LMFDB - Embedded newform 8820.2.d.c.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(881,8820)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8820, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8820.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.d (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:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1134 x^{12} - 3528 x^{11} + 9316 x^{10} - 19960 x^{9} + \cdots + 68 $ x^16 - 8x^15 + 60x^14 - 280x^13 + 1134x^12 - 3528x^11 + 9316x^10 - 19960x^9 + 35421x^8 - 51008x^7 + 58384x^6 - 51840x^5 + 34228x^4 - 15904x^3 + 4848x^2 - 864*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$0.500000 - 0.00906270i$ of defining polynomial
Character	$\chi$	$=$	8820.881
Dual form			8820.2.d.c.881.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} -5.98177i q^{11} -4.53268i q^{13} +5.43762 q^{17} -5.00407i q^{19} -4.86588i q^{23} +1.00000 q^{25} -9.24632i q^{29} -2.59198i q^{31} +9.57814 q^{37} -4.39404 q^{41} -8.38558 q^{43} -5.90132 q^{47} +2.65928i q^{53} +5.98177i q^{55} +7.94392 q^{59} -7.93334i q^{61} +4.53268i q^{65} -13.9760 q^{67} +11.8853i q^{71} +16.6802i q^{73} +4.49359 q^{79} -0.766209 q^{83} -5.43762 q^{85} +13.4360 q^{89} +5.00407i q^{95} +7.39200i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5}+O(q^{10}) $ 16 * q - 16 * q^5	$ 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89}+O(q^{100}) $ 16 * q - 16 * q^5 + 16 * q^25 - 32 * q^41 - 32 * q^43 + 32 * q^47 + 32 * q^59 - 32 * q^67 + 64 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times$.

$n$	$1081$	$4411$	$7057$	$7841$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.98177i	− 1.80357i	−0.432184	−	0.901785i	$-0.642257\pi$
$11$	− 5.98177i	− 1.80357i	0.432184	−	0.901785i	$-0.357743\pi$
$12$	0	0
$13$	− 4.53268i	− 1.25714i	−0.777754	−	0.628569i	$-0.783641\pi$
$13$	− 4.53268i	− 1.25714i	0.777754	−	0.628569i	$-0.216359\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.43762	1.31882	0.659408	−	0.751785i	$-0.270807\pi$
$17$	5.43762	1.31882	0.659408	+	0.751785i	$0.270807\pi$
$18$	0	0
$19$	− 5.00407i	− 1.14801i	−0.818851	−	0.574006i	$-0.805388\pi$
$19$	− 5.00407i	− 1.14801i	0.818851	−	0.574006i	$-0.194612\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.86588i	− 1.01461i	−0.861767	−	0.507304i	$-0.830642\pi$
$23$	− 4.86588i	− 1.01461i	0.861767	−	0.507304i	$-0.169358\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.24632i	− 1.71700i	−0.512815	−	0.858499i	$-0.671397\pi$
$29$	− 9.24632i	− 1.71700i	0.512815	−	0.858499i	$-0.328603\pi$
$30$	0	0
$31$	− 2.59198i	− 0.465534i	−0.972533	−	0.232767i	$-0.925222\pi$
$31$	− 2.59198i	− 0.465534i	0.972533	−	0.232767i	$-0.0747779\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.57814	1.57464	0.787318	−	0.616547i	$-0.211469\pi$
$37$	9.57814	1.57464	0.787318	+	0.616547i	$0.211469\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.39404	−0.686234	−0.343117	−	0.939293i	$-0.611483\pi$
$41$	−4.39404	−0.686234	−0.343117	+	0.939293i	$0.611483\pi$
$42$	0	0
$43$	−8.38558	−1.27879	−0.639394	−	0.768879i	$-0.720815\pi$
$43$	−8.38558	−1.27879	−0.639394	+	0.768879i	$0.720815\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.90132	−0.860796	−0.430398	−	0.902639i	$-0.641627\pi$
$47$	−5.90132	−0.860796	−0.430398	+	0.902639i	$0.641627\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.65928i	0.365281i	0.983180	+	0.182640i	$0.0584644\pi$
$53$	2.65928i	0.365281i	−0.983180	+	0.182640i	$0.941536\pi$
$54$	0	0
$55$	5.98177i	0.806581i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.94392	1.03421	0.517105	−	0.855922i	$-0.327010\pi$
$59$	7.94392	1.03421	0.517105	+	0.855922i	$0.327010\pi$
$60$	0	0
$61$	− 7.93334i	− 1.01576i	−0.861428	−	0.507880i	$-0.830429\pi$
$61$	− 7.93334i	− 1.01576i	0.861428	−	0.507880i	$-0.169571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.53268i	0.562209i
$66$	0	0
$67$	−13.9760	−1.70744	−0.853718	−	0.520736i	$-0.825658\pi$
$67$	−13.9760	−1.70744	−0.853718	+	0.520736i	$0.825658\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.8853i	1.41053i	0.708945	+	0.705264i	$0.249171\pi$
$71$	11.8853i	1.41053i	−0.708945	+	0.705264i	$0.750829\pi$
$72$	0	0
$73$	16.6802i	1.95227i	0.217170	+	0.976134i	$0.430317\pi$
$73$	16.6802i	1.95227i	−0.217170	+	0.976134i	$0.569683\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.49359	0.505569	0.252784	−	0.967523i	$-0.418654\pi$
$79$	4.49359	0.505569	0.252784	+	0.967523i	$0.418654\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.766209	−0.0841023	−0.0420512	−	0.999115i	$-0.513389\pi$
$83$	−0.766209	−0.0841023	−0.0420512	+	0.999115i	$0.513389\pi$
$84$	0	0
$85$	−5.43762	−0.589792
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.4360	1.42421	0.712106	−	0.702072i	$-0.247742\pi$
$89$	13.4360	1.42421	0.712106	+	0.702072i	$0.247742\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.00407i	0.513407i
$96$	0	0
$97$	7.39200i	0.750544i	0.926915	+	0.375272i	$0.122451\pi$
$97$	7.39200i	0.750544i	−0.926915	+	0.375272i	$0.877549\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.97599	−0.495129	−0.247565	−	0.968871i	$-0.579630\pi$
$101$	−4.97599	−0.495129	−0.247565	+	0.968871i	$0.579630\pi$
$102$	0	0
$103$	13.9701i	1.37651i	0.725469	+	0.688255i	$0.241623\pi$
$103$	13.9701i	1.37651i	−0.725469	+	0.688255i	$0.758377\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.36803i	− 0.712294i	−0.934430	−	0.356147i	$-0.884090\pi$
$107$	− 7.36803i	− 0.712294i	0.934430	−	0.356147i	$-0.115910\pi$
$108$	0	0
$109$	15.0333	1.43993	0.719967	−	0.694009i	$-0.244157\pi$
$109$	15.0333	1.43993	0.719967	+	0.694009i	$0.244157\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 19.0792i	− 1.79482i	−0.441199	−	0.897409i	$-0.645447\pi$
$113$	− 19.0792i	− 1.79482i	0.441199	−	0.897409i	$-0.354553\pi$
$114$	0	0
$115$	4.86588i	0.453746i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−24.7815	−2.25287
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.54334	−0.846834	−0.423417	−	0.905935i	$-0.639169\pi$
$127$	−9.54334	−0.846834	−0.423417	+	0.905935i	$0.639169\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.6139	1.88841	0.944207	−	0.329352i	$-0.106830\pi$
$131$	21.6139	1.88841	0.944207	+	0.329352i	$0.106830\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.9264i	− 1.10437i	−0.833720	−	0.552187i	$-0.813793\pi$
$137$	− 12.9264i	− 1.10437i	0.833720	−	0.552187i	$-0.186207\pi$
$138$	0	0
$139$	9.34361i	0.792515i	0.918139	+	0.396257i	$0.129691\pi$
$139$	9.34361i	0.792515i	−0.918139	+	0.396257i	$0.870309\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−27.1134	−2.26734
$144$	0	0
$145$	9.24632i	0.767865i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.11271i	0.173080i	0.996248	+	0.0865401i	$0.0275811\pi$
$149$	2.11271i	0.173080i	−0.996248	+	0.0865401i	$0.972419\pi$
$150$	0	0
$151$	−19.8303	−1.61376	−0.806882	−	0.590713i	$-0.798846\pi$
$151$	−19.8303	−1.61376	−0.806882	+	0.590713i	$0.798846\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.59198i	0.208193i
$156$	0	0
$157$	0.374507i	0.0298889i	0.999888	+	0.0149445i	$0.00475715\pi$
$157$	0.374507i	0.0298889i	−0.999888	+	0.0149445i	$0.995243\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.39995	0.344631	0.172315	−	0.985042i	$-0.444875\pi$
$163$	4.39995	0.344631	0.172315	+	0.985042i	$0.444875\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.38327	−0.107041	−0.0535203	−	0.998567i	$-0.517044\pi$
$167$	−1.38327	−0.107041	−0.0535203	+	0.998567i	$0.517044\pi$
$168$	0	0
$169$	−7.54515	−0.580396
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.04587	0.687745	0.343873	−	0.939016i	$-0.388261\pi$
$173$	9.04587	0.687745	0.343873	+	0.939016i	$0.388261\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 5.06854i	− 0.378840i	−0.981896	−	0.189420i	$-0.939339\pi$
$179$	− 5.06854i	− 0.378840i	0.981896	−	0.189420i	$-0.0606608\pi$
$180$	0	0
$181$	19.4372i	1.44475i	0.691500	+	0.722376i	$0.256950\pi$
$181$	19.4372i	1.44475i	−0.691500	+	0.722376i	$0.743050\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.57814	−0.704199
$186$	0	0
$187$	− 32.5266i	− 2.37858i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.4556i	1.33540i	0.744432	+	0.667698i	$0.232720\pi$
$191$	18.4556i	1.33540i	−0.744432	+	0.667698i	$0.767280\pi$
$192$	0	0
$193$	−5.40795	−0.389273	−0.194636	−	0.980875i	$-0.562353\pi$
$193$	−5.40795	−0.389273	−0.194636	+	0.980875i	$0.562353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.34774i	0.167270i	0.996496	+	0.0836349i	$0.0266529\pi$
$197$	2.34774i	0.167270i	−0.996496	+	0.0836349i	$0.973347\pi$
$198$	0	0
$199$	− 12.0464i	− 0.853946i	−0.904264	−	0.426973i	$-0.859580\pi$
$199$	− 12.0464i	− 0.853946i	0.904264	−	0.426973i	$-0.140420\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.39404	0.306893
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29.9332	−2.07052
$210$	0	0
$211$	−10.4176	−0.717177	−0.358589	−	0.933496i	$-0.616742\pi$
$211$	−10.4176	−0.717177	−0.358589	+	0.933496i	$0.616742\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.38558	0.571892
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 24.6470i	− 1.65793i
$222$	0	0
$223$	20.0374i	1.34181i	0.741545	+	0.670903i	$0.234093\pi$
$223$	20.0374i	1.34181i	−0.741545	+	0.670903i	$0.765907\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.3912	0.888806	0.444403	−	0.895827i	$-0.353416\pi$
$227$	13.3912	0.888806	0.444403	+	0.895827i	$0.353416\pi$
$228$	0	0
$229$	− 13.7112i	− 0.906063i	−0.891495	−	0.453031i	$-0.850343\pi$
$229$	− 13.7112i	− 0.906063i	0.891495	−	0.453031i	$-0.149657\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.33340i	0.611452i	0.952120	+	0.305726i	$0.0988991\pi$
$233$	9.33340i	0.611452i	−0.952120	+	0.305726i	$0.901101\pi$
$234$	0	0
$235$	5.90132	0.384960
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.87524i	0.121299i	0.998159	+	0.0606497i	$0.0193173\pi$
$239$	1.87524i	0.121299i	−0.998159	+	0.0606497i	$0.980683\pi$
$240$	0	0
$241$	− 27.5476i	− 1.77450i	−0.461292	−	0.887249i	$-0.652614\pi$
$241$	− 27.5476i	− 1.77450i	0.461292	−	0.887249i	$-0.347386\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−22.6818	−1.44321
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.92853	−0.311086	−0.155543	−	0.987829i	$-0.549713\pi$
$251$	−4.92853	−0.311086	−0.155543	+	0.987829i	$0.549713\pi$
$252$	0	0
$253$	−29.1066	−1.82992
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.0743385	0.00463711	0.00231855	−	0.999997i	$-0.499262\pi$
$257$	0.0743385	0.00463711	0.00231855	+	0.999997i	$0.499262\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 3.25864i	− 0.200936i	−0.994940	−	0.100468i	$-0.967966\pi$
$263$	− 3.25864i	− 0.200936i	0.994940	−	0.100468i	$-0.0320340\pi$
$264$	0	0
$265$	− 2.65928i	− 0.163358i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.45194	−0.576295	−0.288147	−	0.957586i	$-0.593039\pi$
$269$	−9.45194	−0.576295	−0.288147	+	0.957586i	$0.593039\pi$
$270$	0	0
$271$	− 16.9715i	− 1.03095i	−0.856905	−	0.515474i	$-0.827616\pi$
$271$	− 16.9715i	− 1.03095i	0.856905	−	0.515474i	$-0.172384\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.98177i	− 0.360714i
$276$	0	0
$277$	−2.01845	−0.121277	−0.0606385	−	0.998160i	$-0.519314\pi$
$277$	−2.01845	−0.121277	−0.0606385	+	0.998160i	$0.519314\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6.70012i	− 0.399696i	−0.979827	−	0.199848i	$-0.935955\pi$
$281$	− 6.70012i	− 0.399696i	0.979827	−	0.199848i	$-0.0640448\pi$
$282$	0	0
$283$	31.9956i	1.90194i	0.309283	+	0.950970i	$0.399911\pi$
$283$	31.9956i	1.90194i	−0.309283	+	0.950970i	$0.600089\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.5677	0.739275
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.8437	1.04244	0.521221	−	0.853422i	$-0.325477\pi$
$293$	17.8437	1.04244	0.521221	+	0.853422i	$0.325477\pi$
$294$	0	0
$295$	−7.94392	−0.462513
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−22.0555	−1.27550
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.93334i	0.454262i
$306$	0	0
$307$	− 10.1767i	− 0.580816i	−0.956903	−	0.290408i	$-0.906209\pi$
$307$	− 10.1767i	− 0.580816i	0.956903	−	0.290408i	$-0.0937911\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.3286	0.755793	0.377896	−	0.925848i	$-0.376648\pi$
$311$	13.3286	0.755793	0.377896	+	0.925848i	$0.376648\pi$
$312$	0	0
$313$	− 6.71848i	− 0.379751i	−0.981808	−	0.189876i	$-0.939192\pi$
$313$	− 6.71848i	− 0.379751i	0.981808	−	0.189876i	$-0.0608085\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.17151i	0.402792i	0.979510	+	0.201396i	$0.0645478\pi$
$317$	7.17151i	0.402792i	−0.979510	+	0.201396i	$0.935452\pi$
$318$	0	0
$319$	−55.3093	−3.09673
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 27.2102i	− 1.51402i
$324$	0	0
$325$	− 4.53268i	− 0.251428i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.7801	0.702455	0.351228	−	0.936290i	$-0.385764\pi$
$331$	12.7801	0.702455	0.351228	+	0.936290i	$0.385764\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.9760	0.763589
$336$	0	0
$337$	0.500617	0.0272704	0.0136352	−	0.999907i	$-0.495660\pi$
$337$	0.500617	0.0272704	0.0136352	+	0.999907i	$0.495660\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.5046	−0.839623
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.35771i	0.180251i	0.995930	+	0.0901256i	$0.0287268\pi$
$347$	3.35771i	0.180251i	−0.995930	+	0.0901256i	$0.971273\pi$
$348$	0	0
$349$	2.01110i	0.107651i	0.998550	+	0.0538257i	$0.0171415\pi$
$349$	2.01110i	0.107651i	−0.998550	+	0.0538257i	$0.982858\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.4633	0.876255	0.438128	−	0.898913i	$-0.355642\pi$
$353$	16.4633	0.876255	0.438128	+	0.898913i	$0.355642\pi$
$354$	0	0
$355$	− 11.8853i	− 0.630807i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 3.48922i	− 0.184154i	−0.995752	−	0.0920769i	$-0.970649\pi$
$359$	− 3.48922i	− 0.184154i	0.995752	−	0.0920769i	$-0.0293506\pi$
$360$	0	0
$361$	−6.04074	−0.317933
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 16.6802i	− 0.873081i
$366$	0	0
$367$	6.83911i	0.356998i	0.983940	+	0.178499i	$0.0571242\pi$
$367$	6.83911i	0.356998i	−0.983940	+	0.178499i	$0.942876\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.8710	1.54666	0.773331	−	0.634003i	$-0.218589\pi$
$373$	29.8710	1.54666	0.773331	+	0.634003i	$0.218589\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−41.9106	−2.15850
$378$	0	0
$379$	−18.1923	−0.934476	−0.467238	−	0.884132i	$-0.654751\pi$
$379$	−18.1923	−0.934476	−0.467238	+	0.884132i	$0.654751\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.3037	0.526496	0.263248	−	0.964728i	$-0.415206\pi$
$383$	10.3037	0.526496	0.263248	+	0.964728i	$0.415206\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 10.1944i	− 0.516877i	−0.966028	−	0.258438i	$-0.916792\pi$
$389$	− 10.1944i	− 0.516877i	0.966028	−	0.258438i	$-0.0832079\pi$
$390$	0	0
$391$	− 26.4588i	− 1.33808i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.49359	−0.226097
$396$	0	0
$397$	− 19.5560i	− 0.981485i	−0.871305	−	0.490743i	$-0.836726\pi$
$397$	− 19.5560i	− 0.981485i	0.871305	−	0.490743i	$-0.163274\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.86791i	0.243092i	0.992586	+	0.121546i	$0.0387852\pi$
$401$	4.86791i	0.243092i	−0.992586	+	0.121546i	$0.961215\pi$
$402$	0	0
$403$	−11.7486	−0.585240
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 57.2942i	− 2.83997i
$408$	0	0
$409$	27.2186i	1.34587i	0.739700	+	0.672937i	$0.234967\pi$
$409$	27.2186i	1.34587i	−0.739700	+	0.672937i	$0.765033\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.766209	0.0376117
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.97827	0.438617	0.219309	−	0.975656i	$-0.429620\pi$
$419$	8.97827	0.438617	0.219309	+	0.975656i	$0.429620\pi$
$420$	0	0
$421$	−23.3561	−1.13831	−0.569154	−	0.822231i	$-0.692729\pi$
$421$	−23.3561	−1.13831	−0.569154	+	0.822231i	$0.692729\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.43762	0.263763
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 17.3138i	− 0.833976i	−0.908912	−	0.416988i	$-0.863086\pi$
$431$	− 17.3138i	− 0.833976i	0.908912	−	0.416988i	$-0.136914\pi$
$432$	0	0
$433$	33.2959i	1.60010i	0.599933	+	0.800050i	$0.295194\pi$
$433$	33.2959i	1.60010i	−0.599933	+	0.800050i	$0.704806\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−24.3492	−1.16478
$438$	0	0
$439$	6.44247i	0.307482i	0.988111	+	0.153741i	$0.0491322\pi$
$439$	6.44247i	0.307482i	−0.988111	+	0.153741i	$0.950868\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 38.3223i	− 1.82075i	−0.413787	−	0.910374i	$-0.635794\pi$
$443$	− 38.3223i	− 1.82075i	0.413787	−	0.910374i	$-0.364206\pi$
$444$	0	0
$445$	−13.4360	−0.636927
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 23.1527i	− 1.09264i	−0.837575	−	0.546322i	$-0.816027\pi$
$449$	− 23.1527i	− 1.09264i	0.837575	−	0.546322i	$-0.183973\pi$
$450$	0	0
$451$	26.2841i	1.23767i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.1005	0.799929	0.399965	−	0.916531i	$-0.369022\pi$
$457$	17.1005	0.799929	0.399965	+	0.916531i	$0.369022\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.9704	−1.16299	−0.581494	−	0.813550i	$-0.697532\pi$
$461$	−24.9704	−1.16299	−0.581494	+	0.813550i	$0.697532\pi$
$462$	0	0
$463$	8.73938	0.406153	0.203077	−	0.979163i	$-0.434906\pi$
$463$	8.73938	0.406153	0.203077	+	0.979163i	$0.434906\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.1032	1.90203	0.951014	−	0.309149i	$-0.100044\pi$
$467$	41.1032	1.90203	0.951014	+	0.309149i	$0.100044\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	50.1606i	2.30639i
$474$	0	0
$475$	− 5.00407i	− 0.229603i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.3145	−0.699737	−0.349868	−	0.936799i	$-0.613774\pi$
$479$	−15.3145	−0.699737	−0.349868	+	0.936799i	$0.613774\pi$
$480$	0	0
$481$	− 43.4146i	− 1.97954i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.39200i	− 0.335653i
$486$	0	0
$487$	24.2790	1.10018	0.550092	−	0.835104i	$-0.314592\pi$
$487$	24.2790	1.10018	0.550092	+	0.835104i	$0.314592\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 35.8933i	− 1.61984i	−0.586538	−	0.809922i	$-0.699509\pi$
$491$	− 35.8933i	− 1.61984i	0.586538	−	0.809922i	$-0.300491\pi$
$492$	0	0
$493$	− 50.2779i	− 2.26440i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.9604	−0.938316	−0.469158	−	0.883114i	$-0.655442\pi$
$499$	−20.9604	−0.938316	−0.469158	+	0.883114i	$0.655442\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.29187	−0.414304	−0.207152	−	0.978309i	$-0.566419\pi$
$503$	−9.29187	−0.414304	−0.207152	+	0.978309i	$0.566419\pi$
$504$	0	0
$505$	4.97599	0.221429
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.07505	0.357920	0.178960	−	0.983856i	$-0.442727\pi$
$509$	8.07505	0.357920	0.178960	+	0.983856i	$0.442727\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 13.9701i	− 0.615594i
$516$	0	0
$517$	35.3003i	1.55251i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.7464	0.646050	0.323025	−	0.946390i	$-0.395300\pi$
$521$	14.7464	0.646050	0.323025	+	0.946390i	$0.395300\pi$
$522$	0	0
$523$	20.7293i	0.906429i	0.891401	+	0.453215i	$0.149723\pi$
$523$	20.7293i	0.906429i	−0.891401	+	0.453215i	$0.850277\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 14.0942i	− 0.613953i
$528$	0	0
$529$	−0.676832	−0.0294275
$530$	0	0
$531$	0	0
$532$	0	0
$533$	19.9168i	0.862690i
$534$	0	0
$535$	7.36803i	0.318548i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.9126	0.727129	0.363565	−	0.931569i	$-0.381560\pi$
$541$	16.9126	0.727129	0.363565	+	0.931569i	$0.381560\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.0333	−0.643958
$546$	0	0
$547$	−17.3516	−0.741901	−0.370950	−	0.928653i	$-0.620968\pi$
$547$	−17.3516	−0.741901	−0.370950	+	0.928653i	$0.620968\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−46.2692	−1.97114
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.6854i	− 1.04595i	−0.852346	−	0.522977i	$-0.824821\pi$
$557$	− 24.6854i	− 1.04595i	0.852346	−	0.522977i	$-0.175179\pi$
$558$	0	0
$559$	38.0091i	1.60761i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.6929	−0.534944	−0.267472	−	0.963566i	$-0.586188\pi$
$563$	−12.6929	−0.534944	−0.267472	+	0.963566i	$0.586188\pi$
$564$	0	0
$565$	19.0792i	0.802667i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 4.33418i	− 0.181698i	−0.995865	−	0.0908492i	$-0.971042\pi$
$569$	− 4.33418i	− 0.181698i	0.995865	−	0.0908492i	$-0.0289581\pi$
$570$	0	0
$571$	19.1138	0.799889	0.399944	−	0.916539i	$-0.369029\pi$
$571$	19.1138	0.799889	0.399944	+	0.916539i	$0.369029\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 4.86588i	− 0.202921i
$576$	0	0
$577$	0.476765i	0.0198480i	0.999951	+	0.00992398i	$0.00315895\pi$
$577$	0.476765i	0.0198480i	−0.999951	+	0.00992398i	$0.996841\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	15.9072	0.658810
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.7630	0.774433	0.387216	−	0.921989i	$-0.373437\pi$
$587$	18.7630	0.774433	0.387216	+	0.921989i	$0.373437\pi$
$588$	0	0
$589$	−12.9705	−0.534438
$590$	0	0
$591$	0	0
$592$	0	0
$593$	46.6600	1.91610	0.958048	−	0.286608i	$-0.0925279\pi$
$593$	46.6600	1.91610	0.958048	+	0.286608i	$0.0925279\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 22.4770i	− 0.918385i	−0.888337	−	0.459192i	$-0.848139\pi$
$599$	− 22.4770i	− 0.918385i	0.888337	−	0.459192i	$-0.151861\pi$
$600$	0	0
$601$	6.41057i	0.261492i	0.991416	+	0.130746i	$0.0417373\pi$
$601$	6.41057i	0.261492i	−0.991416	+	0.130746i	$0.958263\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.7815	1.00751
$606$	0	0
$607$	22.4739i	0.912189i	0.889931	+	0.456094i	$0.150752\pi$
$607$	22.4739i	0.912189i	−0.889931	+	0.456094i	$0.849248\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.7488i	1.08214i
$612$	0	0
$613$	−38.5097	−1.55539	−0.777696	−	0.628641i	$-0.783612\pi$
$613$	−38.5097	−1.55539	−0.777696	+	0.628641i	$0.783612\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	47.1856i	1.89962i	0.312826	+	0.949810i	$0.398724\pi$
$617$	47.1856i	1.89962i	−0.312826	+	0.949810i	$0.601276\pi$
$618$	0	0
$619$	34.3555i	1.38086i	0.723398	+	0.690431i	$0.242579\pi$
$619$	34.3555i	1.38086i	−0.723398	+	0.690431i	$0.757421\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	52.0823	2.07666
$630$	0	0
$631$	−8.79970	−0.350310	−0.175155	−	0.984541i	$-0.556043\pi$
$631$	−8.79970	−0.350310	−0.175155	+	0.984541i	$0.556043\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.54334	0.378716
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.7509i	0.938105i	0.883170	+	0.469053i	$0.155405\pi$
$641$	23.7509i	0.938105i	−0.883170	+	0.469053i	$0.844595\pi$
$642$	0	0
$643$	26.9775i	1.06389i	0.846779	+	0.531944i	$0.178538\pi$
$643$	26.9775i	1.06389i	−0.846779	+	0.531944i	$0.821462\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.4158	−1.58891	−0.794454	−	0.607325i	$-0.792243\pi$
$647$	−40.4158	−1.58891	−0.794454	+	0.607325i	$0.792243\pi$
$648$	0	0
$649$	− 47.5187i	− 1.86527i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 8.28623i	− 0.324265i	−0.986769	−	0.162133i	$-0.948163\pi$
$653$	− 8.28623i	− 0.324265i	0.986769	−	0.162133i	$-0.0518373\pi$
$654$	0	0
$655$	−21.6139	−0.844525
$656$	0	0
$657$	0	0
$658$	0	0
$659$	38.5737i	1.50262i	0.659952	+	0.751308i	$0.270577\pi$
$659$	38.5737i	1.50262i	−0.659952	+	0.751308i	$0.729423\pi$
$660$	0	0
$661$	18.1305i	0.705195i	0.935775	+	0.352597i	$0.114701\pi$
$661$	18.1305i	0.705195i	−0.935775	+	0.352597i	$0.885299\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−44.9915	−1.74208
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−47.4554	−1.83199
$672$	0	0
$673$	0.624778	0.0240834	0.0120417	−	0.999927i	$-0.496167\pi$
$673$	0.624778	0.0240834	0.0120417	+	0.999927i	$0.496167\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.2482	−0.970366	−0.485183	−	0.874413i	$-0.661247\pi$
$677$	−25.2482	−0.970366	−0.485183	+	0.874413i	$0.661247\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14.5988i	− 0.558607i	−0.960203	−	0.279303i	$-0.909896\pi$
$683$	− 14.5988i	− 0.558607i	0.960203	−	0.279303i	$-0.0901035\pi$
$684$	0	0
$685$	12.9264i	0.493891i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.0537	0.459208
$690$	0	0
$691$	− 20.7047i	− 0.787646i	−0.919186	−	0.393823i	$-0.871152\pi$
$691$	− 20.7047i	− 0.787646i	0.919186	−	0.393823i	$-0.128848\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 9.34361i	− 0.354423i
$696$	0	0
$697$	−23.8931	−0.905016
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.8408i	0.787147i	0.919293	+	0.393573i	$0.128761\pi$
$701$	20.8408i	0.787147i	−0.919293	+	0.393573i	$0.871239\pi$
$702$	0	0
$703$	− 47.9297i	− 1.80770i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−49.7224	−1.86736	−0.933681	−	0.358105i	$-0.883423\pi$
$709$	−49.7224	−1.86736	−0.933681	+	0.358105i	$0.883423\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.6123	−0.472334
$714$	0	0
$715$	27.1134	1.01398
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.7601	0.774222	0.387111	−	0.922033i	$-0.373473\pi$
$719$	20.7601	0.774222	0.387111	+	0.922033i	$0.373473\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.24632i	− 0.343400i
$726$	0	0
$727$	− 15.9306i	− 0.590834i	−0.955368	−	0.295417i	$-0.904541\pi$
$727$	− 15.9306i	− 0.590834i	0.955368	−	0.295417i	$-0.0954586\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.5976	−1.68649
$732$	0	0
$733$	− 41.6863i	− 1.53972i	−0.638213	−	0.769860i	$-0.720326\pi$
$733$	− 41.6863i	− 1.53972i	0.638213	−	0.769860i	$-0.279674\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	83.6010i	3.07948i
$738$	0	0
$739$	−7.98623	−0.293778	−0.146889	−	0.989153i	$-0.546926\pi$
$739$	−7.98623	−0.293778	−0.146889	+	0.989153i	$0.546926\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.68587i	− 0.355340i	−0.984090	−	0.177670i	$-0.943144\pi$
$743$	− 9.68587i	− 0.355340i	0.984090	−	0.177670i	$-0.0568560\pi$
$744$	0	0
$745$	− 2.11271i	− 0.0774038i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−19.2114	−0.701033	−0.350516	−	0.936557i	$-0.613994\pi$
$751$	−19.2114	−0.701033	−0.350516	+	0.936557i	$0.613994\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	19.8303	0.721697
$756$	0	0
$757$	−17.2007	−0.625172	−0.312586	−	0.949889i	$-0.601195\pi$
$757$	−17.2007	−0.625172	−0.312586	+	0.949889i	$0.601195\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.8337	−1.26272	−0.631360	−	0.775490i	$-0.717503\pi$
$761$	−34.8337	−1.26272	−0.631360	+	0.775490i	$0.717503\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 36.0072i	− 1.30014i
$768$	0	0
$769$	− 25.2694i	− 0.911237i	−0.890175	−	0.455619i	$-0.849418\pi$
$769$	− 25.2694i	− 0.911237i	0.890175	−	0.455619i	$-0.150582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.5406	1.13444	0.567218	−	0.823568i	$-0.308020\pi$
$773$	31.5406	1.13444	0.567218	+	0.823568i	$0.308020\pi$
$774$	0	0
$775$	− 2.59198i	− 0.0931067i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21.9881i	0.787805i
$780$	0	0
$781$	71.0952	2.54399
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 0.374507i	− 0.0133667i
$786$	0	0
$787$	0.0942065i	0.00335810i	0.999999	+	0.00167905i	$0.000534458\pi$
$787$	0.0942065i	0.00335810i	−0.999999	+	0.00167905i	$0.999466\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−35.9592	−1.27695
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.5330	−1.50660	−0.753299	−	0.657679i	$-0.771539\pi$
$797$	−42.5330	−1.50660	−0.753299	+	0.657679i	$0.771539\pi$
$798$	0	0
$799$	−32.0891	−1.13523
$800$	0	0
$801$	0	0
$802$	0	0
$803$	99.7770	3.52105
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.14027i	0.286197i	0.989708	+	0.143098i	$0.0457065\pi$
$809$	8.14027i	0.286197i	−0.989708	+	0.143098i	$0.954293\pi$
$810$	0	0
$811$	− 14.9493i	− 0.524941i	−0.964940	−	0.262470i	$-0.915463\pi$
$811$	− 14.9493i	− 0.524941i	0.964940	−	0.262470i	$-0.0845372\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.39995	−0.154123
$816$	0	0
$817$	41.9620i	1.46807i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 23.6516i	− 0.825447i	−0.910856	−	0.412724i	$-0.864578\pi$
$821$	− 23.6516i	− 0.825447i	0.910856	−	0.412724i	$-0.135422\pi$
$822$	0	0
$823$	−32.3644	−1.12815	−0.564075	−	0.825723i	$-0.690767\pi$
$823$	−32.3644	−1.12815	−0.564075	+	0.825723i	$0.690767\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.2488i	0.530252i	0.964214	+	0.265126i	$0.0854136\pi$
$827$	15.2488i	0.530252i	−0.964214	+	0.265126i	$0.914586\pi$
$828$	0	0
$829$	− 32.4821i	− 1.12815i	−0.825724	−	0.564075i	$-0.809233\pi$
$829$	− 32.4821i	− 1.12815i	0.825724	−	0.564075i	$-0.190767\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.38327	0.0478700
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−47.2019	−1.62959	−0.814796	−	0.579748i	$-0.803151\pi$
$839$	−47.2019	−1.62959	−0.814796	+	0.579748i	$0.803151\pi$
$840$	0	0
$841$	−56.4944	−1.94808
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.54515	0.259561
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 46.6061i	− 1.59764i
$852$	0	0
$853$	17.5636i	0.601366i	0.953724	+	0.300683i	$0.0972146\pi$
$853$	17.5636i	0.601366i	−0.953724	+	0.300683i	$0.902785\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.7944	0.402888	0.201444	−	0.979500i	$-0.435437\pi$
$857$	11.7944	0.402888	0.201444	+	0.979500i	$0.435437\pi$
$858$	0	0
$859$	39.2078i	1.33775i	0.743373	+	0.668877i	$0.233225\pi$
$859$	39.2078i	1.33775i	−0.743373	+	0.668877i	$0.766775\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.29236i	0.0780328i	0.999239	+	0.0390164i	$0.0124225\pi$
$863$	2.29236i	0.0780328i	−0.999239	+	0.0390164i	$0.987578\pi$
$864$	0	0
$865$	−9.04587	−0.307569
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 26.8796i	− 0.911829i
$870$	0	0
$871$	63.3485i	2.14648i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	40.9487	1.38274	0.691369	−	0.722502i	$-0.257008\pi$
$877$	40.9487	1.38274	0.691369	+	0.722502i	$0.257008\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.65057	0.190373	0.0951863	−	0.995459i	$-0.469655\pi$
$881$	5.65057	0.190373	0.0951863	+	0.995459i	$0.469655\pi$
$882$	0	0
$883$	24.9073	0.838197	0.419098	−	0.907941i	$-0.362346\pi$
$883$	24.9073	0.838197	0.419098	+	0.907941i	$0.362346\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.5961	−1.22878	−0.614389	−	0.789004i	$-0.710597\pi$
$887$	−36.5961	−1.22878	−0.614389	+	0.789004i	$0.710597\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29.5306i	0.988204i
$894$	0	0
$895$	5.06854i	0.169422i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−23.9663	−0.799320
$900$	0	0
$901$	14.4602i	0.481738i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.4372i	− 0.646113i
$906$	0	0
$907$	14.3627	0.476907	0.238453	−	0.971154i	$-0.423360\pi$
$907$	14.3627	0.476907	0.238453	+	0.971154i	$0.423360\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 22.3332i	− 0.739933i	−0.929045	−	0.369966i	$-0.879369\pi$
$911$	− 22.3332i	− 0.739933i	0.929045	−	0.369966i	$-0.120631\pi$
$912$	0	0
$913$	4.58328i	0.151685i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	23.6460	0.780008	0.390004	−	0.920813i	$-0.372474\pi$
$919$	23.6460	0.780008	0.390004	+	0.920813i	$0.372474\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	53.8723	1.77323
$924$	0	0
$925$	9.57814	0.314927
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.2835	0.567052	0.283526	−	0.958964i	$-0.408496\pi$
$929$	17.2835	0.567052	0.283526	+	0.958964i	$0.408496\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32.5266i	1.06373i
$936$	0	0
$937$	− 5.12363i	− 0.167382i	−0.996492	−	0.0836909i	$-0.973329\pi$
$937$	− 5.12363i	− 0.167382i	0.996492	−	0.0836909i	$-0.0266708\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.0416	0.783735	0.391867	−	0.920022i	$-0.371829\pi$
$941$	24.0416	0.783735	0.391867	+	0.920022i	$0.371829\pi$
$942$	0	0
$943$	21.3809i	0.696258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	54.0349i	1.75590i	0.478755	+	0.877949i	$0.341088\pi$
$947$	54.0349i	1.75590i	−0.478755	+	0.877949i	$0.658912\pi$
$948$	0	0
$949$	75.6059	2.45427
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.1011i	0.845499i	0.906247	+	0.422749i	$0.138935\pi$
$953$	26.1011i	0.845499i	−0.906247	+	0.422749i	$0.861065\pi$
$954$	0	0
$955$	− 18.4556i	− 0.597208i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	24.2816	0.783279
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.40795	0.174088
$966$	0	0
$967$	−39.1182	−1.25796	−0.628978	−	0.777423i	$-0.716526\pi$
$967$	−39.1182	−1.25796	−0.628978	+	0.777423i	$0.716526\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.3678	−1.23128	−0.615641	−	0.788027i	$-0.711103\pi$
$971$	−38.3678	−1.23128	−0.615641	+	0.788027i	$0.711103\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.0268i	0.512743i	0.966578	+	0.256371i	$0.0825270\pi$
$977$	16.0268i	0.512743i	−0.966578	+	0.256371i	$0.917473\pi$
$978$	0	0
$979$	− 80.3710i	− 2.56867i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.7125	1.29853	0.649263	−	0.760564i	$-0.275078\pi$
$983$	40.7125	1.29853	0.649263	+	0.760564i	$0.275078\pi$
$984$	0	0
$985$	− 2.34774i	− 0.0748053i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.8032i	1.29747i
$990$	0	0
$991$	−12.3814	−0.393308	−0.196654	−	0.980473i	$-0.563008\pi$
$991$	−12.3814	−0.393308	−0.196654	+	0.980473i	$0.563008\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.0464i	0.381896i
$996$	0	0
$997$	− 12.8605i	− 0.407297i	−0.979044	−	0.203648i	$-0.934720\pi$
$997$	− 12.8605i	− 0.407297i	0.979044	−	0.203648i	$-0.0652799\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.d.c.881.2	✓	16
3.2	odd	2		8820.2.d.d.881.15	yes	16
7.6	odd	2		8820.2.d.d.881.2	yes	16
21.20	even	2	inner	8820.2.d.c.881.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8820.2.d.c.881.2	✓	16	1.1	even	1	trivial
8820.2.d.c.881.15	yes	16	21.20	even	2	inner
8820.2.d.d.881.2	yes	16	7.6	odd	2
8820.2.d.d.881.15	yes	16	3.2	odd	2

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:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} -5.98177i q^{11} -4.53268i q^{13} +5.43762 q^{17} -5.00407i q^{19} -4.86588i q^{23} +1.00000 q^{25} -9.24632i q^{29} -2.59198i q^{31} +9.57814 q^{37} -4.39404 q^{41} -8.38558 q^{43} -5.90132 q^{47} +2.65928i q^{53} +5.98177i q^{55} +7.94392 q^{59} -7.93334i q^{61} +4.53268i q^{65} -13.9760 q^{67} +11.8853i q^{71} +16.6802i q^{73} +4.49359 q^{79} -0.766209 q^{83} -5.43762 q^{85} +13.4360 q^{89} +5.00407i q^{95} +7.39200i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5}+O(q^{10}) \) 16 * q - 16 * q^5	\( 16 q - 16 q^{5} + 16 q^{25} - 32 q^{41} - 32 q^{43} + 32 q^{47} + 32 q^{59} - 32 q^{67} + 64 q^{89}+O(q^{100}) \) 16 * q - 16 * q^5 + 16 * q^25 - 32 * q^41 - 32 * q^43 + 32 * q^47 + 32 * q^59 - 32 * q^67 + 64 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more