LMFDB - Embedded newform 2025.2.b.n.649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.34810603776.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 $ x^8 + 12x^6 + 42x^4 + 49*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$-1.47325i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.n.649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47325i q^{2} -0.170479 q^{4} -3.86583i q^{7} -2.69535i q^{8} +O(q^{10})$	$q-1.47325i q^{2} -0.170479 q^{4} -3.86583i q^{7} -2.69535i q^{8} +0.260278 q^{11} -4.07881i q^{13} -5.69535 q^{14} -4.31189 q^{16} -3.26028i q^{17} -4.24928 q^{19} -0.383456i q^{22} +8.69535i q^{23} -6.00912 q^{26} +0.659042i q^{28} -4.22210 q^{29} +2.65285 q^{31} +0.961818i q^{32} -4.80322 q^{34} +2.27559i q^{37} +6.26028i q^{38} +5.64186 q^{41} -9.07262i q^{43} -0.0443719 q^{44} +12.8105 q^{46} +1.42888i q^{47} -7.94463 q^{49} +0.695350i q^{52} +11.3816i q^{53} -10.4198 q^{56} +6.22022i q^{58} +7.12423 q^{59} +2.52487 q^{61} -3.90833i q^{62} -7.20679 q^{64} -11.2856i q^{67} +0.555809i q^{68} -8.38158 q^{71} +0.403568i q^{73} +3.35252 q^{74} +0.724413 q^{76} -1.00619i q^{77} +3.04250 q^{79} -8.31189i q^{82} +4.58024i q^{83} -13.3663 q^{86} -0.701540i q^{88} -7.17772 q^{89} -15.7680 q^{91} -1.48237i q^{92} +2.10511 q^{94} -3.11511i q^{97} +11.7045i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74} + 22 q^{76} + 14 q^{79} - 68 q^{86} + 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$.

$n$	$326$	$1702$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.47325i	− 1.04175i	−0.853633	−	0.520874i	$-0.825606\pi$
$2$	− 1.47325i	− 1.04175i	0.853633	−	0.520874i	$-0.174394\pi$
$3$	0	0
$3$	0	0
$4$	−0.170479	−0.0852394
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.86583i	− 1.46115i	−0.682835	−	0.730573i	$-0.739253\pi$
$7$	− 3.86583i	− 1.46115i	0.682835	−	0.730573i	$-0.260747\pi$
$8$	− 2.69535i	− 0.952950i
$9$	0	0
$10$	0	0
$11$	0.260278	0.0784767	0.0392384	−	0.999230i	$-0.487507\pi$
$11$	0.260278	0.0784767	0.0392384	+	0.999230i	$0.487507\pi$
$12$	0	0
$13$	− 4.07881i	− 1.13126i	−0.824660	−	0.565629i	$-0.808634\pi$
$13$	− 4.07881i	− 1.13126i	0.824660	−	0.565629i	$-0.191366\pi$
$14$	−5.69535	−1.52215
$15$	0	0
$16$	−4.31189	−1.07797
$17$	− 3.26028i	− 0.790734i	−0.918523	−	0.395367i	$-0.870617\pi$
$17$	− 3.26028i	− 0.790734i	0.918523	−	0.395367i	$-0.129383\pi$
$18$	0	0
$19$	−4.24928	−0.974853	−0.487426	−	0.873164i	$-0.662064\pi$
$19$	−4.24928	−0.974853	−0.487426	+	0.873164i	$0.662064\pi$
$20$	0	0
$21$	0	0
$22$	− 0.383456i	− 0.0817530i
$23$	8.69535i	1.81311i	0.422092	+	0.906553i	$0.361296\pi$
$23$	8.69535i	1.81311i	−0.422092	+	0.906553i	$0.638704\pi$
$24$	0	0
$25$	0	0
$26$	−6.00912	−1.17849
$27$	0	0
$28$	0.659042i	0.124547i
$29$	−4.22210	−0.784023	−0.392012	−	0.919960i	$-0.628221\pi$
$29$	−4.22210	−0.784023	−0.392012	+	0.919960i	$0.628221\pi$
$30$	0	0
$31$	2.65285	0.476466	0.238233	−	0.971208i	$-0.423432\pi$
$31$	2.65285	0.476466	0.238233	+	0.971208i	$0.423432\pi$
$32$	0.961818i	0.170027i
$33$	0	0
$34$	−4.80322	−0.823745
$35$	0	0
$36$	0	0
$37$	2.27559i	0.374104i	0.982350	+	0.187052i	$0.0598933\pi$
$37$	2.27559i	0.374104i	−0.982350	+	0.187052i	$0.940107\pi$
$38$	6.26028i	1.01555i
$39$	0	0
$40$	0	0
$41$	5.64186	0.881110	0.440555	−	0.897726i	$-0.354782\pi$
$41$	5.64186	0.881110	0.440555	+	0.897726i	$0.354782\pi$
$42$	0	0
$43$	− 9.07262i	− 1.38356i	−0.722108	−	0.691780i	$-0.756827\pi$
$43$	− 9.07262i	− 1.38356i	0.722108	−	0.691780i	$-0.243173\pi$
$44$	−0.0443719	−0.00668931
$45$	0	0
$46$	12.8105	1.88880
$47$	1.42888i	0.208424i	0.994555	+	0.104212i	$0.0332320\pi$
$47$	1.42888i	0.208424i	−0.994555	+	0.104212i	$0.966768\pi$
$48$	0	0
$49$	−7.94463	−1.13495
$50$	0	0
$51$	0	0
$52$	0.695350i	0.0964277i
$53$	11.3816i	1.56338i	0.623667	+	0.781690i	$0.285642\pi$
$53$	11.3816i	1.56338i	−0.623667	+	0.781690i	$0.714358\pi$
$54$	0	0
$55$	0	0
$56$	−10.4198	−1.39240
$57$	0	0
$58$	6.22022i	0.816755i
$59$	7.12423	0.927496	0.463748	−	0.885967i	$-0.346504\pi$
$59$	7.12423	0.927496	0.463748	+	0.885967i	$0.346504\pi$
$60$	0	0
$61$	2.52487	0.323277	0.161638	−	0.986850i	$-0.448322\pi$
$61$	2.52487	0.323277	0.161638	+	0.986850i	$0.448322\pi$
$62$	− 3.90833i	− 0.496358i
$63$	0	0
$64$	−7.20679	−0.900848
$65$	0	0
$66$	0	0
$67$	− 11.2856i	− 1.37875i	−0.724402	−	0.689377i	$-0.757884\pi$
$67$	− 11.2856i	− 1.37875i	0.724402	−	0.689377i	$-0.242116\pi$
$68$	0.555809i	0.0674017i
$69$	0	0
$70$	0	0
$71$	−8.38158	−0.994711	−0.497355	−	0.867547i	$-0.665695\pi$
$71$	−8.38158	−0.994711	−0.497355	+	0.867547i	$0.665695\pi$
$72$	0	0
$73$	0.403568i	0.0472340i	0.999721	+	0.0236170i	$0.00751823\pi$
$73$	0.403568i	0.0472340i	−0.999721	+	0.0236170i	$0.992482\pi$
$74$	3.35252	0.389722
$75$	0	0
$76$	0.724413	0.0830959
$77$	− 1.00619i	− 0.114666i
$78$	0	0
$79$	3.04250	0.342308	0.171154	−	0.985244i	$-0.445250\pi$
$79$	3.04250	0.342308	0.171154	+	0.985244i	$0.445250\pi$
$80$	0	0
$81$	0	0
$82$	− 8.31189i	− 0.917895i
$83$	4.58024i	0.502746i	0.967890	+	0.251373i	$0.0808821\pi$
$83$	4.58024i	0.502746i	−0.967890	+	0.251373i	$0.919118\pi$
$84$	0	0
$85$	0	0
$86$	−13.3663	−1.44132
$87$	0	0
$88$	− 0.701540i	− 0.0747844i
$89$	−7.17772	−0.760837	−0.380419	−	0.924814i	$-0.624220\pi$
$89$	−7.17772	−0.760837	−0.380419	+	0.924814i	$0.624220\pi$
$90$	0	0
$91$	−15.7680	−1.65293
$92$	− 1.48237i	− 0.154548i
$93$	0	0
$94$	2.10511	0.217125
$95$	0	0
$96$	0	0
$97$	− 3.11511i	− 0.316292i	−0.987416	−	0.158146i	$-0.949448\pi$
$97$	− 3.11511i	− 0.316292i	0.987416	−	0.158146i	$-0.0505516\pi$
$98$	11.7045i	1.18233i
$99$	0	0
$100$	0	0
$101$	3.84572	0.382663	0.191332	−	0.981525i	$-0.438719\pi$
$101$	3.84572	0.382663	0.191332	+	0.981525i	$0.438719\pi$
$102$	0	0
$103$	− 4.20679i	− 0.414507i	−0.978287	−	0.207254i	$-0.933548\pi$
$103$	− 4.20679i	− 0.414507i	0.978287	−	0.207254i	$-0.0664525\pi$
$104$	−10.9938	−1.07803
$105$	0	0
$106$	16.7680	1.62865
$107$	1.62655i	0.157245i	0.996904	+	0.0786223i	$0.0250521\pi$
$107$	1.62655i	0.157245i	−0.996904	+	0.0786223i	$0.974948\pi$
$108$	0	0
$109$	12.9021	1.23580	0.617900	−	0.786256i	$-0.287984\pi$
$109$	12.9021	1.23580	0.617900	+	0.786256i	$0.287984\pi$
$110$	0	0
$111$	0	0
$112$	16.6690i	1.57508i
$113$	1.32908i	0.125029i	0.998044	+	0.0625146i	$0.0199120\pi$
$113$	1.32908i	0.125029i	−0.998044	+	0.0625146i	$0.980088\pi$
$114$	0	0
$115$	0	0
$116$	0.719778	0.0668297
$117$	0	0
$118$	− 10.4958i	− 0.966217i
$119$	−12.6037	−1.15538
$120$	0	0
$121$	−10.9323	−0.993841
$122$	− 3.71978i	− 0.336773i
$123$	0	0
$124$	−0.452255	−0.0406137
$125$	0	0
$126$	0	0
$127$	1.65285i	0.146667i	0.997307	+	0.0733335i	$0.0233637\pi$
$127$	1.65285i	0.146667i	−0.997307	+	0.0733335i	$0.976636\pi$
$128$	12.5411i	1.10848i
$129$	0	0
$130$	0	0
$131$	−13.1777	−1.15134	−0.575672	−	0.817681i	$-0.695259\pi$
$131$	−13.1777	−1.15134	−0.575672	+	0.817681i	$0.695259\pi$
$132$	0	0
$133$	16.4270i	1.42440i
$134$	−16.6266	−1.43632
$135$	0	0
$136$	−8.78759	−0.753530
$137$	− 20.2928i	− 1.73373i	−0.498539	−	0.866867i	$-0.666130\pi$
$137$	− 20.2928i	− 1.73373i	0.498539	−	0.866867i	$-0.333870\pi$
$138$	0	0
$139$	−3.06880	−0.260292	−0.130146	−	0.991495i	$-0.541545\pi$
$139$	−3.06880	−0.260292	−0.130146	+	0.991495i	$0.541545\pi$
$140$	0	0
$141$	0	0
$142$	12.3482i	1.03624i
$143$	− 1.06162i	− 0.0887774i
$144$	0	0
$145$	0	0
$146$	0.594558	0.0492060
$147$	0	0
$148$	− 0.387939i	− 0.0318884i
$149$	−4.06162	−0.332741	−0.166371	−	0.986063i	$-0.553205\pi$
$149$	−4.06162	−0.332741	−0.166371	+	0.986063i	$0.553205\pi$
$150$	0	0
$151$	−13.6199	−1.10837	−0.554185	−	0.832394i	$-0.686970\pi$
$151$	−13.6199	−1.10837	−0.554185	+	0.832394i	$0.686970\pi$
$152$	11.4533i	0.928986i
$153$	0	0
$154$	−1.48237	−0.119453
$155$	0	0
$156$	0	0
$157$	− 2.06261i	− 0.164614i	−0.996607	−	0.0823071i	$-0.973771\pi$
$157$	− 2.06261i	− 0.164614i	0.996607	−	0.0823071i	$-0.0262288\pi$
$158$	− 4.48237i	− 0.356598i
$159$	0	0
$160$	0	0
$161$	33.6147	2.64921
$162$	0	0
$163$	− 3.50525i	− 0.274552i	−0.990533	−	0.137276i	$-0.956165\pi$
$163$	− 3.50525i	− 0.274552i	0.990533	−	0.137276i	$-0.0438347\pi$
$164$	−0.961818	−0.0751054
$165$	0	0
$166$	6.74785	0.523735
$167$	− 20.5349i	− 1.58904i	−0.607240	−	0.794518i	$-0.707723\pi$
$167$	− 20.5349i	− 1.58904i	0.607240	−	0.794518i	$-0.292277\pi$
$168$	0	0
$169$	−3.63666	−0.279743
$170$	0	0
$171$	0	0
$172$	1.54669i	0.117934i
$173$	− 7.75177i	− 0.589356i	−0.955597	−	0.294678i	$-0.904788\pi$
$173$	− 7.75177i	− 0.589356i	0.955597	−	0.294678i	$-0.0952124\pi$
$174$	0	0
$175$	0	0
$176$	−1.12229	−0.0845958
$177$	0	0
$178$	10.5746i	0.792601i
$179$	10.7632	0.804477	0.402238	−	0.915535i	$-0.368232\pi$
$179$	10.7632	0.804477	0.402238	+	0.915535i	$0.368232\pi$
$180$	0	0
$181$	−7.84572	−0.583168	−0.291584	−	0.956545i	$-0.594182\pi$
$181$	−7.84572	−0.583168	−0.291584	+	0.956545i	$0.594182\pi$
$182$	23.2302i	1.72194i
$183$	0	0
$184$	23.4370	1.72780
$185$	0	0
$186$	0	0
$187$	− 0.848578i	− 0.0620542i
$188$	− 0.243594i	− 0.0177659i
$189$	0	0
$190$	0	0
$191$	−5.73255	−0.414792	−0.207396	−	0.978257i	$-0.566499\pi$
$191$	−5.73255	−0.414792	−0.207396	+	0.978257i	$0.566499\pi$
$192$	0	0
$193$	− 8.48237i	− 0.610575i	−0.952260	−	0.305287i	$-0.901248\pi$
$193$	− 8.48237i	− 0.610575i	0.952260	−	0.305287i	$-0.0987525\pi$
$194$	−4.58936	−0.329497
$195$	0	0
$196$	1.35439	0.0967423
$197$	− 10.6266i	− 0.757110i	−0.925579	−	0.378555i	$-0.876421\pi$
$197$	− 10.6266i	− 0.757110i	0.925579	−	0.378555i	$-0.123579\pi$
$198$	0	0
$199$	18.5784	1.31699	0.658495	−	0.752585i	$-0.271193\pi$
$199$	18.5784	1.31699	0.658495	+	0.752585i	$0.271193\pi$
$200$	0	0
$201$	0	0
$202$	− 5.66572i	− 0.398639i
$203$	16.3219i	1.14557i
$204$	0	0
$205$	0	0
$206$	−6.19767	−0.431812
$207$	0	0
$208$	17.5874i	1.21947i
$209$	−1.10599	−0.0765033
$210$	0	0
$211$	10.4533	0.719636	0.359818	−	0.933023i	$-0.382839\pi$
$211$	10.4533	0.719636	0.359818	+	0.933023i	$0.382839\pi$
$212$	− 1.94032i	− 0.133262i
$213$	0	0
$214$	2.39632	0.163809
$215$	0	0
$216$	0	0
$217$	− 10.2555i	− 0.696187i
$218$	− 19.0081i	− 1.28739i
$219$	0	0
$220$	0	0
$221$	−13.2980	−0.894523
$222$	0	0
$223$	3.93120i	0.263253i	0.991299	+	0.131626i	$0.0420199\pi$
$223$	3.93120i	0.263253i	−0.991299	+	0.131626i	$0.957980\pi$
$224$	3.71822	0.248434
$225$	0	0
$226$	1.95807	0.130249
$227$	− 4.83140i	− 0.320671i	−0.987063	−	0.160335i	$-0.948742\pi$
$227$	− 4.83140i	− 0.320671i	0.987063	−	0.160335i	$-0.0512576\pi$
$228$	0	0
$229$	18.8530	1.24584	0.622919	−	0.782286i	$-0.285946\pi$
$229$	18.8530	1.24584	0.622919	+	0.782286i	$0.285946\pi$
$230$	0	0
$231$	0	0
$232$	11.3800i	0.747135i
$233$	− 11.9021i	− 0.779735i	−0.920871	−	0.389867i	$-0.872521\pi$
$233$	− 11.9021i	− 0.779735i	0.920871	−	0.389867i	$-0.127479\pi$
$234$	0	0
$235$	0	0
$236$	−1.21453	−0.0790593
$237$	0	0
$238$	18.5684i	1.20361i
$239$	21.6295	1.39909	0.699547	−	0.714586i	$-0.253385\pi$
$239$	21.6295	1.39909	0.699547	+	0.714586i	$0.253385\pi$
$240$	0	0
$241$	3.89832	0.251113	0.125556	−	0.992086i	$-0.459928\pi$
$241$	3.89832	0.251113	0.125556	+	0.992086i	$0.459928\pi$
$242$	16.1060i	1.03533i
$243$	0	0
$244$	−0.430437	−0.0275559
$245$	0	0
$246$	0	0
$247$	17.3320i	1.10281i
$248$	− 7.15037i	− 0.454049i
$249$	0	0
$250$	0	0
$251$	−30.1033	−1.90010	−0.950052	−	0.312092i	$-0.898970\pi$
$251$	−30.1033	−1.90010	−0.950052	+	0.312092i	$0.898970\pi$
$252$	0	0
$253$	2.26321i	0.142287i
$254$	2.43507	0.152790
$255$	0	0
$256$	4.06261	0.253913
$257$	16.4141i	1.02389i	0.859019	+	0.511943i	$0.171074\pi$
$257$	16.4141i	1.02389i	−0.859019	+	0.511943i	$0.828926\pi$
$258$	0	0
$259$	8.79703	0.546621
$260$	0	0
$261$	0	0
$262$	19.4141i	1.19941i
$263$	25.8072i	1.59134i	0.605730	+	0.795670i	$0.292881\pi$
$263$	25.8072i	1.59134i	−0.605730	+	0.795670i	$0.707119\pi$
$264$	0	0
$265$	0	0
$266$	24.2012	1.48387
$267$	0	0
$268$	1.92396i	0.117524i
$269$	12.5206	0.763392	0.381696	−	0.924288i	$-0.375340\pi$
$269$	12.5206	0.763392	0.381696	+	0.924288i	$0.375340\pi$
$270$	0	0
$271$	19.6462	1.19342	0.596710	−	0.802457i	$-0.296474\pi$
$271$	19.6462	1.19342	0.596710	+	0.802457i	$0.296474\pi$
$272$	14.0580i	0.852390i
$273$	0	0
$274$	−29.8965	−1.80611
$275$	0	0
$276$	0	0
$277$	20.8301i	1.25156i	0.780000	+	0.625779i	$0.215219\pi$
$277$	20.8301i	1.25156i	−0.780000	+	0.625779i	$0.784781\pi$
$278$	4.52112i	0.271159i
$279$	0	0
$280$	0	0
$281$	4.72441	0.281835	0.140917	−	0.990021i	$-0.454995\pi$
$281$	4.72441	0.281835	0.140917	+	0.990021i	$0.454995\pi$
$282$	0	0
$283$	23.1525i	1.37627i	0.725582	+	0.688136i	$0.241571\pi$
$283$	23.1525i	1.37627i	−0.725582	+	0.688136i	$0.758429\pi$
$284$	1.42888	0.0847886
$285$	0	0
$286$	−1.56404	−0.0924837
$287$	− 21.8105i	− 1.28743i
$288$	0	0
$289$	6.37059	0.374740
$290$	0	0
$291$	0	0
$292$	− 0.0687998i	− 0.00402620i
$293$	− 16.8793i	− 0.986097i	−0.870002	−	0.493049i	$-0.835882\pi$
$293$	− 16.8793i	− 0.986097i	0.870002	−	0.493049i	$-0.164118\pi$
$294$	0	0
$295$	0	0
$296$	6.13350	0.356503
$297$	0	0
$298$	5.98380i	0.346632i
$299$	35.4666	2.05109
$300$	0	0
$301$	−35.0732	−2.02158
$302$	20.0655i	1.15464i
$303$	0	0
$304$	18.3225	1.05087
$305$	0	0
$306$	0	0
$307$	22.7177i	1.29657i	0.761398	+	0.648285i	$0.224513\pi$
$307$	22.7177i	1.29657i	−0.761398	+	0.648285i	$0.775487\pi$
$308$	0.171534i	0.00977406i
$309$	0	0
$310$	0	0
$311$	31.5936	1.79151	0.895754	−	0.444551i	$-0.146637\pi$
$311$	31.5936	1.79151	0.895754	+	0.444551i	$0.146637\pi$
$312$	0	0
$313$	− 30.4996i	− 1.72394i	−0.506959	−	0.861970i	$-0.669230\pi$
$313$	− 30.4996i	− 1.72394i	0.506959	−	0.861970i	$-0.330770\pi$
$314$	−3.03875	−0.171487
$315$	0	0
$316$	−0.518682	−0.0291781
$317$	− 22.0890i	− 1.24064i	−0.784349	−	0.620320i	$-0.787003\pi$
$317$	− 22.0890i	− 1.24064i	0.784349	−	0.620320i	$-0.212997\pi$
$318$	0	0
$319$	−1.09892	−0.0615276
$320$	0	0
$321$	0	0
$322$	− 49.5231i	− 2.75981i
$323$	13.8538i	0.770849i
$324$	0	0
$325$	0	0
$326$	−5.16412	−0.286014
$327$	0	0
$328$	− 15.2068i	− 0.839654i
$329$	5.52382	0.304538
$330$	0	0
$331$	−29.6047	−1.62722	−0.813612	−	0.581409i	$-0.802502\pi$
$331$	−29.6047	−1.62722	−0.813612	+	0.581409i	$0.802502\pi$
$332$	− 0.780834i	− 0.0428538i
$333$	0	0
$334$	−30.2531	−1.65538
$335$	0	0
$336$	0	0
$337$	− 12.5311i	− 0.682610i	−0.939953	−	0.341305i	$-0.889131\pi$
$337$	− 12.5311i	− 0.682610i	0.939953	−	0.341305i	$-0.110869\pi$
$338$	5.35772i	0.291422i
$339$	0	0
$340$	0	0
$341$	0.690479	0.0373915
$342$	0	0
$343$	3.65180i	0.197179i
$344$	−24.4539	−1.31846
$345$	0	0
$346$	−11.4203	−0.613961
$347$	17.0974i	0.917839i	0.888478	+	0.458919i	$0.151763\pi$
$347$	17.0974i	0.917839i	−0.888478	+	0.458919i	$0.848237\pi$
$348$	0	0
$349$	18.4046	0.985177	0.492588	−	0.870262i	$-0.336051\pi$
$349$	18.4046	0.985177	0.492588	+	0.870262i	$0.336051\pi$
$350$	0	0
$351$	0	0
$352$	0.250340i	0.0133432i
$353$	− 31.7188i	− 1.68822i	−0.536169	−	0.844110i	$-0.680129\pi$
$353$	− 31.7188i	− 1.68822i	0.536169	−	0.844110i	$-0.319871\pi$
$354$	0	0
$355$	0	0
$356$	1.22365	0.0648533
$357$	0	0
$358$	− 15.8569i	− 0.838062i
$359$	11.4533	0.604483	0.302241	−	0.953231i	$-0.402265\pi$
$359$	11.4533	0.604483	0.302241	+	0.953231i	$0.402265\pi$
$360$	0	0
$361$	−0.943580	−0.0496621
$362$	11.5587i	0.607514i
$363$	0	0
$364$	2.68811	0.140895
$365$	0	0
$366$	0	0
$367$	− 2.49238i	− 0.130101i	−0.997882	−	0.0650506i	$-0.979279\pi$
$367$	− 2.49238i	− 0.130101i	0.997882	−	0.0650506i	$-0.0207209\pi$
$368$	− 37.4934i	− 1.95448i
$369$	0	0
$370$	0	0
$371$	43.9992	2.28433
$372$	0	0
$373$	− 15.0374i	− 0.778605i	−0.921110	−	0.389303i	$-0.872716\pi$
$373$	− 15.0374i	− 0.778605i	0.921110	−	0.389303i	$-0.127284\pi$
$374$	−1.25017	−0.0646448
$375$	0	0
$376$	3.85134	0.198618
$377$	17.2211i	0.886932i
$378$	0	0
$379$	−6.27273	−0.322208	−0.161104	−	0.986937i	$-0.551505\pi$
$379$	−6.27273	−0.322208	−0.161104	+	0.986937i	$0.551505\pi$
$380$	0	0
$381$	0	0
$382$	8.44550i	0.432109i
$383$	− 22.1888i	− 1.13379i	−0.823789	−	0.566897i	$-0.808144\pi$
$383$	− 22.1888i	− 1.13379i	0.823789	−	0.566897i	$-0.191856\pi$
$384$	0	0
$385$	0	0
$386$	−12.4967	−0.636065
$387$	0	0
$388$	0.531061i	0.0269605i
$389$	30.0922	1.52574	0.762869	−	0.646554i	$-0.223790\pi$
$389$	30.0922	1.52574	0.762869	+	0.646554i	$0.223790\pi$
$390$	0	0
$391$	28.3493	1.43368
$392$	21.4136i	1.08155i
$393$	0	0
$394$	−15.6556	−0.788718
$395$	0	0
$396$	0	0
$397$	− 29.2313i	− 1.46708i	−0.679648	−	0.733538i	$-0.737868\pi$
$397$	− 29.2313i	− 1.46708i	0.679648	−	0.733538i	$-0.262132\pi$
$398$	− 27.3708i	− 1.37197i
$399$	0	0
$400$	0	0
$401$	24.2341	1.21020	0.605098	−	0.796151i	$-0.293134\pi$
$401$	24.2341	1.21020	0.605098	+	0.796151i	$0.293134\pi$
$402$	0	0
$403$	− 10.8205i	− 0.539006i
$404$	−0.655614	−0.0326180
$405$	0	0
$406$	24.0463	1.19340
$407$	0.592285i	0.0293585i
$408$	0	0
$409$	−2.33990	−0.115701	−0.0578504	−	0.998325i	$-0.518425\pi$
$409$	−2.33990	−0.115701	−0.0578504	+	0.998325i	$0.518425\pi$
$410$	0	0
$411$	0	0
$412$	0.717168i	0.0353323i
$413$	− 27.5411i	− 1.35521i
$414$	0	0
$415$	0	0
$416$	3.92307	0.192344
$417$	0	0
$418$	1.62941i	0.0796971i
$419$	−22.8425	−1.11593	−0.557964	−	0.829865i	$-0.688417\pi$
$419$	−22.8425	−1.11593	−0.557964	+	0.829865i	$0.688417\pi$
$420$	0	0
$421$	11.8758	0.578793	0.289396	−	0.957209i	$-0.406545\pi$
$421$	11.8758	0.578793	0.289396	+	0.957209i	$0.406545\pi$
$422$	− 15.4004i	− 0.749679i
$423$	0	0
$424$	30.6773	1.48982
$425$	0	0
$426$	0	0
$427$	− 9.76072i	− 0.472354i
$428$	− 0.277293i	− 0.0134034i
$429$	0	0
$430$	0	0
$431$	8.86916	0.427212	0.213606	−	0.976920i	$-0.431479\pi$
$431$	8.86916	0.427212	0.213606	+	0.976920i	$0.431479\pi$
$432$	0	0
$433$	9.37059i	0.450322i	0.974322	+	0.225161i	$0.0722908\pi$
$433$	9.37059i	0.450322i	−0.974322	+	0.225161i	$0.927709\pi$
$434$	−15.1089	−0.725252
$435$	0	0
$436$	−2.19954	−0.105339
$437$	− 36.9490i	− 1.76751i
$438$	0	0
$439$	−19.4231	−0.927014	−0.463507	−	0.886093i	$-0.653409\pi$
$439$	−19.4231	−0.927014	−0.463507	+	0.886093i	$0.653409\pi$
$440$	0	0
$441$	0	0
$442$	19.5914i	0.931868i
$443$	10.8793i	0.516889i	0.966026	+	0.258445i	$0.0832100\pi$
$443$	10.8793i	0.516889i	−0.966026	+	0.258445i	$0.916790\pi$
$444$	0	0
$445$	0	0
$446$	5.79166	0.274243
$447$	0	0
$448$	27.8602i	1.31627i
$449$	−1.34014	−0.0632451	−0.0316225	−	0.999500i	$-0.510067\pi$
$449$	−1.34014	−0.0632451	−0.0316225	+	0.999500i	$0.510067\pi$
$450$	0	0
$451$	1.46845	0.0691467
$452$	− 0.226580i	− 0.0106574i
$453$	0	0
$454$	−7.11787	−0.334058
$455$	0	0
$456$	0	0
$457$	− 20.1113i	− 0.940767i	−0.882462	−	0.470383i	$-0.844116\pi$
$457$	− 20.1113i	− 0.940767i	0.882462	−	0.470383i	$-0.155884\pi$
$458$	− 27.7752i	− 1.29785i
$459$	0	0
$460$	0	0
$461$	−33.7533	−1.57205	−0.786024	−	0.618196i	$-0.787864\pi$
$461$	−33.7533	−1.57205	−0.786024	+	0.618196i	$0.787864\pi$
$462$	0	0
$463$	− 5.24537i	− 0.243773i	−0.992544	−	0.121886i	$-0.961106\pi$
$463$	− 5.24537i	− 0.243773i	0.992544	−	0.121886i	$-0.0388944\pi$
$464$	18.2052	0.845157
$465$	0	0
$466$	−17.5349	−0.812288
$467$	− 14.2120i	− 0.657652i	−0.944390	−	0.328826i	$-0.893347\pi$
$467$	− 14.2120i	− 0.657652i	0.944390	−	0.328826i	$-0.106653\pi$
$468$	0	0
$469$	−43.6282	−2.01456
$470$	0	0
$471$	0	0
$472$	− 19.2023i	− 0.883858i
$473$	− 2.36140i	− 0.108577i
$474$	0	0
$475$	0	0
$476$	2.14866	0.0984837
$477$	0	0
$478$	− 31.8657i	− 1.45750i
$479$	−20.4833	−0.935907	−0.467954	−	0.883753i	$-0.655009\pi$
$479$	−20.4833	−0.935907	−0.467954	+	0.883753i	$0.655009\pi$
$480$	0	0
$481$	9.28168	0.423208
$482$	− 5.74322i	− 0.261596i
$483$	0	0
$484$	1.86372	0.0847145
$485$	0	0
$486$	0	0
$487$	− 31.3554i	− 1.42085i	−0.703772	−	0.710425i	$-0.748503\pi$
$487$	− 31.3554i	− 1.42085i	0.703772	−	0.710425i	$-0.251497\pi$
$488$	− 6.80541i	− 0.308067i
$489$	0	0
$490$	0	0
$491$	−10.3921	−0.468988	−0.234494	−	0.972118i	$-0.575343\pi$
$491$	−10.3921	−0.468988	−0.234494	+	0.972118i	$0.575343\pi$
$492$	0	0
$493$	13.7652i	0.619954i
$494$	25.5345	1.14885
$495$	0	0
$496$	−11.4388	−0.513618
$497$	32.4018i	1.45342i
$498$	0	0
$499$	3.82570	0.171262	0.0856310	−	0.996327i	$-0.472709\pi$
$499$	3.82570	0.171262	0.0856310	+	0.996327i	$0.472709\pi$
$500$	0	0
$501$	0	0
$502$	44.3498i	1.97943i
$503$	1.00236i	0.0446931i	0.999750	+	0.0223466i	$0.00711372\pi$
$503$	1.00236i	0.0446931i	−0.999750	+	0.0223466i	$0.992886\pi$
$504$	0	0
$505$	0	0
$506$	3.33428	0.148227
$507$	0	0
$508$	− 0.281776i	− 0.0125018i
$509$	−4.56322	−0.202261	−0.101131	−	0.994873i	$-0.532246\pi$
$509$	−4.56322	−0.202261	−0.101131	+	0.994873i	$0.532246\pi$
$510$	0	0
$511$	1.56012	0.0690158
$512$	19.0969i	0.843971i
$513$	0	0
$514$	24.1822	1.06663
$515$	0	0
$516$	0	0
$517$	0.371907i	0.0163564i
$518$	− 12.9603i	− 0.569441i
$519$	0	0
$520$	0	0
$521$	39.3708	1.72486	0.862432	−	0.506173i	$-0.168940\pi$
$521$	39.3708	1.72486	0.862432	+	0.506173i	$0.168940\pi$
$522$	0	0
$523$	− 10.3998i	− 0.454749i	−0.973807	−	0.227375i	$-0.926986\pi$
$523$	− 10.3998i	− 0.454749i	0.973807	−	0.227375i	$-0.0730142\pi$
$524$	2.24652	0.0981398
$525$	0	0
$526$	38.0206	1.65778
$527$	− 8.64904i	− 0.376758i
$528$	0	0
$529$	−52.6091	−2.28735
$530$	0	0
$531$	0	0
$532$	− 2.80046i	− 0.121415i
$533$	− 23.0120i	− 0.996762i
$534$	0	0
$535$	0	0
$536$	−30.4186	−1.31388
$537$	0	0
$538$	− 18.4460i	− 0.795262i
$539$	−2.06781	−0.0890670
$540$	0	0
$541$	13.7093	0.589408	0.294704	−	0.955589i	$-0.404779\pi$
$541$	13.7093	0.589408	0.294704	+	0.955589i	$0.404779\pi$
$542$	− 28.9438i	− 1.24324i
$543$	0	0
$544$	3.13579	0.134446
$545$	0	0
$546$	0	0
$547$	− 22.7847i	− 0.974205i	−0.873345	−	0.487102i	$-0.838054\pi$
$547$	− 22.7847i	− 0.974205i	0.873345	−	0.487102i	$-0.161946\pi$
$548$	3.45950i	0.147783i
$549$	0	0
$550$	0	0
$551$	17.9409	0.764307
$552$	0	0
$553$	− 11.7618i	− 0.500162i
$554$	30.6880	1.30381
$555$	0	0
$556$	0.523166	0.0221872
$557$	18.2341i	0.772605i	0.922372	+	0.386303i	$0.126248\pi$
$557$	18.2341i	0.772605i	−0.922372	+	0.386303i	$0.873752\pi$
$558$	0	0
$559$	−37.0054	−1.56516
$560$	0	0
$561$	0	0
$562$	− 6.96026i	− 0.293601i
$563$	24.1112i	1.01617i	0.861308	+	0.508083i	$0.169646\pi$
$563$	24.1112i	1.01617i	−0.861308	+	0.508083i	$0.830354\pi$
$564$	0	0
$565$	0	0
$566$	34.1095	1.43373
$567$	0	0
$568$	22.5913i	0.947910i
$569$	32.0049	1.34171	0.670857	−	0.741587i	$-0.265926\pi$
$569$	32.0049	1.34171	0.670857	+	0.741587i	$0.265926\pi$
$570$	0	0
$571$	−19.7808	−0.827802	−0.413901	−	0.910322i	$-0.635834\pi$
$571$	−19.7808	−0.827802	−0.413901	+	0.910322i	$0.635834\pi$
$572$	0.180984i	0.00756733i
$573$	0	0
$574$	−32.1324	−1.34118
$575$	0	0
$576$	0	0
$577$	35.4119i	1.47422i	0.675775	+	0.737108i	$0.263809\pi$
$577$	35.4119i	1.47422i	−0.675775	+	0.737108i	$0.736191\pi$
$578$	− 9.38550i	− 0.390385i
$579$	0	0
$580$	0	0
$581$	17.7064	0.734586
$582$	0	0
$583$	2.96237i	0.122689i
$584$	1.08776	0.0450117
$585$	0	0
$586$	−24.8675	−1.02726
$587$	32.3851i	1.33668i	0.743858	+	0.668338i	$0.232994\pi$
$587$	32.3851i	1.33668i	−0.743858	+	0.668338i	$0.767006\pi$
$588$	0	0
$589$	−11.2727	−0.464485
$590$	0	0
$591$	0	0
$592$	− 9.81209i	− 0.403274i
$593$	29.2504i	1.20117i	0.799561	+	0.600585i	$0.205066\pi$
$593$	29.2504i	1.20117i	−0.799561	+	0.600585i	$0.794934\pi$
$594$	0	0
$595$	0	0
$596$	0.692421	0.0283627
$597$	0	0
$598$	− 52.2514i	− 2.13672i
$599$	−4.06162	−0.165953	−0.0829767	−	0.996551i	$-0.526443\pi$
$599$	−4.06162	−0.165953	−0.0829767	+	0.996551i	$0.526443\pi$
$600$	0	0
$601$	46.9076	1.91340	0.956700	−	0.291076i	$-0.0940132\pi$
$601$	46.9076	1.91340	0.956700	+	0.291076i	$0.0940132\pi$
$602$	51.6717i	2.10598i
$603$	0	0
$604$	2.32190	0.0944768
$605$	0	0
$606$	0	0
$607$	40.9466i	1.66197i	0.556293	+	0.830987i	$0.312223\pi$
$607$	40.9466i	1.66197i	−0.556293	+	0.830987i	$0.687777\pi$
$608$	− 4.08704i	− 0.165751i
$609$	0	0
$610$	0	0
$611$	5.82813	0.235781
$612$	0	0
$613$	− 33.3827i	− 1.34831i	−0.738588	−	0.674157i	$-0.764507\pi$
$613$	− 33.3827i	− 1.34831i	0.738588	−	0.674157i	$-0.235493\pi$
$614$	33.4690	1.35070
$615$	0	0
$616$	−2.71203	−0.109271
$617$	− 2.24880i	− 0.0905332i	−0.998975	−	0.0452666i	$-0.985586\pi$
$617$	− 2.24880i	− 0.0905332i	0.998975	−	0.0452666i	$-0.0144137\pi$
$618$	0	0
$619$	34.2934	1.37837	0.689184	−	0.724586i	$-0.257969\pi$
$619$	34.2934	1.37837	0.689184	+	0.724586i	$0.257969\pi$
$620$	0	0
$621$	0	0
$622$	− 46.5454i	− 1.86630i
$623$	27.7479i	1.11169i
$624$	0	0
$625$	0	0
$626$	−44.9337	−1.79591
$627$	0	0
$628$	0.351631i	0.0140316i
$629$	7.41904	0.295817
$630$	0	0
$631$	−18.7552	−0.746633	−0.373316	−	0.927704i	$-0.621779\pi$
$631$	−18.7552	−0.746633	−0.373316	+	0.927704i	$0.621779\pi$
$632$	− 8.20060i	− 0.326202i
$633$	0	0
$634$	−32.5427	−1.29244
$635$	0	0
$636$	0	0
$637$	32.4046i	1.28392i
$638$	1.61899i	0.0640963i
$639$	0	0
$640$	0	0
$641$	20.6350	0.815034	0.407517	−	0.913198i	$-0.366395\pi$
$641$	20.6350	0.815034	0.407517	+	0.913198i	$0.366395\pi$
$642$	0	0
$643$	27.1939i	1.07242i	0.844084	+	0.536212i	$0.180145\pi$
$643$	27.1939i	1.07242i	−0.844084	+	0.536212i	$0.819855\pi$
$644$	−5.73060	−0.225817
$645$	0	0
$646$	20.4102	0.803030
$647$	− 16.7316i	− 0.657787i	−0.944367	−	0.328893i	$-0.893324\pi$
$647$	− 16.7316i	− 0.657787i	0.944367	−	0.328893i	$-0.106676\pi$
$648$	0	0
$649$	1.85428	0.0727869
$650$	0	0
$651$	0	0
$652$	0.597571i	0.0234027i
$653$	− 45.7331i	− 1.78967i	−0.446392	−	0.894837i	$-0.647291\pi$
$653$	− 45.7331i	− 1.78967i	0.446392	−	0.894837i	$-0.352709\pi$
$654$	0	0
$655$	0	0
$656$	−24.3271	−0.949814
$657$	0	0
$658$	− 8.13799i	− 0.317252i
$659$	−18.6109	−0.724976	−0.362488	−	0.931988i	$-0.618073\pi$
$659$	−18.6109	−0.724976	−0.362488	+	0.931988i	$0.618073\pi$
$660$	0	0
$661$	16.7960	0.653288	0.326644	−	0.945148i	$-0.394082\pi$
$661$	16.7960	0.653288	0.326644	+	0.945148i	$0.394082\pi$
$662$	43.6153i	1.69516i
$663$	0	0
$664$	12.3453	0.479092
$665$	0	0
$666$	0	0
$667$	− 36.7126i	− 1.42152i
$668$	3.50076i	0.135449i
$669$	0	0
$670$	0	0
$671$	0.657168	0.0253697
$672$	0	0
$673$	49.9480i	1.92535i	0.270656	+	0.962676i	$0.412760\pi$
$673$	49.9480i	1.92535i	−0.270656	+	0.962676i	$0.587240\pi$
$674$	−18.4614	−0.711108
$675$	0	0
$676$	0.619973	0.0238451
$677$	− 10.8311i	− 0.416272i	−0.978100	−	0.208136i	$-0.933260\pi$
$677$	− 10.8311i	− 0.416272i	0.978100	−	0.208136i	$-0.0667396\pi$
$678$	0	0
$679$	−12.0425	−0.462149
$680$	0	0
$681$	0	0
$682$	− 1.01725i	− 0.0389526i
$683$	0.429870i	0.0164485i	0.999966	+	0.00822426i	$0.00261789\pi$
$683$	0.429870i	0.0164485i	−0.999966	+	0.00822426i	$0.997382\pi$
$684$	0	0
$685$	0	0
$686$	5.38003	0.205410
$687$	0	0
$688$	39.1202i	1.49144i
$689$	46.4233	1.76859
$690$	0	0
$691$	34.7036	1.32019	0.660093	−	0.751184i	$-0.270517\pi$
$691$	34.7036	1.32019	0.660093	+	0.751184i	$0.270517\pi$
$692$	1.32151i	0.0502364i
$693$	0	0
$694$	25.1889	0.956157
$695$	0	0
$696$	0	0
$697$	− 18.3940i	− 0.696724i
$698$	− 27.1147i	− 1.02631i
$699$	0	0
$700$	0	0
$701$	1.84808	0.0698010	0.0349005	−	0.999391i	$-0.488889\pi$
$701$	1.84808	0.0698010	0.0349005	+	0.999391i	$0.488889\pi$
$702$	0	0
$703$	− 9.66962i	− 0.364696i
$704$	−1.87577	−0.0706956
$705$	0	0
$706$	−46.7299	−1.75870
$707$	− 14.8669i	− 0.559127i
$708$	0	0
$709$	6.30676	0.236855	0.118428	−	0.992963i	$-0.462215\pi$
$709$	6.30676	0.236855	0.118428	+	0.992963i	$0.462215\pi$
$710$	0	0
$711$	0	0
$712$	19.3465i	0.725040i
$713$	23.0675i	0.863884i
$714$	0	0
$715$	0	0
$716$	−1.83489	−0.0685731
$717$	0	0
$718$	− 16.8736i	− 0.629719i
$719$	−18.0129	−0.671770	−0.335885	−	0.941903i	$-0.609035\pi$
$719$	−18.0129	−0.671770	−0.335885	+	0.941903i	$0.609035\pi$
$720$	0	0
$721$	−16.2627	−0.605655
$722$	1.39013i	0.0517354i
$723$	0	0
$724$	1.33753	0.0497089
$725$	0	0
$726$	0	0
$727$	− 26.2823i	− 0.974758i	−0.873190	−	0.487379i	$-0.837953\pi$
$727$	− 26.2823i	− 0.974758i	0.873190	−	0.487379i	$-0.162047\pi$
$728$	42.5002i	1.57516i
$729$	0	0
$730$	0	0
$731$	−29.5792	−1.09403
$732$	0	0
$733$	47.6633i	1.76049i	0.474524	+	0.880243i	$0.342620\pi$
$733$	47.6633i	1.76049i	−0.474524	+	0.880243i	$0.657380\pi$
$734$	−3.67191	−0.135533
$735$	0	0
$736$	−8.36334	−0.308277
$737$	− 2.93739i	− 0.108200i
$738$	0	0
$739$	10.0273	0.368859	0.184429	−	0.982846i	$-0.440956\pi$
$739$	10.0273	0.368859	0.184429	+	0.982846i	$0.440956\pi$
$740$	0	0
$741$	0	0
$742$	− 64.8221i	− 2.37969i
$743$	8.27266i	0.303494i	0.988419	+	0.151747i	$0.0484900\pi$
$743$	8.27266i	0.303494i	−0.988419	+	0.151747i	$0.951510\pi$
$744$	0	0
$745$	0	0
$746$	−22.1539	−0.811111
$747$	0	0
$748$	0.144665i	0.00528946i
$749$	6.28797	0.229757
$750$	0	0
$751$	−5.79760	−0.211557	−0.105779	−	0.994390i	$-0.533734\pi$
$751$	−5.79760	−0.211557	−0.105779	+	0.994390i	$0.533734\pi$
$752$	− 6.16119i	− 0.224676i
$753$	0	0
$754$	25.3711	0.923960
$755$	0	0
$756$	0	0
$757$	25.2804i	0.918830i	0.888222	+	0.459415i	$0.151941\pi$
$757$	25.2804i	0.918830i	−0.888222	+	0.459415i	$0.848059\pi$
$758$	9.24132i	0.335660i
$759$	0	0
$760$	0	0
$761$	19.4638	0.705562	0.352781	−	0.935706i	$-0.385236\pi$
$761$	19.4638	0.705562	0.352781	+	0.935706i	$0.385236\pi$
$762$	0	0
$763$	− 49.8775i	− 1.80569i
$764$	0.977278	0.0353567
$765$	0	0
$766$	−32.6897	−1.18113
$767$	− 29.0584i	− 1.04924i
$768$	0	0
$769$	49.3431	1.77936	0.889678	−	0.456588i	$-0.150929\pi$
$769$	49.3431	1.77936	0.889678	+	0.456588i	$0.150929\pi$
$770$	0	0
$771$	0	0
$772$	1.44607i	0.0520450i
$773$	20.8502i	0.749930i	0.927039	+	0.374965i	$0.122345\pi$
$773$	20.8502i	0.749930i	−0.927039	+	0.374965i	$0.877655\pi$
$774$	0	0
$775$	0	0
$776$	−8.39632	−0.301410
$777$	0	0
$778$	− 44.3335i	− 1.58943i
$779$	−23.9739	−0.858953
$780$	0	0
$781$	−2.18154	−0.0780616
$782$	− 41.7657i	− 1.49354i
$783$	0	0
$784$	34.2564	1.22344
$785$	0	0
$786$	0	0
$787$	44.1883i	1.57514i	0.616223	+	0.787571i	$0.288662\pi$
$787$	44.1883i	1.57514i	−0.616223	+	0.787571i	$0.711338\pi$
$788$	1.81160i	0.0645357i
$789$	0	0
$790$	0	0
$791$	5.13799	0.182686
$792$	0	0
$793$	− 10.2985i	− 0.365709i
$794$	−43.0651	−1.52832
$795$	0	0
$796$	−3.16723	−0.112259
$797$	− 31.0374i	− 1.09940i	−0.835361	−	0.549701i	$-0.814742\pi$
$797$	− 31.0374i	− 1.09940i	0.835361	−	0.549701i	$-0.185258\pi$
$798$	0	0
$799$	4.65855	0.164808
$800$	0	0
$801$	0	0
$802$	− 35.7031i	− 1.26072i
$803$	0.105040i	0.00370677i
$804$	0	0
$805$	0	0
$806$	−15.9413	−0.561509
$807$	0	0
$808$	− 10.3656i	− 0.364659i
$809$	−14.6229	−0.514114	−0.257057	−	0.966396i	$-0.582753\pi$
$809$	−14.6229	−0.514114	−0.257057	+	0.966396i	$0.582753\pi$
$810$	0	0
$811$	26.7177	0.938187	0.469093	−	0.883149i	$-0.344581\pi$
$811$	26.7177	0.938187	0.469093	+	0.883149i	$0.344581\pi$
$812$	− 2.78254i	− 0.0976480i
$813$	0	0
$814$	0.872586	0.0305841
$815$	0	0
$816$	0	0
$817$	38.5521i	1.34877i
$818$	3.44727i	0.120531i
$819$	0	0
$820$	0	0
$821$	−18.5981	−0.649077	−0.324538	−	0.945873i	$-0.605209\pi$
$821$	−18.5981	−0.649077	−0.324538	+	0.945873i	$0.605209\pi$
$822$	0	0
$823$	− 3.06204i	− 0.106736i	−0.998575	−	0.0533680i	$-0.983004\pi$
$823$	− 3.06204i	− 0.106736i	0.998575	−	0.0533680i	$-0.0169957\pi$
$824$	−11.3388	−0.395005
$825$	0	0
$826$	−40.5750	−1.41178
$827$	− 7.27526i	− 0.252985i	−0.991968	−	0.126493i	$-0.959628\pi$
$827$	− 7.27526i	− 0.252985i	0.991968	−	0.126493i	$-0.0403720\pi$
$828$	0	0
$829$	−10.5211	−0.365411	−0.182706	−	0.983168i	$-0.558486\pi$
$829$	−10.5211	−0.365411	−0.182706	+	0.983168i	$0.558486\pi$
$830$	0	0
$831$	0	0
$832$	29.3951i	1.01909i
$833$	25.9017i	0.897441i
$834$	0	0
$835$	0	0
$836$	0.188549	0.00652109
$837$	0	0
$838$	33.6528i	1.16252i
$839$	−15.1807	−0.524094	−0.262047	−	0.965055i	$-0.584398\pi$
$839$	−15.1807	−0.524094	−0.262047	+	0.965055i	$0.584398\pi$
$840$	0	0
$841$	−11.1739	−0.385307
$842$	− 17.4961i	− 0.602956i
$843$	0	0
$844$	−1.78207	−0.0613413
$845$	0	0
$846$	0	0
$847$	42.2622i	1.45215i
$848$	− 49.0762i	− 1.68528i
$849$	0	0
$850$	0	0
$851$	−19.7870	−0.678290
$852$	0	0
$853$	10.4862i	0.359040i	0.983754	+	0.179520i	$0.0574545\pi$
$853$	10.4862i	0.359040i	−0.983754	+	0.179520i	$0.942545\pi$
$854$	−14.3800	−0.492074
$855$	0	0
$856$	4.38412	0.149846
$857$	8.84075i	0.301994i	0.988534	+	0.150997i	$0.0482484\pi$
$857$	8.84075i	0.301994i	−0.988534	+	0.150997i	$0.951752\pi$
$858$	0	0
$859$	2.06831	0.0705699	0.0352849	−	0.999377i	$-0.488766\pi$
$859$	2.06831	0.0705699	0.0352849	+	0.999377i	$0.488766\pi$
$860$	0	0
$861$	0	0
$862$	− 13.0665i	− 0.445048i
$863$	− 22.4434i	− 0.763984i	−0.924166	−	0.381992i	$-0.875238\pi$
$863$	− 22.4434i	− 0.763984i	0.924166	−	0.381992i	$-0.124762\pi$
$864$	0	0
$865$	0	0
$866$	13.8053	0.469122
$867$	0	0
$868$	1.74834i	0.0593426i
$869$	0.791895	0.0268632
$870$	0	0
$871$	−46.0317	−1.55973
$872$	− 34.7758i	− 1.17766i
$873$	0	0
$874$	−54.4353	−1.84130
$875$	0	0
$876$	0	0
$877$	− 27.8932i	− 0.941886i	−0.882164	−	0.470943i	$-0.843914\pi$
$877$	− 27.8932i	− 0.941886i	0.882164	−	0.470943i	$-0.156086\pi$
$878$	28.6152i	0.965715i
$879$	0	0
$880$	0	0
$881$	−9.22153	−0.310681	−0.155341	−	0.987861i	$-0.549647\pi$
$881$	−9.22153	−0.310681	−0.155341	+	0.987861i	$0.549647\pi$
$882$	0	0
$883$	49.2436i	1.65718i	0.559858	+	0.828589i	$0.310856\pi$
$883$	49.2436i	1.65718i	−0.559858	+	0.828589i	$0.689144\pi$
$884$	2.26704	0.0762486
$885$	0	0
$886$	16.0279	0.538469
$887$	− 10.7681i	− 0.361556i	−0.983524	−	0.180778i	$-0.942138\pi$
$887$	− 10.7681i	− 0.361556i	0.983524	−	0.180778i	$-0.0578616\pi$
$888$	0	0
$889$	6.38965	0.214302
$890$	0	0
$891$	0	0
$892$	− 0.670187i	− 0.0224395i
$893$	− 6.07173i	− 0.203183i
$894$	0	0
$895$	0	0
$896$	48.4816	1.61966
$897$	0	0
$898$	1.97437i	0.0658854i
$899$	−11.2006	−0.373561
$900$	0	0
$901$	37.1071	1.23622
$902$	− 2.16340i	− 0.0720334i
$903$	0	0
$904$	3.58233	0.119147
$905$	0	0
$906$	0	0
$907$	31.9703i	1.06156i	0.847510	+	0.530779i	$0.178101\pi$
$907$	31.9703i	1.06156i	−0.847510	+	0.530779i	$0.821899\pi$
$908$	0.823651i	0.0273338i
$909$	0	0
$910$	0	0
$911$	−10.0802	−0.333972	−0.166986	−	0.985959i	$-0.553403\pi$
$911$	−10.0802	−0.333972	−0.166986	+	0.985959i	$0.553403\pi$
$912$	0	0
$913$	1.19213i	0.0394539i
$914$	−29.6291	−0.980042
$915$	0	0
$916$	−3.21403	−0.106195
$917$	50.9428i	1.68228i
$918$	0	0
$919$	29.7976	0.982932	0.491466	−	0.870897i	$-0.336461\pi$
$919$	29.7976	0.982932	0.491466	+	0.870897i	$0.336461\pi$
$920$	0	0
$921$	0	0
$922$	49.7272i	1.63768i
$923$	34.1868i	1.12527i
$924$	0	0
$925$	0	0
$926$	−7.72776	−0.253950
$927$	0	0
$928$	− 4.06089i	− 0.133305i
$929$	12.3855	0.406355	0.203178	−	0.979142i	$-0.434873\pi$
$929$	12.3855	0.406355	0.203178	+	0.979142i	$0.434873\pi$
$930$	0	0
$931$	33.7590	1.10641
$932$	2.02906i	0.0664642i
$933$	0	0
$934$	−20.9379	−0.685108
$935$	0	0
$936$	0	0
$937$	44.4280i	1.45140i	0.688012	+	0.725699i	$0.258484\pi$
$937$	44.4280i	1.45140i	−0.688012	+	0.725699i	$0.741516\pi$
$938$	64.2754i	2.09867i
$939$	0	0
$940$	0	0
$941$	15.3323	0.499820	0.249910	−	0.968269i	$-0.419599\pi$
$941$	15.3323	0.499820	0.249910	+	0.968269i	$0.419599\pi$
$942$	0	0
$943$	49.0579i	1.59755i
$944$	−30.7189	−0.999816
$945$	0	0
$946$	−3.47894	−0.113110
$947$	21.6997i	0.705144i	0.935785	+	0.352572i	$0.114693\pi$
$947$	21.6997i	0.705144i	−0.935785	+	0.352572i	$0.885307\pi$
$948$	0	0
$949$	1.64607	0.0534338
$950$	0	0
$951$	0	0
$952$	33.9713i	1.10102i
$953$	− 36.9099i	− 1.19563i	−0.801634	−	0.597815i	$-0.796036\pi$
$953$	− 36.9099i	− 1.19563i	0.801634	−	0.597815i	$-0.203964\pi$
$954$	0	0
$955$	0	0
$956$	−3.68737	−0.119258
$957$	0	0
$958$	30.1772i	0.974980i
$959$	−78.4486	−2.53324
$960$	0	0
$961$	−23.9624	−0.772980
$962$	− 13.6743i	− 0.440876i
$963$	0	0
$964$	−0.664581	−0.0214047
$965$	0	0
$966$	0	0
$967$	− 21.3382i	− 0.686190i	−0.939301	−	0.343095i	$-0.888525\pi$
$967$	− 21.3382i	− 0.686190i	0.939301	−	0.343095i	$-0.111475\pi$
$968$	29.4663i	0.947081i
$969$	0	0
$970$	0	0
$971$	−42.5851	−1.36662	−0.683311	−	0.730128i	$-0.739460\pi$
$971$	−42.5851	−1.36662	−0.683311	+	0.730128i	$0.739460\pi$
$972$	0	0
$973$	11.8635i	0.380325i
$974$	−46.1946	−1.48017
$975$	0	0
$976$	−10.8870	−0.348484
$977$	48.9392i	1.56570i	0.622209	+	0.782852i	$0.286236\pi$
$977$	48.9392i	1.56570i	−0.622209	+	0.782852i	$0.713764\pi$
$978$	0	0
$979$	−1.86820	−0.0597080
$980$	0	0
$981$	0	0
$982$	15.3102i	0.488567i
$983$	36.1089i	1.15170i	0.817557	+	0.575848i	$0.195328\pi$
$983$	36.1089i	1.15170i	−0.817557	+	0.575848i	$0.804672\pi$
$984$	0	0
$985$	0	0
$986$	20.2797	0.645836
$987$	0	0
$988$	− 2.95474i	− 0.0940028i
$989$	78.8896	2.50854
$990$	0	0
$991$	32.0054	1.01669	0.508343	−	0.861155i	$-0.330258\pi$
$991$	32.0054	1.01669	0.508343	+	0.861155i	$0.330258\pi$
$992$	2.55156i	0.0810121i
$993$	0	0
$994$	47.7360	1.51410
$995$	0	0
$996$	0	0
$997$	− 51.7680i	− 1.63951i	−0.572716	−	0.819754i	$-0.694110\pi$
$997$	− 51.7680i	− 1.63951i	0.572716	−	0.819754i	$-0.305890\pi$
$998$	− 5.63624i	− 0.178412i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.n.649.3		8
3.2	odd	2		2025.2.b.o.649.6		8
5.2	odd	4		2025.2.a.y.1.3		4
5.3	odd	4		2025.2.a.q.1.2		4
5.4	even	2	inner	2025.2.b.n.649.6		8
9.2	odd	6		675.2.k.c.199.6		16
9.4	even	3		225.2.k.c.124.6		16
9.5	odd	6		675.2.k.c.424.3		16
9.7	even	3		225.2.k.c.49.3		16
15.2	even	4		2025.2.a.p.1.2		4
15.8	even	4		2025.2.a.z.1.3		4
15.14	odd	2		2025.2.b.o.649.3		8
45.2	even	12		675.2.e.e.226.3		8
45.4	even	6		225.2.k.c.124.3		16
45.7	odd	12		225.2.e.c.76.2	✓	8
45.13	odd	12		225.2.e.e.151.3	yes	8
45.14	odd	6		675.2.k.c.424.6		16
45.22	odd	12		225.2.e.c.151.2	yes	8
45.23	even	12		675.2.e.c.451.2		8
45.29	odd	6		675.2.k.c.199.3		16
45.32	even	12		675.2.e.e.451.3		8
45.34	even	6		225.2.k.c.49.6		16
45.38	even	12		675.2.e.c.226.2		8
45.43	odd	12		225.2.e.e.76.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.e.c.76.2	✓	8	45.7	odd	12
225.2.e.c.151.2	yes	8	45.22	odd	12
225.2.e.e.76.3	yes	8	45.43	odd	12
225.2.e.e.151.3	yes	8	45.13	odd	12
225.2.k.c.49.3		16	9.7	even	3
225.2.k.c.49.6		16	45.34	even	6
225.2.k.c.124.3		16	45.4	even	6
225.2.k.c.124.6		16	9.4	even	3
675.2.e.c.226.2		8	45.38	even	12
675.2.e.c.451.2		8	45.23	even	12
675.2.e.e.226.3		8	45.2	even	12
675.2.e.e.451.3		8	45.32	even	12
675.2.k.c.199.3		16	45.29	odd	6
675.2.k.c.199.6		16	9.2	odd	6
675.2.k.c.424.3		16	9.5	odd	6
675.2.k.c.424.6		16	45.14	odd	6
2025.2.a.p.1.2		4	15.2	even	4
2025.2.a.q.1.2		4	5.3	odd	4
2025.2.a.y.1.3		4	5.2	odd	4
2025.2.a.z.1.3		4	15.8	even	4
2025.2.b.n.649.3		8	1.1	even	1	trivial
2025.2.b.n.649.6		8	5.4	even	2	inner
2025.2.b.o.649.3		8	15.14	odd	2
2025.2.b.o.649.6		8	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.47325i q^{2} -0.170479 q^{4} -3.86583i q^{7} -2.69535i q^{8} +O(q^{10})\)	\(q-1.47325i q^{2} -0.170479 q^{4} -3.86583i q^{7} -2.69535i q^{8} +0.260278 q^{11} -4.07881i q^{13} -5.69535 q^{14} -4.31189 q^{16} -3.26028i q^{17} -4.24928 q^{19} -0.383456i q^{22} +8.69535i q^{23} -6.00912 q^{26} +0.659042i q^{28} -4.22210 q^{29} +2.65285 q^{31} +0.961818i q^{32} -4.80322 q^{34} +2.27559i q^{37} +6.26028i q^{38} +5.64186 q^{41} -9.07262i q^{43} -0.0443719 q^{44} +12.8105 q^{46} +1.42888i q^{47} -7.94463 q^{49} +0.695350i q^{52} +11.3816i q^{53} -10.4198 q^{56} +6.22022i q^{58} +7.12423 q^{59} +2.52487 q^{61} -3.90833i q^{62} -7.20679 q^{64} -11.2856i q^{67} +0.555809i q^{68} -8.38158 q^{71} +0.403568i q^{73} +3.35252 q^{74} +0.724413 q^{76} -1.00619i q^{77} +3.04250 q^{79} -8.31189i q^{82} +4.58024i q^{83} -13.3663 q^{86} -0.701540i q^{88} -7.17772 q^{89} -15.7680 q^{91} -1.48237i q^{92} +2.10511 q^{94} -3.11511i q^{97} +11.7045i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 2 q^{11} - 6 q^{14} + 8 q^{16} - 4 q^{19} - 20 q^{26} - 2 q^{29} - 8 q^{31} - 18 q^{34} - 10 q^{41} - 44 q^{44} + 6 q^{49} - 60 q^{56} - 34 q^{59} - 26 q^{61} - 38 q^{64} - 16 q^{71} - 80 q^{74} + 22 q^{76} + 14 q^{79} - 68 q^{86} + 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 - 2 * q^11 - 6 * q^14 + 8 * q^16 - 4 * q^19 - 20 * q^26 - 2 * q^29 - 8 * q^31 - 18 * q^34 - 10 * q^41 - 44 * q^44 + 6 * q^49 - 60 * q^56 - 34 * q^59 - 26 * q^61 - 38 * q^64 - 16 * q^71 - 80 * q^74 + 22 * q^76 + 14 * q^79 - 68 * q^86 + 18 * q^89 - 34 * q^91 - 6 * q^94

Introduction

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