LMFDB - Embedded newform 4012.2.b.b.237.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(237,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.237");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			237.13
Character	$\chi$	$=$	4012.237
Dual form			4012.2.b.b.237.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.58213i q^{3} -3.58424i q^{5} -1.89303i q^{7} +0.496852 q^{9} +O(q^{10})$	$q-1.58213i q^{3} -3.58424i q^{5} -1.89303i q^{7} +0.496852 q^{9} +1.56283i q^{11} +4.45085 q^{13} -5.67074 q^{15} +(4.09490 + 0.481483i) q^{17} -5.20490 q^{19} -2.99502 q^{21} +3.19080i q^{23} -7.84674 q^{25} -5.53249i q^{27} -5.05381i q^{29} -3.34568i q^{31} +2.47260 q^{33} -6.78505 q^{35} -10.4521i q^{37} -7.04183i q^{39} +9.35341i q^{41} -7.48993 q^{43} -1.78083i q^{45} -8.79104 q^{47} +3.41645 q^{49} +(0.761771 - 6.47867i) q^{51} -2.62104 q^{53} +5.60154 q^{55} +8.23485i q^{57} +1.00000 q^{59} -12.4686i q^{61} -0.940554i q^{63} -15.9529i q^{65} +1.37777 q^{67} +5.04827 q^{69} -9.38491i q^{71} +0.343768i q^{73} +12.4146i q^{75} +2.95847 q^{77} -0.0938078i q^{79} -7.26258 q^{81} -5.61501 q^{83} +(1.72575 - 14.6771i) q^{85} -7.99581 q^{87} +7.62341 q^{89} -8.42557i q^{91} -5.29331 q^{93} +18.6556i q^{95} +17.3041i q^{97} +0.776494i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 54 q^{9}+O(q^{10}) $ 46 * q - 54 * q^9	$ 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) $ 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times$.

$n$	$2007$	$3129$	$3777$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		−	1.58213i		−	0.913446i	−0.889609	−	0.456723i	$-0.849023\pi$
$3$		−	1.58213i		−	0.913446i	0.889609	−	0.456723i	$-0.150977\pi$
$4$		0			0
$5$		−	3.58424i		−	1.60292i	−0.598049	−	0.801459i	$-0.704057\pi$
$5$		−	3.58424i		−	1.60292i	0.598049	−	0.801459i	$-0.295943\pi$
$6$		0			0
$7$		−	1.89303i		−	0.715497i	−0.933818	−	0.357748i	$-0.883545\pi$
$7$		−	1.89303i		−	0.715497i	0.933818	−	0.357748i	$-0.116455\pi$
$8$		0			0
$9$	0.496852			0.165617
$10$		0			0
$11$			1.56283i			0.471210i	0.971849	+	0.235605i	$0.0757072\pi$
$11$			1.56283i			0.471210i	−0.971849	+	0.235605i	$0.924293\pi$
$12$		0			0
$13$	4.45085			1.23444			0.617221	−	0.786790i	$-0.288258\pi$
$13$	4.45085			1.23444			0.617221	+	0.786790i	$0.288258\pi$
$14$		0			0
$15$	−5.67074			−1.46418
$16$		0			0
$17$	4.09490	+	0.481483i	0.993158	+	0.116777i
$17$	4.09490	+	0.481483i	0.993158	+	0.116777i
$18$		0			0
$19$	−5.20490			−1.19409			−0.597043	−	0.802209i	$-0.703658\pi$
$19$	−5.20490			−1.19409			−0.597043	+	0.802209i	$0.703658\pi$
$20$		0			0
$21$	−2.99502			−0.653567
$22$		0			0
$23$			3.19080i			0.665327i	0.943046	+	0.332663i	$0.107947\pi$
$23$			3.19080i			0.665327i	−0.943046	+	0.332663i	$0.892053\pi$
$24$		0			0
$25$	−7.84674			−1.56935
$26$		0			0
$27$		−	5.53249i		−	1.06473i
$28$		0			0
$29$		−	5.05381i		−	0.938469i	−0.883073	−	0.469235i	$-0.844530\pi$
$29$		−	5.05381i		−	0.938469i	0.883073	−	0.469235i	$-0.155470\pi$
$30$		0			0
$31$		−	3.34568i		−	0.600901i	−0.953797	−	0.300451i	$-0.902863\pi$
$31$		−	3.34568i		−	0.600901i	0.953797	−	0.300451i	$-0.0971371\pi$
$32$		0			0
$33$	2.47260			0.430425
$34$		0			0
$35$	−6.78505			−1.14688
$36$		0			0
$37$		−	10.4521i		−	1.71832i	−0.511708	−	0.859159i	$-0.670987\pi$
$37$		−	10.4521i		−	1.71832i	0.511708	−	0.859159i	$-0.329013\pi$
$38$		0			0
$39$		−	7.04183i		−	1.12760i
$40$		0			0
$41$			9.35341i			1.46076i	0.683042	+	0.730379i	$0.260656\pi$
$41$			9.35341i			1.46076i	−0.683042	+	0.730379i	$0.739344\pi$
$42$		0			0
$43$	−7.48993			−1.14220			−0.571102	−	0.820879i	$-0.693484\pi$
$43$	−7.48993			−1.14220			−0.571102	+	0.820879i	$0.693484\pi$
$44$		0			0
$45$		−	1.78083i		−	0.265471i
$46$		0			0
$47$	−8.79104			−1.28230			−0.641152	−	0.767413i	$-0.721543\pi$
$47$	−8.79104			−1.28230			−0.641152	+	0.767413i	$0.721543\pi$
$48$		0			0
$49$	3.41645			0.488064
$50$		0			0
$51$	0.761771	−	6.47867i	0.106669	−	0.907196i
$52$		0			0
$53$	−2.62104			−0.360027			−0.180014	−	0.983664i	$-0.557614\pi$
$53$	−2.62104			−0.360027			−0.180014	+	0.983664i	$0.557614\pi$
$54$		0			0
$55$	5.60154			0.755312
$56$		0			0
$57$			8.23485i			1.09073i
$58$		0			0
$59$	1.00000			0.130189
$59$	1.00000			0.130189
$60$		0			0
$61$		−	12.4686i		−	1.59644i	−0.602365	−	0.798221i	$-0.705775\pi$
$61$		−	12.4686i		−	1.59644i	0.602365	−	0.798221i	$-0.294225\pi$
$62$		0			0
$63$		−	0.940554i		−	0.118499i
$64$		0			0
$65$		−	15.9529i		−	1.97871i
$66$		0			0
$67$	1.37777			0.168322			0.0841610	−	0.996452i	$-0.473179\pi$
$67$	1.37777			0.168322			0.0841610	+	0.996452i	$0.473179\pi$
$68$		0			0
$69$	5.04827			0.607740
$70$		0			0
$71$		−	9.38491i		−	1.11378i	−0.830585	−	0.556892i	$-0.811994\pi$
$71$		−	9.38491i		−	1.11378i	0.830585	−	0.556892i	$-0.188006\pi$
$72$		0			0
$73$			0.343768i			0.0402350i	0.999798	+	0.0201175i	$0.00640403\pi$
$73$			0.343768i			0.0402350i	−0.999798	+	0.0201175i	$0.993596\pi$
$74$		0			0
$75$			12.4146i			1.43351i
$76$		0			0
$77$	2.95847			0.337149
$78$		0			0
$79$		−	0.0938078i		−	0.0105542i	−0.999986	−	0.00527710i	$-0.998320\pi$
$79$		−	0.0938078i		−	0.0105542i	0.999986	−	0.00527710i	$-0.00167976\pi$
$80$		0			0
$81$	−7.26258			−0.806954
$82$		0			0
$83$	−5.61501			−0.616327			−0.308164	−	0.951333i	$-0.599714\pi$
$83$	−5.61501			−0.616327			−0.308164	+	0.951333i	$0.599714\pi$
$84$		0			0
$85$	1.72575	−	14.6771i	0.187184	−	1.59195i
$86$		0			0
$87$	−7.99581			−0.857241
$88$		0			0
$89$	7.62341			0.808080			0.404040	−	0.914741i	$-0.367606\pi$
$89$	7.62341			0.808080			0.404040	+	0.914741i	$0.367606\pi$
$90$		0			0
$91$		−	8.42557i		−	0.883240i
$92$		0			0
$93$	−5.29331			−0.548891
$94$		0			0
$95$			18.6556i			1.91402i
$96$		0			0
$97$			17.3041i			1.75696i	0.477778	+	0.878481i	$0.341442\pi$
$97$			17.3041i			1.75696i	−0.477778	+	0.878481i	$0.658558\pi$
$98$		0			0
$99$			0.776494i			0.0780406i
$100$		0			0
$101$	−6.25828			−0.622722			−0.311361	−	0.950292i	$-0.600785\pi$
$101$	−6.25828			−0.622722			−0.311361	+	0.950292i	$0.600785\pi$
$102$		0			0
$103$	−5.23906			−0.516220			−0.258110	−	0.966116i	$-0.583100\pi$
$103$	−5.23906			−0.516220			−0.258110	+	0.966116i	$0.583100\pi$
$104$		0			0
$105$			10.7349i			1.04762i
$106$		0			0
$107$		−	15.5877i		−	1.50692i	−0.657493	−	0.753461i	$-0.728383\pi$
$107$		−	15.5877i		−	1.50692i	0.657493	−	0.753461i	$-0.271617\pi$
$108$		0			0
$109$			10.3432i			0.990702i	0.868693	+	0.495351i	$0.164960\pi$
$109$			10.3432i			0.990702i	−0.868693	+	0.495351i	$0.835040\pi$
$110$		0			0
$111$	−16.5367			−1.56959
$112$		0			0
$113$			2.39401i			0.225209i	0.993640	+	0.112605i	$0.0359193\pi$
$113$			2.39401i			0.225209i	−0.993640	+	0.112605i	$0.964081\pi$
$114$		0			0
$115$	11.4366			1.06646
$116$		0			0
$117$	2.21141			0.204445
$118$		0			0
$119$	0.911460	−	7.75175i	0.0835534	−	0.710602i
$120$		0			0
$121$	8.55757			0.777961
$122$		0			0
$123$	14.7984			1.33432
$124$		0			0
$125$			10.2034i			0.912619i
$126$		0			0
$127$	5.00038			0.443712			0.221856	−	0.975079i	$-0.428789\pi$
$127$	5.00038			0.443712			0.221856	+	0.975079i	$0.428789\pi$
$128$		0			0
$129$			11.8501i			1.04334i
$130$		0			0
$131$			12.3002i			1.07467i	0.843368	+	0.537336i	$0.180569\pi$
$131$			12.3002i			1.07467i	−0.843368	+	0.537336i	$0.819431\pi$
$132$		0			0
$133$			9.85302i			0.854365i
$134$		0			0
$135$	−19.8297			−1.70667
$136$		0			0
$137$	10.8131			0.923826			0.461913	−	0.886925i	$-0.347163\pi$
$137$	10.8131			0.923826			0.461913	+	0.886925i	$0.347163\pi$
$138$		0			0
$139$			12.0120i			1.01884i	0.860517	+	0.509422i	$0.170141\pi$
$139$			12.0120i			1.01884i	−0.860517	+	0.509422i	$0.829859\pi$
$140$		0			0
$141$			13.9086i			1.17132i
$142$		0			0
$143$			6.95590i			0.581682i
$144$		0			0
$145$	−18.1141			−1.50429
$146$		0			0
$147$		−	5.40528i		−	0.445820i
$148$		0			0
$149$	−5.55621			−0.455183			−0.227591	−	0.973757i	$-0.573085\pi$
$149$	−5.55621			−0.455183			−0.227591	+	0.973757i	$0.573085\pi$
$150$		0			0
$151$	1.65649			0.134803			0.0674017	−	0.997726i	$-0.478529\pi$
$151$	1.65649			0.134803			0.0674017	+	0.997726i	$0.478529\pi$
$152$		0			0
$153$	2.03456	+	0.239226i	0.164484	+	0.0193403i
$154$		0			0
$155$	−11.9917			−0.963196
$156$		0			0
$157$	−23.3905			−1.86677			−0.933384	−	0.358879i	$-0.883159\pi$
$157$	−23.3905			−1.86677			−0.933384	+	0.358879i	$0.883159\pi$
$158$		0			0
$159$			4.14683i			0.328865i
$160$		0			0
$161$	6.04026			0.476039
$162$		0			0
$163$			16.0466i			1.25687i	0.777863	+	0.628434i	$0.216304\pi$
$163$			16.0466i			1.25687i	−0.777863	+	0.628434i	$0.783696\pi$
$164$		0			0
$165$		−	8.86239i		−	0.689936i
$166$		0			0
$167$		−	15.2908i		−	1.18324i	−0.806218	−	0.591619i	$-0.798489\pi$
$167$		−	15.2908i		−	1.18324i	0.806218	−	0.591619i	$-0.201511\pi$
$168$		0			0
$169$	6.81003			0.523848
$170$		0			0
$171$	−2.58607			−0.197761
$172$		0			0
$173$		−	18.0988i		−	1.37602i	−0.725699	−	0.688012i	$-0.758484\pi$
$173$		−	18.0988i		−	1.37602i	0.725699	−	0.688012i	$-0.241516\pi$
$174$		0			0
$175$			14.8541i			1.12286i
$176$		0			0
$177$		−	1.58213i		−	0.118920i
$178$		0			0
$179$	−3.81731			−0.285319			−0.142660	−	0.989772i	$-0.545565\pi$
$179$	−3.81731			−0.285319			−0.142660	+	0.989772i	$0.545565\pi$
$180$		0			0
$181$		−	14.6812i		−	1.09125i	−0.838030	−	0.545623i	$-0.816293\pi$
$181$		−	14.6812i		−	1.09125i	0.838030	−	0.545623i	$-0.183707\pi$
$182$		0			0
$183$	−19.7270			−1.45826
$184$		0			0
$185$	−37.4629			−2.75433
$186$		0			0
$187$	−0.752475	+	6.39961i	−0.0550264	+	0.467986i
$188$		0			0
$189$	−10.4731			−0.761810
$190$		0			0
$191$	24.0400			1.73947			0.869736	−	0.493516i	$-0.164289\pi$
$191$	24.0400			1.73947			0.869736	+	0.493516i	$0.164289\pi$
$192$		0			0
$193$		−	7.17007i		−	0.516113i	−0.966130	−	0.258056i	$-0.916918\pi$
$193$		−	7.17007i		−	0.516113i	0.966130	−	0.258056i	$-0.0830820\pi$
$194$		0			0
$195$	−25.2396			−1.80744
$196$		0			0
$197$			26.2724i			1.87183i	0.352223	+	0.935916i	$0.385426\pi$
$197$			26.2724i			1.87183i	−0.352223	+	0.935916i	$0.614574\pi$
$198$		0			0
$199$			18.6922i			1.32505i	0.749039	+	0.662526i	$0.230516\pi$
$199$			18.6922i			1.32505i	−0.749039	+	0.662526i	$0.769484\pi$
$200$		0			0
$201$		−	2.17982i		−	0.153753i
$202$		0			0
$203$	−9.56700			−0.671472
$204$		0			0
$205$	33.5248			2.34148
$206$		0			0
$207$			1.58535i			0.110190i
$208$		0			0
$209$		−	8.13436i		−	0.562666i
$210$		0			0
$211$		−	3.31656i		−	0.228321i	−0.993462	−	0.114161i	$-0.963582\pi$
$211$		−	3.31656i		−	0.228321i	0.993462	−	0.114161i	$-0.0364179\pi$
$212$		0			0
$213$	−14.8482			−1.01738
$214$		0			0
$215$			26.8457i			1.83086i
$216$		0			0
$217$	−6.33346			−0.429943
$218$		0			0
$219$	0.543887			0.0367525
$220$		0			0
$221$	18.2258	+	2.14301i	1.22600	+	0.144154i
$222$		0			0
$223$	23.9721			1.60529			0.802646	−	0.596455i	$-0.203425\pi$
$223$	23.9721			1.60529			0.802646	+	0.596455i	$0.203425\pi$
$224$		0			0
$225$	−3.89867			−0.259911
$226$		0			0
$227$		−	5.60333i		−	0.371906i	−0.982559	−	0.185953i	$-0.940463\pi$
$227$		−	5.60333i		−	0.371906i	0.982559	−	0.185953i	$-0.0595373\pi$
$228$		0			0
$229$	−22.4728			−1.48504			−0.742522	−	0.669822i	$-0.766371\pi$
$229$	−22.4728			−1.48504			−0.742522	+	0.669822i	$0.766371\pi$
$230$		0			0
$231$		−	4.68070i		−	0.307968i
$232$		0			0
$233$		−	1.42422i		−	0.0933040i	−0.998911	−	0.0466520i	$-0.985145\pi$
$233$		−	1.42422i		−	0.0933040i	0.998911	−	0.0466520i	$-0.0148552\pi$
$234$		0			0
$235$			31.5092i			2.05543i
$236$		0			0
$237$	−0.148416			−0.00964068
$238$		0			0
$239$	2.15838			0.139614			0.0698069	−	0.997561i	$-0.477762\pi$
$239$	2.15838			0.139614			0.0698069	+	0.997561i	$0.477762\pi$
$240$		0			0
$241$			3.58435i			0.230888i	0.993314	+	0.115444i	$0.0368292\pi$
$241$			3.58435i			0.230888i	−0.993314	+	0.115444i	$0.963171\pi$
$242$		0			0
$243$		−	5.10709i		−	0.327620i
$244$		0			0
$245$		−	12.2454i		−	0.782327i
$246$		0			0
$247$	−23.1662			−1.47403
$248$		0			0
$249$			8.88369i			0.562981i
$250$		0			0
$251$	27.1042			1.71080			0.855402	−	0.517965i	$-0.173310\pi$
$251$	27.1042			1.71080			0.855402	+	0.517965i	$0.173310\pi$
$252$		0			0
$253$	−4.98666			−0.313509
$254$		0			0
$255$	−23.2211	−	2.73037i	−1.45416	−	0.170982i
$256$		0			0
$257$	0.0133424			0.000832275			0.000416138	−	1.00000i	$-0.499868\pi$
$257$	0.0133424			0.000832275			0.000416138		1.00000i	$0.499868\pi$
$258$		0			0
$259$	−19.7862			−1.22945
$260$		0			0
$261$		−	2.51100i		−	0.155427i
$262$		0			0
$263$	24.6287			1.51867			0.759336	−	0.650699i	$-0.225524\pi$
$263$	24.6287			1.51867			0.759336	+	0.650699i	$0.225524\pi$
$264$		0			0
$265$			9.39441i			0.577094i
$266$		0			0
$267$		−	12.0613i		−	0.738137i
$268$		0			0
$269$			20.8960i			1.27405i	0.770842	+	0.637026i	$0.219836\pi$
$269$			20.8960i			1.27405i	−0.770842	+	0.637026i	$0.780164\pi$
$270$		0			0
$271$	4.90014			0.297662			0.148831	−	0.988863i	$-0.452449\pi$
$271$	4.90014			0.297662			0.148831	+	0.988863i	$0.452449\pi$
$272$		0			0
$273$	−13.3304			−0.806791
$274$		0			0
$275$		−	12.2631i		−	0.739493i
$276$		0			0
$277$			6.82078i			0.409821i	0.978781	+	0.204911i	$0.0656904\pi$
$277$			6.82078i			0.409821i	−0.978781	+	0.204911i	$0.934310\pi$
$278$		0			0
$279$		−	1.66231i		−	0.0995196i
$280$		0			0
$281$	13.6085			0.811813			0.405907	−	0.913915i	$-0.366956\pi$
$281$	13.6085			0.811813			0.405907	+	0.913915i	$0.366956\pi$
$282$		0			0
$283$			0.0157633i			0.000937032i	1.00000		0.000468516i	$0.000149133\pi$
$283$			0.0157633i			0.000937032i	−1.00000		0.000468516i	$0.999851\pi$
$284$		0			0
$285$	29.5157			1.74836
$286$		0			0
$287$	17.7063			1.04517
$288$		0			0
$289$	16.5363	+	3.94325i	0.972726	+	0.231956i
$290$		0			0
$291$	27.3774			1.60489
$292$		0			0
$293$	19.9213			1.16382			0.581909	−	0.813254i	$-0.302306\pi$
$293$	19.9213			1.16382			0.581909	+	0.813254i	$0.302306\pi$
$294$		0			0
$295$		−	3.58424i		−	0.208682i
$296$		0			0
$297$	8.64632			0.501711
$298$		0			0
$299$			14.2017i			0.821308i
$300$		0			0
$301$			14.1786i			0.817244i
$302$		0			0
$303$			9.90143i			0.568822i
$304$		0			0
$305$	−44.6904			−2.55897
$306$		0			0
$307$	−26.9494			−1.53808			−0.769041	−	0.639200i	$-0.779266\pi$
$307$	−26.9494			−1.53808			−0.769041	+	0.639200i	$0.779266\pi$
$308$		0			0
$309$			8.28890i			0.471539i
$310$		0			0
$311$		−	26.3066i		−	1.49171i	−0.666107	−	0.745856i	$-0.732041\pi$
$311$		−	26.3066i		−	1.49171i	0.666107	−	0.745856i	$-0.267959\pi$
$312$		0			0
$313$		−	4.40205i		−	0.248819i	−0.992231	−	0.124409i	$-0.960296\pi$
$313$		−	4.40205i		−	0.248819i	0.992231	−	0.124409i	$-0.0397036\pi$
$314$		0			0
$315$	−3.37117			−0.189944
$316$		0			0
$317$		−	31.6804i		−	1.77935i	−0.456596	−	0.889674i	$-0.650931\pi$
$317$		−	31.6804i		−	1.77935i	0.456596	−	0.889674i	$-0.349069\pi$
$318$		0			0
$319$	7.89824			0.442216
$320$		0			0
$321$	−24.6619			−1.37649
$322$		0			0
$323$	−21.3135	−	2.50607i	−1.18592	−	0.139442i
$324$		0			0
$325$	−34.9246			−1.93727
$326$		0			0
$327$	16.3644			0.904952
$328$		0			0
$329$			16.6417i			0.917485i
$330$		0			0
$331$	2.43983			0.134105			0.0670526	−	0.997749i	$-0.478640\pi$
$331$	2.43983			0.134105			0.0670526	+	0.997749i	$0.478640\pi$
$332$		0			0
$333$		−	5.19316i		−	0.284583i
$334$		0			0
$335$		−	4.93827i		−	0.269806i
$336$		0			0
$337$			6.74925i			0.367655i	0.982959	+	0.183827i	$0.0588488\pi$
$337$			6.74925i			0.367655i	−0.982959	+	0.183827i	$0.941151\pi$
$338$		0			0
$339$	3.78764			0.205716
$340$		0			0
$341$	5.22871			0.283151
$342$		0			0
$343$		−	19.7186i		−	1.06471i
$344$		0			0
$345$		−	18.0942i		−	0.974158i
$346$		0			0
$347$		−	9.92260i		−	0.532673i	−0.963880	−	0.266336i	$-0.914187\pi$
$347$		−	9.92260i		−	0.532673i	0.963880	−	0.266336i	$-0.0858132\pi$
$348$		0			0
$349$	5.02529			0.268997			0.134499	−	0.990914i	$-0.457058\pi$
$349$	5.02529			0.268997			0.134499	+	0.990914i	$0.457058\pi$
$350$		0			0
$351$		−	24.6243i		−	1.31435i
$352$		0			0
$353$	17.5183			0.932404			0.466202	−	0.884678i	$-0.345622\pi$
$353$	17.5183			0.932404			0.466202	+	0.884678i	$0.345622\pi$
$354$		0			0
$355$	−33.6377			−1.78531
$356$		0			0
$357$	−12.2643	−	1.44205i	−0.649096	−	0.0763215i
$358$		0			0
$359$	21.3101			1.12471			0.562353	−	0.826897i	$-0.309896\pi$
$359$	21.3101			1.12471			0.562353	+	0.826897i	$0.309896\pi$
$360$		0			0
$361$	8.09101			0.425843
$362$		0			0
$363$		−	13.5392i		−	0.710625i
$364$		0			0
$365$	1.23215			0.0644934
$366$		0			0
$367$			11.3182i			0.590808i	0.955372	+	0.295404i	$0.0954542\pi$
$367$			11.3182i			0.590808i	−0.955372	+	0.295404i	$0.904546\pi$
$368$		0			0
$369$			4.64726i			0.241927i
$370$		0			0
$371$			4.96169i			0.257598i
$372$		0			0
$373$	−27.2356			−1.41021			−0.705104	−	0.709104i	$-0.749100\pi$
$373$	−27.2356			−1.41021			−0.705104	+	0.709104i	$0.749100\pi$
$374$		0			0
$375$	16.1431			0.833628
$376$		0			0
$377$		−	22.4937i		−	1.15849i
$378$		0			0
$379$			15.8756i			0.815473i	0.913100	+	0.407737i	$0.133682\pi$
$379$			15.8756i			0.815473i	−0.913100	+	0.407737i	$0.866318\pi$
$380$		0			0
$381$		−	7.91127i		−	0.405307i
$382$		0			0
$383$	−31.5223			−1.61072			−0.805358	−	0.592788i	$-0.798027\pi$
$383$	−31.5223			−1.61072			−0.805358	+	0.592788i	$0.798027\pi$
$384$		0			0
$385$		−	10.6039i		−	0.540423i
$386$		0			0
$387$	−3.72139			−0.189169
$388$		0			0
$389$	0.593284			0.0300807			0.0150403	−	0.999887i	$-0.495212\pi$
$389$	0.593284			0.0300807			0.0150403	+	0.999887i	$0.495212\pi$
$390$		0			0
$391$	−1.53631	+	13.0660i	−0.0776947	+	0.660775i
$392$		0			0
$393$	19.4605			0.981654
$394$		0			0
$395$	−0.336229			−0.0169175
$396$		0			0
$397$		−	23.0032i		−	1.15450i	−0.816568	−	0.577249i	$-0.804126\pi$
$397$		−	23.0032i		−	1.15450i	0.816568	−	0.577249i	$-0.195874\pi$
$398$		0			0
$399$	15.5888			0.780416
$400$		0			0
$401$			11.7157i			0.585052i	0.956258	+	0.292526i	$0.0944959\pi$
$401$			11.7157i			0.585052i	−0.956258	+	0.292526i	$0.905504\pi$
$402$		0			0
$403$		−	14.8911i		−	0.741778i
$404$		0			0
$405$			26.0308i			1.29348i
$406$		0			0
$407$	16.3349			0.809689
$408$		0			0
$409$	19.8877			0.983383			0.491692	−	0.870769i	$-0.336379\pi$
$409$	19.8877			0.983383			0.491692	+	0.870769i	$0.336379\pi$
$410$		0			0
$411$		−	17.1078i		−	0.843865i
$412$		0			0
$413$		−	1.89303i		−	0.0931498i
$414$		0			0
$415$			20.1255i			0.987922i
$416$		0			0
$417$	19.0046			0.930658
$418$		0			0
$419$		−	13.1372i		−	0.641796i	−0.947114	−	0.320898i	$-0.896015\pi$
$419$		−	13.1372i		−	0.641796i	0.947114	−	0.320898i	$-0.103985\pi$
$420$		0			0
$421$	−16.9753			−0.827326			−0.413663	−	0.910430i	$-0.635751\pi$
$421$	−16.9753			−0.827326			−0.413663	+	0.910430i	$0.635751\pi$
$422$		0			0
$423$	−4.36785			−0.212372
$424$		0			0
$425$	−32.1316	−	3.77807i	−1.55861	−	0.183263i
$426$		0			0
$427$	−23.6034			−1.14225
$428$		0			0
$429$	11.0052			0.531335
$430$		0			0
$431$			25.5419i			1.23031i	0.788407	+	0.615155i	$0.210906\pi$
$431$			25.5419i			1.23031i	−0.788407	+	0.615155i	$0.789094\pi$
$432$		0			0
$433$	−9.83130			−0.472462			−0.236231	−	0.971697i	$-0.575912\pi$
$433$	−9.83130			−0.472462			−0.236231	+	0.971697i	$0.575912\pi$
$434$		0			0
$435$			28.6589i			1.37409i
$436$		0			0
$437$		−	16.6078i		−	0.794458i
$438$		0			0
$439$		−	2.76931i		−	0.132172i	−0.997814	−	0.0660859i	$-0.978949\pi$
$439$		−	2.76931i		−	0.132172i	0.997814	−	0.0660859i	$-0.0210511\pi$
$440$		0			0
$441$	1.69747			0.0808319
$442$		0			0
$443$	12.6505			0.601046			0.300523	−	0.953775i	$-0.402839\pi$
$443$	12.6505			0.601046			0.300523	+	0.953775i	$0.402839\pi$
$444$		0			0
$445$		−	27.3241i		−	1.29529i
$446$		0			0
$447$			8.79067i			0.415784i
$448$		0			0
$449$		−	27.6353i		−	1.30419i	−0.758136	−	0.652096i	$-0.773890\pi$
$449$		−	27.6353i		−	1.30419i	0.758136	−	0.652096i	$-0.226110\pi$
$450$		0			0
$451$	−14.6178			−0.688324
$452$		0			0
$453$		−	2.62079i		−	0.123136i
$454$		0			0
$455$	−30.1992			−1.41576
$456$		0			0
$457$	6.73621			0.315106			0.157553	−	0.987510i	$-0.449639\pi$
$457$	6.73621			0.315106			0.157553	+	0.987510i	$0.449639\pi$
$458$		0			0
$459$	2.66380	−	22.6550i	0.124335	−	1.05744i
$460$		0			0
$461$	35.4289			1.65009			0.825044	−	0.565068i	$-0.191150\pi$
$461$	35.4289			1.65009			0.825044	+	0.565068i	$0.191150\pi$
$462$		0			0
$463$	−30.8991			−1.43600			−0.718001	−	0.696042i	$-0.754943\pi$
$463$	−30.8991			−1.43600			−0.718001	+	0.696042i	$0.754943\pi$
$464$		0			0
$465$			18.9725i			0.879827i
$466$		0			0
$467$	−42.9151			−1.98588			−0.992938	−	0.118635i	$-0.962148\pi$
$467$	−42.9151			−1.98588			−0.992938	+	0.118635i	$0.962148\pi$
$468$		0			0
$469$		−	2.60816i		−	0.120434i
$470$		0			0
$471$			37.0070i			1.70519i
$472$		0			0
$473$		−	11.7055i		−	0.538218i
$474$		0			0
$475$	40.8415			1.87394
$476$		0			0
$477$	−1.30227			−0.0596267
$478$		0			0
$479$		−	19.0325i		−	0.869619i	−0.900522	−	0.434810i	$-0.856816\pi$
$479$		−	19.0325i		−	0.869619i	0.900522	−	0.434810i	$-0.143184\pi$
$480$		0			0
$481$		−	46.5208i		−	2.12117i
$482$		0			0
$483$		−	9.55650i		−	0.434836i
$484$		0			0
$485$	62.0218			2.81627
$486$		0			0
$487$		−	12.8292i		−	0.581346i	−0.956823	−	0.290673i	$-0.906121\pi$
$487$		−	12.8292i		−	0.581346i	0.956823	−	0.290673i	$-0.0938791\pi$
$488$		0			0
$489$	25.3879			1.14808
$490$		0			0
$491$	41.6194			1.87826			0.939128	−	0.343568i	$-0.111636\pi$
$491$	41.6194			1.87826			0.939128	+	0.343568i	$0.111636\pi$
$492$		0			0
$493$	2.43332	−	20.6948i	0.109591	−	0.932049i
$494$		0			0
$495$	2.78314			0.125093
$496$		0			0
$497$	−17.7659			−0.796909
$498$		0			0
$499$			13.1730i			0.589704i	0.955543	+	0.294852i	$0.0952703\pi$
$499$			13.1730i			0.589704i	−0.955543	+	0.294852i	$0.904730\pi$
$500$		0			0
$501$	−24.1921			−1.08082
$502$		0			0
$503$		−	19.1004i		−	0.851645i	−0.904807	−	0.425822i	$-0.859985\pi$
$503$		−	19.1004i		−	0.851645i	0.904807	−	0.425822i	$-0.140015\pi$
$504$		0			0
$505$			22.4311i			0.998172i
$506$		0			0
$507$		−	10.7744i		−	0.478507i
$508$		0			0
$509$	7.61049			0.337329			0.168665	−	0.985674i	$-0.446055\pi$
$509$	7.61049			0.337329			0.168665	+	0.985674i	$0.446055\pi$
$510$		0			0
$511$	0.650762			0.0287880
$512$		0			0
$513$			28.7961i			1.27138i
$514$		0			0
$515$			18.7780i			0.827459i
$516$		0			0
$517$		−	13.7389i		−	0.604235i
$518$		0			0
$519$	−28.6347			−1.25692
$520$		0			0
$521$			12.2618i			0.537200i	0.963252	+	0.268600i	$0.0865609\pi$
$521$			12.2618i			0.537200i	−0.963252	+	0.268600i	$0.913439\pi$
$522$		0			0
$523$	12.8885			0.563576			0.281788	−	0.959477i	$-0.409073\pi$
$523$	12.8885			0.563576			0.281788	+	0.959477i	$0.409073\pi$
$524$		0			0
$525$	23.5012			1.02567
$526$		0			0
$527$	1.61089	−	13.7002i	0.0701713	−	0.596790i
$528$		0			0
$529$	12.8188			0.557340
$530$		0			0
$531$	0.496852			0.0215615
$532$		0			0
$533$			41.6306i			1.80322i
$534$		0			0
$535$	−55.8700			−2.41547
$536$		0			0
$537$			6.03949i			0.260623i
$538$		0			0
$539$			5.33932i			0.229981i
$540$		0			0
$541$			3.06145i			0.131622i	0.997832	+	0.0658110i	$0.0209634\pi$
$541$			3.06145i			0.131622i	−0.997832	+	0.0658110i	$0.979037\pi$
$542$		0			0
$543$	−23.2277			−0.996795
$544$		0			0
$545$	37.0726			1.58801
$546$		0			0
$547$			11.3138i			0.483742i	0.970308	+	0.241871i	$0.0777611\pi$
$547$			11.3138i			0.483742i	−0.970308	+	0.241871i	$0.922239\pi$
$548$		0			0
$549$		−	6.19505i		−	0.264398i
$550$		0			0
$551$			26.3046i			1.12061i
$552$		0			0
$553$	−0.177581			−0.00755150
$554$		0			0
$555$			59.2713i			2.51593i
$556$		0			0
$557$	16.2845			0.689997			0.344998	−	0.938603i	$-0.387879\pi$
$557$	16.2845			0.689997			0.344998	+	0.938603i	$0.387879\pi$
$558$		0			0
$559$	−33.3365			−1.40999
$560$		0			0
$561$	10.1250	+	1.19052i	0.427480	+	0.0502636i
$562$		0			0
$563$	10.8913			0.459015			0.229507	−	0.973307i	$-0.426288\pi$
$563$	10.8913			0.459015			0.229507	+	0.973307i	$0.426288\pi$
$564$		0			0
$565$	8.58068			0.360992
$566$		0			0
$567$			13.7483i			0.577373i
$568$		0			0
$569$	−5.94516			−0.249234			−0.124617	−	0.992205i	$-0.539770\pi$
$569$	−5.94516			−0.249234			−0.124617	+	0.992205i	$0.539770\pi$
$570$		0			0
$571$			33.2531i			1.39160i	0.718236	+	0.695800i	$0.244950\pi$
$571$			33.2531i			1.39160i	−0.718236	+	0.695800i	$0.755050\pi$
$572$		0			0
$573$		−	38.0345i		−	1.58891i
$574$		0			0
$575$		−	25.0373i		−	1.04413i
$576$		0			0
$577$	33.8801			1.41045			0.705223	−	0.708986i	$-0.250847\pi$
$577$	33.8801			1.41045			0.705223	+	0.708986i	$0.250847\pi$
$578$		0			0
$579$	−11.3440			−0.471441
$580$		0			0
$581$			10.6294i			0.440980i
$582$		0			0
$583$		−	4.09623i		−	0.169648i
$584$		0			0
$585$		−	7.92622i		−	0.327709i
$586$		0			0
$587$	−8.40852			−0.347057			−0.173528	−	0.984829i	$-0.555517\pi$
$587$	−8.40852			−0.347057			−0.173528	+	0.984829i	$0.555517\pi$
$588$		0			0
$589$			17.4139i			0.717528i
$590$		0			0
$591$	41.5665			1.70982
$592$		0			0
$593$	8.05925			0.330954			0.165477	−	0.986214i	$-0.447084\pi$
$593$	8.05925			0.330954			0.165477	+	0.986214i	$0.447084\pi$
$594$		0			0
$595$	−27.7841	−	3.26689i	−1.13904	−	0.133929i
$596$		0			0
$597$	29.5735			1.21036
$598$		0			0
$599$	−12.7884			−0.522521			−0.261261	−	0.965268i	$-0.584138\pi$
$599$	−12.7884			−0.522521			−0.261261	+	0.965268i	$0.584138\pi$
$600$		0			0
$601$		−	27.7382i		−	1.13147i	−0.824588	−	0.565733i	$-0.808593\pi$
$601$		−	27.7382i		−	1.13147i	0.824588	−	0.565733i	$-0.191407\pi$
$602$		0			0
$603$	0.684550			0.0278770
$604$		0			0
$605$		−	30.6723i		−	1.24701i
$606$		0			0
$607$			21.8395i			0.886437i	0.896414	+	0.443218i	$0.146163\pi$
$607$			21.8395i			0.886437i	−0.896414	+	0.443218i	$0.853837\pi$
$608$		0			0
$609$			15.1363i			0.613353i
$610$		0			0
$611$	−39.1276			−1.58293
$612$		0			0
$613$	−4.60258			−0.185896			−0.0929482	−	0.995671i	$-0.529629\pi$
$613$	−4.60258			−0.185896			−0.0929482	+	0.995671i	$0.529629\pi$
$614$		0			0
$615$		−	53.0408i		−	2.13881i
$616$		0			0
$617$			20.2849i			0.816638i	0.912839	+	0.408319i	$0.133885\pi$
$617$			20.2849i			0.816638i	−0.912839	+	0.408319i	$0.866115\pi$
$618$		0			0
$619$		−	13.1774i		−	0.529643i	−0.964297	−	0.264822i	$-0.914687\pi$
$619$		−	13.1774i		−	0.529643i	0.964297	−	0.264822i	$-0.0853131\pi$
$620$		0			0
$621$	17.6530			0.708392
$622$		0			0
$623$		−	14.4313i		−	0.578179i
$624$		0			0
$625$	−2.66235			−0.106494
$626$		0			0
$627$	−12.8697			−0.513964
$628$		0			0
$629$	5.03252	−	42.8004i	0.200660	−	1.70656i
$630$		0			0
$631$	−29.2829			−1.16573			−0.582867	−	0.812567i	$-0.698069\pi$
$631$	−29.2829			−1.16573			−0.582867	+	0.812567i	$0.698069\pi$
$632$		0			0
$633$	−5.24724			−0.208559
$634$		0			0
$635$		−	17.9225i		−	0.711234i
$636$		0			0
$637$	15.2061			0.602487
$638$		0			0
$639$		−	4.66291i		−	0.184462i
$640$		0			0
$641$			4.14950i			0.163895i	0.996637	+	0.0819477i	$0.0261141\pi$
$641$			4.14950i			0.163895i	−0.996637	+	0.0819477i	$0.973886\pi$
$642$		0			0
$643$		−	0.693712i		−	0.0273573i	−0.999906	−	0.0136787i	$-0.995646\pi$
$643$		−	0.693712i		−	0.0273573i	0.999906	−	0.0136787i	$-0.00435419\pi$
$644$		0			0
$645$	42.4735			1.67239
$646$		0			0
$647$	−42.9107			−1.68699			−0.843497	−	0.537133i	$-0.819507\pi$
$647$	−42.9107			−1.68699			−0.843497	+	0.537133i	$0.819507\pi$
$648$		0			0
$649$			1.56283i			0.0613463i
$650$		0			0
$651$			10.0204i			0.392729i
$652$		0			0
$653$			44.3141i			1.73415i	0.498182	+	0.867073i	$0.334001\pi$
$653$			44.3141i			1.73415i	−0.498182	+	0.867073i	$0.665999\pi$
$654$		0			0
$655$	44.0867			1.72261
$656$		0			0
$657$			0.170802i			0.00666361i
$658$		0			0
$659$	−25.3683			−0.988208			−0.494104	−	0.869403i	$-0.664504\pi$
$659$	−25.3683			−0.988208			−0.494104	+	0.869403i	$0.664504\pi$
$660$		0			0
$661$	26.6011			1.03466			0.517331	−	0.855785i	$-0.326926\pi$
$661$	26.6011			1.03466			0.517331	+	0.855785i	$0.326926\pi$
$662$		0			0
$663$	3.39052	−	28.8356i	0.131677	−	1.11988i
$664$		0			0
$665$	35.3155			1.36948
$666$		0			0
$667$	16.1257			0.624389
$668$		0			0
$669$		−	37.9271i		−	1.46635i
$670$		0			0
$671$	19.4863			0.752260
$672$		0			0
$673$		−	23.3769i		−	0.901114i	−0.892748	−	0.450557i	$-0.851225\pi$
$673$		−	23.3769i		−	0.901114i	0.892748	−	0.450557i	$-0.148775\pi$
$674$		0			0
$675$			43.4120i			1.67093i
$676$		0			0
$677$			5.91913i			0.227491i	0.993510	+	0.113745i	$0.0362848\pi$
$677$			5.91913i			0.227491i	−0.993510	+	0.113745i	$0.963715\pi$
$678$		0			0
$679$	32.7571			1.25710
$680$		0			0
$681$	−8.86522			−0.339716
$682$		0			0
$683$		−	39.2331i		−	1.50121i	−0.660749	−	0.750607i	$-0.729761\pi$
$683$		−	39.2331i		−	1.50121i	0.660749	−	0.750607i	$-0.270239\pi$
$684$		0			0
$685$		−	38.7567i		−	1.48082i
$686$		0			0
$687$			35.5550i			1.35651i
$688$		0			0
$689$	−11.6658			−0.444433
$690$		0			0
$691$			0.796860i			0.0303140i	0.999885	+	0.0151570i	$0.00482481\pi$
$691$			0.796860i			0.0303140i	−0.999885	+	0.0151570i	$0.995175\pi$
$692$		0			0
$693$	1.46992			0.0558378
$694$		0			0
$695$	43.0538			1.63312
$696$		0			0
$697$	−4.50351	+	38.3013i	−0.170583	+	1.45076i
$698$		0			0
$699$	−2.25331			−0.0852281
$700$		0			0
$701$	−51.4629			−1.94373			−0.971863	−	0.235547i	$-0.924312\pi$
$701$	−51.4629			−1.94373			−0.971863	+	0.235547i	$0.924312\pi$
$702$		0			0
$703$			54.4023i			2.05182i
$704$		0			0
$705$	49.8517			1.87752
$706$		0			0
$707$			11.8471i			0.445555i
$708$		0			0
$709$		−	9.03296i		−	0.339240i	−0.985510	−	0.169620i	$-0.945746\pi$
$709$		−	9.03296i		−	0.339240i	0.985510	−	0.169620i	$-0.0542540\pi$
$710$		0			0
$711$		−	0.0466086i		−	0.00174796i
$712$		0			0
$713$	10.6754			0.399796
$714$		0			0
$715$	24.9316			0.932389
$716$		0			0
$717$		−	3.41484i		−	0.127530i
$718$		0			0
$719$		−	32.4318i		−	1.20950i	−0.796414	−	0.604751i	$-0.793273\pi$
$719$		−	32.4318i		−	1.20950i	0.796414	−	0.604751i	$-0.206727\pi$
$720$		0			0
$721$			9.91768i			0.369354i
$722$		0			0
$723$	5.67093			0.210904
$724$		0			0
$725$			39.6560i			1.47279i
$726$		0			0
$727$	−2.10920			−0.0782260			−0.0391130	−	0.999235i	$-0.512453\pi$
$727$	−2.10920			−0.0782260			−0.0391130	+	0.999235i	$0.512453\pi$
$728$		0			0
$729$	−29.8678			−1.10622
$730$		0			0
$731$	−30.6705	−	3.60628i	−1.13439	−	0.133383i
$732$		0			0
$733$	43.0998			1.59193			0.795965	−	0.605343i	$-0.206964\pi$
$733$	43.0998			1.59193			0.795965	+	0.605343i	$0.206964\pi$
$734$		0			0
$735$	−19.3738			−0.714613
$736$		0			0
$737$			2.15322i			0.0793150i
$738$		0			0
$739$	20.3126			0.747213			0.373606	−	0.927587i	$-0.378121\pi$
$739$	20.3126			0.747213			0.373606	+	0.927587i	$0.378121\pi$
$740$		0			0
$741$			36.6521i			1.34645i
$742$		0			0
$743$		−	17.0484i		−	0.625444i	−0.949845	−	0.312722i	$-0.898759\pi$
$743$		−	17.0484i		−	0.625444i	0.949845	−	0.312722i	$-0.101241\pi$
$744$		0			0
$745$			19.9148i			0.729621i
$746$		0			0
$747$	−2.78983			−0.102074
$748$		0			0
$749$	−29.5080			−1.07820
$750$		0			0
$751$		−	12.4080i		−	0.452774i	−0.974037	−	0.226387i	$-0.927309\pi$
$751$		−	12.4080i		−	0.452774i	0.974037	−	0.226387i	$-0.0726914\pi$
$752$		0			0
$753$		−	42.8825i		−	1.56273i
$754$		0			0
$755$		−	5.93726i		−	0.216079i
$756$		0			0
$757$	44.1856			1.60595			0.802977	−	0.596010i	$-0.203248\pi$
$757$	44.1856			1.60595			0.802977	+	0.596010i	$0.203248\pi$
$758$		0			0
$759$			7.88957i			0.286373i
$760$		0			0
$761$	−43.6006			−1.58052			−0.790260	−	0.612771i	$-0.790055\pi$
$761$	−43.6006			−1.58052			−0.790260	+	0.612771i	$0.790055\pi$
$762$		0			0
$763$	19.5800			0.708844
$764$		0			0
$765$	0.857441	−	7.29233i	0.0310009	−	0.263655i
$766$		0			0
$767$	4.45085			0.160711
$768$		0			0
$769$	−34.3200			−1.23761			−0.618805	−	0.785545i	$-0.712383\pi$
$769$	−34.3200			−1.23761			−0.618805	+	0.785545i	$0.712383\pi$
$770$		0			0
$771$		−	0.0211094i		−	0.000760238i
$772$		0			0
$773$	17.7230			0.637453			0.318727	−	0.947847i	$-0.396745\pi$
$773$	17.7230			0.637453			0.318727	+	0.947847i	$0.396745\pi$
$774$		0			0
$775$			26.2527i			0.943023i
$776$		0			0
$777$			31.3043i			1.12304i
$778$		0			0
$779$		−	48.6836i		−	1.74427i
$780$		0			0
$781$	14.6670			0.524826
$782$		0			0
$783$	−27.9602			−0.999215
$784$		0			0
$785$			83.8372i			2.99228i
$786$		0			0
$787$		−	11.9869i		−	0.427286i	−0.976912	−	0.213643i	$-0.931467\pi$
$787$		−	11.9869i		−	0.427286i	0.976912	−	0.213643i	$-0.0685330\pi$
$788$		0			0
$789$		−	38.9659i		−	1.38722i
$790$		0			0
$791$	4.53192			0.161137
$792$		0			0
$793$		−	55.4959i		−	1.97072i
$794$		0			0
$795$	14.8632			0.527144
$796$		0			0
$797$	14.8688			0.526679			0.263340	−	0.964703i	$-0.415176\pi$
$797$	14.8688			0.526679			0.263340	+	0.964703i	$0.415176\pi$
$798$		0			0
$799$	−35.9984	−	4.23274i	−1.27353	−	0.149743i
$800$		0			0
$801$	3.78771			0.133832
$802$		0			0
$803$	−0.537250			−0.0189591
$804$		0			0
$805$		−	21.6497i		−	0.763052i
$806$		0			0
$807$	33.0603			1.16378
$808$		0			0
$809$		−	6.78859i		−	0.238674i	−0.992854	−	0.119337i	$-0.961923\pi$
$809$		−	6.78859i		−	0.238674i	0.992854	−	0.119337i	$-0.0380769\pi$
$810$		0			0
$811$			48.3778i			1.69877i	0.527771	+	0.849387i	$0.323028\pi$
$811$			48.3778i			1.69877i	−0.527771	+	0.849387i	$0.676972\pi$
$812$		0			0
$813$		−	7.75268i		−	0.271898i
$814$		0			0
$815$	57.5148			2.01466
$816$		0			0
$817$	38.9844			1.36389
$818$		0			0
$819$		−	4.18626i		−	0.146280i
$820$		0			0
$821$		−	36.2609i		−	1.26552i	−0.774350	−	0.632758i	$-0.781923\pi$
$821$		−	36.2609i		−	1.26552i	0.774350	−	0.632758i	$-0.218077\pi$
$822$		0			0
$823$			42.0596i			1.46611i	0.680171	+	0.733053i	$0.261905\pi$
$823$			42.0596i			1.46611i	−0.680171	+	0.733053i	$0.738095\pi$
$824$		0			0
$825$	−19.4019			−0.675486
$826$		0			0
$827$		−	45.6803i		−	1.58846i	−0.607618	−	0.794229i	$-0.707875\pi$
$827$		−	45.6803i		−	1.58846i	0.607618	−	0.794229i	$-0.292125\pi$
$828$		0			0
$829$	33.7905			1.17359			0.586797	−	0.809734i	$-0.300389\pi$
$829$	33.7905			1.17359			0.586797	+	0.809734i	$0.300389\pi$
$830$		0			0
$831$	10.7914			0.374349
$832$		0			0
$833$	13.9900	+	1.64496i	0.484725	+	0.0569946i
$834$		0			0
$835$	−54.8059			−1.89664
$836$		0			0
$837$	−18.5099			−0.639796
$838$		0			0
$839$			22.5082i			0.777070i	0.921434	+	0.388535i	$0.127019\pi$
$839$			22.5082i			0.777070i	−0.921434	+	0.388535i	$0.872981\pi$
$840$		0			0
$841$	3.45898			0.119275
$842$		0			0
$843$		−	21.5304i		−	0.741547i
$844$		0			0
$845$		−	24.4087i		−	0.839686i
$846$		0			0
$847$		−	16.1997i		−	0.556629i
$848$		0			0
$849$	0.0249397			0.000855928
$850$		0			0
$851$	33.3506			1.14324
$852$		0			0
$853$		−	29.8377i		−	1.02162i	−0.859693	−	0.510812i	$-0.829345\pi$
$853$		−	29.8377i		−	1.02162i	0.859693	−	0.510812i	$-0.170655\pi$
$854$		0			0
$855$			9.26907i			0.316995i
$856$		0			0
$857$		−	41.5786i		−	1.42030i	−0.704051	−	0.710149i	$-0.748628\pi$
$857$		−	41.5786i		−	1.42030i	0.704051	−	0.710149i	$-0.251372\pi$
$858$		0			0
$859$	44.8165			1.52912			0.764559	−	0.644553i	$-0.222957\pi$
$859$	44.8165			1.52912			0.764559	+	0.644553i	$0.222957\pi$
$860$		0			0
$861$		−	28.0137i		−	0.954704i
$862$		0			0
$863$	49.6589			1.69041			0.845204	−	0.534445i	$-0.179479\pi$
$863$	49.6589			1.69041			0.845204	+	0.534445i	$0.179479\pi$
$864$		0			0
$865$	−64.8702			−2.20565
$866$		0			0
$867$	6.23874	−	26.1627i	0.211879	−	0.888533i
$868$		0			0
$869$	0.146605			0.00497324
$870$		0			0
$871$	6.13226			0.207784
$872$		0			0
$873$			8.59756i			0.290983i
$874$		0			0
$875$	19.3153			0.652976
$876$		0			0
$877$			33.2814i			1.12383i	0.827194	+	0.561917i	$0.189936\pi$
$877$			33.2814i			1.12383i	−0.827194	+	0.561917i	$0.810064\pi$
$878$		0			0
$879$		−	31.5182i		−	1.06308i
$880$		0			0
$881$			42.2188i			1.42239i	0.702995	+	0.711194i	$0.251845\pi$
$881$			42.2188i			1.42239i	−0.702995	+	0.711194i	$0.748155\pi$
$882$		0			0
$883$	21.4815			0.722908			0.361454	−	0.932390i	$-0.382280\pi$
$883$	21.4815			0.722908			0.361454	+	0.932390i	$0.382280\pi$
$884$		0			0
$885$	−5.67074			−0.190620
$886$		0			0
$887$		−	8.65510i		−	0.290610i	−0.989387	−	0.145305i	$-0.953584\pi$
$887$		−	8.65510i		−	0.290610i	0.989387	−	0.145305i	$-0.0464163\pi$
$888$		0			0
$889$		−	9.46585i		−	0.317475i
$890$		0			0
$891$		−	11.3502i		−	0.380245i
$892$		0			0
$893$	45.7565			1.53118
$894$		0			0
$895$			13.6821i			0.457343i
$896$		0			0
$897$	22.4691			0.750220
$898$		0			0
$899$	−16.9084			−0.563927
$900$		0			0
$901$	−10.7329	−	1.26198i	−0.357564	−	0.0420428i
$902$		0			0
$903$	22.4325			0.746508
$904$		0			0
$905$	−52.6210			−1.74918
$906$		0			0
$907$			20.3331i			0.675148i	0.941299	+	0.337574i	$0.109606\pi$
$907$			20.3331i			0.675148i	−0.941299	+	0.337574i	$0.890394\pi$
$908$		0			0
$909$	−3.10944			−0.103133
$910$		0			0
$911$			22.1710i			0.734557i	0.930111	+	0.367278i	$0.119710\pi$
$911$			22.1710i			0.734557i	−0.930111	+	0.367278i	$0.880290\pi$
$912$		0			0
$913$		−	8.77529i		−	0.290420i
$914$		0			0
$915$			70.7062i			2.33748i
$916$		0			0
$917$	23.2846			0.768924
$918$		0			0
$919$	10.4400			0.344382			0.172191	−	0.985064i	$-0.444915\pi$
$919$	10.4400			0.344382			0.172191	+	0.985064i	$0.444915\pi$
$920$		0			0
$921$			42.6375i			1.40495i
$922$		0			0
$923$		−	41.7708i		−	1.37490i
$924$		0			0
$925$			82.0151i			2.69664i
$926$		0			0
$927$	−2.60304			−0.0854950
$928$		0			0
$929$			32.0755i			1.05236i	0.850372	+	0.526181i	$0.176377\pi$
$929$			32.0755i			1.05236i	−0.850372	+	0.526181i	$0.823623\pi$
$930$		0			0
$931$	−17.7823			−0.582791
$932$		0			0
$933$	−41.6206			−1.36260
$934$		0			0
$935$	22.9377	+	2.69705i	0.750144	+	0.0882028i
$936$		0			0
$937$	−44.5251			−1.45457			−0.727286	−	0.686334i	$-0.759219\pi$
$937$	−44.5251			−1.45457			−0.727286	+	0.686334i	$0.759219\pi$
$938$		0			0
$939$	−6.96463			−0.227282
$940$		0			0
$941$		−	5.50560i		−	0.179477i	−0.995965	−	0.0897387i	$-0.971397\pi$
$941$		−	5.50560i		−	0.179477i	0.995965	−	0.0897387i	$-0.0286032\pi$
$942$		0			0
$943$	−29.8448			−0.971881
$944$		0			0
$945$			37.5382i			1.22112i
$946$		0			0
$947$			14.9363i			0.485365i	0.970106	+	0.242682i	$0.0780273\pi$
$947$			14.9363i			0.485365i	−0.970106	+	0.242682i	$0.921973\pi$
$948$		0			0
$949$			1.53006i			0.0496678i
$950$		0			0
$951$	−50.1226			−1.62534
$952$		0			0
$953$	51.0150			1.65254			0.826269	−	0.563275i	$-0.190459\pi$
$953$	51.0150			1.65254			0.826269	+	0.563275i	$0.190459\pi$
$954$		0			0
$955$		−	86.1650i		−	2.78823i
$956$		0			0
$957$		−	12.4961i		−	0.403940i
$958$		0			0
$959$		−	20.4695i		−	0.660995i
$960$		0			0
$961$	19.8064			0.638918
$962$		0			0
$963$		−	7.74479i		−	0.249572i
$964$		0			0
$965$	−25.6992			−0.827287
$966$		0			0
$967$	29.6627			0.953888			0.476944	−	0.878934i	$-0.341745\pi$
$967$	29.6627			0.953888			0.476944	+	0.878934i	$0.341745\pi$
$968$		0			0
$969$	−3.96494	+	33.7209i	−0.127372	+	1.08327i
$970$		0			0
$971$	31.6626			1.01610			0.508051	−	0.861327i	$-0.330366\pi$
$971$	31.6626			1.01610			0.508051	+	0.861327i	$0.330366\pi$
$972$		0			0
$973$	22.7390			0.728979
$974$		0			0
$975$			55.2555i			1.76959i
$976$		0			0
$977$	−43.0522			−1.37736			−0.688682	−	0.725064i	$-0.741810\pi$
$977$	−43.0522			−1.37736			−0.688682	+	0.725064i	$0.741810\pi$
$978$		0			0
$979$			11.9141i			0.380776i
$980$		0			0
$981$			5.13905i			0.164077i
$982$		0			0
$983$			40.7923i			1.30107i	0.759475	+	0.650536i	$0.225456\pi$
$983$			40.7923i			1.30107i	−0.759475	+	0.650536i	$0.774544\pi$
$984$		0			0
$985$	94.1665			3.00040
$986$		0			0
$987$	26.3294			0.838073
$988$		0			0
$989$		−	23.8988i		−	0.759939i
$990$		0			0
$991$			6.81018i			0.216332i	0.994133	+	0.108166i	$0.0344979\pi$
$991$			6.81018i			0.216332i	−0.994133	+	0.108166i	$0.965502\pi$
$992$		0			0
$993$		−	3.86014i		−	0.122498i
$994$		0			0
$995$	66.9971			2.12395
$996$		0			0
$997$		−	39.8118i		−	1.26085i	−0.776249	−	0.630427i	$-0.782880\pi$
$997$		−	39.8118i		−	1.26085i	0.776249	−	0.630427i	$-0.217120\pi$
$998$		0			0
$999$	−57.8263			−1.82954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.b.b.237.13	✓	46
17.16	even	2	inner	4012.2.b.b.237.34	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.b.b.237.13	✓	46	1.1	even	1	trivial
4012.2.b.b.237.34	yes	46	17.16	even	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.58213i q^{3} -3.58424i q^{5} -1.89303i q^{7} +0.496852 q^{9} +O(q^{10})\)	\(q-1.58213i q^{3} -3.58424i q^{5} -1.89303i q^{7} +0.496852 q^{9} +1.56283i q^{11} +4.45085 q^{13} -5.67074 q^{15} +(4.09490 + 0.481483i) q^{17} -5.20490 q^{19} -2.99502 q^{21} +3.19080i q^{23} -7.84674 q^{25} -5.53249i q^{27} -5.05381i q^{29} -3.34568i q^{31} +2.47260 q^{33} -6.78505 q^{35} -10.4521i q^{37} -7.04183i q^{39} +9.35341i q^{41} -7.48993 q^{43} -1.78083i q^{45} -8.79104 q^{47} +3.41645 q^{49} +(0.761771 - 6.47867i) q^{51} -2.62104 q^{53} +5.60154 q^{55} +8.23485i q^{57} +1.00000 q^{59} -12.4686i q^{61} -0.940554i q^{63} -15.9529i q^{65} +1.37777 q^{67} +5.04827 q^{69} -9.38491i q^{71} +0.343768i q^{73} +12.4146i q^{75} +2.95847 q^{77} -0.0938078i q^{79} -7.26258 q^{81} -5.61501 q^{83} +(1.72575 - 14.6771i) q^{85} -7.99581 q^{87} +7.62341 q^{89} -8.42557i q^{91} -5.29331 q^{93} +18.6556i q^{95} +17.3041i q^{97} +0.776494i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 54 q^{9}+O(q^{10}) \) 46 * q - 54 * q^9	\( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) 46 * q - 54 * q^9 + 8 * q^13 - 10 * q^15 + q^17 - 20 * q^19 - 24 * q^21 - 54 * q^25 + 2 * q^33 + 26 * q^35 - 38 * q^43 + 6 * q^47 - 66 * q^49 + 26 * q^51 + 18 * q^53 - 20 * q^55 + 46 * q^59 + 48 * q^67 + 28 * q^69 + 22 * q^77 + 70 * q^81 - 52 * q^83 - 2 * q^85 + 44 * q^87 - 76 * q^89 - 26 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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