774384	−	0.632716i	$-0.781940\pi$
$281$	−25.9621	−1.54877	−0.774384	+	0.632716i	$0.781940\pi$
$282$	0	0
$283$	28.5776i	1.69876i	0.527780	+	0.849381i	$0.323025\pi$
$283$	28.5776i	1.69876i	−0.527780	+	0.849381i	$0.676975\pi$
$284$	0	0
$285$	− 16.7633i	− 0.992971i
$286$	0	0
$287$	11.7902	0.695954
$288$	0	0
$289$	−13.3058	−0.782693
$290$	0	0
$291$	17.1449i	1.00505i
$292$	0	0
$293$	12.1719i	0.711087i	0.934660	+	0.355544i	$0.115704\pi$
$293$	12.1719i	0.711087i	−0.934660	+	0.355544i	$0.884296\pi$
$294$	0	0
$295$	6.97307	0.405988
$296$	0	0
$297$	16.4217	0.952882
$298$	0	0
$299$	29.8703i	1.72744i
$300$	0	0
$301$	2.00000i	0.115278i
$302$	0	0
$303$	−27.7364	−1.59341
$304$	0	0
$305$	14.7633	0.845343
$306$	0	0
$307$	12.8412i	0.732889i	0.930440	+	0.366444i	$0.119425\pi$
$307$	12.8412i	0.732889i	−0.930440	+	0.366444i	$0.880575\pi$
$308$	0	0
$309$	− 19.0829i	− 1.08559i
$310$	0	0
$311$	17.1829	0.974350	0.487175	−	0.873304i	$-0.338027\pi$
$311$	17.1829	0.974350	0.487175	+	0.873304i	$0.338027\pi$
$312$	0	0
$313$	−15.2927	−0.864393	−0.432197	−	0.901779i	$-0.642261\pi$
$313$	−15.2927	−0.864393	−0.432197	+	0.901779i	$0.642261\pi$
$314$	0	0
$315$	5.38164i	0.303221i
$316$	0	0
$317$	− 30.8972i	− 1.73536i	−0.497123	−	0.867680i	$-0.665610\pi$
$317$	− 30.8972i	− 1.73536i	0.497123	−	0.867680i	$-0.334390\pi$
$318$	0	0
$319$	−13.9759	−0.782500
$320$	0	0
$321$	0.155928	0.00870304
$322$	0	0
$323$	− 11.1290i	− 0.619234i
$324$	0	0
$325$	− 3.40857i	− 0.189074i
$326$	0	0
$327$	−27.3547	−1.51272
$328$	0	0
$329$	5.40857	0.298184
$330$	0	0
$331$	26.4514i	1.45390i	0.686689	+	0.726951i	$0.259063\pi$
$331$	26.4514i	1.45390i	−0.686689	+	0.726951i	$0.740937\pi$
$332$	0	0
$333$	15.1609i	0.830810i
$334$	0	0
$335$	0.209787	0.0114619
$336$	0	0
$337$	−0.553497	−0.0301509	−0.0150754	−	0.999886i	$-0.504799\pi$
$337$	−0.553497	−0.0301509	−0.0150754	+	0.999886i	$0.504799\pi$
$338$	0	0
$339$	36.3437i	1.97392i
$340$	0	0
$341$	2.44578i	0.132446i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−25.3706	−1.36591
$346$	0	0
$347$	30.0801i	1.61478i	0.590016	+	0.807391i	$0.299121\pi$
$347$	30.0801i	1.61478i	−0.590016	+	0.807391i	$0.700879\pi$
$348$	0	0
$349$	23.2147i	1.24266i	0.783551	+	0.621328i	$0.213406\pi$
$349$	23.2147i	1.24266i	−0.783551	+	0.621328i	$0.786594\pi$
$350$	0	0
$351$	23.5025	1.25447
$352$	0	0
$353$	−30.7654	−1.63748	−0.818738	−	0.574167i	$-0.805326\pi$
$353$	−30.7654	−1.63748	−0.818738	+	0.574167i	$0.805326\pi$
$354$	0	0
$355$	13.7902i	0.731909i
$356$	0	0
$357$	5.56450i	0.294505i
$358$	0	0
$359$	11.4245	0.602962	0.301481	−	0.953472i	$-0.402519\pi$
$359$	11.4245	0.602962	0.301481	+	0.953472i	$0.402519\pi$
$360$	0	0
$361$	−14.5266	−0.764556
$362$	0	0
$363$	15.4245i	0.809576i
$364$	0	0
$365$	− 14.7633i	− 0.772746i
$366$	0	0
$367$	20.9890	1.09562	0.547808	−	0.836604i	$-0.315462\pi$
$367$	20.9890	1.09562	0.547808	+	0.836604i	$0.315462\pi$
$368$	0	0
$369$	63.4507	3.30311
$370$	0	0
$371$	6.97307i	0.362024i
$372$	0	0
$373$	− 23.5804i	− 1.22095i	−0.792036	−	0.610474i	$-0.790979\pi$
$373$	− 23.5804i	− 1.22095i	0.792036	−	0.610474i	$-0.209021\pi$
$374$	0	0
$375$	2.89511	0.149503
$376$	0	0
$377$	−20.0021	−1.03016
$378$	0	0
$379$	24.3976i	1.25322i	0.779333	+	0.626609i	$0.215558\pi$
$379$	24.3976i	1.25322i	−0.779333	+	0.626609i	$0.784442\pi$
$380$	0	0
$381$	− 8.60736i	− 0.440968i
$382$	0	0
$383$	0.311856	0.0159351	0.00796754	−	0.999968i	$-0.497464\pi$
$383$	0.311856	0.0159351	0.00796754	+	0.999968i	$0.497464\pi$
$384$	0	0
$385$	2.38164	0.121380
$386$	0	0
$387$	10.7633i	0.547128i
$388$	0	0
$389$	0.0779639i	0.00395293i	0.999998	+	0.00197646i	$0.000629128\pi$
$389$	0.0779639i	0.00395293i	−0.999998	+	0.00197646i	$0.999371\pi$
$390$	0	0
$391$	−16.8433	−0.851805
$392$	0	0
$393$	31.3168	1.57972
$394$	0	0
$395$	13.4486i	0.676673i
$396$	0	0
$397$	26.8813i	1.34913i	0.738214	+	0.674566i	$0.235669\pi$
$397$	26.8813i	1.34913i	−0.738214	+	0.674566i	$0.764331\pi$
$398$	0	0
$399$	16.7633	0.839214
$400$	0	0
$401$	23.1449	1.15580	0.577901	−	0.816107i	$-0.303872\pi$
$401$	23.1449	1.15580	0.577901	+	0.816107i	$0.303872\pi$
$402$	0	0
$403$	3.50036i	0.174365i
$404$	0	0
$405$	3.81714i	0.189675i
$406$	0	0
$407$	6.70942	0.332574
$408$	0	0
$409$	16.5535	0.818518	0.409259	−	0.912418i	$-0.365787\pi$
$409$	16.5535	0.818518	0.409259	+	0.912418i	$0.365787\pi$
$410$	0	0
$411$	8.00000i	0.394611i
$412$	0	0
$413$	6.97307i	0.343122i
$414$	0	0
$415$	−5.79021	−0.284230
$416$	0	0
$417$	8.15593	0.399398
$418$	0	0
$419$	5.44650i	0.266079i	0.991111	+	0.133040i	$0.0424737\pi$
$419$	5.44650i	0.266079i	−0.991111	+	0.133040i	$0.957526\pi$
$420$	0	0
$421$	3.92204i	0.191148i	0.995422	+	0.0955742i	$0.0304687\pi$
$421$	3.92204i	0.191148i	−0.995422	+	0.0955742i	$0.969531\pi$
$422$	0	0
$423$	29.1070	1.41523
$424$	0	0
$425$	1.92204	0.0932324
$426$	0	0
$427$	14.7633i	0.714445i
$428$	0	0
$429$	− 23.5025i	− 1.13471i
$430$	0	0
$431$	29.1367	1.40347	0.701734	−	0.712439i	$-0.252410\pi$
$431$	29.1367	1.40347	0.701734	+	0.712439i	$0.252410\pi$
$432$	0	0
$433$	40.3976	1.94138	0.970692	−	0.240328i	$-0.0772551\pi$
$433$	40.3976	1.94138	0.970692	+	0.240328i	$0.0772551\pi$
$434$	0	0
$435$	− 16.9890i	− 0.814560i
$436$	0	0
$437$	50.7413i	2.42728i
$438$	0	0
$439$	−25.7902	−1.23090	−0.615450	−	0.788176i	$-0.711026\pi$
$439$	−25.7902	−1.23090	−0.615450	+	0.788176i	$0.711026\pi$
$440$	0	0
$441$	−5.38164	−0.256269
$442$	0	0
$443$	36.7413i	1.74563i	0.488050	+	0.872815i	$0.337708\pi$
$443$	36.7413i	1.74563i	−0.488050	+	0.872815i	$0.662292\pi$
$444$	0	0
$445$	− 1.23672i	− 0.0586260i
$446$	0	0
$447$	−5.94614	−0.281243
$448$	0	0
$449$	−21.5425	−1.01665	−0.508327	−	0.861164i	$-0.669736\pi$
$449$	−21.5425	−1.01665	−0.508327	+	0.861164i	$0.669736\pi$
$450$	0	0
$451$	− 28.0801i	− 1.32224i
$452$	0	0
$453$	− 28.1180i	− 1.32110i
$454$	0	0
$455$	3.40857	0.159796
$456$	0	0
$457$	−17.8441	−0.834710	−0.417355	−	0.908743i	$-0.637043\pi$
$457$	−17.8441	−0.834710	−0.417355	+	0.908743i	$0.637043\pi$
$458$	0	0
$459$	13.2526i	0.618580i
$460$	0	0
$461$	− 3.07514i	− 0.143224i	−0.997433	−	0.0716118i	$-0.977186\pi$
$461$	− 3.07514i	− 0.143224i	0.997433	−	0.0716118i	$-0.0228143\pi$
$462$	0	0
$463$	−21.1070	−0.980925	−0.490463	−	0.871462i	$-0.663172\pi$
$463$	−21.1070	−0.980925	−0.490463	+	0.871462i	$0.663172\pi$
$464$	0	0
$465$	−2.97307	−0.137873
$466$	0	0
$467$	3.31468i	0.153385i	0.997055	+	0.0766926i	$0.0244360\pi$
$467$	3.31468i	0.153385i	−0.997055	+	0.0766926i	$0.975564\pi$
$468$	0	0
$469$	0.209787i	0.00968706i
$470$	0	0
$471$	−7.00492	−0.322770
$472$	0	0
$473$	4.76328	0.219016
$474$	0	0
$475$	− 5.79021i	− 0.265673i
$476$	0	0
$477$	37.5266i	1.71822i
$478$	0	0
$479$	−19.3168	−0.882606	−0.441303	−	0.897358i	$-0.645484\pi$
$479$	−19.3168	−0.882606	−0.441303	+	0.897358i	$0.645484\pi$
$480$	0	0
$481$	9.60243	0.437833
$482$	0	0
$483$	− 25.3706i	− 1.15440i
$484$	0	0
$485$	5.92204i	0.268906i
$486$	0	0
$487$	−3.65629	−0.165682	−0.0828412	−	0.996563i	$-0.526399\pi$
$487$	−3.65629	−0.165682	−0.0828412	+	0.996563i	$0.526399\pi$
$488$	0	0
$489$	−36.3437	−1.64352
$490$	0	0
$491$	4.32778i	0.195310i	0.995220	+	0.0976550i	$0.0311342\pi$
$491$	4.32778i	0.195310i	−0.995220	+	0.0976550i	$0.968866\pi$
$492$	0	0
$493$	− 11.2788i	− 0.507974i
$494$	0	0
$495$	12.8171	0.576088
$496$	0	0
$497$	−13.7902	−0.618576
$498$	0	0
$499$	− 29.1229i	− 1.30372i	−0.758339	−	0.651860i	$-0.773989\pi$
$499$	− 29.1229i	− 1.30372i	0.758339	−	0.651860i	$-0.226011\pi$
$500$	0	0
$501$	− 71.1311i	− 3.17790i
$502$	0	0
$503$	30.1719	1.34530	0.672648	−	0.739962i	$-0.265157\pi$
$503$	30.1719	1.34530	0.672648	+	0.739962i	$0.265157\pi$
$504$	0	0
$505$	−9.58043	−0.426323
$506$	0	0
$507$	4.00000i	0.177646i
$508$	0	0
$509$	12.5535i	0.556424i	0.960520	+	0.278212i	$0.0897418\pi$
$509$	12.5535i	0.556424i	−0.960520	+	0.278212i	$0.910258\pi$
$510$	0	0
$511$	14.7633	0.653089
$512$	0	0
$513$	39.9241	1.76269
$514$	0	0
$515$	− 6.59143i	− 0.290453i
$516$	0	0
$517$	− 12.8813i	− 0.566518i
$518$	0	0
$519$	4.23389	0.185847
$520$	0	0
$521$	−8.66121	−0.379455	−0.189727	−	0.981837i	$-0.560760\pi$
$521$	−8.66121	−0.379455	−0.189727	+	0.981837i	$0.560760\pi$
$522$	0	0
$523$	9.79021i	0.428096i	0.976823	+	0.214048i	$0.0686649\pi$
$523$	9.79021i	0.428096i	−0.976823	+	0.214048i	$0.931335\pi$
$524$	0	0
$525$	2.89511i	0.126353i
$526$	0	0
$527$	−1.97380	−0.0859799
$528$	0	0
$529$	53.7951	2.33892
$530$	0	0
$531$	37.5266i	1.62851i
$532$	0	0
$533$	− 40.1878i	− 1.74073i
$534$	0	0
$535$	0.0538591	0.00232853
$536$	0	0
$537$	−25.5266	−1.10155
$538$	0	0
$539$	2.38164i	0.102585i
$540$	0	0
$541$	− 31.5025i	− 1.35440i	−0.735801	−	0.677198i	$-0.763194\pi$
$541$	− 31.5025i	− 1.35440i	0.735801	−	0.677198i	$-0.236806\pi$
$542$	0	0
$543$	−50.1339	−2.15145
$544$	0	0
$545$	−9.44860	−0.404734
$546$	0	0
$547$	30.7633i	1.31534i	0.753305	+	0.657672i	$0.228458\pi$
$547$	30.7633i	1.31534i	−0.753305	+	0.657672i	$0.771542\pi$
$548$	0	0
$549$	79.4507i	3.39087i
$550$	0	0
$551$	−33.9780	−1.44751
$552$	0	0
$553$	−13.4486	−0.571893
$554$	0	0
$555$	8.15593i	0.346200i
$556$	0	0
$557$	17.1070i	0.724847i	0.932014	+	0.362423i	$0.118051\pi$
$557$	17.1070i	0.724847i	−0.932014	+	0.362423i	$0.881949\pi$
$558$	0	0
$559$	6.81714	0.288334
$560$	0	0
$561$	13.2526	0.559527
$562$	0	0
$563$	2.62936i	0.110814i	0.998464	+	0.0554072i	$0.0176457\pi$
$563$	2.62936i	0.110814i	−0.998464	+	0.0554072i	$0.982354\pi$
$564$	0	0
$565$	12.5535i	0.528130i
$566$	0	0
$567$	−3.81714	−0.160305
$568$	0	0
$569$	−28.7413	−1.20490	−0.602449	−	0.798158i	$-0.705808\pi$
$569$	−28.7413	−1.20490	−0.602449	+	0.798158i	$0.705808\pi$
$570$	0	0
$571$	7.18286i	0.300593i	0.988641	+	0.150297i	$0.0480229\pi$
$571$	7.18286i	0.300593i	−0.988641	+	0.150297i	$0.951977\pi$
$572$	0	0
$573$	44.8813i	1.87494i
$574$	0	0
$575$	−8.76328	−0.365454
$576$	0	0
$577$	20.2878	0.844590	0.422295	−	0.906458i	$-0.361225\pi$
$577$	20.2878	0.844590	0.422295	+	0.906458i	$0.361225\pi$
$578$	0	0
$579$	− 24.3119i	− 1.01037i
$580$	0	0
$581$	− 5.79021i	− 0.240219i
$582$	0	0
$583$	16.6074	0.687806
$584$	0	0
$585$	18.3437	0.758419
$586$	0	0
$587$	− 43.7364i	− 1.80519i	−0.430488	−	0.902596i	$-0.641659\pi$
$587$	− 43.7364i	− 1.80519i	0.430488	−	0.902596i	$-0.358341\pi$
$588$	0	0
$589$	5.94614i	0.245006i
$590$	0	0
$591$	40.9192	1.68319
$592$	0	0
$593$	−3.86818	−0.158847	−0.0794235	−	0.996841i	$-0.525308\pi$
$593$	−3.86818	−0.158847	−0.0794235	+	0.996841i	$0.525308\pi$
$594$	0	0
$595$	1.92204i	0.0787958i
$596$	0	0
$597$	59.3486i	2.42898i
$598$	0	0
$599$	−41.8682	−1.71069	−0.855344	−	0.518061i	$-0.826654\pi$
$599$	−41.8682	−1.71069	−0.855344	+	0.518061i	$0.826654\pi$
$600$	0	0
$601$	22.2955	0.909452	0.454726	−	0.890631i	$-0.349737\pi$
$601$	22.2955	0.909452	0.454726	+	0.890631i	$0.349737\pi$
$602$	0	0
$603$	1.12900i	0.0459764i
$604$	0	0
$605$	5.32778i	0.216605i
$606$	0	0
$607$	11.6984	0.474824	0.237412	−	0.971409i	$-0.423701\pi$
$607$	11.6984	0.474824	0.237412	+	0.971409i	$0.423701\pi$
$608$	0	0
$609$	16.9890	0.688429
$610$	0	0
$611$	− 18.4355i	− 0.745821i
$612$	0	0
$613$	− 29.4784i	− 1.19062i	−0.803496	−	0.595310i	$-0.797029\pi$
$613$	− 29.4784i	− 1.19062i	0.803496	−	0.595310i	$-0.202971\pi$
$614$	0	0
$615$	34.1339	1.37641
$616$	0	0
$617$	29.7682	1.19842	0.599211	−	0.800591i	$-0.295481\pi$
$617$	29.7682	1.19842	0.599211	+	0.800591i	$0.295481\pi$
$618$	0	0
$619$	34.0857i	1.37002i	0.728533	+	0.685010i	$0.240202\pi$
$619$	34.0857i	1.37002i	−0.728533	+	0.685010i	$0.759798\pi$
$620$	0	0
$621$	− 60.4238i	− 2.42472i
$622$	0	0
$623$	1.23672	0.0495480
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 39.9241i	− 1.59442i
$628$	0	0
$629$	5.41465i	0.215896i
$630$	0	0
$631$	30.1318	1.19953	0.599764	−	0.800177i	$-0.295261\pi$
$631$	30.1318	1.19953	0.599764	+	0.800177i	$0.295261\pi$
$632$	0	0
$633$	−75.1630	−2.98746
$634$	0	0
$635$	− 2.97307i	− 0.117983i
$636$	0	0
$637$	3.40857i	0.135053i
$638$	0	0
$639$	−74.2140	−2.93586
$640$	0	0
$641$	−41.1609	−1.62576	−0.812878	−	0.582434i	$-0.802100\pi$
$641$	−41.1609	−1.62576	−0.812878	+	0.582434i	$0.802100\pi$
$642$	0	0
$643$	− 20.1098i	− 0.793054i	−0.918023	−	0.396527i	$-0.870215\pi$
$643$	− 20.1098i	− 0.793054i	0.918023	−	0.396527i	$-0.129785\pi$
$644$	0	0
$645$	5.79021i	0.227989i
$646$	0	0
$647$	−4.31186	−0.169517	−0.0847583	−	0.996402i	$-0.527012\pi$
$647$	−4.31186	−0.169517	−0.0847583	+	0.996402i	$0.527012\pi$
$648$	0	0
$649$	16.6074	0.651896
$650$	0	0
$651$	− 2.97307i	− 0.116524i
$652$	0	0
$653$	− 26.8972i	− 1.05257i	−0.850309	−	0.526285i	$-0.823585\pi$
$653$	− 26.8972i	− 1.05257i	0.850309	−	0.526285i	$-0.176415\pi$
$654$	0	0
$655$	10.8171	0.422661
$656$	0	0
$657$	79.4507	3.09967
$658$	0	0
$659$	− 28.4674i	− 1.10893i	−0.832207	−	0.554465i	$-0.812923\pi$
$659$	− 28.4674i	− 1.10893i	0.832207	−	0.554465i	$-0.187077\pi$
$660$	0	0
$661$	− 5.12900i	− 0.199495i	−0.995013	−	0.0997475i	$-0.968197\pi$
$661$	− 5.12900i	− 0.199495i	0.995013	−	0.0997475i	$-0.0318035\pi$
$662$	0	0
$663$	18.9670	0.736617
$664$	0	0
$665$	5.79021	0.224535
$666$	0	0
$667$	51.4245i	1.99116i
$668$	0	0
$669$	− 71.1311i	− 2.75009i
$670$	0	0
$671$	35.1609	1.35737
$672$	0	0
$673$	46.3437	1.78642	0.893209	−	0.449641i	$-0.148448\pi$
$673$	46.3437	1.78642	0.893209	+	0.449641i	$0.148448\pi$
$674$	0	0
$675$	6.89511i	0.265393i
$676$	0	0
$677$	13.8061i	0.530613i	0.964164	+	0.265307i	$0.0854732\pi$
$677$	13.8061i	0.530613i	−0.964164	+	0.265307i	$0.914527\pi$
$678$	0	0
$679$	−5.92204	−0.227267
$680$	0	0
$681$	60.3376	2.31214
$682$	0	0
$683$	− 0.817143i	− 0.0312671i	−0.999878	−	0.0156335i	$-0.995023\pi$
$683$	− 0.817143i	− 0.0312671i	0.999878	−	0.0156335i	$-0.00497651\pi$
$684$	0	0
$685$	2.76328i	0.105580i
$686$	0	0
$687$	−5.33879	−0.203687
$688$	0	0
$689$	23.7682	0.905497
$690$	0	0
$691$	− 21.1070i	− 0.802948i	−0.915870	−	0.401474i	$-0.868498\pi$
$691$	− 21.1070i	− 0.802948i	0.915870	−	0.401474i	$-0.131502\pi$
$692$	0	0
$693$	12.8171i	0.486883i
$694$	0	0
$695$	2.81714	0.106860
$696$	0	0
$697$	22.6612	0.858355
$698$	0	0
$699$	− 3.26857i	− 0.123629i
$700$	0	0
$701$	38.5556i	1.45622i	0.685458	+	0.728112i	$0.259602\pi$
$701$	38.5556i	1.45622i	−0.685458	+	0.728112i	$0.740398\pi$
$702$	0	0
$703$	16.3119	0.615213
$704$	0	0
$705$	15.6584	0.589729
$706$	0	0
$707$	− 9.58043i	− 0.360309i
$708$	0	0
$709$	40.9213i	1.53683i	0.639951	+	0.768416i	$0.278955\pi$
$709$	40.9213i	1.53683i	−0.639951	+	0.768416i	$0.721045\pi$
$710$	0	0
$711$	−72.3756	−2.71430
$712$	0	0
$713$	8.99927	0.337025
$714$	0	0
$715$	− 8.11800i	− 0.303596i
$716$	0	0
$717$	− 17.7523i	− 0.662971i
$718$	0	0
$719$	−40.9511	−1.52722	−0.763609	−	0.645680i	$-0.776574\pi$
$719$	−40.9511	−1.52722	−0.763609	+	0.645680i	$0.776574\pi$
$720$	0	0
$721$	6.59143	0.245478
$722$	0	0
$723$	74.0581i	2.75425i
$724$	0	0
$725$	− 5.86818i	− 0.217939i
$726$	0	0
$727$	42.6336	1.58119	0.790596	−	0.612339i	$-0.209771\pi$
$727$	42.6336	1.58119	0.790596	+	0.612339i	$0.209771\pi$
$728$	0	0
$729$	39.3437	1.45717
$730$	0	0
$731$	3.84407i	0.142178i
$732$	0	0
$733$	− 26.8813i	− 0.992883i	−0.868070	−	0.496441i	$-0.834640\pi$
$733$	− 26.8813i	− 0.992883i	0.868070	−	0.496441i	$-0.165360\pi$
$734$	0	0
$735$	−2.89511	−0.106788
$736$	0	0
$737$	0.499637	0.0184044
$738$	0	0
$739$	28.2201i	1.03809i	0.854746	+	0.519046i	$0.173713\pi$
$739$	28.2201i	1.03809i	−0.854746	+	0.519046i	$0.826287\pi$
$740$	0	0
$741$	− 57.1388i	− 2.09905i
$742$	0	0
$743$	−35.6605	−1.30826	−0.654128	−	0.756384i	$-0.726964\pi$
$743$	−35.6605	−1.30826	−0.654128	+	0.756384i	$0.726964\pi$
$744$	0	0
$745$	−2.05386	−0.0752476
$746$	0	0
$747$	− 31.1609i	− 1.14012i
$748$	0	0
$749$	0.0538591i	0.00196797i
$750$	0	0
$751$	28.7654	1.04966	0.524832	−	0.851206i	$-0.324128\pi$
$751$	28.7654	1.04966	0.524832	+	0.851206i	$0.324128\pi$
$752$	0	0
$753$	−8.15593	−0.297219
$754$	0	0
$755$	− 9.71225i	− 0.353465i
$756$	0	0
$757$	50.7413i	1.84422i	0.386924	+	0.922112i	$0.373538\pi$
$757$	50.7413i	1.84422i	−0.386924	+	0.922112i	$0.626462\pi$
$758$	0	0
$759$	−60.4238	−2.19324
$760$	0	0
$761$	21.8923	0.793595	0.396797	−	0.917906i	$-0.370122\pi$
$761$	21.8923	0.793595	0.396797	+	0.917906i	$0.370122\pi$
$762$	0	0
$763$	− 9.44860i	− 0.342062i
$764$	0	0
$765$	10.3437i	0.373978i
$766$	0	0
$767$	23.7682	0.858220
$768$	0	0
$769$	−43.8221	−1.58026	−0.790132	−	0.612937i	$-0.789988\pi$
$769$	−43.8221	−1.58026	−0.790132	+	0.612937i	$0.789988\pi$
$770$	0	0
$771$	− 41.5266i	− 1.49554i
$772$	0	0
$773$	− 40.9351i	− 1.47233i	−0.676800	−	0.736167i	$-0.736634\pi$
$773$	− 40.9351i	− 1.47233i	0.676800	−	0.736167i	$-0.263366\pi$
$774$	0	0
$775$	−1.02693	−0.0368884
$776$	0	0
$777$	−8.15593	−0.292592
$778$	0	0
$779$	− 68.2678i	− 2.44595i
$780$	0	0
$781$	32.8433i	1.17523i
$782$	0	0
$783$	40.4617	1.44598
$784$	0	0
$785$	−2.41957	−0.0863583
$786$	0	0
$787$	− 10.3196i	− 0.367854i	−0.982940	−	0.183927i	$-0.941119\pi$
$787$	− 10.3196i	− 0.367854i	0.982940	−	0.183927i	$-0.0588810\pi$
$788$	0	0
$789$	− 29.5584i	− 1.05231i
$790$	0	0
$791$	−12.5535	−0.446351
$792$	0	0
$793$	50.3217	1.78698
$794$	0	0
$795$	20.1878i	0.715987i
$796$	0	0
$797$	38.9890i	1.38106i	0.723303	+	0.690531i	$0.242623\pi$
$797$	38.9890i	1.38106i	−0.723303	+	0.690531i	$0.757377\pi$
$798$	0	0
$799$	10.3955	0.367765
$800$	0	0
$801$	6.65557	0.235163
$802$	0	0
$803$	− 35.1609i	− 1.24080i
$804$	0	0
$805$	− 8.76328i	− 0.308865i
$806$	0	0
$807$	24.7633	0.871709
$808$	0	0
$809$	−19.6722	−0.691638	−0.345819	−	0.938301i	$-0.612399\pi$
$809$	−19.6722	−0.691638	−0.345819	+	0.938301i	$0.612399\pi$
$810$	0	0
$811$	33.8703i	1.18935i	0.803968	+	0.594673i	$0.202719\pi$
$811$	33.8703i	1.18935i	−0.803968	+	0.594673i	$0.797281\pi$
$812$	0	0
$813$	56.6874i	1.98811i
$814$	0	0
$815$	−12.5535	−0.439730
$816$	0	0
$817$	11.5804	0.405148
$818$	0	0
$819$	18.3437i	0.640981i
$820$	0	0
$821$	24.8731i	0.868077i	0.900894	+	0.434039i	$0.142912\pi$
$821$	24.8731i	0.868077i	−0.900894	+	0.434039i	$0.857088\pi$
$822$	0	0
$823$	−1.44650	−0.0504219	−0.0252110	−	0.999682i	$-0.508026\pi$
$823$	−1.44650	−0.0504219	−0.0252110	+	0.999682i	$0.508026\pi$
$824$	0	0
$825$	6.89511	0.240057
$826$	0	0
$827$	24.2416i	0.842964i	0.906837	+	0.421482i	$0.138490\pi$
$827$	24.2416i	0.842964i	−0.906837	+	0.421482i	$0.861510\pi$
$828$	0	0
$829$	− 40.5854i	− 1.40959i	−0.709412	−	0.704794i	$-0.751040\pi$
$829$	− 40.5854i	− 1.40959i	0.709412	−	0.704794i	$-0.248960\pi$
$830$	0	0
$831$	36.9511	1.28182
$832$	0	0
$833$	−1.92204	−0.0665946
$834$	0	0
$835$	− 24.5694i	− 0.850260i
$836$	0	0
$837$	− 7.08079i	− 0.244748i
$838$	0	0
$839$	12.0482	0.415950	0.207975	−	0.978134i	$-0.433313\pi$
$839$	12.0482	0.415950	0.207975	+	0.978134i	$0.433313\pi$
$840$	0	0
$841$	−5.43550	−0.187431
$842$	0	0
$843$	− 75.1630i	− 2.58875i
$844$	0	0
$845$	1.38164i	0.0475299i
$846$	0	0
$847$	−5.32778	−0.183065
$848$	0	0
$849$	−82.7352	−2.83946
$850$	0	0
$851$	− 24.6874i	− 0.846274i
$852$	0	0
$853$	− 36.3217i	− 1.24363i	−0.783164	−	0.621816i	$-0.786395\pi$
$853$	− 36.3217i	− 1.24363i	0.783164	−	0.621816i	$-0.213605\pi$
$854$	0	0
$855$	31.1609	1.06568
$856$	0	0
$857$	39.9780	1.36562	0.682811	−	0.730595i	$-0.260757\pi$
$857$	39.9780	1.36562	0.682811	+	0.730595i	$0.260757\pi$
$858$	0	0
$859$	46.9454i	1.60176i	0.598827	+	0.800878i	$0.295634\pi$
$859$	46.9454i	1.60176i	−0.598827	+	0.800878i	$0.704366\pi$
$860$	0	0
$861$	34.1339i	1.16328i
$862$	0	0
$863$	−16.2955	−0.554705	−0.277353	−	0.960768i	$-0.589457\pi$
$863$	−16.2955	−0.554705	−0.277353	+	0.960768i	$0.589457\pi$
$864$	0	0
$865$	1.46243	0.0497241
$866$	0	0
$867$	− 38.5216i	− 1.30826i
$868$	0	0
$869$	32.0298i	1.08653i
$870$	0	0
$871$	0.715074	0.0242294
$872$	0	0
$873$	−31.8703	−1.07865
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	36.9511i	1.24775i	0.781525	+	0.623874i	$0.214442\pi$
$877$	36.9511i	1.24775i	−0.781525	+	0.623874i	$0.785558\pi$
$878$	0	0
$879$	−35.2388	−1.18858
$880$	0	0
$881$	11.9780	0.403549	0.201774	−	0.979432i	$-0.435329\pi$
$881$	11.9780	0.403549	0.201774	+	0.979432i	$0.435329\pi$
$882$	0	0
$883$	− 26.6556i	− 0.897031i	−0.893775	−	0.448516i	$-0.851953\pi$
$883$	− 26.6556i	− 0.897031i	0.893775	−	0.448516i	$-0.148047\pi$
$884$	0	0
$885$	20.1878i	0.678605i
$886$	0	0
$887$	−4.31186	−0.144778	−0.0723890	−	0.997376i	$-0.523062\pi$
$887$	−4.31186	−0.144778	−0.0723890	+	0.997376i	$0.523062\pi$
$888$	0	0
$889$	2.97307	0.0997136
$890$	0	0
$891$	9.09107i	0.304562i
$892$	0	0
$893$	− 31.3168i	− 1.04798i
$894$	0	0
$895$	−8.81714	−0.294725
$896$	0	0
$897$	−86.4776	−2.88740
$898$	0	0
$899$	6.02620i	0.200985i
$900$	0	0
$901$	13.4025i	0.446502i
$902$	0	0
$903$	−5.79021	−0.192686
$904$	0	0
$905$	−17.3168	−0.575629
$906$	0	0
$907$	− 11.0269i	− 0.366143i	−0.983100	−	0.183072i	$-0.941396\pi$
$907$	− 11.0269i	− 0.366143i	0.983100	−	0.183072i	$-0.0586040\pi$
$908$	0	0
$909$	− 51.5584i	− 1.71008i
$910$	0	0
$911$	−23.0049	−0.762187	−0.381094	−	0.924536i	$-0.624452\pi$
$911$	−23.0049	−0.762187	−0.381094	+	0.924536i	$0.624452\pi$
$912$	0	0
$913$	−13.7902	−0.456389
$914$	0	0
$915$	42.7413i	1.41298i
$916$	0	0
$917$	10.8171i	0.357214i
$918$	0	0
$919$	−5.13675	−0.169446	−0.0847228	−	0.996405i	$-0.527000\pi$
$919$	−5.13675	−0.169446	−0.0847228	+	0.996405i	$0.527000\pi$
$920$	0	0
$921$	−37.1768	−1.22502
$922$	0	0
$923$	47.0049i	1.54719i
$924$	0	0
$925$	2.81714i	0.0926271i
$926$	0	0
$927$	35.4727	1.16508
$928$	0	0
$929$	15.0588	0.494063	0.247031	−	0.969007i	$-0.420545\pi$
$929$	15.0588	0.494063	0.247031	+	0.969007i	$0.420545\pi$
$930$	0	0
$931$	5.79021i	0.189767i
$932$	0	0
$933$	49.7462i	1.62862i
$934$	0	0
$935$	4.57760	0.149704
$936$	0	0
$937$	6.76118	0.220878	0.110439	−	0.993883i	$-0.464774\pi$
$937$	6.76118	0.220878	0.110439	+	0.993883i	$0.464774\pi$
$938$	0	0
$939$	− 44.2739i	− 1.44482i
$940$	0	0
$941$	− 43.5584i	− 1.41996i	−0.704220	−	0.709982i	$-0.748703\pi$
$941$	− 43.5584i	− 1.41996i	0.704220	−	0.709982i	$-0.251297\pi$
$942$	0	0
$943$	−103.321	−3.36459
$944$	0	0
$945$	−6.89511	−0.224298
$946$	0	0
$947$	− 29.2091i	− 0.949167i	−0.880210	−	0.474583i	$-0.842599\pi$
$947$	− 29.2091i	− 0.949167i	0.880210	−	0.474583i	$-0.157401\pi$
$948$	0	0
$949$	− 50.3217i	− 1.63351i
$950$	0	0
$951$	89.4507	2.90064
$952$	0	0
$953$	−37.3168	−1.20881	−0.604405	−	0.796678i	$-0.706589\pi$
$953$	−37.3168	−1.20881	−0.604405	+	0.796678i	$0.706589\pi$
$954$	0	0
$955$	15.5025i	0.501648i
$956$	0	0
$957$	− 40.4617i	− 1.30794i
$958$	0	0
$959$	−2.76328	−0.0892311
$960$	0	0
$961$	−29.9454	−0.965981
$962$	0	0
$963$	0.289850i	0.00934030i
$964$	0	0
$965$	− 8.39757i	− 0.270327i
$966$	0	0
$967$	6.62936	0.213186	0.106593	−	0.994303i	$-0.466006\pi$
$967$	6.62936	0.213186	0.106593	+	0.994303i	$0.466006\pi$
$968$	0	0
$969$	32.2196	1.03504
$970$	0	0
$971$	− 40.4996i	− 1.29970i	−0.760065	−	0.649848i	$-0.774833\pi$
$971$	− 40.4996i	− 1.29970i	0.760065	−	0.649848i	$-0.225167\pi$
$972$	0	0
$973$	2.81714i	0.0903134i
$974$	0	0
$975$	9.86818	0.316035
$976$	0	0
$977$	−33.0531	−1.05746	−0.528732	−	0.848789i	$-0.677332\pi$
$977$	−33.0531	−1.05746	−0.528732	+	0.848789i	$0.677332\pi$
$978$	0	0
$979$	− 2.94542i	− 0.0941359i
$980$	0	0
$981$	− 50.8490i	− 1.62348i
$982$	0	0
$983$	9.40857	0.300087	0.150043	−	0.988679i	$-0.452059\pi$
$983$	9.40857	0.300087	0.150043	+	0.988679i	$0.452059\pi$
$984$	0	0
$985$	14.1339	0.450344
$986$	0	0
$987$	15.6584i	0.498412i
$988$	0	0
$989$	− 17.5266i	− 0.557312i
$990$	0	0
$991$	15.0049	0.476647	0.238324	−	0.971186i	$-0.423402\pi$
$991$	15.0049	0.476647	0.238324	+	0.971186i	$0.423402\pi$
$992$	0	0
$993$	−76.5797	−2.43018
$994$	0	0
$995$	20.4996i	0.649882i
$996$	0	0
$997$	36.1719i	1.14557i	0.819704	+	0.572787i	$0.194138\pi$
$997$	36.1719i	1.14557i	−0.819704	+	0.572787i	$0.805862\pi$
$998$	0	0
$999$	−19.4245	−0.614564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.e.1121.6	yes	6
4.3	odd	2		2240.2.b.f.1121.1	yes	6
8.3	odd	2		2240.2.b.f.1121.6	yes	6
8.5	even	2	inner	2240.2.b.e.1121.1	✓	6
16.3	odd	4		8960.2.a.bn.1.3		3
16.5	even	4		8960.2.a.bk.1.3		3
16.11	odd	4		8960.2.a.bh.1.1		3
16.13	even	4		8960.2.a.bq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.e.1121.1	✓	6	8.5	even	2	inner
2240.2.b.e.1121.6	yes	6	1.1	even	1	trivial
2240.2.b.f.1121.1	yes	6	4.3	odd	2
2240.2.b.f.1121.6	yes	6	8.3	odd	2
8960.2.a.bh.1.1		3	16.11	odd	4
8960.2.a.bk.1.3		3	16.5	even	4
8960.2.a.bn.1.3		3	16.3	odd	4
8960.2.a.bq.1.1		3	16.13	even	4

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.89511i q^{3} +1.00000i q^{5} -1.00000 q^{7} -5.38164 q^{9} +O(q^{10})\)	\(q+2.89511i q^{3} +1.00000i q^{5} -1.00000 q^{7} -5.38164 q^{9} +2.38164i q^{11} +3.40857i q^{13} -2.89511 q^{15} -1.92204 q^{17} +5.79021i q^{19} -2.89511i q^{21} +8.76328 q^{23} -1.00000 q^{25} -6.89511i q^{27} +5.86818i q^{29} +1.02693 q^{31} -6.89511 q^{33} -1.00000i q^{35} -2.81714i q^{37} -9.86818 q^{39} -11.7902 q^{41} -2.00000i q^{43} -5.38164i q^{45} -5.40857 q^{47} +1.00000 q^{49} -5.56450i q^{51} -6.97307i q^{53} -2.38164 q^{55} -16.7633 q^{57} -6.97307i q^{59} -14.7633i q^{61} +5.38164 q^{63} -3.40857 q^{65} -0.209787i q^{67} +25.3706i q^{69} +13.7902 q^{71} -14.7633 q^{73} -2.89511i q^{75} -2.38164i q^{77} +13.4486 q^{79} +3.81714 q^{81} +5.79021i q^{83} -1.92204i q^{85} -16.9890 q^{87} -1.23672 q^{89} -3.40857i q^{91} +2.97307i q^{93} -5.79021 q^{95} +5.92204 q^{97} -12.8171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 10 * q^9	\( 6 q - 6 q^{7} - 10 q^{9} - 4 q^{17} + 8 q^{23} - 6 q^{25} + 16 q^{31} - 24 q^{33} - 32 q^{39} - 36 q^{41} - 20 q^{47} + 6 q^{49} + 8 q^{55} - 56 q^{57} + 10 q^{63} - 8 q^{65} + 48 q^{71} - 44 q^{73} - 16 q^{79} - 2 q^{81} - 20 q^{87} - 52 q^{89} + 28 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 10 * q^9 - 4 * q^17 + 8 * q^23 - 6 * q^25 + 16 * q^31 - 24 * q^33 - 32 * q^39 - 36 * q^41 - 20 * q^47 + 6 * q^49 + 8 * q^55 - 56 * q^57 + 10 * q^63 - 8 * q^65 + 48 * q^71 - 44 * q^73 - 16 * q^79 - 2 * q^81 - 20 * q^87 - 52 * q^89 + 28 * q^97

Introduction

L-functions

Modular forms

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Database

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