LMFDB - Embedded newform 608.3.e.b.417.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,3,Mod(417,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.417"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	608.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,-52,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.5668000731$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 $ x^20 - 8x^19 - 196x^18 + 1676x^17 + 16346x^16 - 161824x^15 - 667200x^14 + 8850712x^13 + 9918758x^12 - 288078272x^11 + 224256138x^10 + 5383493580x^9 - 11116604703x^8 - 53755955160x^7 + 166723956672x^6 + 262105513072x^5 - 1053812084224x^4 - 855143026944x^3 + 3318150008320x^2 + 3262428344448*x + 1135285065792
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.17
Root			$2.72330 - 2.14790i$ of defining polynomial
Character	$\chi$	$=$	608.417
Dual form			608.3.e.b.417.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.29580i q^{3} +2.32882 q^{5} -2.95740 q^{7} -9.45390 q^{9} +19.4859 q^{11} +14.0235i q^{13} +10.0041i q^{15} -22.2731 q^{17} +(-10.5756 + 15.7847i) q^{19} -12.7044i q^{21} +41.7289 q^{23} -19.5766 q^{25} -1.94987i q^{27} +22.9343i q^{29} +30.9806i q^{31} +83.7075i q^{33} -6.88725 q^{35} -62.1601i q^{37} -60.2423 q^{39} +20.2283i q^{41} -53.2356 q^{43} -22.0164 q^{45} +21.3514 q^{47} -40.2538 q^{49} -95.6807i q^{51} -12.4978i q^{53} +45.3791 q^{55} +(-67.8080 - 45.4305i) q^{57} -19.2741i q^{59} +4.96429 q^{61} +27.9590 q^{63} +32.6583i q^{65} -58.9495i q^{67} +179.259i q^{69} -61.6822i q^{71} +14.5807 q^{73} -84.0972i q^{75} -57.6276 q^{77} +113.346i q^{79} -76.7089 q^{81} -48.9861 q^{83} -51.8699 q^{85} -98.5213 q^{87} -27.8155i q^{89} -41.4733i q^{91} -133.087 q^{93} +(-24.6285 + 36.7597i) q^{95} +144.982i q^{97} -184.218 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) $ 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			4.29580i			1.43193i	0.698134	+	0.715967i	$0.254014\pi$
$3$			4.29580i			1.43193i	−0.698134	+	0.715967i	$0.745986\pi$
$4$		0			0
$5$	2.32882			0.465763			0.232882	−	0.972505i	$-0.425184\pi$
$5$	2.32882			0.465763			0.232882	+	0.972505i	$0.425184\pi$
$6$		0			0
$7$	−2.95740			−0.422486			−0.211243	−	0.977434i	$-0.567751\pi$
$7$	−2.95740			−0.422486			−0.211243	+	0.977434i	$0.567751\pi$
$8$		0			0
$9$	−9.45390			−1.05043
$10$		0			0
$11$	19.4859			1.77144			0.885722	−	0.464216i	$-0.153664\pi$
$11$	19.4859			1.77144			0.885722	+	0.464216i	$0.153664\pi$
$12$		0			0
$13$			14.0235i			1.07873i	0.842071	+	0.539367i	$0.181336\pi$
$13$			14.0235i			1.07873i	−0.842071	+	0.539367i	$0.818664\pi$
$14$		0			0
$15$			10.0041i			0.666942i
$16$		0			0
$17$	−22.2731			−1.31018			−0.655090	−	0.755551i	$-0.727369\pi$
$17$	−22.2731			−1.31018			−0.655090	+	0.755551i	$0.727369\pi$
$18$		0			0
$19$	−10.5756	+	15.7847i	−0.556608	+	0.830775i
$19$	−10.5756	+	15.7847i	−0.556608	+	0.830775i
$20$		0			0
$21$		−	12.7044i		−	0.604972i
$22$		0			0
$23$	41.7289			1.81430			0.907150	−	0.420808i	$-0.138253\pi$
$23$	41.7289			1.81430			0.907150	+	0.420808i	$0.138253\pi$
$24$		0			0
$25$	−19.5766			−0.783064
$26$		0			0
$27$		−	1.94987i		−	0.0722174i
$28$		0			0
$29$			22.9343i			0.790839i	0.918501	+	0.395420i	$0.129401\pi$
$29$			22.9343i			0.790839i	−0.918501	+	0.395420i	$0.870599\pi$
$30$		0			0
$31$			30.9806i			0.999375i	0.866206	+	0.499688i	$0.166552\pi$
$31$			30.9806i			0.999375i	−0.866206	+	0.499688i	$0.833448\pi$
$32$		0			0
$33$			83.7075i			2.53659i
$34$		0			0
$35$	−6.88725			−0.196779
$36$		0			0
$37$		−	62.1601i		−	1.68000i	−0.542584	−	0.840001i	$-0.682554\pi$
$37$		−	62.1601i		−	1.68000i	0.542584	−	0.840001i	$-0.317446\pi$
$38$		0			0
$39$	−60.2423			−1.54468
$40$		0			0
$41$			20.2283i			0.493373i	0.969095	+	0.246686i	$0.0793417\pi$
$41$			20.2283i			0.493373i	−0.969095	+	0.246686i	$0.920658\pi$
$42$		0			0
$43$	−53.2356			−1.23804			−0.619019	−	0.785376i	$-0.712470\pi$
$43$	−53.2356			−1.23804			−0.619019	+	0.785376i	$0.712470\pi$
$44$		0			0
$45$	−22.0164			−0.489254
$46$		0			0
$47$	21.3514			0.454285			0.227143	−	0.973861i	$-0.427062\pi$
$47$	21.3514			0.454285			0.227143	+	0.973861i	$0.427062\pi$
$48$		0			0
$49$	−40.2538			−0.821505
$50$		0			0
$51$		−	95.6807i		−	1.87609i
$52$		0			0
$53$		−	12.4978i		−	0.235808i	−0.993025	−	0.117904i	$-0.962383\pi$
$53$		−	12.4978i		−	0.235808i	0.993025	−	0.117904i	$-0.0376174\pi$
$54$		0			0
$55$	45.3791			0.825074
$56$		0			0
$57$	−67.8080	−	45.4305i	−1.18961	−	0.797026i
$58$		0			0
$59$		−	19.2741i		−	0.326680i	−0.986570	−	0.163340i	$-0.947773\pi$
$59$		−	19.2741i		−	0.326680i	0.986570	−	0.163340i	$-0.0522267\pi$
$60$		0			0
$61$	4.96429			0.0813817			0.0406909	−	0.999172i	$-0.487044\pi$
$61$	4.96429			0.0813817			0.0406909	+	0.999172i	$0.487044\pi$
$62$		0			0
$63$	27.9590			0.443794
$64$		0			0
$65$			32.6583i			0.502435i
$66$		0			0
$67$		−	58.9495i		−	0.879844i	−0.898036	−	0.439922i	$-0.855006\pi$
$67$		−	58.9495i		−	0.879844i	0.898036	−	0.439922i	$-0.144994\pi$
$68$		0			0
$69$			179.259i			2.59796i
$70$		0			0
$71$		−	61.6822i		−	0.868763i	−0.900729	−	0.434382i	$-0.856967\pi$
$71$		−	61.6822i		−	0.868763i	0.900729	−	0.434382i	$-0.143033\pi$
$72$		0			0
$73$	14.5807			0.199736			0.0998680	−	0.995001i	$-0.468158\pi$
$73$	14.5807			0.199736			0.0998680	+	0.995001i	$0.468158\pi$
$74$		0			0
$75$		−	84.0972i		−	1.12130i
$76$		0			0
$77$	−57.6276			−0.748411
$78$		0			0
$79$			113.346i			1.43475i	0.696685	+	0.717377i	$0.254658\pi$
$79$			113.346i			1.43475i	−0.696685	+	0.717377i	$0.745342\pi$
$80$		0			0
$81$	−76.7089			−0.947023
$82$		0			0
$83$	−48.9861			−0.590194			−0.295097	−	0.955467i	$-0.595352\pi$
$83$	−48.9861			−0.590194			−0.295097	+	0.955467i	$0.595352\pi$
$84$		0			0
$85$	−51.8699			−0.610234
$86$		0			0
$87$	−98.5213			−1.13243
$88$		0			0
$89$		−	27.8155i		−	0.312534i	−0.987715	−	0.156267i	$-0.950054\pi$
$89$		−	27.8155i		−	0.312534i	0.987715	−	0.156267i	$-0.0499460\pi$
$90$		0			0
$91$		−	41.4733i		−	0.455750i
$92$		0			0
$93$	−133.087			−1.43104
$94$		0			0
$95$	−24.6285	+	36.7597i	−0.259248	+	0.386945i
$96$		0			0
$97$			144.982i			1.49466i	0.664452	+	0.747331i	$0.268665\pi$
$97$			144.982i			1.49466i	−0.664452	+	0.747331i	$0.731335\pi$
$98$		0			0
$99$	−184.218			−1.86078
$100$		0			0
$101$	150.334			1.48845			0.744226	−	0.667927i	$-0.232818\pi$
$101$	150.334			1.48845			0.744226	+	0.667927i	$0.232818\pi$
$102$		0			0
$103$			74.2695i			0.721063i	0.932747	+	0.360532i	$0.117405\pi$
$103$			74.2695i			0.721063i	−0.932747	+	0.360532i	$0.882595\pi$
$104$		0			0
$105$		−	29.5863i		−	0.281774i
$106$		0			0
$107$			139.704i			1.30564i	0.757511	+	0.652822i	$0.226415\pi$
$107$			139.704i			1.30564i	−0.757511	+	0.652822i	$0.773585\pi$
$108$		0			0
$109$			42.5178i			0.390072i	0.980796	+	0.195036i	$0.0624823\pi$
$109$			42.5178i			0.390072i	−0.980796	+	0.195036i	$0.937518\pi$
$110$		0			0
$111$	267.027			2.40565
$112$		0			0
$113$		−	109.368i		−	0.967862i	−0.875106	−	0.483931i	$-0.839209\pi$
$113$		−	109.368i		−	0.967862i	0.875106	−	0.483931i	$-0.160791\pi$
$114$		0			0
$115$	97.1790			0.845034
$116$		0			0
$117$		−	132.577i		−	1.13314i
$118$		0			0
$119$	65.8705			0.553533
$120$		0			0
$121$	258.700			2.13802
$122$		0			0
$123$	−86.8966			−0.706477
$124$		0			0
$125$	−103.811			−0.830486
$126$		0			0
$127$		−	40.2362i		−	0.316821i	−0.987373	−	0.158410i	$-0.949363\pi$
$127$		−	40.2362i		−	0.316821i	0.987373	−	0.158410i	$-0.0506369\pi$
$128$		0			0
$129$		−	228.690i		−	1.77279i
$130$		0			0
$131$	65.6183			0.500903			0.250452	−	0.968129i	$-0.419421\pi$
$131$	65.6183			0.500903			0.250452	+	0.968129i	$0.419421\pi$
$132$		0			0
$133$	31.2762	−	46.6818i	0.235159	−	0.350991i
$134$		0			0
$135$		−	4.54089i		−	0.0336362i
$136$		0			0
$137$	168.934			1.23310			0.616549	−	0.787316i	$-0.288530\pi$
$137$	168.934			1.23310			0.616549	+	0.787316i	$0.288530\pi$
$138$		0			0
$139$	137.539			0.989487			0.494743	−	0.869039i	$-0.335262\pi$
$139$	137.539			0.989487			0.494743	+	0.869039i	$0.335262\pi$
$140$		0			0
$141$			91.7214i			0.650506i
$142$		0			0
$143$			273.261i			1.91092i
$144$		0			0
$145$			53.4099i			0.368344i
$146$		0			0
$147$		−	172.922i		−	1.17634i
$148$		0			0
$149$	−50.0197			−0.335703			−0.167851	−	0.985812i	$-0.553683\pi$
$149$	−50.0197			−0.335703			−0.167851	+	0.985812i	$0.553683\pi$
$150$		0			0
$151$			204.762i			1.35604i	0.735045	+	0.678019i	$0.237161\pi$
$151$			204.762i			1.35604i	−0.735045	+	0.678019i	$0.762839\pi$
$152$		0			0
$153$	210.567			1.37626
$154$		0			0
$155$			72.1482i			0.465472i
$156$		0			0
$157$	57.6193			0.367002			0.183501	−	0.983020i	$-0.441257\pi$
$157$	57.6193			0.367002			0.183501	+	0.983020i	$0.441257\pi$
$158$		0			0
$159$	53.6881			0.337661
$160$		0			0
$161$	−123.409			−0.766517
$162$		0			0
$163$	97.9480			0.600908			0.300454	−	0.953796i	$-0.402862\pi$
$163$	97.9480			0.600908			0.300454	+	0.953796i	$0.402862\pi$
$164$		0			0
$165$			194.939i			1.18145i
$166$		0			0
$167$		−	84.7060i		−	0.507222i	−0.967306	−	0.253611i	$-0.918382\pi$
$167$		−	84.7060i		−	0.507222i	0.967306	−	0.253611i	$-0.0816182\pi$
$168$		0			0
$169$	−27.6597			−0.163667
$170$		0			0
$171$	99.9803	−	149.227i	0.584680	−	0.872674i
$172$		0			0
$173$			22.3618i			0.129259i	0.997909	+	0.0646294i	$0.0205865\pi$
$173$			22.3618i			0.129259i	−0.997909	+	0.0646294i	$0.979413\pi$
$174$		0			0
$175$	57.8959			0.330834
$176$		0			0
$177$	82.7977			0.467784
$178$		0			0
$179$			72.7922i			0.406661i	0.979110	+	0.203330i	$0.0651765\pi$
$179$			72.7922i			0.406661i	−0.979110	+	0.203330i	$0.934823\pi$
$180$		0			0
$181$		−	350.286i		−	1.93528i	−0.252326	−	0.967642i	$-0.581196\pi$
$181$		−	350.286i		−	1.93528i	0.252326	−	0.967642i	$-0.418804\pi$
$182$		0			0
$183$			21.3256i			0.116533i
$184$		0			0
$185$		−	144.760i		−	0.782484i
$186$		0			0
$187$	−434.011			−2.32091
$188$		0			0
$189$			5.76656i			0.0305109i
$190$		0			0
$191$	−107.952			−0.565194			−0.282597	−	0.959239i	$-0.591196\pi$
$191$	−107.952			−0.565194			−0.282597	+	0.959239i	$0.591196\pi$
$192$		0			0
$193$		−	267.636i		−	1.38672i	−0.720593	−	0.693359i	$-0.756130\pi$
$193$		−	267.636i		−	1.38672i	0.720593	−	0.693359i	$-0.243870\pi$
$194$		0			0
$195$	−140.293			−0.719453
$196$		0			0
$197$	290.715			1.47571			0.737854	−	0.674960i	$-0.235839\pi$
$197$	290.715			1.47571			0.737854	+	0.674960i	$0.235839\pi$
$198$		0			0
$199$	302.296			1.51907			0.759537	−	0.650464i	$-0.225426\pi$
$199$	302.296			1.51907			0.759537	+	0.650464i	$0.225426\pi$
$200$		0			0
$201$	253.235			1.25988
$202$		0			0
$203$		−	67.8261i		−	0.334119i
$204$		0			0
$205$			47.1080i			0.229795i
$206$		0			0
$207$	−394.501			−1.90580
$208$		0			0
$209$	−206.074	+	307.579i	−0.986001	+	1.47167i
$210$		0			0
$211$			73.5756i			0.348700i	0.984684	+	0.174350i	$0.0557824\pi$
$211$			73.5756i			0.348700i	−0.984684	+	0.174350i	$0.944218\pi$
$212$		0			0
$213$	264.974			1.24401
$214$		0			0
$215$	−123.976			−0.576633
$216$		0			0
$217$		−	91.6222i		−	0.422222i
$218$		0			0
$219$			62.6359i			0.286009i
$220$		0			0
$221$		−	312.347i		−	1.41334i
$222$		0			0
$223$		−	225.342i		−	1.01050i	−0.862973	−	0.505251i	$-0.831400\pi$
$223$		−	225.342i		−	1.01050i	0.862973	−	0.505251i	$-0.168600\pi$
$224$		0			0
$225$	185.075			0.822557
$226$		0			0
$227$		−	19.4056i		−	0.0854872i	−0.999086	−	0.0427436i	$-0.986390\pi$
$227$		−	19.4056i		−	0.0854872i	0.999086	−	0.0427436i	$-0.0136099\pi$
$228$		0			0
$229$	285.505			1.24675			0.623374	−	0.781924i	$-0.285762\pi$
$229$	285.505			1.24675			0.623374	+	0.781924i	$0.285762\pi$
$230$		0			0
$231$		−	247.557i		−	1.07167i
$232$		0			0
$233$	−359.060			−1.54103			−0.770515	−	0.637422i	$-0.780001\pi$
$233$	−359.060			−1.54103			−0.770515	+	0.637422i	$0.780001\pi$
$234$		0			0
$235$	49.7235			0.211590
$236$		0			0
$237$	−486.910			−2.05447
$238$		0			0
$239$	182.395			0.763160			0.381580	−	0.924336i	$-0.375380\pi$
$239$	182.395			0.763160			0.381580	+	0.924336i	$0.375380\pi$
$240$		0			0
$241$		−	119.880i		−	0.497429i	−0.968577	−	0.248714i	$-0.919992\pi$
$241$		−	119.880i		−	0.497429i	0.968577	−	0.248714i	$-0.0800081\pi$
$242$		0			0
$243$		−	347.075i		−	1.42829i
$244$		0			0
$245$	−93.7437			−0.382627
$246$		0			0
$247$	−221.358	−	148.307i	−0.896185	−	0.600432i
$248$		0			0
$249$		−	210.434i		−	0.845118i
$250$		0			0
$251$	449.306			1.79007			0.895033	−	0.446000i	$-0.147152\pi$
$251$	449.306			1.79007			0.895033	+	0.446000i	$0.147152\pi$
$252$		0			0
$253$	813.125			3.21393
$254$		0			0
$255$		−	222.823i		−	0.873815i
$256$		0			0
$257$			469.173i			1.82558i	0.408432	+	0.912789i	$0.366076\pi$
$257$			469.173i			1.82558i	−0.408432	+	0.912789i	$0.633924\pi$
$258$		0			0
$259$			183.833i			0.709778i
$260$		0			0
$261$		−	216.819i		−	0.830724i
$262$		0			0
$263$	−265.109			−1.00802			−0.504010	−	0.863698i	$-0.668142\pi$
$263$	−265.109			−1.00802			−0.504010	+	0.863698i	$0.668142\pi$
$264$		0			0
$265$		−	29.1051i		−	0.109831i
$266$		0			0
$267$	119.490			0.447528
$268$		0			0
$269$			303.491i			1.12822i	0.825700	+	0.564109i	$0.190780\pi$
$269$			303.491i			1.12822i	−0.825700	+	0.564109i	$0.809220\pi$
$270$		0			0
$271$	200.301			0.739117			0.369559	−	0.929207i	$-0.379509\pi$
$271$	200.301			0.739117			0.369559	+	0.929207i	$0.379509\pi$
$272$		0			0
$273$	178.161			0.652604
$274$		0			0
$275$	−381.468			−1.38716
$276$		0			0
$277$	−160.105			−0.577998			−0.288999	−	0.957329i	$-0.593322\pi$
$277$	−160.105			−0.577998			−0.288999	+	0.957329i	$0.593322\pi$
$278$		0			0
$279$		−	292.888i		−	1.04978i
$280$		0			0
$281$			383.009i			1.36302i	0.731808	+	0.681511i	$0.238677\pi$
$281$			383.009i			1.36302i	−0.731808	+	0.681511i	$0.761323\pi$
$282$		0			0
$283$	226.723			0.801140			0.400570	−	0.916266i	$-0.368812\pi$
$283$	226.723			0.801140			0.400570	+	0.916266i	$0.368812\pi$
$284$		0			0
$285$	−157.913	−	105.799i	−0.554079	−	0.371226i
$286$		0			0
$287$		−	59.8232i		−	0.208443i
$288$		0			0
$289$	207.090			0.716573
$290$		0			0
$291$	−622.815			−2.14026
$292$		0			0
$293$		−	429.962i		−	1.46745i	−0.679447	−	0.733724i	$-0.737780\pi$
$293$		−	429.962i		−	1.46745i	0.679447	−	0.733724i	$-0.262220\pi$
$294$		0			0
$295$		−	44.8859i		−	0.152155i
$296$		0			0
$297$		−	37.9950i		−	0.127929i
$298$		0			0
$299$			585.187i			1.95715i
$300$		0			0
$301$	157.439			0.523054
$302$		0			0
$303$			645.804i			2.13137i
$304$		0			0
$305$	11.5609			0.0379046
$306$		0			0
$307$		−	208.273i		−	0.678414i	−0.940712	−	0.339207i	$-0.889841\pi$
$307$		−	208.273i		−	0.678414i	0.940712	−	0.339207i	$-0.110159\pi$
$308$		0			0
$309$	−319.047			−1.03251
$310$		0			0
$311$	−135.179			−0.434659			−0.217329	−	0.976098i	$-0.569735\pi$
$311$	−135.179			−0.434659			−0.217329	+	0.976098i	$0.569735\pi$
$312$		0			0
$313$	−412.048			−1.31645			−0.658224	−	0.752823i	$-0.728692\pi$
$313$	−412.048			−1.31645			−0.658224	+	0.752823i	$0.728692\pi$
$314$		0			0
$315$	65.1114			0.206703
$316$		0			0
$317$		−	386.723i		−	1.21995i	−0.792422	−	0.609973i	$-0.791180\pi$
$317$		−	386.723i		−	1.21995i	0.792422	−	0.609973i	$-0.208820\pi$
$318$		0			0
$319$			446.896i			1.40093i
$320$		0			0
$321$	−600.140			−1.86960
$322$		0			0
$323$	235.550	−	351.574i	0.729257	−	1.08847i
$324$		0			0
$325$		−	274.533i		−	0.844718i
$326$		0			0
$327$	−182.648			−0.558557
$328$		0			0
$329$	−63.1447			−0.191929
$330$		0			0
$331$			143.670i			0.434047i	0.976166	+	0.217024i	$0.0696349\pi$
$331$			143.670i			0.434047i	−0.976166	+	0.217024i	$0.930365\pi$
$332$		0			0
$333$			587.655i			1.76473i
$334$		0			0
$335$		−	137.283i		−	0.409799i
$336$		0			0
$337$		−	162.296i		−	0.481591i	−0.970576	−	0.240795i	$-0.922592\pi$
$337$		−	162.296i		−	0.481591i	0.970576	−	0.240795i	$-0.0774082\pi$
$338$		0			0
$339$	469.825			1.38591
$340$		0			0
$341$			603.685i			1.77034i
$342$		0			0
$343$	263.959			0.769561
$344$		0			0
$345$			417.461i			1.21003i
$346$		0			0
$347$	429.273			1.23710			0.618549	−	0.785746i	$-0.287721\pi$
$347$	429.273			1.23710			0.618549	+	0.785746i	$0.287721\pi$
$348$		0			0
$349$	463.869			1.32914			0.664568	−	0.747227i	$-0.268615\pi$
$349$	463.869			1.32914			0.664568	+	0.747227i	$0.268615\pi$
$350$		0			0
$351$	27.3441			0.0779034
$352$		0			0
$353$	78.3436			0.221937			0.110968	−	0.993824i	$-0.464605\pi$
$353$	78.3436			0.221937			0.110968	+	0.993824i	$0.464605\pi$
$354$		0			0
$355$		−	143.647i		−	0.404638i
$356$		0			0
$357$			282.966i			0.792623i
$358$		0			0
$359$	243.233			0.677528			0.338764	−	0.940871i	$-0.389991\pi$
$359$	243.233			0.677528			0.338764	+	0.940871i	$0.389991\pi$
$360$		0			0
$361$	−137.315	−	333.865i	−0.380374	−	0.924833i
$362$		0			0
$363$			1111.32i			3.06150i
$364$		0			0
$365$	33.9558			0.0930297
$366$		0			0
$367$	323.083			0.880335			0.440167	−	0.897916i	$-0.354919\pi$
$367$	323.083			0.880335			0.440167	+	0.897916i	$0.354919\pi$
$368$		0			0
$369$		−	191.236i		−	0.518255i
$370$		0			0
$371$			36.9610i			0.0996255i
$372$		0			0
$373$			86.8197i			0.232761i	0.993205	+	0.116380i	$0.0371291\pi$
$373$			86.8197i			0.232761i	−0.993205	+	0.116380i	$0.962871\pi$
$374$		0			0
$375$		−	445.950i		−	1.18920i
$376$		0			0
$377$	−321.621			−0.853105
$378$		0			0
$379$			20.8856i			0.0551071i	0.999620	+	0.0275535i	$0.00877168\pi$
$379$			20.8856i			0.0551071i	−0.999620	+	0.0275535i	$0.991228\pi$
$380$		0			0
$381$	172.847			0.453666
$382$		0			0
$383$			310.036i			0.809492i	0.914429	+	0.404746i	$0.132640\pi$
$383$			310.036i			0.809492i	−0.914429	+	0.404746i	$0.867360\pi$
$384$		0			0
$385$	−134.204			−0.348582
$386$		0			0
$387$	503.285			1.30048
$388$		0			0
$389$	−483.760			−1.24360			−0.621799	−	0.783177i	$-0.713598\pi$
$389$	−483.760			−1.24360			−0.621799	+	0.783177i	$0.713598\pi$
$390$		0			0
$391$	−929.430			−2.37706
$392$		0			0
$393$			281.883i			0.717260i
$394$		0			0
$395$			263.961i			0.668256i
$396$		0			0
$397$	−114.551			−0.288543			−0.144271	−	0.989538i	$-0.546084\pi$
$397$	−114.551			−0.288543			−0.144271	+	0.989538i	$0.546084\pi$
$398$		0			0
$399$	200.536	+	134.356i	0.502596	+	0.336733i
$400$		0			0
$401$			20.2091i			0.0503968i	0.999682	+	0.0251984i	$0.00802175\pi$
$401$			20.2091i			0.0503968i	−0.999682	+	0.0251984i	$0.991978\pi$
$402$		0			0
$403$	−434.458			−1.07806
$404$		0			0
$405$	−178.641			−0.441089
$406$		0			0
$407$		−	1211.24i		−	2.97603i
$408$		0			0
$409$		−	137.213i		−	0.335485i	−0.985831	−	0.167742i	$-0.946352\pi$
$409$		−	137.213i		−	0.335485i	0.985831	−	0.167742i	$-0.0536477\pi$
$410$		0			0
$411$			725.709i			1.76571i
$412$		0			0
$413$			57.0013i			0.138018i
$414$		0			0
$415$	−114.080			−0.274891
$416$		0			0
$417$			590.839i			1.41688i
$418$		0			0
$419$	69.2314			0.165230			0.0826151	−	0.996582i	$-0.473673\pi$
$419$	69.2314			0.165230			0.0826151	+	0.996582i	$0.473673\pi$
$420$		0			0
$421$			96.9103i			0.230191i	0.993354	+	0.115095i	$0.0367174\pi$
$421$			96.9103i			0.230191i	−0.993354	+	0.115095i	$0.963283\pi$
$422$		0			0
$423$	−201.854			−0.477197
$424$		0			0
$425$	436.031			1.02596
$426$		0			0
$427$	−14.6814			−0.0343827
$428$		0			0
$429$	−1173.88			−2.73631
$430$		0			0
$431$		−	473.629i		−	1.09891i	−0.835524	−	0.549453i	$-0.814836\pi$
$431$		−	473.629i		−	1.09891i	0.835524	−	0.549453i	$-0.185164\pi$
$432$		0			0
$433$			371.336i			0.857588i	0.903402	+	0.428794i	$0.141061\pi$
$433$			371.336i			0.857588i	−0.903402	+	0.428794i	$0.858939\pi$
$434$		0			0
$435$	−229.438			−0.527444
$436$		0			0
$437$	−441.306	+	658.679i	−1.00985	+	1.50727i
$438$		0			0
$439$			576.675i			1.31361i	0.754060	+	0.656806i	$0.228093\pi$
$439$			576.675i			1.31361i	−0.754060	+	0.656806i	$0.771907\pi$
$440$		0			0
$441$	380.555			0.862937
$442$		0			0
$443$	436.152			0.984542			0.492271	−	0.870442i	$-0.336167\pi$
$443$	436.152			0.984542			0.492271	+	0.870442i	$0.336167\pi$
$444$		0			0
$445$		−	64.7773i		−	0.145567i
$446$		0			0
$447$		−	214.875i		−	0.480704i
$448$		0			0
$449$		−	875.653i		−	1.95023i	−0.221699	−	0.975115i	$-0.571160\pi$
$449$		−	875.653i		−	1.95023i	0.221699	−	0.975115i	$-0.428840\pi$
$450$		0			0
$451$			394.166i			0.873982i
$452$		0			0
$453$	−879.615			−1.94176
$454$		0			0
$455$		−	96.5837i		−	0.212272i
$456$		0			0
$457$	−135.889			−0.297350			−0.148675	−	0.988886i	$-0.547501\pi$
$457$	−135.889			−0.297350			−0.148675	+	0.988886i	$0.547501\pi$
$458$		0			0
$459$			43.4296i			0.0946179i
$460$		0			0
$461$	−258.134			−0.559943			−0.279972	−	0.960008i	$-0.590325\pi$
$461$	−258.134			−0.559943			−0.279972	+	0.960008i	$0.590325\pi$
$462$		0			0
$463$	−83.7030			−0.180784			−0.0903920	−	0.995906i	$-0.528812\pi$
$463$	−83.7030			−0.180784			−0.0903920	+	0.995906i	$0.528812\pi$
$464$		0			0
$465$	−309.934			−0.666526
$466$		0			0
$467$	−210.142			−0.449984			−0.224992	−	0.974361i	$-0.572236\pi$
$467$	−210.142			−0.449984			−0.224992	+	0.974361i	$0.572236\pi$
$468$		0			0
$469$			174.338i			0.371722i
$470$		0			0
$471$			247.521i			0.525522i
$472$		0			0
$473$	−1037.34			−2.19312
$474$		0			0
$475$	207.034	−	309.011i	0.435860	−	0.650550i
$476$		0			0
$477$			118.153i			0.247700i
$478$		0			0
$479$	−226.280			−0.472400			−0.236200	−	0.971704i	$-0.575902\pi$
$479$	−226.280			−0.472400			−0.236200	+	0.971704i	$0.575902\pi$
$480$		0			0
$481$	871.705			1.81228
$482$		0			0
$483$		−	530.141i		−	1.09760i
$484$		0			0
$485$			337.637i			0.696159i
$486$		0			0
$487$		−	899.572i		−	1.84717i	−0.383394	−	0.923585i	$-0.625245\pi$
$487$		−	899.572i		−	1.84717i	0.383394	−	0.923585i	$-0.374755\pi$
$488$		0			0
$489$			420.765i			0.860460i
$490$		0			0
$491$	405.337			0.825533			0.412767	−	0.910837i	$-0.364562\pi$
$491$	405.337			0.825533			0.412767	+	0.910837i	$0.364562\pi$
$492$		0			0
$493$		−	510.818i		−	1.03614i
$494$		0			0
$495$	−429.009			−0.866686
$496$		0			0
$497$			182.419i			0.367041i
$498$		0			0
$499$	−194.024			−0.388825			−0.194413	−	0.980920i	$-0.562280\pi$
$499$	−194.024			−0.388825			−0.194413	+	0.980920i	$0.562280\pi$
$500$		0			0
$501$	363.880			0.726308
$502$		0			0
$503$	−623.065			−1.23870			−0.619348	−	0.785116i	$-0.712603\pi$
$503$	−623.065			−1.23870			−0.619348	+	0.785116i	$0.712603\pi$
$504$		0			0
$505$	350.100			0.693267
$506$		0			0
$507$		−	118.821i		−	0.234360i
$508$		0			0
$509$		−	321.176i		−	0.630995i	−0.948926	−	0.315497i	$-0.897829\pi$
$509$		−	321.176i		−	0.630995i	0.948926	−	0.315497i	$-0.102171\pi$
$510$		0			0
$511$	−43.1211			−0.0843857
$512$		0			0
$513$	30.7782	+	20.6210i	0.0599964	+	0.0401968i
$514$		0			0
$515$			172.960i			0.335845i
$516$		0			0
$517$	416.051			0.804741
$518$		0			0
$519$	−96.0618			−0.185090
$520$		0			0
$521$			535.331i			1.02751i	0.857938	+	0.513753i	$0.171745\pi$
$521$			535.331i			1.02751i	−0.857938	+	0.513753i	$0.828255\pi$
$522$		0			0
$523$		−	873.472i		−	1.67012i	−0.550160	−	0.835059i	$-0.685433\pi$
$523$		−	873.472i		−	1.67012i	0.550160	−	0.835059i	$-0.314567\pi$
$524$		0			0
$525$			248.709i			0.473732i
$526$		0			0
$527$		−	690.034i		−	1.30936i
$528$		0			0
$529$	1212.30			2.29168
$530$		0			0
$531$			182.215i			0.343155i
$532$		0			0
$533$	−283.672			−0.532218
$534$		0			0
$535$			325.345i			0.608122i
$536$		0			0
$537$	−312.701			−0.582311
$538$		0			0
$539$	−784.380			−1.45525
$540$		0			0
$541$	302.002			0.558229			0.279114	−	0.960258i	$-0.409959\pi$
$541$	302.002			0.558229			0.279114	+	0.960258i	$0.409959\pi$
$542$		0			0
$543$	1504.76			2.77120
$544$		0			0
$545$			99.0162i			0.181681i
$546$		0			0
$547$		−	1004.10i		−	1.83564i	−0.396994	−	0.917821i	$-0.629946\pi$
$547$		−	1004.10i		−	1.83564i	0.396994	−	0.917821i	$-0.370054\pi$
$548$		0			0
$549$	−46.9319			−0.0854861
$550$		0			0
$551$	−362.012	−	242.543i	−0.657009	−	0.440188i
$552$		0			0
$553$		−	335.209i		−	0.606164i
$554$		0			0
$555$	621.858			1.12046
$556$		0			0
$557$	449.650			0.807271			0.403635	−	0.914920i	$-0.367746\pi$
$557$	449.650			0.807271			0.403635	+	0.914920i	$0.367746\pi$
$558$		0			0
$559$		−	746.552i		−	1.33551i
$560$		0			0
$561$		−	1864.42i		−	3.32339i
$562$		0			0
$563$			829.613i			1.47356i	0.676133	+	0.736779i	$0.263654\pi$
$563$			829.613i			1.47356i	−0.676133	+	0.736779i	$0.736346\pi$
$564$		0			0
$565$		−	254.699i		−	0.450795i
$566$		0			0
$567$	226.859			0.400104
$568$		0			0
$569$		−	46.4574i		−	0.0816475i	−0.999166	−	0.0408238i	$-0.987002\pi$
$569$		−	46.4574i		−	0.0816475i	0.999166	−	0.0408238i	$-0.0129982\pi$
$570$		0			0
$571$	−788.963			−1.38172			−0.690860	−	0.722988i	$-0.742768\pi$
$571$	−788.963			−1.38172			−0.690860	+	0.722988i	$0.742768\pi$
$572$		0			0
$573$		−	463.740i		−	0.809320i
$574$		0			0
$575$	−816.910			−1.42071
$576$		0			0
$577$	−25.1604			−0.0436055			−0.0218028	−	0.999762i	$-0.506941\pi$
$577$	−25.1604			−0.0436055			−0.0218028	+	0.999762i	$0.506941\pi$
$578$		0			0
$579$	1149.71			1.98569
$580$		0			0
$581$	144.872			0.249349
$582$		0			0
$583$		−	243.531i		−	0.417720i
$584$		0			0
$585$		−	308.748i		−	0.527774i
$586$		0			0
$587$	−779.591			−1.32809			−0.664047	−	0.747691i	$-0.731162\pi$
$587$	−779.591			−1.32809			−0.664047	+	0.747691i	$0.731162\pi$
$588$		0			0
$589$	−489.021	−	327.638i	−0.830256	−	0.556261i
$590$		0			0
$591$			1248.85i			2.11312i
$592$		0			0
$593$	−657.878			−1.10941			−0.554703	−	0.832048i	$-0.687168\pi$
$593$	−657.878			−1.10941			−0.554703	+	0.832048i	$0.687168\pi$
$594$		0			0
$595$	153.400			0.257816
$596$		0			0
$597$			1298.60i			2.17521i
$598$		0			0
$599$			761.121i			1.27065i	0.772244	+	0.635326i	$0.219134\pi$
$599$			761.121i			1.27065i	−0.772244	+	0.635326i	$0.780866\pi$
$600$		0			0
$601$			798.347i			1.32837i	0.747570	+	0.664183i	$0.231220\pi$
$601$			798.347i			1.32837i	−0.747570	+	0.664183i	$0.768780\pi$
$602$		0			0
$603$			557.303i			0.924218i
$604$		0			0
$605$	602.465			0.995810
$606$		0			0
$607$			752.935i			1.24042i	0.784436	+	0.620210i	$0.212953\pi$
$607$			752.935i			1.24042i	−0.784436	+	0.620210i	$0.787047\pi$
$608$		0			0
$609$	291.367			0.478436
$610$		0			0
$611$			299.422i			0.490053i
$612$		0			0
$613$	536.480			0.875171			0.437586	−	0.899177i	$-0.355834\pi$
$613$	536.480			0.875171			0.437586	+	0.899177i	$0.355834\pi$
$614$		0			0
$615$	−202.366			−0.329051
$616$		0			0
$617$	−317.064			−0.513879			−0.256940	−	0.966427i	$-0.582714\pi$
$617$	−317.064			−0.513879			−0.256940	+	0.966427i	$0.582714\pi$
$618$		0			0
$619$	351.867			0.568445			0.284222	−	0.958758i	$-0.408265\pi$
$619$	351.867			0.568445			0.284222	+	0.958758i	$0.408265\pi$
$620$		0			0
$621$		−	81.3660i		−	0.131024i
$622$		0			0
$623$			82.2617i			0.132041i
$624$		0			0
$625$	247.659			0.396254
$626$		0			0
$627$	−1321.30	−	885.254i	−2.10734	−	1.41189i
$628$		0			0
$629$			1384.50i			2.20111i
$630$		0			0
$631$	371.411			0.588606			0.294303	−	0.955712i	$-0.404912\pi$
$631$	371.411			0.588606			0.294303	+	0.955712i	$0.404912\pi$
$632$		0			0
$633$	−316.066			−0.499315
$634$		0			0
$635$		−	93.7029i		−	0.147564i
$636$		0			0
$637$		−	564.500i		−	0.886186i
$638$		0			0
$639$			583.137i			0.912578i
$640$		0			0
$641$		−	256.605i		−	0.400319i	−0.979763	−	0.200160i	$-0.935854\pi$
$641$		−	256.605i		−	0.400319i	0.979763	−	0.200160i	$-0.0641461\pi$
$642$		0			0
$643$	−759.280			−1.18084			−0.590420	−	0.807096i	$-0.701038\pi$
$643$	−759.280			−1.18084			−0.590420	+	0.807096i	$0.701038\pi$
$644$		0			0
$645$		−	532.577i		−	0.825700i
$646$		0			0
$647$	398.410			0.615780			0.307890	−	0.951422i	$-0.400377\pi$
$647$	398.410			0.615780			0.307890	+	0.951422i	$0.400377\pi$
$648$		0			0
$649$		−	375.573i		−	0.578695i
$650$		0			0
$651$	393.591			0.604594
$652$		0			0
$653$	−453.692			−0.694781			−0.347391	−	0.937721i	$-0.612932\pi$
$653$	−453.692			−0.694781			−0.347391	+	0.937721i	$0.612932\pi$
$654$		0			0
$655$	152.813			0.233302
$656$		0			0
$657$	−137.845			−0.209809
$658$		0			0
$659$		−	193.340i		−	0.293383i	−0.989182	−	0.146692i	$-0.953138\pi$
$659$		−	193.340i		−	0.293383i	0.989182	−	0.146692i	$-0.0468625\pi$
$660$		0			0
$661$		−	902.276i		−	1.36502i	−0.730878	−	0.682508i	$-0.760889\pi$
$661$		−	902.276i		−	1.36502i	0.730878	−	0.682508i	$-0.239111\pi$
$662$		0			0
$663$	1341.78			2.02380
$664$		0			0
$665$	72.8366	−	108.713i	0.109529	−	0.163479i
$666$		0			0
$667$			957.024i			1.43482i
$668$		0			0
$669$	968.023			1.44697
$670$		0			0
$671$	96.7335			0.144163
$672$		0			0
$673$			486.217i			0.722463i	0.932476	+	0.361231i	$0.117644\pi$
$673$			486.217i			0.722463i	−0.932476	+	0.361231i	$0.882356\pi$
$674$		0			0
$675$			38.1719i			0.0565509i
$676$		0			0
$677$		−	1034.98i		−	1.52877i	−0.644759	−	0.764386i	$-0.723042\pi$
$677$		−	1034.98i		−	1.52877i	0.644759	−	0.764386i	$-0.276958\pi$
$678$		0			0
$679$		−	428.771i		−	0.631474i
$680$		0			0
$681$	83.3626			0.122412
$682$		0			0
$683$			100.984i			0.147853i	0.997264	+	0.0739267i	$0.0235531\pi$
$683$			100.984i			0.147853i	−0.997264	+	0.0739267i	$0.976447\pi$
$684$		0			0
$685$	393.417			0.574332
$686$		0			0
$687$			1226.47i			1.78526i
$688$		0			0
$689$	175.263			0.254374
$690$		0			0
$691$	774.181			1.12038			0.560189	−	0.828365i	$-0.310728\pi$
$691$	774.181			1.12038			0.560189	+	0.828365i	$0.310728\pi$
$692$		0			0
$693$	544.806			0.786156
$694$		0			0
$695$	320.302			0.460867
$696$		0			0
$697$		−	450.546i		−	0.646407i
$698$		0			0
$699$		−	1542.45i		−	2.20665i
$700$		0			0
$701$	1010.71			1.44181			0.720906	−	0.693033i	$-0.243726\pi$
$701$	1010.71			1.44181			0.720906	+	0.693033i	$0.243726\pi$
$702$		0			0
$703$	981.180	+	657.378i	1.39570	+	0.935104i
$704$		0			0
$705$			213.602i			0.302982i
$706$		0			0
$707$	−444.598			−0.628851
$708$		0			0
$709$	−542.239			−0.764794			−0.382397	−	0.923998i	$-0.624901\pi$
$709$	−542.239			−0.764794			−0.382397	+	0.923998i	$0.624901\pi$
$710$		0			0
$711$		−	1071.56i		−	1.50711i
$712$		0			0
$713$			1292.79i			1.81317i
$714$		0			0
$715$			636.375i			0.890035i
$716$		0			0
$717$			783.534i			1.09280i
$718$		0			0
$719$	−105.476			−0.146699			−0.0733493	−	0.997306i	$-0.523369\pi$
$719$	−105.476			−0.146699			−0.0733493	+	0.997306i	$0.523369\pi$
$720$		0			0
$721$		−	219.645i		−	0.304639i
$722$		0			0
$723$	514.982			0.712285
$724$		0			0
$725$		−	448.977i		−	0.619278i
$726$		0			0
$727$	−859.614			−1.18241			−0.591206	−	0.806520i	$-0.701348\pi$
$727$	−859.614			−1.18241			−0.591206	+	0.806520i	$0.701348\pi$
$728$		0			0
$729$	800.584			1.09820
$730$		0			0
$731$	1185.72			1.62205
$732$		0			0
$733$	1170.01			1.59619			0.798097	−	0.602528i	$-0.205840\pi$
$733$	1170.01			1.59619			0.798097	+	0.602528i	$0.205840\pi$
$734$		0			0
$735$		−	402.704i		−	0.547897i
$736$		0			0
$737$		−	1148.68i		−	1.55859i
$738$		0			0
$739$	742.350			1.00453			0.502267	−	0.864713i	$-0.332500\pi$
$739$	742.350			1.00453			0.502267	+	0.864713i	$0.332500\pi$
$740$		0			0
$741$	637.096	−	950.909i	0.859779	−	1.28328i
$742$		0			0
$743$		−	1306.08i		−	1.75784i	−0.476967	−	0.878921i	$-0.658264\pi$
$743$		−	1306.08i		−	1.75784i	0.476967	−	0.878921i	$-0.341736\pi$
$744$		0			0
$745$	−116.487			−0.156358
$746$		0			0
$747$	463.109			0.619959
$748$		0			0
$749$		−	413.161i		−	0.551617i
$750$		0			0
$751$			324.212i			0.431707i	0.976426	+	0.215853i	$0.0692533\pi$
$751$			324.212i			0.431707i	−0.976426	+	0.215853i	$0.930747\pi$
$752$		0			0
$753$			1930.13i			2.56325i
$754$		0			0
$755$			476.853i			0.631593i
$756$		0			0
$757$	−1118.35			−1.47734			−0.738672	−	0.674065i	$-0.764547\pi$
$757$	−1118.35			−1.47734			−0.738672	+	0.674065i	$0.764547\pi$
$758$		0			0
$759$			3493.02i			4.60214i
$760$		0			0
$761$	696.142			0.914772			0.457386	−	0.889268i	$-0.348786\pi$
$761$	696.142			0.914772			0.457386	+	0.889268i	$0.348786\pi$
$762$		0			0
$763$		−	125.742i		−	0.164800i
$764$		0			0
$765$	490.373			0.641010
$766$		0			0
$767$	270.291			0.352400
$768$		0			0
$769$	55.8346			0.0726067			0.0363034	−	0.999341i	$-0.488442\pi$
$769$	55.8346			0.0726067			0.0363034	+	0.999341i	$0.488442\pi$
$770$		0			0
$771$	−2015.48			−2.61411
$772$		0			0
$773$		−	1113.47i		−	1.44045i	−0.693740	−	0.720226i	$-0.744038\pi$
$773$		−	1113.47i		−	1.44045i	0.693740	−	0.720226i	$-0.255962\pi$
$774$		0			0
$775$		−	606.496i		−	0.782575i
$776$		0			0
$777$	−789.708			−1.01635
$778$		0			0
$779$	−319.298	−	213.925i	−0.409882	−	0.274615i
$780$		0			0
$781$		−	1201.93i		−	1.53897i
$782$		0			0
$783$	44.7190			0.0571124
$784$		0			0
$785$	134.185			0.170936
$786$		0			0
$787$		−	104.121i		−	0.132301i	−0.997810	−	0.0661504i	$-0.978928\pi$
$787$		−	104.121i		−	0.132301i	0.997810	−	0.0661504i	$-0.0210717\pi$
$788$		0			0
$789$		−	1138.86i		−	1.44342i
$790$		0			0
$791$			323.447i			0.408908i
$792$		0			0
$793$			69.6169i			0.0877892i
$794$		0			0
$795$	125.030			0.157270
$796$		0			0
$797$		−	292.129i		−	0.366536i	−0.983063	−	0.183268i	$-0.941332\pi$
$797$		−	292.129i		−	0.366536i	0.983063	−	0.183268i	$-0.0586677\pi$
$798$		0			0
$799$	−475.561			−0.595196
$800$		0			0
$801$			262.965i			0.328296i
$802$		0			0
$803$	284.118			0.353821
$804$		0			0
$805$	−287.397			−0.357015
$806$		0			0
$807$	−1303.74			−1.61553
$808$		0			0
$809$	−304.463			−0.376344			−0.188172	−	0.982136i	$-0.560256\pi$
$809$	−304.463			−0.376344			−0.188172	+	0.982136i	$0.560256\pi$
$810$		0			0
$811$		−	816.016i		−	1.00618i	−0.864233	−	0.503092i	$-0.832196\pi$
$811$		−	816.016i		−	1.00618i	0.864233	−	0.503092i	$-0.167804\pi$
$812$		0			0
$813$			860.452i			1.05837i
$814$		0			0
$815$	228.103			0.279881
$816$		0			0
$817$	562.997	−	840.310i	0.689102	−	1.02853i
$818$		0			0
$819$			392.084i			0.478735i
$820$		0			0
$821$	721.591			0.878917			0.439458	−	0.898263i	$-0.355170\pi$
$821$	721.591			0.878917			0.439458	+	0.898263i	$0.355170\pi$
$822$		0			0
$823$	−414.065			−0.503117			−0.251558	−	0.967842i	$-0.580943\pi$
$823$	−414.065			−0.503117			−0.251558	+	0.967842i	$0.580943\pi$
$824$		0			0
$825$		−	1638.71i		−	1.98631i
$826$		0			0
$827$			100.878i			0.121981i	0.998138	+	0.0609903i	$0.0194259\pi$
$827$			100.878i			0.121981i	−0.998138	+	0.0609903i	$0.980574\pi$
$828$		0			0
$829$			621.755i			0.750006i	0.927024	+	0.375003i	$0.122358\pi$
$829$			621.755i			0.750006i	−0.927024	+	0.375003i	$0.877642\pi$
$830$		0			0
$831$		−	687.781i		−	0.827655i
$832$		0			0
$833$	896.575			1.07632
$834$		0			0
$835$		−	197.265i		−	0.236245i
$836$		0			0
$837$	60.4082			0.0721723
$838$		0			0
$839$		−	115.138i		−	0.137232i	−0.997643	−	0.0686160i	$-0.978142\pi$
$839$		−	115.138i		−	0.137232i	0.997643	−	0.0686160i	$-0.0218583\pi$
$840$		0			0
$841$	315.016			0.374573
$842$		0			0
$843$	−1645.33			−1.95176
$844$		0			0
$845$	−64.4144			−0.0762300
$846$		0			0
$847$	−765.080			−0.903282
$848$		0			0
$849$			973.956i			1.14718i
$850$		0			0
$851$		−	2593.87i		−	3.04803i
$852$		0			0
$853$	762.387			0.893772			0.446886	−	0.894591i	$-0.352533\pi$
$853$	762.387			0.893772			0.446886	+	0.894591i	$0.352533\pi$
$854$		0			0
$855$	232.836	−	347.523i	0.272323	−	0.406460i
$856$		0			0
$857$			537.227i			0.626869i	0.949610	+	0.313435i	$0.101480\pi$
$857$			537.227i			0.626869i	−0.949610	+	0.313435i	$0.898520\pi$
$858$		0			0
$859$	−645.182			−0.751084			−0.375542	−	0.926805i	$-0.622544\pi$
$859$	−645.182			−0.751084			−0.375542	+	0.926805i	$0.622544\pi$
$860$		0			0
$861$	256.988			0.298477
$862$		0			0
$863$		−	892.080i		−	1.03370i	−0.856077	−	0.516848i	$-0.827105\pi$
$863$		−	892.080i		−	1.03370i	0.856077	−	0.516848i	$-0.172895\pi$
$864$		0			0
$865$			52.0765i			0.0602041i
$866$		0			0
$867$			889.615i			1.02608i
$868$		0			0
$869$			2208.64i			2.54159i
$870$		0			0
$871$	826.681			0.949118
$872$		0			0
$873$		−	1370.65i		−	1.57004i
$874$		0			0
$875$	307.010			0.350869
$876$		0			0
$877$			873.903i			0.996469i	0.867042	+	0.498235i	$0.166018\pi$
$877$			873.903i			0.996469i	−0.867042	+	0.498235i	$0.833982\pi$
$878$		0			0
$879$	1847.03			2.10129
$880$		0			0
$881$	−687.622			−0.780502			−0.390251	−	0.920709i	$-0.627612\pi$
$881$	−687.622			−0.780502			−0.390251	+	0.920709i	$0.627612\pi$
$882$		0			0
$883$	−1305.56			−1.47855			−0.739276	−	0.673403i	$-0.764832\pi$
$883$	−1305.56			−1.47855			−0.739276	+	0.673403i	$0.764832\pi$
$884$		0			0
$885$	192.821			0.217876
$886$		0			0
$887$		−	57.7531i		−	0.0651106i	−0.999470	−	0.0325553i	$-0.989635\pi$
$887$		−	57.7531i		−	0.0651106i	0.999470	−	0.0325553i	$-0.0103645\pi$
$888$		0			0
$889$			118.995i			0.133852i
$890$		0			0
$891$	−1494.74			−1.67760
$892$		0			0
$893$	−225.803	+	337.026i	−0.252859	+	0.377409i
$894$		0			0
$895$			169.520i			0.189408i
$896$		0			0
$897$	−2513.85			−2.80250
$898$		0			0
$899$	−710.520			−0.790345
$900$		0			0
$901$			278.364i			0.308951i
$902$		0			0
$903$			676.328i			0.748979i
$904$		0			0
$905$		−	815.753i		−	0.901385i
$906$		0			0
$907$		−	837.489i		−	0.923362i	−0.887046	−	0.461681i	$-0.847247\pi$
$907$		−	837.489i		−	0.923362i	0.887046	−	0.461681i	$-0.152753\pi$
$908$		0			0
$909$	−1421.24			−1.56352
$910$		0			0
$911$			662.892i			0.727653i	0.931467	+	0.363826i	$0.118530\pi$
$911$			662.892i			0.727653i	−0.931467	+	0.363826i	$0.881470\pi$
$912$		0			0
$913$	−954.537			−1.04550
$914$		0			0
$915$			49.6634i			0.0542769i
$916$		0			0
$917$	−194.060			−0.211625
$918$		0			0
$919$	1000.18			1.08834			0.544169	−	0.838976i	$-0.316845\pi$
$919$	1000.18			1.08834			0.544169	+	0.838976i	$0.316845\pi$
$920$		0			0
$921$	894.699			0.971443
$922$		0			0
$923$	865.003			0.937164
$924$		0			0
$925$			1216.88i			1.31555i
$926$		0			0
$927$		−	702.137i		−	0.757429i
$928$		0			0
$929$	1504.37			1.61934			0.809671	−	0.586885i	$-0.199646\pi$
$929$	1504.37			1.61934			0.809671	+	0.586885i	$0.199646\pi$
$930$		0			0
$931$	425.706	−	635.395i	0.457257	−	0.682486i
$932$		0			0
$933$		−	580.701i		−	0.622402i
$934$		0			0
$935$	−1010.73			−1.08100
$936$		0			0
$937$	−961.503			−1.02615			−0.513075	−	0.858344i	$-0.671494\pi$
$937$	−961.503			−1.02615			−0.513075	+	0.858344i	$0.671494\pi$
$938$		0			0
$939$		−	1770.08i		−	1.88506i
$940$		0			0
$941$			681.880i			0.724633i	0.932055	+	0.362317i	$0.118014\pi$
$941$			681.880i			0.724633i	−0.932055	+	0.362317i	$0.881986\pi$
$942$		0			0
$943$			844.103i			0.895126i
$944$		0			0
$945$			13.4293i			0.0142109i
$946$		0			0
$947$	1441.83			1.52253			0.761263	−	0.648443i	$-0.224580\pi$
$947$	1441.83			1.52253			0.761263	+	0.648443i	$0.224580\pi$
$948$		0			0
$949$			204.473i			0.215462i
$950$		0			0
$951$	1661.29			1.74688
$952$		0			0
$953$		−	495.317i		−	0.519745i	−0.965643	−	0.259872i	$-0.916320\pi$
$953$		−	495.317i		−	0.519745i	0.965643	−	0.259872i	$-0.0836805\pi$
$954$		0			0
$955$	−251.400			−0.263247
$956$		0			0
$957$	−1919.78			−2.00604
$958$		0			0
$959$	−499.607			−0.520967
$960$		0			0
$961$	1.20041			0.00124912
$962$		0			0
$963$		−	1320.75i		−	1.37149i
$964$		0			0
$965$		−	623.276i		−	0.645882i
$966$		0			0
$967$	−1020.51			−1.05534			−0.527670	−	0.849449i	$-0.676934\pi$
$967$	−1020.51			−1.05534			−0.527670	+	0.849449i	$0.676934\pi$
$968$		0			0
$969$	1510.29	+	1011.88i	1.55861	+	1.04425i
$970$		0			0
$971$			563.750i			0.580587i	0.956938	+	0.290293i	$0.0937529\pi$
$971$			563.750i			0.580587i	−0.956938	+	0.290293i	$0.906247\pi$
$972$		0			0
$973$	−406.757			−0.418045
$974$		0			0
$975$	1179.34			1.20958
$976$		0			0
$977$			786.557i			0.805073i	0.915404	+	0.402537i	$0.131871\pi$
$977$			786.557i			0.805073i	−0.915404	+	0.402537i	$0.868129\pi$
$978$		0			0
$979$		−	542.010i		−	0.553636i
$980$		0			0
$981$		−	401.959i		−	0.409744i
$982$		0			0
$983$		−	337.555i		−	0.343393i	−0.985150	−	0.171696i	$-0.945075\pi$
$983$		−	337.555i		−	0.343393i	0.985150	−	0.171696i	$-0.0549248\pi$
$984$		0			0
$985$	677.021			0.687331
$986$		0			0
$987$		−	271.257i		−	0.274830i
$988$		0			0
$989$	−2221.46			−2.24617
$990$		0			0
$991$			1048.22i			1.05774i	0.848703	+	0.528870i	$0.177384\pi$
$991$			1048.22i			1.05774i	−0.848703	+	0.528870i	$0.822616\pi$
$992$		0			0
$993$	−617.176			−0.621527
$994$		0			0
$995$	703.991			0.707529
$996$		0			0
$997$	1041.54			1.04467			0.522335	−	0.852741i	$-0.325061\pi$
$997$	1041.54			1.04467			0.522335	+	0.852741i	$0.325061\pi$
$998$		0			0
$999$	−121.204			−0.121325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.3.e.b.417.17	yes	20
4.3	odd	2	inner	608.3.e.b.417.4	yes	20
8.3	odd	2		1216.3.e.p.1025.18		20
8.5	even	2		1216.3.e.p.1025.3		20
19.18	odd	2	inner	608.3.e.b.417.3	✓	20
76.75	even	2	inner	608.3.e.b.417.18	yes	20
152.37	odd	2		1216.3.e.p.1025.17		20
152.75	even	2		1216.3.e.p.1025.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.3	✓	20	19.18	odd	2	inner
608.3.e.b.417.4	yes	20	4.3	odd	2	inner
608.3.e.b.417.17	yes	20	1.1	even	1	trivial
608.3.e.b.417.18	yes	20	76.75	even	2	inner
1216.3.e.p.1025.3		20	8.5	even	2
1216.3.e.p.1025.4		20	152.75	even	2
1216.3.e.p.1025.17		20	152.37	odd	2
1216.3.e.p.1025.18		20	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	608.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.29580i q^{3} +2.32882 q^{5} -2.95740 q^{7} -9.45390 q^{9} +19.4859 q^{11} +14.0235i q^{13} +10.0041i q^{15} -22.2731 q^{17} +(-10.5756 + 15.7847i) q^{19} -12.7044i q^{21} +41.7289 q^{23} -19.5766 q^{25} -1.94987i q^{27} +22.9343i q^{29} +30.9806i q^{31} +83.7075i q^{33} -6.88725 q^{35} -62.1601i q^{37} -60.2423 q^{39} +20.2283i q^{41} -53.2356 q^{43} -22.0164 q^{45} +21.3514 q^{47} -40.2538 q^{49} -95.6807i q^{51} -12.4978i q^{53} +45.3791 q^{55} +(-67.8080 - 45.4305i) q^{57} -19.2741i q^{59} +4.96429 q^{61} +27.9590 q^{63} +32.6583i q^{65} -58.9495i q^{67} +179.259i q^{69} -61.6822i q^{71} +14.5807 q^{73} -84.0972i q^{75} -57.6276 q^{77} +113.346i q^{79} -76.7089 q^{81} -48.9861 q^{83} -51.8699 q^{85} -98.5213 q^{87} -27.8155i q^{89} -41.4733i q^{91} -133.087 q^{93} +(-24.6285 + 36.7597i) q^{95} +144.982i q^{97} -184.218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} + 64 q^{45} + 332 q^{49} + 88 q^{57} - 32 q^{61} - 152 q^{73} + 360 q^{77} - 476 q^{81} - 552 q^{85} + 336 q^{93}+O(q^{100}) \) 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 + 64 * q^45 + 332 * q^49 + 88 * q^57 - 32 * q^61 - 152 * q^73 + 360 * q^77 - 476 * q^81 - 552 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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