LMFDB - Embedded newform 2151.2.b.a.2150.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(2150,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2151.2150");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2150.8
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.73

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53879i q^{2} -4.44547 q^{4} +1.00261i q^{5} +0.202697i q^{7} +6.20855i q^{8} +O(q^{10})$	\(q-2.53879i q^{2} -4.44547 q^{4} +1.00261i q^{5} +0.202697i q^{7} +6.20855i q^{8} +2.54543 q^{10} -2.12902i q^{11} -1.58049i q^{13} +0.514606 q^{14} +6.87129 q^{16} +4.05688i q^{17} +7.85123i q^{19} -4.45709i q^{20} -5.40514 q^{22} -7.60643 q^{23} +3.99477 q^{25} -4.01255 q^{26} -0.901084i q^{28} -6.23054i q^{29} +6.20533 q^{31} -5.02768i q^{32} +10.2996 q^{34} -0.203227 q^{35} -7.62601i q^{37} +19.9327 q^{38} -6.22478 q^{40} +8.37515 q^{41} +9.12480i q^{43} +9.46450i q^{44} +19.3112i q^{46} +11.9025 q^{47} +6.95891 q^{49} -10.1419i q^{50} +7.02605i q^{52} -0.380892 q^{53} +2.13458 q^{55} -1.25845 q^{56} -15.8181 q^{58} +7.37074 q^{59} +7.43937 q^{61} -15.7540i q^{62} +0.978334 q^{64} +1.58462 q^{65} -14.3082 q^{67} -18.0347i q^{68} +0.515950i q^{70} -11.4320i q^{71} +2.03284i q^{73} -19.3609 q^{74} -34.9024i q^{76} +0.431546 q^{77} +14.3619i q^{79} +6.88924i q^{80} -21.2628i q^{82} -6.46271i q^{83} -4.06748 q^{85} +23.1660 q^{86} +13.2181 q^{88} +2.13672 q^{89} +0.320361 q^{91} +33.8142 q^{92} -30.2181i q^{94} -7.87174 q^{95} +3.90034i q^{97} -17.6672i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 80 q^{4}+O(q^{10}) $ 80 * q - 80 * q^4	$ 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) $ 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

$n$	$479$	$1441$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.53879i		−	1.79520i	−0.440813	−	0.897599i	$-0.645310\pi$
$2$		−	2.53879i		−	1.79520i	0.440813	−	0.897599i	$-0.354690\pi$
$3$		0			0
$3$		0			0
$4$	−4.44547			−2.22274
$5$			1.00261i			0.448382i	0.974545	+	0.224191i	$0.0719740\pi$
$5$			1.00261i			0.448382i	−0.974545	+	0.224191i	$0.928026\pi$
$6$		0			0
$7$			0.202697i			0.0766122i	0.999266	+	0.0383061i	$0.0121962\pi$
$7$			0.202697i			0.0766122i	−0.999266	+	0.0383061i	$0.987804\pi$
$8$			6.20855i			2.19506i
$9$		0			0
$10$	2.54543			0.804935
$11$		−	2.12902i		−	0.641924i	−0.947092	−	0.320962i	$-0.895994\pi$
$11$		−	2.12902i		−	0.641924i	0.947092	−	0.320962i	$-0.104006\pi$
$12$		0			0
$13$		−	1.58049i		−	0.438350i	−0.975686	−	0.219175i	$-0.929663\pi$
$13$		−	1.58049i		−	0.438350i	0.975686	−	0.219175i	$-0.0703366\pi$
$14$	0.514606			0.137534
$15$		0			0
$16$	6.87129			1.71782
$17$			4.05688i			0.983937i	0.870613	+	0.491969i	$0.163723\pi$
$17$			4.05688i			0.983937i	−0.870613	+	0.491969i	$0.836277\pi$
$18$		0			0
$19$			7.85123i			1.80120i	0.434654	+	0.900598i	$0.356871\pi$
$19$			7.85123i			1.80120i	−0.434654	+	0.900598i	$0.643129\pi$
$20$		−	4.45709i		−	0.996635i
$21$		0			0
$22$	−5.40514			−1.15238
$23$	−7.60643			−1.58605			−0.793026	−	0.609188i	$-0.791495\pi$
$23$	−7.60643			−1.58605			−0.793026	+	0.609188i	$0.791495\pi$
$24$		0			0
$25$	3.99477			0.798953
$26$	−4.01255			−0.786926
$27$		0			0
$28$		−	0.901084i		−	0.170289i
$29$		−	6.23054i		−	1.15698i	−0.815689	−	0.578491i	$-0.803642\pi$
$29$		−	6.23054i		−	1.15698i	0.815689	−	0.578491i	$-0.196358\pi$
$30$		0			0
$31$	6.20533			1.11451			0.557255	−	0.830342i	$-0.311855\pi$
$31$	6.20533			1.11451			0.557255	+	0.830342i	$0.311855\pi$
$32$		−	5.02768i		−	0.888777i
$33$		0			0
$34$	10.2996			1.76636
$35$	−0.203227			−0.0343516
$36$		0			0
$37$		−	7.62601i		−	1.25371i	−0.779136	−	0.626855i	$-0.784342\pi$
$37$		−	7.62601i		−	1.25371i	0.779136	−	0.626855i	$-0.215658\pi$
$38$	19.9327			3.23350
$39$		0			0
$40$	−6.22478			−0.984224
$41$	8.37515			1.30798			0.653989	−	0.756504i	$-0.273094\pi$
$41$	8.37515			1.30798			0.653989	+	0.756504i	$0.273094\pi$
$42$		0			0
$43$			9.12480i			1.39152i	0.718275	+	0.695759i	$0.244932\pi$
$43$			9.12480i			1.39152i	−0.718275	+	0.695759i	$0.755068\pi$
$44$			9.46450i			1.42683i
$45$		0			0
$46$			19.3112i			2.84728i
$47$	11.9025			1.73616			0.868082	−	0.496421i	$-0.165353\pi$
$47$	11.9025			1.73616			0.868082	+	0.496421i	$0.165353\pi$
$48$		0			0
$49$	6.95891			0.994131
$50$		−	10.1419i		−	1.43428i
$51$		0			0
$52$			7.02605i			0.974337i
$53$	−0.380892			−0.0523195			−0.0261598	−	0.999658i	$-0.508328\pi$
$53$	−0.380892			−0.0523195			−0.0261598	+	0.999658i	$0.508328\pi$
$54$		0			0
$55$	2.13458			0.287827
$56$	−1.25845			−0.168168
$57$		0			0
$58$	−15.8181			−2.07701
$59$	7.37074			0.959588			0.479794	−	0.877381i	$-0.340711\pi$
$59$	7.37074			0.959588			0.479794	+	0.877381i	$0.340711\pi$
$60$		0			0
$61$	7.43937			0.952514			0.476257	−	0.879306i	$-0.341993\pi$
$61$	7.43937			0.952514			0.476257	+	0.879306i	$0.341993\pi$
$62$		−	15.7540i		−	2.00077i
$63$		0			0
$64$	0.978334			0.122292
$65$	1.58462			0.196548
$66$		0			0
$67$	−14.3082			−1.74802			−0.874011	−	0.485907i	$-0.838489\pi$
$67$	−14.3082			−1.74802			−0.874011	+	0.485907i	$0.838489\pi$
$68$		−	18.0347i		−	2.18703i
$69$		0			0
$70$			0.515950i			0.0616678i
$71$		−	11.4320i		−	1.35673i	−0.734726	−	0.678364i	$-0.762689\pi$
$71$		−	11.4320i		−	1.35673i	0.734726	−	0.678364i	$-0.237311\pi$
$72$		0			0
$73$			2.03284i			0.237926i	0.992899	+	0.118963i	$0.0379569\pi$
$73$			2.03284i			0.237926i	−0.992899	+	0.118963i	$0.962043\pi$
$74$	−19.3609			−2.25066
$75$		0			0
$76$		−	34.9024i		−	4.00358i
$77$	0.431546			0.0491792
$78$		0			0
$79$			14.3619i			1.61584i	0.589293	+	0.807920i	$0.299407\pi$
$79$			14.3619i			1.61584i	−0.589293	+	0.807920i	$0.700593\pi$
$80$			6.88924i			0.770241i
$81$		0			0
$82$		−	21.2628i		−	2.34808i
$83$		−	6.46271i		−	0.709375i	−0.934985	−	0.354687i	$-0.884587\pi$
$83$		−	6.46271i		−	0.709375i	0.934985	−	0.354687i	$-0.115413\pi$
$84$		0			0
$85$	−4.06748			−0.441180
$86$	23.1660			2.49805
$87$		0			0
$88$	13.2181			1.40906
$89$	2.13672			0.226492			0.113246	−	0.993567i	$-0.463875\pi$
$89$	2.13672			0.226492			0.113246	+	0.993567i	$0.463875\pi$
$90$		0			0
$91$	0.320361			0.0335830
$92$	33.8142			3.52537
$93$		0			0
$94$		−	30.2181i		−	3.11676i
$95$	−7.87174			−0.807624
$96$		0			0
$97$			3.90034i			0.396020i	0.980200	+	0.198010i	$0.0634478\pi$
$97$			3.90034i			0.396020i	−0.980200	+	0.198010i	$0.936552\pi$
$98$		−	17.6672i		−	1.78466i
$99$		0			0
$100$	−17.7586			−1.77586
$101$		−	11.8944i		−	1.18353i	−0.806110	−	0.591766i	$-0.798431\pi$
$101$		−	11.8944i		−	1.18353i	0.806110	−	0.591766i	$-0.201569\pi$
$102$		0			0
$103$			8.69316i			0.856562i	0.903646	+	0.428281i	$0.140881\pi$
$103$			8.69316i			0.856562i	−0.903646	+	0.428281i	$0.859119\pi$
$104$	9.81258			0.962203
$105$		0			0
$106$			0.967006i			0.0939240i
$107$	11.3388			1.09616			0.548080	−	0.836426i	$-0.315359\pi$
$107$	11.3388			1.09616			0.548080	+	0.836426i	$0.315359\pi$
$108$		0			0
$109$	−5.40752			−0.517947			−0.258973	−	0.965884i	$-0.583384\pi$
$109$	−5.40752			−0.517947			−0.258973	+	0.965884i	$0.583384\pi$
$110$		−	5.41927i		−	0.516707i
$111$		0			0
$112$			1.39279i			0.131606i
$113$			1.82968i			0.172122i	0.996290	+	0.0860608i	$0.0274279\pi$
$113$			1.82968i			0.172122i	−0.996290	+	0.0860608i	$0.972572\pi$
$114$		0			0
$115$		−	7.62631i		−	0.711157i
$116$			27.6977i			2.57167i
$117$		0			0
$118$		−	18.7128i		−	1.72265i
$119$	−0.822317			−0.0753816
$120$		0			0
$121$	6.46727			0.587934
$122$		−	18.8870i		−	1.70995i
$123$		0			0
$124$	−27.5856			−2.47726
$125$			9.01827i			0.806619i
$126$		0			0
$127$	−0.911182			−0.0808543			−0.0404271	−	0.999182i	$-0.512872\pi$
$127$	−0.911182			−0.0808543			−0.0404271	+	0.999182i	$0.512872\pi$
$128$		−	12.5392i		−	1.10831i
$129$		0			0
$130$		−	4.02303i		−	0.352843i
$131$	1.17333			0.102515			0.0512574	−	0.998685i	$-0.483677\pi$
$131$	1.17333			0.102515			0.0512574	+	0.998685i	$0.483677\pi$
$132$		0			0
$133$	−1.59142			−0.137994
$134$			36.3255i			3.13804i
$135$		0			0
$136$	−25.1873			−2.15980
$137$	−2.54603			−0.217522			−0.108761	−	0.994068i	$-0.534688\pi$
$137$	−2.54603			−0.217522			−0.108761	+	0.994068i	$0.534688\pi$
$138$		0			0
$139$		−	20.0894i		−	1.70396i	−0.523572	−	0.851982i	$-0.675401\pi$
$139$		−	20.0894i		−	1.70396i	0.523572	−	0.851982i	$-0.324599\pi$
$140$	0.903438			0.0763545
$141$		0			0
$142$	−29.0235			−2.43560
$143$	−3.36490			−0.281387
$144$		0			0
$145$	6.24682			0.518770
$146$	5.16095			0.427124
$147$		0			0
$148$			33.9013i			2.78667i
$149$	−22.0120			−1.80330			−0.901648	−	0.432470i	$-0.857642\pi$
$149$	−22.0120			−1.80330			−0.901648	+	0.432470i	$0.857642\pi$
$150$		0			0
$151$			10.0323i			0.816416i	0.912889	+	0.408208i	$0.133846\pi$
$151$			10.0323i			0.816416i	−0.912889	+	0.408208i	$0.866154\pi$
$152$	−48.7448			−3.95372
$153$		0			0
$154$		−	1.09561i		−	0.0882864i
$155$			6.22154i			0.499726i
$156$		0			0
$157$	−9.99829			−0.797951			−0.398975	−	0.916962i	$-0.630634\pi$
$157$	−9.99829			−0.797951			−0.398975	+	0.916962i	$0.630634\pi$
$158$	36.4619			2.90075
$159$		0			0
$160$	5.04082			0.398512
$161$		−	1.54180i		−	0.121511i
$162$		0			0
$163$	24.2176			1.89687			0.948433	−	0.316978i	$-0.102668\pi$
$163$	24.2176			1.89687			0.948433	+	0.316978i	$0.102668\pi$
$164$	−37.2315			−2.90729
$165$		0			0
$166$	−16.4075			−1.27347
$167$	4.16262			0.322113			0.161057	−	0.986945i	$-0.448510\pi$
$167$	4.16262			0.322113			0.161057	+	0.986945i	$0.448510\pi$
$168$		0			0
$169$	10.5020			0.807849
$170$			10.3265i			0.792005i
$171$		0			0
$172$		−	40.5641i		−	3.09298i
$173$	−7.86304			−0.597816			−0.298908	−	0.954282i	$-0.596622\pi$
$173$	−7.86304			−0.597816			−0.298908	+	0.954282i	$0.596622\pi$
$174$		0			0
$175$			0.809727i			0.0612096i
$176$		−	14.6291i		−	1.10271i
$177$		0			0
$178$		−	5.42470i		−	0.406598i
$179$	10.4869			0.783828			0.391914	−	0.920002i	$-0.371813\pi$
$179$	10.4869			0.783828			0.391914	+	0.920002i	$0.371813\pi$
$180$		0			0
$181$		−	15.4370i		−	1.14742i	−0.819058	−	0.573711i	$-0.805503\pi$
$181$		−	15.4370i		−	1.14742i	0.819058	−	0.573711i	$-0.194497\pi$
$182$		−	0.813331i		−	0.0602881i
$183$		0			0
$184$		−	47.2250i		−	3.48147i
$185$	7.64594			0.562141
$186$		0			0
$187$	8.63717			0.631613
$188$	−52.9124			−3.85903
$189$		0			0
$190$			19.9847i			1.44984i
$191$	−2.69508			−0.195009			−0.0975046	−	0.995235i	$-0.531086\pi$
$191$	−2.69508			−0.195009			−0.0975046	+	0.995235i	$0.531086\pi$
$192$		0			0
$193$	−8.42167			−0.606205			−0.303103	−	0.952958i	$-0.598022\pi$
$193$	−8.42167			−0.606205			−0.303103	+	0.952958i	$0.598022\pi$
$194$	9.90216			0.710934
$195$		0			0
$196$	−30.9357			−2.20969
$197$			9.47576i			0.675120i	0.941304	+	0.337560i	$0.109602\pi$
$197$			9.47576i			0.675120i	−0.941304	+	0.337560i	$0.890398\pi$
$198$		0			0
$199$			22.3448i			1.58398i	0.610534	+	0.791990i	$0.290955\pi$
$199$			22.3448i			1.58398i	−0.610534	+	0.791990i	$0.709045\pi$
$200$			24.8017i			1.75375i
$201$		0			0
$202$	−30.1973			−2.12468
$203$	1.26291			0.0886390
$204$		0			0
$205$			8.39703i			0.586474i
$206$	22.0701			1.53770
$207$		0			0
$208$		−	10.8600i		−	0.753008i
$209$	16.7154			1.15623
$210$		0			0
$211$	19.4828			1.34125			0.670626	−	0.741795i	$-0.266025\pi$
$211$	19.4828			1.34125			0.670626	+	0.741795i	$0.266025\pi$
$212$	1.69325			0.116293
$213$		0			0
$214$		−	28.7868i		−	1.96782i
$215$	−9.14864			−0.623932
$216$		0			0
$217$			1.25780i			0.0853851i
$218$			13.7286i			0.929817i
$219$		0			0
$220$	−9.48923			−0.639764
$221$	6.41187			0.431309
$222$		0			0
$223$			23.1325i			1.54906i	0.632535	+	0.774532i	$0.282014\pi$
$223$			23.1325i			1.54906i	−0.632535	+	0.774532i	$0.717986\pi$
$224$	1.01910			0.0680912
$225$		0			0
$226$	4.64517			0.308992
$227$	19.5986			1.30081			0.650403	−	0.759590i	$-0.274600\pi$
$227$	19.5986			1.30081			0.650403	+	0.759590i	$0.274600\pi$
$228$		0			0
$229$			20.4918i			1.35414i	0.735920	+	0.677068i	$0.236750\pi$
$229$			20.4918i			1.35414i	−0.735920	+	0.677068i	$0.763250\pi$
$230$	−19.3616			−1.27667
$231$		0			0
$232$	38.6827			2.53964
$233$	−6.51297			−0.426679			−0.213339	−	0.976978i	$-0.568434\pi$
$233$	−6.51297			−0.426679			−0.213339	+	0.976978i	$0.568434\pi$
$234$		0			0
$235$			11.9336i			0.778465i
$236$	−32.7664			−2.13291
$237$		0			0
$238$			2.08769i			0.135325i
$239$	−11.0060	−	10.8567i	−0.711921	−	0.702259i
$239$	−11.0060	−	10.8567i	−0.711921	−	0.702259i
$240$		0			0
$241$	13.2011			0.850358			0.425179	−	0.905109i	$-0.360211\pi$
$241$	13.2011			0.850358			0.425179	+	0.905109i	$0.360211\pi$
$242$		−	16.4191i		−	1.05546i
$243$		0			0
$244$	−33.0715			−2.11719
$245$			6.97710i			0.445750i
$246$		0			0
$247$	12.4088			0.789554
$248$			38.5261i			2.44641i
$249$		0			0
$250$	22.8955			1.44804
$251$			2.12359i			0.134040i	0.997752	+	0.0670198i	$0.0213491\pi$
$251$			2.12359i			0.134040i	−0.997752	+	0.0670198i	$0.978651\pi$
$252$		0			0
$253$			16.1942i			1.01812i
$254$			2.31330i			0.145149i
$255$		0			0
$256$	−29.8777			−1.86735
$257$		−	5.16958i		−	0.322469i	−0.986916	−	0.161235i	$-0.948452\pi$
$257$		−	5.16958i		−	0.322469i	0.986916	−	0.161235i	$-0.0515476\pi$
$258$		0			0
$259$	1.54577			0.0960494
$260$	−7.04440			−0.436875
$261$		0			0
$262$		−	2.97885i		−	0.184034i
$263$			10.5890i			0.652949i	0.945206	+	0.326474i	$0.105861\pi$
$263$			10.5890i			0.652949i	−0.945206	+	0.326474i	$0.894139\pi$
$264$		0			0
$265$		−	0.381887i		−	0.0234591i
$266$			4.04029i			0.247726i
$267$		0			0
$268$	63.6066			3.88539
$269$			2.96094i			0.180532i	0.995918	+	0.0902658i	$0.0287717\pi$
$269$			2.96094i			0.180532i	−0.995918	+	0.0902658i	$0.971228\pi$
$270$		0			0
$271$	31.4055			1.90775			0.953875	−	0.300204i	$-0.0970547\pi$
$271$	31.4055			1.90775			0.953875	+	0.300204i	$0.0970547\pi$
$272$			27.8760i			1.69023i
$273$		0			0
$274$			6.46384i			0.390495i
$275$		−	8.50494i		−	0.512867i
$276$		0			0
$277$		−	20.5114i		−	1.23241i	−0.787585	−	0.616206i	$-0.788669\pi$
$277$		−	20.5114i		−	1.23241i	0.787585	−	0.616206i	$-0.211331\pi$
$278$	−51.0029			−3.05895
$279$		0			0
$280$		−	1.26174i		−	0.0754036i
$281$	−21.8595			−1.30403			−0.652015	−	0.758206i	$-0.726076\pi$
$281$	−21.8595			−1.30403			−0.652015	+	0.758206i	$0.726076\pi$
$282$		0			0
$283$	23.6384			1.40516			0.702578	−	0.711607i	$-0.252032\pi$
$283$	23.6384			1.40516			0.702578	+	0.711607i	$0.252032\pi$
$284$			50.8206i			3.01565i
$285$		0			0
$286$			8.54280i			0.505146i
$287$			1.69762i			0.100207i
$288$		0			0
$289$	0.541743			0.0318672
$290$		−	15.8594i		−	0.931295i
$291$		0			0
$292$		−	9.03693i		−	0.528846i
$293$			0.144377i			0.00843462i	0.999991	+	0.00421731i	$0.00134242\pi$
$293$			0.144377i			0.00843462i	−0.999991	+	0.00421731i	$0.998658\pi$
$294$		0			0
$295$			7.39000i			0.430262i
$296$	47.3465			2.75196
$297$		0			0
$298$			55.8840i			3.23727i
$299$			12.0219i			0.695246i
$300$		0			0
$301$	−1.84957			−0.106607
$302$	25.4699			1.46563
$303$		0			0
$304$			53.9481i			3.09413i
$305$			7.45881i			0.427090i
$306$		0			0
$307$	−8.36308			−0.477306			−0.238653	−	0.971105i	$-0.576706\pi$
$307$	−8.36308			−0.477306			−0.238653	+	0.971105i	$0.576706\pi$
$308$	−1.91843			−0.109312
$309$		0			0
$310$	15.7952			0.897108
$311$		−	28.3312i		−	1.60652i	−0.595632	−	0.803258i	$-0.703098\pi$
$311$		−	28.3312i		−	1.60652i	0.595632	−	0.803258i	$-0.296902\pi$
$312$		0			0
$313$			13.5520i			0.766005i	0.923747	+	0.383003i	$0.125110\pi$
$313$			13.5520i			0.766005i	−0.923747	+	0.383003i	$0.874890\pi$
$314$			25.3836i			1.43248i
$315$		0			0
$316$		−	63.8454i		−	3.59159i
$317$	25.0488			1.40688			0.703441	−	0.710754i	$-0.251646\pi$
$317$	25.0488			1.40688			0.703441	+	0.710754i	$0.251646\pi$
$318$		0			0
$319$	−13.2649			−0.742694
$320$			0.980890i			0.0548334i
$321$		0			0
$322$	−3.91431			−0.218136
$323$	−31.8515			−1.77226
$324$		0			0
$325$		−	6.31371i		−	0.350221i
$326$		−	61.4834i		−	3.40525i
$327$		0			0
$328$			51.9976i			2.87109i
$329$			2.41261i			0.133011i
$330$		0			0
$331$		−	12.9207i		−	0.710184i	−0.934831	−	0.355092i	$-0.884449\pi$
$331$		−	12.9207i		−	0.710184i	0.934831	−	0.355092i	$-0.115551\pi$
$332$			28.7298i			1.57675i
$333$		0			0
$334$		−	10.5680i		−	0.578257i
$335$		−	14.3456i		−	0.783781i
$336$		0			0
$337$	−7.28557			−0.396870			−0.198435	−	0.980114i	$-0.563586\pi$
$337$	−7.28557			−0.396870			−0.198435	+	0.980114i	$0.563586\pi$
$338$		−	26.6625i		−	1.45025i
$339$		0			0
$340$	18.0819			0.980627
$341$		−	13.2113i		−	0.715430i
$342$		0			0
$343$			2.82943i			0.152775i
$344$	−56.6518			−3.05446
$345$		0			0
$346$			19.9626i			1.07320i
$347$			30.4202i			1.63304i	0.577316	+	0.816521i	$0.304100\pi$
$347$			30.4202i			1.63304i	−0.577316	+	0.816521i	$0.695900\pi$
$348$		0			0
$349$	−27.6480			−1.47996			−0.739981	−	0.672627i	$-0.765166\pi$
$349$	−27.6480			−1.47996			−0.739981	+	0.672627i	$0.765166\pi$
$350$	2.05573			0.109883
$351$		0			0
$352$	−10.7040			−0.570527
$353$	27.6712			1.47279			0.736395	−	0.676552i	$-0.236526\pi$
$353$	27.6712			1.47279			0.736395	+	0.676552i	$0.236526\pi$
$354$		0			0
$355$	11.4619			0.608332
$356$	−9.49875			−0.503432
$357$		0			0
$358$		−	26.6241i		−	1.40713i
$359$		−	2.34504i		−	0.123767i	−0.998083	−	0.0618834i	$-0.980289\pi$
$359$		−	2.34504i		−	0.123767i	0.998083	−	0.0618834i	$-0.0197107\pi$
$360$		0			0
$361$	−42.6418			−2.24430
$362$	−39.1913			−2.05985
$363$		0			0
$364$	−1.42416			−0.0746461
$365$	−2.03815			−0.106682
$366$		0			0
$367$	−1.82990			−0.0955201			−0.0477600	−	0.998859i	$-0.515208\pi$
$367$	−1.82990			−0.0955201			−0.0477600	+	0.998859i	$0.515208\pi$
$368$	−52.2660			−2.72455
$369$		0			0
$370$		−	19.4115i		−	1.00915i
$371$		−	0.0772056i		−	0.00400832i
$372$		0			0
$373$	1.24517			0.0644724			0.0322362	−	0.999480i	$-0.489737\pi$
$373$	1.24517			0.0644724			0.0322362	+	0.999480i	$0.489737\pi$
$374$		−	21.9280i		−	1.13387i
$375$		0			0
$376$			73.8975i			3.81098i
$377$	−9.84733			−0.507163
$378$		0			0
$379$			1.41319i			0.0725909i	0.999341	+	0.0362955i	$0.0115557\pi$
$379$			1.41319i			0.0725909i	−0.999341	+	0.0362955i	$0.988444\pi$
$380$	34.9936			1.79514
$381$		0			0
$382$			6.84225i			0.350080i
$383$		−	3.82834i		−	0.195619i	−0.995205	−	0.0978095i	$-0.968816\pi$
$383$		−	3.82834i		−	0.195619i	0.995205	−	0.0978095i	$-0.0311836\pi$
$384$		0			0
$385$			0.432673i			0.0220511i
$386$			21.3809i			1.08826i
$387$		0			0
$388$		−	17.3389i		−	0.880247i
$389$			10.3854i			0.526561i	0.964719	+	0.263280i	$0.0848045\pi$
$389$			10.3854i			0.526561i	−0.964719	+	0.263280i	$0.915196\pi$
$390$		0			0
$391$		−	30.8584i		−	1.56058i
$392$			43.2048i			2.18217i
$393$		0			0
$394$	24.0570			1.21197
$395$	−14.3994			−0.724513
$396$		0			0
$397$		−	5.66801i		−	0.284469i	−0.989833	−	0.142235i	$-0.954571\pi$
$397$		−	5.66801i		−	0.284469i	0.989833	−	0.142235i	$-0.0454288\pi$
$398$	56.7288			2.84356
$399$		0			0
$400$	27.4492			1.37246
$401$			2.95268i			0.147450i	0.997279	+	0.0737250i	$0.0234887\pi$
$401$			2.95268i			0.147450i	−0.997279	+	0.0737250i	$0.976511\pi$
$402$		0			0
$403$		−	9.80748i		−	0.488546i
$404$			52.8760i			2.63068i
$405$		0			0
$406$		−	3.20627i		−	0.159125i
$407$	−16.2359			−0.804785
$408$		0			0
$409$	2.66664			0.131857			0.0659284	−	0.997824i	$-0.478999\pi$
$409$	2.66664			0.131857			0.0659284	+	0.997824i	$0.478999\pi$
$410$	21.3183			1.05284
$411$		0			0
$412$		−	38.6452i		−	1.90391i
$413$			1.49403i			0.0735162i
$414$		0			0
$415$	6.47960			0.318071
$416$	−7.94622			−0.389596
$417$		0			0
$418$		−	42.4370i		−	2.07566i
$419$			19.6810i			0.961479i	0.876863	+	0.480739i	$0.159632\pi$
$419$			19.6810i			0.961479i	−0.876863	+	0.480739i	$0.840368\pi$
$420$		0			0
$421$	−20.7305			−1.01034			−0.505171	−	0.863019i	$-0.668571\pi$
$421$	−20.7305			−1.01034			−0.505171	+	0.863019i	$0.668571\pi$
$422$		−	49.4629i		−	2.40781i
$423$		0			0
$424$		−	2.36479i		−	0.114844i
$425$			16.2063i			0.786120i
$426$		0			0
$427$			1.50794i			0.0729742i
$428$	−50.4062			−2.43647
$429$		0			0
$430$			23.2265i			1.12008i
$431$		−	8.84447i		−	0.426023i	−0.977050	−	0.213012i	$-0.931673\pi$
$431$		−	8.84447i		−	0.426023i	0.977050	−	0.213012i	$-0.0683272\pi$
$432$		0			0
$433$			32.2277i			1.54876i	0.632718	+	0.774382i	$0.281939\pi$
$433$			32.2277i			1.54876i	−0.632718	+	0.774382i	$0.718061\pi$
$434$	3.19330			0.153283
$435$		0			0
$436$	24.0390			1.15126
$437$		−	59.7199i		−	2.85679i
$438$		0			0
$439$	−12.6052			−0.601612			−0.300806	−	0.953685i	$-0.597256\pi$
$439$	−12.6052			−0.601612			−0.300806	+	0.953685i	$0.597256\pi$
$440$			13.2527i			0.631796i
$441$		0			0
$442$		−	16.2784i		−	0.774285i
$443$			27.5996i			1.31130i	0.755066	+	0.655649i	$0.227605\pi$
$443$			27.5996i			1.31130i	−0.755066	+	0.655649i	$0.772395\pi$
$444$		0			0
$445$			2.14231i			0.101555i
$446$	58.7285			2.78088
$447$		0			0
$448$			0.198305i			0.00936904i
$449$	−21.6297			−1.02077			−0.510384	−	0.859947i	$-0.670497\pi$
$449$	−21.6297			−1.02077			−0.510384	+	0.859947i	$0.670497\pi$
$450$		0			0
$451$		−	17.8309i		−	0.839623i
$452$		−	8.13378i		−	0.382581i
$453$		0			0
$454$		−	49.7568i		−	2.33520i
$455$			0.321198i			0.0150580i
$456$		0			0
$457$	32.7289			1.53099			0.765497	−	0.643440i	$-0.222493\pi$
$457$	32.7289			1.53099			0.765497	+	0.643440i	$0.222493\pi$
$458$	52.0245			2.43094
$459$		0			0
$460$			33.9026i			1.58071i
$461$	−17.6311			−0.821163			−0.410582	−	0.911824i	$-0.634674\pi$
$461$	−17.6311			−0.821163			−0.410582	+	0.911824i	$0.634674\pi$
$462$		0			0
$463$		−	13.6995i		−	0.636668i	−0.947979	−	0.318334i	$-0.896877\pi$
$463$		−	13.6995i		−	0.636668i	0.947979	−	0.318334i	$-0.103123\pi$
$464$		−	42.8119i		−	1.98749i
$465$		0			0
$466$			16.5351i			0.765973i
$467$	8.09427			0.374558			0.187279	−	0.982307i	$-0.440033\pi$
$467$	8.09427			0.374558			0.187279	+	0.982307i	$0.440033\pi$
$468$		0			0
$469$		−	2.90022i		−	0.133920i
$470$	30.2970			1.39750
$471$		0			0
$472$			45.7616i			2.10635i
$473$	19.4269			0.893249
$474$		0			0
$475$			31.3638i			1.43907i
$476$	3.65559			0.167554
$477$		0			0
$478$	−27.5628	+	27.9421i	−1.26069	+	1.27804i
$479$		−	13.0666i		−	0.597028i	−0.954405	−	0.298514i	$-0.903509\pi$
$479$		−	13.0666i		−	0.597028i	0.954405	−	0.298514i	$-0.0964910\pi$
$480$		0			0
$481$	−12.0529			−0.549564
$482$		−	33.5149i		−	1.52656i
$483$		0			0
$484$	−28.7501			−1.30682
$485$	−3.91053			−0.177568
$486$		0			0
$487$	−15.7929			−0.715646			−0.357823	−	0.933790i	$-0.616481\pi$
$487$	−15.7929			−0.715646			−0.357823	+	0.933790i	$0.616481\pi$
$488$			46.1878i			2.09082i
$489$		0			0
$490$	17.7134			0.800210
$491$	−5.95718			−0.268844			−0.134422	−	0.990924i	$-0.542918\pi$
$491$	−5.95718			−0.268844			−0.134422	+	0.990924i	$0.542918\pi$
$492$		0			0
$493$	25.2765			1.13840
$494$		−	31.5034i		−	1.41741i
$495$		0			0
$496$	42.6386			1.91453
$497$	2.31723			0.103942
$498$		0			0
$499$			33.6073i			1.50447i	0.658896	+	0.752234i	$0.271024\pi$
$499$			33.6073i			1.50447i	−0.658896	+	0.752234i	$0.728976\pi$
$500$		−	40.0905i		−	1.79290i
$501$		0			0
$502$	5.39135			0.240628
$503$			28.1240i			1.25399i	0.779024	+	0.626994i	$0.215715\pi$
$503$			28.1240i			1.25399i	−0.779024	+	0.626994i	$0.784285\pi$
$504$		0			0
$505$	11.9254			0.530675
$506$	41.1139			1.82773
$507$		0			0
$508$	4.05063			0.179718
$509$			31.4716i			1.39495i	0.716608	+	0.697476i	$0.245694\pi$
$509$			31.4716i			1.39495i	−0.716608	+	0.697476i	$0.754306\pi$
$510$		0			0
$511$	−0.412050			−0.0182280
$512$			50.7749i			2.24395i
$513$		0			0
$514$	−13.1245			−0.578897
$515$	−8.71587			−0.384067
$516$		0			0
$517$		−	25.3407i		−	1.11448i
$518$		−	3.92439i		−	0.172428i
$519$		0			0
$520$			9.83822i			0.431435i
$521$	16.3465			0.716152			0.358076	−	0.933692i	$-0.383433\pi$
$521$	16.3465			0.716152			0.358076	+	0.933692i	$0.383433\pi$
$522$		0			0
$523$	−34.6019			−1.51304			−0.756518	−	0.653973i	$-0.773101\pi$
$523$	−34.6019			−1.51304			−0.756518	+	0.653973i	$0.773101\pi$
$524$	−5.21603			−0.227863
$525$		0			0
$526$	26.8834			1.17217
$527$			25.1743i			1.09661i
$528$		0			0
$529$	34.8578			1.51556
$530$	−0.969533			−0.0421138
$531$		0			0
$532$	7.07461			0.306723
$533$		−	13.2369i		−	0.573353i
$534$		0			0
$535$			11.3684i			0.491498i
$536$		−	88.8331i		−	3.83700i
$537$		0			0
$538$	7.51721			0.324090
$539$		−	14.8157i		−	0.638156i
$540$		0			0
$541$			0.186320i			0.00801052i	0.999992	+	0.00400526i	$0.00127492\pi$
$541$			0.186320i			0.00801052i	−0.999992	+	0.00400526i	$0.998725\pi$
$542$		−	79.7322i		−	3.42479i
$543$		0			0
$544$	20.3967			0.874501
$545$		−	5.42165i		−	0.232238i
$546$		0			0
$547$			5.56344i			0.237875i	0.992902	+	0.118938i	$0.0379489\pi$
$547$			5.56344i			0.237875i	−0.992902	+	0.118938i	$0.962051\pi$
$548$	11.3183			0.483494
$549$		0			0
$550$	−21.5923			−0.920698
$551$	48.9174			2.08395
$552$		0			0
$553$	−2.91111			−0.123793
$554$	−52.0743			−2.21242
$555$		0			0
$556$			89.3070i			3.78746i
$557$	7.44256			0.315351			0.157676	−	0.987491i	$-0.449600\pi$
$557$	7.44256			0.315351			0.157676	+	0.987491i	$0.449600\pi$
$558$		0			0
$559$	14.4217			0.609973
$560$	−1.39643			−0.0590099
$561$		0			0
$562$			55.4968i			2.34099i
$563$			14.1577i			0.596674i	0.954461	+	0.298337i	$0.0964320\pi$
$563$			14.1577i			0.596674i	−0.954461	+	0.298337i	$0.903568\pi$
$564$		0			0
$565$	−1.83446			−0.0771762
$566$		−	60.0130i		−	2.52253i
$567$		0			0
$568$	70.9761			2.97809
$569$		−	32.9425i		−	1.38102i	−0.723322	−	0.690511i	$-0.757386\pi$
$569$		−	32.9425i		−	1.38102i	0.723322	−	0.690511i	$-0.242614\pi$
$570$		0			0
$571$	15.9247			0.666429			0.333214	−	0.942851i	$-0.391867\pi$
$571$	15.9247			0.666429			0.333214	+	0.942851i	$0.391867\pi$
$572$	14.9586			0.625450
$573$		0			0
$574$	4.30990			0.179892
$575$	−30.3859			−1.26718
$576$		0			0
$577$	7.46735			0.310870			0.155435	−	0.987846i	$-0.450322\pi$
$577$	7.46735			0.310870			0.155435	+	0.987846i	$0.450322\pi$
$578$		−	1.37537i		−	0.0572080i
$579$		0			0
$580$	−27.7701			−1.15309
$581$	1.30997			0.0543468
$582$		0			0
$583$			0.810927i			0.0335852i
$584$	−12.6210			−0.522260
$585$		0			0
$586$	0.366544			0.0151418
$587$			22.1901i			0.915885i	0.888982	+	0.457943i	$0.151413\pi$
$587$			22.1901i			0.915885i	−0.888982	+	0.457943i	$0.848587\pi$
$588$		0			0
$589$			48.7194i			2.00745i
$590$	18.7617			0.772406
$591$		0			0
$592$		−	52.4006i		−	2.15365i
$593$	13.2092			0.542436			0.271218	−	0.962518i	$-0.412574\pi$
$593$	13.2092			0.542436			0.271218	+	0.962518i	$0.412574\pi$
$594$		0			0
$595$		−	0.824465i		−	0.0337998i
$596$	97.8539			4.00825
$597$		0			0
$598$	30.5212			1.24810
$599$		−	28.3276i		−	1.15743i	−0.815529	−	0.578716i	$-0.803554\pi$
$599$		−	28.3276i		−	1.15743i	0.815529	−	0.578716i	$-0.196446\pi$
$600$		0			0
$601$		−	17.2614i		−	0.704106i	−0.935980	−	0.352053i	$-0.885484\pi$
$601$		−	17.2614i		−	0.704106i	0.935980	−	0.352053i	$-0.114516\pi$
$602$			4.69567i			0.191381i
$603$		0			0
$604$		−	44.5983i		−	1.81468i
$605$			6.48417i			0.263619i
$606$		0			0
$607$		−	26.8071i		−	1.08807i	−0.839063	−	0.544034i	$-0.816896\pi$
$607$		−	26.8071i		−	1.08807i	0.839063	−	0.544034i	$-0.183104\pi$
$608$	39.4735			1.60086
$609$		0			0
$610$	18.9364			0.766712
$611$		−	18.8119i		−	0.761048i
$612$		0			0
$613$	−10.7908			−0.435836			−0.217918	−	0.975967i	$-0.569927\pi$
$613$	−10.7908			−0.435836			−0.217918	+	0.975967i	$0.569927\pi$
$614$			21.2321i			0.856859i
$615$		0			0
$616$			2.67927i			0.107951i
$617$	−35.4732			−1.42810			−0.714049	−	0.700096i	$-0.753141\pi$
$617$	−35.4732			−1.42810			−0.714049	+	0.700096i	$0.753141\pi$
$618$		0			0
$619$		−	37.3500i		−	1.50122i	−0.660744	−	0.750612i	$-0.729759\pi$
$619$		−	37.3500i		−	1.50122i	0.660744	−	0.750612i	$-0.270241\pi$
$620$		−	27.6577i		−	1.11076i
$621$		0			0
$622$	−71.9271			−2.88401
$623$			0.433107i			0.0173521i
$624$		0			0
$625$	10.9320			0.437280
$626$	34.4058			1.37513
$627$		0			0
$628$	44.4471			1.77363
$629$	30.9378			1.23357
$630$		0			0
$631$	−32.9840			−1.31307			−0.656537	−	0.754294i	$-0.727979\pi$
$631$	−32.9840			−1.31307			−0.656537	+	0.754294i	$0.727979\pi$
$632$	−89.1666			−3.54686
$633$		0			0
$634$		−	63.5938i		−	2.52563i
$635$		−	0.913562i		−	0.0362536i
$636$		0			0
$637$		−	10.9985i		−	0.435777i
$638$			33.6770i			1.33328i
$639$		0			0
$640$	12.5719			0.496949
$641$		−	26.5093i		−	1.04705i	−0.852009	−	0.523527i	$-0.824616\pi$
$641$		−	26.5093i		−	1.04705i	0.852009	−	0.523527i	$-0.175384\pi$
$642$		0			0
$643$	−10.3683			−0.408886			−0.204443	−	0.978878i	$-0.565538\pi$
$643$	−10.3683			−0.408886			−0.204443	+	0.978878i	$0.565538\pi$
$644$			6.85403i			0.270087i
$645$		0			0
$646$			80.8643i			3.18156i
$647$			31.1998i			1.22659i	0.789854	+	0.613296i	$0.210157\pi$
$647$			31.1998i			1.22659i	−0.789854	+	0.613296i	$0.789843\pi$
$648$		0			0
$649$		−	15.6924i		−	0.615983i
$650$	−16.0292			−0.628717
$651$		0			0
$652$	−107.659			−4.21623
$653$	31.7988			1.24438			0.622192	−	0.782865i	$-0.286242\pi$
$653$	31.7988			1.24438			0.622192	+	0.782865i	$0.286242\pi$
$654$		0			0
$655$			1.17640i			0.0459658i
$656$	57.5481			2.24688
$657$		0			0
$658$	6.12511			0.238782
$659$	0.997098			0.0388414			0.0194207	−	0.999811i	$-0.493818\pi$
$659$	0.997098			0.0388414			0.0194207	+	0.999811i	$0.493818\pi$
$660$		0			0
$661$	−5.49685			−0.213803			−0.106901	−	0.994270i	$-0.534093\pi$
$661$	−5.49685			−0.213803			−0.106901	+	0.994270i	$0.534093\pi$
$662$	−32.8029			−1.27492
$663$		0			0
$664$	40.1241			1.55712
$665$		−	1.59558i		−	0.0618739i
$666$		0			0
$667$			47.3922i			1.83503i
$668$	−18.5048			−0.715973
$669$		0			0
$670$	−36.4204			−1.40704
$671$		−	15.8386i		−	0.611441i
$672$		0			0
$673$		−	20.2932i		−	0.782246i	−0.920338	−	0.391123i	$-0.872087\pi$
$673$		−	20.2932i		−	0.782246i	0.920338	−	0.391123i	$-0.127913\pi$
$674$			18.4966i			0.712461i
$675$		0			0
$676$	−46.6865			−1.79564
$677$	−41.3640			−1.58975			−0.794873	−	0.606776i	$-0.792463\pi$
$677$	−41.3640			−1.58975			−0.794873	+	0.606776i	$0.792463\pi$
$678$		0			0
$679$	−0.790587			−0.0303399
$680$		−	25.2532i		−	0.968414i
$681$		0			0
$682$	−33.5407			−1.28434
$683$	1.76522			0.0675444			0.0337722	−	0.999430i	$-0.489248\pi$
$683$	1.76522			0.0675444			0.0337722	+	0.999430i	$0.489248\pi$
$684$		0			0
$685$		−	2.55268i		−	0.0975329i
$686$	7.18334			0.274261
$687$		0			0
$688$			62.6991i			2.39038i
$689$			0.601998i			0.0229343i
$690$		0			0
$691$	8.63431			0.328465			0.164232	−	0.986422i	$-0.447485\pi$
$691$	8.63431			0.328465			0.164232	+	0.986422i	$0.447485\pi$
$692$	34.9549			1.32879
$693$		0			0
$694$	77.2306			2.93163
$695$	20.1419			0.764027
$696$		0			0
$697$			33.9770i			1.28697i
$698$			70.1925i			2.65683i
$699$		0			0
$700$		−	3.59962i		−	0.136053i
$701$	25.2485			0.953621			0.476811	−	0.879006i	$-0.341793\pi$
$701$	25.2485			0.953621			0.476811	+	0.879006i	$0.341793\pi$
$702$		0			0
$703$	59.8736			2.25818
$704$		−	2.08289i		−	0.0785019i
$705$		0			0
$706$		−	70.2515i		−	2.64395i
$707$	2.41095			0.0906731
$708$		0			0
$709$		−	42.2655i		−	1.58731i	−0.608366	−	0.793656i	$-0.708175\pi$
$709$		−	42.2655i		−	1.58731i	0.608366	−	0.793656i	$-0.291825\pi$
$710$		−	29.0993i		−	1.09208i
$711$		0			0
$712$			13.2660i			0.497163i
$713$	−47.2004			−1.76767
$714$		0			0
$715$		−	3.37370i		−	0.126169i
$716$	−46.6192			−1.74224
$717$		0			0
$718$	−5.95358			−0.222186
$719$			35.0526i			1.30724i	0.756822	+	0.653621i	$0.226751\pi$
$719$			35.0526i			1.30724i	−0.756822	+	0.653621i	$0.773249\pi$
$720$		0			0
$721$	−1.76208			−0.0656231
$722$			108.259i			4.02897i
$723$		0			0
$724$			68.6247i			2.55042i
$725$		−	24.8896i		−	0.924375i
$726$		0			0
$727$	16.9131			0.627271			0.313635	−	0.949543i	$-0.398453\pi$
$727$	16.9131			0.627271			0.313635	+	0.949543i	$0.398453\pi$
$728$			1.98898i			0.0737165i
$729$		0			0
$730$			5.17444i			0.191515i
$731$	−37.0182			−1.36917
$732$		0			0
$733$	−3.13835			−0.115917			−0.0579587	−	0.998319i	$-0.518459\pi$
$733$	−3.13835			−0.115917			−0.0579587	+	0.998319i	$0.518459\pi$
$734$			4.64574i			0.171478i
$735$		0			0
$736$			38.2427i			1.40965i
$737$			30.4624i			1.12210i
$738$		0			0
$739$	11.4927			0.422765			0.211383	−	0.977403i	$-0.432203\pi$
$739$	11.4927			0.422765			0.211383	+	0.977403i	$0.432203\pi$
$740$	−33.9898			−1.24949
$741$		0			0
$742$	−0.196009			−0.00719572
$743$	−25.9160			−0.950767			−0.475383	−	0.879779i	$-0.657691\pi$
$743$	−25.9160			−0.950767			−0.475383	+	0.879779i	$0.657691\pi$
$744$		0			0
$745$		−	22.0696i		−	0.808566i
$746$		−	3.16123i		−	0.115741i
$747$		0			0
$748$	−38.3963			−1.40391
$749$			2.29833i			0.0839792i
$750$		0			0
$751$	35.2543			1.28645			0.643224	−	0.765678i	$-0.277596\pi$
$751$	35.2543			1.28645			0.643224	+	0.765678i	$0.277596\pi$
$752$	81.7858			2.98242
$753$		0			0
$754$			25.0003i			0.910459i
$755$	−10.0585			−0.366066
$756$		0			0
$757$	36.9842			1.34421			0.672107	−	0.740454i	$-0.265389\pi$
$757$	36.9842			1.34421			0.672107	+	0.740454i	$0.265389\pi$
$758$	3.58781			0.130315
$759$		0			0
$760$		−	48.8721i		−	1.77278i
$761$		−	28.2033i		−	1.02237i	−0.859471	−	0.511185i	$-0.829207\pi$
$761$		−	28.2033i		−	1.02237i	0.859471	−	0.511185i	$-0.170793\pi$
$762$		0			0
$763$		−	1.09609i		−	0.0396811i
$764$	11.9809			0.433454
$765$		0			0
$766$	−9.71937			−0.351175
$767$		−	11.6494i		−	0.420636i
$768$		0			0
$769$		−	13.3866i		−	0.482732i	−0.970434	−	0.241366i	$-0.922405\pi$
$769$		−	13.3866i		−	0.482732i	0.970434	−	0.241366i	$-0.0775955\pi$
$770$	1.09847			0.0395860
$771$		0			0
$772$	37.4383			1.34743
$773$	26.2722			0.944945			0.472473	−	0.881345i	$-0.343362\pi$
$773$	26.2722			0.944945			0.472473	+	0.881345i	$0.343362\pi$
$774$		0			0
$775$	24.7888			0.890441
$776$	−24.2155			−0.869285
$777$		0			0
$778$	26.3664			0.945281
$779$			65.7552i			2.35593i
$780$		0			0
$781$	−24.3389			−0.870916
$782$	−78.3430			−2.80154
$783$		0			0
$784$	47.8167			1.70774
$785$		−	10.0244i		−	0.357787i
$786$		0			0
$787$			9.23638i			0.329241i	0.986357	+	0.164621i	$0.0526400\pi$
$787$			9.23638i			0.329241i	−0.986357	+	0.164621i	$0.947360\pi$
$788$		−	42.1242i		−	1.50061i
$789$		0			0
$790$			36.5572i			1.30065i
$791$	−0.370870			−0.0131866
$792$		0			0
$793$		−	11.7579i		−	0.417535i
$794$	−14.3899			−0.510679
$795$		0			0
$796$		−	99.3332i		−	3.52077i
$797$		−	10.1091i		−	0.358082i	−0.983842	−	0.179041i	$-0.942700\pi$
$797$		−	10.1091i		−	0.358082i	0.983842	−	0.179041i	$-0.0572995\pi$
$798$		0			0
$799$			48.2871i			1.70828i
$800$		−	20.0844i		−	0.710091i
$801$		0			0
$802$	7.49626			0.264702
$803$	4.32795			0.152730
$804$		0			0
$805$	1.54583			0.0544833
$806$	−24.8992			−0.877036
$807$		0			0
$808$	73.8467			2.59792
$809$	−37.2298			−1.30893			−0.654465	−	0.756092i	$-0.727106\pi$
$809$	−37.2298			−1.30893			−0.654465	+	0.756092i	$0.727106\pi$
$810$		0			0
$811$			25.9861i			0.912496i	0.889853	+	0.456248i	$0.150807\pi$
$811$			25.9861i			0.912496i	−0.889853	+	0.456248i	$0.849193\pi$
$812$	−5.61424			−0.197021
$813$		0			0
$814$			41.2197i			1.44475i
$815$			24.2808i			0.850521i
$816$		0			0
$817$	−71.6409			−2.50640
$818$		−	6.77004i		−	0.236709i
$819$		0			0
$820$		−	37.3288i		−	1.30358i
$821$	−37.7003			−1.31575			−0.657875	−	0.753127i	$-0.728544\pi$
$821$	−37.7003			−1.31575			−0.657875	+	0.753127i	$0.728544\pi$
$822$		0			0
$823$		−	12.3129i		−	0.429201i	−0.976702	−	0.214600i	$-0.931155\pi$
$823$		−	12.3129i		−	0.429201i	0.976702	−	0.214600i	$-0.0688449\pi$
$824$	−53.9719			−1.88020
$825$		0			0
$826$	3.79302			0.131976
$827$		−	8.16207i		−	0.283823i	−0.989879	−	0.141912i	$-0.954675\pi$
$827$		−	8.16207i		−	0.283823i	0.989879	−	0.141912i	$-0.0453249\pi$
$828$		0			0
$829$			21.4138i			0.743730i	0.928287	+	0.371865i	$0.121282\pi$
$829$			21.4138i			0.743730i	−0.928287	+	0.371865i	$0.878718\pi$
$830$		−	16.4504i		−	0.571001i
$831$		0			0
$832$		−	1.54625i		−	0.0536066i
$833$			28.2315i			0.978162i
$834$		0			0
$835$			4.17350i			0.144430i
$836$	−74.3080			−2.56999
$837$		0			0
$838$	49.9660			1.72605
$839$			37.9872i			1.31146i	0.754993	+	0.655732i	$0.227640\pi$
$839$			37.9872i			1.31146i	−0.754993	+	0.655732i	$0.772360\pi$
$840$		0			0
$841$	−9.81964			−0.338608
$842$			52.6304i			1.81376i
$843$		0			0
$844$	−86.6104			−2.98125
$845$			10.5295i			0.362225i
$846$		0			0
$847$			1.31090i			0.0450429i
$848$	−2.61722			−0.0898757
$849$		0			0
$850$	41.1444			1.41124
$851$			58.0068i			1.98845i
$852$		0			0
$853$	−16.9365			−0.579893			−0.289947	−	0.957043i	$-0.593638\pi$
$853$	−16.9365			−0.579893			−0.289947	+	0.957043i	$0.593638\pi$
$854$	3.82834			0.131003
$855$		0			0
$856$			70.3973i			2.40613i
$857$	−41.2105			−1.40772			−0.703862	−	0.710337i	$-0.748543\pi$
$857$	−41.2105			−1.40772			−0.703862	+	0.710337i	$0.748543\pi$
$858$		0			0
$859$	−46.3831			−1.58257			−0.791286	−	0.611446i	$-0.790588\pi$
$859$	−46.3831			−1.58257			−0.791286	+	0.611446i	$0.790588\pi$
$860$	40.6700			1.38684
$861$		0			0
$862$	−22.4543			−0.764796
$863$	−24.3727			−0.829656			−0.414828	−	0.909900i	$-0.636158\pi$
$863$	−24.3727			−0.829656			−0.414828	+	0.909900i	$0.636158\pi$
$864$		0			0
$865$		−	7.88359i		−	0.268050i
$866$	81.8195			2.78034
$867$		0			0
$868$		−	5.59152i		−	0.189789i
$869$	30.5768			1.03725
$870$		0			0
$871$			22.6140i			0.766245i
$872$		−	33.5729i		−	1.13692i
$873$		0			0
$874$	−151.616			−5.12850
$875$	−1.82798			−0.0617968
$876$		0			0
$877$	−44.9401			−1.51752			−0.758759	−	0.651371i	$-0.774194\pi$
$877$	−44.9401			−1.51752			−0.758759	+	0.651371i	$0.774194\pi$
$878$			32.0020i			1.08001i
$879$		0			0
$880$	14.6673			0.494436
$881$	35.3645			1.19146			0.595730	−	0.803185i	$-0.296863\pi$
$881$	35.3645			1.19146			0.595730	+	0.803185i	$0.296863\pi$
$882$		0			0
$883$	−7.76537			−0.261326			−0.130663	−	0.991427i	$-0.541711\pi$
$883$	−7.76537			−0.261326			−0.130663	+	0.991427i	$0.541711\pi$
$884$	−28.5038			−0.958687
$885$		0			0
$886$	70.0698			2.35404
$887$			7.38788i			0.248061i	0.992278	+	0.124030i	$0.0395820\pi$
$887$			7.38788i			0.248061i	−0.992278	+	0.124030i	$0.960418\pi$
$888$		0			0
$889$		−	0.184694i		−	0.00619443i
$890$	5.43887			0.182311
$891$		0			0
$892$		−	102.835i		−	3.44316i
$893$			93.4495i			3.12717i
$894$		0			0
$895$			10.5143i			0.351454i
$896$	2.54165			0.0849105
$897$		0			0
$898$			54.9133i			1.83248i
$899$		−	38.6625i		−	1.28947i
$900$		0			0
$901$		−	1.54523i		−	0.0514792i
$902$	−45.2689			−1.50729
$903$		0			0
$904$	−11.3597			−0.377816
$905$	15.4773			0.514484
$906$		0			0
$907$		−	43.0501i		−	1.42945i	−0.699403	−	0.714727i	$-0.746551\pi$
$907$		−	43.0501i		−	1.42945i	0.699403	−	0.714727i	$-0.253449\pi$
$908$	−87.1251			−2.89135
$909$		0			0
$910$	0.815456			0.0270321
$911$	−34.5957			−1.14621			−0.573103	−	0.819483i	$-0.694261\pi$
$911$	−34.5957			−1.14621			−0.573103	+	0.819483i	$0.694261\pi$
$912$		0			0
$913$	−13.7592			−0.455365
$914$		−	83.0920i		−	2.74844i
$915$		0			0
$916$		−	91.0958i		−	3.00989i
$917$			0.237831i			0.00785388i
$918$		0			0
$919$	45.0931			1.48748			0.743742	−	0.668466i	$-0.233049\pi$
$919$	45.0931			1.48748			0.743742	+	0.668466i	$0.233049\pi$
$920$	47.3484			1.56103
$921$		0			0
$922$			44.7618i			1.47415i
$923$	−18.0682			−0.594722
$924$		0			0
$925$		−	30.4642i		−	1.00166i
$926$	−34.7801			−1.14295
$927$		0			0
$928$	−31.3252			−1.02830
$929$	22.1120			0.725472			0.362736	−	0.931892i	$-0.381843\pi$
$929$	22.1120			0.725472			0.362736	+	0.931892i	$0.381843\pi$
$930$		0			0
$931$			54.6360i			1.79062i
$932$	28.9532			0.948395
$933$		0			0
$934$		−	20.5497i		−	0.672406i
$935$			8.65974i			0.283204i
$936$		0			0
$937$	8.19854			0.267835			0.133917	−	0.990993i	$-0.457244\pi$
$937$	8.19854			0.267835			0.133917	+	0.990993i	$0.457244\pi$
$938$	−7.36307			−0.240413
$939$		0			0
$940$		−	53.0507i		−	1.73032i
$941$	12.6796			0.413343			0.206671	−	0.978410i	$-0.433737\pi$
$941$	12.6796			0.413343			0.206671	+	0.978410i	$0.433737\pi$
$942$		0			0
$943$	−63.7050			−2.07452
$944$	50.6465			1.64840
$945$		0			0
$946$		−	49.3208i		−	1.60356i
$947$	−7.26631			−0.236123			−0.118062	−	0.993006i	$-0.537668\pi$
$947$	−7.26631			−0.236123			−0.118062	+	0.993006i	$0.537668\pi$
$948$		0			0
$949$	3.21289			0.104295
$950$	79.6263			2.58342
$951$		0			0
$952$		−	5.10540i		−	0.165467i
$953$	24.0628			0.779470			0.389735	−	0.920927i	$-0.372567\pi$
$953$	24.0628			0.779470			0.389735	+	0.920927i	$0.372567\pi$
$954$		0			0
$955$		−	2.70212i		−	0.0874386i
$956$	48.9270	+	48.2630i	1.58241	+	1.56094i
$957$		0			0
$958$	−33.1734			−1.07178
$959$		−	0.516072i		−	0.0166648i
$960$		0			0
$961$	7.50608			0.242132
$962$			30.5998i			0.986576i
$963$		0			0
$964$	−58.6852			−1.89012
$965$		−	8.44367i		−	0.271811i
$966$		0			0
$967$	13.0184			0.418644			0.209322	−	0.977847i	$-0.432874\pi$
$967$	13.0184			0.418644			0.209322	+	0.977847i	$0.432874\pi$
$968$			40.1524i			1.29055i
$969$		0			0
$970$			9.92803i			0.318770i
$971$		−	4.53779i		−	0.145625i	−0.997346	−	0.0728124i	$-0.976803\pi$
$971$		−	4.53779i		−	0.145625i	0.997346	−	0.0728124i	$-0.0231974\pi$
$972$		0			0
$973$	4.07207			0.130544
$974$			40.0950i			1.28473i
$975$		0			0
$976$	51.1181			1.63625
$977$	−36.4107			−1.16488			−0.582441	−	0.812873i	$-0.697902\pi$
$977$	−36.4107			−1.16488			−0.582441	+	0.812873i	$0.697902\pi$
$978$		0			0
$979$		−	4.54912i		−	0.145391i
$980$		−	31.0165i		−	0.990786i
$981$		0			0
$982$			15.1241i			0.482628i
$983$		−	44.8689i		−	1.43109i	−0.698564	−	0.715547i	$-0.746177\pi$
$983$		−	44.8689i		−	1.43109i	0.698564	−	0.715547i	$-0.253823\pi$
$984$		0			0
$985$	−9.50052			−0.302712
$986$		−	64.1719i		−	2.04365i
$987$		0			0
$988$	−55.1631			−1.75497
$989$		−	69.4072i		−	2.20702i
$990$		0			0
$991$		−	24.1554i		−	0.767321i	−0.923474	−	0.383661i	$-0.874663\pi$
$991$		−	24.1554i		−	0.767321i	0.923474	−	0.383661i	$-0.125337\pi$
$992$		−	31.1984i		−	0.990551i
$993$		0			0
$994$		−	5.88297i		−	0.186596i
$995$	−22.4032			−0.710228
$996$		0			0
$997$			5.83319i			0.184739i	0.995725	+	0.0923695i	$0.0294441\pi$
$997$			5.83319i			0.184739i	−0.995725	+	0.0923695i	$0.970556\pi$
$998$	85.3219			2.70082
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.8	yes	80
3.2	odd	2	inner	2151.2.b.a.2150.74	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.7	✓	80
717.716	even	2	inner	2151.2.b.a.2150.73	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.7	✓	80	239.238	odd	2	inner
2151.2.b.a.2150.8	yes	80	1.1	even	1	trivial
2151.2.b.a.2150.73	yes	80	717.716	even	2	inner
2151.2.b.a.2150.74	yes	80	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.53879i q^{2} -4.44547 q^{4} +1.00261i q^{5} +0.202697i q^{7} +6.20855i q^{8} +O(q^{10})\)	\(q-2.53879i q^{2} -4.44547 q^{4} +1.00261i q^{5} +0.202697i q^{7} +6.20855i q^{8} +2.54543 q^{10} -2.12902i q^{11} -1.58049i q^{13} +0.514606 q^{14} +6.87129 q^{16} +4.05688i q^{17} +7.85123i q^{19} -4.45709i q^{20} -5.40514 q^{22} -7.60643 q^{23} +3.99477 q^{25} -4.01255 q^{26} -0.901084i q^{28} -6.23054i q^{29} +6.20533 q^{31} -5.02768i q^{32} +10.2996 q^{34} -0.203227 q^{35} -7.62601i q^{37} +19.9327 q^{38} -6.22478 q^{40} +8.37515 q^{41} +9.12480i q^{43} +9.46450i q^{44} +19.3112i q^{46} +11.9025 q^{47} +6.95891 q^{49} -10.1419i q^{50} +7.02605i q^{52} -0.380892 q^{53} +2.13458 q^{55} -1.25845 q^{56} -15.8181 q^{58} +7.37074 q^{59} +7.43937 q^{61} -15.7540i q^{62} +0.978334 q^{64} +1.58462 q^{65} -14.3082 q^{67} -18.0347i q^{68} +0.515950i q^{70} -11.4320i q^{71} +2.03284i q^{73} -19.3609 q^{74} -34.9024i q^{76} +0.431546 q^{77} +14.3619i q^{79} +6.88924i q^{80} -21.2628i q^{82} -6.46271i q^{83} -4.06748 q^{85} +23.1660 q^{86} +13.2181 q^{88} +2.13672 q^{89} +0.320361 q^{91} +33.8142 q^{92} -30.2181i q^{94} -7.87174 q^{95} +3.90034i q^{97} -17.6672i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 80 * q - 80 * q^4	\( 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) \) 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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